Can it be that some physicists believe in the actual infinite?
In other words, in something that philosophers have always shunned and still shun.
Brian Greene seems to believe in an infinite and at the same time filled space.
The following link should immediately show the relevant passage (18:32).
https://youtu.be/fJqpNudIss4?t=1111
He brings the example of a spaceship flying into space and asks what would happen if it went on and on. Is there an end point or does one eventually loop back to the starting point? These possibilities seem rather implausible.
The interviewer expresses skepticism about this. He says that you just can't keep going on and on.
Since Aristotle, the philosophers say that there is only the potentially infinite. That means in Greene's example that the spaceship always "extracts" actual spaces from the potentially infinite during its journey. In other words, create new space.
But that doesn't make any sense either. After all, I don't assume that my next room only exists when I enter it. It is the same with the spaceship, only in other orders of magnitude.
So do philosophers have to accept the actual infinite?
Brian Greene seems to believe in an infinite and at the same time filled space.
The following link should immediately show the relevant passage (18:32).
https://youtu.be/fJqpNudIss4?t=1111
He brings the example of a spaceship flying into space and asks what would happen if it went on and on. Is there an end point or does one eventually loop back to the starting point? These possibilities seem rather implausible.
The interviewer expresses skepticism about this. He says that you just can't keep going on and on.
Since Aristotle, the philosophers say that there is only the potentially infinite. That means in Greene's example that the spaceship always "extracts" actual spaces from the potentially infinite during its journey. In other words, create new space.
But that doesn't make any sense either. After all, I don't assume that my next room only exists when I enter it. It is the same with the spaceship, only in other orders of magnitude.
So do philosophers have to accept the actual infinite?
Comments (606)
So the spaceship isn't creating new space, it is using the space the expansion is creating.
Even as a concept, 'that location over there' must exist unless you can explain why it is impossible for anything to get there.
Some physicists tried to argue that space (the dimensional lines) were therefore curved and self-contained within the boundaries of the universe, so you couldn't reach any place beyond the bounds of the universe because it truly didn't exist.
However the 9 year results of NASA's WMAP survey concluded that ....
"The universe is flat, with a 0.4% margin of error, and that Euclidean geometry probably applies".
In other words, the dimensional lines are straight and therefore the Universe is potentially infinite.
That is an exact analogy for the flat-earth belief that ships would sail off the edge of the Earth if they went far enough.
No matter where the spaceship flies, it will always be receiving light from billions of years ago/away. The Universe is finite but unbounded.
(@apokrisis please correct me if I’m wrong.)
So if you could create a spaceship that is faster than the expansion of the universe, then that spaceship would hit a provisional end of the universe.
Thanks for the reply, it seems to me then that Brian Greene explanations are very misleading.
"If space is now infinite, then it always was infinite. Even at the Big Bang. A finite universe can’t expand to become infinite."
https://twitter.com/bgreene/status/839112447923486720?lang=de
He seems to advocate the quilted multiverse model:
"A universe with infinite spatial extent will contain infinitely many mini-universes. An infinite number of these mini-universes will be exactly like our own. Welcome to the mind-blowing nature of infinity – and the sometimes equally mind-blowing nature of the multiverse, which is a common theme among the books in this month’s column. First up is Brian Greene’s The Hidden Reality, which explores nine variations on the multiverse theme. Of these, the type of multiverse that arises as a consequence of infinite space – Greene calls it the “quilted multiverse” because regions of space will repeat like patterns in a quilt – is actually one of the easiest to comprehend." https://physicsworld.com/a/between-the-lines-multiverse-special/
That is a gross oversimplification. Philosophers argued for (or at least insisted on) the existence of infinities both before and after Aristotle. The extension of the universe is the most common type of purported infinity, and it was widely believed in the ancient world until the Church made Aristotelian position on the matter something of a theological dogma. For example, both Atomists/Epicureans and Stoics believed in the infinity of space - they only disagreed on whether it was uniformly populated with matter and even other worlds (Atomists/Epicureans) or whether it was void beyond our world (Stoics).
In the West Aristotelian dogma began to crumble during the Enlightenment, and in modern times the infinity of space, at least, was thought to be pretty much self-evident. That only started to change with the development of topology and differential geometry in mathematics and of General Relativity in physics.
Nowadays you would be hard-pressed to find a physicist who denies the possibility of some type of infinity on principle. Even those cosmologists and astrophysicists who propose that the universe is finite in extent do so on contingent empirical grounds, and would readily admit that there is no decisive evidence one way or the other.
Quoting spirit-salamander
Why is this "philosophically irritating"? (He is stating the mainstream position on the matter, BTW.)
Okay, you're right. I was going by what I assumed was a consensus that may have existed in philosophy since Aristotle. In fact, I think if a survey were done today with academic philosophers, most would "abhor" the infinite mundane.
Quoting SophistiCat
Giordano Bruno could also be mentioned. Not directly enlightenment, but strongly influenced the Enlightenment.
Quoting SophistiCat
My point was about philosophers.
Quoting SophistiCat
And therefore possibly philosophically irritating.
Depends on who you ask. I would expect that philosophers of physics, and generally those who have a handle on the mathematical and physical concepts of the last three centuries would, for the most part, be comfortable with the idea of physical infinity in some form, particularly the infinity of space. Classicists and medievalists (such as might use the words "infinite mundane") may well exhibit the prejudices of their subjects.
Quoting spirit-salamander
Indeed, the influence goes all the way back to Lucretius, who praised Epicurus and his doctrine of infinite worlds - which, of course, was considered heretical in Bruno's time.
Quoting spirit-salamander
Well, you did ask about physicists. But so far I have found that scientifically and mathematically literate philosophers are largely on the same page with physicists on this. There are, however, both respectable philosophers and physicists who are skeptical about even the least controversial forms of infinity: cosmologist George Ellis, for instance, who has been making inroads into philosophy in his later years. But they seem to be in the minority.
If you divorce yourself from the myth that is mathematics, it's easy.
On the scale of the universe, it's hard to disentangle the infinite from temporality. For instance, if the universe were infinite, then how is it that we arose as a species? It would take an infinite number of years to get to this point, yet here we are.
Then again, who knows.
We use infinite in math. That for sure
Sure. Any human language is infinite. One can write an infinite number of sentences in any language and never run out of things to say.
One can write an infinite number of sentences and never run out of things to say at all.
I said that one can write an infinite number of sentences and never run out of things to say at all.
Etc.
Yeah, I never said that math doesn't use the concept of infinite.
Consider the following...
If you would accept the notion that each object in The Universe occupies unique coordinates and is subject to unique Universal forces, then one might conclude that each object in The Universe is "one of a kind," that is, unique in and of itself. If this is indeed the case, then what exactly does "2" mean?
You have to accept the fact that mathematics sort of avoids this reality and "pretends" that 2 or 3 (or whatever number you choose) exists because it works (until it does not). So, as we go towards 0 and infinity, are we just supposed to say, "Oh well?" Apparently.
If would appear to me that man is quite a ways off from coming anywhere close to "understanding" much of anything, so coming up with concepts like infinity might be similar to tossing a dart across the bar room (backwards and standing on your head) after your twenty-third beer and hoping for a perfect bullseye.
There are unique coordinates? I take it you mean two objects don't occupy the same space at the same time. What does "one of a kind" have to do with counting two apples, one red and one green?
Quoting synthesis
No I don't. Nor should you. But we each choose our paths. You might consider joining forces with Metaphysician Undercover. His concern is the supposed equality between 2+2 and 4. :roll:
Maths, set theoretical infinities, kind courtesy of Georg Cantor, is an altogther different story as maths is essentially an axiomatic system, anything goes so long as you don't contradict yourself within one.
There is a fundamental problem with the concept of numbers. The numeral "1" represents a basic unity. an individual. The "2" represents two of those individuals together, and "3" represents three, etc. But then we want "2" and "3", each to represent a distinct unity as well. So we have to allow that "1" represents a different type of unity than "2" does, or else we'd have the contradiction of "2" representing both one and also two of the same type of unity.
Each object in The Universe is unique because it occupies its own space. What more do you need to know?
Quoting jgill
You are correct, you don't. Although it is easy to see the illusions that people cling to for whatever the reasons, sometimes it harder to acknowledge the ones to create the very foundations of our society. Mathematics is a system made up over time that works sometimes for some things, but has no real existence outside of this space.
If you cannot tell me what happens as zero and infinity is approached, what kind of system is that? And (again) the entire system is based on the idea that identical objects exist (when it is clear that this is not the case).
I don't believe you even need to go there. The idea that more than one (of anything) exists is simply untrue.
The convention to use multiples is just the lazy person's way of arranging things together instead of dealing with each thing as a unique individual. Maybe it's the ultimate form of identity group-think?
Really, numbers exist independent from physical reality? If you happen to be visiting a Universe that had no objects, what does "10" mean?
Quoting Alexandros
"We?" Who are we?
Mathematics is to reality as words are to feeling. The closest you can get to reality is through direct experience.
1. Set theory does not have, in this context, formal terms 'actual infinity' or 'completed infinity'. So for formal concerns we don't need to vindicate those notions that are not even used.
2. Set theory, in this context, does not use 'infinity' as a noun, but instead uses the adjective 'infinite'. This is important since, in this context, set theory does not point to a set named 'infinity' but rather mentions that various sets have the attribute of being infinite.
3. The non-formal notion of actual infinity does not need to resort to the term 'completed'.
4. Moreover, where does one find even a non-formal definition of 'infinity' that includes 'that which can't be completed'? If 'cannot be completed' is not part of the actual definition, then it is question begging to use this to argue that 'completed infinity' is a contradiction in terms.
6. Where does one find a definition of 'actual' that includes 'completed'?
7. Cantor's work was not axiomatized by him. It was only later mathematicians who axiomatized infinitistic mathematics.
8. The set theoretic definition of 'is infinite' is given this way.
x is finite iff there is a natural number n such that there is a 1-1 correspondence between x and n.
x if infinite iff x is not finite.
9. Quoting TheMadFool
Yes, in a broad sense, and from a certain point of view, that is correct:
"It is not our business to set up prohibitions, but rather to arrive at conventions." - Rudolph Carnap
What exactly do you mean by "go towards 0 and infinity"? And who apparently says "oh well" in this context?
Whose concept is that? Where can I actualy read anyone explaining the concept of numbers that way?
Didn't you just read it?
Good troll thread, eh doctor? :cool:
The post to which you responded didn't say it. So would you please link to a post or reference anything on the Internet or anywhere else?
I am in agreement with this. Whatever it is that physicists mean by infinity, they surely don't mean mathematical infinity. The set of positive integers is infinite by construction in Peano arithmetic, or by the axiom of infinity in set theory. But I do not believe such an infinite set can be instantiated in the real world.
I have another idle thought ... that the next revolution in physics will be the discovery of the actual infinite in the real world. By analogy, non-Euclidean geometry was thought to be a mathematical parlor trick of no use to physicists. Then Einstein came along and non-Euclidean geometry became real to the physicists.
Physically realized actual infinity has the same status in physics today as non-Euclidean geometry had in physics in the 1840s. The future genius to make this next breakthrough hasn't been born yet. Perhaps.
What do you need a link for? If you don't understand what I said, just show me what you do not understand, and I'll explain. If you understand but disagree, just tell me what you disagree with, and maybe we can hash it out.
I thought you meant that there is a fundamental problem with:
"The numeral "1" represents a basic unity. an individual. The "2" represents two of those individuals together, and "3" represents three, etc. But then we want "2" and "3", each to represent a distinct unity as well."
And that your supposed solution to the supposed problem is:
"[...] we have to allow that "1" represents a different type of unity than "2" does [...]"
So I was wondering who you have in mind as having said or written anywhere that:
"The numeral "1" represents a basic unity. an individual. The "2" represents two of those individuals together, and "3" represents three, etc. But then we want "2" and "3", each to represent a distinct unity as well."
Or are you saying that you yourself holds that:
"The numeral "1" represents a basic unity. an individual. The "2" represents two of those individuals together, and "3" represents three, etc. But then we want "2" and "3", each to represent a distinct unity as well."?
Or perhaps you would make clear which parts of your passage are ones you are critiquing and which parts are ones you are claiming.
My friend, just because it doesn't happen to make sense to you doesn't mean that you should not consider it.
I would bet that if we compared credentials (and experience), you just might end up on the short end. Fortunately, none of that matters in the least as even the least educated among us have attained great realization.
What's interesting about this is that whereas it is quite easy to see how mathematics (at its extremes) makes no sense, everything else knowable is EXACTLY the same. It's just more difficult to see.
I totally agree. Bertrand Russell nailed it:
Mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true..
Tru dat.
There is no proposed solution. The issue was stated as a fundamental problem with numbers, without a solution.
Quoting TonesInDeepFreeze
I am not critiquing anything, the whole thing is what I am claiming. I am claiming that there is a fundamental problem with numbers. If "1", "2", "3", etc. , are used to represent unities, then "2" and "3" must represent a different type of unity from "1", for the reason I explained.
Now here is a proposal for a solution. If "2" and "3" are said to represent numbers, then maybe we ought to say that "1" represents something other than a number.
Yes, as I thought, you find that there is a problem with the notion (whatever it means) that 'the numeral "1" represents a basic unity. an individual. The "2" represents two of those individuals together, and "3" represents three, etc".
But (aside from even trying to parse the broken phrases) I don't know who says anything along the lines of 'the numeral "1" represents a basic unity. an individual. The "2" represents two of those individuals together, and "3" represents three, etc". So I don't see why you think it is a problem that needs to be addressed.
No, I see no problem with that in itself. The problem is when we want to say that, and also that "2" and "3" represent a type of unity.
Quoting TonesInDeepFreeze
I don't know about you, but I always use "1", "2", and "3" in that way. If you don't ever talk about 1 chair, 2 or 3, or any number of other things like that, then I guess you don't use them the same way. But if someone asks you how old you are, do you answer with a number?
I understood that; I thought you meant that you do want to take '2' and '3' as representing a type of unity, while you think that that is contradicted by 'the numeral "1" represents a basic unity. an individual. The "2" represents two of those individuals together, and "3" represents three, etc" so that it needs correction .
Am I not correct that that is your view?
More basically, I don't know why one would fret over any of this, since I don't know anyone who claims "the numeral "1" represents a basic unity. an individual. The "2" represents two of those individuals together, and "3" represents three, etc" or anyone who claims "'2" and "3" represent some kind of unity". Moreover, aside from the ill-formed English, I don't know that you're not using such terms as 'unity' and 'represents' other than in a personal idiosyncratic way. Also, I would wonder what are your rigorous mathematical or philosophical definitions of 'basic unity', 'individuals together'. In sum, I can't make sense of what you're trying to say.
Suggestion: You could reference some actual piece of mathematical or philosophical writing that you disagree with and show how you think you can correct it.
That's pretty close, except I do not necessarily want to take "2" and "3" as representing unities, that's why I said "if" we want to. I see the numeral as representing a group with a specific number of things in that group, but the unity of that group is questionable.
.Quoting TonesInDeepFreeze
I find that very strange I hear them used that way all the time. I suppose I didn't explain very well. Isn't this how we count? One represents one unit, two represents two of those, etc.. If you're put off by the terminology, "unity", "represent", etc., that's understandable, but why don't you just relax and enjoy the simplicity of the terms. It seems to me like there's always some people who get really flustered, and then have difficulty understanding simple terms, as soon as you mention any sort of problems within the systems of mathematics.
Quoting TonesInDeepFreeze
Yes, I can see that. You haven't really ever thought about such fundamental issues as how we use numerals, and you don't really understand why anyone else would. Why did you engage me, if what I was saying appeared so foreign to you?
Quoting TonesInDeepFreeze
I've addressed particular pieces of mathematical writing which I disagree with before, in the past, but I cannot think of any way to correct these issues. So people have told me that if I don't have a solution, then don't point out a problem. But I think that's nonsense. I think we have to find the problems, and get a good clear understanding of why and how they are problems, before we can move toward an adequate solution. Solutions don't come easily, they require a thorough understanding of the problems.
But neither P nor Q are stated coherently by you. And there's no reason to think anyone wants P or Q anyway.
Quoting Metaphysician Undercover
You cut off the rest of my quote of you.
Quoting TonesInDeepFreeze
Of course the notion of 'one' is related to that of a unity. But even aside from parsing, I don't know who in particular you think holds that "The "2" represents two of those individuals together, and "3" represents three, etc". It would help if you would cite at least one particular written passage by someone that you think is properly rendered as "the numeral "1" represents a basic unity. an individual. The "2" represents two of those individuals together, and "3" represents three, etc" and "'2" and "3" represent some kind of unity".
Quoting Metaphysician Undercover
You don't know that I haven't thought about numerals. Actually in another thread, in posts directly with you, I posted a fair amount on the subject. And, most pointedly, I have never written that I don't understand why anyone would be interested in the subject.
Then I conclude that what needs to be discussed is clarification of P and Q.
Quoting TonesInDeepFreeze
Whenever we count something it is like this. Count the books on the shelf for example. "Book" signifies the type of unity being counted, "1" signifies that a unity called "a book" has been identified, and a first one has been counted , "2" signifies two of these units, etc..
Quoting TonesInDeepFreeze
I assume you know how to use Google or some other search facility. You could simply search this if you need such a confirmation, instead of asking me to do your research for you. Here is the first paragraph from the Wikipedia entry on "1":
[quote=Wikipedia]1 (one, also called unit, and unity) is a number and a numerical digit used to represent that number in numerals. It represents a single entity, the unit of counting or measurement. For example, a line segment of unit length is a line segment of length 1. In conventions of sign where zero is considered neither positive nor negative, 1 is the first and smallest positive integer.[1] It is also sometimes considered the first of the infinite sequence of natural numbers, followed by 2, although by other definitions 1 is the second natural number, following 0.[/quote]
Also, if you actually are interested (which you don't seem to be by your half-hearted replies, and refusal to do any research yourself, and near complete denial of the relation between one and unity), you could look into number theory, and the reason why 1 is generally determined as not a prime number. Here's the first entry I get when I Google that question, is 1 a prime number: https://blogs.scientificamerican.com/roots-of-unity/why-isnt-1-a-prime-number/
Here's a passage from that article:
[quote=* reference above]In the very most basic example, we can ask whether the number -2 is prime. The question may seem nonsensical, but it can motivate us to put into words the unique role of 1 in the whole numbers. The most unusual aspect of 1 in the whole numbers is that it has a multiplicative inverse that is also an integer. (A multiplicative inverse of the number x is a number that when multiplied by x gives 1. The number 2 has a multiplicative inverse in the set of the rational or real numbers, 1/2: 1/2×2=1, but 1/2 is not an integer.) The number 1 happens to be its own multiplicative inverse. No other positive integer has a multiplicative inverse within the set of integers.* The property of having a multiplicative inverse is called being a unit. The number -1 is also a unit within the set of integers: again, it is its own multiplicative inverse. We don’t consider units to be either prime or composite because you can multiply them by certain other units without changing much. We can then think of the number -2 as not so different from 2; from the point of view of multiplication, -2 is just 2 times a unit. If 2 is prime, -2 should be as well.
*This sentence was edited after publication to clarify that no other positive integer has a multiplicative inverse that is also an integer.[/quote]
I had said in the very post to which you now replied, "Of course the notion of 'one' is related to that of a unity." So I couldn't possibly be in denial about it. You are very confused.
Quoting Metaphysician Undercover
Then you need to clarify them, since you are the one who mentioned them, and you have not shown what anyone else said that is properly rendered as you did. But now you do clarify:
Quoting Metaphysician Undercover
That might be sharpened a little, but I guess it's okay on its own. However, "The "2" represents two of those individuals together, and "3" represents three, etc" is garbled, so I asked for an example of anything said by anyone that you think is properly rendered that way. First you said that I had just read it. But, as far as I can tell, it's not in the posts of the poster you quoted. Then you said that it's actually your own notion. Then you came back around to telling me that I can find it on the Internet somewhere.
Quoting Metaphysician Undercover
I didn't ask you to research for me. I just wanted to know of a particular example that YOU consider to be properly rendered by "The "2" represents two of those individuals together, and "3" represents three, etc" and "'2" and "3" represent some kind of unity". I can find all kinds of things about units and unity on the Internet. But that wouldn't tell me what in particular YOU have in mind as being properly rendered by "The "2" represents two of those individuals together, and "3" represents three, etc" and "'2" and "3" represent some kind of unity".
Quoting Metaphysician Undercover
I said in the very post to which you now replied:
Quoting TonesInDeepFreeze
So I made clear I wasn't asking about 1 but about the part where you write, "The "2" represents two of those individuals together, and "3" represents three, etc" and "'2" and "3" represent some kind of unity".
So your example about 1 not being a prime number does not address this.
Anyway, after first telling me that I had already read someone who said this, then saying instead that it was your own notion, then saying it is on the Internet (but citing an example not addressing it), we did get something of a better statement by you as I mentioned earlier in this post. Such long and unnecessary detours by you. As well as you flatly claimed the opposite about me when you said that I'm in "near denial" about the relation between the notions of one and unity when I had actually explicitly stated that there is a relation.
Now that you've somewhat clarified that, here's the best I can make of it (I don't claim to represent what you have in mind, but this is the best I can make sense of it):
1 is the count at the first member of the set, a particular unity (whatever it is). 2 is the count at the second member of the set. Etc. And '1' and '2' name different individual numbers. And 1 is the count of the members of the set with one unit. And 2 is the count of the members of a unity that is a set with two members. And a set with one member is a different kind of unity from a set with two members.
I think that's all okay. But then you conclude:
"we'd have the contradiction of "2" representing both one and also two of the same type of unity."
'2' denotes the number 2. The number 2 is the count of a set with two members. And a set of two members is itself a unity as a set. But '2' does not denote a unity; it does not denote the set that it counts. It denotes the COUNT of a set that is itself a unity. When we say that a set is a unity, we mean that it is one set, while we recognize that the number of members of the set may be greater than one.
{'War And Peace' 'Portnoy's Complaint'} is one set, which is a unity. But the count of {'War And Peace' 'Portnoy's Complaint'} is 2.
I don't see a contradiction.
The universe is not expanding, objects in the universe are simply moving away from each other and the space within which they are moving, in is infinite.
I do not assume any sets, or numbers, to begin with. Numerals are used fundamentally for counting things, objects like chairs, cars, etc.. There is no such thing as "the count", without things that are counted. So in that situation "1" signifies the existence of one object counted, "2" signifies two, etc..
Quoting TonesInDeepFreeze
The inconsistency arises now, if we say that numerals signify numbers rather than the things being counted. Let's call the number, "the count" which seems acceptable to both of us. Let me look at the difference between a count of one, and a count of two.
To have a count of one, there must be an object which is counted. In order for the count to be a valid count, there must be something which is counted. This is not the number 1 which is counted. It is something independent, an object like a chair, or a car, one of the things which is going to be counted. What validates the count of one, is an independent object, what I call a fundamental unity, which is counted.
Now let's consider a count of two. The count of two is justified by the existence of two such objects. But you want to say that "the count" itself is an object, the number two. So we have two distinct types of objects referred to with "a count of two". We have the two material objects, which have been counted, justifying the count of two as a valid count, and also we have the count itself, as an abstract object, which is called the number.
So, if we assume the reality of abstract objects, numbers, then when we use "2", there is always, if it is a valid use of "2", two distinct types of objects referred to. There is a number, 2, which as a unified object, as "the count", and there is also two of another type of unity, being the things counted, in order that the count is a true and valid count. In the case of "1" however, we can say that the number is the fundamental unity, the thing being counted, and also the abstract unity, represented as "the count", because they are each one simple unity. Therefore we would have consistency saying that the number 1 is both the thing being counted, making a valid count, and an abstract object itself.
To summarize now. Let's say that "1" refers to the number 1, which represents the count, and is also the thing counted, abstract numbers. We cannot use "2" in the same way. "2" might refer to a number, which represents "the count", as an object, or it might refer to the two distinct objects which are counted. It cannot refer to both, due to the inconsistency of one being one object, and the other two. My contention is, that if we use "2" to refer to "the count" itself, as the number 2, an abstract object, and this is what you are doing in your post, then the count itself is rendered false or invalid, because "2" cannot refer to both one object and two objects at the same time without contradiction.
Quoting Present awareness
So there wasn't a big bang that started an expanding universe?
No, let's not say that 1 is the thing counted.
The things that are counted are, in this case, the books.
'1' is a numeral.
1 is the number denoted by the numeral '1'.
1 is the count of the books.
1 is not a book.
Right, but do you agree that it is necessary that there is a thing counted, a book in this case? So as much as the numeral "1" "denotes" what you call the number 1, which is a property of "the count", it must also refer to the one book, or else the count is not a true count. For "the count" to qualify as a true count, there must be something which is counted. If "1" does not refer to the book, as well as what you call the number, then there is nothing being counted, and therefore no count, because if there is nothing being counted, this does not qualify as a count.
Therefore, we cannot dispense with the fact that "1" must refer to the object being counted, a book, as well as what you call the number 1, or else we have annihilated "the count" as false because we cannot have a count with nothing being counted. But we cannot annihilate the count, because that is what gives logical coherency to the numbers.
If this is not clear to you, imagine that you go to count the books, and you count the same book over and over again, 1, 2, 3, 4, etc., such that you could have an infinite number of books, by counting the same book over and over again. That is not a valid or true count. To have a true count, "1" must refer to the first book, "2" refers to the first and second together, "3" refers to those two with a third, etc.. If we remove the need to have distinct objects being counted, then the count is not a valid or true count.
The Creator of the Universe, who is revealed only to the pure in heart.
I think this just implies that if Greene's spaceship was pimped out with an odometer, we wouldn't expect it to ever read "infinite.". It's always going to show a finite number no matter how far we go.
I don't think Greene would deny that. I don't think it's relevant to his stance.
BTW, did you see the Space Time episode where he explained why they think the universe is flat?
(1) A count is an instance of counting. "Do a count of the books."
(2) A count is the result of counting. "The count of the books is five."
A number (we're talking about natural numbers in this context) is a count in sense (2). That doesn't preclude that a number is a mathematical object.
Quoting Metaphysician Undercover
To have a count (in sense (1)), you need something to count. (Except in the base case, there is the empty count.)
Quoting Metaphysician Undercover
Numbers also have other aspects than being counts (or 'results of counts' depending on how exactly we might define 'count').
Quoting Metaphysician Undercover
We better dispense with that notion. It's nuts. A number is not a book.
Quoting Metaphysician Undercover
'1' does not denote a book. And 'together' is not defined by you.
The most common mathematical understanding of counting is bijection.
For a set with only one book - 'War And Peace' - in the set, the count is the greatest number in the range of the bijection between the set {'War And Peace'} and the set {1}, and that greatest number is 1.
The bijection is {<'War And Peace' 1>}.
For a set with only two books - 'War And Peace' and 'Portnoy's Complaint' - in the set, the count is the greatest number in the range of the bijection between the set {'War And Peace' and 'Portnoy's Complaint'} and the set {1 2}, and that greatest number is 2.
The bijection is {<'War And Peace' 1> <'Portnoy's Complaint' 2>}.
So the numeral does not denote a book, but rather it denotes the number that is paired to the book in the bijection (or, in everyday terms, in the pairing off procedure we call 'counting').
We pair 'War And Peace' to 1, then we pair 'Portnoy's Complaint' to 2. That's counting.
We don't say "''1' denotes 'War And Peace' and '2' denotes 'War And Peace' together with 'Portnoy's Complaint'". That's crazy.
At the moment of conception, there is a rapid expansion of cells, like a Big Bang, only on a biological scale if you will. Looking at our observable universe, we see what conception looks like on a cosmic scale. Since space is infinite, there may be an infinite number of Big Bangs, but we’ll never observe them from earth because of the enormous distances involved. The light from a universe 100 billion light years away, won’t arrive on earth for another 86 billion years.
Yes. It's a contradiction.
Quoting TheMadFool
It might go the same way it came (per Mary Tiles).
Quoting TonesInDeepFreeze
Quoting frank
What do you mean by that? And are you referring to her book 'The Philosophy Of Set Theory'? If so, what in particular do you have in mind from that book?
In the introduction she maps out the intellectual landscape pretty straightforwardly. The whole book is good though (for lay people like me.)
Did she say that?
Where did she say that?
Page 3, last paragraph.
If you read that again, you'll see she's laying out an existing viewpoint. It's not hers.
Read page 4 where she explains the problems that arise from the fact that set theory is unproven: problems with how we might assess it's truth and problems with set theory's relationship to the real world.
She isn't advocating the downfall of set theory. She's explaining its unresolved and pretty significant philosophical problems.
I didn't say that it is her view that talk about infinite sets is not to be taken seriously. I said that she mentions that the difficulties '"MIGHT" be taken as evidence that talk about infinite numbers is not to be taken seriously.
Oh. You called it "her idea", so I misunderstood.
I don't see anything there about set theory being unproven. I don't know what sense of 'unproven' you have in mind. The theorems of set theory are provable from the axioms of set theory, while of course the axioms are not proven except in the trivial sense that an axiom on a line alone is a derivation. However, neither the continuum hypothesis nor its negation are provable from ZFC (if ZFC is consistent, which is a "background" assumption in discussion of independence), which she mentions earlier, so maybe that's what you have in mind.
You're misunderstanding that passage. Is English not your first language?
"Finally, the whole situation might be interpreted as evidence that talk of infinite numbers is not really to be taken seriously"
Then she goes on to mention how some people argue for the position that talk of infinite numbers is not to be taken seriously. Clearly, she is presenting that argument not necessarily as her own position but rather to explain the views of those who do ascribe to the argument.
As is typical in such writing, she temporarily argues on behalf of others in order to explain their views but later goes on to examine those views from outside.
I understand it well.
Quoting frank
it should be 'its' there.
Quoting frank
The period should be after the right parenthesis.
That sounded better.
So she lays out the conflict between finitists and set theory advocates and talks about how we might resolve the conflict.
It seems that the only route available is an indirect one: explaining the costs of going with finitism and understanding the value set theory has to math. The rest of the book fleshes all this out.
With this kind of approach, it's pretty clear that set theory is different from the kind of math most people learn. For instance arithmetic goes hand in hand with clear intuitions. There's no alternative viewpoint. There are no suspicions that arithmetic is contrived to provide some value internal to math itself.
That wasn't a cheap shot. I really thought English wasn't your first language.
It's basically what I said when you first took exception:
Quoting TonesInDeepFreeze
Then you're ridiculous.
Wow. You really got snagged on that one.
Certified, buddy.
I didn't write that. I wrote:
Quoting TonesInDeepFreeze
I don't know what snagging you think there is. You made an unnecessary rally about the matter even after I gave you ample clarification.
Perhaps you went off course when you overlooked that I included the word 'MIGHT' [emphasis added here] just as she did.
And I still don't know what you mean by
Quoting frank
or what you mean by
Quoting frank
since she doesn't say anything about "set theory is unproven" or even what one would mean by "set theory is unproven".
Read the whole introduction. I still think English isn't your first language. You're doing great, though.
I read it. She doesn't say anything on page 4 about set theory not being proven. At an earlier point, she does mention that the continuum hypothesis is not provable from the axioms. If that's what you have in mind, then it is not even close to saying that "set theory is unproven". You seem not to understand what set theory is when you say "set theory is unproven".
Quoting frank
Your sophomoric sarcasm is misplaced.
Just a tidbit: completely proper grammar gives you away as a non-native. Native speakers are usually slack. That's how it is with Spanish, anyway.
I don't think I have that certification. What agency takes care of that?
Oh I see I missed your point the other day and this is a good point. Yes even hard science is ultimately nonsense. There are no quarks. If you drill it down far enough you get to something that can't be quite right. Is that what you meant?
And social sciences like history are like that too. Something incredibly complex happens out of the interactions of thousands of people, then we label it "The Peloponnesian war" or "The industrial revolution," but these are just abstractions that summarize so many individual actions and events that in the end the abstraction must be a lie.
That's right, it doesn't preclude that the number is a mathematical object. But the point is that your definition (2) stipulates "the result of counting". So correct use of "5" is dependent on the count of the books, that there are five books. Therefore the number 5 loses its meaning if it does not refer to five of something counted, books in this case. Anytime we use "5" regardless of whether you think it refers to a mathematical object or not, it necessarily refers to five distinct units, or else you are using it incorrectly.
Quoting TonesInDeepFreeze
I'm not saying a number is a book, that's nonsense. But when we use "5" it is necessary that there are five distinct units indicated in that usage or else you are using "5" in an unacceptable way. Do you agree?
Quoting TonesInDeepFreeze
Strictly speaking this (bijection in the way you describe it) is not a valid count. Suppose we say that there are two books. "War and Peace" is numbered as 1, and "Portnoy's Complaint" is numbered as 2. The relation between "Portnoy's Complaint" and the number 2 is not a simple pairing. This is evident from the fact that if we remove "War and Peace", there is no longer two books, and the pairing is invalidated. You might still use "2" to name the book, but it is not a valid count of two, because there is only one book.
So we cannot say that "Portnoy's Complaint" is paired with 2. That is a false representation because it does not include the necessary requirement of another book. "Portnoy's Complaint" can only be paired with 2 in a valid count, if there is another book paired with one. Furthermore, neither Portnoy's complaint nor "War and Peace" need to be paired with either 1 or 2, for there to be a valid count of 2. Do you recognize this point? There is no need for a pairing to have a valid count. We can have two objects, and say that there are two, without naming either as one or two, they are simply two.
This latter point is something which is very important to understand, especially when we count things like electrons which are difficult to distinguish from one another. We can have a count of 2 without establishing the principles required to distinguish one from the other. We can say that there are two electrons in the same orbit, without the need of distinguishing one from the other. We have principles which say they are distinguishable, but we need not distinguish them. Likewise, we can talk about 12 volts, without the need to distinguish and label each unit of electrical potential, as 1,2,3, etc..
So it is very clear that your method of representing "a count", as pairing a number with a unit (bijection) is a totally inadequate representation of what a count really is.
Quoting TonesInDeepFreeze
You think it's crazy, but it's what's required to have a valid count. If "2" denotes "Portnoy's Complaint", unconditionally, and you have no other books, then obviously your count of 2 books is invalid. If you deny this requirement them you allow for invalid counts. You look at your bookshelf, number "Portnoy's Complaint" as 2, and bring it in to me, telling me you have two books in your hand, because "Portnoy's Complaint" is identified as two books. That's what's really crazy.
It doesn't.
Quoting Metaphysician Undercover
If I'm not mistaken, in another thread, you were using the word 'refer' in the sense of 'denote'. So if not 'denote' what exactly do you mean by 'refer' in this thread?
The numeral '5' has meaning. The number 5 is not the numeral '5'.
5 is the count of a set of five books. 5 is the count of a set of five apples. 5 is the count of the set with two books, one apple, one house, and one person.
The fact that 5 is a count doesn't contradict that 5 also is a number no matter what it happens to count.
5 is the successor of 4. 4 is the successor of 3. 3 is the successor of 2. 2 is the successor of 1. 1 is the successor of 0.
No matter what the numbers count, they exist by virtue of successorship or by being 0.
Bijections are not 'validated' or 'invalidated'.
The bijections
{<'War And Peace' 1> <'Portnoy's Complaint' 2>}
{<'Portnoy's Complaint' 1>}
of course are different, but nothing is "invalidated". Saying the pairings are "invalidated" is not even sensical.
You're doing it again! We do not use '2' to name a book. '2' does not denote a book.
Quoting Metaphysician Undercover
We can switch them so that we have:
{<'Portnoy's Complaint' 1> <'War And Peace' 2>}
But the greatest number in the range is still 2.
Quoting Metaphysician Undercover
We may infer, by whatever means, that there are a certain number of electrons or volts. That doesn't contradict that when we see discrete objects then we may count them.
Scientific measurement may have its special considerations. Or even everyday situations such as one glass of water having 8 ounces. But the question here is simple counting. How we use the concept of counting is a matter of practical approach, such as putting the water in a beaker with lines and counting the lines in the beaker to the point the water level ends or whatever. Whatever difficulties there may be conceptually with that, they don't negate the more basic notion of counting by bijection.
I don't do that.
You present as so confused that I wonder whether you are posting as some kind of stunt or dumb cluck character.
Quoting Present awareness
Nothing within the universe is supposed to be able to travel faster than the speed of light - it's called the cosmic speed limit. As galaxies are moving away from us faster than the speed of light, physicists say that as nothing within the universe can move faster than the speed of light, it is the universe itself expanding.
Yes it is. Unfortunately, most people freak-out when you suggest this, but to come to terms with this idea is the most intellectually liberating thing there is. Imagine not having the pressure of trying to figure everything out, instead, just going with the flow of ideas, allowing them to come and go as do all things.
Well, people say a lot of things and none of it is true (albeit, it might be the best bullshit currently available).
"Refer" is more general than denote, such that to denote is a specific type of referring. So when we say that a word refers to something, whether that something is a thing, an activity, an idea, a concept, or whatever, it means that we must direct our attention toward whatever it is which is referred to, in order to understand the use of the word.
Quoting TonesInDeepFreeze
The number 5 is a concept, therefore it has meaning, like any other concept.
Quoting TonesInDeepFreeze
My point is that 5 must count something, or else we forfeit its meaning. There is no sense to saying that there is a count of five which does not have five things.
Quoting TonesInDeepFreeze
This is simply not true. Numbers are defined by quantity, not order. If you want to define numbers by order, then you assign temporality as the difference between 1 ,2,3 and 4. But this is not at all how numbers are used. We might assign numbers to units of time, like first second, third, fourth, but it's really not true to say that numbers derive their value from order or succession, rather than from quantity.
Quoting TonesInDeepFreeze
As I explained, what is invalidated is your representation of the count as a pairing. The pairing you described is not a valid representation of a count, for the reasons explained.
Quoting TonesInDeepFreeze
If your count is nothing but a pairing, then that is all you are doing, assigning a number to a book, naming a book, with a number. This is why your representation of a count, as a pairing, or bijection, is incorrect. That is not what a count is.
Quoting TonesInDeepFreeze
If you switch them, then your original pairing is invalidated. Which pair is the true representation of the count? It can't be both at the same time. But in a true count, neither book is paired with 1 or 2, because a count is not a paring. There are two books, and neither one is number 1 or number 2, they are equivalent as books.
Quoting TonesInDeepFreeze
Sure, we can count discrete objects (units), that's what I've been arguing is necessary for a count, to have discrete units which are counted. What is incorrect, for the reasons explained, is your representation of a count, as an act of pairing a discrete unit with a number. Do you understand those reasons given?
Quoting TonesInDeepFreeze
Again, this is completely untrue. If we want to know what a number is, within a count, then we must produce a true representation of what a count is. To simply produce a false representation of a count, for the sake of supporting your claim of what a number is, is to just beg the question with a false premise.
Quoting TonesInDeepFreeze
You present yourself as someone who has not yet learned how to count.
Then:
(1)
Quoting Metaphysician Undercover
doesn't belong here. We do not use '2' to name a book.
(2) It seems your 'refer' might be close to what I mean by 'to pair with' or, more everyday, 'to associate with'.
In everyday understanding, when we count, we associate one thing with 1, then the next thing with 2, etc. Literally. We say the numbers, one for each object as we count the objects. Mathematically. this is expressed as a function from the set of things counted to a set of numbers:
{<'Portnoy's Complaint' 1> <'War And Peace' 2>}
That's a mathematical rendering of picking up 'Portnoy's Complaint' and saying '1', then picking up 'War And Peace' and saying '2', and if those are the only books, then saying 'The count is 2'.
Quoting Metaphysician Undercover
It is the very point that you can count more than one way.
You can count 'War And Peace as the first, then 'Portnoy's Complaint' as the second. Or you can count 'Portnoy's Complaint' as the first, then 'War And Peace' as the second. In either case, both counts show that there's a first and second, thus there are two.
Everybody knows that but you.
Quoting synthesis
Like a leaf in a stream, floating quietly in sluggish waters, but skimming past whirlpools to be on its way, frivolous and ethereal. :cool:
We might say the numbers "one for each object as we count the objects", but that does not mean that we associate "2" solely with the object pointed to when "two" is said. In reality we associate "2" with having counted two objects, so the first object is also associated with the spoken "2". It is imperative to the count that the other object counted is remembered, and is an integral part of the meaning of "2'" when it is spoken. If the other object is not remembered as a part of the 2, then we could go back to the first object and say "3", but that's not a valid count.
Quoting TonesInDeepFreeze
Right you can count the objects in any order that you want. Therefore "pairing", or bijection, which represents the count as assigning a specific order to the objects is a false representation of counting. In instances when there is a small number of objects we can look at them and see the number of objects, without giving them any order at all. So ordering them, or "pairing" them is accidental to the count, it is not an essential aspect of counting. We simply do it as an aid, to ensure that we are not making a mistake and producing a false count.
You ought to accept and understand this fact, because it is fundamental to many forms of measurement, and how we actually count something in reality. When we weigh something, we do not pair a different part of the object with each gram counted, and when we measure the electrical potential we do not pair each part of it with a volt. This is clear evidence, that in general practice, counting something is not a matter of pairing objects with numbers. In modern practice, we deal with billions, trillions, and numbers so high, that if counting something was a matter of pairing, we'd never get done counting any of these astronomically high numbers which we deal with.
Quoting jgill
That's an arbitrary designation, dependent on a stipulation that there is a left to right order to the sequence. "a" could just as easily be letter five, or we could assume an ordering which makes any of the letters number five. The point being that even though we order things when counting, (first, second, third, counted, etc.) because it facilitates distinguishing between what has been counted, and what has not been counted, helping to ensure certainty, ordering is not essential to counting. We can count things without ordering them.
You just don't get it.
No. I have implied the order of the sequence. It's not arbitrary. A more mathematical format would be
(a,b,c,d,e). This omission may have confused you. :roll:
Again, arbitrary. That you designate "a" as first in that sequence, is arbitrary.
The tuple notation is defined in mathematics.
And, of course, for sequences of length at least 2, there are different permutations. That there are permutations does not affect the count, since the count is the greatest number in the range, which remains constant under permutation.
You don't know anything.
The point is that to describe a count as a tuple is not a correct description of a count. You just don't get it.
You're failing to distinguish between cardinals and ordinals.
Let me give you a standard example. Consider the positive integers in their usual order:
1, 2, 3, 4, 5, 6, 7, 8, ...
Now consider the exact same set of positive integers, but with the number 3 moved to the very end:
1, 2, 4, 5, 6, 7, ..., 3
We can implement this new order relation by defining the "funny less than" symbol [math]\prec[/math] as follows:
* If [math]m[/math] is any positive integer and [math]m \neq 3[/math], we say that [math]m \prec 3[/math]. So [math]2 \prec 3[/math] and [math]47 \prec 3[/math] and so forth.
* If [math]m,n[/math] are positive integers we say that [math]m \prec n[/math] if [math]m \neq 3[/math] and [math]m < n[/math] where [math]<[/math] is the usual less-than relationship.
Now the quantity of positive integers is exactly the same in either case, since the ordered set [math](\{1, 2, 3, \dots \}, <)[/math] and the ordered set [math](\{1, 2, 4\dots, 3 \}, \prec)[/math] have the exact same elements, just slightly permuted. There is a one-to-one correspondence between the elements of the two ordered sets.
But the two ordered sets have a different order type, since 1, 2, 3, ... has no last element, and 1, 2, 4, ..., 3 does.
Both sets have cardinality [math]\aleph_0[/math]. But 1, 2, 3, ... has order type [math]\omega[/math]; and 1, 2, 4, ..., 3 has order type [math]\omega + 1[/math].
Numbers that denote quantity are called cardinals; and numbers that denote order are called ordinals. This insightful distinction goes back to Cantor in 1883.
In grade school they teach this distinction as "one, two, three ..." versus, "first, second, third, ..." The distinction turns out to have more substantive content in the transfinite domain.
https://en.wikipedia.org/wiki/Ordinal_number
MU knows so little of mathematics and yet is so confident. It's almost an admirable trait . . . but not quite.
We don't describe a count as a tuple. You don't even know what it is that you don't get.
The point is that we were talking about a count, which is a measure of quantity, not an order. To use numbers to indicate an order is a different matter. So to demonstrate the use of numbers in ordering now, is to equivocate, because an order does not necessarily imply a count
Quoting fishfry
That is not true. These sets do not have the same elements. If "..." implies an infinite extension of the order, then 3 does not exist in the second set. Therefore they do not have the same elements. The symbol "3" is there, but the number is excluded by the infinite order which must occur prior to it. That's an obvious problem with your mode of equivocation, and conflating counting and ordering, it allows for contradiction. You can describe an order which is never ending (infinite), then say that there is a 3 after the end of it. And for you, that 3 is there. But of course you've just accepted the contradiction.
Quoting TonesInDeepFreeze
Well of course. If I knew what it is was that I didn't get, that would mean I was getting it.
Try again, maybe after you explain an infinite number of times, I'll get it.
That's like when the judge hands down the guilty verdict and thinks: 'that guy was so persistent in his claims of innocence, that I almost feel like letting him go free'. But in this case lack of knowledge is innocence, so there's no guilty verdict to be handed out. Why not just pure admiration then?
I doubt it.
In this context, there are two senses of 'count':
(1) A count is an instance of counting. "Do a count of the books."
(2) A count is the result of counting. "The count of the books is five."
A count(1) implies an ordering and a result that is a cardinality ("quantity", i.e. a count(2)). Different orderings may determine the same cardinality, so different counts(1) whose result is the same count(2) may imply different orderings .
Quoting Metaphysician Undercover
R as defined below is a well ordering of the set of natural numbers:
R = {
And let 'w' stand for the set of natural numbesrs. w+1 is the ordinal of
If lack of knowledge is innocence, then you are a saint.
I only replied to what you said and did not investigate the context. You wrote: "Numbers are defined by quantity, not order ..." If you didn't mean that you should not have written that.
Quoting Metaphysician Undercover
My God, you wield your ignorance like a cudgel. I could have just as easily notated the two ordered sets as:
* [math](\{1,2,3,4, \dots \}, < )[/math] and
* [math](\{1,2,3,4, \dots \}, \prec )[/math]
which shows that these two ordered sets consist of the exact same underlying set of elements but different linear orders. Remember that sets have no inherent order. So {1,2,3,4,...} has no inherent order. The order is given by [math]<[/math] or [math]\prec[/math].
More concisely:
* [math](\mathbb N, < )[/math] and
* [math](\mathbb N, \prec )[/math]
Tell me again how you think these two ordered sets don't have the same elements.
Quoting Metaphysician Undercover
Crap sandwich. Nonsense. Garbage. And I'm being restrained. In the two formulations [math](\mathbb N, < )[/math] and [math](\mathbb N, \prec )[/math], explain to me how [math]\mathbb N[/math] has two different meanings. I really want to hear this.
On the contrary, sets have no inherent order. Given a set, we can put many different orders on it; just as if we have a class of school kids we can line them up by height or we can line them up by alphabetic order of their last name. Two different orders on the same underlying set. Of course both these orders have the same order type, as all orders on a finite set do. In the transfinite case, the same underlying set may have multiple distinct order types imposed on it.
I will agree with you that the notation {1,2,4,...,3} is meant to be suggestive. But in fact this is exactly the same set as {1,2,3,...} since sets have no inherent order. Order is something imposed on top of an existing set; and given a set, many different orders and order types may be imposed on it.
Why don't you have a look at the Wiki page on ordinal numbers and learn something instead of continually arguing from your lack of mathematical knowledge?
https://en.wikipedia.org/wiki/Ordinal_number
Right, one is a verb signifying an action, the other is a noun, signifying the result of the action.
Quoting TonesInDeepFreeze
This is what I have been telling you is incorrect. A count does not imply an order. You might order things to facilitate your activity of counting, but as you agreed, there's more than one way to count, and as I've been telling you, they are not all necessarily instances of ordering. Therefore you cannot define, or describe counting as ordering. That's why you can weigh a sac of flour and see that it's 5 kg. without ordering each kg of flour. And, you can see that there are five books on the shelf without placing them in any order. "A count" only implies a quantity, five, and there is no necessity of any particular order, or any order at all, only a quantity.
Quoting TonesInDeepFreeze
It requires more than innocence to be a saint.
Quoting fishfry
That's what I meant, and though you can use numbers in ordering, it is not what defines them, quantity does.
Quoting fishfry
OK, so doesn't this support my point, order is not what defines a number? If not, then I really don't know what you are trying to demonstrate, and how it is relevant. Perhaps you could explain.
Quoting fishfry
Exactly what I've been arguing, a count is a quantity, not an order, hence what I said "numbers are defined by quantity, not order".
Quoting fishfry
As I said, you can use numbers to order things, but this is not what defines numbers.
Here's an example by analogy. Ordinal numbers are a type of numbers which are used for ordering. Ordering is what defines the "ordinal" aspect, not the "number" aspect. In a similar way, human beings are a type of animal said to be rational. Rational defines the human aspect but it does not define the "animal" aspect.
'count' is also a verb. But here I am mentioning two nouns.
Quoting Metaphysician Undercover
You don't even know what I'm saying. You don't even have the mathematical vocabulary.
You have no standing to tell me what is incorrect in this matter.
Quoting Metaphysician Undercover
I showed you how it does. And less formally, even a child understands that when you count, there's the first item counted then the second item counted ...
Quoting Metaphysician Undercover
And I didn't.
Quoting Metaphysician Undercover
There's measuring and there's counting. A measurement might not itself be a (human) count. For example, a digital scale may measure the flour without a person actually counting. On the other hand, counting would be to count the marks on a scale up to the mark where the needle landed. Your argument is grasping at straws.
Quoting Metaphysician Undercover
We're not talking about taking in at a glance a quantity. We're talking about counting. You're grasping at straws. I notice you tend to do that after a while in a thread.
Quoting Metaphysician Undercover
count(2) is a number (quantity, if you like). count(1) is not a number or quantity. We're talking about count(1).
That's kind of okay in a very informal sense. But, just to be clear, it is not the definition of 'ordinal'.
Quoting Metaphysician Undercover
I don't know what you mean exactly by "the ordinal aspect" and "number aspect".
If R is a well ordering of S, then there is a unique ordinal L such that
. That implies that the cardinality of S equals the cardinality of L.
Anyway, I don't know what point you're trying to make. You disagreed with what fishfry wrote, then he explained how your disagreement is incorrect. You seem not to understand his explanation, though it was eminently clear.
I know what you said. You said "A count (1) implies an ordering". And I'm telling you that this is false for the reasons I explained. There is more than one way to carry out that action which is counting, and not all ways require ordering. Therefore it is false to say a count (1) implies an ordering.
Quoting TonesInDeepFreeze
You showed me one way of counting, which involved ordering, but you also admitted that there are other ways of counting. So clearly you use invalid logic when you say that counting implies ordering. Only that one way of counting, which you demonstrated, implies an ordering, not all ways of counting. You can see that there are five books on the shelf without ordering them at all, just like I can see that there are two chairs in front of me right now, without ordering them at all. That is counting them without ordering them.
Quoting TonesInDeepFreeze
Why does the action of counting have to be a human count? We have, as humans, devised all sorts of mechanisms to make counting easier, or even do our counting for us. This is the important point here, the essence of counting (what is necessary to the act), is to determine the quantity, no matter how this is done, by machine or whatever. That we commonly do this by ordering is accidental, not an essential aspect of counting.
Quoting TonesInDeepFreeze
Actually it's you who is grasping at straws. My OED defines "count" definition #1 as "determine the total number or amount of, esp. by assigning successive numbers". Notice that it says "esp.", which means mostly, or more often than not, but it does not mean necessarily. Therefore, to determine the total number or amount of, in a way which is not assigning successive numbers, though it might be a less common use of "count", it is still an act of counting.
Quoting TonesInDeepFreeze
Right, I don't understand how what fishfry was saying is relevant.
And you don't understand what that means.
Quoting Metaphysician Undercover
Whatever you have in mind, I didn't say that one first declares an ordering. I said that the count itself implies an ordering. The ordering I have in mind is the ordering by the number associated to each item.
You may pick up 'War And Peace' and say (or think) '1', then 'Portnoy's Complaint' and say '2' etc. The ordering implied by that count is {<'War And Peace' 'Portnoy's Complaint'>} because 1<2.
or
You may pick up 'Portnoy's Complaint' and say '1', then 'War And Peace' and say '2'. The ordering implied by that count is {<'Portnoy's Complaint' 'War And Peace'>} again because 1<2.
Quoting Metaphysician Undercover
You see my response in this post about ordering. And in my previous post I refuted the argument about seeing things at a glance. But you skip what I said. Again, immediate instantaneous impression of a quantity is not at stake in a discussion about counting. Counting is different from immediate instantaneous impression of a quantity. You really make yourself look like a dishonest interlocutor with such a grasping at straws argument.
Quoting Metaphysician Undercover
It doesn't. Indeed I mentioned a purely mathematical formulation of counting that doesn't require consideration of human features. But when you get into certain kinds of measurements of quantities, it may be murky whether it's what we mean by 'count'. By 'count' in this context, we mean consideration of discrete objects that are recognized each one at a time, one after another, just as your original example of books on a shelf.
Quoting Metaphysician Undercover
Quoting Metaphysician Undercover
But this is how you started this tangent on counting:
Quoting Metaphysician Undercover
That's talk about "a first" and "units". That sets a context that is a far cry from the far broader "determine the total number". You can't blame me for addressing the kind of counting that you mentioned yourself (counting chairs, one after another, or books, one after another) and then switch the context to something far wider.
Either you are being intentionally sneaky or you just forgot the context that you set up yourself.
Quoting Metaphysician Undercover
Right, it is common that you lose your place in the discussion.
Clearly, to see that there are two chairs in front of me, does not require that I associate a number to each of them. Therefore "the count", the determination that there are two chairs, does not imply an order. We can count (determine the number) without associating a number with each item. Therefore associating a number with each item is not an essential aspect of counting, or the count itself.
Quoting TonesInDeepFreeze
All you said is "We're not talking about taking in at a glance a quantity". That's your idea of a refutation? The definition of counting is to determine the number, clearly "taking in at a glance" qualifies.
From my experience with you, your mode of argument is to define the term in an unacceptable, false way (in the sense of correspondence with how the word is actually used), which begs the question. So, you define counting in a way which excludes any form of determining the quantity without any ordering, to support your conclusion that counting implies ordering. Obviously your so-called refutation is fallacious because you're just begging the question.
Do you accept the OED definition, that to count is to determine the number? And do you accept the fact that we can determine the number without ordering as you said here?
Quoting TonesInDeepFreeze
The important point, which I'll return to, is that when we have a count, it is necessary that there are as many objects as the count indicates, but it is not necessary that any object is paired with any number. When you recognize this, you'll see that the act, which is counting (determining the count), is not necessarily an ordering, or pairing. Counting, the act which produces a count, is not necessarily an ordering.
Quoting TonesInDeepFreeze
I was giving an example of counting. Did you or did you not agree that there is more than one way to count? If so , then you ought to be able to understand that a count does not imply an ordering.
Now that we have somewhat of an idea about what each other thinks about this matter, let's return to the issue at hand. Let's look at the numeral "2", and see if we can agree on the valid use of it. When we use "2" within the act of counting, do you agree that it signifies that a quantity of two objects have been counted. or do you believe that the numeral pairs with one particular object as "the second"?
If you choose the latter as the use of "2", then I would argue that you are talking about an act of ordering, not an act of counting and these two are distinct. Do you recognize the difference between such ordering, and counting? When we say or write "2" it is implied that there is a quantity of objects, two, which is referred to. When we say "second", it is not necessary that there is such a quantity, because when we say "second", the first may have already disappeared, like counting the hours. So "second" refers directly to one object, and there is no necessity that the prior object still exists, because we are not saying that there are two objects. But when we say "2" it implies that there is a quantity of two objects, or else it's not a valid use of "2".
So when we are counting the hours, and we assign "2" to the second hour, what is really being said is that it is the second hour, not that there are two hours. And these two ways of "counting" one being determining the number or quantity, the other being assigning numerals to an order of things, are very distinct and ought not be conflated by reason of equivocation.
Quoting Metaphysician Undercover
And later you have said:
Quoting Metaphysician Undercover
Quoting Metaphysician Undercover
Quoting Metaphysician Undercover
Quoting Metaphysician Undercover
Ordinarily, when someone says "I counted the books on the shelf", we understand that he used numbers (indeed as the positive natural numbers are sometimes called 'the counting numbers'), numbering in increasing order as he looked individually at each book, and not that just that he immediately perceived a quantity. That is the ordinary sense of counting I have been talking about.
Also, for example, if I see an 8 oz glass and that it's full of water, then I may say that the quantity of water is 8 ounces, without counting in the sense of numbering each ounce one by one. But that's not what people ordinarily mean by 'counting'.
Again, if you mean some wider sense, then of course certain of my remarks would not pertain.
Quoting Metaphysician Undercover
That's the case where there is immediate recognition of the number of objects, but when immediate recognition is not possible, then we count with numbers. Counting in the sense of numbering is what would be understood in the context of this discussion.
Quoting Metaphysician Undercover
I don't doubt that you quoted part of an OED definition:
Quoting Metaphysician Undercover
And the sense I have been using is indeed the one that is relevant - assigning successive numbers. You are only retroactively saying that the sense we should use is the widest sense. That widest sense is not what one would ordinarily and fairly understand by "count the books on the shelf".
I suggest, going forward, that if you wish to use 'count' in the wider sense, you would say 'count(wide)'.
If there is nothing beyond, why wouldn't it come back to its beginning?
Both. That is entailed by remarks in an earlier post of mine.
Quoting Metaphysician Undercover
There is no equivocation. The second book is mapped to 2. And 2 is also the greatest element in the range of the mapping,
What would that look like exactly?
In a way this describes this thread. :roll:
OK, let's proceed using your sense of counting, "assigning successive numbers". Do you agree with me, that when you assign "2" indicating the second object, the first object is also implied, as necessary to make your assignment of second a valid and truthful assignment? The "2" does not simply pair with the second object, because "second" implies that there was a first, so this is more than a straight pairing, because there is necessarily implied another pairing between "1" and the first object. Therefore "2", in this count, of assigning successive numbers, refers to or signifies, two objects, the first and the second.
Quoting TonesInDeepFreeze
It is your representation of counting as a simple pairing which I objected to. Even when restricted to a "successive numbering", counting is not a simple pairing. This is because, as I explained, when you pair the second, the pairing of the first is also implied, therefore referred to within the mention of "second". To say "second" refers to the first pairing and the second paring, as two distinct pairings.
Quoting TonesInDeepFreeze
OK, I agree that this is the "ordinary way" that a person counts, so we have a pretty good understanding between us as to what counting is, so let's go back to the fundamental problem I mentioned in the first place. When you say "2" if you are counting (ordering in your sense), and there are two objects referred to by "2", the fist and the second (the first is necessary to validate the notion of "second"), by what principle do we say that "2" refers to one object, the number 2?
I think you agree with me on the necessity of having two objects to make the use of "2" or "second", a true or valid use. So if we say that "2" also refers to one object, a number, then this type of object must be completely distinct from the other type of object, or else we'd have contradiction, because now there are three objects indicated, the first, the second, and the number 2. If this is the case, then "2" refers to the two objects counted, and a third object, the number 2.
Now, do you see the need to say that the number 2, if it is to be considered an object, must be a distinct type of object, or else we'd have three objects being referred to by "2"? If you see this need, to say that the number 2, if it is supposed to be an object, must be a very distinct type of object from the type of objects which we count, or order when counting, then you ought to also see the need to ask whether it is even possible to count this type of object. I think it is impossible to count these so-called objects because the fact that they are the count, rather than what is counted, is what distinguishes them from the objects which are counted. Therefore, as "the count" , and distinguished from what is counted as "not what is counted", they are by definition not countable. So the simple solution (I offered already), is to recognize that they are not really objects and therefore not countable.
That the numbers, proposed as objects, are not countable, is also evident from the problem of infinite regress. If we wanted to count the numbers, as objects, it would require a different numbering system from the one we use to count ordinary objects, to avoid equivocation. For example, when we have two ordinary objects, we have the number 2 which is another object that would be counted as 1,object if numbers are counted. So we cannot have both "1" and "2" describing how many objects are there unless the "1" was part of a distinct numbering system from the "2". However, then these numbers in the distinct system would be proposed as objects as well, and we'd want to count them alao, so we'd need another numbering system to count them. then we'd proceed toward an infinite number of numbering systems, in the attempt to count all the numbers which count the numbers which count the numbers, ad infinitum..
The simple solution again, is to recognize the truth of the fact, that the numbers are simply not countable. They are infinite and this renders them as not countable, by definition. So we ought not even attempt to count them as this is known to be impossible. Also, we can clearly see that the numbers are not objects, and so they are not something which is countable.
I don't speak of objects being implied. What are implied are statements (or propositions). And the mathematical representation I have mentioned doesn't even need to involve such things as "it is implied that there exist [insert the members of the field of the bijection here]."
As I mentioned, there are two senses of 'count' here:
(1) A count is an instance of counting. "Do a count of the books."
(2) A count is the result of counting. "The count of the books is five."
In order not to have to continually specify which sense I mean, I'll use 'count' in sense (1) and 'result' for sense (2).
Again, here is the mathematical representation I have told you about:
A (non-empty) count is a bijection form a set onto a set of natural numbers (where 1 is in the set and there are no gaps). The result is the greatest number in the range of the count.
Here is a count:
{<'War And Peace' 1> <'Portnoy's Complaint' 2>}
The result of that count is 2.
The ordinary order induced by that count is <'War And Peace' 'Portnoy's Complaint'>
Here is another count:
{<'Portnoy's Complaint' 1> <'War And Peace' 2> }
The result of that count is 2.
The ordinary order induced by that count is <'Portnoy's Complaint' 'War And Peace''>
This involves nothing about "implying objects" or "signifying objects".
Of course, though, it is already assumed that there are objects (books on a shelf in this case) named 'War And Peace' and 'Portnoy's Complaint'. But that's not a mathematical concern. It's just a given from the physical world example.
Quoting Metaphysician Undercover
By the principle of stipulative definition. Anyway, your question doesn't weigh on the mathematical notion of counting.
Quoting Metaphysician Undercover
You are using 'refer' without specifying in what you sense of the word you mean. Here is what obtains:
2 is the cardinality of a set of two objects.
'2' names 2.
'2' names the cardinality of a set of two objects.
2 is the cardinality of {'War And Peace' 'Portnoy's Complaint'}.
'2' names the cardinality of {'War And Peace' 'Portnoy's Complaint'}.
In the bijection, 'Portnoy's Complaint' is mapped to 2.
'2' names the number that 'Portnoy's Complaint' is mapped to in the bijection.
The result of the count is 2.
'2' names the result of the count.
There is no equivocation or contradiction in any of that.
Quoting Metaphysician Undercover
Setting aside your other confusions, I will address the term 'countable' as used in a mathematics, to prevent misunderstanding that might arise:
'countable' is a technical term in mathematics that does not adhere to the way 'countable' is often used in non-mathematical contexts.
In non-mathematical contexts, people might use 'countable' in the sense that that a set can be counted as in a finite human count.
But in mathematics 'countable' doesn't have that meaning. Instead, in mathematics the definition of 'countable' is given by:
x is countable iff (there is a bijection between x and a natural number or there is a bijection between x and the set of natural numbers).
I like that! Or as the late, great Howard Cosell once said to a reporter: "If ignorance is bliss, you, my friend, must be ecstatic!"
Quoting Metaphysician Undercover
Quoting Metaphysician Undercover
First, there is no general definition of number in mathematics. We can define real numbers, rational numbers, p-adic numbers, hyperreal numbers, and so forth, but there is no general definition by which we can say what a number is. Therefore it's possible that you are working from an entirely different definition; in which case you could well be correct via your definition, but not correct mathematically.
What is your definition of number?
In math, there are ordinal numbers and cardinal numbers. In fact in modern set theory, the cardinals are defined in terms of ordinals. That is, a cardinal number is a special type of ordinal. So in math, ordinals are logically prior to cardinals.
But as I say, if you are working from a different definition, your point of view could be consistent with that. But not with the usage in math.
Quoting Metaphysician Undercover
Not in math. After all, some numbers have neither quantity nor order, like [math]3 + 5i[/math] in the complex numbers. No quantity, no order, but a perfectly respectable number. You take this point, I hope. And are you claiming a philosopher would deny the numbertude of [math]3 + 5i[/math]? You won't be able to support that claim.
Quoting Metaphysician Undercover
There is no general definition of number; and complex numbers have neither order nor quantity.
Quoting Metaphysician Undercover
You're wrong mathematically, as I've pointed out. But what is your definition of number then? And how do you account for [math]3 + 5i[/math]? What about familiar real numbers like [math]\pi[/math]? No quantity except by stretching the term. You're using an extremely restrictive concept of number.
Quoting Metaphysician Undercover
I'm sure you don't need to explain that to me. But number and order are not an instance of this phenomenon. And as I noted, a cardinal is actually a special kind of ordinal; not the other way 'round.
The statement is not implied, it is explicit, stated as "first", "second", etc... What is implied, in order that your count be a true count, is that there are objects counted . Otherwise, as I said it is not a true or valid count. You can state "first", "second", "third", "fourth", but unless there is something referred to, you are not counting anything and it's not a true or valid count.
Quoting TonesInDeepFreeze
I like that, instead of calling (2) the count, we'll call it the result of the count. We might even call it the conclusion, Then I can say that the conclusion is unsound if there aren't any objects counted, because to say "that is the second", or "there are two", is not true unless there are objects which have been counted. To count "1", or "first", without counting anything is to make a false statement.
Quoting TonesInDeepFreeze
As I explained in my last post, we ought not consider that a number is a countable object, for the reasons I described. So I consider such a count to be a false count.
Quoting TonesInDeepFreeze
Of course it implies objects. You have mentioned things being counted. I deny that natural numbers are things which can be counted. Therefore I conclude that your result is unsound, by this false premise that natural numbers are things which can be counted.
Quoting TonesInDeepFreeze
Truth and falsity may not be a mathematical concern, but it is a philosophical concern.
Quoting TonesInDeepFreeze
Stipulation does not make truth.
Quoting TonesInDeepFreeze
Obviously, I do not accept this stipulative definition of "countable", for the reasons explained in my last post. Principally, if we use numbers to count numbers, the numbering system which counts numbers will need to be different than the numbers being counted (by the reasons explained), then we'll want another numbering system to count those numbers, and another to count those numbers, etc', ad infinitum.
There is really no reason to attempt to count the natural numbers, when we know that this is impossible because they are infinite. And numbers are not even countable objects in the first place, they are imaginary, so such a count, counting imaginary things, is a false count. Therefore natural numbers ought not be thought of as countable.
Quoting fishfry
That's because numbers are not objects, and therefore they cannot be described or identified as such. And since they cannot be identified, they cannot be counted.
Quoting fishfry
It is a value representing a quantity.
Quoting fishfry
Yes, that's a symptom of the problem I explained to TIDF. Once we decide that numbers are objects which can be counted, then we need to devise a numbering system to count them. So we create a new type of number. Then we might want to count these numbers, as objects as well, so we need to devise another numbering system, and onward, ad infinitum. Instead of falling into this infinite regress of creating new types of imaginary objects (numbers), mathemajicians ought to just recognize that numbers are not countable, and work on something useful.
Quoting fishfry
Of course I'm wrong mathematically, I'm arguing against accepted mathematical principles. But the question is one of truth and falsity. Are numbers objects which can be counted, rendering a true result to a count, or are they just something in your imagination, and if you count them and say "I have ten", you don't really have ten, a false count is what you really have?
How do you feel your campaign is doing?
Has it been worth the struggle?
Have there been casualties?
Are you holding up? :chin:
When I say 'P is implied', then P is a statement, not an object.
So I don't say
'War And Peace' is implied.
But I do say
That 'War And Peace' is on the bookshelf is implied.
This is just a matter of being very careful in usage that may be critical in discussions about mathematics.
Regarding this example of counting, I take it as a given assumption that
'War And Peace' is on the bookshelf and 'Portnoy's Complaint' is on the bookshelf.
I am not deriving ''War And Peace' is on the bookshelf and 'Portnoy's Complaint' is on the bookshelf' as implied by anything other than the initial assumption of the example.
And, of course, I am not showing an example of a non-empty count on the empty set. It is a given assumption of the example that:
the set of books on shelf = {'War And Peace' 'Portnoy's Complaint'}
/
Quoting Metaphysician Undercover
I knew you would respond in a way that would evince that you don't understand the concept of definition.
Quoting Metaphysician Undercover
No, your belief that numbers are not objects is not the reason that mathematics doesn't provide a definition of 'is a number'.
Actually, I'm starting to get a real feel for the problem now, and I sincerely want to thank TIDF and fishfry for helping me come to this realization. I now see that there is a fundamental difference between using numerals to signify quantities, and using them to signify orders. The former requires distinct entities, objects counted, for truth in the usage, while the truth or falsity of the latter is dependent on spatial-temporal relations. So the truth of a determined quantity depends on the criteria for what qualifies as an object to be counted, while the truth of a determined order is dependent only on our concepts of space and time. So, in the case of quantity, truth or falsity is dependent on the truth of our concept of distinct, individual objects, but in the case of ordering, truth or falsity is dependent on the truth of our concepts of space and time. Since we think of space and time as continuous, non-discrete, we have two very different, and incompatible uses of the same numerals.
Quoting TonesInDeepFreeze
Sorry, I don't follow this at all. If you count "1", then it is implied that there is one thing (an object) counted. Do you, or do you not agree with this? If you do not agree, then what are you counting when you count "1"? If you are counting books, then aren't books objects? And you could be counting any type of objects, or maybe just objects in general. But don't you agree that if you count "1", it is necessary that an object has been counted? Therefore an object is implied by any count of 1?
Quoting TonesInDeepFreeze
I don't see how this is relevant. You seem to have changed the subject. We were not talking about sets. We were talking about (1) the act of counting, and (2) the result of this act. When did a "set" enter the picture?
For physical world matters. However, in the mathematics itself, ordinals don't refer to space and time.
Quoting Metaphysician Undercover
Agree.
Quoting Metaphysician Undercover
Yes.
Quoting Metaphysician Undercover
I just told you that I don't use the 'implied' that way.
In your post you said, "it is implied that there is one thing". And that is how I use 'imply' too. I use 'imply' to say 'It is implied that [fill in statement here].
Then you said, "an object is implied".
I don't use 'implied' to say '[fill in noun phrase here] is implied'.
Quoting Metaphysician Undercover
When I gave a mathematical representation of a count.
For your next trick, do one of an earl. :cool:
I'll do one of Earl Hines's "Blues In Thirds".
I was talking about truth and falsity in the use of mathematics, and I use these terms in the sense of correspondence with reality. So it's not necessarily the "physical world" we are talking about, it's "reality" in general. If mathematics talks about an order which is not temporally, nor spatially grounded, then I think such a mathematics would be nonsensical. I've seen some people argue for a "logical order" which is neither temporal nor spatial, but this so-called logical order, which is usually expressed in terms of first and second, is always reducible to a temporal order.
Quoting TonesInDeepFreeze
When you agree that "it is implied that there is one thing", do you not agree that the "thing" is an object? Can we go to my original term, a "unity". Do you agree that the thing is a "unity"? I mean, we could stick to calling it a "thing", as you seem to agree that there is something which is referred to as "thing" here, but why quibble about terms? Like I said in the last post, what we call the thing is irrelevant; we could call it "object", "entity", "unity", "particular", "individual", "book", "War and Peace", whatever, so long as there is something counted. What is important is that this name refers to something or else you are not truly counting. Do you agree? Even if you are counting names or titles, "War and Peace", etc., those are still "things" which are being counted
If you simply say "1,2,3,4,5" , you might say "I am counting", but it's not a true count, because nothing is counted, therefore the symbols actually refer to nothing whatsoever, and the count itself is invalidated because that sequence of symbols does not have any meaning at all. Suppose someone memorizes that sequence of symbols, 1-5, and repeats them saying "I can count to five". Unless the person knows what the symbols mean they are not really counting to five, they are just repeating symbols. If they know what the symbols mean, then they know that there must be five things (objects, unities, individuals, or whatever you want to call them), or else the count is false. Do you agree? If not how do you validate the meaning of the symbols?
Quoting TonesInDeepFreeze
Please, do not jump ahead like that. You spent days differentiating between (1) the act of counting, and (2) the result of that act. As far as I can see, the "mathematical representation" of both (1) and (2) consists of numerals, "1", "2", "3", etc.. There is no need to represent (2), the result of the act of counting, as a "set", or whatever your intent is. Let's just adhere to these defined principles, and maintain clarity.
The mathematics of ordering and ordinals may be applied to study of space and time, but the mathematics itself doesn't mention space and time.
Quoting Metaphysician Undercover
I agree that things are objects. In my previous post, I answered essentially the same question, when I said 'Yes'.
Quoting Metaphysician Undercover
I already shared my thoughts about 'unity' earlier in this thread.
(3) Quoting Metaphysician Undercover
That would be another sense of the English word 'count', and it may be represented mathematically as
<1 2 3 4 5>
But it was not the sense in your bookshelf example, which may be represented mathematicaly as the bijection I mentioned.
Quoting Metaphysician Undercover
I don't have an opinion about that.
Quoting Metaphysician Undercover
What? I'm not jumping ahead. I'm referring back. You asked me where the notion of 'set' came from in this discussion, so I told you.
Quoting Metaphysician Undercover
You are critically confused on the very point here, and one that previously you even said you understood. That point is that the result is different from the count. I didn't represent the result as a set*. I explicity said (several times) that the result is a number. Meanwhile I represented the count (not the result) as a bijection, which is a certain kind of set.
(* Putting aside the technical sense of all objects as sets in formal set theory.)
I explained to you already why bijection (paring) is an inadequate representation of counting, as defined by you (1). This effort required a number of posts. I assume you didn't understand.
I'm sorry Tones, but you've really lost me now. You don't seem to be directly addressing any of the points I make, and we do not seem to be understanding each other at all, at this point.
Quoting TonesInDeepFreeze
I don't know what a "set" is, you haven't defined it. But you seemed to be using it as if it meant the result of the count, i.e. the number. I asked where did the notion of a set come from, and you said "When I gave a mathematical representation of a count." Isn't it the case, that the mathematical representation of a count, is the number, which is the result of the count? Or, you might give a mathematical representation of the activity of counting as "1+1+1+1...". However you've already agreed that there's more than one way to count, so there is probably a number of different acceptable mathematical representations of counting. Bijection though, as described by you as pairing, is not an acceptable representation, for the reasons I already explained.
And at every juncture I pointed out where you are wrong or confused.
Quoting Metaphysician Undercover
x is a set iff (x is the empty class or (x is a non-empty class and there is a y such x is a member of y)).
Or, the sets are objects that satisfy the set theory axioms.
Or, the sets are the objects that the quantifier ranges over.
Quoting Metaphysician Undercover
No, I did not. I have always been completely clear that the bijection represents the count, not the result. You are terribly terribly confused.
Quoting Metaphysician Undercover
No! And I've told you this already. What is wrong with you? The mathematical representation of the count is a representation of the count, not of the result. The representation of the count is the bijection. The result of the count is a number.
In my naivete I once thought of a set as a collection of things called elements. Then I learned the error of my ways. Now I try to avoid thinking of them at all. It's a refreshing experience, like standing at the beach with the soft winds off the ocean caressing your body. :cool:
So, a set is a class. How's that relevant? Say we're counting books, the set is called "books" then. Do you agree that there must be some of these things (objects) which are classed as "books", for us to have a true count. If there aren't any books, we do not have any counting of books at all.
Quoting TonesInDeepFreeze
And I've been completely clear, that bijection is unacceptable as a representation of counting. Therefore one or both of us misunderstands what the activity of counting is, so we are stuck here, unable to proceed until we find some agreement or compromise on this. Do you agree that there is no activity of counting if there is no objects counted?
I've answered that already a few times. To have a non-empty count, of course there exist the objects counted, and in you example, these objects are books.
Quoting Metaphysician Undercover
Now I'm answering yet again, there is no no-empty count if there are not objects counted.
Now, are you going to continue asking me this over and over again?
Quoting Metaphysician Undercover
I don't seek agreement or compromise. I'm interested in showing where your remarks are incorrect, especially ignorant and/or confused, and sometimes also to add explanations about mathematics, whether you ever understand them.
That's silly. I can count the captains of the starship Enterprise even though they're imaginary.
https://screenrant.com/star-trek-movies-shows-enterprise-captains-kirk-picard/
I can count the harpooneers on the Pequod even though they're imaginary. What ever are you talking about?
Quoting Metaphysician Undercover
This is "not even wrong." As Truman Capote once said of an inferior writer: "That's not writing, that's typing."
Quoting Metaphysician Undercover
A moment ago you Meta-splained why there can be no general definition of number, and now you give one. Don't you even read your own posts?
But quantity is only one thing numbers represent. Integers (zeri, positive and negative whole numbers) represent signed or directed quantities. Rational numbers represent ratios of whole numbers. Irrationals and complex numbers, p-adics and hypperreals, each represent some other aspect of number-hood. Aristotle said that the reason bowling balls fall to earth is that earth is the "natural place" of a bowling ball. Surely you are aware that we have more modern explanations now. Why do you deny the historical development of our understanding of the concept of number?
Curious to know: If you deny complex numbers do you likewise deny quantum physics, which has the imaginary unit i in its core equation?
Quoting Metaphysician Undercover
Not following that convo, but do I take it that you deny complex numbers? Do you likewise deny negative numbers, zero, rationals and irrationals? Is your physics likewise stuck in the days of Aristotle? Why exactly do YOU think bowling balls fall down?
Quoting Metaphysician Undercover
You're a trivial sophist with no insight or awareness of intellectual history.
Quoting Metaphysician Undercover
If only you were, we could have a conversation. But you have no actual principles or arguments, only nihilism and denial.
Quoting Metaphysician Undercover
Guess we're done here. Again. I'd like to say something more substantive, but what can I say to someone who rejects the role of numbers as expressing order, or numbers as used in quantum physics, or even fractions for dividing up a pumpkin pie? What words besides nihilism fairly describes your mathematical perspective?
Not in the real world. I eat the whole pie all at once.
Math needs to correspond to reality!
The question was whether there could be a count if there are no books.. If no books are counted, do you consider this to be a count? I think that if no books are counted then there is no activity, of counting, therefore no result of counting either.
Quoting TonesInDeepFreeze
I'm asking you if you believe there is such a thing as an empty count. That would be contradiction, obviously, to have an activity of counting when nothing is being counted. Do you agree? You did say that a set could be an empty class. Do you agree, that by your definition of "count" (1) the act of counting, an empty set is not countable? There seems to be discrepancy between how you define the count (1), and and how you say "countable" is defined in the mathematical sense.
Quoting fishfry
That's what I would call a false count, because it's hypothetical. It's like if you look at an architect's blueprints, and count how many doors are on the first floor of a planned building. You are not really counting doors, you are counting hypothetical doors, symbolic representations of doors, in the architect's design. Likewise, if you count how many people are in a work of fiction, these people are hypothetical people, so you are not really counting people, you are counting symbolic representations. We can count representations, but they are counted as symbols, like the architect's representation of a door, may be counted as a specific type of symbol. And when you count captains of the Enterprise, you are likewise counting symbolic representations. If you present this as a true count of actual captains of an actual starship, you'd be engaged in deception. You are not counting captains of a starship, only symbolic representations.
Quoting fishfry
Yes, I think quantum physics uses a very primitive, and completely mistaken representation of space and time. That's why it has so many interpretative difficulties.
That was a while ago. But you're still asking!
Quoting Metaphysician Undercover
There you even correctly posted yourself that you surmise that I agree that if we count 2 objects then there exist 2 objects.
The context here has been of a shelf that has books on it. I've said more than once that if you count 1 book or 2 books, then, yes, there are books on the shelf.
From this thread, this is the context in which we are talking about a shelf that has books on it:
Quoting Metaphysician Undercover
Quoting TonesInDeepFreeze
Quoting TonesInDeepFreeze
Quoting Metaphysician Undercover
The context is not a shelf with no books, but a shelf with 5 books.
Quoting TonesInDeepFreeze
There I keep in context of having at least one book on the shelf.
Quoting Metaphysician Undercover
Again, the context is that there are books on the shelf.
Quoting Metaphysician Undercover
Again, the context is that there are books on the shelf.
Now:
Quoting Metaphysician Undercover
I already addressed that. If there are no books, then it's not a non-empty count. It's not the kind of count we're talking about in your example.
Quoting TonesInDeepFreeze
And that parenthetical is simply to make clear that in this context we're not talking about the technical notion of an empty count. We're talking about counts that start at 1.
/
I'll say it one more time in this forrm: If there is a count that reaches 1, then there exists at least one object counted, and if there is a count that reaches 2, then there exist at least two objects counted.
And that reflects the representation with a bijection. If a natural number n is in the range, then there must be at least n objects in the domain.
You don't read my posts adequately to register in your mind what I wrote, let alone understanding them.
And you're even more ridiculous, since the question of whether there are objects counted - already answered by me - is answered right in the representation with a bijection itself. You can see for yourself that the two books are right there in the domain of the bijection.
/
Quoting Metaphysician Undercover
Your original and ongoing question regarded the context in which there are books on the shelf. You didn't ask me about the notion of an empty count.
I mentioned the empty count only to avoid a pedantic, technical hitch. I am not talking about empty counts in the context where there are books on the shelf.
But about the empty count: It's a technical set theoretical matter. It's not intended that the use of the word 'count' in 'empty count' corresponds to our everyday English senses of 'count'. I happily agree that it's an odd use of the word 'count'. If you don't like the notion, then that's okay in this context, because the representation with a bijection doesn't depend on the notion.
/
Quoting Metaphysician Undercover
But it's still counting.
Quoting Metaphysician Undercover
Ah, you resort to the strawman. We are not claiming it is a count of actual captains.
We've been talking about what it means to count. And we've determine that the count starts at one. If you know of some other way of counting which is based in something else, let me know please.
Quoting TonesInDeepFreeze
If the count does not reach one, then it is not a count, because one is the beginning of the count. We could count by twos, or fives, or tens, but I don't think you've even accepted this yet, insisting that counting is a bijection with individuals. How do you ever get to the idea that the count "reaches" one when it necessarily starts at one and there is no count prior to one?
Quoting TonesInDeepFreeze
Why do you keep avoiding the question? We're moving on from my original question, because I want to know how you come up with your notion of "countable". This is relevant to the topic of the thread, infinity. How do you proceed from the notion that "a count" is the activity of counting, to the conclusion that zero objects are countable, or that an infinite amount of objects are countable? It seems to me, that to do this you would need to change the definition of "a count".
Quoting TonesInDeepFreeze
Do you realize, that within a logical system you cannot change the "sense" of a word without the fallacy of equivocation? I think therefore, that we have started with a faulty definition of "a count", your definition (1). If we are going to say that zero objects is a countable number of objects, then we need a definition of "count" which is consistent with this.
Should we try definition (2), the result of a count? How many books are on the shelf? None. We know that there are zero, without counting any. It's an observation, there is nothing which satisfies the criteria for "book", so we make an empirical claim that there is zero books. This is similar to what I said about seeing two chairs, or seeing that there are five books, without pairing them individually with a number (bijection). To derive the number of a specified object, we do not need to count (def 1) the objects. Cleary then, 0 is not the result of an act of counting Can we assume that numbers do not represent "a count" at all, nor do they represent the result of a count, they represent empirical observations? Otherwise, we need a definition of "count" which could be consistently applied, and this doesn't seem possible.
Quoting TonesInDeepFreeze
You defined "count" with the activity of counting. And we described counting as requiring objects to be counted. I distinguished a true count from a false count on this basis, as requiring objects to be counted. Clearly, if the objects counted are not actual objects, but imaginary objects, it is not a true count.
I think this helps to demonstrate that we cannot define numbers with counting. So, my original assumption that "2" implies a specified quantity of objects, must be false. But now we have the question of what does "2" mean? I think it is a sort of value, and by my statement above, a value we assign to empirical observations. However, if we can assign such a value to imaginary things in a similar way, we need a principle to establish equality, or compatibility, between observed things and imaginary things. This is required to use negative numbers.
Why do you keep saying I've avoided the question when I have not, when, indeed, I have answered several times and with copious explanation and detail? Possible answers: (1) You are dishonest, (2) You have cognitive problems.
Quoting Metaphysician Undercover
In a formal system, a terminology can be defined only once, so it is not possible to have equivocation.
Quoting Metaphysician Undercover
I didn't use the phrase "countable number of objects".
Quoting Metaphysician Undercover
Correct. I told you that the pedantic technical mention I made does not pertain to the everyday English sense of 'count'.
Quoting Metaphysician Undercover
That is an extraordinary statement, even for you.
We are in a context of everyday English. Then, I have given a mathematical representation of that everyday English sense. A mathematical representation:
A count is a bijection f from a finite set onto a set of successive positive numbers that includes 1. The result of the count is the greatest number in the range of the bijection. The count induces an order on the domain by: x precedes y iff f(x) < f(y).
{<'War And Peace' 1> <'Portnoy's Complaint' 2>} is a count.
The domain is {'War And Peace' 'Portnoy's Complaint'}.
The range is {1 2}.
The result is 2.
The order induced is {<'War And Peace' 'Portnoy's Complaint'>}.
I'm unmoved by your argument. I can't respond at all. I don't think you've said anything meaningful here. You can't count the harpooneers on the Pequod without engaging in deception? If I'm building a house and the architect shows me the plans and I count the doors, I'm not really counting the doors? This is an intellectual point you want people to take seriously?
Quoting Metaphysician Undercover
You have a better idea? You reject it wholesale? You disagree with the famous measurement of the magnetic moment of the electron, the most accurate physical experiment ever done, accurate to within 7.6 parts in [math]10^{13}[/math]? When shown this result you say, "Pish tosh, those quantum mechanics don't know jack."
I want to be clear in my mind. Is this your position on the subject?
Read my last post.
I'm thinking that I've read your last post.
Then why do you ask me to repeat myself?
Look, I think it's very important for a rigorous mathematics to distinguish between counting real things, and counting imaginary things. This is because we have no empirical criteria by which we can determine what qualifies as a thing or not, when the things are imaginary. Therefore we can only count representations of the imaginary things, which exist as symbols. So we are not really counting the imaginary things, but symbols or representations of them, and we have empirical criteria by which we judge the symbols and pretend to count the imaginary things represented by the symbols. But this is not really counting because there are no things being counted. We simply assume that the symbol represents a thing, or a number of things, so we count them as things when there really aren't any things there at all.
So counting imaginary things by means of symbols is completely different from counting real things because one symbol can represent numerous things, like "5" represents a number of things. And we aren't really counting things, we are inferring from the symbol that there is an imaginary thing, or number of things represented by the symbol, to be counted. So it's a matter of faith, that the imaginary things represented by the symbol, are really there to counted. But of course they really are not there, because they are imaginary, so it's false faith.
To begin with in all that, what's your definition of "real thing"?
He didn't.
LOL. First of all, I did actually scroll back to read your last post, and it totally failed to address the question I asked you, which was whether your claimed disbelief in quantum physics causes you to reject the most accurate physical experiment ever done, namely the calculation and experimental verification, good to 13 decimal places, of the magnetic moment of the electron. You simply ignored the question.
And where the question is coming from is that since your understanding of the concept of number is stuck back about 2500 years ago, it makes me wonder if your understanding of physics is similarly ancient.
But that's actually not what my most recent remark meant. When I wrote,
Quoting fishfry
I meant it sarcastically. As, "I have read your posts for the last time." Funny that you entirely missed that.
Quoting Metaphysician Undercover
Actually you couldn't be more wrong about that. Pure math doesn't care what you count. If you count chickens, you're a farmer. If you count harpooneers in Moby Dick, you're a professor of English literature or a high school student reading the Cliff notes. If you count molecules you're a chemist; if you count quarks your a physicist. But if you study the act of counting itself, utterly without regard to the thing being counted, then you are a mathematician.
Once again you utterly fail to understand the nature of mathematics yet wield your ignorance like a cudgel.
Quoting Metaphysician Undercover
To the chemist, physicists, or professor of English literature, this may well be true. But to the mathematician, it's utterly irrelevant. Mathematicians study the natural numbers; in particular their properties of quantity (cardinals) or order (ordinals). What they are counting or ordering is not important. And to the extent that it ever is, the things that mathematicians count are ALWAYS imaginary. We count the rational numbers (same quantity as the naturals) or the reals (a higher cardinality). There's no claim that these things "exist" like rocks or planets or even quarks. How you fail to understand this yet regard yourself as having insight into the philosophy of mathematics, I can't figure out.
Quoting Metaphysician Undercover
It's perfectly true (or at least I'm willing to stipulate for sake of conversation) that the things mathematicians count are imaginary. Though I could easily make the opposite argument. The number of ways I can arrange 5 objects is 5! = 120. This is a true fact about the world, even though it's an abstract mathematical fact. If you're not sure about this you can count by hand the number of distinct ways to arrange 3 items, and you'll find that there are exactly 3! = 6. This is a truth about the world, as concrete as kicking a rock. Yet it involves counting abstractions, namely permutations on a set.
But when you say that imaginary things "exist as" symbols, you conflate abstract objects with their symbolic representations. A rookie mistake for the philosopher of math, I'd have thought you'd have figured this out by now.
Quoting Metaphysician Undercover
Really? You don't think that counting the 120 distinct permutations of five objects is counting imaginary things? I don't believe you actually think that. Rather, I believe that if you gave the matter some actual thought, you'd realize that many of the things mathematicians count are very real, even though abstract. Others aren't. But it doesn't matter, math is in the business of dealing with conceptual abstractions. Math is about the counting, not the things. Farming or chemistry or literature are about the things. The farmer cares about three chickens. The mathematician only cares about three.
How do you not get this?
Quoting Metaphysician Undercover
It's hard to take this line of thought seriously since mathematical practice so obviously falsifies your claim.
Quoting Metaphysician Undercover
Well that's my first point above. To a pure mathematician there is no difference between counting 120 rocks and counting the 120 distinct permutations of five objects. Why don't you understand that?
Quoting Metaphysician Undercover
Nonsense. Abject bullpucky.
Quoting Metaphysician Undercover
No it's not. One need not reify abstract things in order to talk about them. YOU continually try to reify things that need not and should not be reified. I'm coming to see that this is your core error.
Quoting Metaphysician Undercover
Yeah right.
Imaginary things only exist as symbols or representations; that's what makes them imaginary. You therefore acknowledge that we can count imaginary things.
Quoting Metaphysician Undercover
Counting symbols or representations is really counting. If you're not counting imaginary sheep to help you sleep, then what would you call it instead of "counting"?
Let' just say, it's existence is supported by empirical evidence. But we could go to the law of identity for our definition if you want.
Quoting fishfry
Sorry, your question wasn't clear. I'll answer, though it is already answered in the other post. Physicists work with an inadequate representation of space and time. They can't even figure out whether an electron exists as a wave or a particle. When I look up the magnetic moment of an electron on a google search, I get an approximation. So much for your "most accurate" experiment.
Quoting fishfry
See why I didn't answer your question? You don't make yourself clear.
Quoting fishfry
Thanks for providing support to what I am arguing. Counting possibilities, or "possible ways", is completely different from counting things, and therefore ought not be represented by the same word in a rigorous system of logic, to avoid equivocation. Furthermore, since it is a distinct activity, giving the symbols used, (numerals), a distinct meaning, we ought not even use those same symbols. If both, counting real things, and counting imaginary things (possible ways), are understood as the same way of using "counting" then the logical fallacy of equivocation will result. Since the very same numerals are used for both of these very distinct activities, such fallacy is inevitable.
Quoting fishfry
This is false. What the mathematicians are counting with their use of symbols, numerals, is important, because it determines the validity of the logical system they are structuring. Mathematical systems are structured on the principles of the meaning of the symbols, which is derived from how they are, or may be used. So, the study of quantity and order, is restricted by those possibilities.
In the case of quantity for instance the mathematician is restricted by the assumption of discrete units necessary for the count of a quantity. In the case of ordering there is a more complex problem because we need to distinguish which is prior, order or unity. If we can work with all possible orders, with complete disregard for the need of discrete units to be ordered, placing order as prior to unity, then order appears to be unrestricted. But if it is necessary that there must be something which is ordered, for an order to be valid, then we have a set of restrictions which may be applied to order, derived from the principles of quantity.
Therefore, what they are counting or ordering is very important to mathematicians, because order is always dependent on a judgement of logical priority, and this judgement will be reflected in the logical structure produced. The mathematician cannot proceed without any such judgements, and pretending that no such judgements are involved turns the mathematician into a mathemajician.
Quoting fishfry
Possible things are not real things, and this makes a big difference in how numerals are used. In the one case, we can start with the assumption of infinite possibilities, and restrict the infinite through the use of numerals. In the other case we start with what is real, actual, based on empirical observation, and the principles derived from these observations, to provide the necessary restrictions. Notice the difference. In the former case the restrictions on the possibilities for the use of numerals have no necessity, being completely arbitrary. In the latter case, we have restrictions based in real empirical evidence, and inductive reasoning.
Quoting fishfry
Sure, but how "3" is used is a judgement which the mathematician must make. We can say that it refers to the result of a count, a group of three units, or we can say that it refers to the third in an order. These two uses of "3" are fundamentally different and equivocation produces logical fallacy. If the two are conflated in equivocation the mathematician is a mathemajician. Therefore the honest mathematician must make a judgement of priority in defining what "3" means. Is it referring to a quantity or to an order?
Quoting fishfry
That's exactly why I've argued that there is no such thing as the "pure mathematician". If there is such a thing in the world, we ought to call that person by a better name, the mathemajician, to reveal that this person actually operates with smoke and mirror illusions.
Quoting fishfry
Talking about things is completely different from counting things. When we count things it is implied that rigorous principle of logic are being followed. There is no such implication in talking about things.
Quoting Luke
Call it counting then if you want, but we just spent pages discussing the criteria for "counting",
Quoting Luke
I'd say it's ordering, not counting.
For the intuitionist, truth is synonymous with verification of some sort, meaning that according to this stance there are no unknowable true propositions. This implies that if it is unknowable in principle as to whether the universe is finite or infinite, then there is no transcendental matter-of-fact as to which is the case.
Also, it should be mentioned that the commonest use-case of potential infinity involves an agent querying nature for the value of an unbounded variable and accepting the received response (if any). Therefore it is false to claim that denial of actual infinity entails denial of external reality.
Wait, NOW you believe in ordinals?
I'm going to skip responding to your points. Earlier you said numbers were for quantity and I pointed out that there are other kinds of numbers (in the finite case, the exact same numbers viewed differently) for order. I made my point. Then we got off onto other things. You say you don't think there are pure mathematicians, that you can't count abstract things, that you don't believe in experimental science (since you apparently reject the example I gave) and so forth. I'm out of enthusiasm to continue. Till next time.
Oh dear. Did you not read that section of the thread, where I described the difference between quantity and order? It's odd that you wouldn't read those posts, because they were mostly in reply to you. Here's what I said:
Quoting Metaphysician Undercover
To say that something is a "different matter", from what we were discussing, is not to say that I do not believe in it. I'd ask you to go back and read that section again, but I think it's rather pointless because you do not seem at all inclined to make any effort toward understanding. TonesInDeepFreeze was equivocating, or at best, creating ambiguity between quantity and order, using "2" to mean "second", when counting a quantity of two.
Anyway, here is a further post I made, a few days ago:
Quoting Metaphysician Undercover
TonesInDeepFreeze objected saying that ordering in mathematics requires no spatial or temporal relations, but I disagree with that as I think it can be demonstrated that each and every order imaginable is dependent on a spatial or temporal relation. To the right, left, or any such pattern, is spatial, and any intelligible sense of "prior" is reducible to a temporal relation. I really do not think there is any type of order which is not based in a spatial or temporal relation.
I didn't call it counting; you did. You said: "So we are not really counting the imaginary things, but symbols or representations of them."
If imaginary things only exist as their symbols or representations, and if we are really counting those symbols or representations, then we are really counting the imaginary things.
Quoting Metaphysician Undercover
Why?
It's called "counting sheep" not "ordering sheep". Are you saying we cannot count imaginary things but we can order imaginary things?
How do you account for The Magnificent Seven, The Famous Five, 12 Angry Men, etc.?
They're not called The Magnificent Seventh, The Famous Fifth and 12th Angry Men.
Symbols are not imaginary.
That's right. I never said they were.
Nothing exists as it's representation, or else we would not call it a representation, it would be the thing itself..
Quite possibly I didn't. I only read the posts that generated my mentions. I did not read anything else in the thread. And as I believe I admitted, my cardinal/ordinal remark was only in reference to a single sentence you wrote, and not at all in reference to the larger context of the discussion, of which I was and still remain ignorant.
That also explains why I ran out of steam for engaging in further convo. I made a small point, that there are ordinals as well as cardinals. I was not intending to engage at any deeper level.
Quoting Metaphysician Undercover
I would say that I've made a considerable effort the past several years to understand your point of view. But I agree that I prefer to make the effort in small doses; and on this thread, I reached my local limit. I'm sure we'll do this again in some other thread. But in truth you made so many strange statements here that I saw no basis to continue. When you pooh-poohed the 13-digit accuracy of the measurement of the magnetic moment of the electron, you indicated a dismissal of all experimental science. This is perfectly consistent with your 2200 year old view of math. You have a 2200 year old view of physics as well. I don't wish to argue that point with you.
Quoting Metaphysician Undercover
Thank you for the kind words. I did not see this, as it did not contain an @ before my handle. As I said, I've only looked at posts that contain a mention of my handle.
Quoting Metaphysician Undercover
But since I'm here, let me note that this could not be more false. I already gave the counterexample One can order the natural numbers [math]\mathbb N[/math] with the linear order [math]<[/math], the usual order; or [math]\prec[/math], the "funny order" in which everything is the same as the standard order except that [math]n \prec 3[/math] for all [math]n \neq 3[/math]. This is a purely abstract order relation on the natural numbers. There is no spacial or temporal referent involved. One abstract set, two distinct abstract orders. Absolutely no referents in the physical world (that we yet know of) but of critical importance in mathematical logic, proof theory, and various other abstract branches of math.
You can't claim ignorance of this illustration of the distinction between quantity and order, since I already showed it to you in this thread. So whence comes your claim, which is false on its face, and falls on its face as well?
Quoting Metaphysician Undercover
This also is wrong, since there is no mathematical difference between counting abstract or imaginary objects (sheep, for example, as someone noted) and counting rocks.
Quoting Metaphysician Undercover
Please show me space or time in the [math]\prec[/math] order on the natural numbers.
Quoting Metaphysician Undercover
Those harpooneers don't exist, yet there are three of them; four if you count Ahab's personal harpooneer Fedallah. They can be counted sure as the planets. Even more surely, since there's no committee removing harpooneer-hood from Queequeg, as there is for removing planet-hood from poor Pluto. I remember being on vacation, sitting in the Portland airport reading the New Yorker, and discovering that Pluto was no longer a planet. Counting is not as sure a thing as you'd think. The "truth of our concept of distinct, individual objects," as you put it, turns out to be subject to the vote of a committee. Such is reality these days. A lot more tenuous than you'd think.
Quoting Metaphysician Undercover
I ask again for you to please show me space and time in [math]\prec[/math]. It's an alternate ordering of the natural numbers, an alternate order type in fact, but there is no space or time involved. It's purely abstract.
Quoting Metaphysician Undercover
Who is this "we?" Surely there are many who can argue the opposite. Planck scale and all that. Simulation theory and all that. Of course we "think" of space and time as continuous if we are Newtonians, but that worldview's been paradigm-shifted as you know.
Quoting Metaphysician Undercover
Of course quantity and order are two distinct aspects of the "same numerals" in the finite case. In the transfinite case we use different numerals; [math]\aleph_0[/math] for the cardinal representing the natural numbers; and [math]\omega[/math] to represent the exact same set with its usual order [math]<[/math].
But I don't see your point. Cardinals refer to quantity and ordinals to order. The number 5 may be the cardinal 5 or the ordinal 5. The symbology is overloaded but the meaning is always clear from context; and in any event, the order type of a finite set never changes even if its order does. The distinction between cardinals and ordinals only gets interesting in the transfinite case.
Quoting Metaphysician Undercover
You have just been shown one, namely [math]\prec[/math]. But you haven't actually "just" been shown one. I showed you this example several days ago.
You know, to sum this all up, I understand that it's difficult to give a coherent account of abstract objects. But that doesn't mean they're not important. You use the former to utterly reject the latter, and that forces you into positions that are impossible to defend.
Then what is (represented by) an "imaginary thing"?
There is no equivocation or ambiguity in what I said, Your confusions and implacable dedication to remaining ignorant of mathematics and dreadfully misunderstanding it are your own.
All I saw in you demonstration was a spatial ordering of symbols. I really do not see how to derive a purely abstract order from this. If you truly think that there is some type of order which is intelligible without any spatial or temporal reference, you need to do a better job demonstrating and explaining it.
I assure you, I am very interested to see this demonstration, because I've been looking for such a thing for a long time, because it would justify a pure form of "a priori". Of course, I'll be very harsh in my criticism because I used to believe in the pure a priori years ago, but when such a believe could not ever be justified I've since changed my mind. To persuade me back, would require what I would apprehend your demonstration as a faultless proof.
There is an issue though, that I'll warn you of. Any such demonstration which you can make, will be an empirical demonstration, using symbols to represent the abstract. So the onus will be on you, to demonstrate how the proposed "purely abstract order" could exist without the use of the empirical symbols, or else to show that the empirical symbols could exist in some sort of order which is grounded or understood neither through temporal nor spatial ideas.
I'll tell you something else though, I have opted for a sort of compromise to this problem of justifying the pure a priori, by concluding that time itself is non-empirical, thus justifying the temporal order of first, second, third, etc., as purely a priori. However, this requires that I divorce myself from the conventional idea of time which sees time as derived from spatial change. Instead, we need to see time as required, necessary for spatial change, and this places the passing of time as prior to all spatial existence. This is why I said what I did about modern physics, this position is completely incompatible with the representation of time employed in physics. In conceiving of time in this way we have the means for a sort of compromised pure a priori order. It is compromised because it divides "experience" into two parts, associated with the internal and external intuitions. The internal being the intuition of time, must be separated from "experience" to maintain the status of "a priori", free from experience, for the temporal order. So it's a compromised pure a priori.
Quoting fishfry
I didn't deny the distinction between quantity and order, I emphasized it to accuse Tones of equivocation between the two in his representation of a count as bijection.
Quoting fishfry
That is exactly why I attack the principles of mathematics as faulty. There are empirical principles based in the law of identity, by which a physical, and sensible object is designated as an individual unit, a distinct particular, which can be counted as one discrete entity. There are no such principles for imaginary things. Imaginary things have vague and fuzzy boundaries as evidenced from the sorites paradox. so the fact that "there is no mathematical difference between counting abstract or imaginary objects...and counting rocks", is evidence of faulty mathematics.
Quoting fishfry
As I said, all you've given me is a representation of a spatial ordering of symbols. If you are presenting me with something more than this you'll have to provide me with a better demonstration.
Quoting fishfry
I go both ways on this. Of space and time, one is continuous, the other discrete. But this is another reason why I think physics has a faulty representation of space and time, they tend to class the two together, as both either one or the other.
Quoting fishfry
You might think, that "the meaning is always clear from context", but if you go back and reread TIDF's discussion of counting a quantity, you'll see the equivocation with order.
Quoting Luke
A faulty, self-contradicting set of ideas, which has found a place of acceptance in common parlance. Unfortunately, our language is full of these.
How is an imaginary thing a self-contradicting set of ideas?
That is the second time you've made that false claim. Moreover, I did not couch anything as "counting a quantity".
Yes, I've apprehend this, and I respect it. I know that's why you keep on engaging me. it's not easy to understand unorthodox and unconventional ways of thinking like mine though, so I've seen your frustration. But I do appreciate the effort. I've see the same effort to understand from jgill. I don't think TonesInDeepFreeze quite has that attitude though, and Luke just seems to be always looking for the easiest ways (mostly fallacious) of making me appear to be wrong, no matter what I say.
Quoting fishfry
Let me tell you something. The magnetic moment of an electron is a defining feature of how magnetism effects a massive object. Therefore it is not measured it is a stipulation based in specific assumptions such as a circular orbit. But if the electron's orbit is really not circular, then the stipulated number is incorrect.
You don't need my help with that.
You asserted that natural numbers are not countable because they are imaginary:
Quoting Metaphysician Undercover
You then stated that "we can only count representations of the imaginary things, which exist as symbols."
I pointed out that imaginary things have no existence other than their symbols or representations, so counting the symbols or representations is, in fact, counting the imaginary things. For example, counting imaginary sheep is really counting.
You tried arguing that imaginary things cannot exist only as their symbols or representations, and implied that they must have some real existence despite being imaginary. That is, you implied that all representations or symbols (including those of imaginary things) must necessarily represent or symbolise something real.
You then claimed that imaginary things are "a self-contradicting set of ideas". (Perhaps others will appreciate the irony that ideas are also imaginary things.) Presumably this claim was based on your implied, unfounded stipulation that all representations or symbols must necessarily represent or symbolise something real. So it's no wonder that you failed to tell us why imaginary things are self-contradicting ideas.
After a little lighthearted back-and-forth (/s) over on the Israel/Palestine thread it's nice to return here to the things that really matter!
I have taken your point a long time ago that I can't actually \give a logically coherent definition of a mathematical object that doesn't depend on the contingent opinions of mathematicians. But I can't agree with your apparent extrapolation from that to an apparent rejection of all abstract math.
Quoting Metaphysician Undercover
I'm not enough of a physicist to comment. My point was only that you seemed to reject QM for some reason. I noted that you can't dismiss it so trivially, since QM has a theory -- admittedly fictional in some sense -- but that nevertheless corresponds with actual physical experiment to 13 decimal places. That's impressive, and one has to account for the way in which a fictional story about electrons can so accurately correspond to reality. Of course all science consists of historically contingent approximations. But lately some of the approximations are getting really good. Your dismissal seems excessive.
FWIW I don't think anyone thinks the orbits are circular anymore. They're quantum fields, sort of everywhere at once, and in any particular place only with a calculable probability. Or even worse, we can calculate the probability of where we'll find it if we look. Where it "really" is, is a matter of metaphysics. The question lies outside of science.
Now I agree with you (if this is what you are saying) that this makes the whole enterprise metaphysically suspect in some sense. But you still have to account for the amazing agreement of theory with experiment. We might almost talk about the unreasonable effectiveness of physics in the physical sciences!
I'm taking this from the end of your post and addressing it first to get it out of the way. As I mentioned, I didn't read any posts in this thread that didn't mention my handle. I only responded to one single sentence of yours to the effect that numbers are about quantity. I simply pointed out that there is another completely distinct use of numbers, namely order. Anything else going on in this thread I have no comment on.
Quoting Metaphysician Undercover
I may not be fully aware of the philosophical context of your use of "a priori." Do you mean mathematical abstraction? Because I am talking about, and you seem to be objecting to, the essentially abstract nature of math. The farmer has five cows but the mathematician only cares about the five. The referent of the quantity or order is unimportant. If you don't believe in abstraction at all (a theme of yours) then there's no hope. In elementary physics problems a vector has a length of 3 meters; but the exact same problem in calculus class presents the length as 3. There are no units in math other than with reference to the arbitrarily stipulated unit of 1. There aren't grams and meters and seconds. There's no time or space, just abstract numbers. I don't know how to say it better than that, and it's frustrating to me that you either pretend to not believe in mathematical abstraction, or really don't.
Looking ahead a bit I will try to explain abstract order after I respond to your other remarks.
Quoting Metaphysician Undercover
You can't get civilization off the ground without abstract symbolic reasoning, from language to math. Even music has a notation. But the notation is not the music, I hope you'll agree. Likewise the notational explanation of ordinals won't be the ordinals. I can show you the symbols but you have to hear the music on your own. That's the tricky bit, right?
This is a good point so I'll repeat it. You would not confuse music with its notation; so I hope you'll indulge my mathematical notation in the service of communicating abstract ideas. The ideas aren't the notation. We agree on that. You seem to want to deny the ideas themselves simply because they're abstract. That's the part of your viewpoint I don't understand.
Quoting Metaphysician Undercover
To explain to you the theory of ordinal numbers? I don't know if I can do that but I'll give it a shot.
Quoting Metaphysician Undercover
No. That can't be right. You are correct that I believe that there is a notion of purely abstract order; and that I'm constrained to talk about it using symbols that are not the actual things being talked about. But you can't reject my presentation because of that. You can't deny the quest for justice just because it's an abstract idea that is not literally present in the string "justice." If you reject all human abstractions, there's no point in my starting, because you'll just say, "Oh that's only symbolic representation of things that don't exist." I can't overcome that level of nihiism.
Quoting Metaphysician Undercover
Well, as it happens, Cantor discovered ordinal numbers when he was studying the zeros of Fourier's trigonometric series; which were an abstract mathematical model of heat distribution. That is, one can make the case that if you heat up one end of an iron bar under lab conditions, and carefully measure how the heat travels to the rest of the bar; you will inevitably discover the transfinite ordinals. There is physics behind abstract order theory. Nevertheless, the theory stands on its own as an expression of the notion of purely abstract order.
Quoting Metaphysician Undercover
I see your annoyance. I want to talk about first, second, third, but I don't want to relate them to first base, second base, or third base. I want to regard ordinals as pure ideas that can be arranged in purely abstract order. If you utterly reject that then we're done. All I can do is try to explain how mathematicians view the subject of abstract order. I can't convince you that such a thing exists. But I don't need to. I can fall back on formalism and say that even if it doesn't exist, it's a fun mental pastime, like chess. There's no physical referent for chess but it's fun and educational and some practitioners take it very seriously indeed. But it's not real. I'm sure I've made this analogy before.
Quoting Metaphysician Undercover
There is no need for time or space in math. I can't talk or argue or logic you out of your disbelief in human abstraction.
Quoting Metaphysician Undercover
And yet they get 13 decimal places of agreement between theory and experiment. That has to count for something. It's all we've got. It's helped us to crawl out of caves and build all this. For whatever that's worth.
Quoting Metaphysician Undercover
I can't comment on your conversations in this thread that I didn't read. But as a technical matter, in cardinality theory we care about bijections. In order theory we care about order-preserving bijections.
Quoting Metaphysician Undercover
Doesn't Captain Kirk = Captain Kirk? Look, we're never going to get to ordinals at this rate. I don't know what you mean that the law of identity doesn't apply to fictional entities but there's a whole philosophy of fictional entities that I don't know much about.
Quoting Metaphysician Undercover
You just phrase things like that to annoy me. How can you utterly deny human abstractions? Language is an abstraction. Law, property, traffic lights are abstractions. So is math.
Quoting Metaphysician Undercover
This thread's already long. Do you want me to talk about ordinals or not?
Quoting Metaphysician Undercover
Ok, you reject math, you reject physics. And you miss the distinction between physics and metaphysics, between a mathematical model and the thing being modeled. Whatever. Let me talk about ordinals.
Ordinals as abstract order types
========================
I'll keep this relatively brief since the rest of the post is long. I hope we can talk about ordinal numbers and not have any more endless disagreements about human abstraction. Our capacity for abstraction is one of the foundations of civilization, along with the opposable thumb. It's pointless to argue about it.
Ok first finite sets. You have a class full of school kids. You line them up by height. Or you line them up alphabetically by last name. Two distinct ways of ordering the same set. One cardinality but two distinct orders.
However, you will observe that these two distinct orderings nevertheless have the same order type. By that I mean that there is an order-preserving bijection between the set of kids in height order, and the set of kids in alpha order. You just match the first to the first, the second to the second, and so forth.
It's not hard to believe, and not hard to prove, that any two distinct orderings of a finite set have the same order type; in other words, that there is an order-preserving bijection. or order isomorphism, between the two orders. So orders on finite sets aren't very interesting.
So now, infinite sets. In fact only one infinite set is of interest to us at the moment, the natural numbers [math]\mathbb N = \{0, 1, 2, 3, 4, \dots \}[/math].
I hope you will grant me the abstract existence of this set, else there's nothing to talk about.
And I hope you won't be so tedious as to complain, "Well those dots are bullshit and they don't really stand for anything or mean anything blah blah blah." I pretty much agree with you, literally. The notation is only suggestive of a deeper abstract truth, that of the idea of an endless progression of things, one after the next, with no end, such that each thing has an immediate successor. Again if you want to stand on a soapbox and deny that, there's no point in this conversation. You have to at some level believe -- or at least accept, for purposes of playing the game -- the reality of such an endless progression. [math]\mathbb N[/math] is not the symbol or the list in brackets with the mysterious dots at the end. That's only a notation for the music. I want you to imagine the music, and form an association in your mind between the symbols, and the deeper abstract idea they represent. Surely you must be able to do this, after all you do it just fine using the English language. It's not really any different. Meaningless symbols that stand for abstract ideas. You do it every day with letters and words. It's no different in math.
Now the set of natural numbers [math]\mathbb N = \{0, 1, 2, 3, 4, \dots \}[/math] has no inherent order. You may recall that sets have no order. The set {a,b,c} and the set {b,c,a} are the exact same set. This is in fact the axiom of extensionality in set theory. It says that one of the rules of the game of set theory is that two sets are the same if and only if they have the exact same elements.
So given this set [math]\mathbb N[/math], we would like to put an order on it. What is an order? Well again, we play a symbolic game. We say that an order is a binary relation on a set that's reflexive, antisymmetric, and transitive. These terms are defined in the Wiki article I linked but they're not important. What is important is that they characterize the binary relation that we usually call [math]\leq[/math], the "less than or equal" relationship.
In the present case we also require that the order be total, in the sense that given two elements [math]x[/math] and [math]y[/math], either [math]x \leq[/math] or [math]y \leq x[/math].
And finally for convenience in this context, we prefer to work with the strict order [math]<[/math], which works like "less than or equal" but we disallow the equal; that is, [math]n < n[/math] is disallowed. Again this is all common sense that you already know, the details aren't important.details aren't important.
What IS important is that we have defined order without regard to any external meaning. It's all a formal symbolic game.
Then, we can formally define the symbols 0, 1, 2, 3, ... according to von Neumann's clever idea such that [math]0 \in 1 \in 2 \in 3 \in \dots[/math]
Having done this, we now have a formal definition of each natural number within the rules of set theory; and then we can make the definition: [math]n < m[/math] just in case [math]n \in m[/math].
The point of all this is that we can define the '<' relation without regard to quantity and without regard to time or space or anything physical or meaningful. It's just an arbitrary symbol in the formal game of set theory; in principle no different to saying how the knight moves in chess.
Having done that, we have defined what's known as the usual order on [math]\mathbb N[/math]. It's the order you learned in grade school, the one everyone knows. But the point is that I have defined '<' in such a way that this order relation means nothing at all other than the formal relation of set membership; which in set theory actually has no definition at all. [math]\in[/math] is an undefined symbol, just as point and line are in Euclidean geometry.
Having now stripped the usual order of any meaning, I'm free to define alternate orders like [math]\prec[/math] that I defined earlier; which is the same order as [math]<[/math] except that [math]n \prec 3[/math] for all [math]n \neq 3[/math].
This is clearly an alternate order on [math]\mathbb N[/math], just like lining the kids up by height versus by alpha last name. But in this case, these two orders represent distinct order types. There is in fact no conceivable way to create an order-isomorphism between these two ordered sets [math](\mathbb N, <)[/math] and [math](\mathbb N, \prec)[/math].
We can see this because [math](\mathbb N, <)[/math] has no largest element, and and [math](\mathbb N, \prec)[/math] does: namely, 3.
Now what I have outlined is the mathematical point of view in which:
* Order is an entirely arbitrary and meaningless binary relation on a set;
* That there are multiple possible orders on a given set; and that
* In the case of infinite sets, there can be distinct orders that are also distinct order types; that is, there are distinct orders that can not be put into order-isomorphism with each other.
This is the foundation of the idea of ordinal numbers, which are just order types of sets. And in passing, I hope I have made the point that while you object to the meaninglessness of math; on the contrary, it's the very meaninglessness of math that is essential! We have stripped all notion of external meaning from the order relation; in order to be able to investigate the properties of order without regard to the things that may be ordered.
What you call a vice, math calls a virtue. Meaninglessness, or lack of reference to anything tangible, is the heart of the power of mathematical abstraction.
As Wiles said when he proved Fermat's last theorem at a conference: "I think I'll stop now."
Mote and Beam!
As Miles Davis said to producer Alfred Lion, "Is that what you wanted, Alfred?"
Or, just after Davis, Hancock, Carter and Williams laid down "Thisness" - an exceptionally gorgeous, introspective, haunting modern and abstract ballad - producer Teo Macero said to the group, "Your sandwiches are here."
That's a false quote. I said "we are not really counting the imaginary things, but symbols or representations of them". You said they only exist as symbols, not I.
Quoting fishfry
What I look for, is points within abstract math where improvement is warranted.
Quoting fishfry
The capacity to use mathematics to make very precise predictions does not necessarily indicate an understanding of the activity which is predictable. I often use as an example the capacity of ancient people to predict the position of the sun, moon, and planets, without understanding the orbits of the solar system. Thales apparently predicted a solar eclipse. So an ancient 'scientist' could predict the exact location the sun would rise on the horizon, and one could insist that this justifies a model which assumes that a dragon carries the sun in it's mouth around the earth from sun set to sunrise. Predicting the appearance of objects is completely different from understanding the activity involved.
Quoting fishfry
I know, that's the point. The concept of the magnetic moment of an electron is based in the assumption of a circular orbit, which is an idea known to be faulty. And the whole idea of "spin" in fundamental particles is not any sort of spin at all, because the particles cannot be shown to have any proper spatial area, within which to be spinning. The physicists simply apply the appropriate mathematics which produces the desired predictions, but the models which explain what the mathematics is doing are completely unacceptable, indicating that the physicists are capable of making predictions without knowing what is going on.
Quoting fishfry
That's the power of mathematics. But the experiments and the mathematics are designed for one another, so that the experiments show how good the mathematics is, and the mathematics shows how good the experimenters are. But they are only working with a very small portion of the microscale world, because of limited capacity for experimentation, and attempts to extrapolate show just how inadequate the mathematics, experiments, or both, are for producing a wider understanding.
Quoting fishfry
That is where I started in this thread, with the assertion that numbers are about quantity, but I've changed my mind twice since then. You got me to see the difference between quantity and order, and this difference is why I could not understand Tones' representation of counting as bijection. So, the act of "counting" may be an act of determining a quantity or it may be an act expressing an order. The two are very distinct as you say, but both are commonly referred to as "counting".
But if there are two distinct but related ways of using numerals, and each relates to the same concept "number", then we must proceed toward something further, some other idea which synthesizes the two, into one concept, "number". So I changed my mind again, in that post I asked you to reread I believe. What I said is that I think a number is "a value". This allows that the same value, expressed as "2" for example, can be assigned to a quantity of two, and also to the second in order. Remember what I said about a value. When we say "the same value", there is an equality between two distinct things without saying that the two things are the same thing (as implied by the law of identity).
Here's the quote from that post:
Quoting Metaphysician Undercover
Quoting fishfry
Surely I believe in abstraction, but all abstractions are derived (abstracted) from somewhere, unless they are completely innate. So the abstractions "quantity", and "order" must have some sort of referent themselves which give meaning to the concept. It does not make any sense, even in the context of pure mathematics, to say that there is a quantity of 5 which does not consist of five units. That's the meaning of "a quantity of 5". Even in abstraction there are necessary aspects of the concept which must be fulfilled to account for the meaning of the abstraction. If you are simply talking about "the number 5", and not a quantity of five, then we must look to see what gives "the number 5" its meaning., when it is supposed that it does not signify 5 discrete units. We might suppose that the meaning of "the number 5" is found in an order, it's the fifth. But the number 5 is not necessarily the fifth, and that's why I turned at first, to quantity to see how "the number 5" gets its meaning.
Quoting fishfry
This is not true, because the numbers have meaning, that's the point. You cannot use the number 5 however you please, and say "5+5=8". That is restricted by the meaning. So it's not simply "abstract numbers", it's specific numbers. Each number has its specific meaning or else all numbers would be the same. And when I look at the meaning of any specific number I find that the number either refers to spatially distinct units, 5 of them, or it refers to a temporal order, the fifth. Clearly there is time and space implied with abstract numbers, or else each number would lose its meaning which is specific to it.
Quoting fishfry
I do not want to deny the ideas, I want to understand them. And understanding them is what requires spatial and temporal reference. The number 5 has no meaning, and cannot be understood without such reference.
Quoting fishfry
An abstraction must be intelligible or else it is meaningless, useless. If it can't be understood without spatial or temporal reference, then there clearly is a need for space and time in math, or else all mathematics would be simply unintelligible.
Quoting fishfry
I do not deny human abstractions, I just insist that they are fundamentally distinct, different from objects. An object is a unique particular. An abstraction is a generalization. The two are very different from each other, and ought not be both classed together in the same category as "objects".
Quoting fishfry
Do you agree that this order, "an endless progression of things, one after the next" is a temporal order?
Quoting fishfry
You'll have to do better than a simple assertion here. To say that the natural numbers have no inherent order, is to remove "order" as a defining feature of the natural numbers. Now we are left with quantity as the defining feature. Do you agree? There must be something which gives 5 and 6 meaning, if it's not a specific order, it must be a quantity. So we are not talking about an endless progression of things when we talk about the natural numbers in this way, we are talking about specific symbols, "1", "2", "3", etc., which represent specific quantities. Now we don't have a set of natural numbers, because we have no things, only symbols representing quantities. So the rest of your discussion of order is irrelevant. You have nothing to order, and no order to offer.
Nope.
Quoting Metaphysician Undercover
Luke, learn how to read! The representations, (which is what we count), exist as symbols. I did not say that the imaginary things exist as symbols. You've taken the sentence out of its context so that it appears possible that I might be saying what you claim to interpret. Though context clearly shows otherwise. This is exactly what I mean, you interpret, and represent what I say, in a totally incorrect (not what I intended), strawman way, solely for the purpose of knocking it down. Your MO, to ridicule, is itself ridiculous.
You're just going to gloss over your accusation that I misquoted you, and the fact that I didn't?
I never took your position to be that "imaginary things exist as symbols [and representations]". Quite the opposite. This was, instead, the point I was trying to make and what I was trying to get you to understand. Perhaps you should learn to read.
When counting sheep, the imaginary sheep have no existence other than as imagined sheep. The imagined sheep are representations of real sheep, but they only exist as representations. That is to say, for example:
Quoting Luke
Quoting Luke
There don't need to be any real sheep in order to make the count. One could as easily count unicorns instead of sheep. Or Enterprise captains. Or any other fictional entities.
If you ask me, we need to look at scientifically-confirmed physical limits. Take, for example, the speed of light. It is finite of course at 186,000 mph but nothing can attain that speed. Doesn't this bear an uncanny resemblance to infinity as something unattainable. This is, in my humble opinion, one of the many ways, infinity manifests itself in the universe. I like to call it, as oxymoronic as it sounds, finite infinity - the speed of light is finite but it's infinity as nothing can attain it.
As I said, that's an order, one imagined thing after the other, it's not a quantity.
How is it that we can (really) order imaginary things, but we cannot (really) count imaginary things?
https://en.wikipedia.org/wiki/Natural_numbers_object
And so the notion of elements isn't needed for the pure purpose of constructing abstract numbers.
Does that help?
Category theory is a popular mathematical area. An offshoot of algebra, it can be used as an alternative to establish the foundations of math. It searches for so-called universal properties in various categories. Personally, I find it alien and entirely non-productive in the nitty gritty stuff I study in complex variables.
The Wikipedia page for Category theory gets 575 views/day, a respectable number. The page you linked gets 5 views/day and is classified as low priority (like my math page). So it may not help. But good try.
If "count" is defined as determining the quantity of, then it is an act of measuring. We can't measure imaginary things. But we can describe an order without requiring that measurable things exist in that order, the order itself is imaginary.
You are saying that counting is the same as measuring, but that can’t be right. Otherwise, what unit of measurement do we use to count? Litres? Metres? Hours? Bananas?
Immediately before JFK was assassinated on November 22, 1963, Nellie Connolly, wife of Texas governor John Connolly, sitting in the seat in front of JFK in the presidential limo, turned around and said to JFK, "You can't say that Dallas doesn't love you, Mr. President."
LOL There's a famous off-color joke with a very similar punchline, but nevermind.
Quoting Metaphysician Undercover
I would use the word "abstract" rather than "remove." The point is that by abstracting the concept of order from any particular meaning, we can better study order. That's the entire point of my post, which, if it didn't help, didn't help. The point of abstraction is to take away meaning such as first base, second base, so that we can study first and second abstracted from meaning. That doesn't make abstraction meaningless, it just means that we use abstraction to study concrete things by abstracting away the concreteness.
Quoting Metaphysician Undercover
Well, yes and no. Von Neumann's coding of the natural numbers has the feature that the cardinality of the number n is n. But there are other codings in which this isn't true, for example 0 = {}, 1 = {{}}, etc. So we can abstract away quantity too if we like. But that wasn't the point, Even if I grant you that cardinality provides a natural way of ordering the natural numbers, it's still not the only way.
Anyway I did my best and don't think I can add anything.
Quoting Metaphysician Undercover
I get that you don't believe in mathematical "objects." What do you call them then? What do you call numbers, sets, topological spaces, and the like?
Quoting Metaphysician Undercover
Obviously the abstract mathematical object (oops there's that phrase again) 5 is derived or inspired by the familiar concept of 5. Sort of like Moby Dick, which is entirely fictional, was inspired by an actual whaleboat, the Essex, that was sunk by a whale.
But the 5 that mathematicians study is indeed an abstract object. It's not 5 oranges or 5 planets or 5 anything. It's just 5. That's mathematical abstraction. I guess I'm all out of explanations.
Quoting Metaphysician Undercover
There is no space or time in math. Why can't you accept abstraction? There's space and time in physics, an application of math. There's no space or time in math itself. Is this really a point I need to explain? What am I missing that would let me get through to you on this point? A physicist cares about 5 meters or 5 seconds. The mathematician only cares about 5. It's cardinality, its ordinality. It's role as a natural number, an integer, a rational, or a real number. Its primality. What is the thing on which you and I disagree regarding this?
Quoting sime
It's certainly interesting that one can do set theory without elements.
I'm unsurprised that you dislike CT. Abandoning elements feels a bit like abandoning nouns in ordinary language. It isn't a coincidence that the Turing Machine has been the predominant model of computing over the Lambda calculus - the modular conceptualisation of systems in terms of reusable and independently existing entities or elements is cognitively and practically expedient relative to the holistic structuralism of type theory/CT, albeit at the mere cost of potential philosophical confusion.
Although in the case of comp-sci, the practical expedience has recently moved towards CT due the shift to multi-core parallel programming, where control flows can be containerized, layered and equalized in a logically concise fashion using Monads. I also suspect that some generalisation of CT will become important to AI and machine learning over the next few years, due to it's notational emphasis on processes and interaction.
If nothing else, CT serves as an elitist language for getting ahead in the programming jobs market.
Counting is not "the same as measuring", it's a form of measuring. What is required for measuring is a standard, The standard for counting is "the unit", which is defined as an individual, a single, a particular. So in measuring a quantity (counting) we must make a judgement as to what qualifies as a unit to be counted.
Quoting fishfry
OK, so you define "order" as "having no meaning". That is your starting premise? What's the point? Any meaning you give to it will be logically invalid, as contradictory to that definition. There is nothing to study in a concept which has no meaning.
Quoting fishfry
Of course it makes it meaningless, you just said you take away meaning from it. If you take away all the meaning from "first" and "second", you just have symbols without meaning. If you leave some sort of meaning as a ground, a base, you have a temporal reference, first is before, (prior to) second.
You are using "abstract" in a way opposite to convention. We do not "take away meaning" through abstraction, abstraction is how we construct meaning. There is a process called "abstraction", by which we remove accidental properties to give us essentials, what is necessary to the concept. We do not abstract away the meaning, we abstract what is judged as "necessary" from the concreteness, leaving behind what is unnecessary, "accidental".
Quoting fishfry
Sure, cardinality is not the only possible way of ordering numbers, but if the point is, as you described, to allow for any possible order, then we have to deny the necessity of all possible orders. That is to say that there is no specific order which is necessary. This removes "order" as a defining feature of numbers, because no order is necessary, so numbers do not inherently have order. Therefore order is not essential to the concept of numbers Then, we need something else to say what makes a number a number, or else we just have symbols without meaning.
We could try saying that it is necessary that numbers have an order, but the specific order which they have is not necessary, like we might say a certain type of thing must have a colour, but it could be any colour. But this will prove to be a logical quagmire because it's really just a way of smuggling in a contradiction. It is impossible, by way of contradiction, that something must be a specific colour, and at the same time is possibly any colour. It is only possible that it is the colour that it is. Likewise, it is impossible that numbers must have a specific order, but could possibly be any order, because the order that they currently have, would restrict the possibility of another order.
The point was, that if remove all order, to say that numbers are not necessarily in any order, then we must define the essence of numbers in something other than order. If this is cardinality, then cardinality is not an order.
Quoting fishfry
They are concepts, abstractions. I apprehend a difference between concepts and objects, because concepts are universals and objects are particulars. There is an incompatibility between the two, and to confuse them, or conflate them is known as a category mistake.
Quoting fishfry
It's an idea, and ideas are not objects. I have an idea to post this comment, and this idea exists as a goal. Goals are "objects", or objectives, in a completely different sense of the word. So if you want to say that numbers, as ideas are "objects", we'd have to look at this sense of the word, goals. But it doesn't make too much sense to say that they are objects in this sense, nor does it make any sense at all, to say that numbers, as ideas, are objects in the sense of particulars, because they are universals.
Quoting fishfry
Space and time are themselves abstractions, and these concepts very clearly enter into, and are fundamental to mathematics. Are a circle and a square not a spatial concept, which are mathematical? Is the order of first, second, third, fourth, not a temporal order whish is mathematical? If you seriously think that you can separate mathematical concepts from spatial and temporal concepts, then yes, this is something you really need to explain, because I've been trying to do it for many years and cannot figure out how it's possible. So please oblige me, and explain.
Quoting fishfry
The problem is that "5" means nothing without a spatial or temporal reference. If you think that the mathematician believes that "5" refers simply to the number 5, without any further reference to give the concept which you call the number 5 meaning, then you must believe that mathematicians think that the number 5 is a concept of nothing.
If I'm not mistaken, Von Neumann formalized without 'element' as primitive in 1925.
Would be most interested in a reference or more context.
You have it backwards.
The standard for counting is "the unit".
The standard for measuring is "the unit of measurement".
What unit of measurement is required for counting the natural numbers? Metres? Litres? Hours? Bananas? Obviously, no unit of measurement is required. You can count to ten without having to determine any unit of measurement. Therefore, counting is independent of measuring. Counting is not a "form of" measuring.
@jgill does complex analysis, where CT has made few if any inroads. The theory of fields doesn't make for a good category, as I've heard it put, so the analysts haven't been categorified as the algebraists have.
@Metaphysician Undercover, What he said.
Excellent job of misquoting me, attributing to me things I didn't say, and launching yourself into another irrelevant tirade against math, or abstraction, or the ordinal numbers, or whatever it is you're against.
Perhaps you're right that meaning isn't the correct word. If I said we remove a concept from its worldly or physical referent, would that be better? We care about first, second, third, and not first base, second base, third base. So how would you describe that? I'm focusing on ordinality itself and not the things ordered. So you're right, meaning was an imprecise word.
Quoting Metaphysician Undercover
We have order, without reference to the things ordered. We still have meaning, I'll concede that meaning was the wrong word. What would would you use?
Quoting Metaphysician Undercover
There is no temporal reference.
Quoting Metaphysician Undercover
No not at all. One meter, one fish, one planet is meaning. One by itself is a mathematical abstraction. I'm not entirely sure that it means anything now that you mention it. But we can still study it, and then apply what we learn in the abstract setting to any particularities of interest.
Quoting Metaphysician Undercover
Ok. I agree that I'm having trouble precisely defining abstraction and I sort of see your point. But ordinal numbers are purely about order, but they're not about any particular things being ordered. How would you describe that? It's not meaningless, yet it refers to nothing in the world at all other than the pure concept of order. Which you don't seem to believe in.
Quoting Metaphysician Undercover
No, although we do deny the primacy of any particular order. That is, in order theory, the usual order 1, 2, 3, ... is no more important or special than one of the funny orders like 1 2, 4, ..., 3. Although the 1, 2, 3, ... order is important enough to give it a name, the "standard order" or "usual order" on the natural numbers. But you are correct that in order theory, the process of abstraction does put us in the position of regarding all possible orders as equally valid. Not unlike lining up the schoolkids by height, by alpha last name, by reverse alpha first name, by date of birth, by test score, etc. Each of those orders is equally valid in some particular context, and none is inherently preferred over any other. Right? Surely you'll grant me that. And then further grant me that mathematically, sometimes the usual order on the natural numbers is useful (like in most ordinary usages of math), and other times alternate orders are (like when studying or using the higher ordinals).
Quoting Metaphysician Undercover
Correct correct correct. Although I suspect you're about to object to that! But yes, that is exactly right. One order is as valid as another if we're studying pure order theory; although we DO honor the grade school teachers of the world by giving 1, 2, 3, ... a special name, the standard or usual order.
Quoting Metaphysician Undercover
Absolutely correct. I think you're trying to disprove or invalidate the idea, but actually you're understanding the process perfectly.
Quoting Metaphysician Undercover
Correct correct correct. One order is as good as another, though the standard order has considerable mindshare among the general public and of course among mathematicians too. I'd be pushing the point too far if I denied that the standard order is special. After all in the standard order, the numbers are arranged by cardinality, which important; and the ordinals are arranged by set membership, whichis also important. So yes there IS in fact a "natural" way of characterizing the standard order is important.
Quoting Metaphysician Undercover
Just as being lined up by height is not essential to the concept of school children. Being orderable in one or many ways is an attributed of children and numbers, but it is not essential to the concept.
Quoting Metaphysician Undercover
The question of what is the meaning of numbers is an interesting one. I'm not sure mathematicians concern themselves about it, just like biologists don't spend much time talking about the meaning of cells, or physicists (when they are doing physics) talk about the meaning of quarks. When physicists are doing philosophy, they talk about the meaning of quarks. And when mathematicians are doing philosophy, perhaps they talk about the meaning of numbers. But even on that last point, I'm not too sure.
I don't know what numbers "mean." I had dinner earlier and I don't know what my dinner meant. I know it tasted good. Is that a problem, that I don't know the meaning of dinner? What do you even mean by meaning in this context?
Quoting Metaphysician Undercover
It isn't. We could consider the set [math]\mathbb N[/math] of natural numbers, which has no particular order at all, or that implicitly comes along by convention with its standard order. But order is not essential to numbers, it's imposed afterward. At least in the mathematical formalism. I get that you are drawing a distinction between the mathematical formalism, in which order is secondary to the existence of numbers; and philosophy, in which order is an essential aspect of numbers.
But in the Peano formulation, order is inherent via the successor relation. In the past you've rejected the Peano axioms, but now it seems that you should be happy with them. Because in Peano arithmetic we have 0, and we have S0, and then SS0, and then SSS0, and so forth, and there is an inherent order to the process. Happy now?
Quoting Metaphysician Undercover
A schoolkid must have a height, but it could be any height. But with numbers it's even worse than that. A set of numbers, like a set of anything, has no inherent order. Order is a relation imposed upon a set. The set is logically prior to the order. Yes you are right about that, and I get that you're unhappy about that, but that's how it is. At least in the modern formulation of these matters.
Quoting Metaphysician Undercover
You see it that way. I see it as providing beautifully logical clarity. We have the set of natural numbers, and we have the standard order and we have a lot of other orders, and we can even consider the entire collection of all possible orders, which itself turns out to be a very interesting mathematical object. It's quite a lovely intellectual structure. I'm sorry it gives you such distress.
Quoting Metaphysician Undercover
A contradiction is a proposition P such that both P and not-P may be proven from the axioms. Perhaps you would CLEALY state some proposition whose assertion and negation are provable from the concept of order as I've presented it. I don't think you can.
I would believe that you have some philosophical unease. That's not the same as a contradiction. Can you see that?
Quoting Metaphysician Undercover
But I have not asserted that a set must have any order at all. The set [math]\mathbb N[/math] has no inherent order at all. Just like a classroom full of kids has no inherent order till the teacher tells them to line up by height or by alpha firstname or reverse alpha lastname or age or test score or age. Why can't you see that?
A set has no inherent order. Order is imposed on a set afterward, and only for our own convenience in a given context. Sometimes one order, sometimes another, depending on what we're trying to achieve or express.
Quoting Metaphysician Undercover
Terrible analogy. Physical objects have color, but sets don't have an inherent order. Besides I could play the game of pointing out that while physical objects reflect light of a particular wavelength, their color is a function of the physiology of the visual system of the perceiver. So the color isn't really inherent in the object.
But I won't go there. Rather, I will just note that physical objects do have color (or at least a wavelength that gets reflected when it is hit with white light), and mass, and electric charge, and various other physical parameters. But sets do not have inherent order and this is absolutely fundamental to the nature of sets. Axiom of extensionality again: A set is completely characterized by its elements. Order has nothing to do with it, and a set by itself has no inherent order at all.
Quoting Metaphysician Undercover
Not at all, and I've shown you the example several times. The order sets [math](\mathbb N, <)[/math] and [math](\mathbb N, \prec)[/math] are the same underlying set of elements, each with a different order. Neither order is inherent to the underlying set.
Quoting Metaphysician Undercover
Absolutely agreed. Yes. The essence of a set of numbers is NOT in their order, since we can easily impose many different orders on the same underlying set. Just as the ordering by height is not essential to the classroom of kids, since we can impose a different order; or by letting them loose in the playground at recess, we can remove all semblance of order! Surely you must take this point.
Quoting Metaphysician Undercover
Of course cardinality is not an order, I thought that was abundantly clear long ago. But yes we can order a set by cardinality, if the set consists of elements of distinct cardinality. We can do that. We can order the kids by height, if in fact their heights are all distinct. If two kids have the exact same height then we can't linearly order the class by height.
Quoting Metaphysician Undercover
Ok. But that's not good enough. I asked how do you call mathematical objects like topological spaces. But justice and property are concepts and abstractions, yet they are not mathematical objects.
If you don't like the phrase, "mathematical object," what do you call them? Sure they're an abstraction, but that's way too general. You see that I'm sure.
Quoting Metaphysician Undercover
Ok. I call numbers, sets, topological spaces, Abelian groups, etc., by the collective name mathematical objects. What do you call them? You can't say "abstractions," because justice and property are abstractions that are not mathematical objects. Consider yourself challenged to come up with a better name, if you don't like "mathematical object."
Quoting Metaphysician Undercover
Ok. this is a difference between us. I say 5 is a mathematical object, a very familiar one. Relatively few people know what a topological space is, but every child knows what 5 is. You say it's an idea and not an object. I think you're wrong about that. But we've been arguing this point for a long time.
Quoting Metaphysician Undercover
An object is not a goal. An (American) football is an object, and the goal is to get it across the goal line. You would not say the football is a goal. I think you're way off the mark with your claim that an object is a goal or objective. 5 has no object or purpose. It's just the number 5. A mathematical object. An abstract object, as all mathematical objects are.
Quoting Metaphysician Undercover
We disagree, since I say 5 is a mathematical object. And I don't think you have a good theory to the contrary. And 5 is a PARTICULAR mathematical object. A universal is a "... class of mind-independent entities ..." (Internet Encyclopedia of Philosophy). 5 is not a class of entities, it's a single entity. A mathematical object.
Quoting Metaphysician Undercover
They don't exist anywhere in mathematics. Of course "space" is a technical term very commonly used in math, as in a topological space or Euclidean space or a Banach space etc. But space as conceived in physics, as well as time, do not exist in math. If you would carefully study the axioms of set theory, you will see no references to time or space. Of this I am quite certain.
Quoting Metaphysician Undercover
They're idealized geometric mathematical objects. There are no circles or squares in the world, only approximations to the mathematical ideal.
Quoting Metaphysician Undercover
No, not in the least. How can you say that? That's not even the meaning of the words in everyday speech in the real world. The winner takes first place and the runner up takes second place sometimes (as in a foot race) but not always (as in a weight lifting contest) by being temporally first. You must know this, why are you using such a weak argument? First place in golf goes to the player with the lowest score, not to the player who finishes the course first. This is a TERRIBLE argument you're making here.
Quoting Metaphysician Undercover
No, the onus is on you, as space and time play no role in the axioms and principle of mathematics.
Quoting Metaphysician Undercover
If you stop confusing math and physics you will be enlightened.
Quoting Metaphysician Undercover
Space and time have nothing to do with mathematics. Show me where it says they do. Look up the axioms of set theory (here for instance) and show me time and space. They're not there.
Nor are they even philosophically a part of math. A physicist has 5 meters or 5 seconds. Math just has the number 5. I hope you weren't fooled by grade school when they tell you that Sally had 5 apples and Fred had 3 apples, how many apples do they have together? Once you get past that level, they just ask you to add 5 plus 3. The apples are part of physics or biology or grocery store management. Applications of math, not math.
Quoting Metaphysician Undercover
If I ask you if 5 is prime, can you answer? How did you do that? Did you really have to mentally imagine 5 apples?
Quoting Metaphysician Undercover
No, it's the mathematical object 5. It's the third prime number. It's the order of the group of integers mod 5, and the number of sides of a pentagon. It's the largest positive integer such that all groups of order less than or equal to it are Abelian. That property uniquely characterizes the number 5. Martian mathematicians must necessarily have the same theorem. Likewise the Martians must know that there is no algebraic formula to solve a polynomial equation of degree 5 having integer coefficients. Any sentient race anywhere in the universe must necessarily discover this sooner or later.
Mathematicians have many concrete, if you'll permit that use of the word, ideas and concepts about the abstract mathematical object 5. They can represent it within set theory as a natural number, an integer, a rational, a real, a complex number, or a quaternion. Each such representation is a distinct set. We can identify them all via standard conventions. These representations might NOT be used by Martian mathematicians. The representations are contingent, but the facts about the number 5 are not. Mathematicians could write a dissertation on 5. Wikipedia has a relatively long article on the number 5. And that ain't nothing!
The original paper is in Jean van Heijenoorts's 'From Frege To Godel'.
And it's even classified as Vital. There must be 10K pages on math on Wiki. I wonder how many are added each day?
"Counting the natural numbers", as described here, is a matter of established an order. It is not an instance of counting in the sense of determining a quantity. There are no objects (numbers) being counted.
You are equivocating between these two senses of "counting". To count, in the sense of determining a quantity, is an act of measuring. To "count" in the sense of counting up to ten, is a case of expressing an order, two comes after one, three comes after two, etc.. To call this "counting the natural numbers" is a misnomer because this is nothing being counted, no quantity being determined. That is why we can theoretically "count the natural numbers" infinitely, without end, because we are just stating an order, not determining a quantity.
Quoting fishfry
Let's get this straight. I am not talking physical referents here. I am talking space and time, which are conceptual. The issue is that when we remove the physical referents (required for "counting" in the sense of determining a quantity, as the things counted), for the sake of what you might call purely abstract numbers, the meaning of the numbers is grounded in the abstract concepts of space and time. Numbers no longer refer to physical objects being counted, they refer to these abstract concepts of space and time.
Now, we have only deferred the need to refer to physical existence, because if our conceptions of space and time are inaccurate, and the ordering of our numbers is based in these conception of space and time, then our ordering of the numbers will be faulty as well. You seem to think that in pure mathematics, a logician is free to establish whatever one wants as "an order", but this is not true, because the logician is bound by the precepts of "logic" in order that the order be logical. For example, a self-contradicting premise is not allowed. So there are fundamental rules as to the criteria for "order" which cannot be broken. And even if you argue that the order could be a completely random ordering of numbers, the rule here is that each thing in the order must be a number. And every time a logician tries to escape the rule, by establishing a principle allowing oneself to go outside that rule, there must be a new rule created, or else the logician goes outside the field of logic. And the point, is that if the rule is not grounded in empirical fact (physical existence) the logic produced is faulty, and the proposed rule ought to be rejected as a false premise.
Quoting fishfry
Surely, "first" does not mean "highest quality", or "best", in mathematics, so if it's not a temporal reference, what is it?
Quoting fishfry
Yes that is my point as to how counting order is different from counting a quantity. To count a quantity requires particular things, but to count an order requires only time. However, time is something in the world, and that's why I don't believe in what you call "the pure concept of order".
Quoting fishfry
If order is not essential to numbers, then something else must be, because to be a concept is to be definable according to essential properties. I propose, then that quantity is essential to numbers. Do you agree? If for example you make an order, or a category, of odd numbers, or even numbers, or prime numbers, it is something about the quantity represented by the number which makes it belong in one or more of these categories. If it's not quantity which is essential to numbers, as the defining feature of "number" then what do you think is? You've already ruled out order.
Quoting fishfry
No, I am saying that if order is secondary to the existence of numbers, then quantity must be primary.
Quoting fishfry
That's not true at all, it's the fallacy I referred to. The schoolkid must have height, and that height must be the height that the schoolkid has. Therefore it is impossible that the schoolkid has a height other than the height that the schoolkid has, and very obviously impossible that "it could be any height". To make such a claim is clearly fallacious, in violation of the law of identity, because you are implying that a thing could have properties other than those that it has, saying it could have any property. Obviously this is not true because a thing can only have the properties that it has, otherwise it is not the thing that it is.
Quoting fishfry
It gives me distress to see you describe something so obviously fallacious as "providing beautiful logical clarity". If you consider circumventing the law of identity as beautiful logical clarity, I have pity.
Quoting fishfry
Again, you're continuing with your fallacy. A classroom full of kids must have an order, or else the kids have no spatial positions in the classroom. Clearly though, they are within the classroom, and whatever position they are in is the order which they have. To deny that they have an order is to deny that they have spatial existence within the room, but that contradicts your premise "a classroom full of kids".
Quoting fishfry
Above, is your CLEAR example of contradiction "a classroom full of kids has no inherent order". By saying "there is a classroom full of kids", you are saying that there is an order to these kids, they exist with determinate positions, in a defined space. You contradict this by saying they have no inherent order.
Quoting fishfry
So, if "a set" is like the kids in the classroom, then it must have an order to exist as a set. We can say that the order is accidental, it is not an essential feature, so that the same set could change from one order to another, just like the kids in the class, and still maintain its status as the same set. However, we cannot say that a set could have any order by reason of the fallacy described above, because this is to say that it has no actual order which implies that it does not exist.
Quoting fishfry
No, I don't see that at all. They are all concepts, ideas. By what principle do you say that mathematical concepts are "objects", but concepts like "justice" are not objects. I mean where is your criteria as to what constitutes a conceptual "object". I know it's not the law of identity.
Quoting fishfry
You've never heard "the object of the game"?
Quoting fishfry
So in this context, "first" means best. Clearly this is not how "first" is used in mathematics. In mathematics, "first" has a temporal reference of prior to, as I said, not a qualitative reference as "best". Your attempt at equivocation is not very good, I'm happy to say, for your sake. Ask Luke who is the master of equivocation for guidance, if you want to learn. I think you ought to stay away from that though.
Quoting fishfry
.The problem obviously, is that you, and mathematicians in general, according to what you said above, haven't got a clue as to what a number is. It's just an imaginary thing which you claim is an object. It appears like you can't even tell me how to distinguish the number 4 from the number 5, because you refuse to recognize the importance of quantity. And if you would recognize that it is by means of quantity that we distinguish 4 from 5, then you would see that "4", and "5" cannot each represent an object, because one represents four objects, and the other five objects. Why do you take numbers for granted?
.
You neglected adding, "And you don't care!" :scream:
If I spent my time brooding over this issue I'd not get much math done. It's good there are gurus like you who are willing to navel gaze into this profound mystery and lay the foundations while we flitter about, inconsequential moths circling your flame.
Or, as Jerry Seinfeld reminds us, taking Silver in the Olympics just means you're the best of all the losers.
If a flame be a dumpster fire.
You're the only one who thinks it's a misnomer. Everyone else considers "counting up to ten" to be counting (you also called it "counting", by the way). Why should we care about your unjustified stipulation that counting the natural numbers is not real counting or that real counting must involve "determining a quantity"?
Yes, i call it "counting", but the point is that there's two very distinct senses of "counting" and to avoid ambiguity and equivocation we ought to have two distinct names for the activity, like fishfry explained with the distinct names for the numerals used, cardinals and ordinals.
Quoting Luke
Don't mathematicians and other logicians harbour a goal of of maintaining validity, and avoiding fallacies such as equivocation? If it is the case, that when a person expresses the order of numerals, one to ten, and the person calls this "counting", it is interpreted that the person has counted a quantity of objects, a bunch of numbers, rather than having expressed an ordering of numerals, then the interpretation is fallacious due to equivocation between the distinct meanings of "counting".
The issue which fishfry and I have now approached is the idea of a set without any order. I have argued that this is a contradictory idea because if the set exists as a set, its members must have existence in the order which they have in the existing set. It is only by removing existence from the set that we can say the members of the set have any possible order. But then the set itself is not an actual set, it's just the possibility of a set. This would be like a definition without the necessity of anything fulfilling that definition. We could say it's an imaginary set, whereas a real set has real existing members and therefore a real existing order.
In the case of mathematics the question becomes what is supposed to be in the set, the symbols (numerals) or what the symbols represent (numbers). If it is the latter, then the set can be defined with the symbols, and the members within the set, being imaginary, have no existence, and therefore can be said to have no order, or any possible order. But such a set is necessarily non-existent and imaginary, and it cannot be used to represent any real things in the physical world, because real things have an order.
So we have a distinction to be made between two different uses of "set". We can refer to a group of existing objects which necessarily have an order, as a "set". And this type of set is "countable" in the sense that we can determine the quantity of objects within the set. And we can also can use "set" to refer to an imaginary group of objects, having no order because they have no existence. But this type of "set" is not "countable" in the sense that we cannot determine the quantity of objects within such a set. In other words, any set which is stated as having no order, but only possible orders, ought to be considered as imaginary and therefore of indeterminate quantity.
That's me, the dumpster arsonist. Easiest way to dispose of garbage is to burn it. Not so good for the environment though. But neither is garbage.
You call it "counting" even though you consider it a misnomer to call it "counting" (since there is "nothing being counted")?
Quoting Metaphysician Undercover
You introduced this distinction solely to make the point that one side of the distinction is not real counting. But sure, let's avoid ambiguity and equivocation over the two distinct types of counting (P.S. you think one of them isn't real counting).
Quoting Metaphysician Undercover
You're repeating yourself. This is just another way of saying that real counting must involve "determining a quantity". But what is the justification for your stipulation that counting natural numbers is not real counting or that real counting must involve "determining a quantity"?
The point is to avoid equivocation which is a logical fallacy. Since one sense of "counting" involves counting real things, then why not call this "real counting"?
That's not your point, though. Your point is not merely to avoid equivocation; your point in drawing the distinction between the two senses of "counting" is to discount the sense of "counting the natural numbers", "counting from one to ten", or "counting imaginary things" as not a true sort of counting. You have attempted to argue that the only true sort of counting is "determining a quantity" and/or counting "actual objects". For example:
Quoting Metaphysician Undercover
Quoting Metaphysician Undercover
Quoting Metaphysician Undercover
Quoting Metaphysician Undercover
You have not been using the term "real counting"; I have. My use of "real counting" does not denote "counting real things". "Real counting" denotes genuine counting, as opposed to non-genuine counting.
You have attempted to argue that counting natural numbers, or counting imaginary things, is not true counting, and that to call this "counting" is a misnomer. Your only "argument" has been that true counting must involve "determining a quantity" and/or counting "actual objects". However, you have still provided no justification for this so-called "argument" (i.e. stipulation).
Or in a two-person race, the loser finished second, and the winner finished next to last.
What's the smallest positive integer that Wiki doesn't have an article on? That would make it deserving of its own article!
I see that I can buy a copy, but I didn't find a pdf. I'm wondering if you could summarize. Did von Neumann anticipate the categorical approach to set theory way back on 1925? Wouldn't be surprised, just curious to know, but not curious enough to buy the book. I have half a dozen physical books already stacked up to read. It's so much easer to buy books than to read them. Someday we'll just be able to upload via neural interface.
Right, and the reason why I argued this is that we ought not have two distinct activities going by the same name in a rigorous logical system, because equivocation is inevitable. So, one ought to be called "counting" and the other something else. I propose the obvious, for the other, expressing an order.
And, I provided all the required justification. You just do not accept it. So in your mind it has not been justified. That's the nature of justification, regardless of how sound the argument is, if it is not accepted the proposal does not qualify as "justified".
I however, expect nothing less from you. This is consistent with your previous behaviour. No matter what explanation I provide, as to why specific words ought to be restricted in certain ways, to enhance the epistemic capacity of a logical system, you'll reject it. It's quite clear to me that you reject these proposals because they would incapacitate your principal means of argumentation, which is equivocation.
Now that is very interesting. When you say space and time, I've understood you to be referring to those words as understood in physics. The space and time of the physical world. Which makes sense. You would be claiming that 4 comes before 5 in terms of physical space or time.
But now you are saying that space and time have "conceptual" meaning; at the same time you deny that 5 or other numbers can have conceptual meaning. I confess you've lost me and perhaps lost your own point as well. If space and time are abstract conceptual things, then why can't numbers be also?
Quoting Metaphysician Undercover
How about "inspired by" rather than grounded? As in Moby Dick being a work of fiction nevertheless inspired by a real historical event. Of course we get our concept of number from real, physical things. Nobody's denying that.
Quoting Metaphysician Undercover
Ok, but now I'm confused by your claim that space and time are no longer physical things, but rather conceptual things. Why can't numbers be conceptual things inspired by physical things too?
Quoting Metaphysician Undercover
You've swapped out mathematicians for logicians, and I'm not sure I can accept that. I'm talking about mathematical practice, which goes far beyond logic. Logicism's dead, right?
I am making the point that in order theory, one order is as good as another. An order is ANY relation that's reflexive, antisymmetric, and transitive. That's what an order is in order theory. I didn't make this up, it's on Wikipedia and as someone with a (little) bit of mathematical training, I can confirm that Wiki got this one right. https://en.wikipedia.org/wiki/Order_theory
Of course you are correct that the natural order of the positive integers is 1, 2, 3, ... but that is not the ONLY possible order relation on them, there are many others.
Quoting Metaphysician Undercover
Yes there are. Reflexive, anti-symmetric, and transitive. Those are the rules. Or sometimes we required a strict order for convenience, and deny reflexivity (ie < rather than <=) but that's a small point.
But there ARE rules, and the rules are documented on Wikipedia, and I've pointed them out to you.
Quoting Metaphysician Undercover
Even that's not true. I can put an order on red, green, blue. Say lex order: blue, green, red. Or length order: red, blue, green. In each case the order is reflexive, anti-symmetric, and transitive.
I'm sorry your intuition is challenged on this point, but a big part of learning mathematics is having our intuitions challenged, so that we come out the other end with better intuitions.
Quoting Metaphysician Undercover
I have no idea why you've swapped in logicians. I'm talking about mathematicians. I'm explaining to you how mathematicians define order. I can't help your naive intuitions, I'm trying to dispel them in favor of more clarifying concepts.
Quoting Metaphysician Undercover
The only thing faulty is your intuition about what an order relation is.
Quoting Metaphysician Undercover
Well the "first" element of a total order is an element that is less than any other element. Some orders have a first element, such as 1 in the positive integers. Some orders don't. There's no first positive rational number.
That's what first means.
Quoting Metaphysician Undercover
Now that's funny, as we got off onto this conversation by pointing out to you that numbers can indicate order as well as quantity. But of course ordinals are different than cardinals. Two distinct ordinals can have the same cardinal.
Quoting Metaphysician Undercover
red, blue, green. Three words ordered by length. There is no time involved. You are stuck on this point through stubborness, not rational discourse. The player who finishes first in a golf tournament is the one with the lowest score, NOT the one who races around the course first.
Quoting Metaphysician Undercover
Your belief in mathematics is not required by mathematics. Mathematics can exist in the world side-by-side with your willful ignorance and obfuscation.
Quoting Metaphysician Undercover
I have already given many counterexamples such as rationals, reals, complex numbers, p-adics, hyperreals, and various other exotic classes of numbers studied by mathematicians. What quantity or order does [math]3 + 5 i[/math] represent?
There is no general definition of number in math. That's kind of a curiosity, and it's kind of an interesting philosophical point, and it's also factually true.
Quoting Metaphysician Undercover
There is no particular attributed that's a defining feature of number. The concept of number is a historically contingent opinion of mathematicians. Zero didn't used to be a number, neither did [math]5 + i[/math], and neither did [math]\aleph_{47}[/math]. Today they're all regarded as numbers.
There is no general definition of number; nor is there any particular defining property by which we can say, "This thing is a number," and "That thing isn't." The concept of number is whatever the mathematicians of a given era agree is a number.
That's how it is.
Quoting Metaphysician Undercover
There's no general attribute that uniquely characterizes a thing as a number. What is or is not a number is a matter of historically contingent opinion of mathematicians.
Quoting Metaphysician Undercover
I think we're at a point of diminishing returns in this convo. You're flailing and not saying anything I find interesting enough to even argue with.
Quoting Metaphysician Undercover
Why don't we table this till next time. I've made my point and all you have is mathematical ignorance. That's all you ever have. You've even fatally undermined your own thesis by agreeing that space and time aren't even the space and time of physics, but are rather "conceptual," while denying the same status to numbers.
Quoting Metaphysician Undercover
You haven't seen them in the playground at recess. Of course that's only when I was a kid. These days I gather they don't let the kids run around randomly at recess.
Quoting Metaphysician Undercover
You're flailing and no longer even trying to make a coherent point.
Quoting Metaphysician Undercover
That's manifestly false.
Quoting Metaphysician Undercover
If you don't know that sets have no inherent order, there is no point in my arguing with your willful mathematical ignorance.
https://en.wikipedia.org/wiki/Axiom_of_extensionality
Quoting Metaphysician Undercover
No that is not true. It's entirely contrary to the concept of set. A set has no inherent order. An order is a binary relation that's imposed on a given set. If I have a set and don't bother to supply an order relation, then the set has no order. Sets inherently have no order. That's what a set is. You can sit here all day long and make up your own definitions, but that's of no use or interest to anyone.
Quoting Metaphysician Undercover
It can have any order as long as the order satisfies the properties of a partial or total order.
Quoting Metaphysician Undercover
I'm asking you, if you don't accept the phrase mathematical object, what phrase do you use to name or label conceptual entities that are mathematical, as opposed to conceptual entities like justice that are not mathematical?
Quoting Metaphysician Undercover
That's a different meaning. You're being silly now, unserious.
Quoting Metaphysician Undercover
You're wrong. If you have a set, and you impose an order on the set, and in that order there's an element that's less than every other element, that order may be called the first element. That's the definition.
Quoting Metaphysician Undercover
Ad hominems is all you've got left, I see. I hope you will not mind if I'm done here, nothing new can be said at this point.
Quoting Metaphysician Undercover
Project much?
Quoting Metaphysician Undercover
I distinguish them just fine.
Quoting Metaphysician Undercover
It's been fun chatting. I'm done with this topic. Till next time.
I propose instead that we reserve the term "counting" for counting the natural numbers and counting imaginary things, and that we should use the term "measuring" (instead of "counting") for "determining a quantity".
I trust you will have no problem with this as it avoids any equivocation.
It's too many technicalities to easily summarize. Roughly speaking, primitives:
2-place operation
(x y)
"pairing"
2-place operation
[f x]
"value of the function f at argument x"
constant
A
constant
B
predicate
I-object
"is a function"
predicate
II-object
"is an argument"
predicate
I-II-object
"is a function that is itself also an argument"
Then a lot of axioms with those.
Quoting fishfry
It doesn't seem to me to be a precursor to category theory, but I don't opine.
I surely have not denied that "5" has conceptual meaning. To say that the numeral "5", when it is properly used, must refer to five distinct particular things, is to give it conceptual meaning. It is a universal statement, therefore conceptual. I am not saying that it must refer to one specific group of five, as a name of that group, I am saying that it could refer to any group of five, therefore it is a universal, and this indicates that the "5" in my usage refers to a concept, what you've called an abstraction, rather than any particular group of five.
For example, if I said that to properly use "square", it must refer to an equilateral rectangle, or "circle" must refer to a plane round figure with a circumference which has each point equidistant from its center point, I give these terms conceptual meaning, because I do not say that the words must refer to a particular figure, I allow them to refer to a class or category of figures.
Even if I said that "5" must refer only to one particular group of five, or that "square" must refer only to one particular figure, it could still be argued that this is "conceptual meaning", because to understand this phrase "must refer only to one particular", is to understand something conceptual. In reality any meaning assigned to word usage is conceptual, so this position you've thrust at me, that I deny the conceptual meaning of 5, is nonsense. What I say is that the conceptual meaning given to "5", in some situations, namely that it refers to a type of object called a number (as described by platonic realism), ought to be considered as wrong. Do you accept the fact that concepts can be wrong? For instance, your example of "justice". A group of people could have a wrong idea about what "justice" means. Likewise, a group of people could have a wrong idea about what "5" means.
Quoting fishfry
Why would you want to make this change to "inspired" rather than "grounded"? Logic is grounded in true premises, and this is an important aspect of soundness. If your desire is to remove that requirement, and insist that the axioms of mathematics need not be true, they need only to be "inspired", like a work of fiction, the result would be unsound mathematics. Sure this unsound mathematics might be fun to play with for these people whom you call "pure mathematicians", and I call "mathemagicians", but unsound mathematics can't be said to provide acceptable principles for a discipline like physics.
Quoting fishfry
OK, I assume that "less than" refers to quantity. So we're right back to my original argument then. Numerals like "1", "2", "3", "4", refer to a quantity of objects, "3' indicating a quantity which is less than that indicated by "4", and "first" indicates a lower quantity. How do you propose to remove the quantitative reference to produce a pure order, not grounded in a physical quantity?
Quoting fishfry
If you are grounding your definition of "order" in "less than", as you have, then numbers simply indicate quantity, and your "order" is just implied. It is not the case that "2" indicates "first" in relation to "3" and "4", it is the case that "2" indicates a quantity which is less than the quantity indicated by "3" and by "4". And by your premise, that the "first "is the one which is less than the others, you conclude that "2" is first.
Therefore "order" as you have presented it is not indicated by the numbers, only quantity is indicated by the numbers. Order is indicated by something other than the numbers, it's indicated by your premise that the numeral signifying a quantity less than the others, is first.
Quoting fishfry
You clearly haven't followed what I've been saying, and I realize that I did not make myself clear at all. The point is that if we remove the reference to a quantity of individual objects, from numerals, then the ordering of numbers requires a spatial or temporal reference. You seemed to believe that we could remove the quantitative reference, and have numbers with their meanings understood in reference to order only. Clearly, "less than" does not provide this for us. And your example of the length of the word here, is a spatial reference.
Your other example, of the best score being first is only made relevant through a quantitative interpretation. How is 3 better than 4? Because it's less than. So you have not removed the reference to quantity as the necessary aspect of numerals, to provide a purely ordinal definition. Therefore I am still waiting for you to prove your claims.
Quoting fishfry
You have shown me absolutely nothing in the sense of a number not dependent on quantity for its meaning.
Quoting fishfry
If your point is that "order" is defined by " less than", and this is supposed to be an order which is independent from quantity, then you've failed miserably at making your point.
Quoting fishfry
Obviously, "in the playground" is not "in the classroom", and you're clutching at straws in defense of a lost cause.
Quoting fishfry
Instead of addressing my argument you portray me as mathematically ignorant. It's not a matter of ignorance on my part, it's a refusal to accept a mathematical axiom which is clearly false. So I'd correct this to say that this is an instance of your denial, and willful ignorance of the truth, for the sake of supporting a false mathematical axiom.
Quoting fishfry
Show me that set which has no order then. And remember, there is a difference between a thing itself, and the description of a thing. Therefore to describe a set which has no order is not to show me a set which has no order.
I think you need to make clear what "set" means. Does it refer to a group of things, or does it refer to the category which those things are classed into? The two are completely different. Take your example of "schoolkids" for instance. Does "set" refer to the actual kids, in which case there is necessarily an order which they are in, even if they are running around and changing their order? Or, does "set" refer to the concept, the category "schoolkids", in which case there are no particular individuals being referred to, and no necessary order? Which is it that "set" refers to, the particulars or the universal? Or is "set" just a clusterfuck, a massive category mistake?
Quoting fishfry
You just named it for me. "Mathematical" is the word I use to refer to mathematical concepts. In ethics there are ethical concepts like justice, in biology there are biological concepts like evolution, and in physics there are physical concepts like mass. Why do you think mathematics ought to be afforded the luxury of treating their concepts as if they are objects?
Now you just have a vicious circle. What does the numeral "2" refer to? The imaginary object which is the number 2. What is the number 2? The imaginary object which the numeral "2" refers to. See, vicious circle.
If you give the number 2 meaning, a definition, to validate its existence as a conceptual object, you might say that it means a quantity of two, but then you justify my argument, that counting is counting a quantity of objects, and "2" refers to two objects, not one object, the number 2. If you give the number 2 meaning by saying that it is the number which comes after 1, then you justify my argument that what you are doing is expressing an order, rather than counting. Either way, you'd be validating what I called justification, and you refused to acknowledge as justification. Or, would you like to give the number 2 some other type of definition, to validate its existence as a conceptual object? Prefer just remain within your vicious circle?
Furthermore, you ought to see that there is no need to assume "an object", or "number", as the intermediary between the sign "2", and its definition. When we say "square" there is no need to assume a conceptual object which is a square, as an intermediary between the word "square", and its definition, "equilateral rectangle".
Hi, I didn't read the rest of your post yet but I realized I needed to clarify what I wrote last night.
4 is indeed the cardinal number 4 or the ordinal 4, and 5 is the cardinal or ordinal 5. So 4 and 5 do indeed represent quantities, or orders. I misspoke myself, or rather I failed to adequately address your point.
What I was saying about order is that the usual or standard order on the entire set of natural numbers is not the only possible order. So if I decide to reorder the naturals as 1, 2, 3, 5, 4, 6, ..., where 5 comes before 4, it is still the case that 5 represents a set of five elements. So each individual natural number can be seen as representing a quantity, or an order. For example in the Peano construction, 5 is the successor of 4, which is the successor of 3, etc.
So it's the SET of natural numbers that have no inherent order. But an individual natural number does represent a quantity or order.
On the other hand, rational, real, and complex numbers can't be seen as representing quantity in the same way that natural numbers do. Quaternions are little known, but it turns out that game developers use quaternions because they're the most natural tool for representing 3-D rotations. So you can add "rotation" to quantity and order. Every number represents some quality of interest, but there are more of those than just quantity and order.
Hope this clarifies a point of confusion that I didn't adequately address last night. I'll get to the rest of your post later.
Thanks much.
If the number 2 means "a quantity of two", then how could counting the natural numbers be "expressing an order", as you claim?
Quoting Metaphysician Undercover
If you give the number 2 meaning by saying that it is the number which comes after 1, then you justify my argument that what you are doing is measuring (determining a quantity), rather than counting.
That is, 2 comes after 1 in either case.
Yes ok. I misspoke myself when I was unclear that a SET can be given an arbitrary order, and that no one order is to be preferred above any other; but that nevertheless you are correct that the natural numbers individually are either cardinals or ordinals, referring to quantity or order. You're right about that I should have been more clear.
Quoting Metaphysician Undercover
You're right, 5 refers to the class of sets having 5 elements, or it refers to a canonical representation in set theory of the number 5, or in Peano arithmetic it refers to the successor of 4. Quantity and order are essential aspects of natural numbers. I was wrong not to realize earlier that I should have noted that.
Quoting Metaphysician Undercover
Have I clarified my earlier inaccuracy enough yet? 5 refers to fiveness, quantity or order. I agree.
Quoting Metaphysician Undercover
Does our disagreement on this point go away now that I've clarified my inaccuracy?
Quoting Metaphysician Undercover
Sigh. Less than refers to whatever is on the left side of x < y if '<' denotes a strict order relation.
Quoting Metaphysician Undercover
Ok. You're right, up to a point. Natural numbers refer to quantity or order. But the set of natural number may nonetheless be ordered in many alternative ways.
Quoting Metaphysician Undercover
I hope I've clarified my exposition here. I see that I caused myself trouble by not being more clear earlier.
Quoting Metaphysician Undercover
Sorry that one lost me.
Quoting Metaphysician Undercover
Not for lack of trying.
Quoting Metaphysician Undercover
Ok let's skip all this and hopefully go forward with my admission that natural numbers individually do refer to quantity or order; but (imperative you get this) the SET of natural numbers may be reordered at will.
Quoting Metaphysician Undercover
I agree that natural numbers have a quantitative reference. I don't know why I obfuscated that earlier. My bad.
Quoting Metaphysician Undercover
You swapped out temporal for quantity. Sneaky sneaky.
Quoting Metaphysician Undercover
Quaternions? Transcendentals? p-adics?
Quoting Metaphysician Undercover
Or you've failed miserably to understand my point.
Quoting Metaphysician Undercover
You're hanging on to the other end of the straw.
Quoting Metaphysician Undercover
I based my statement on your general mathematical ignorance, and the way you use it as a weapon in debate.
Quoting Metaphysician Undercover
{a, b, c}.
Or [math]\mathbb N]/math] without any particular order being implied.
Quoting Metaphysician Undercover
But no set has order. That's the axiom of extensionality. Will you kindly engage with this point?
Quoting Metaphysician Undercover
LOL. For purposes of our discussion, anything that satisfies the ZF or ZFC axioms. The very first of which is the axiom of extensionality which says that sets have no inherent order, being completely characterized by their elements.
https://en.wikipedia.org/wiki/Axiom_of_extensionality
Quoting Metaphysician Undercover
A set is a mathematical set.
Quoting Metaphysician Undercover
Schoolkids are not mathematical sets. I'm using them purely as an analogy.
Quoting Metaphysician Undercover
So I can use the phrase mathematical, but not mathematical objects? But mathematical is an adjective and mathematical object is a noun. You've still not answered the question.
But are you saying that if I call 5 a "mathematical concept" you're ok with that, but NOT with my calling it a mathematical object? Ok, I can almost live with that. Although to me, it's a mathematical object.
I think you need to reread my post. I have no desire to respond to your misinterpretation.
Thanks for the clarification fishfry, but here's a couple more things still to clear up.
To me, the following statements contradict each other.
Quoting fishfry
Quoting fishfry
Which is the case, no set has order, or a set may be ordered in many different ways. Do you apprehend the contradiction? Which is it, ordered in different ways, or not ordered?
Let me go back to my question from the last post. What exactly constitutes "the set"? Is it the description, or is it the elements which are the members of the set. If it is the description, or definition, then order is excluded by the definition. But if the set is the actual participants, then as I explained already they cannot exist without having an order. If the supposed participants have no existence then they cannot constitute the set.
That's why I ask, which is it? Can a set be ordered, or is it inherently without order? Surely it cannot be both.
Quoting fishfry
Let's look at "concept" as a noun, as if a concept is a thing. Do you agree that a concept is the product, or result of conception, which is a mental activity? There's different mental activity involved, understanding, judgement, conclusion, and effort to remember. Would you agree that the effort to remember is what maintains the concept as a static thing, So if a "concept" is used as a noun, and is said to be a thing, it is in the same sense that a memory is said to be a thing. Would you agree that if a mathematical concept is "a thing", it is a thing in the same sense that a memory is a thing?
I see. Allow me to try again.
Quoting Metaphysician Undercover
If you give the number 2 meaning, a definition, to validate its existence as a conceptual object, you might say that it means the number which comes after 1, but then you justify my argument, that counting (e.g counting the natural numbers) is expressing an order, and “2” refers to one of the numbers in that order, the number 2.
Quoting Metaphysician Undercover
If you give the number 2 meaning by saying that it means a quantity of two, then you justify my argument that what you are doing is measuring, rather than counting.
In plain terms, your argument is like saying that there is a difference in meaning between beating a drum and beating eggs, therefore we shouldn’t use the same word “beating” for both of these, and eggs are not genuinely beatable. But of course you can “beat” both eggs and a drum despite the difference in meaning, and despite your protestations, and attempts to restrict language, to the contrary. We don’t “beat” eggs the same way that we “beat” a drum, but neither do we “beat” a drum the same way that we “beat” eggs. One meaning is not superior to the other.
You might argue that “counting” in the sense of reciting the natural numbers in ascending order is not the proper meaning of the word, but why is it not? Why is “counting” in the sense of determining a quantity the only proper meaning of the word? These are both counting.
In a logical proceeding, it is imperative that the symbol employed maintains the same meaning, to avoid the fallacy of equivocation. If "beating" means something different when used to describe beating eggs, from what it means when used to describe beating drums, and we proceed with a logic process, there could be a fallacious conclusion. For example, after the eggs are beaten, the internal parts are all mixed up into a new order, therefore if I beat the drums the internal parts will become all mixed up into a new order.
Quoting Luke
I explained this already, I think more than once. It is my opinion that there is no such thing as numbers which serve as a medium between the numeral (symbol) and its meaning, or what it represents. So this sense of "counting", which you describe, or define, as "reciting the natural numbers in ascending order", has a false description, or definition. This false definition would act as an unsound premise.
When we "count" in that sense, we are making an expression of symbols. As fishfry has explained, there is no inherent order to those symbols. To say that a particular order is "ascending order", is simply to make a reference to quantity. Therefore the meaning of that sense off "counting" is derived from, or based in that other sense, which is determining a quantity. So determining a quantity is the primary, and proper sense. If we remove that reference to quantity then there is no basis for any specific order, and you cannot say that "counting" involves an "ascending order", because "ascending" is not justified.
What I've been trying to tell fishfry, is that there is better sense of "order" which is not based in quantity, but it is temporal. If we refer to a temporal order, then we need some reason other than quantity to support any proposed order, showing why one symbol ought to be prior in time to another, when we "count" in the sense of expressing an order. This reason for ordering in this manner would provided the alternative name.
I welcome you to provide a non-circular reason for why "determining a quantity" is (true) counting and why "reciting the natural numbers in ascending order" is not (true) counting.
Quoting Metaphysician Undercover
I have no idea what this is supposed to mean or how you think it relates to anything I've said.
Quoting Metaphysician Undercover
To determine a quantity is equally to make reference to an ascending order. Counting in one sense is no different to counting in the other sense. They are both the same sense of "counting".
I explained this already. Your "ascending order" is based on quantity, therefore your supposed "count" of ascending order means nothing unless it is determining a quantity. This is why "numbers" as objects are assumed, so that when you count up to ten you have counted ten objects, (numbers).
Without this assumption that the symbol represents a quantity, you have a meaningless order of symbols which cannot be said to be "ascending". But if you allow that the symbol represents a quantity, then you have an ascending order. However, if the symbol does represent a quantity, then there must be objects which are counted, to validate the use of the symbols. Therefore, it is proposed that numbers are the objects which are counted, to validate the fact that the symbol must represent a quantity.
Therefore your "reciting the natural numbers in ascending order" is nothing but an act of determining a quantity of numbers. And, if numbers are not true objects, as I argue is the case, then this is not a true act of counting at all.
Quoting Luke
This is not true, as I argued with TIDF earlier in the thread. There are many ways to determine a quantity without referencing an ascending order.
I explained this already. Your “quantity” is based on ascending order, therefore your supposed “count” of quantity means nothing unless it is reciting an ascending order.
Quoting Metaphysician Undercover
Quoting Metaphysician Undercover
You assume that numbers are objects but argue that numbers are not objects? Sounds about right given your confusion.
I don't know what specifically MU has in mind that I said, but I have not said anything that could be correctly paraphrased as "There are not many ways to determine a quantity without referencing an ascending order".
The concepts are built up in layers, like a burrito.
At the bottom is the concept of set. A set is informally a collection of objects. Formally, set is an undefined term, just as point and line are undefined terms in Euclidean geometry. We all "know" what the intended meaning is, but when reasoning formally, we can only use their properties as stated by the axioms. Likewise with sets.
By the axiom of extensionality, a set is entirely characterized by its elements, without regard to order. So the set {a,b,c} is the exact same set as {b,c,a} or {c,b,a}.
That's at the lowest layer. Now we want to layer on the concept of order. To do that, we define a binary relation, which I'll call <, and we list or designate all the true pairs x < y in our set. So for example to designate the order relation a,b,c, we would take the base set {a,b,c}, and pair it with the set of ordered pairs {a < b, a < c, b < c}. Then the ordered set is designated as the PAIR ({a,b,c,}, {a < b, a < c, b < c}). I hope this is clear.
For example we have the unordered set [math]\mathbb N[/math] consisting of all the natural numbers 0, 1, 2, ... Then we layer on top of it the usual order <, so that the ordered sets is now [math](\mathbb N, <)[/math]. In other words a set is an unordered collection. An ordered set is a PAIR consisting of an unordered set, along with an order relation.
In the case of the usual order relation on the natural numbers, the order relation < is actually the set of all the true order statements: {0 < 1, 0 < 2, 0 < 3, ..., 1 < 2, 1 < 3, 1 < 4, ..., 2 < 3, 2 < 4, ...}.
In order to remove the apparent ambiguity of using the symbol < as both the relation and as specific instances of it, formally a relation is a set of ordered pairs; so the usual < relation on the natural numbers is actually the set {(0,1), (0,2), (0, 3), ..., (1,2), (1,3), ...}. Again I hope this is clear, it's basic stuff for a math major but is definitely a little formalism-heavy if you haven't seen it before.
The basic takeaway is that a set has no inherent order. We impose an order on a set by PAIRING the set with an order relation. That's why earlier I noted that we can start with the set [math]\mathbb N[/math] and then form two distinct ordered sets [math](\mathbb N, <)[/math] and [math](\mathbb N, \prec)[/math], where [math]<[/math] and [math]\prec[/math] are distinct order relations.
I'll mention in passing that this is a very common pattern in math. We start with a set [math]X[/math], which has no inherent structure at all. Then we let [math]\tau[/math] be a topology on [math]X[/math], and we call the pair [math](X, \tau)[/math] a topological space. Or we have an unstructured set [math]X[/math] and pair it with an operation [math]+[/math], subject to some rules for how [math]+[/math] behaves, and we call the pair [math](X, +)[/math] an Abelian group.
Pretty much everything in math is defined as some set, along with some other structures that impose whatever attributes on the set that we're interested in.
Finally, there's always some notational ambiguity, because when we say, for example, [math]<\mathbb N[/math], we very often mean the set of natural numbers along with its usual order. The meaning is always clear from context. If we were being precise we would always write [math](\mathbb N, <)[/math] for the set [math]\mathbb N[/math] with its usual order; and we would write [math](\mathbb N, <, + *)[/math] for the natural numbers with their usual order and standard arithmetic operations of addition and multiplication.
You don't have to care about the details. What is important is that any "structured set" actually consists, formally, of an unstructured set combined with whatever additional structure we care about: an order relation, a topology, arithmetic operations, and the like. I truly hope this is clear, and if not please ask.
Quoting Metaphysician Undercover
Of course the description is just a representation, as 5 is a representation of the abstract number 5 (whatever that is!) and "snow" is a representation of snow.
A set is entirely characterized by its elements; but a set is more than just its elements. It's the elements along with the collecting of the objects into a set. Maybe that's a bit philosophical, I'm not sure if I can really explain it any better than that. A set is a collection of elements, regarded as an individual thing, a set. So in Peano arithmetic we have numbers 0, 1, 2, 3, 4, ... But the axiom of infinity is much stronger. It says that there is a SET that contains all the numbers. It's the difference between 0, 1, 2, 3, ... and {0, 1, 2, 3, ...}. I hope that's clear, but if you think there's a lot of philosophical mystery that I haven't adequately explained, I'd be inclined to agree. Perhaps it's the distinction between a bunch of athletes and a team, or a collection of birds and a flock. I'm sure some philosophers have found ways to describe this. A set is a collection of elements, along with the concept of their set-hood. That's the best I can do!
Quoting Metaphysician Undercover
Agreed. The order is imposed by PAIRING the set with a separate order relation. Hope I made that clear in my longwinded exposition a moment ago.
Quoting Metaphysician Undercover
The order relation is technically a separate set, consisting of the collection of ordered pairs that define the order. Hope I made that clear already.
Quoting Metaphysician Undercover
A set is inherently without order and without any kind of structure. We impose order and other structures (topology, arithmetic, etc.) on a set by pairing the set with additional relations, which are themselves formally defined as other sets.
And as I've mentioned, we often CASUALLY say "the ordered set X," or "the topological space X", when we REALLY mean the pair (X, <) or the pair (X, [math]\tau)[/math]. That's perhaps the source of some confusion, as I can sometimes mean X the unstructured set, or X the ordered set, or X the topological space, because I'm implicitly leaving out the addition structure with which X is paired.
But formally, every set is inherently unstructured and unordered. We impose structure, order, arithmetic operations, etc., after the fact, by associating the set with other sets that represent order relations, topologies, arithmetic operations, and so forth.
Quoting Metaphysician Undercover
You lost me a bit here. The original question is that you object to my use of the phrase mathematical object, and I'm just asking what to call it instead. If you want me to call numbers, topological spaces, groups, rings, and fields "mathematical concepts" instead of mathematical objects, I'll do that if it makes you happy, but really, they're mathematical objects and universally recognized as such by people trained in math.
Do you understand the meaning of the word "if"? I don't think it's me who's the confused one.
Quoting fishfry
That "elements" may exist without an order is the falsity I've explained to you already. And if we say that "element" indicates an abstraction, then it is a universal, not a particular, and to assume that an abstraction is a particular is a category mistake.
Let's assume a special type of "element", created, or imagined specifically for set theory. This type of element can exist in a multitude without that multitude having any order. Each of these elements would have no spatial or temporal relation to any other element, or else there would be an order, according to that relation. We could say that they are like points, but without a spatial reference, so that we cannot draw lines between them etc., because there is no order to them. But if they were like points, without spatial relations constituting order, there would be no way to distinguish one from another
Unlike points though, there is something which distinguishes one element from another, so that in the set of (a,b,c,), "a" does not represent the same thing as "b" does. Can I conclude, that the distinct elements are separated from one another, and distinguished one from another, by something other than space? To make them distinct and individual, they must have separation, but the separation cannot be spatial or else they would have an order, by that spatial relation.
Do you see, that from this premise alone, we cannot give any order to any set? To give a set an order would be a violation of the fundamental meaning of "element" which allows that elements can exist as particulars without any spatial temporal; relations. To be able to talk about an order within a set, would require that we transform the elements into something other than "elements", something which could have spatial or temporal relations and therefore an order. Remember, even quantity requires spatial-temporal separation between one and the other, to distinguish separate individuals.
Quoting fishfry
No, sorry, it's not clear at all. You have imagined distinct "elements" which exist without any spatial or temporal relations, thereby having no order, though they are somehow distinct individuals. Now you want to add order. You have already defined order out of the set, to add it in, is blatant contradiction.
Quoting fishfry
What I need, is a clear explanation of what an "order relation" is. What type of relation are you attempting to give to these elements, which gives them an order, when you've already stipulated the premise that they have no order?
The point is, that to give them existence without order requires a special conceptualization which I described above. Now if we want to proceed with that conceptualization, and now bring in principles of order, we must do so in a consistent way. So, we need to describe what separates one element from another, since it's clearly not space, and what makes it distinct as an element, in terms which do not give it a relationship to the others, to allow that the multitude of them do not already have any order, Then we need a principle by which order can be initiated within this non-ordered type of separation.
Quoting fishfry
Clearly, for a group of things to be regarded as an individual thing, "unity" is implied. And, it is quite clear that for a group of parts to form a unity, it is necessary that the parts exist with some sort of order. So this statement directly contradicts you assertion that a set has no inherent order.
Quoting fishfry
Do you see the contradiction? You describe a "set" as a thing, a unity, like a team, but then you say that there is no inherent order to this unified thing. Do you see how ridiculous this is, to say that there exists a unified thing, composed of parts, but there is no order to the parts? How in the world are we supposed to conceive of a unity of parts which have no order? To say that they are a unity is to say that they have order.
Quoting fishfry
Take a look again. You are proposing a type of unity, a "set", without any structure. By what principle do you say that it is a collection?
Quoting fishfry
What is this act which you call "the collecting of the objects into a set"? Wouldn't such an act necessarily create an order, if only just a temporal order according to which ones are collected first? I'm trying to figure out how you get around the need for the elements to have an order. I mean, it's one thing to assert, "I've got this collection of elements and they have no inherent order" (to which I'd say you're lying), and another thing to demonstrate how you've collected a group of elements into a unified whole, without them having any order.
As in, if you were arguing that numbers are not objects? But you already told us that you were. You also told us that you assume numbers are objects.
For someone so rabid about logic, you seem quite content with your contradiction.
It's not that sets don't have orderings. It's that sets have many orderings (though in some cases we need a choice axiom or an axiom weaker than choice but still implying linear ordering). So the point is that there is no single ordering that is "the ordering".
/
Typically, in an informal sense, the notion of 'set' is taken as undefined. But just a technical note: In set theory and in class theory, we can formally define 'set' from the primitive 'element'.
Notice, the quoted passage says numbers are assumed when "you" count. And, it's your count that I argue is false. .
You are back to your pathetic strawman misinterpretation, for the sake of ridicule.
Quoting TonesInDeepFreeze
If we assume that a set necessarily has an ordering, but it could be one of many possible orderings, by what principle can we say that each of these many possible orderings constitutes the same set? What type of entity is an "element", such that the identity of a unity of numerous elements is based solely in the identity of its parts with complete disregard for the relations between those parts? Isn't this a sort of fallacy of composition?
I'm sure you meant the impersonal pronoun. This is more obvious with the preceding sentence provided for context:
Quoting Metaphysician Undercover
You assert your stipulation/argument that "ascending order" is based on quantity, and then "This is why "numbers" as objects are assumed" by you. It's your argument and your assumption.
Otherwise, if you were arguing that "my" count is false, then it would also be false that "when you count up to ten you have counted ten objects, (numbers)." This would be odd, since it's the antithesis of your argument in earlier posts. But you are no stranger to contradiction, since you said in the very same post:
Quoting Metaphysician Undercover
Keep blowing smoke trying to hide your contradiction. You clearly cannot account for it.
We prove from axioms.
Quoting Metaphysician Undercover
"constitutes" is your word.
Quoting Metaphysician Undercover
An element is an x such that there exists a y such that xey. In set theory, every x is an element of some y.
Quoting Metaphysician Undercover
No.
Your questions reflect your complete ignorance of set theory. You could remedy your ignorance by getting a book and reading it.
If an axiom is false then the proof is unsound.
Is it possible for an axiom to be false? Please explain. Don't refer to inconsistency. :roll:
Which axioms of finite set theory do you think are false?
Sorry, that was a stupid question. You don't know any axioms.
I think you are just not cut out for mathematical abstraction and should pick another major.
Quoting Metaphysician Undercover
Ditto.
Quoting Metaphysician Undercover
Ditto. I'm not going to bother. You are obfuscatory in the extreme. The only thing I can't figure out is why someone with zero aptitude for mathematical abstraction is so interested in it, yet so utterly unwilling to engage with it.
Quoting Metaphysician Undercover
Yes exactly.
Quoting Metaphysician Undercover
Enough. You win. You wore me out.
Quoting Metaphysician Undercover
I have explained this many times. I linked you to the Wiki page on mathematical order theory. An order is a binary relation; that is, a function that outputs True or False for every pair of elements in a set; that has certain properties as I've mentioned several times already.
Here is the page. Come back when you've made a sincere effort to understand the material.
https://en.wikipedia.org/wiki/Order_theory
Quoting Metaphysician Undercover
I explained that it's a process of abstraction, where we start from no assumptions and layer on the structure we want. If you don't get it, you don't get it.
Quoting Metaphysician Undercover
Pick another major.
Skipping the rest. I've done what I can. I do recommend that you read the page on order theory.
I do get that you reject the mathematical concept of set. Not much anyone can do about that. It's like saying you want to learn physics and then arguing with the concepts of space, time, force, motion, energy, and temperature. You may well have a philosophical point to make, but you are preventing yourself from learning the subject. And it's learning the subject that would allow you to make more substantive rather than naive and obfuscatory objections.
I have explained that set is an undefined term entirely characterized by its behavior under the axioms. You insist on imposing your own incorrect conceptions. So there's no conversation to be had.
For sure it's possible, the difficulty would be to demonstrate falsity, and this would require reference to some sort of inconsistency. What else could demonstrate falsity other than a reference to some form of inconsistency?.
An axiom is expressed as a bunch of symbols, so it must be interpreted. Interpretation requires that it be related to something else, and here we can have inconsistency and contradiction. So the author of an axiom will intentionally avoid internal inconsistency, or contradiction, but to understand, or employ the axiom it must be related to something external to it. If in interpretation, there is a contradiction with another principle then one or both must be false. If the other is a principle one holds to be true, then the axiom must be viewed as false.
Take the axiom of extensionality for example. Here's how Wikipedia states it:
".Given any set A and any set B, if for every set X, X is a member of A if and only if X is a member of B, then A is equal to B.
(It is not really essential that X here be a set — but in ZF, everything is. See Ur-elements below for when this is violated.)"
Further, Wikipedia says it is interpreted like this:
" To understand this axiom, note that the clause in parentheses in the symbolic statement above simply states that A and B have precisely the same members. Thus, what the axiom is really saying is that two sets are equal if and only if they have precisely the same members."
Notice there is an exchange of "equal" and "same". As I've argued in other threads, if we adhere to the law of identity, this is a false use of "same". To resolve this issue, one might deny the law of identity, or insist on a faulty interpretation of that law. I think that approach is futile, so we must look directly at the axiom of extensionality and see what "equal" means in that context. If we can interpret in a way which does not employ "same" we might avoid the falsity.
Quoting TonesInDeepFreeze
Read above.
Quoting fishfry
I found that out at about tenth grade, despite living in an extremely mathematically inclined family. Prior to that though, I had difficulty even in grade school, when the teachers insisted on distinguishing numbers from numerals. Where are these "numbers" that the teacher kept trying to tell us about, I thought. All I could see is the numerals, and the quantity of objects referred to by the numeral. But the teacher insisted no, the numeral is not the number. So it took me very long to figure out that the numeral was not the "number" which the teacher was talking about, and that the number was just some fictitious thing existing in the teacher's mind, so I shouldn't even bother looking for it because I have to make up that fiction in my own mind, for there to be a number for me to "see". Of course, I chose philosophy as a major instead, because philosophy provides a solid, grounded understanding of abstraction, rather than simply insisting on the existence of fictitious "numbers" existing in people's minds, and trying to convince people to create those fictitious things in their minds.
Quoting fishfry
Simply put, I'm right and you're wrong. Nah, nah, let's go back to grade school. You should have chosen philosophy instead of math, if you wanted to learn the truth about abstraction.
Quoting fishfry
I have picked another major, philosophy. That's why I'm discussing this in a philosophy forum. Care to join me? Or will you simply assert that mathematics is far superior to philosophy, then run off and hide under some numbers somewhere when the unintelligibility of your principles is demonstrated to you?
Quoting fishfry
You seem to have a very naive outlook. How do you propose that one proceed toward "learning the subject", when the most basic principles in that subject do not make any sense to the person? To me, the distinction between a numeral and a number is fundamentally unintelligible, as a falsity, because it requires producing a fictitious thing in my mind, and then talking about that fictitious thing as if it is a truth. Therefore proceeding into "learning the subject" requires an initial step of dishonesty, self-deception, then deceiving others in talking about this issue I have deceived myself about. I am not prepared to make that step of dishonesty. Making that initial step of self-deception is the first step toward misunderstanding, not toward understanding.
Falsity is semantic; inconsistency is syntactical.
Given a model M of a theory T, a sentence may be false in M but not inconsistent with T.
Quoting Metaphysician Undercover
Formulas don't have to be interpreted, though usually they are when they are substantively motivated.
Quoting Metaphysician Undercover
It might not be a matter of principles but of framework. Frameworks don't have to be evaluated as true or false, but may be regarded by their uselfulness in providing a conceptual context or their productivity in other ways.
Quoting Metaphysician Undercover
Even though there is nothing wrong with taking 'equal' to mean 'same', the axiom of extensionality doesn't require such mention.
Az(zex <-> zey) -> x=y.
"=' is mentioned, but not "same".
Quoting Metaphysician Undercover
As you argued unsuccessfully, ignorantly and incoherently.
Quoting Metaphysician Undercover
The education system let you down. They should have given you proper cognitve tests to investigate your learning disability.
Quoting Metaphysician Undercover
We don't have to assert such a thing. But understanding mathematics is prerquisite to philosophizing about it.
Quoting Metaphysician Undercover
By the person at least reading the first chapters in a textbook on the subject. If the person cannot comprehend those first basics, then we might have to admit that the person is simply ineducable.
Well in fact that is exactly how one learns math! Once you get past calculus, the early upper division math classes make no sense to students. Abstract algebra and real analysis typically baffle students, until at some point they either get it or give up. It's like saying that learning to play a musical instrument is tremendously difficult at first so people should just give up. On the contrary, you do your scales over and over and over and one day you find that you can play music passably well. It's called learning.
It's true of virtually EVERYTHING that at first, the subject makes no sense. You just do as you're told, do the exercises, do the homework, do the problem sets without comprehension, till one day you wake up and realize you've learned something. It must be that you've learned nothing at all in your life, having given up the moment something doesn't make immediate sense to you.
If you truly wish to criticize the foundations of math, wouldn't it make sense for you to temporarily put aside your objections, and learn the material on its own terms? Especially as you've found someone willing to explain it to you. Then after you have grasped the basic methodologies, such as abstraction and layering properties on top of formless sets -- THEN you are in a better position to make substantive criticisms rather than childish ones.
When you learned to play chess, or any game -- bridge, poker, whist -- do you say, "Oh this is nonsense, no knight REALLY moves this way," and quit? Why can't you learn a formal game on its own terms? If for no other reason than to be able to criticize it from a base of knowledge rather than ignorance? If you've never seen a baseball game, it makes no sense. As you watch, especially if you are lucky enough to have a companion who is willing to teach you the fine points of the game, you develop appreciation. Is that not the human activity called LEARNING? Why are you morally opposed to it?
Finally, even your basic objection to unordered sets is wrong. Imagine a bunch (infinitely many, even) of points randomly distributed on the plane or in 3-space. Can't you see that there is no inherent order? Then you come by and say, "Order them left to right, top to bottom." Or, "Order them by distance from the origin, and break ties by flipping a coin." Or, "Call this one 1, call this one 2, etc."
Where is the inherent order in an otherwise random assemblage of points?
Quoting Metaphysician Undercover
I feel for you. LOL. But you see, you WERE capable of getting it. Or you could just take the formalist perspective and say that the entire thing is a fictional game made up of marks on paper. In which case you couldn't object to math any more than you can object to chess.
Quoting Metaphysician Undercover
So just adopt the formalist perspective. There are only numerals and the rules for manipulating them. It's a game. What on earth is your objection? Were you like this when you learned to play chess? "There is no knight!" "The Queen has her hands full with Harry and that witch Meghan!" etc. Surely you're not like this all the time, are you?
Quoting Metaphysician Undercover
If you can find any instance of my ever asserting such a thing on this forum, then show it. Otherwise you have lied, claiming I said something I never said nor implied in any way.
Quoting Metaphysician Undercover
On the contrary. I'm doing you the service of attempting to explain to you how modern abstract mathematics is set up. And I'm giving up, since it's clear that you'd rather cling to your naive and incorrect beliefs about the subject rather than learn anything.
That the subject at first makes little sense is probably usually true. And it's true for me for many different subjects. But symbolic logic is one subject that made perfect sense to me immediately. Then, after learning the predicate calculus I found that there is a mathematical analysis of it and theorems not just in the predicate calculus but about the predicate calculus. I was blown away with admiration of the human intelligence that would devise such a calculus but also move on to prove its consistency and completeness.
But when I was a child, I was not interested in having to learn arithmetic by rote stipulations. But I was intrigued by some of the other math I was introduced to when I was 10 years old, including sets, Venn diagrams, base number systems and things like that. Those ideas - the abstraction involved - immediately impressed me as like pure cool water of abstract invention. That's what I most admire about mathematics - that it combines full rigor with free exploration of abstract imagination. Later, I was not interested in high school algebra - again just a bunch of stipulations. But in the back of my mind, I wondered what kinds of problems have algorithms (though I didn't know the word 'algrorithm' then) for solving, even if only in principle. Again, years later when I discovered mathematical logic, I found that this question had been investigated thoroughly, and is still being investigated.
Mathematics is not at all one of my intellectual strengths; I'm really not very good at it. But I love it.
The other area that I understood immediately is jazz. The very first time I happend to put on a jazz record to give it real attention, I loved it, understand it, and embraced it for a lifetime.
Sorry Tones, but we're so far apart on these principles of truth and falsity, that I see no place to start, or any point to it. I look at truth as corresponding with reality.
Quoting fishfry
I don't think the analogy is good. I learned to play a musical instrument, and it always made sense to me, right from the start. I learned philosophy and it always made sense to me. The point is that I proceeded because it made sense to. If mathematics requires self-deception, then this does not make sense to me, and so I will not proceed. Many people would not become athletes because there is much physical pain involved in the practice. This might be similar to my refusal to learn math because physical pain, and self-deception may both be viewed as harmful. Some people though will put up with the physical pain because they see a greater good in being an athlete. Maybe you put up with the self-deception of mathematics because you apprehend a greater good.
Quoting fishfry
Again I don't agree with this. Many things I've learned made sense to me right from the start. Even learning the numerals, how to count, and simple arithmetic, addition, subtraction multiplication, division made sense to me right from the start. It was only later, when they started insisting that there existed a number, distinct from the numeral, that things started not making sense.
I had a similar experience later with physics. We learned basic physics, then we learned about waves, and got to experiment in wave tanks. We learned that waves were an activity within a medium and we were shown through diagrams how the particles of the medium moved to formulate such an activity. All of this made very much sense to me. Then we were shown empirical proof that light existed as waves, and we were told that light waves had no medium. Of course this made no sense to me.
Quoting fishfry
When I learn a game, I must learn the rules before I play. If the rules are such that I have no desire to play the game, I do not play. It's not a question of whether the game makes sense or not, so the analogy is not a good one.
Quoting fishfry
Finally, you decided to address the issue. If there are points distributed on a plane, or 3d space, the positioning of those points relative to each other is describable, therefore there is an inherent order to them. If there was no order their positioning relative to each other could not be described..
You say that they are "randomly distributed", to create the illusion that there is no order. But the fact is that they must have been distributed by some activity, and their positioning posterior to that activity is a reflection of that activity, therefore their positioning is necessarily ordered, by that activity.
Quoting fishfry
If you think you can interpret the rules as we go, then I'd advise you not to play any games with me.
Quoting fishfry
The inherent order is the exact spatial positioning shown in the diagram. If any point changes location, then the order is broken. Is that so difficult to understand? A spatial ordering is not a matter of first and second, that is a temporal ordering.
That about sums it up. Math is like religion, a whole bunch of bullshit which we are told to accept on faith.
Whatever your personal meaning of "reality", or lack of meaning, might be.
Meanwhile, you're not even familiar with the distinction between semantics and syntax and the notion of model theoretic truth. So you don't know anything about the mathematics you disdain.
Quoting Metaphysician Undercover
It doesn't.
Quoting Metaphysician Undercover
Exactly, you could perform calculations by rote adherence, but as soon as you confronted actual abstract thought, you couldn't handle it and dealt with it by reviling it.
Quoting Metaphysician Undercover
He didn't suggest anything like that. You're back to one of your favorite tactics again: the strawman argument.
Quoting Metaphysician Undercover
No one but you uses the word "order' that way. But it does allow you to evade the challenge in the example.
I was thinking you'd respond to that with a little self-aware sense of humor.
Quoting Metaphysician Undercover
Could there be some unresolved psychological dynamics at play?
In any event, can you please respond to my point about chess? Surely if you learned to play chess, or any other artificial game -- monopoly, bridge, checkers, baseball -- you were willing to simply accept the rules as given, without objecting that they don't have proper referents in the real world or that they make unwarranted philosophical assumptions. If you could see math that way, even temporarily, for sake of discussion, you might learn a little about it. And then your criticisms would have more punch, because they'd be based on knowledge. I wonder if you can respond to this point. Why can't you just treat math like chess? Take it on its own terms and shelve your philosophical objections in favor of the pleasure of the game.
Mathematics is the opposite of religion. In another post I completely demolished the comparison.
Ok. I can understand that. But I'm offering you a way out. Just take the position of formalism. There are no numbers, only numerals and the rules for manipulating them. Then you can enjoy the game without reifying it. Like chess. Could that be a philosophical viewpoint that would allow you to get past your current impasse?
Quoting Metaphysician Undercover
It makes no sense to anyone else either. This is well known. Especially in terms of quantum fields being "probability waves." That makes no sense to me. Physics has perhaps lost its way. Many argue so. You and I might well be in agreement on this.
Quoting Metaphysician Undercover
Ok. I get that. And I've asked you this many times. You don't want to play the game of math. So then why the energetic objection to it? After all if someone invites me to play Parcheesi and I prefer not to, I don't then go on an anti-Parcheesi crusade to convince the enthusiasts of the game that they are mis-allocating their time on a philosophically wrong pursuit. So there must be more to it than that. With respect to a perfectly harmless pastime like Parcheesi or modern math, one can be for, against, or indifferent. You have explained why you are indifferent; but NOT why you are so vehemently against.
Quoting Metaphysician Undercover
Makes no sense. It's perfectly clear that you can order a random assemblage of disordered points any way you like, and that no one order is to be preferred over any other.
Quoting Metaphysician Undercover
Yes I did say that, and for that reason. We understand each other on this point. The points are placed randomly, so there is no inherent order to their position.
Quoting Metaphysician Undercover
Well yes, the random number generator I used was actually determined at the moment of the big bang, if one believes in determinism. But you're making a point about randomness, not about the order of the points. You are not persuading me with your claim that a completely random collection of points has an inherent order.
Quoting Metaphysician Undercover
You're flailing.
Quoting Metaphysician Undercover
Ok. Perhaps we can put this one to bed now and meet again some other time. You don't want to read the Wiki piece on order theory. You don't want to learn any math, even for the sake of sharpening your own arguments against it. I do believe we've gone as far as we can here. My conscience is clear as to my having made a good faith effort to inform you as to how mathematicians regard the subject of order.
Actually it doesn't make initial sense. Moving from one letter to the next is always a whole step, except from B to C and from E to F. And then double flats move you down a letter except from C to Cbb and from F to Fbb, and double sharps move you up a step except from B to B## and from E to E##.
And on some instruments, when you play at note it's called one particular letter, but on another instrument it's called a different letter, and usually with a flat sign too.
And a 7th chord is not actually the 7th of the scale of the key, but rather it is the minor 7th. But we don't call it a 'minor 7th chord' because that's yet another different chord.
And some intervals in the scale that are not minor nor diminished are called 'major' but others are called 'perfect'.
And a crank (such as you are crank in logic and math) when first confronted with musical notation could come up with nonsense like "a minor second is supposed to be the most dissonant interval, but it's actually two notes that are the closest! It makes no sense!" And the fact that it's enharmonic with an augmented unison. The crank may say, "augmented unison? it's an oxymoron, like empty set!".
And a crank can say, "The major 6th chord is an inversion of a minor 7th chord, so it's two different things! Can't be both major and minor! Doesn't "correspond to reality"! So I'm not going to learn music - it requires that I decieve myself!"
Etc.
Me too. But I well remember my experience in abstract algebra. At first it made no sense whatsoever, and it was that way several weeks into the course, until the textbook mentioned a particular example that related to something I was already familiar with. I said, "Ah, this stuff is actually ABOUT something!" That was a great revelation.
And we are so lucky that people who did actually go on to learn mathematics were not arrested in development as you are. You wouldn't be typing on your computer or enjoying all the other comforts of science and technology if all the mathematicians dropped out of the subject at the mere suggestion that there is a difference between numbers and numerals.
You could see the quantity of objects but not the number of objects?
Quoting Metaphysician Undercover
You must have already understood that the number is not the numeral in order to do simple arithmetic. Otherwise, the addition of any two numbers (i.e. numerals) would always equal 2 (numerals).
I didn't need a teacher to make me aware that numerals are not numbers. '2' and 'two' refer to the same thing. But '2' is not 'two'. So whatever they refer to is something else, which is a number, which is an abstraction. Rather than be a benighted bloviating ignoramus (such as you), I could see that thought uses concepts and abstraction and our explanations, reasoning and knowledge are not limited to always merely pointing at physical objects.
That's not true. I simply don't accept it as a realistic notion of "truth", and don't want to waste my time discussing it.
Quoting fishfry
I didn't answer, because it's not relevant. Philosophy is not a game in which you either accept the rules of play or you don't,, neither is theoretical physics such a game, nor is what you call "pure mathematics" (or as close to "pure" as is possible). In these fields we determine, and create rules which are deemed applicable. So your analogy is not relevant, because the issue here is not a matter of "will you follow the rules or not", it's a matter of making up the rules. And there's no point to arguing that people must follow rules in the act of making up rules because this is circular, and does not account for how rules come into existence in the first place.
Quoting fishfry
Ok, we've found a point of agreement, physics has lost it's way. Do you ever think that there must be a reason for this? And, since physics is firmly based in mathematics, don't you see the implication, that perhaps the root of the problem is actually that mathematics has lost its way.
Quoting fishfry
Physicists, engineers, and others, applying mathematics in the world have a huge impact on the world in which I live, unlike Parcheesi players. Despite arguments that mathematical objects exist in some realm of eternal truth where they are ineffectual, non-causal, I think it is undeniable, that the mathematical principles which are applied, have an impact on our world. I believe it is inevitable that bad mathematics will have a bad effect.
That people vehemently support and defend fundamental axioms which may or may not be true, refusing to analyze and understand the meaning of these axioms, simply accepting them on faith, and applying them in the conventional way, in new situations, with little or no understanding of the situation, or the axioms, to me is a clear indication that bad results are inevitable.
Quoting fishfry
You do not seem to be making any effort to understand this fundamental principle, which is the key to understanding what I am arguing. A group of particles, or dots (we cannot really use "points" here because they are imaginary) existing in a spatial layout, have an order by that very fact that they are existing in a spatial arrangement. Yes, they can be "ordered any way you like", but not without changing the order that they already have. The order which they have is their actual order, whereas all those others are possible orders.
Do you understand and accept this? Or do you dispute it, and know some way to demonstrate how a spatial arrangement of dots or particles could exist without any order? It's one thing to move to imaginary points, and claim to have a specific number of imaginary points, in your mind, which have no spatial arrangement, but once you give them a spatial arrangement you give them order. Even if we just claim "a specific number of points", we need to validate that imaginary number of points without ordering them. This is what Tones and I discussed earlier. How can we count a specific number of points without assigning some sort of order to them? To count them we need to distinguish one from the other by some means or else we do not know which ones have been counted and which have not been counted. So even to have "a specific number of points", imaginary, in your mind, requires that they have an order, or else that specific number cannot be validated.
Quoting fishfry
Yes, I'm making a point about "randomness" because you are using the term "random" to justify your claim that a bunch of dots in a spatial arrangement could have no order. You simply say, the points are "randomly distributed" and you think that just because you say "randomly", this means that there actually could be existing dots in a spatial assemblage, without any order. But your use of the term does not support your claim. There was a process which placed the dots where they are, therefore they were ordered by that process, regardless of whether you call that process "random" or not.
Quoting fishfry
I looked at the Wikipedia entry, and it does not appear to cover the issue of whether existing things necessarily have an order or not. So it seems to provide nothing which bears on the point which I am trying to get you to understand.
Quoting TonesInDeepFreeze
All this, what you say, comes later, it's not "initial". What is "initial" is that you learn a specific fingering, and it sounds good, therefore it makes sense. The part that doesn't make sense is when you move deeper into the theory. This is just like mathematics. The initial practice makes sense, learning addition, multiplication, pi, Pythagorean theorem, etc.. All these simple procedures make perfect sense, you learn a procedure, apply it, and it works. However, then there is layers of theory piled on after the fact, and this is where the sense gets lost, because the theory doesn't necessarily follow what is actually the case.
Quoting Luke
Right, I don't look at two chairs and see the number 2 there.
Quoting Luke
No, the numeral represents a quantity, and a quantity must consist of particulars, or individual things. So "2"" represents a quantity, or number of individuals, two, and "1" represents a quantity of one individual. What is added or multiplied is the quantity or number of individuals. The number is of the individual, a predication, and what is added or subtracted is the individuals, not the number. Have two individuals, add two more individuals, and you have four individuals. See, the operation is a manipulation of individuals, not a manipulation of some imaginary "numbers". And, the fact that we can make the individuals imaginary, such that the manipulation of individuals involves imaginary individuals, does not change the reality that the individuals are what is manipulated, not the numbers.
Quoting TonesInDeepFreeze
When I was seven years old I had no idea what an abstraction is, or what a concept is. I didn't understand this until much later when I studied philosophy. This is why mathematics really is like religion. We are required just to accept the rules, on faith, follow and obey, without any real understanding. 'There really is a number two there, accept and obey'. 'There really is a God there, accept and obey'.
Right, so numbers are not objects?
Quoting Metaphysician Undercover
You said your teacher insisted that "the numeral is not the number" and that you couldn't understand it. But you also said that you had no problem with basic arithmetic. My point was that you must have understood that "the numeral is not the number" in order to do basic arithmetic.
In order to do basic arithmetic you must have already understood what you say here - that ""2" represents a quantity, or number of individuals"; or that "2" represents something other than the symbol itself. What I don't understand is why you had a problem with the distinction between numeral and number if you already understood basic arithmetic.
Quoting Metaphysician Undercover
"Two" and "four" do not refer to numbers? How many is an "individual"?
You want to say that an individual is 1, and add another one to get 2. Therefore, the objects are themselves numbers, or numbers are their objects, right? But "1" or "2" are the number of individuals, not the individuals. You want to pretend that you don't need the abstract concept of number, but you are still using it. Also, you are still vacillating between numbers being objects and numbers not being objects.
That's just a plain contradiction from one sentence to the next.
Dollars to donuts that, without copy/paste from Wikipedia, you could not in your own words state the distinction bewteen syntax and semantics and the notion of truth in a model.
Quoting Metaphysician Undercover
The computer you're typing on and the science and technology that makes your world better are enabled by mathematics. What is an example of a bad effect from mathematics?
Quoting Metaphysician Undercover
But in the philosophy of mathematics, which includes many mathematicians themselves, people do investigate, question, and debate the axioms - giving reasoned arguments for and against axioms. It's just that you are ignorant of that.
Quoting Metaphysician Undercover
That is one of the best, most risible, evasions of a challenge I've ever read. What is "the order they actually have" as opposed to all the others? Saying that they have the order they "actually" have is not telling us what you contend to be the order nor how other orderings are not the "actual" ordering. You are so transparently evading and obfuscating here.
When confronted with the challenge of points in a plane, a reasonable response by you would be "Let me think about that." But instead you reflexively resort to the first specious and evasive reply that comes to you and post it twice with supposed serious intent. That indicates once again your lack of intellectual curiosity, honesty or credibility.
Quoting Metaphysician Undercover
'random' in this context need not have anything other than an informal sense. One could just as well say 'unstated'. You're harping on the word 'random' to evade the heart of the argument against you.
You were presented with points in a plane, without being given a stated particular ordering. You were asked to say what is the "inherent order". You reply by saying, essentially, that their order is the postions they have. But that is not ordering. Ordering, such as a linear order, is a relation in which each object is determined to be before or after another object. And not necessarily temorally or physically. And you can't say what is the "inherent order" even temporally or physically! You fail.
Quoting Metaphysician Undercover
And you comprehended nothing from it.
Quoting Metaphysician Undercover
I want to understand: Are you saying that music theory is wrong just as you say mathematics is wrong? And, by the way, do you know any music theory?
Quoting Metaphysician Undercover
When I was seven I didn't know about abstraction, but I used abstraction. Later, I didn't have to wait until studying philosophy to know about abstraction. You seem to have a condition that prevents you from grasping the notion of abstraction and therefore to revile it.
Quoting Metaphysician Undercover
That is false. It's the opposite. That describes the grade school memorization and regurgitation of tables and rules for basic addition, subtraction, multiplication, and division that you find so suitable. Mathematics though provides understanding of the bases for those rules.
Meanwhile, your own presentation is not by reason but from your own very personal and subjective misundertstandings and dogma. And your use of even common language terms is wildly personal and impossible to negotiate with common meanings. You insist on confused, incoherent, illogical, and self-contradictiory concepts, meanings, and flat out unsupported assertions, expecting that others should accept them while you are ignorant of even the basics of the subject as it has been developed and offered to open scrutiny in a rich peer-reviewed literature.
/
And aside from you specious (essentially vacuous) argument about the points in plane, here are some of the other points still unanswered by you (and these are just the most recent):
Quoting TonesInDeepFreeze
Quoting TonesInDeepFreeze
Quoting TonesInDeepFreeze
Quoting TonesInDeepFreeze
That's a beauty.
Quoting Metaphysician Undercover
It's ironic that if you knew any mathematics, you could have given an answer:
is before
That is a linear ordering.
(But it can be called the 'standard ordering' only by convention. It is no more "inherent" than any other ordering.)
The crank is not necessarily unintelligent. He (it seems that virtually all cranks have male names if their username does suggest gender) may be adept at peforming complicated mathematical operations, computer programming, applied mathematics, engineering and physics. Some cranks got good grades in high school math and even into college. This was a point of pride for the crank. But when the crank was confronted by more abstraction, there was a breakdown. He cannot understand such things as the empty set, material implication (with the FT and FF truth table rows mapping to T), infinite sets, diagonalization, uncountability, incompleteness, and the unsolvability of the halting problem. So when the crank sees other people understanding what he cannot understand, to avoid feeling inadquate, he lashes out with sour grapes that logic and math are all a bunch of nonsense. And the more you try to help him with information and explanation, the more entrenched he becomes in his own world of "they're all wrong; I'll show them who is right!" Then, for him, not only are logicians and mathematicians wrong, but they are knaves and scoundrels (one crank on another forum didn't just want to defund and abolish univerersity mathematics, but he (seriously) advocated mass executions). Internet discussion forums are where the crank lives out his pathetic agenda, and once he claims his perch, he will howl from it forever.
You are obfuscating by sliding between adressing "order" and "actual order" (or "inherent order"). That's typical of your intellectual sloppiness.
It is not the case that there are not orderings. The point though is that there is not a single ordering that is "THE actual ordering". There are many orderings and they are actual even though 'actual' is gratuitious.
You are mentioning me yet again, without quotation or context. This is about the fifth time you've done it.
I never claimed that we can count things that are not distinguishable.
First, of course, is that we may take a collection of dots as given, without stipulating that a particular person placed the dots herself.
Second, let's even suppose that "actual order" is a function of a person placing the dots. Say that Joe places the dots in temporal succession and Val places the dots in a different temporal succession. But that both collection of dots look exactly the same to us. So there's "Joes actual (temporal) order" and "Val's actual (temporal) order", but no one can say which is THE actual order of the collection of dots we are looking at without Joe and Val there to tell us (if they even remember) the different order of placement they used.
This is the magical ideation of crank mathematics. That for all the possible formulas, statements, objects, and states-of-affairs, there are actual people running around creating each of them individually. It's so ludicrous that even a child would know it makes no sense; and it surely does not "correspond to reality".
No, as I explained. The numeral 2 represents how many objects there are. We could also call that symbol the number 2, which represents how many objects there are. There is no need to assume that the number 2 is distinct from the symbol, to do basic arithmetic..
Quoting Luke
If that were the case, I'd be fine with it, but it's not what I was told. I was told that "1" and "2" are numerals, symbols, and there is also something else, called the numbers 1 and 2. The numbers are distinct from the numerals, as what is represented by the numerals. So, I was told that "1" and "2" are symbols, which represent the numbers 1 and 2, and the number represent how many individuals there are. Why not just say that the symbols "'1" and "2" represent how many individuals there are, directly?
Quoting TonesInDeepFreeze
Fishfry posted the order, it's right here:
What more do you want?
Quoting TonesInDeepFreeze
Look, if the dots exist on a plane, they have positions on that plane, and therefore an exact order which is specific to that particular positioning. They do not have any other order, or else they would not be those same dots on that plane. Take a look at that posting of fishfry's and see the order which the dots have, on that plane, and tell me how they could have a different order, or no order at all, and still be those same dots on that same plane.
If you cannot apprehend this simple fact, then tell me what is so difficult for you.
Quoting TonesInDeepFreeze
How can you not see that 'points in a plane without a particular ordering' is a blatant contradiction? If the points exist on a plane, then they each have a particular position on that plane, as demonstrated in fishfry's post, and it is impossible that they have no particular order, because the particular order has been posted. Can you grasp this fact?
Quoting TonesInDeepFreeze
The problem is, that it is stated. It is stated that they exist on a plane. Therefore each point has a position on that plane unique to itself. Not one of these points makes a line, nor occupies a section of the plane, they each have a specific position. Therefore there is necessarily an order to these points, their positions on that plane, according to what is stated. To give them no order you'd have to remove them from their positions on the plane.
Suppose we just assume a multitude of points, without any spatial reference, no dimensions or anything, just points. Then we have the question of what distinguishes one point from another. It is stipulated that there is a multitude of points. If there is no spatial reference, therefore no space separating one point from another, what makes them distinct from one another? How can we assume a multitude of points when we posit no principle whereby one point is distinguished from another point? And if we posit a principle of separation other than space, (suppose one is later in time than another, or something like that), then isn't this a principle of order. it is impossible to posit a multitude of points without implying order.
It isn't I who is evading the issue. All those people who simply assume that it is possible for a multitude of points to exist without any order, are the one's evading the issue, because such a scenario is logically impossible.
Quoting TonesInDeepFreeze
Again, look at fishfry's post:
Do you not see that there is an actual order to those dots on the plane? How could there be "many orderings" if to give them a different order would be to change their positions? Then it would no longer be those dots on that plane. And if your intent is to abstract them, remove them from that plane, then they are no longer those dots on that plane. Why is something so simple so difficult for you to understand?
Quoting TonesInDeepFreeze
OK, but do you agree that something must have caused those dots to be where they are, i.e. given them that order?
Quoting TonesInDeepFreeze
I am talking about their spatial ordering, their positioning on the plane, like what is described by a Cartesian system. Do you not apprehend spatial arrangements as order?
Quoting Metaphysician Undercover
Suppose the number 2 is not distinct from the numeral '2'. Suppose also that the number 2 is not distinct from the Hebrew numeral for 2. Then both the numeral '2' and the Hebrew numeral for 2 are the same. But they are not.
The numerals are not the numbers. If they were, then anybody who used different numerals would be naming different numbers. But they're not. Everybody is naming the same number 2 whether they use the numeral '2' or the Hebrew numeral of the Roman numeral or the word 'two' or any of many other names for the number 2. A child can understand that.
Quoting Metaphysician Undercover
We can, and we do. But also we wish to mention in particular that the number of individuals is 1 or 2 as may be the case.
Quoting Metaphysician Undercover
You are totally confused. That's a picture of dots in a disk. It's not an ordering.
Quoting Metaphysician Undercover
For you to state what you claim to be the "actual ordering of the dots" - as I asked about five times already. And to give reason why that is the "actual ordering" as opposed to other orderings. That means for you to state which dots come before other dots, for each dot. And then say why that ordering you chose is "actual" while other orderings we can choose are "not actual".
Quoting Metaphysician Undercover
A contradiction is a statement and its negation.
One things virtually all cranks have in common is claiming to point out a contradiction when all they're doing is pointing out something that they happen to disagree with. There should be a name for that fallacy.
Anyway, you cannot see that there are many orderings of that finite set of points, but no one of those many ordefings is "THE actual ordering". They ALL are actual orderings. And "actual" is gratuitious anyway.
One more time: There are many different orderings. But not one of them is privileged as being more "actual" or "inherent" than the others. Do really still not understand that?
The rest of your post is just different ways of you repeating your misunderstanding sourced in your not knowing what an ordering is.
Again, you use the word 'ordering' or 'order' in a way that is neither their use in mathematics nor even in everday speech. You assert an entirely personal notion and usage. And your own usage is not even coherent onto itself nor consistently applied by you. Yet you expect everyone else to come around to adopt your personal usage and then also to revise their clear and common notions about basic mathematics to conform to your ignorant, uneducated, and incoherent concept of mathematics. Classic crank to the core.
Quoting TonesInDeepFreeze
Quoting TonesInDeepFreeze
Quoting TonesInDeepFreeze
Quoting TonesInDeepFreeze
/
And new ones:
Quoting Luke
Quoting TonesInDeepFreeze
Quoting TonesInDeepFreeze
Quoting TonesInDeepFreeze
Quoting TonesInDeepFreeze
/
Also, you continue to mention me sometimes without quote or context, placing me in certain roles in the dialectic that I have not taken.
The numeral/symbol represents the number, or “how many”. The symbol is not the number, it is the numeral.
Similarly, the word “tree” represents a tree but the word/symbol is not a tree. The symbol is not the tree, it is the word.
Quoting Metaphysician Undercover
The symbols do represent how many individuals there are. What do you mean by “directly”?
Do you recognize that the word 'tree' is not a tree?
That the word 'Chicago' is not the city of Chicago?
That the word 'courageousness' is not the atrribute courageousness?
That the word 'Ahab' is not a fictional character?
That the word 'liberty' is not liberty itself nor the concept of liberty?
Yes?
But you fail to recognize that the word 'two' or the symbol '2' are not the number 2.
I overlooked this.
The number does not represent how many individuals there are.
The number is how many individuals there are.
Come on TIDF, don't you see that as a ridiculous question? If one could predict the bad things that were going to happen, before they happened, then we could take the necessary measures to ensure that they don't happen. It's like asking me what accident are you going to have today. It's a matter of risk management. If the mathematics employed in any given situation is faulty, the risk is increased. The biggest problem, I think, is the complete denial of the faults, from people like you. This creates a false sense of certainty. That's why it's like religion, you completely submit to the power of the mathematics, with your faith, believing that your omnibenevolent "God", the mathematics would never mislead you.
Quoting TonesInDeepFreeze
The symbols are not the same, nor ought they be said to be the same, or to say the same thing. They ought not be said to say the same thing, because different cultures have different ways of looking at the world. Where's the problem with that? If someone translates a passage of philosophy from ancient Greece, we ought not say that the translation says the same thing as the original. Something is always lost in translation. Likewise, we ought not say that the numeral 2 says the same thing as the Hebrew symbol. This would be very clear to you if you would consider all the different numbering systems discussed on this forum, natural, rational, real, cardinals, ordinals, etc.. The same symbol has a different meaning depending on the system. If we do not keep these distinguished, and adhere to the rules of the specific system, we have equivocation.
Quoting TonesInDeepFreeze
If you refuse to acknowledge that there's an order to those dots, then I don't see any point in proceeding with this discussion.
Quoting TonesInDeepFreeze
Order is not necessarily temporal. And, modern physics looks at time as the fourth dimension of space. So if you cannot see order in an arrangement on a two dimensional plane, I don't see any point in discussing "order" with you.
Quoting Luke
If you follow what is taught in math, the symbol "2" represents a mathematical object which is called a number. The number represents how many individuals there are.
Quoting TonesInDeepFreeze
Of course, the word "tree" might be used as a symbol, to represent a tree.
Quoting TonesInDeepFreeze
You misunderstand. What I am asking is why can't the symbol "2" be used to represent a quantity of two individuals, just like the word "tree" is used to represent a tree? Why must the symbol "2" represent a mathematical object, the number two, and the number two represents a quantity of two individuals? We don't say that the word "tree" represents a conceptual object, tree, and this concept represents the individual tree.
In reality we simply use the word "tree" to represent a tree, and we use the symbol "2" to represent a quantity of two individuals. There is no conceptual object, or mathematical object in between. So if someone states as a premise, that "2" represents a mathematical object, the number two, this would be a false premise.
Quoting Luke
Well no, this is not true. The number is how many individuals it is said that there are. The number is supposed to be what the numeral stands for. It is conceptual, and a representation of a particular quantity of individuals. Being universal, we cannot say that it is actually a feature of the individuals involved, but a feature of our description, therefore a representation. That's why the OED defines "number" as "an arithmetical value representing a particular quantity and used in counting and making calculations." If the number is not a representation of how many individuals there are, but actually "how many individuals there are", there would be no possibility of error, or falsity. If I said "there are 2 chairs", and the supposed mathematical object, the number 2 which is said to be signified by the numeral "2" was "how many individuals there are", rather than how many there are said to be, how could I possibly lie?
A bit of vaudeville relief :lol:
We don't? To which particular object does the word "tree" refer, then?
Quoting Metaphysician Undercover
We seem to have been using "individuals" differently. I was trying to explain to you the concept/meaning of number, and I was considering "individuals" as abstract units, e.g. the (number of) individuals/units represented by the numeral "2". You seem to be using "individuals" to refer to individual objects, or in the application of numbers to particular objects.
Regardless, haven't you answered your own question:
Quoting Metaphysician Undercover
You've been asking why a number must represent objects, yet here you are telling me why a number must represent objects.
Quoting Metaphysician Undercover
Your concern is that if you say that there are 2 chairs, even if there are not 2 chairs, then the world will somehow magically become whatever you say? And, therefore, you will be unable to lie or make any errors because whatever you say will always become true? Oh no. Luckily, that's not how language or the world works.
This was in reference to my question, Why don't you treat math like chess, and accept it on its own terms? And I see no answer here. Math is a game that has standard rules, and many varieties of nonstandard rules. The essence of creativity in math is to make up new rules. That's the history of math, the creation of new rules that violated the old. Because you won't put aside your naive objections long enough to understand math, your objections have no force, because they come across as petulant rather than informed. As far as how the rules came into being in the first place, there is a huge, extensive literature on the subject from Frege and Russell and Zermelo through the modern philosophers of math. A history you have no interest in, because you prefer to remain ignorant. The problem is that you can't make a good case, because your ignorance of the subject shines through above all.
Quoting Metaphysician Undercover
Yes, primarily two. One, we have reached the point where experiments are so expensive as to command a large share of the public treasury. Bill Clinton came into office in 1993 and killed the Superconducting Supercollider. a project that would have reached far higher energies than the Large Hadron collider at CERN. And the next generation of particle accelerators is estimated to come in at over $20B. The expenditures have become a matter of politics, and there are always more worthy and immediate causes to be funded.
Secondly, just as there were a couple of thousand years between Aristotle and Newton, and 250 or so years between Newton and Einstein, it may well be that physics needs to tread water for another couple of centuries before the next breakthrough. We can't hope to have a major revolution every year or even every century.
Quoting Metaphysician Undercover
Not in the least. Math stands on its own. Just as math invented non-Euclidean geometry 70 years before Einstein had any use for it, math today as always is full of meaningless and useless curiosities that may or may not find practical application in the future. Math stands on its own and needs no applicability or practicality to justify itself. This is your fundamental conceptual error. It's not the fault of math that physics is lost. That's the physicists' problem. The mathematicians are doing just fine; and for that matter, are in the midst of a great period of revolutionary turmoil and development in its foundations, to wit category theory, homotopy type theory, computerized proof assistants, and neo-intuitionism. Developments far ahead of anything the physicists care about or even know about.
Quoting Metaphysician Undercover
Then your complaint is with the physicists, engineers, and others; and not the mathematicians, who frankly are harmless. This is your core error. You have no idea what math is about, so you think it's engineering.
Quoting Metaphysician Undercover
Again, your complaint is with those mis-applying math or applying math to bad ends. The mathematicians themselves do work that is so far out there that the only reason you think it has any applicability to the real world is that you have no idea what modern math is or does. If you're upset with applications of math, then your complaint is with those applying it. The math exists on its own, and must be understood and comprehended on its own terms. If you can't do that, your ire is greatly misdirected.
Quoting Metaphysician Undercover
Again, your ignorance betrays you and makes talking to you tedious. There's been 170 years of intensive research into mathematical foundations, starting with the revolution of non-Euclidean geometry. I could point you to Zermelo, or Mac Lane, or Maddy, but what good would it do? You'd rather be ignorant than learn anything. You say that people have accepted the axioms on faith "with little or no understanding," which betrays an ignorance that would deeply embarrass you, if you had any self-awareness of your mathematical and philosophical ignorance.
Quoting Metaphysician Undercover
Particles? Dots? What are those? In math, the elements of sets are other sets. There are no particles or dots. Again, you confuse math with physics.
Quoting Metaphysician Undercover
The very conception of a mathematical set does not include any inherent order. You're just making that up and I'm supposed to sit here several times a week and argue with you about it. It's pointless and tedious. Why don't you learn something about sets instead of showing off your ignorance?
Quoting Metaphysician Undercover
Your delusions about mathematics? Of course not.
Quoting Metaphysician Undercover
I have no idea what you could possibly mean by dots or particles. A set is defined by the axioms of set theory. The axiom of extensionality says that a set is entirely characterized by its elements. Period. That's all anyone needs to know about sets, but you prefer to live in your own fantasy world of dots and particles. You're making it up.
Quoting Metaphysician Undercover
There is no spatial arrangement. You're just throwing an ignorant tantrum about things you refuse to learn.
Quoting Metaphysician Undercover
Validate the imaginary number of points? What does that mean? I have no idea.
Quoting Metaphysician Undercover
I'm not reading much of this thread, only my mentions.
Quoting Metaphysician Undercover
I can't argue with the fantasies in your head. Set theory is what it is.
Quoting Metaphysician Undercover
Counting is a much more sophisticated operation than merely positing the existence of a set. To count, we must have the cardinal or ordinal numbers. To have the cardinal or ordinal numbers, we must conceptually build them up from the basic concept of set, which is as I've tried to describe to you.
[quote="Metaphysician Undercover;544250]
Yes, I'm making a point about "randomness" because you are using the term "random" to justify your claim that a bunch of dots in a spatial arrangement could have no order. [/quote]
There are no dots. I don't know what dots are. I tried to give you a visual example but perhaps that was yet another rhetorical error. I should just refer you to the axiom of extensionality and be done with it, because in truth that is all there is to the matter.
Quoting Metaphysician Undercover
Forget the visual analogy. Now you're just arguing with the analogy and not with the concept of set. A set is entirely determined by its elements. That's rule one of the game. Take it or leave it. I don't care.
Quoting Metaphysician Undercover
You surely did not engage with it.
Quoting Metaphysician Undercover
https://en.wikipedia.org/wiki/Axiom_of_extensionality
Contradiction may be implied. Here's Wikipedia's opening statement:
'In traditional logic, a contradiction consists of a logical incompatibility or incongruity between two or more propositions."
The problem is that you refuse to recognize that an arrangement of points on a plane, logically implies order, therefore "an arrangement of points on a plane without order" is contradictory.
Quoting fishfry
Don't you see that I said math is not like chess. Therefore I do not treat math like chess. I answered your question.
Quoting fishfry
Obviously not, as you've already noticed,
Quoting fishfry
No, my complaint is with the fundamental principles of mathematicians, As explained already to you, violation of the law of identity, contradiction, and falsity. You, and Tones alike (please excuse me Tones, but I love to mention you, and see your response. Still counting?), are simply in denial of these logical fallacies existing in the fundamental principles of mathematics, and you say truth and falsity is irrelevant to the pure mathematicians.
Quoting fishfry
In case you forgot, you posted a diagram with dots, intended to represent a plane with an arrangement of points without any order. This is what I argued is contradictory, "an arrangement... without order". And this was representative of our disagreement about the ordering of sets. You insisted that it is possible to have a set in which the elements have no order. You implied that there was some special, magical act of "collection" by which the elements could be collected together, and exist without any order. What you are in denial of, is that if the elements exist, in any way, shape, or form, then they necessarily have order, because that's what existence is, to be endowed with some type of order.
You tell me, just imagine a plane, with points on the plane, without any order, and I tell you I can't imagine such a thing because it's clearly contradictory. If the points are on the plane, then they have order. And you just want to pretend that it has been imagined and proceed into your smoke and mirrors tricks of the mathemajicians. I'm sorry, but I refuse to follow such sophistry.
Quoting fishfry
Why not give it a try? I can argue with the fantasies in your head, demonstrating that they are contradictory. So please explain to me how you think you can have a collection of elements, points, or anything, and that collection has no order. Take this fantasy out of your head and demonstrate the reality of it.
Quoting fishfry
The dots. I believe, were supposed to be a representation of points on a plane. The points on a plane, I believe, were supposed to be a representation of elements in a set. And you were using these representations in an attempt to show me that there is no inherent order within a set. So, are you ready to give it another try? Demonstrate to me how there could be a set with elements, and no order to these elements.
I've explained to you the problem. You describe the set as a sort of unity. And you want to say that the parts which compose this unity have no inherent order. Do you recognize that to be a unity, the parts must be ordered? There is no unity in disordered parts. Or are you going to continue with your denial and refusal to recognize the fundamental flaws of set theory?
Hopefully others will correct me if I'm wrong but, as I understand it, the point iof the diagram of "dots" is that the elements of the set have no inherent numerical order or sequence. Otherwise, you should be able to number the elements from 1 to n and explain why that is their inherent order.
And now we have another installment of ignorance, confusion, illogic, dishonesty, and trolling from him.
Let's start with the dishonesty:
Quoting Metaphysician Undercover
I don't speak for fishfry, but the second 'you' above appears to include both of us. But I have never said, implied, or remotely suggested, that truth and falsity are irrelevant to pure mathematics. So you are lying to suggest that I did.
Quoting Metaphysician Undercover
I have never claimed that mathematics, or classical mathematics, is exempt from criticisms. Indeed I have said at least a few times that I am interested in discussion of criticisms, including from such tenets as predicativism, constructivism, finitism, formalism, relevance logic, and even paraconsistent logic. Also, the question of infinite regress in meta-theory. Also, objections that set theory is too rococo and overshoots the target of mathematics for the sciences. Also, reverse mathematics. And I have not opined that all these critiques are incorrect and especially I have not opined that they are not worthwhile. So you are lying about me.
Quoting Metaphysician Undercover
I pointed out that continually you mention me without stating the context or quotes, thus making it seem that I have played a certain role or taken a certain position in an unspecified exchange with you. And above you admit that you do this to provoke my response regarding that. That is the very definition of 'trolling'. And you admit it. You're an obnoxious bane.
Quoting TonesInDeepFreeze
Quoting Metaphysician Undercover
I'm not asking you what particular bad things you think will happen, but what kind of bad things. At least you could say what is the general nature of the bad things you think will happen (you do below, I'll get to it).
Quoting Metaphysician Undercover
No it's not. That's a strawman argument by you. I'm not asking you to predict that Joe Blow in Paducah will burn his toast tomorrow. Obviously, that would be ridiculous. So obviously it's not what I'm asking. I'm asking what is the general category of bad things you are warning against (you do below, I'll get to it). .
Quoting Metaphysician Undercover
I mentioned in a post above that I don't have such a denial. But at least you do mention a general kind of bad effect you have in mind.
(1) The most present example is your denial of the utter ludicrousness of you your ignorant, confused, illogical, and deceptive ideas about mathematics. No matter how clearly those are pointed out to you, you evade and deny.
(2) Any subject can have people in denial about its faults. If we deleted intellectual work on the basis that there are certain people that have too rigid adherence, then we'd have virtually no intellectual work to refer to for all human history.
(3) So when you said "bad things", it turns out that for the most part, these bad things are that people explain to you how mathematics actually works so that they may disabuse you of your ignorant and confused imaginings about it.
Quoting Metaphysician Undercover
I have addressed the "religion" claim in detail in other posts. You ignore what I said. That is your favorite argument tactic: Don't recognize the points others make and instead just keep repeating your false and confused claims.
Also, you are virtually lying about me again by claiming that I believe that mathematics is an omnibevolent "God" that would never be misleading.
Quoting Metaphysician Undercover
We sure better say that '2' and 'bet' name the same number. Otherwise, translation would be impossible. If 2' and 'bet' named different numbers then English speakers and Hebrew speakers could never agree on such ordinary observations as that the quantity (you like the word 'quantity') of apples in the bag is the same whether you say it in English or in Hebrew.
Quoting Metaphysician Undercover
Ah, red herring.
The point is whether the English numeral and the Hebrew numeral name the same number. That is unproblematic. It is not a contradiction or illogical for an object to have different words denoting it.
It is an unrelated point that there are different kinds of numbers.
(By the way, for naturals, ordinals and cardinals, they are the same.)
Quoting Metaphysician Undercover
You have it reversed, as you often do.
Yes, by making clear that certain symbols are used differently in different contexts, we avoid equivocation. Using a symbol in more than one way is one-to-many: one (one symbol) to many (many different meanings). And one-to-many is a problem if we don't make clear contexts.
But with the English numeral and Hebrew numeral, we're not talking about one-to-many. Rather, we are talking many-to-one: many (two symbols) to one (one number).
Either you are actually so confused that you can't help but reversing or you are dishonest trying to make the reversal work for you as an argument. I'm guessing the former, since, even though you are often dishonest, more often it is apparent that you are just pathetically confused.
Quoting Metaphysician Undercover
YOU were the one harping on temporality and saying that things were place in order temporally by people. I don't rely on temporality. I didn't say that 'before' is 'before' only in a temporal sense.
Still you are evading the challenge: What is "THE INHERENT" order you claim that the dots have?
Whatever you like - temporal or not - you claim that sets have "AN INHERENT" order. So what is the inherent order of those dots?
Quoting Metaphysician Undercover
STOP
Stop driving right past the point I have told you over and over.
Just read this. Take a moment. And try to understand:
I have said several times that there ARE orderings. There are MANY orderings. So there is not a single ordering that can be called "THE INHERENT" ordering.
Indeed, for a finite set of cardinality n, the number of (total linear) orderings of is n factorial ('n!' in math notation). And when n>1, n!>1, so there are MORE THAN ONE orderings of the set.
And stop trying to make it seem that I deny that there are no orderings of a set.
Quoting Metaphysician Undercover
It is used to denote the quantity two.
Quoting Metaphysician Undercover
Because 'tree' is not a proper noun.
Quoting Metaphysician Undercover
Quoting Metaphysician Undercover
Well yes, it is true.
Start with what people say in everyday language. Jack says, "What is the number of students in the class?" and Sue says, "The number of students in the class is two".
The number IS the number of students. That is everyday language.
And mathematics captures that thinking.
You want for us to regard everyday language working differently, working according to your own nut-case confusions.
2. If for any moment in time, we can always ask what the time was before that then, the past is infinite
Ergo,
3. The past is infinite [1, 2 MP]
4. If the past is infinite and we're in the present then, the infinite past is an actual infinity
5. We're in the present
6. The past is infinite and we're in the present [3, 5 Conj]
7. The infinite past is an actual infinity [4, 6 MP]
8. If the infinite past is an actual infinity then, there are actual infinities
9. There are actual infinities [7, 8 MP]
Quoting Metaphysician Undercover
(1) Again, I have said at least a few times already that sets have orderings. Sets of cardinality greater than 1 have more than one ordering.
You even put in QUOTE MARKS "an arrangement of points on a plane without order", which is something I never said. You are lying about me.
(2) A set of statements is inconsistent if and only if it implies a contradiction. A contradiction is a statement and its negation.
You claim that I have advanced a contradiction. So, you should be able to show that anything I've said implies a contradiction. But these two statements are not a contradiction:
* For every set, there are orderings of the set.
* For sets of cardinality greater than 1, there is no single ordering that is "THE INHERENT ORDERING".
(I'll add that if one want to define 'the inherent ordering' for certain sets, such as the ordering by membership for ordinals, or the standard ordering of the reals, etc., then that's okay with me. But the point is that there is not such a definition for sets in general.)
Devising new frameworks and systems is an important aspect of creativity in mathematics. But, while I can't properly quantify, it seems to me that most of mathematical creativity is in proving theorems.
Quoting Luke
I don't speak for the originator of the illustration, but I take it to mean that there is not an "inherent" order (there is not one particular ordering that is the "inherent ordering"), whether numerical or of any kind.
Said yet another way: Saying (1) "there is not one particular ordering that is 'the inherent ordering'" is not saying (2) "there is no ordering".
A logical fallacy is an improper argument form. You've not shown any fallacy in mathematics. Of course, you may reject the axioms, and you may reject the rules of inference and claim that the rules are not proper. But you've not shown any argument why we should consider the rules improper.
Moreover, the consistency of the rules is proven finitistically. (I forgot, but I think that finitistically proves the soundness of the rules too.) That is how intellectually thorough and honest mathematics is. Mathematics even proves its own proof methods are consistent and sound, and does it finitistically. And mathematical logic even investigates alternate proof methods that incorporate alternative views of mathematics and logic such as constructivism, paraconsistency, relevance logic, etc.
Isn't "important aspect" weaselly enough? I didn't say "all" or "most," just an important aspect. A lot of the big breakthroughs do involve radically new ways of seeing things. I'm not quantifying it, was just pointing out to @Metaphysician Undercover that there's a lot more to math than just following arbitrary rules.
You seem to want to place math on some kind of pedestal, as if someone is claiming it's absolute truth, then you point out that it's not absolute truth. But that's a classic strawman. Nobody is claiming math is absolute truth but you. Math is just a way that we formalize certain intuitive notions. But it's a formalization and not intended to be the "real thing." So your metaphysical points may be right, but you are wrong in believing that anyone is claiming that math is stating metaphysical truths. On the contrary, all mathematical truths are relative. IF this THEN that. Math takes no position on whether "this" is true or even meaningful. Math only says that if you accept this then you can prove that. You are the one trying to make more of this than anyone intends. Can you see that?
Quoting Metaphysician Undercover
"Mathematics considered harmful," LOL. The reference is to the early days of programming, when GOTO statements were prevalent and led to messy, "spaghetti" code. Computer scientists Edsgar Dijkstra published an article called Go To Statement Considered Harmful. Ever since then, "X considered harmful" is an inside joke in CS. This is the first time I've ever seen anyone claim that mathematics is considered harmful! Hence my amusement. Perhaps you can 'splain yourself.
Quoting Metaphysician Undercover
Nonsense, nonsense, and nonsense. You haven't made your case in three years of trying.
Quoting Metaphysician Undercover
But truth and falsity ARE irrelevant to pure mathematicians. Russell famously quipped, "Mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true." Don't you think he was recognizing and responding to exactly the point you are making? What do you make of old Bertie's remark?
Quoting Metaphysician Undercover
And as I already said, my diagram was intended to help make a point, but it clearly didn't work very well, so forget it. Sets are defined by the rules of set theory, nothing more and nothing less. And once again I pointed you to the axiom of extensionality. So nevermind the dots in the circle. If the analogy was lost on you, then forget it. You can't still be arguing against an example. Try understanding the axiom of extensionality.
Quoting Metaphysician Undercover
No more magic that the rules of chess are magic when they say how the various pieces move. The rules of set theory say how sets behave. The analogy is perfect, whether you get it or not.
Quoting Metaphysician Undercover
Existence is to be endowed with order? Now that is utter nonsense. One example that comes to mind is the famous counterexample to the identity of indiscernibles, in which we posit a universe consisting of two identical spheres. You can't distinguish one from the other by any property, even though the spheres are distinct. This example also serves as a pair of objects without any inherent order.
But "existence is to be endowed with order?" Man you are just flailing, making things up instead of honestly engaging.
Quoting Metaphysician Undercover
Well ok. I can't add much. As I've noted, we're kind of done here. I have nothing to add and you aren't interested in engaging with math. You prefer to reject it wholesale and that's your privilege. I can't do anything about it. Not for lack of trying.
Quoting Metaphysician Undercover
Reality? I make no claims of reality of math. I've said that a hundred times. YOU are the one who claims math is real then complains that this can't possibly true. The solution is that nobody claims math is "real" in the sense that you mean. Of course math is real in terms of implications: if this, then that. But math makes no claims as to the truth of "this." Only IF this THEN that. Nobody says sets are "real" in any meaningful sense. Sets are abstract thingies (you don't like the word objects) that behave according to rules; just like chess pieces. You don't want to get that but your refusal to get that is the root cause of your misunderstandings. Nobody claims math is real in the sense that YOU use the word real.
But on the other hand math IS real, in the sense that, for example, the rules of group theory fully encapsulate the behavior of reversible transformations, like addition and subtraction, or multiplication and division, or rotating counterclockwise by 47 degrees and rotating clockwise by 47 degrees. The math of group theory is abstract and not "real" as YOU use the word real, but the relationships encoded by group theory ARE real. And you simply won't grapple with this. I don't know why.
Quoting Metaphysician Undercover
https://en.wikipedia.org/wiki/Axiom_of_extensionality
https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory
That's all there is too it, since you take pot shots rather than attempt to derive understanding from visual analogies. The ZF axioms fully characterize what sets are, by specifying how sets behave. As to what sets actually are, nobody has the slightest idea. Set is an undefined term, just as point and line are undefined terms in Euclidean geometry.
Quoting Metaphysician Undercover
No. I describe a set as whatever obeys the ZF axioms. There are alternative axiom systems that describe a different notion of set. We do have intuitive, casual, naive examples of what we mean, but presenting these to you (collections of things, bags of groceries, circles containing red dots) are lost on you and only give you ammunition for childish objections.
So a set is whatever ZF says a set is. Just like the knight moves the way the rules of chess say the knight moves. They are both formal systems with no referent in the real world.
I said 'important aspect'. I don't see anything "weaselly" about that.
Quoting fishfry
What you said is:
Quoting fishfry
I didn't claim that you said "all" or "most". Rather, I shared my impression that most mathematical creativity is in theorem proving. I don't take either one of devising new systems or theorem proving to be the essence of mathematical creativity, but would be happy to agree that together they combine to make the essence of mathematical creativity.
There are mathematicians and philosophers who do claim that mathematics states metaphysical (platonic, or however it may be couched) truths.
Quoting fishfry
There are many mathematicians to whom truth and falsity are very relevant. And not just model-theoretic truth and falsity.
Quoting fishfry
It didn't work to bring Metaphysician Undercover to reason. But it was a fine illustration for anyone with the ability and willingness to comprehend.
Quoting fishfry
Not quite. The only primitive of set theory is 'element of'. We don't need 'set' for set theory as we need 'point' and 'line' for Euclidean geometry.
Of course, in our background understanding we also take the notion of a 'set' as a given. But in actual formality, 'set' can be defined from 'element of'.
Quoting fishfry
But there are important properties of sets that are not settled by the axioms, so many set theorists do not believe that the axioms fully characterize the sets.
Quoting fishfry
I have an exact idea, relative to the the undefined 'element of'. For me, 'set' is not the notion itself of which I could not explicate, but rather the actual primitive 'element of'.
Actually "numerical order" (whatever that is supposed to mean in reference to a diagram of dots) was not specified. It was simply asserted that the elements have no inherent order.
Quoting TonesInDeepFreeze
Are you taking lessons from Luke on how to make strawman interpretations? Read the quoted passage. Fishfry and Tones are in denial of the logical fallacies, and "you" (directed at fishfry only) talk about truth and falsity not being relevant to pure mathematics. How is that sentence so difficult for you to read properly?
Quoting TonesInDeepFreeze
Yes, I am finding that to be the best tactic in dealing with the type of nonsense you throw at me.
Quoting TonesInDeepFreeze
The one in the diagram. Take a look at it yourself, and see it.
Quoting TonesInDeepFreeze
I have no problem with what people say in everyday language, about the number of students in the class, the number of chairs in the room, the number of trees in the forest, etc., where I have the problem is with what mathematicians say about numbers alone, without referring to "the number of ..."
Quoting TonesInDeepFreeze
Look at it this way Tones. As you describe sets, order is an attribute, or property of the set. How is it possible that a set can be ordered in one way and in a contrary way at the same time, without contradiction? Fishfry resolves this by saying that a set has no order, so order is not a property of a set. But then it appears like fishfry wants to smuggle order in, with some notion of possible orders. However the set is already defined as not having the property of order, therefore order is impossible.
Between you and fishfry, the two of you do not even seem to be in agreement as to whether a set has order or not. Fishfry says that a set has no inherent order. You say that not only does a set have order, but it has a multitude of different orders at the same time. See what happens when you employ contradictory axioms? Total confusion.
Quoting fishfry
Wow, that's an even worse interpretation of what I'm saying than TIDF's terrible interpretation. I argue to demonstrate untruths in math, and you say I'm claiming math is absolute truth. This thread has gone too far. I think you're cracking up.
Quoting fishfry
No, I have read a fair bit of Russel and he was in no way responding to the same points I'm making. More precisely he was helping to establish the situation which I am so critical of. Remarks like that only inspire mathematicians to produce more nonsense.
Quoting fishfry
We've been through the axiom of extensionality you and I, in case you've forgotten. It's where you get the faulty idea that equal to, means the same as.
Quoting fishfry
Contradiction again. If you cannot distinguish one from the other, you cannot say that there are two. To count two, you need to apprehend two distinct things. But to say that you cannot distinguish one from the other means that you cannot apprehend two distinct things. Therefore it is false to say that there are two. So you are just proposing a contradictory scenario, that there are two distinct spheres which cannot be distinguished as two distinct spheres (therefore they are not two distinct spheres), hoping that someone will fall for your contradiction. Obviously, if one cannot be distinguished from the other, they are simply two instances of the same sphere, and you cannot say that there are two. And to count one and the same sphere as two spheres is a false count.
Quoting fishfry
Coming from the Platonic realist who claims the reality of "mathematical objects".
Quoting fishfry
Right, just like the statement from Russel. That's why there is a real need for metaphysicians to rid mathematics of falsity. The mathematicians obviously do not care about festering falsities.
.
I think we can defer to Gowers's great essay, The Two Cultures of Math, which he identifies as theory builders and problem solvers.
https://www.dpmms.cam.ac.uk/~wtg10/2cultures.pdf
Gowers: "The “two cultures” I wish to discuss will be familiar to all professional mathematicians.Loosely speaking, I mean the distinction between mathematicians who regard their central aim as being to solve problems, and those who are more concerned with building and understanding theories"
This is very well said. Of course the distinctions are not clean cut. And the following resonates:
Atiyah: "Some people may sit back and say, “I want to solve this problem” and they sit down and say, “How do I solve this problem?” I don’t. I just move around in the mathematical waters, thinking about things, being curious, interested, talking to people, stirring up ideas; things emerge and I follow them up. "
It is, at best, ambiguous whether the third 'you' applies only to fishfry or "fishry and Tones alike". If you meant to be clear, then you would have been clear which part of your claim is meant for only fishfry and and which for both of us "alike".
And why, other for the purpose of trolling (as you admit to mentioning me only to provoke response) drag me into your scattershot claim?
Moreover, you have not shown that a logical fallacy, let alone one that I have denied. And, you skip my rebuttal to you regarding fallacies.
Quoting Metaphysician Undercover
You evade even the most clear refutations.
Quoting Metaphysician Undercover
You repeat your evasion yet again.
The set of points has many different orderings. No one ordering is privileged as "the inherent ordering".
If there is a certain ordering that you think is "the inherent ordering" then tell us what it is. Point to each dot and tell us which dots it comes before and which dots it comes after. That is what is meant by an ordering in this discussion (a total linear ordering).
I have a jar of sand in front of me. There are many orderings of the grains, but there is no single particular ordering that is "THE inherent ordering".
But you will evade that point yet again.
Quoting Metaphysician Undercover
You are welcome to change your position, but it's not what it was:
Quoting TonesInDeepFreeze
In everyday language, the number is how many individuals there are.
Quoting Metaphysician Undercover
I didn't say that.
Quoting Metaphysician Undercover
First, aside from answering you, to make clear, there is no "same time" term in set theory.
There is no contradiction in the existence of more than one ordering of a set. Example:
{1 2} has two linear orderings:
{<1 2>}
and
{<2 1>}
Or, using concretes:
{Bob, Sue} has two linear orderings:
(
and
{Sue Bob}
See, I answer your questions. Howzabout you answer mine?:
Do you think music theory is wrong like mathematics? And, if so, what are the bad things things that happen from music theory? And do you know any music theory?
Quoting Metaphysician Undercover
I speak for myself and not for fishfry.
"Set has order" is not a rubric I would use except loosely.
I've stated exactly the case. Refer to what I have said, not tangling it up with another poster.
For every set S with cardinality greater than 1, there exists more than one total linear ordering of S.
Quoting Metaphysician Undercover
Whether or not that is a fair characterization of fishfry's view, it does not bear on my comments; I speak only for myself. I will say that "possible" is sometimes used informally in mathematics, but in a context such as this, we can dispense with 'possible'. We prove existence statements about sets and orderings of the sets. We don't have to say they are "possible" orderings.
Quoting Metaphysician Undercover
Wrong. You have not shown any contradiction in the axioms. People take liberties from the strict formulations and discuss set theory in informal ways that may be misunderstand by an ignorant, stubborn and confused crank. But a reasonable person informed in the subject would very well understand that phrases such as "lack of inherent order" or "possible other orders" can be resolved to instead more definite formalisms without loose terminology such as "inherent" or "possible".
I'll give it to you again:
If S has cardinality greater than 1, then there are more than 1 total linear orderings of S.
Well I'm a formalist sometimes and a Platonist other times. I made the point earlier to @Metaphysician Undercover that while math isn't "true" in the sense he thinks it's supposed to be, on the other hand it DOES express certain relational or structural truths. For example group theory expresses everything we could ever know about invertible transformations. Group theory expresses truths about such things. Yet formally, groups are sets, and sets have an very tenuous claim on being real. So somehow, math is fiction yet expresses deep structural truths. A lot of philosophers have said clever things about this.
Quoting TonesInDeepFreeze
Yes, but in what sense? I like my example. I don't think sets are particularly real. I don't ever try to defend the reality of sets. I don't believe there is an empty set or a set containing the empty set and the set that contains the empty set. But groups are defined as particular types of sets, and group theory expresses deep truths about invertible transformations. Out of nonsense, we get sense.
Quoting TonesInDeepFreeze
It was a fine example indeed! But @Meta cleverly turned it against me and made me regret mentioning it. He does that so well! :-)
Quoting TonesInDeepFreeze
From the very first paragraph of the introduction to Kunen's Set Theory: An Introduction to Independence Proofs, he says: "All mathematical concepts are defined in terms of the primitive notions of set and membership." Then as the text evolves, he builds up the usual hierarchy of sets starting from the empty set, and never seems to say what a set is; except that a set is one of the things built up. That's what I know about it, but I'm not equipped to dispute the fine points. I think it's fair to say that set is an undefined term; but there's no actual axiom that says, "Set is an undefined term." The whole business is kind of vague, actually.
Quoting TonesInDeepFreeze
Yes ok, I think I agree with your point. The larger point that I made to @Meta still holds. Informally a set is a bag of groceries or a circle with some red dots in it; but formally, a set has no meaning of its own outside of its behaviors under the axioms.
Quoting TonesInDeepFreeze
Yes of course. Skolem thought that his Lowenheim-Skolem theorem showed that the notion of set isn't nearly as definite as we think. And of course there are various different axiomatic systems such as Von Neumann–Bernays–Gödel set theory, or Morse-Kelley set theory. And each such system is incomplete, so there are always true statements about sets that we can't prove. So it's fair to say that nobody knows for sure what a set is. Even the greatest set theorists; especially the greatest set theorists.
Quoting TonesInDeepFreeze
I don't think I'm in a position to argue that point one way or the other. Not entirely sure I'm following.
If not specified, then at least strongly implied in the same post:
Quoting fishfry
Did you not read this when you went on to argue that the diagram has the order it has?
Just another example of your wilful ignorance and dishonest argumentation.
In a realist sense, whatever the mathematician's or philosopher's concept of mathematical realism. In particular, many mathematicians believe that the continuum hypothesis is true or false, in a real sense.
Quoting fishfry
Yes, that is quite common. However, as I said, in a strict technical sense, we don't need to regard 'set' as primitive. 'set' does not occur in the axioms, and is not even a primitive in the language.
Quoting fishfry
I can give you a definition using only 'member of'.
As previous:
https://thephilosophyforum.com/discussion/comment/544630
And now:
Quoting TonesInDeepFreeze
Quoting TonesInDeepFreeze
Quoting TonesInDeepFreeze
Quoting TonesInDeepFreeze
Quoting TonesInDeepFreeze
You wrote 'Russel' twice. It's 'Russell'.
Yes ok I think you're right about that.
I wasn't born wearing a hat but I can go buy one. The idea that a thing can't gain stuff it didn't have before is false. But I've already shown you how we impose order on a set mathematically. We start with the bare (newborn if you like) set. Then we pair it with another, entirely different set that consists of all the ordered pairs of elements of the first set that define the desired order.
So we start with the unordered set {a,b,c}. And we PAIR it with a different set {(a,b), (a,c), (b,c)} that defines the order a < b < c.
Now note well please. I am not saying that this process corresponds to any aspect of reality. It plainly doesn't. It is rather the way we formalize the idea of ordering a set.
Perhaps a key distinction I haven't explicitly called attention to is this:
I am not talking about reality. I am talking about how we use math to MODEL certain aspects of reality . With that distinction, I believe all our problems are solved. Math is a tool kit for modeling things we may be interested in. It's not the things themselves.
Also: "How is it possible that a set can be ordered in one way and in a contrary way at the same time, without contradiction?" Like ordering schoolkids by height, or alphabetically by name. Two different ways to order the same set. I can't understand why this isn't obvious to you.
Quoting Metaphysician Undercover
A set has no inherent order. That's the axiom of extensionality. A set may be ordered in many different ways, by pairing it with another set consisting of the ordered pairs that defined the order we're interested in.
Quoting Metaphysician Undercover
If you wish to argue the untruth of math, you won't get an argument from me. Math isn't supposed to be about the truth of particular things. Math says IF this THEN that. Or it reveals structural truths, as in group theory. But the contradictions you think you see, are only contradictions between math and what you THINK math is. But if your ideas are wrong, your argument fails.
Quoting Metaphysician Undercover
Not for nothin' is category theory called abstract nonsense. Never let it be said that mathematicians don't have a self-deprecating and self-aware sense of humor. Something you might aspire to emulate.
Quoting Metaphysician Undercover
Well your objection to it is wrong. Two sets are the same set if they have the same element. They are not two identical copies of the set. They're the same set. I don't see what your objection is, nor did I follow your objection several years ago. Jeez has it been that long? LOL.
Quoting Metaphysician Undercover
You can't visualize two identical spheres floating in an otherwise empty universe? How can you not be able to visualize that? I don't believe you.
Here, look.
https://en.wikipedia.org/wiki/Identity_of_indiscernibles
I did not make this up. I'm not clever enough to have made it up. I read it somewhere. Now you've read it.
Quoting Metaphysician Undercover
I claim their mathematical existence and not their physical existence. But I admit that I'm not sufficiently knowledgable in philosophy to verbalize these subtle distinctions. Does the number 5 exist? Yes, as an abstract object. What does that mean? I can't personally say, but plenty of smart people have made the attempt.
Quoting Metaphysician Undercover
No, on the contrary. Mathematicians love festering falsities. And of course even this point of view is historically contingent and relatively recent. In the time of Newton and Kant, Euclidean geometry was the true geometry of the world, and Euclidean math was true in an absolute, physical sense. Then Riemann and others showed that non-Euclidean geometry was at least logically consistent, so that math itself could not distinguish truth from falsity. Then Einstein (or more properly Minkowski and Einstein's buddy Marcel Grossman) realized that Riemann's crazy and difficult ideas were exactly what was needed to give general relativity a beautiful mathematical formalism. The resulting "loss of certainty" was documented in Morris Kline's book, Mathematics: The Loss of Certainty. Surely the title alone must tell you that mathematicians are not unaware of the issues you raise.
Yes, nice article.
I should revise what I said. Maybe something like this (not necessarily in this order):
(1) Creating new systems.
(2) Ingeniously proving theorems.
(3) Proving theorems in a way that engenders new techniques (such as Cohen's forcing).
(4) Posing questions or conjectures.
(5) Critiquing (such as Brouwer's prosecution against classical mathematics).
(6) Thinking up good pedagogical explanations by which people can better understand mathematics.
(7) Writing scorching ripostes to cranks. (Just kidding about that one.)
That book gets important technical points wrong and it's a deplorably tendentious hatchet job. (I don't have the book, and it's been a long time since I read it, so I admit I can't supply specifics right now for my criticism.)
LOL. So I've heard. But the larger point remains, that in the past century math has experienced a loss of certainty.
Quoting TonesInDeepFreeze
You say that like it's a bad thing.
I really don't see how the qualification "numerical" is relevant , or even meaningful in the context of dots on a plane. So I don't see why you think it was implied. Fishfry is not sloppy and would not have forgotten to mention a special type of order was meant when "no inherent order" was said numerous times.
Quoting TonesInDeepFreeze
Sorry "Bertie", as fishfry says.
Quoting TonesInDeepFreeze
I believe all the relevant points were addressed. You don't seem to know how to read very well.
Quoting fishfry
The problem is that your demonstration was unacceptable because you claimed to start with a set that had no order. A newborn is not a thing without order, so the newborn analogy doesn't help you.
Quoting fishfry
You are showing me an order, "a" is to the right of "b" which is to the right of "c". And even if you state that there is a set which consists of these three letters without any order, that would be unacceptable because it's impossible that three letters could exist without any order.
If you insist that it's not the letters you are talking about, but what the letters stand for, or symbolize, then I ask you what kind of things do these letters stand for, which allows them to be free from any order? To me, "a", "b", and "c" signify sounds. How can you have sounds without an order? Maybe have them all at the same time like a musical chord? No, that constitutes an order. Maybe suppose they are non-existent sounds? But then they are not sounds. So the result is contradiction.
I really do not understand, and need an explanation, if you think you understand how these things in the set can exist without any order. What do "a", "b", and "c" signify, if it's something which can exist without any order? Do you know of some type of magical "element" which has the quality of existing in a multitude without any order? I don't think so. I think it's just a ploy to avoid the fundamental laws of logic, just like your supposed "two spheres" which cannot be distinguished one from the other, because they are really just one sphere.
Quoting fishfry
Those are different ways, but not contrary ways.
Quoting fishfry
Huh, all my research into the axiom of extensionality indicates that it is concerned with equality. I really don't see it mentioned anywhere that the axiom states that a set has no inherent order. Are you sure you interpret the axiom in the conventional way? I have no problem admitting that two equal things might consist of the same elements in different orders. We might say that they are equal on the basis of having the same elements, but then we cannot say that the two are the same set, because they have different orders to those elements, making them different sets, by that fact.
Quoting fishfry
Are you serious? If I can imagine them as distinct things, I know that they cannot be identical. That's the law of identity, the uniqueness of an individual,. A fundamental law of logic which you clearly have no respect for.
Quoting fishfry
The argument of Max Black fails because pi is irrational. There is no such thing as a perfectly symmetrical sphere. The irrationality of pi indicates that there cannot be a center point to a perfect circle. Therefore we cannot even imagine an ideal sphere, let alone two of them.
Quoting fishfry
Yes, it seems like mathematics has really taken a turn for the worse. If you really believe that mathematicians are aware of this problem, why do you think they keep heading deeper and deeper in this direction of worse? I really don't think they are aware of the depth of the problem.
Perhaps “numerical” wasn’t the right word. The context of the post and the preceding discussion indicates that a “before and after” ordinality was implied, of the sort Tones recently mentioned:
Quoting TonesInDeepFreeze
Your assertion that the diagram has an inherent order which can be discerned simply by looking at the diagram does not specify what that order is.
I read fine. But with your lack of replies to many crucial points, I admit that I can't read what doesn't exist.
Here is a previous post tracking your recent evasions:
https://thephilosophyforum.com/discussion/comment/545568
And now the points in this post too:
https://thephilosophyforum.com/discussion/comment/545558
Yes, it certainly looks like that faulty brick in the foundations of math will surely cause the giant structure to collapse. I wish I had known this before becoming a mathematician. :cry:
Before and after, are temporal terms. Fishfry had rejected the notion that "order" is based in spatial-temporal relations, and wanted an order based in quantity. But a quantity based order is what produces the problem I first referred to. If "2" refers to a quantity of objects in the context of "order", then it does not refer to a single object, the number 2. In any case, what distinguishes one thing from another, allowing for individuals, and quantity itself is spatial relations. So we're back to spatial relations as the bases of order. It's very clear that the subject was order of any sort, when "no inherent order" was mentioned.
It was my suggestion that "order" is fundamentally temporal, but fishfry produced examples of an order based on a judgement of better and worse, to discount that theory. So we really haven't agreed on any specific type of order yet. This is probably because we haven't agreed on what type of existence the things which are supposed to have no inherent order, but are capable of being ordered, have.
Quoting jgill
I don't foresee any imminent collapse, but the structure ought to be dismantled because it supports the faulty worldview which is prevalent today, which is a sort of scientism. Fishfry perceives that physics has reached a sort of dead end in its endeavours, but refuses to acknowledge that the dead end is brought about by the principles employed (mathematics included) rather than the unintelligibility of the world itself.
The use of "infinity" which is the topic of this thread (believe it or not) is a very good example. I apprehend, that at the base of the idea of infinity in natural numbers, is the desire, or intention to allow that numbers can be used to count anything. There will never be something which cannot be counted because the numbering system has been designed to allow that the numbers can always go higher. This gives the appearance that everything is measurable.
The only drawback is that this renders infinity itself, (the principle which provides this capacity, that everything is measurable) as immeasurable. So in reality everything is measurable except the principles we use to measure with. What this means is that to understand the nature of measurement and infinity, we must place these into a different category from the category of things which are measurable, and understand it on those terms, as immeasurable. If we attempt to bring infinity into the category of things which are measurable, as is the trend in modern mathematics, because we want mathematics to enable science to be applicable everything, even the thing which by its own design is immeasurable, then we introduce contradiction (the immeasurable is measurable) and therefore unintelligibility into our principles. That is where we are today, we have allowed unintelligibility to inhere within our principles of measurement. As fishfry said "math has experienced a loss of certainty".
Fishfry and I really agree that the big picture is very hazy (uncertain). But when it comes down to determining the specific points where the haziness arises from, fishfry refuses to follow. It's like seeing smoke on the horizon and wondering why it's there. But when I point to the fire, fishfry refuses to acknowledge a relationship between the fire and the smoke on the horizon. Maybe if fishfry would accept the possibility of a relationship, a closer look would reveal smoke rising from the fire.
How does this account for the order of rank of military officers, or of suits in a game of bridge? Or the order of values of playing cards in Blackjack? Or the order of the letters of the alphabet? Or monetary value?
'before' and 'after' are often in a temporal sense, but clearly not exclusively. Not in English. And surely not in math that doesn't mention temporality.
Yes, when 'we' includes you. But with math, we do specify specific kinds of order.
Quoting Metaphysician Undercover
The notion of infinite sets is used calculus, which is mathematics for the sciences, which is mathematics for the technology you enjoy.
That’s a cop out. You claimed that the diagram has an inherent order. Specify that order. Which dots are the start and end points of that order? This needn’t imply a temporal start and end. For example, winning poker hands have a rank from lowest to highest; from a pair to a royal flush. This is not a spatial or temporal order of rank.
A newborn doesn't have a hat but it can acquire one.
But the deeper point is that YOUR CONCEPT of a set has inherent order, and that's fine. But the mathematical concept of a set has no inherent order. So you have your own private math. I certainly can't argue with you about it.
Quoting Metaphysician Undercover
You're confusing presentation with the set itself. The sets {a,b,c}, {c,b,a}, and {a,c,b} are exactly the same set. Just as Sonny and Cher are the same singing group as Cher and Sonny. They're the same two people. You and I are the same two people whether we're described as Meta and fishfry or fishfry and Meta. If you can't see that, what the heck could I ever say and why would I bother?
Quoting Metaphysician Undercover
Whatever man. You're talking nonsense. The Sun, the Moon, and the stars are the same collection of astronomical objects as the Moon, the stars, and the Sun.
Quoting Metaphysician Undercover
Whatever. I can't add anything. The points you're making are too silly to require response.
Quoting Metaphysician Undercover
Yes.
Quoting Metaphysician Undercover
They're different as ordered sets, but the same as sets.
Quoting Metaphysician Undercover
If I stopped responding would that be ok? I've been done with this for a while. You're not making any points worthy of response.
Quoting Metaphysician Undercover
That might be the dumbest thing you've ever said. If you said that no two physical spheres could be identical, that would be true. But why can't I have two conceptual, abstract spheres? There can't be a center point to a perfect circle? Look @Meta if you deny the unit circle in the Euclidean plane, with its center at the origin, we're done here. We're done here anyway, you are not making any points that seem reasonable to me. Pi is a computable real number anyway, so even if the universe is a simulation, the great computer in the sky would know about pi.
Quoting Metaphysician Undercover
How would you fix math?
Examples like that is how fishfry convinced me otherwise.
Quoting Luke
I believe I already did. It's a spatial order, each dot has its own specific position on the plane. To change the position of one would change the order, requiring a different diagram. So that order is inherent to that diagram.
Quoting Luke
There is no need to specify a start and end. After giving me examples of order which is not a temporal order, you cannot now turn around and insist that "order" implies a known start and end. "Order" is defined as "the condition in which every part, unit, etc., is in its right place". That's why the diagram has an inherent order. If any of the dots were in a different place it would not be the diagram which it is, because some part would be in the wrong place for it to be that particular diagram.
Quoting fishfry
Right, and do you also see now, that the mathematical concept of a set is incoherent? I hope so, after all the time I've spent explaining that to you.
Quoting fishfry
Now, do you see that Sonny and Cher, Meta and fishfry, as individual people, have spatial temporal positioning, therefore an inherent order? I am here, you are there, etc.. We can change the order, and switch places, or move to other places, but at no time is there not an order.
So, you propose a set [a,b.c], [c,b,a], or phrase it however you like. You have these three elements. Do you agree that the three things referred to by "a", "b", and "c", must have an order, just like three people must have an order, or else the set is really not a set of anything? There is nothing which could fulfill the condition of having no order.
I know, you'll probably say it's abstract objects, mathematical objects, referred to by the letters as members of the set, therefore there is no spatial-temporal order. But even this type of "thing" must have an order as defined by, or as being part of a logical system, or else they can't even qualify as conceptions or abstract objects. Without any order, they cannot be logical, and are simply nothings, not even abstract objects. It appears like you want to abstract the order out of the thing, but that's completely incoherent. Order is what is intelligible to us, so to remove the order is to render the concept unintelligible. What's the point to an unintelligible concept of "set"?
Quoting fishfry
Yes. Now do you see that these three things have order, regardless of the order in which you name them? And all things have some sort of order regardless of whether you recognize the order, or not. If there was something without any order it would not be sensible, cognizable or recognizable at all. In fact it makes no sense whatsoever to assume something without any order, or even to claim that such a thing is a real possibility. So to propose that there could be a complete lack of order, and start with this as a premise, whereby you might claim infinite possibility for order, you'd be making a false proposition. It's false because a complete lack of order would be absolute nothing, therefore nothing to be ordered, and absolutely zero possibility for order. But you want to say that there is "something" which has no order, and this something provides the possibility of order. By insisting that there is no order to this "something" you presume it to be unintelligible.
Quoting fishfry
If you think about what it means to be a conceptual abstract sphere, the answer ought to become apparent to you. What makes one sphere different from another is their physical presence. If you have two distinct concepts of a sphere, then they are only both the exact same concept of sphere through the fallacy of equivocation. If you have one concept of an abstract sphere then it is false to say that this is two concepts. It is simply impossible to have two distinct abstract concepts which are exactly the same, because you could not tell them apart. It's just one concept.
It is not I who is making the dumb propositions.
That is not specification of an order, let alone of "THE inherent order".
Of course, we could move left to right and also up and down, zigzagging to specify an ordering of the dots. But we could also do it right to left, or from spiraling from the center outward, or from one particular dot anywhere. So there is not one single order that is "THE inherent order".
It's amazing that you don't get this. Or that you simply evade, as you are Metaphysician Undercover from Evasionsville.
That's not the sense of the word 'order' we're talking about! You have yourself even being using 'order' in a sense not expressed as "in its right place". Wow, you really do have a cognitive problem.
There are two issues here: "order" and "inherent".
You claim that the elements of the diagram have a "spatial order". Effectively, this is to say that the elements "have the order they have" without specifying what that order is.
You also presume that the elements of the diagram have an inherent order. You have provided no explanation as to why the diagram has the spatial order it has instead of any other possible spatial ordering of the the same elements.
Quoting Metaphysician Undercover
A more suitable definition in the context of this discussion might be “the arrangement or disposition of people or things in relation to each other according to a particular sequence, pattern, or method.”
Quoting Metaphysician Undercover
Those "start" and "end" points would be non-temporal. That was the point of the counterexamples.
Nearly all of those counterexamples have elements which can be ordered first to last, lowest to highest, worst to best, etc. Those orderings (e.g. poker hands, military ranks, letters of the alphabet) therefore have "start" and "end" points in terms of the arrangement of their elements.
I specified the order. It is a spatial order, the one demonstrated by the diagram. Why is this difficult for you to understand? When a diagram shows us an arrangement of dots, it shows us the spatial order of those dots, where the dots must be on a spatial plane to fulfill the order being demonstrated. What is the diagram? An arrangement of dots. What does it demonstrate to us? An ordering of those dots. Someone could proceed with that diagram to lay out the same pattern with other objects, with the ground, or some other surface as the plane. Just because fishfry called it a "random" arrangement doesn't mean that it does not demonstrate an order. Fishfry used "random" deceptively, as I explained already.
Quoting Luke
Yes I did explain that. There was a process which put those dots where they are, a cause, therefore a reason for them being as they are and not in any other possible ordering. That is why it is not true to call it a random arrangement, unless you are using "random" to signify something other than no order.
This is an important constituent of the distinction between actual order, and possible order. A distinction which fishfry rejected as not with principle. But there is such a principle, which fishfry simply denied, that things must have an actual order, to have existence. I believe it's called the principle of sufficient reason. And this principle renders "the set", as being a unity composed of parts, without any inherent order, as an incoherent notion. We could say that there are many possible orders which the parts could have, but if they do not have an actual order, the supposition of "unity" and therefore "set" is unjustified.
You continue to assert that the elements have an order without specifying what that order is. "What is that order? An arrangement. And what is that arrangement? The order it has."
Quoting Metaphysician Undercover
What algorithm might someone use to re-create the same pattern?
Quoting Metaphysician Undercover
"It's not random because it has an order and it has an order so it isn't random." Empty words.
Quoting Metaphysician Undercover
"They have that order and not some other order because there was a cause of that order."
Seriously? That does not explain why the elements must have this particular order instead of some other order. Why wasn't some other order or arrangement caused instead?
That's just another undefined term by you as is 'inherent order'. It adds nothing to your incorrect argument.
There are many orders. You have not defined what it means to say that one of the orders is 'AN actual order' or 'THE inherent order' while none of the other orders are an actual order or the inherent order. It is a remarkable feat of stubbornness for you to persist in not recognizing that clear and simple fact.
And we still have this classic:
Quoting Metaphysician Undercover
Clearly, we have not been talking about that sense of 'order'. Credit to you though for your sophistical resourcefulness in looking at a dictionary to find a different sense of a word to distract from the sense that has been used (even by you) throughout the discussion.
From earlier in the discussion, “inherent” is being used to mean “non-arbitrary”.
what is the mathematical definition of 'abritrary'?
Anyway, I don't think we need 'inherent', 'actual' or 'non-arbitrary'.
It is enough to observe that sets of cardinality greater than 1 have more than 1 ordering, and that any other privileged designation is only by stipulation (such as 'the standard ordering' for specific sets is fine, since it is stipulated and defined), and there is no overall rubric of individuation for the orderings of sets in general.
No, it's your own private concept of a set that's incoherent. But what I do find noteworthy is that you genuinely believe (unless everything you post on this site is an elaborate troll, which I do suspect) that you are "explaining" anything to me. On the contrary, you're demonstrating your mathematical ignorance, which I labor mightily, and without hope of succeeding, to correct. Like Sisyphus rolling his boulder uphill, only to watch it roll down again; in vain do I endlessly explain to you that mathematical sets have no inherent order, only to suffer yet more sophistry from you.
Quoting Metaphysician Undercover
As sets, they have no order. If you ADD IN their spatio-temporal position, that gives them order. The positioning is something added in on top of their basic setness. For some reason this is lost on you.
Quoting Metaphysician Undercover
Yes. That is true. But the SETNESS of these elements has no order. Not for any deep metaphysical reason, but rather because that is simply how mathematical sets are conceived. It is no different, in principle, than the way the knight moves in chess. Do you similarly argue with that? Why not?
Quoting Metaphysician Undercover
No. Consider for example the vertices of an equilateral triangle. We may call them v1, v2, and v3, realizing that this labeling is completely arbitrary and that labels could be assigned in many different ways. Six different ways in fact. Now we have a SET of vertices which we can denote {v1, v2, v3} or {v2, v1, v3} or {v3, v2, v1}. In each case the set of vertices doesn't change. There are three vertices and they are the same set of vertices regardless of how we list them.
Now consider. You claim that their position in space defines an inherent order. But what if I rotate the triangle so that the formerly leftmost vertex is now on the bottom, and the uppermost vertex is now on the left? The set of vertices hasn't changed but YOUR order has. So therefore order was not an inherent part of the set, but rather depends on the spatial orientation of the triangle.
Quoting Metaphysician Undercover
I just gave you a nice example, but I'm sure you'll argue. I push the boulder up the hill, it rolls down again.
Quoting Metaphysician Undercover
Well, I take heart in your at least acknowledging my position. Just as the three vertices of a triangle have no inherent order. And that if you do assign them an order based on "leftmost" or some such, that order is contingently based on the orientation of the triangle. But a triangle's orientation is not an inherent part of its trianglitude. It's the same triangle no matter how we spin it.
Or would you say that the earth right now isn't the same as the earth five minutes from now, because it's spun on its axis? I think you either have to admit that a triangle is the same triangle no matter how it's oriented; OR you have to claim that the earth isn't the same earth from moment to moment because it's spinning. As if you could rearrange your living room by moving your couch, and it somehow becomes a different couch.
Quoting Metaphysician Undercover
Vertices of a triangle. Inherently without order. Any spatial order is a function of the triangle's contingent orientation. I think this is a good example.
Quoting Metaphysician Undercover
The vertices of a triangle are not nothing, they're the vertices of a triangle.
Quoting Metaphysician Undercover
Well no, not really. I do abstract out order, for the purpose of formalizing our notions of order. I'm not making metaphysical claims. I'm showing you how mathematicians conceive of abstract order, which they do so that they can study order, in the abstract. But you utterly reject abstract thinking, for purposes of trolling or contrariness or for some other motive that I cannot discern.
Quoting Metaphysician Undercover
It clarifies our thinking, by showing us how to separate the collection-ness of some objects from any of the many different ways to order it.
Quoting Metaphysician Undercover
Vertices of an equilateral triangle. Let's drill down on that. It's a good example.
But take the sun, the earth, and the moon. Today we might say they have an inherent order because the sun is the center of the solar system, the earth is a planet, and the moon is a satellite of the earth.
But the ancients thought it was more like the earth, sun, and moon. The earth is the center, the sun is really bright, and the moon only comes out at night.
Is "inherent order" historically contingent? What exactly do YOU think is the "inherent order" of the sun, the earth, and the moon? You can't make a case.
Quoting Metaphysician Undercover
Vertices of an equilateral triangle.
Quoting Metaphysician Undercover
A mathematical set is such a thing. And even if you claim that your own private concept of a set has inherent order, you still have to admit that the mathematical concept of a set doesn't.
Quoting Metaphysician Undercover
You're wrong. Or a troll. Lately you're starting to convince me of the latter.
Quoting Metaphysician Undercover
Repeating the same ignorant falsehood doesn't work in mathematics. Only in politics.
Quoting Metaphysician Undercover
Repetitive and wrong. And boring. At least say something interesting once in a while.
Quoting Metaphysician Undercover
Take two identical sphere, of radius 1, say, in Euclidean three-space. You might say that one is to the "left" or "above" the other, as the case may be; but that is only a function of the coordinate system. And changing the coordinate system doesn't change the essential nature of an object. So if you had a universe consisting of two identical unit spheres and nothing else, how would you tell them apart? For ease of visualization, take them as two unit circles in the plane. How do you tell them apart without reference to a coordinate system?
(Edit) -- Ah, I see your point. Let me rephrase that. I'll stipulate that if they are identical, they are the same sphere. You have corrected me and I stand corrected. Consider two congruent spheres of radius one. The rest of my argument stands as stated.
Quoting Metaphysician Undercover
What I referred to as a dumb proposition is your claim that you can't have two identical spheres [Edit -- congruent] because pi is irrational. That's just such a bad argument that you should be embarrassed. The unit circle in Euclidean space has a circumference of 2 pi. I am sorry to have to be the one to break that news to you. But why do you care? Pi is a computable real number. We have many finite-length algorithms that exactly and uniquely characterize it.
That should become a classic. Perfectly said.
What more needs to be said? (but I am a victim of naive set theory)
What is the set then? You already said it's not the names. If it's the individual people named, then they necessarily have spatial temporal positioning. You cannot remove the necessity of spatial-temporal positioning for those individuals, and still claim that the name refers to the individuals. That would be a falsity. So I ask you again, what constitutes "the set"? It's not the symbols, and it's not the individuals named by the symbol (which necessarily have order). What is it?
Quoting fishfry
Well, it might be the case, that this "is simply how mathematical sets are conceived", but the question is whether this is a misconception.
Quoting fishfry
This is not true, the order has not changed . The vertices still have the same spatial-temporal relations with each other, and this is what constitutes their order. By rotating the triangle you simply change the relations of the points in relation to something else, something external. So it is nothing but a change in perspective, similar to looking at the triangle from the opposite side of the plane. It appears like there is a different ordering, but this is only a perspective dependent ordering, not the ordering that the object truly has.
Quoting fishfry
Clearly your example fails to give what you desire. We are talking about "inherent order". This is the order which inheres within the group of things. It is not the perspective dependent order, which we assign to the things in an arbitrary manner, which is an extrinsically imposed order, it is the order which the things have independently of such an imposed order.
The issue is whether or not there can be a group of things without any such inherent order. It is only by denying all inherent order that one can claim that an arbitrarily assigned order has any truth, thereby claiming to be able to attribute any possible order to the individuals.
In your example of "equilateral triangle" you have granted the points an inherent order with that designation. You can only remove the order with the assumption that each point is "the same". However, it is necessary that each point is different, because if they were the same you would just have a single point, not a triangle. Therefore, it is necessary to assume that each point is different, with its own unique identity, and cannot be exchanged one for the other as equal things are said to be exchangeable. So when the triangle is rotated, each point maintains its unique identity, and its order in relation to the other points, and there is no change of order. A change of order would destroy the defined triangle.
Quoting fishfry
You think it's a good example, I see it as a contradiction. "Vertices of a triangle" specifies an inherent order.
Quoting fishfry
The point though, is that to remove all order from a group of things is physically impossible. And, "order" is a physically based concept. So the effort to remove all order from a group of things in an attempt to "conceive of abstract order" will produce nothing but misconception. If this is the mathematician's mode of studying order, then the mathematician is lost in misunderstanding.
Quoting fishfry
There is a fundamental principle which must be respected when considering "the many different ways to order" a group of things. That is the fact that such possibility is restricted by the present order. This is a physical principle. Existing physical conditions restrict the possibility for ordering. Therefore whenever we consider "the many different ways to order" a group of things, we must necessarily consider their present order, if we want a true outlook. To claim "no order" and deny the fact that there is a present order, is a simple falsity.
Quoting fishfry
The "inherent order" is the order that the things have independently of the order that we assign to them. This is the reason why the law of identity is an important law to uphold, and why it was introduced in the first place. It assigns identity to the thing itself, rather than what we say about it. We can apply this to the "order" of related things like the ones you mentioned. The sun, earth, and moon, as three unique points, have an order inherent to them, which is distinct from any order which we might assign to them. The order we assign to them is perspective dependent. The order which inheres within them is the assumed true order. In our actions of assigning to them a perspective dependent order, we must pay attention to the fact that they do have an independent, inherent order, and the goal of representing that order truthfully will restrict the possibility for orders which we can assign to them.
Now, you want to assume "a set" of points or some such thing without any inherent order at all. Of course we can all see that such points cannot have any real spatial temporal existence, they are simply abstract tools. To deny them of all inherent order is to deny them of all spatial-temporal existence. The point which you do not seem to grasp, is that once you have abstracted all order away from these points, to grant to them "no inherent order" by denying them all spatial-temporal relations, you cannot now turn around and talk about their possible orders. Order is a spatial-temporal concept, and you have removed this from those points, in your abstraction. That abstraction has removed any possibility of order, so to speak of possible orders now is contradiction.
Quoting fishfry
Assuming you have understood the paragraphs I wrote above, let's say that "a mathematical set is such a thing". It consists of points, or some similar type of thing which have had all principles for ordering removed from them, therefore these points (or whatever) have no inherent order. By what means do you say that there is a possibility for ordering them? They have no spatial-temporal separation, therefore no means for distinguishing one from the other, they are simply assumed to exist as a set. How do you think it is possible to order them when they have been conceived by denying all principles of order.? To introduce a principle of order would contradict the essential nature of these things.
Quoting fishfry
I'm waiting for a demonstration to support this repeated assertion. How would you distinguish one from another if you remove all principles which produce inherent order?
Quoting fishfry
Accepted, and I think that course of two identical spheres is a dead end route not to be pursued.
Quoting jgill
You have stipulated an order, "primes" indicates a relation to each other.
It is the unique object whose members are all and only those specified by the set's definition.
{0 1} is the unique set whose members are all and only 0 and 1.
Quoting Metaphysician Undercover
Set theory adheres to the laws of identity.
Quoting Metaphysician Undercover
What is that "inherent order"? On what basis would you say one of these is the "inherent" order but not the others?:
There are no spatial-temporal parameters for you to reference, just 3 people, with 3!=6 strict linear orderings. Order them temporally by birth date? Temporally by the day you first met them? Spatially east to west? Spatially west to east? Which is "THE inherent order" such that the other orders are not "THE inherent order"?
For a finite set S of cardinality n, there are n! strict linear ordering of S. What is the general definition of "THE inherent ordering" among n! orderings?
Quoting Metaphysician Undercover
We don't assume that sets don't have orderings. Indeed, for a set S with cardinality n, there are n! strict linear orderings of S. But when you say that one of them is "THE inherent ordering" then that requires saying what "THE inherent ordering" means.
Quoting Metaphysician Undercover
'is prime' is a predicate, not an ordering.
Quoting Metaphysician Undercover
Ordering was not "removed". There are n! orderings of the set. Remarking that you have not defined "THE inherent ordering" is not "removing" anything.
Quoting Metaphysician Undercover
There is not just a possibility. There exist n! orderings of the set. We prove that from axioms.
Quoting Metaphysician Undercover
Members of mathematical sets are distinguished by properties other than spatial-temporal. And members of sets in non-mathematical contexts may be distinguished by means other than spatial-temporal. The queen of hearts is distinguished from the ace of spades, without having to refer to their positions in time and space.
Quoting Metaphysician Undercover
Not just "possibility" but existence.
S = {queen-hearts ace-spades} = {ace-spades queen-hearts}
Abstraction has not "removed" orderings. S has two orderings:
{
and
{
Neither is "THE inherent ordering", unless you first give a definition of "THE inherent ordering".
'is prime' is a predicate, not an ordering.
In the case of a collection of things in the physical world, they have spatio-temporal positoin, but there is no inherent order. How would you define it? You can't say leftmost or rightmost or top or bottom-most, because that only depends on the position of the observer. In modern physics you can't even line things up by temporal order since even that depends on one's frame of reference, and there is no frame of reference.
The point is even stronger when considering abstract objects such as the vertices of a triangle, which can not have any inherent order before you arbitrarily impose one. If the triangle is in the plane you might again say leftmost or topmost or whatever, but that depends on the coordinate system; and someone else could just as easily choose a different coordinate system.
Quoting Metaphysician Undercover
Ok!! Well we have made progress. You agree finally that mathematical sets have no inherent order, until we impose one. This point is made more strongly by mathematical objects that may not be familiar to you, such as topological spaces. A set may have many different topologies. A topological space is first a SET with no inherent topological structure. Then we impose a topological structure on it by associating the set with a SECOND set called the "topology," which is a particular collection of subsets of the first set. Given a set there are many different topologies that can be put on it. No one topology has any primacy over any other.
This pattern is so pervasive in math that it soon becomes second nature. You have a bare set with no structure. You impose on it an order to make it an ordered set. Or you impose a topology to make it a topological space. Or you impose a binary operation or two to make it a group or a ring or a field. That's the power of mathematical abstraction. You start with a bare set and toss in the ingredients you want. Like making a salad. You start with a bowl. The bowl is not initially any kind of salad. It's not even inherently a salad, it might turn out to be a bowl of oatmeal. You start with the bowl and add in the ingredients you want to get a particular object that you're interested in.
Quoting Metaphysician Undercover
It's been a long time, but we've made progress. You agree finally that a mathematical set has no inherent order, and you ask whether that's the right conceptual model of a set. THIS at last is a conversation we can have going forward. I don't want to start that now, I want to make sure we're in agreement. You agree that mathematical sets as currently understood have no order, and you question whether Cantor and Zermelo may have gotten it wrong back in the day and led everyone else astray for a century. This is a conversation we can have.
Quoting Metaphysician Undercover
What is the natural, inherent order of the vertices of a triangle? This I really want to hear.
Quoting Metaphysician Undercover
What is the inherent order of the vertices of a triangle? Which one is first, which second, which third? How do you know? I want you to answer this.
Quoting Metaphysician Undercover
I get that. So what is the inherent order, the "order which the things have independently of such an imposed order," of the vertices of an equilateral triangle? I am standing by for your response.
Quoting Metaphysician Undercover
I'd prefer the word "collection," since a group is a specific mathematical object that's not at issue here. But I would say the vertices of an equilateral triangle are a pretty good example of a collection of three things that have no inherent order. If you disagree, tell me which one is first.
Quoting Metaphysician Undercover
Triangle triangle triangle. Please answer.
Quoting Metaphysician Undercover
This is sophistry. Clearly there is more than one point in math. I daresay there's a physical analogy here, because other than position, all electrons in the universe are the same. All points on the real line or in Euclidean space are the same. There's a point here and a point there. You can't deny and wish to retain any intellectual credibility.
Quoting Metaphysician Undercover
Come on, man. The point at (0,0) and the point at (1,1) are two distinct points. Or two distinct locations in the plane, if you like to think of it this way. You can't pretend to throw out analytic geometry by denying there are points.
However I will give you this. We can use the word congruent instead of identical. Two geometric objects are congruent if they have the exact same shape, even if they are in different locations or have different orientations. I trust that handles your objections to saying they are identical.
Quoting Metaphysician Undercover
Tell me what the order is so that I may know.
Quoting Metaphysician Undercover
Mathematical order is inspired by physical order, but goes far beyond it. Graph theory for example is all about partially ordered sets. Big deal in computer science, social media, and machine learning.
Quoting Metaphysician Undercover
Which are the first, second, and third vertices of an equilateral triangle?
Quoting Metaphysician Undercover
Which is what? What is the inherent order of the earth, the sun, and a bowl of spaghetti? What is the inherent order of the vertices of a triangle? Would this order be the same for any observer in the universe? Make your case. You don't seem to be able to grapple with any specific examples.
Quoting Metaphysician Undercover
And what is that order? You keep saying they have an inherent order but you won't say what that order is.
Quoting Metaphysician Undercover
Then what is their inherent order, one that would be recognized by any intelligent observer anywhere in the universe? When we meet Martian mathematicians we expect they will know pi (or one of its multiples such as 2pi or pi/2 etc.) I would not expect them to agree on the order of the vertices of a triangle as you seem to claim they would.
Quoting Metaphysician Undercover
For purposes of founding all the diverse set-based mathematical structures such as totally ordered sets, partially ordered sets, well-ordered sets, topological spaces, measure spaces, groups, rings, and field, vector spaces, yes. Exactly. That's the formalism. You can't argue with a formalism any more than you can argue with how the knight moves in chess.
Quoting Metaphysician Undercover
Yes, has it really taken you this long to understand that?
Quoting Metaphysician Undercover
Mathematical abstractions don't have spacio-temporal existence. This is news to you?
Quoting Metaphysician Undercover
Of course we can. We have a bare set. We order it this way. We order it that way. We put on a partial order, a linear order, a well-order. We make it into a topological space in several different ways. We make it a group or a ring or a field. I'm sorry you haven't seen any modern math but you must recognize your own limitations in this regard.
Quoting Metaphysician Undercover
More repetitive falsehoods.
Quoting Metaphysician Undercover
Ok. Good.
Quoting Metaphysician Undercover
Define a binary relation on the set that is antisymmetric, reflexive, and transitive. As explained in painful detail in the Wiki article on order theory.
Quoting Metaphysician Undercover
Of course the elements can be distinguished from each other, just as the elements of the set {sun, moon, tuna sandwich} can. There's no inherent order on the elements of that set.
Quoting Metaphysician Undercover
How about {ass, elbow}. Can you distinguish your ass from your elbow? That's how. And what is the 'inherent order" of the two? A proctologist would put the asshole first, an orthopedist would put the elbow first.
Quoting Metaphysician Undercover
If I use the word congruent, the example stands. And what it's an example of, is a universe with two congruent -- that is, identical except for location -- objects, which can not be distinguished by any quality that you can name. You can't even distinguish their location, as in "this one's to the left of that one," because they are the only two objects in the universe. This is proposed as a counterexample to identity of indiscernibles. I take no position on the subject. but I propose this thought experiment as a set consisting of two objects that can not possibly have any inherent order.
As I explained, the objects, as existing objects, have an inherent order, so it is wrong to deny that the objects have an inherent order.
Quoting TonesInDeepFreeze
The inherent order is the true order, which inheres in the arrangement of objects. If I stated an order, this would be an order which I assign to those objects, from an external perspective, and therefore not the inherent order.
Quoting TonesInDeepFreeze
I assume that there are three individual human beings indicated by those names. The inherent order, if we were to attempt to describe it, would contain all the truthful relations between those beings. Order is the condition under which every part is in its right place. Therefore everything said about the relations between these people would be true if we were describing the inherent order.
Quoting TonesInDeepFreeze
It is a predicate which refers to relations with others, therefore an order.
Quoting TonesInDeepFreeze
Yes, according to fishfry ordering was removed, abstracted away, to leave the members of the set without any inherent order. If you are having difficulty with "inherent order", it is fishfry's term as well, but I suggest that it means order which inheres within the mentioned object (set). A set has been described by fishfry as a type of unity, but it was also said that this unity has no inherent order to its parts.
Quoting fishfry
In a collection of things in the physical world, everything has the place that it has. This is the order of that collection. The fact that we cannot adequately define that order only indicates that we do not adequately understand the positioning of things in the world.
Quoting fishfry
You're missing the point. If the triangle has any existence, then each of its vertices has a position relative to the others, and therefore an order. You can say "equilateral triangle" and insist that this refers to an abstract object with no inherent order to its vertices, but you'd be speaking falsely. There is clearly an inherent order to the vertices signified by "equilateral triangle", or else you could have something other than an equilateral triangle.
Quoting fishfry
I never denied that this was how you conceived "set". I just argued that it is incoherent. Which I still do. If that's progress, then great.
Quoting fishfry
No, rather than say "currently understood", I would class it as a misunderstanding. The reason, as I explained, I think it is impossible to have a set without order, regardless of what mathematicians believe. You continue to assert that this is possible, but have not addressed my arguments against it, nor have you demonstrated how a group of things could exist as a unity (a set), without any order to that group. so I still believe that your assertions, and those of other mathematicians, if they assert similar things, are reflections of a deep misunderstanding.
Quoting fishfry
That is untrue, just like your claim of multiple identical spheres is untrue. If the point is truly non-dimensional then there is nothing to distinguish one point from another, point therefore only one point in math. We make representations of points, in speaking and drawing diagrams, but these are representations, they are not a true point which must be purely abstract. Once you abstract a pure point, how would you make another?
Quoting fishfry
As I said above in my reply to Tones, if I stated an order, it would be a representation, imposed from my perspective, and therefore not the order which inheres within the object, the inherent order. This is the issue similar to the issue with the law of identity. The identity of a thing is within the thing itself, that's what the law of identity states, a thing is the same as itself, its identity is itself. If you asked, me, tell me, what is the thing's identity, well very clearly I cannot tell you, because I'd be assigning an identity and this is not the true identity which inheres within the thing. Likewise, I cannot tell you the order which inheres within the group of things, because iIwould just be giving you an order which I impose on that group from an external perspective.
Quoting fishfry
Each vertex is distinct from the others, and necessarily unique, separated from the others and having a specific relation with the others, or else it would not be the mentioned object. It is not a matter of "first, second, and third", that is simply how you might order them. However, there must be three distinct vertices each with its own unique identity, and a spatial order between these three, indicated by "equilateral triangle".
Quoting fishfry
That is not a set, it's a category mistake. Sun and moon refer to particular objects existing with an inherent order, but "tuna sandwich" refers not to a particular, it signifies a universal, an abstraction.
Quoting fishfry
You do not think that there is an inherent order between your ass and your elbow? You're just being ridiculous now. Do you even know what "order" means? Look it up in the dictionary please. Maybe then we might proceed. However, it appears like we are far apart as to what that word actually means. I'm starting to see now the misunderstanding in mathematics. You give "order" some special meaning, as a magical thing which you can take away from unities, and give to unities without affecting the unity of that unity.
Quoting fishfry
There's contradiction in your description. You say that they have distinguishable locations, yet you can't distinguish their locations. In any case to have an object here, and an object over there, at the same time, is sufficient to say that they are distinct objects and therefore not identical.
You know, you can simply reject the identity of indiscernibles if you don't like it. You can just say that you do not believe it, and you think that two distinct objects might be exactly the same. It's an ontological principle based in inductive reasoning, so it's not necessarily true. It's just that it's a strong inductive principle.
As you dogmatically claim. You keep skipping the central challenge to your claim. That challenge will be repeated in this post.
Quoting Metaphysician Undercover
Petitio principii!
You just shift the answer among various still undefined terms: "actual", "inherent", and now "true".
What is "THE true order" such that the other orders are not "true" orders?
Quoting Metaphysician Undercover
Before, it was temporal/spatial. But now, for just three people it's an even more complicated big deal with you just to attempt to describe "THE inherent order". And "all the truthful relations". I guess you mean the various comparisons, connections, associations, differences, shared properties, contrary properties, contexts, etc. among those three people. That could be a vast number of things. And still you haven't defined how "THE inherent order" is determined by a vast number of associated properties among the three people.
Here is a set of three living people you know about:
{Angela Merkel, Lance Armstrong, Justin Bieber}
Now, please tell me "THE inherent order" of them. Please tell me how you used "all the truthful relations" to determine "THE inherent order".
Quoting Metaphysician Undercover
"The true order that inheres", but you can't say what it is. Sounds pretty "abstract", nay mystical, to me.
Quoting Metaphysician Undercover
From "inherent" to "actual" to "true" to "in its right place". None of them defined by you.
Quoting Metaphysician Undercover
So any predicate that involves "relations with others" is an order?
'the queen of hearts is a red card' and 'the four of clubs is a black card' are statements about predicates of the two cards. So what is "THE inherent order" -
By the way, what do you mean by 'truthful' that wouldn't be said by just 'true'? And why even say 'truthful relations' when you could just say 'relations'?
Quoting Metaphysician Undercover
I don't speak for him, but I would imagine he's using such locutions as figures of speech.
I'll say it for you again, without recourse to figures of speech such as "removing" as I did before without recourse to figures of speech such as "removing":
{Bob Sue Tom} is the set whose members are all and only Bob, Sue and Tom.
There are 6 strict linear orderings of the set {Bob Sue Tom}.
If you want to say there is one of those orderings that is "THE inherent/actual/true/truthful/tutti-fruiti" ordering while the others are not "THE inherent/actual/true/truthful/tutti-fruiti" ordering, then you need to DEFINE the terminology "THE inherent/actual/true/truthful/tutti-fruiti" ordering".
Quoting Metaphysician Undercover
Again, I don't speak for him, but I understood him to be making a correct point that does not depend on whether he used a figure of speech such as "inherent". His point was that the set is determined by its members and not by the orderings of the set. I don't speak for him, but I have little doubt that he would agree (without using "inherent"):
(1) A set is determined by its members. Axiomatically, S =T if and only if every member of S is a member of T and every member of T is a member of S.
(2) A finite set of cardinality n has n! strict linear orderings. So for sets with cardinality greater than 1, there are more than 1 strict linear orderings of the set.
See, no need to use the word 'inherent' and especially not "THE inherent ordering".
Saying "the set has no inherent ordering" boils down as a locution to saying (1) and (2).
But then YOU jumped to say there IS "THE inherent ordering", and you are floundering to meet the challenge of defining it.
And you chided ME for my interest in mathematics that you deem not empirically justified. Now you're resorting to a mystical "true order that inheres in an object" but such that if we even attempted to state what that order would be, then we would be wrong!
Just by saying which you think is "THE inherent order" you would necessarily not be choosing "THE inherent order"? So just by saying which ordering you think is "THE inherent order" you would necessarily be wrong!?
If the true order cannot be assigned from an external perspective, then what is the "internal perspective" of an arrangement of objects? Will I know the "true order" of its vertices if I stand in the middle of a triangle?
In other words, how could the inherent order be known? If it cannot be known then how do you know there is one?
Temporal/spatial was just one type of order, fishfry and Lluke gave examples of many other types. So we're not restricted to temporal/spatial order in our attempts at understanding the nature of inherent order.
Quoting TonesInDeepFreeze
I explained very clearly in the last post why i cannot tell you the inherent order. It's not something that can be spoken,. Just like the identity of a thing, as stated by the law of identity, is not something that can be spoken. It is what is proper to the thing itself, not what is said about the thing. The order which inheres within the thing is proper to the thing itself, and not what is said about the thing. Do you understand this principle of identity?
Quoting TonesInDeepFreeze
Of course, any relation with another is an order.
Quoting TonesInDeepFreeze
If you wish to view the law of identity as a "mystical" principle, I have no problem with that. I would consider that most good ontology is based in mysticism.
Quoting Luke
Are you aware of Kant;s distinction between phenomena and noumena? As human beings, we do not know the thing itself, we only know how it appears to us. Kant seems to describe the noumena as fundamentally unknowable. Others argue that it is fundamentally knowable, but only to a divine intellect, and not a lower intellect like the human being.
Quoting Luke
We assume that there is a way that the world is (inherent order), because this is what makes sense intuitively. If we assume that there is no such thing, then we assume that the world is fundamentally unintelligible. To the philosophically inclined mind, which has the desire to know, the assumption that the world is fundamentally unintelligible is self-defeating. Therefore the rational choice is to assume that there is a way that the world is (inherent order). So we don't know that there is an inherent order, we assume that there is, because that is the rational choice.
Quoting Metaphysician Undercover
If order is not restricted to "temporal/spatial", then order is not restricted to unknowable noumena.
It was pointed out to you that there are orderings that are not temporal-spatial. But you insisted, over and over, that temporal-spatial position is required for ordering.
But now you enlist the view of other posters, with whom you so strongly disagreed, to wiggle out of your own untenable view!
What has happened is that it has finally gotten through to you that ordering is NOT only temporal-spatial, so you shifted to saying that "THE inherent order" is based on "all the relations". A complete reversal of your position, except you still cling to your notion of "THE inherent order".
And you still can't say what "THE inherent order" is with regard to your new bases of "all relations".
Quoting Metaphysician Undercover
Flat out contradicts your new claim; "we're not restricted to temporal/spatial order".
Quoting Metaphysician Undercover
Flat out contradicts your new claim; "we're not restricted to temporal/spatial order".
Quoting Metaphysician Undercover
Flat out contradicts your new claim; "we're not restricted to temporal/spatial order".
Quoting Metaphysician Undercover
Quoting Metaphysician Undercover
Quoting Metaphysician Undercover
Quoting Metaphysician Undercover
Quoting Metaphysician Undercover
Quoting Metaphysician Undercover
Quoting Metaphysician Undercover
Quoting Metaphysician Undercover
Quoting Metaphysician Undercover
Quoting Metaphysician Undercover
Quoting Metaphysician Undercover
Not only cannot you tell us what "THE inherent order" is for ANY set, but you can't even define the rubric.
You can't even say what is "THE inherent order" of {your-mouse your-keyboard} even though they are right in front of you. And you said a really crazy thing: If you identified "THE inherent order" of a set then we will identified it incorrectly. You have a 50% chance of correctly identifying "THE inherent order" of {your-mouse your-keyboard} but you say that you would be wrong no matter which order you identified as "THE inherent order"!
Metaphysician Undercover fails to distinguish between two facts:
(1) For every set, there is a determined set of strict linear orderings of the set. (That is a simple and intelligible alternative to his confused notion of "THE inherent order" ensuing from identity.)
(2) For a finite set of cardinality n>1, there are n! strict linear orderings of the set. And there has not been given a definition of "THE inherent ordering" from among those n! orderings (expect Metaphysician Undercover's mumbo jumbo about an ordering based (based in what way?) on all relations among the members and such that no one can correctly identify it but that it exists by virtue of identity).
I don't. Quoting Metaphysician Undercover
Non sequitur.Quoting Metaphysician Undercover
I am sympathetic to the idea of assuming frameworks for making sense our experience. And, indeed, the mathematical notion of ordering can be part of that. It doesn't require an undefined notion of "THE inherent order".
Quoting Metaphysician Undercover
To each his own.
See, not dogmatic. You expressed a philosophical preference, which is not itself an ignorant, confused and incorrect claim about mathematical logic and set theory, so I don't begrudge you having that preference.
I changed my mind on that days ago, when fishfry offered an ordering based on best. Then we moved along to "inherent order".
Quoting TonesInDeepFreeze
Right, I cannot say what the inherent order is, for the reasons explained. Do you have a problem with those reasons? Or do you just not understand what I've already repeated? You understand what "inherent" means don't you?
Quoting TonesInDeepFreeze
The question is whether or not it is possible for a set to be free from inherent order, i.e. having no inherent order, as fishfry claimed. You still don't seem to be grasping the issue.
Of course, that's why we have to acknowledge the difference between the order we say that a group of things has, and the inherent order of that group of things. They are both called "order". That they are different accounts for the fact that we make mistakes in understanding the order of things..
That doesn’t work and you’ve misunderstood.
You are trying to draw an analogy between order/inherent order and phenomena/noumena. However, phenomena and noumena are both temporal-spatial, which makes order and inherent order also temporal-spatial by analogy.
You have already conceded that there are “many other types” of order besides temporal-spatial.
If there are “many other types” of order besides temporal-spatial, then order is not necessarily phenomenal or noumenal, so your argument fails.
If order is not limited to temporal-spatial order, then you can no longer hide behind your claim that true order is unknowable. So how do you account for any order which is not temporal-spatial?
Do you mean this post?:
Whatever "change of mind you had" in that pile of confusions, you said inter alia:
(1) Quoting Metaphysician Undercover
and
(2) Quoting Metaphysician Undercover
So there you are, still demanding that order must be temporal-spatial.
However, then, fifteen hours ago (not days ago), yes, you wrote:
Quoting Metaphysician Undercover
And that is the very remark that I just replied to. So I don't see you changing your mind since the post with (1) and (2) except the recent post of which I pointed out that it is inconsistent with your earlier stance.
/
And after so many days on end of you claiming that orderings are necessarily temporal-spatial, now you recognize that orderings do not have to be temporal-spatial, so what took you so long? It's piercingly clear that there are orderings that are not not temporal-spatial, but you could not see that because you are stubborn and obtuse.
I have rebutted great amounts of your confusions. You either skip the most crucial parts of those rebuttals or get them all mixed up in your mind.
Anyway, to say that there is "THE inherent ordering" of a set, but not be able to identify it for a set as simple as two members is, at the least, problematic. But more importantly, you cannot even define the "THE inherent ordering" as a general notion. That is, you cannot provide a definition like:
R is the inherent ordering of S if and only if P
where P is the definiens.
And your notion is so ridiculous that you say that if one did attempt to identify "The inherent ordering" of a set then one would not correctly choose "The inherent ordering". What? For a set with two members, I have a 50% chance of identifying "THE inherent ordering" (if there were such a thing) just by guessing.
Your boorish condescension is stupid.
Quoting Metaphysician Undercover
I never said that the set of orderings of a set is not inherent to the set. I said over and over and over that sets have multiple orderings. The point I have been making to you is that you have not defined what it means for one of those orderings in particular to be "THE inherent ordering". You are the one who doesn't grasp the issue.
There are multiple orderings. Given a reasonable sense of 'inherent', the orderings of the set are all inherent to the set. In set theory and abstract mathematics. EVERY property of an object is inherent to the object. (Mathematical) objects don't change properties. They have the exact properties they have - always - and no other properties - always. There is no "time" operator that allows (mathematical) objects to have different properties at different times, so the properties are inherent. That is just a report on set theory, which doesn't have a "time" operator.
But the point you keep missing is that you have not defined what it means to say that one of the orderings in particular is "THE inherent ordering". They are all orderings of the set, and they are all inherent to the set. I have put 'THE' in all caps about a hundred times now. The reason I do that is obvious, but you still don't get it.
This started with discussion of the axiom of extensionality. With that axiom, sets are equal if they have the same members. In that regard, a set is determined by its members, whatever the set of orderings of the set might be. And, of course, for every set there is the set of all the orderings on that set. That set of all the orderings on the set is "inherent" in the sense that it doesn't change. But the point you don't get is that there is not one of those orderings that is in particular "THE inherent ordering" while the others are not. And it seems the reason you don't get that is because you started out needing to deny the sense of the axiom of extensionality itself, even though you are ignorant of what it does in set theory and you are ignorant of virtually the entire context of logic, set theory and mathematics.
How infinity was defined (from ancient Greeks and Indians till just before Georg Cantor) operationally and thus its conceptualization as an endless process. As is obvious to me now, this idea of the infinite as a task that can't be completed immediately and violently conflicts with infinity as actual defined as ended/completed, leaving only potential infinity as a conceivable mathematical object.
Enter Georg Cantor and he discards, perhaps because he intuits the complication I refer to above, the traditional idea of infinity as an endless task in favor of, surprisingly, an even older understanding of numbers viz. 1-to-1 correspondence. Thus, he defines infinity as a set whose members can be put in a 1-to-1 correspondence with the set of natural numbers. As you can see, defining infinity as such sidesteps the vexing issue of endlessness; that infinity can't be completed is a non issue because all that matters is whether or not we can uniquely match one element of a given set with another element of the set of natural numbers {1, 2, 3,...}.
It's exactly how the first mathematicians, by that I mean to refer to prehistoric times when tally marks were first invented/discovered, solved counting problems. Prehistoric people didn't know how to count, some say, beyond 2 and 3 and more were, for them uncountable which comes very, very close to what infinity is to the modern man. The way they got around this problem was by matching what they wanted to count, their population, livestock, etc. the relevant individuals with counters (tally marks). As you can see, we don't have to know the actual size of what's being counted, all that's required is a unique tally for each member of the set of objects that's being counted. Completing/ending/finishing the counting process of infinity is now a non issue.
To cut to the chase, infinity under this interpretation (1-to-1 correspondence between a set and the set of natural numbers), very ingeniously I must say, avoids the endless nature of infinity and the controversy over actual and potential infinities fails to gain the traction it needs to wreak havoc in set theory that was designed to deal with infinity.
Coming to the matter of an actual infinity in set theory, it becomes patently clear that there are sets whose elements can be put in a 1-to-1 correspondence with the elements of the set of natural numbers which includes itself and hence, in that sense, there are actual infinities.
I don't know how he reads in the original German, but the above is not how the set theory that came from Cantor works.
We don't define "infinity" as a noun. Rather, we define the predicate 'is infinite'. And the definition is NOT
x is infinite iff x is 1-1 with N.
Indeed not, since there are infinite sets that are not 1-1 with N.
Rather, the definition is:
x is finite iff x is 1-1 with some natural number.
x is infinite iff x is not finite.
An alternate definition is equivalent to the above with the axiom of choice:
x is infinite iff there is a proper subset s of x such that x is 1-1 with s.
I admit it's possible that there's more of me in my post about infinity than Cantor. Nevertheless, I'm fairly confident that what I wrote would've brought a smile to his face. He was a deeply troubled man I believe, in no small measure due to Leopold Kroenecker's scathing criticisms of his life's work.
Yes, it's possible he might get a chuckle at your hapless ignorance.
Quoting TheMadFool
It wasn't just that Kronecker criticized the work. But it does seems reasonable to think that his professional difficulties vis-a-vis Kronecker might have contributed to his poor mental condition, but I don't think we know for sure.
Now, hold on a minute. The post I made is clear and to the point and captures the essence of Cantor's views on infinity.
Quoting TonesInDeepFreeze
Indeed, any ideas why Kroenecker was so dead against Cantor? Was there anything more going on then just an academic disagreement on infinity? You know, like a personal grudge, anti-Russian sentiments? Your guess is as good as mine.
The question of whether God is bound by the laws of physics is an old one. I found some references but these are not definitive, I just grabbed them off Google to illustrate that people have been thinking about the matter. I didn't read any of them, just wanted a random sample.
https://www.reddit.com/r/AskAChristian/comments/7br2fb/are_angels_bound_by_the_laws_of_physics_is_god/
https://faithfoundedonfact.com/is-god-bound-by-logic/
https://theconversation.com/can-the-laws-of-physics-disprove-god-146638
https://philosophy.stackexchange.com/questions/47105/are-gods-also-bound-to-the-laws-of-physics
https://www.quora.com/Does-God-obey-the-laws-of-physics
https://consultingbyrpm.com/blog/2011/08/can-god-violate-the-laws-of-physics.html
Now for my own contribution, consider a video game designer who creates an artificial but self-consistent world. The beings in that world are bound by the laws as defined by the designer; but the designer lives in what we call the real world and is not bound by the artificial rules of the game.
Why wouldn't God be exactly the same way? God has created the world, including the laws of physics. God's creatures, namely us, are bound by the laws of physics. But God isn't. Remember, God said, "Let there be light." I've always found it interesting that the ancients who wrote the Bible intuited that electromagnetic radiation was fundamental. And clearly the ancients saw God as existing outside of time and space, outside of the laws of physics.
Another point is that the laws of physics themselves are historically contingent ideas of human beings. It's a philosophical assumption that there are actually any laws that govern the universe, as opposed to science being a collection of theories that just seem to work to a good approximation, but that aren't actually true in any absolute sense. In fact there is a name for the belief that the world studied by science is real: scientific realism. There's no absolute proof that scientific realism is true. It could all be a dream, I could be a brain in a vat, or I could be a Boltzmann brain, a momentary coherence in an otherwise random and formless universe.
The same reasoning applies to simulation theory, which a lot of people take seriously these days. The advocates of simulation theory assume that the Great Programmer in the Sky operates according to the same laws of physics that we do and reason accordingly. But of course such an assumption is unwarranted. The Programmer, if such there be, lives in a completely different world with totally different physics. We can't use reason and logic to figure out what the next level up is like.
I haven't seen much of Brian Greene, but I'm a big fan of Sean Carroll.
Sure, if "captures the essence" means grossly mischaracterizes with ignorant confusions.
You seem to be contradicting yourself. First set your own house in order is advice that I've been given and have heeded. I suggest a similar course of action for your good self. Have a g'day.
Right, inherent order, which I classed as noumenal, appears to be spatial-temporal. But the type of ordering which fishfry demonstrated to me, ordering by best, or better, cannot be inherent order because it is relative to intention, therefore phenomenal.
I don't see the problem.
Quoting TonesInDeepFreeze
Yes, I agree that "order", when reduced in the extreme, seems to require necessarily, spatial-temporal conceptions. .I agree that some types of ordering such as those presented, which I called order by best, or better, appear to be free from spatial-temporal conceptions. But ultimately there must be something which is being ordered, individual objects, and this requires spatial separations. Perhaps however, we can order intentions themselves, as better or worse, as objects in the sense of goals, and we might give ends an ordering which is completely void of spatial-temporal conceptions. But this requires that we determine what type of existence an intention has.
Quoting TonesInDeepFreeze
Earlier in the thread, it was flatly denied as nonsensical, that the type of objects which existed in sets, are intentions. As explained above, that is the only way I can apprehend an "object" which has no inherent order, if it were an intention. So, if the things in sets are said to have no inherent order, the issue remains. How do you conceive of individual things with no spatial-temporal ordering, such that they can exist in a set without inherent ordering if these things are not intentions?
Quoting TonesInDeepFreeze
I have defined "inherent order", in relation to the law of identity. It is you who is skipping the most crucial parts of what I write.
Quoting TonesInDeepFreeze
OK then, two distinct ordering of the same elements constitutes two distinct sets. Do you agree? If order is inherent to the object, as you claim, then two distinct orderings of the same elements constitutes two distinct objects, therefore two distinct sets. Do you see this?
Quoting TonesInDeepFreeze
It's you who keeps missing the point. The "inherent order" is as "inherent" implies, the one which inheres within the object, as its identity, stipulated by the law of identity. It is categorically different from any order which we might assign to the object. Therefore it is not "one of the orderings" which we lay out, it is distinct from these. And the question you keep asking me, which of these orders is the inherent order is nonsensical because i keep telling you it's none of those orders.
Quoting TonesInDeepFreeze
Do you agree with me, that "equal" does not mean "the same"? Therefore equal sets are not the same set. Two sets with the same members in different orders can be said to be equal, but they cannot be said to be the same set. This is the part that fishfry doesn't get. Fishfry believes that if the sets are equal, they are necessarily the same set.
Quoting TonesInDeepFreeze
I am not denying the axiom of extensionality, I am denying a particular interpretation of it, which says that equal sets are the same set. I look at this as a misunderstanding.
How is intention phenomenal (in the relevant Kantian sense)? Or are you no longer talking about Kantian concepts, just like you are no longer talking about the "dots" diagram having the inherent spatial order that it has?
Quoting Metaphysician Undercover
Intention is an integral part of the phenomenal system, the world as it appears to us, as the fulfillment of our wants and needs have shaped the way that we perceive the world evolutionarily, and have much influence over our perceptions on a day to day basis..
Where does Kant say this?
Also, do you have any intention of accounting for your latest blatant contradiction:
Quoting Metaphysician Undercover
Quoting Metaphysician Undercover
Before your claim was that the inherent order is what's shown. Now you claim that the inherent order is what's hidden. It can't be both.
I don't know if Kant ever said, but it's pretty obvious how intention must fit in.
Quoting Luke
No, sorry I must have made a mistake, or perhaps you just misunderstood. More likely, you intentionally misinterpreted, as usual. I'm very well acquainted with your strawman interpretations designed at creating the appearance of contradiction.
Quoting Luke
There is no contradiction in saying that I am showing you an order which I cannot describe. Try again. But I suggest you try to understand rather than trying to misunderstand.
So you don't know whether intention has anything to do with Kant's phenomena-noumena distinction?
And yet you still use this distinction as the basis of your argument regarding inherent order?
You tried to draw an analogy between your supposed inherent order and Kant's noumena. When I pointed out that you had already conceded that "many other types" of order are not spatio-temporal and therefore not noumenal, you said that one other type (best to worst) "is relevant to intention, therefore phenomenal". If you don't know whether intention has anything to do with Kant's phenomena-noumena distinction, as you now admit, then you cannot claim that best-to-worst order is "relevant to intention, therefore phenomenal".
Quoting Metaphysician Undercover
There has been no misunderstanding. It's clear to everyone that you continually change your position and argue out of both sides of your mouth.
Quoting Metaphysician Undercover
What strawman interpretation? Instead of empty accusations, go ahead and explain how or what I have misinterpreted.
Quoting Metaphysician Undercover
That's different to what you were saying earlier in the discussion. Earlier, you were saying that the inherent order can be seen and described. For example:
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Only recently did you invoke Kant's phenomena-noumena distinction, changing your position entirely:
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Contrary to the claims of your earlier posts, we can no longer simply look and see the inherent order which is demonstrated by the diagram. You now claim that what we see is a mere phenomena, and that the true, inherent order cannot be seen, described or known. Pure contradiction.
I don't use that distinction as the basis for my argument, I gave that distinction as an example which i thought you might be able to understand.
Quoting Luke
Come on Luke, use some intelligence. Kant did not have to name every instance of what contributes to phenomena for us to place things in that category. If you think I am wrong, and intention ought not be placed in that category, then just tell me. But please give reasons. Simply saying Kant didn't explicitly say it therefore, you're wrong in your analogy, is pointless.
Quoting Luke
I told you how you misinterpreted., You claimed a contradiction when I said I couldn't describe something which was shown. That is just an indication of the limits to human intelligence, and word use, not a contradiction.
Quoting Luke
Thanks for all the quotes removed from context. To be shown, or demonstrated does not mean to be stated, I went through that in the last post, and again above. And, "the positioning of those points relative to each other is describable" does not mean that I have the capacity to describe them. I do believe I mentioned that it would require an intelligence superior to a human intelligence, like a divine intellect. I was arguing the deficiencies of the human intellect, in being incapable of describing what is inherently describable.
This is exactly the problem which quantum physics actually has. The physicists are incapable of adequately describing the positioning, therefore "order" of the particles. We can either conclude that the particles have no inherent order, because the order cannot be determined by the human techniques, or we can conclude that they have an order, but other principles, and a higher intelligence, are required to figure out the order. As I explained to you already, (which you've left out of your inflammatory interpretation), is that the latter choice is the rational choice.
Therefore I reaffirm my accusation, that you are intentionally misinterpreting what I write for the sake of making it appear as contradictory, instead of putting any effort into trying to understand it. This is very consistent with my observations of your mode of operation at this forum.
That's odd. When I asked you what the "internal perspective" of an arrangement of objects was, you said:
Quoting Metaphysician Undercover
And only a day ago you said:
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But now you say that Kant's phenomena-noumena distinction is not the basis for your argument. How do you expect me to understand your argument about inherent order if one day you say that inherent order is noumenal, and the next day you say that Kant's phenomena-noumena distinction is not the basis for your argument?
Quoting Metaphysician Undercover
You've now told me that you don't use Kant's distinction as the basis for your argument, so I don't know what analogy you're referring to. Either inherent order is noumenal or it isn't. Maybe you meant indirect realism instead of noumena? I don't know.
Quoting Metaphysician Undercover
False. I claimed a contradiction between your position and statements before you introduced Kant, and your position and statements after you introduced Kant.
Quoting Metaphysician Undercover
What context is lacking? Feel free to use the links provided to find the context.
Quoting Metaphysician Undercover
Yes, but in the posts before you introduced Kant, you were clearly saying that the appearances were the reality (i.e. direct realism), as demonstrated by the quotes. Again:
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You asked us here (prior to your introduction of Kant) to take a look at the diagram and see the order the dots have, and that they could not have any other order. Yet now (after your introduction of Kant) you are trying to convince us of the opposite: that there must be another order - the inherent order - which is different to the order we can see in the diagram. Moreover, you have claimed that the appearance of order and the inherent order could not be the same just by chance, despite your admission that you don't know whether or not they could be the same.
The contradiction is more stark here:
Quoting Metaphysician Undercover
To return to my recent point, you have conceded that there are "many other types" of order which are not "temporal-spatial", therefore your references to phenomena-noumena (or indirect realism or whatever) do not apply to these many other types of order. Therefore, you cannot claim that there is some hidden order to these other types. While that might be irrelevant to your claims, it is not irrelevant to the criticisms of your claims made by the other posters here. You are the only one arguing that order must involve spatio-temporal phenomena (and/or noumena).
Right, this principle has a long tradition, it goes back at leas to Aristotle, with the law of identity, so it is definitely not based in Kant. Kant has simply presented the similar principle in his own way. I present it in my way. The principle is not "the same" it is a similar principle, needing to be refined and understood in the unique way of each particular individual mind who desires to understand..
In Aquinas, we see that independent Forms are fundamentally "intelligible" but not intelligible to the human intellect, because that intellect is united with a material body. This position, of being dependent on a body and the sense organs makes the intellect deficient. We find this same principle in Kant. The human intellect produces knowledge from phenomena which is dependent on sensation, and sense appearances. Notice that Kant refers to the noumenon as "intelligible", though it is not intelligible to us human beings, due to this predicament, which is not a contradiction.
Quoting Luke
No, I don't believe I mentioned "appearances". And "inherent" clearly means within the object, as what inheres within. So if you interpreted me as saying the "inherent order" is part of the appearance of the object within a mind, rather than within the thing itself, I think this was a matter of misinterpretation. You did demonstrate some confusion as to what "inherent" means, as if you were somewhat unfamiliar with the word, so perhaps you thought I was talking about an order abstracted from an appearance, rather than an inherent order at that time.. But if you understood what "inherent" means, and what "appearance" means, you would not have interpreted in this contradictory way.
I think perhaps the issue was confused because we were talking about a diagram, which is intended to show something. Therefore there is a number of levels of representation which adds ambiguity. The diagram is an actual thing itself, with an inherent order. But it is also made to represent an order (an apparent order), which a human mind apprehends. This produced the problem with fishfry claiming it was "random", lacking order, because that is the intended (apparent order) which it was made to represent, However, I argued that there is necessarily an inherent order within the thing shown, and fishfry's claim that it did not show an order, that it was "random", is a false claim. If you had understood this argument from me, you would have recognized that I was making the same distinction at that time.
Quoting Luke
That's a misinterpretation. I was asking the same thing both times, to look at the thing, and see that there is an order within the thing itself. What seems to be causing you confusion is the fact that we can look at a thing, and conclude that there is order inherent within (that's what makes a thing intelligible) without actually understanding the order., i.e. we see order without understanding it.
Quoting Luke
I don't understand your point. Your reference to "hidden order" doesn't make sense. I'm not talking about a hidden order, and this idea seems to be the source of your misunderstanding. The order is right there in plain view, as things are, but it is just not understood, because we do not have the capacity to understand it.
If you were talking about the inherent order the entire time, and if the inherent order is not perceived or apprehended, then why did you say:
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It can only be because you were not talking about the inherent order the entire time. You have contradicted yourself.
Furthermore this:
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contradicts this:
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I don't see any problem with those quotes. As I said, the order is right there, in the object, as shown by the object, and seen by you, as you actually see the object, along with the order which inheres within the object, yet it's not apprehended by your mind.
Sorry L:uke, but I find it extremely ridiculous that you are trying to tell me what I was talking about. As I said, you need to go back and reread the entire section, with the understanding, and commitment, that it's all about the inherent order, therefore the order which inheres within the object. And quit trying to force your nonsense interpretation, insisting that you know better than I do, what I was trying to say, simply so that you can say that I was trying to contradict myself. It's foolish of you.
Are these both the inherent order (bolded)? If so, then why do you say "along with the order"?
So, a professional philosopher. At one point in the article he says: "We are indeed rationally justified in thinking 2 plus 3 will always be 5, because 2 plus 3 is not distinct from but rather identical with 5." My emphasis. So at least one professional philosopher would object to your claim that they are not identical.
Yes, you see the object along with the order which inheres within, meaning you see the order, you just do not apprehend it. Consider the dots, we see them, we must see the order because it's there, yet fishfry claimed that the dots were randomly arranged, indicating the order was not apprehended
Quoting fishfry
There are numerous philosophers who argue against the law of identity as stated by Aristotle, Hegel opposed it, as is evident here: https://thephilosophyforum.com/discussion/9078/hegel-versus-aristotle-and-the-law-of-identity/p1
What I see as an issue which arises from rejecting the idea that each particular object has its own unique identity (law of identity), is a failure of the other two interrelated laws, non-contradiction, and excluded middle. Some philosophers in the Hegelian tradition, like dialectical materialists, and dialetheists, openly reject the the law of non-contradiction. When the law of identity is dismissed, and a thing does not have an identity inherent to itself, the law of non-contradiction loses its applicability because things, or "objects" are imaginary, and physical reality has no bearing on how we conceive of objects.
There are specific issues with the nature of the physical world that we observe with our senses, which make aspects of it appear to be unintelligible. There must be a reason why aspects of it appear as unintelligible. We can assume that unintelligibility inheres within the object itself, it violates those fundamental laws of intelligibility, or we can assume that our approach to understanding it is making it appear.as unintelligible. I argue that the latter is the only rational choice, and I look for faults in mathematical axioms, and theories of physics, to account for the reason why aspects appear as unintelligible. I believe this is the only rational choice, because if we take the other option, and assume that there is nothing which distinguishes a thing as itself, making it distinct from everything else (aspects of reality violate the law of identity), or that the same thing has contradictory properties at the same time (aspects of reality violate the law of non-contradiction), we actually assume that it is impossible to understand these aspects of reality. So I say it is the irrational choice, because if we start from the assumption that it is impossible to understand certain aspects of reality, we will not attempt to understand them, even though it may be the case that the appearance of unintelligibility is actually caused by the application of faulty principles. Therefore it is our duty subject all fundamental principles to skeptical practices, to first rule out that possibility before we can conclude that unintelligibility inheres within the object.
Aristotle devised principles whereby the third fundamental law, excluded middle would be suspended under certain circumstances, to account for the appearance of unintelligibility. Ontologically, there is a very big difference between violating the law of excluded middle, and violating the law of non-contradiction. When we allow that excluded middle is violated we admit that the object has not been adequately identified by us. When we allow that non-contradiction is violated we assume that the object has been adequately identified, and it simply is unintelligible.
Do we perceive both the apparent order and the inherent order? Is there a difference between the apparent order and the inherent order? If so, what is the difference between them?
If there is no difference between the apparent order and the inherent order, then why did you draw the analogy with Kant's phenomena-noumena distinction, and why did you state:
Quoting Metaphysician Undercover
If there is a difference between the apparent order and the inherent order, then why did you state:
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Appreciate the reference.
Quoting Metaphysician Undercover
Not a word of this is even on topic relative to whether 2 + 3 and 5 are identical. Since mathematically they are, and as a mathematical expression it must necessarily be interpreted in terms of mathematics, nothing you say can make the slightest difference. Excluded middle? Did Aristotle anticipate intuitionism? That's interesting.
The apparent order is made up, a created order, assigned to the group of things, so it is not perceived, it is produced by the mind.
Quoting Luke
Why not? The order which we assign to things is clearly not the same as the "exact" order which inheres within things or else we'd have an absolutely perfect understanding of the order of the universe.
Quoting fishfry
As I've explained to you already, the idea that 2+3 is mathematically the same as 5, is simply a misunderstanding of the difference between equality and identity. They are equal, but equal is distinct from identity. I've told you this numerous times before, but you do not listen. Nor do you seem to pay any attention to my references, only repeating your misunderstanding in ignorance.
However, I'll reproduce for you the opening lines from the Wikipedia entries on both "equality" and "identity" below, just to remind you of how bad your interpretation really is.
In mathematics, equality is a relationship between two quantities or, more generally two mathematical expressions, asserting that the quantities have the same value, or that the expressions represent the same mathematical object.
In philosophy, identity, from Latin: identitas ("sameness"), is the relation each thing bears only to itself.
On the contrary. 2 + 3 and 5 are mathematically identical. There is not the slightest question, controversy, or doubt about that.
Quoting Metaphysician Undercover
I listen very well, but the problem is that you are factually wrong. Wrong on the facts. 2 + 3 and 5 are mathematically identical. They represent the same thing. The identical thing. There is one single thing, and it has two names, 2 + 3 and 5. It has many other names as well, such as, "The cardinality of the vertices of a pentagon." If you kept telling me the sun rises in the west and I disagreed and you accused me of not listening, that would be an unfair statement on your part, would it not? Likewise 2 + 3.
Quoting Metaphysician Undercover
You did in fact give me a reference, the very first one you've given me in three years, after I've asked you many times for references. And it turned out to be only a reference to another thread on this board, and not a reference to the work of any reputable or even disreputable philosopher. And when I read the reference, I did not find anything that shows that 2 + 3 is anything at all other than 5.
Apparent order is not perceived? Do you know what "apparent" means?
If apparent order is not perceived, then your earlier distinction between "internal" and "external" perspective is irrelevant; it's not a matter of perspective at all. So why did you introduce the distinction between "internal" and "external" perspective?
In that context, "apparent" must mean "seems". If you used "apparent" to mean "perceived by the senses", I would say that you had stated an oxymoron. We apprehend order with the mind, we do not perceive it with the senses.
Quoting Luke
I don't believe I said anything about an internal perspective. I distinguished between the order which inheres within the thing itself (which we assume must be real to account for the consistency we note from observations), and the order which we assign to things, from our external perspective of them. This is the reason why our knowledge of the order of things is fallible, the order which we say something has is not the same as the order which it actually has. The best we can say is that we have created a representation of the order that things have. We create the representations through the means of analysis of empirical observations, which do not provide us with the inherent order, in conjunction with theorizing, hypotheses. So there is a separation, a medium consisting of observation and theory, which lies between the apparent order (the order which things seem to have), and the true order which inheres within the things themselves.
The reason why I introduced this distinction is because fishfry claimed that there is a sort of thing, "a set", which has no inherent order at all. I said that such a thing does not exist, because to exist is to have some sort of inherent order. Fishfry scoffed at this. So I proceeded to ask fishfry to explain this type of unity of parts, within which the parts have no order. How could there be such a unity? The point being, that this is simply an imaginary thing stated, 'parts without order', which doesn't correspond to any reality, which is really a logical inconsistency representing falsity, because it is impossible to have parts without order. To be a part of something implies an order in relation to a whole, without that order it cannot be said to be a part. We would have to call it something other than a "part". The point being, that the whole, which the so-called part is said to be a part of, "the set", is not a true whole because it provides no order relations to the so-called parts. Things existing with absolutely no relations of order cannot be said to form a whole, or unity of any kind.
You spoke of an "external perspective", which implies an internal perspective. You might recall I asked you about it and you responded:
Quoting Metaphysician Undercover
It was when I asked you about the implied "internal perspective" of an arrangement/order that you drew the analogy with Kant's distinction between phenomena and noumena, saying of a thing that "we only know how it appears to us".
Quoting Metaphysician Undercover
Bullshit. You are trying to pretend you were never talking about sense perception? Sense perception is exactly what Kantian phenomena is about, and clearly what you meant when you said that we can only know how a thing "appears to us".
If we apprehend order with the mind and not with the senses, then perhaps you could finally explain this:
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Your current argument is that we do not perceive order with the senses, and that we cannot apprehend inherent order at all. Therefore, how is it possible that the inherent order is the exact spatial positioning shown in the diagram?
As per the quotes above, from Wikipedia, the mathematical notion of identical , as equal, is not consistent with the philosophical notion of identity, described by the law of identity. In other words, mathematicians violate the law of identity to apply a different concept of identity, making two things of equal value mathematically identical. You might accept this, and we could move on to visit the possible consequences of what I believe is an ontological failure of mathematics, or you could continue to deny that mathematicians violate this principle. The latter is rather pointless.
Quoting Luke
If all perspectives are external to the object, then "perspective" is necessarily external, and saying "external perspective" just emphasizes the fact that perspective is external. Internal perspective is not implied, just like saying "cold ice" doesn't imply that there is warm ice. That's why I referred to Kant, to show how our perspective of the thing in itself is external to the thing.
Quoting Luke
Why not? I don't understand your inability to understand. Let me go through each part of your question. 1) We do not perceive order with the senses. No problem so far, as we understand order with the mind, not the senses. 2) We cannot apprehend the inherent order. Correct, because the order which we understand is created by human minds, as principles of mathematics and physics, and we assign this artificially created order to the object, as a representation of the order which inheres within the object, in an attempt to understand the inherent order. But that representation, the created order is inaccurate due to the deficiencies of the human mind. 3)The inherent order is the exact positioning of the parts, which is what we do not understand due to the deficiencies of the human mind.
Does that help? You have emboldened the word "shown". Why? Do you understand that something can be shown to you which you do not have the capacity to understand? The physical world shows us many things which we do not have the capacity to understand. For example, the theologians used to argue that the physical world shows us the existence of God. Most people would claim that the physical world is not evidence of God, and in no way does the physical world show us God. The theologian would say that you just do not understand what is being shown to you. the exact order is being shown to us but we do not understand it.
You're factually wrong.
Is the set {0,1,2,3,4} identical to the set {0,1,2,3,4}? I have to assume you'd say yes.
But 2 + 3 and 5 are both representations of the set {0,1,2,3,4}. So they're identical.
Quoting Metaphysician Undercover
I just showed (as I have probably a hundred times before) that you are wrong about this. Mathematicians call two things identical when those things are identical.
Now I will concede that there are subtle counterexamples. For example the natural number 5 and the real number 5 are distinct as sets. Nevertheless they are identical when viewed structurally. If you wanted to say they are equal but not identical that would be arguable, but it's a subtle point, and could be argued either way.
Quoting Metaphysician Undercover
Well, math does not violate this principle. 2 + 3 and 5 are identical. They are both representations of the set represented by {0,1,2,3,4}, which of course is not actually "the" set, but is rather yet another representation of that abstract concept of 5.
Quoting Metaphysician Undercover
Only to the extent that you don't seem to think a thing is identical to itself. Because when mathematicians use equality, they mean identity, and this is provable from first principles. They either mean identity as sets; which is easy to show; or, they often mean identical structurally. This is a more subtle philosophical point.
You said that "1) We do not perceive order with the senses" and that "2) We cannot apprehend the inherent order". Therefore, how do you know that what's shown in the diagram is the exact positioning of the parts (i.e. the inherent order)?
Isn't the positioning of the dots that I perceive the "perspective dependent order" which you earlier stated was not the inherent order? So how can the diagram show the inherent order to anybody?
The inherent order cannot be perceived by the senses and we can't apprehend it, anyway.
Quoting Metaphysician Undercover
If "We cannot apprehend the inherent order", then how do you know that our representations are inaccurate?
If the inherent order is unintelligible to the human mind by definition, then what makes inherent order preferable to (or distinguishable from) randomness?
The way you described sets in this thread, a set is something which cannot have an identity because it has no inherent order. Therefore I cannot agree that the set {0,1,2,3,4} is identical to the set {0,1,2,3,4}. It seems like a set is an abstraction, a universal, rather than a particular, and therefore does not have an identity as a "thing". It is particulars, individual things, which have identity according to the law of identity. Notice that the law of identity says something about things, a thing is the same as itself.
The law of identity is intended to make that category separation between particular things, and abstractions which are universals, so that we can avoid the category mistake of thinking that abstractions are things. "The set {0,1,2,3,4}" refers to something with no inherent order, so it does not have an identity and is therefore not a thing, by the law of identity, To say that it is a thing with an identity is to violate the law of identity.
Quoting fishfry
This is the whole point of the law of identity, to distinguish an abstract concept from a thing, so that we have a solid principle whereby we can avoid the category mistake of thinking of concepts as if they are things. A thing has an identity which means that it has a form proper to itself as a particular. To have a form is to have an order, because every part of the thing must be in the required order for the thing to have the form that it has. So to talk about something with no inherent order, is to talk about something without a form, and this is to talk about something without an identity, and this is therefore not a thing.
Quoting fishfry
The problem is not that I don't think a thing is the same as itself. That is the law of identity, which I adhere to. The problem is that you make the category mistake of believing that abstract conceptions are things. Because you will not admit that a concept is not a thing, you make great effort to show that two distinct concepts, like what "2+3" means, and what "5" means, which have equal quantitative value, refer to the same "thing". Obviously though, "2+3" refers to a completely different concept from "5".
If you would just recognize the very simple, easy to understand, fact, that "2+3" does not mean the same thing as "5" does, you would understand that the two expressions do not refer to the same concept. So even if concepts were things, we could not say that "2+3" refers to the same thing as "5", because they each have different associated concepts. And it's futile to argue as you do, that the law of identity is upheld in your practice of saying that they refer to the same "mathematical object", because all you are doing is assuming something else, something beyond the concepts of "2+3", and "5", as your "mathematical object". This supposed "object" is not a particular, nor a universal concept, but something conjured up for the sake of saying that there is a thing referred to. But there is no basis for this object. It is not the concept of "2+3" nor is it the concept of "5", it is just a fiction, a false premise you produce for the sake of begging the question in your claim that the law of identity is not violated.
Quoting Luke
It is a fundamental ontological assumption based in the law of identity, that a thing has an identity. In Kant we see it as the assumption that there is noumena, which is intelligible, just not intelligible to us. In Descartes we see skepticism as to whether there even is external objects.
So we assume that there is something, the sensible world, and we assume it to be intelligible, it has an inherent order. To answer your question of how do we "know" this, it is inductive. We sense things, and we conclude that there is reality there. Also, we have some capacity to understand and manipulate what is there, so we conclude that there is intelligibility there, intelligibility being dependent on ordering. We have some degree of reliability in our understanding of the ordering therefore there must be some ordering.
Quoting Luke
Yes, you perception provides for you, the basis for a perspective dependent order, which your mind produces. What the object is showing you, its inherent order, and the order which you are producing towards understanding the inherent order, are two distinct things. As I described, there is some degree of inconsistency, constituting a difference, between what is shown to you, and what you apprehend from that showing. The claim of difference is justified by our failures. The inherent order is shown. It is not perceived by the senses. If you try to understand the inherent order, your mind will produce an order which you think best represents that which inheres in the object.
Consider, that in seeing objects we do not see the molecules, atoms or other fundamental particles, we have to figure those things out as a representation of the order which inheres within. But we cannot completely apprehend that order because our minds are deficient. This doesn't mean that sentient beings will never be able to apprehend it, or that there isn't an omniscient being which already can apprehend it. And even if it is impossible that human beings or any sentient beings will ever be able to understand it in perfection, like an omniscient being is supposed to be able to, we can still improve our understanding, i.e. get a better understanding, and decrease our failures.
Quoting Luke
I think we judge the accuracy of our understanding mostly by the reliability of our predictions. But reliability is perspective dependent and subjective. So where some people see reliability, I see unreliability. It all depends on what type of predictions you are looking for the fulfillment of.
Quoting Luke
Order is fundamentally intelligible. So assuming order is to assume the possibility of being understood, which is to inspire the philosophical mind which has the desire to understand. To assume randomness is to assume unintelligibility which is repugnant to philosophical mind which has the desire to understand.
So, as I explained. If the object appears (seems) to be unintelligible (without inherent order), we need to determine why. Is it our approach (are we applying the wrong principles in our attempt to understand), or is it the reality, that the object truly has no inherent order? The latter is repugnant to the philosophical mind, and even if it were true, it cannot be confirmed until the possibility of the former is excluded. Therefore, when the object appears to be unintelligible (without inherent order), we must assume that our approach is faulty (we are applying the wrong principles in our attempt to understand), and we must subject all principles to extreme skepticism, before we can conclude that this object is truly unintelligible (without inherent order). The rational approach is to assume that we are applying the wrong principles, and to assume that the object has no inherent order is irrational.
This is irrelevant to my question. I did not ask you about the history of philosophy or why there must be inherent order; I asked you specifically about your statement regarding @fishfry's diagram:
Quoting Metaphysician Undercover
Given that (1) we do not perceive order via the senses and that (2) we cannot apprehend inherent order, then how can the inherent order be the exact spatial positioning shown in the diagram? Even if I grant you that there is an inherent order to the universe, how can you say that the inherent order of the diagram is the same as the order that we perceive via the senses, or "the exact spatial positioning shown"?
Quoting Metaphysician Undercover
I note your change in position. You are no longer arguing that the inherent order cannot be apprehended. You have now adopted the weaker claim that the inherent order cannot be completely apprehended.
Quoting Metaphysician Undercover
No, it cannot be confirmed at all. Assuming that the world is random (or not) makes no difference to our ability to find patterns and order in the world. More importantly, as far as I can tell, inherent order is not the kind of order that mathematicians are concerned with.
I think I've answered this about three times, so either I don't understand your question, or you don't understand my answer, or both.
Quoting Luke
I told you, we don't perceive order with the senses, we create orders with the mind. Judging by this statement, I'm thinking it's you who is the one not understanding.
Quoting Luke
Again, you're looking for hidden meaning to make unjustified inferences. The inherent order cannot be apprehended by us. I can't even imagine what it would mean to partially understand an order. If my use of "completely" misled you, I retract it as a mistake, and apologize.
Quoting Luke
Maybe, but fishfry claimed that a set has no inherent order. So if mathematicians are making such assumptions in their axioms, then they are concerned with it; concerned enough to exclude it from the conceptions of set theory. The issue I'm concerned with is the question of whether a thing without inherent order is a logically valid conception.
Exactly, so why did you identify/equate "the inherent order" with "the exact spatial positioning shown in the diagram"?
It sounds very much as though the inherent order is identical with what is apprehended as the "exact spatial positioning shown". Otherwise, why specify the "exact spatial positioning shown"? You have not merely said that the diagram has an inherent order which we are unable to apprehend despite what we see; you have identified the inherent order with the "exact spatial positioning" that we do see. Otherwise, to whom is the exact spatial positioning "shown", and from which perspective?
Do you think that location can be shown to someone without it being sensed?
Anyway, that's not what I said.
Of course, location is intelligible, conceptual, so it's not actually sensed it's something determined by the mind, just like meaning. When you tell someone something, they do not sense the meaning. This is "showing" in the sense of a logical demonstration. And the point is that upon seeing, or hearing what is shown, the mind may or may not produce the required conceptualization, which would qualify for what we could call apprehending the principle. More specifically, the conceptualization produced in the mind being shown the demonstration is distinct and uniquely different from the conceptualization in the mind which is showing the demonstration, therefore they are not the same. That is why people misunderstand each other..
I now see why we have such a hard time understanding each other, we seem to be very far apart on some fundamental ideas, which form the basis for our conceptualizing what is shown by the other. I thought you were intentionally misreading me, in order to say that I contradict myself. But now I see that this is the way you actually understand those words. My apologies for the accusation.
You avoid the question instead of answering it. How can location be shown to someone without it being sensed?
"Shown" in the sense of a logical demonstration is different to "shown" in the sense of your statement: "the exact spatial positioning shown in the diagram". That's obvious.
I answered the question, it's just like a logical demonstration. Like when one person shows another something, and the other makes a logical inference. All instances of being shown something intelligible, i.e. conceptual, bear this same principle. We perceive something with the senses and conclude something with the mind. The things that I sense with my senses are showing me something intelligible, so I produce what serves me as an adequate representation of that intelligible thing, with my mind.
Quoting Luke
Sorry, you've lost me. You appear to be making up a difference in the meaning of "shown", for the sake of saying that I contradict myself. I take back my apology, I'm going back to thinking that you do it intentionally.
Sorry, I haven't kept up. Are you speaking of inherent order or inherent ordering?
I'm glad that you finally acknowledge the role played by the senses in "showing". Only a couple of posts ago, you stated:
Quoting Metaphysician Undercover
But you now concede that sense perception is involved in showing.
Quoting Metaphysician Undercover
I'm not making anything up; the word has more than one meaning. Google's definitions of the verb "show" include:
1. allow or cause (something) to be visible.
2. allow (a quality or emotion) to be perceived; display.
3. demonstrate or prove.
In the context of your statement:
Quoting Metaphysician Undercover
I would say that the word "shown" here means what is visible in, or displayed by, the diagram; not what is demonstrated or proved by the diagram.
Regardless, I never claimed that there is no apprehension or awareness involved in showing. It was immediately before you began on this detour that I said:
Quoting Luke
Hopefully, this now clarifies my original intent. My omission of "[and apprehend]" seems to be what sent you off on a tangent. But I'm glad you now concede that sense perception is involved in "the exact spatial positioning shown in the diagram" (even if you refuse to concede that you intended "shown" in the seemingly more obvious sense that I outlined above).
All that remains for you to explain is your contradictory pair of claims that (i) the inherent order is the exact spatial positioning that we do apprehend in the diagram; and that (ii) we are unable to apprehend the inherent order.
If we are unable to apprehend the inherent order (in any case), then how can the inherent order possibly be (demonstrated or proved by) the exact spatial positioning shown in the diagram?
A = A is false must necessarily be a conversation stopper...awkward pause.
Well this is just nonsense, but it relates to the reason I didn't reply to the last post you wrote to me regarding the subject of order. It finally became clear that by order you mean "where everything is in time and space," so that for example a collection of spatial points or a collection of school kids does have an inherent order.
But this is a total equivocation of the way I defined mathematical order to you, as a binary relation on a set that is reflexive, antisymmetric, and transitive.
Now I gave you that definition several times. So you could (and should) have said something like,
"I don't understand what the words reflexive, antisymmetric, and transitive mean," or "I don't know what you mean by binary relation," or, "I see you're giving the mathematical definition, but I am using the more general definition of "where things are in time and space,"".
But you didn't say any of those things. Instead you just accepted the mathematical definition I repeatedly gave you, and then kept arguing from your own private definition. When I finally figured out what you were doing, I was literally shocked by your bad faith and disingenuousness. I'm willing to have you explain yourself, or put your deliberate confusion-inducing equivocation into context, but failing that I no longer believe you are arguing in good faith at all. You have no interest in communication, but rather prefer to waste people's time by deliberately inducing confusion.
I'm perfectly willing to have you explain yourself. But once I repeatedly gave the mathematical definition of order, and instead of saying, "Oh that's not how I define order," but rather kept arguing from your own private definition without acknowledging that you were doing so, I believe you were arguing in bad faith and I have lost interest in conversing with you further. Like I say I'm perfectly willing to hear your side of this, but I can't tell you how dismayed I was to finally realize what you meant by order, and to realize that you deliberately didn't bother to explain that you were ignoring the mathematical definition. You never said, "No that is not the definition I use." My frustration with this your conversational style is terminal at this point.
Quoting Metaphysician Undercover
Yeah right, whatever.
Quoting Metaphysician Undercover
Uh ... what the hell else could it be? I've told you a dozen times a set is a mathematical abstraction.
Quoting Metaphysician Undercover
Yeah ok. We could have a conversation around this, but I don't believe you're interested in communication, only obfuscation.
Quoting Metaphysician Undercover
You just denied a set is equal to itself.
Quoting Metaphysician Undercover
Word salad. Which I don't mind. But you've convinced me you're not conversating in good faith.
Quoting Metaphysician Undercover
When you put those in quotes of course that is correct, but that has never been the subject of the conversation. More obvious bad faith and sophistry.
Quoting Metaphysician Undercover
I think any reader following this thread can perfectly well see that the quotation marks have never been the subject of the conversation. I think you genuinely argue in bad faith and like to waste people's time. You're a troll.
Quoting Metaphysician Undercover
I'd be more inclined to respond if you hadn't been deliberately obfuscatory about your different use of the word order, after I'd given you the mathematical definition several times. Absent a clear explanation of why you did that, knowing how much confusion you were causing, I don't want to play.
I think I said "inherent order", but I don't quite understand the point to making the difference.
Quoting Luke
Of course sense perception is involved! Where have you been? We've been talking about seeing things and inferring an order. My point was that we do not sense the order which inheres within the thing, we produce an order in the mind. I never said anything ridiculous like we do not use the senses to see the thing, when we produce a representation of order for the thing.
Quoting Luke
But we went through this already. I explained that this is not what I meant by "shown"., and the reason why, being that order is inferred by the mind, it is not visible. I went through a large number of posts explaining this to you. It seemed to be a very difficult thing for you to grasp. And now, when you finally seemed to grasp the meaning associated with the way I used the term, you've gone right back to assuming that this is not the way I used it, despite all those explanations. Why? We just go around in circles, it's stupid. You pretend to have understood my explanation as to what I meant by "shown", then all of a sudden you say but obviously that's not what you meant. It's ridiculous. It' like you're saying 'I would not have used "shown" that way, therefore you did not'. And when i go through extreme lengths to explain that this is actually how I used the term, to the point where you seem to understand, you turn right back to the starting point, claiming but I would not have used it that way, therefore obviously you didn't. What's the point?
Quoting Luke
I stated repeatedly that we do not apprehend the exact spatial positioning, so you have a strawman here.. You don't seem to be capable of understanding any of what I am saying, we're just going around in circles of misunderstanding. it's pointless.
By "shown" you do not mean "displayed". After all, "it is not visible". Therefore, you must not be making the argument - as you did earlier - that we see the order but do not apprehend it.
By "shown" you mean "logically demonstrated". If something is logically demonstrated then it is apprehended, right?
Quoting Metaphysician Undercover
You are saying that the "exact spatial positioning" is logically demonstrated by ("shown" in) the diagram, but it is not apprehended? If the exact spatial positioning is not (or cannot be) apprehended, then how has it been logically demonstrated by the diagram?
Inherent order is a wider concept, applying in particular to biological systems and natural phenomena. Ordering is more specific having to do with listing. I think you are discussing the latter.
Let me remind you. You started in the discussion with the repeated assertion "sets have no inherent order". Check this post, I think you'll see that claim stated a number of times.
Quoting fishfry
When I said order is spatial and temporal, you claimed a completely "abstract order", which I didn't understand, and still don't understand because you haven't yet explained this in a coherent way. So I continued to insist order was spatial or temporal until you gave me examples of first and second place in competition, what I called order relative to best. I accepted this as non-spatial or temporal ordering, but I still don't see it as completely abstract because it still is based in concrete criteria for judgement.
You proceeded to define order in terms of "less than", as if you thought that this is purely abstract. However, I had already explained how "less that" is dependent on, defined in relation to, quantity. So you only contradicted your earlier claim that order is logically prior to quantity, by defining order in relation to quantity. And, since quantity is dependent on spatial separation between individuals you have not really escaped the spatial aspect of order, to get to a purely abstract order.
So this is where we stand. You have claimed a purely abstract order, but given me an order based in "less than" which is based in quantity. And quantity relies on spatial conception, so you have really given me a concept of order based in spatial conceptions..
Quoting Luke
Luke. I very consistently said, over and over again, that we do not see the order.
Quoting Luke
I went through that already, more than once. There is a logical demonstration of an order. The order which the mind apprehends, based on the demonstration is not the same order as that which inheres within the things shown. You keep neglected the principal point of the argument, that the order apprehended in the mind is not the same as the order in the object. Therefore "the exact spatial positioning" is not what is being demonstrated. So, do not ask again, this same strawman question. Check out these quotes:
Quoting Metaphysician Undercover
Quoting Metaphysician Undercover
Quoting jgill
Fishfry claimed that a set has no inherent order, and I questioned whether it is possible that there could be a thing with no inherent order.
i really do not think that "listing" is the subject here, because listing Is an ordering of symbols, not the things represented by the symbols. The list may represent an order, but the reason for the order is something other than the spatial order of the symbols. And fishfry insisted on the reality of a purely abstract order, which could not be a spatial relation of symbols, as listing is. We would need to find a principle of order which is purely abstract.
Here you say that "the exact spatial positioning" is not what is being demonstrated (i.e. shown) in the diagram. However:
Quoting Metaphysician Undercover
Here you said that the exact spatial positioning is what is being demonstrated (i.e. shown) in the diagram.
Which is it?
And as an indicator of how you continually change your position:
Quoting Metaphysician Undercover
No that's not a good interpretation. You need to respect the fact that what is being shown to the observer, as inhering within the physical thing being used in the demonstration, is not the same order as that which exists in the mind of the person performing the demonstration. I said there is a demonstration of "an order". I also said "the order which the mind apprehends, based on the demonstration is not the same order as that which inheres within the things shown." Then I said "the order apprehended in the mind is not the same as the order in the object. Therefore 'the exact spatial positioning' is not what is being demonstrated." The exact spatial positioning is what inheres within the object, and though it is what is being shown by the one doing the demonstration, it is not the same order as what the person is trying to demonstrate. This is why I said fishfry's claim that the order was random is false. That's what the person doing the demonstration was trying to demonstrate, but it was not what the demonstration actually showed.
What is "being demonstrated" is an order which exists in the mind of the person making the demonstration. This is the first line, the "demonstration of an order". What appears to the person making the interpretation, as what is "shown", is the physical object with an inherent order. This is a representation of the order which exists in the mind of the person making the demonstration. It is not the same order, but a representation of it. So the order being demonstrated is not the same as the order which inheres within the representation, (as a representation is different from the thing it represents), and the order in the mind of the person interpreting what is shown, is not the same as the order which inheres in the object. And, because of this medium, which exists between the one demonstrating and the one interpreting, the physical object as symbols, the order on the minds of the two individuals is not the same. That as I said is why we misunderstand each other.
Quoting Luke
As I said, numerous times, the mind creates an order to account for the order assumed to be in the thing Therefore the order in the mind it is not the order shown by the thing. No change of position, just a difficult ontological principle to describe to someone with a different worldview.
What interpretation? It's exactly what you said.
Quoting Metaphysician Undercover
I don't care about your latest position. In case you missed it, my entire point for the last three or four pages is that you changed your position three or four pages ago. This is obvious from the quote that you somehow managed to overlook:
Quoting Metaphysician Undercover
This is so obviously the opposite of your updated position: that we cannot see the inherent order in the diagram. This is all the evidence I need to make my point. Of course, you won't admit to it because that's why you're here: to bullshit and troll people.
As I explained, I haven't changed my position. You have not yet understood it.
Ok. This is a good starting point.
The question is, are you interested in understanding mathematical order in a coherent way? The idea, as with anything else mathematical, is that we have some aspect of the real world, in this case "order"; and we create a mathematical formalism that can be used to study it. And like many mathematical formalisms, it often seems funny or strange compared to our everyday understanding of the aspect of the world we're trying to formally model.
So if you're interested, I can explain that. Or frankly the Wiki article can do the same. If you're interested. If not, not.
After all bowling balls fall down, and the moon orbits the earth. To help us understand why, Newton said things like [math]F = ma[/math], and [math]F = \frac{m_1 m_2} {r^2}[/math]. And [math]E = \frac{1}{2} mv^2[/math], and things like that. And you could just as easily say, "Well this doesn't seem to be about bowling balls. These are highly artificial definitions that Newton just made up." And you'd essentially be right, while at the same time totally missing the point of how we use formalized mathematical models in order to clarify our understanding of various aspects of the real world.
So if you can see the difference between a real world thing like order, on the one hand; and how mathematicians formalize it, on the other; and if you are interested in the latter, if for no other reason than to be better able to throw rocks at it, I'm at your service.
And it's helpful to remember that the mathematical formalisms are not supposed to be reality. Nobody is saying they are. It's like chess. You don't complain about how the knight moves, because you understand that chess is a formal game that must be taken on its own terms.
That's the mindset for understanding how math works. You seem to object to math because it's a formalized model and not the thing itself, but that's how formal models and formal systems like chess work. They are not supposed to be reality and it's no knock agains them that they are not reality. They're formal systems. If you can see your way to taking math on its own terms, you'd be in a better position to understand it. And like I say, for no other reason than to have better arguments when you want to throw rocks at it.
I have understood it. According to your latest position, the inherent order is the true order of actual physical objects which humans are unable to apprehend; akin to Kantian noumena. The order that we are able to apprehend - which is not the inherent order - is the apparent order, which is invented by humans and assigned to those objects "from an external perspective". The inherent order is not something that can be truly spoken, perceived or apprehended.
Although we can neither perceive nor apprehend the inherent order, you claim that it is not hidden.
But this only evolved into your current position at about this post, after you were pressed (and unable) to specify the inherent order. Your change in position is the reason for these contradictory statements:
We both can see and can apprehend the inherent order:
Quoting Metaphysician Undercover
But we cannot see the inherent order:
Quoting Metaphysician Undercover
And we can see the inherent order, but we cannot understand or apprehend it:
Quoting Metaphysician Undercover
Quoting Metaphysician Undercover
An order that is shown can be seen:
Quoting Metaphysician Undercover
But an order that is shown cannot be seen:
Quoting Metaphysician Undercover
Quoting Metaphysician Undercover
The exact spatial positioning is logically demonstrated in the diagram:
Quoting Metaphysician Undercover
Quoting Metaphysician Undercover
But the exact spatial positioning is not logically demonstrated in the diagram:
Quoting Metaphysician Undercover
Quoting Luke
I already explained in what sense we see the inherent order, and do not see it, just like when we look at an object and we see the molecules of that object. The order is there, just like the molecules are there, and what our eyes are seeing, yet we do not distinguish nor apprehend the molecules nor the order, so we cannot say that we see it. We are always seeing things without actually seeing them, because it is a different sense of the word "see".
There was no change to my position, just a need to go deeper in explanation, to clarify the use of common terms, to assist you in understanding.
Quoting fishfry
I follow this, it seems to be exactly what I've been trying to explain to Luke, so we're on the same page here.
Quoting fishfry
These are what I would call universals, generalities produced from inductive reasoning, sometimes people call them laws, because they are meant to have a very wide application. As inductive conclusions they are derived from empirical observations of the physical world
Quoting fishfry
The issue is with what you call the purely abstract. It appears to me, that you believe there are some sort of "abstractions" which are completely unrelated to the physical world. They are not generalizations, not produced from inductive reasoning, therefore not laws, or "artificial definitions", in the sense described above. You seem to think axioms of "pure mathematics" are like this, completely unrelated to, and not derived from, the physical world.
Quoting fishfry
I object to the parts of these formalizations which do not correspond with our observations of the world. These would be faulty inductive conclusions, falsities. You claim that they do not need to correspond, that they a completely unrelated to the physical world. Yet when you go to describe what they are, you describe them as inductive conclusions, above, which are meant to correspond, in order that they might accurately "clarify our understanding of various aspects of the real world.".
So I see a disconnect here, an inconsistency. You describe "pure abstractions" as being related to the world in the sense of being tools, or formalizations intended to help us understand the world. Yet you insist that those who create these formalizations need not pay any attention to truth or falsity, how they correspond with the physical world, in the process of creating them. And you claim that when mathematicians dream up axioms, they do not pay any attention to how these axioms correspond with the world, because they are working within some sort of realm of pure abstraction.
As an example consider what we've discussed in this thread concerning " a set". It appears to me, that mathematicians have dreamed up some sort of imaginary object, a set, which has no inherent order. This supposed object is inconsistent with inductive conclusions which show all existing objects as having an inherent order. You seem to think, that's fine so long as this formalized mathematical system helps us to understand the world. I would agree that falsities, such as the use of counterfactuals, may help us to understand the world in some instances. But if we do not keep a clear demarcation between premises which are factual, and premises which are counterfactual, then the use of such falsities will produce a blurred or vague boundary between understanding and misunderstanding, where we have no principles to distinguish one from the other. If axioms, as the premises for logical formalizations are allowed to be false, then how do we maintain sound conclusions?
And this includes the inherent order? There's no contradiction here, I take it?
You have said that the inherent order can neither be perceived nor apprehended. So how can it be seen?
Language needs to reflect the scope of cosmological questions. It is reasonable that in the very long run, no matter how stable, all particles will decay. Then to make sense of those questions, wouldn't you want to redefine the universe as whatever energetic spacetime left after all ordinary particles have vanished into pure potential energy? Or are you only concerned with material matters and their relative forms?
Ok good. Progress is being made. One point, I am not reading the entire thread. From your side it must seem like you're being tag-teamed by @Luke and myself, but I'm not reading his posts. I'm not aware of that half of the conversation.
Quoting Metaphysician Undercover
Oh that's what you call universals. Physical laws? Ok. I'm not sure if that's standard but no matter. At least I have an idea now what you mean by that.
But "generalities produced from inductive reasoning?" I'm not sure if I agree with that. Surely F - ma is not a "generality" at all. On the contrary, Newton had to first define what he meant by force and mass. F = ma has sometimes been called a definition. It's an abstraction intended to formalize an aspect of nature. If you think it's a generalization of something, you might be missing the point. Hard to say.
Quoting Metaphysician Undercover
I think you are missing the point. If I drop a hundred bowling balls and I say, "Bowling balls fall down. That's a law of nature," then THAT is an inductive conclusion.
But if you see 100 bowling balls fall down and you go, F = ma, that is an abstraction and a mathematical formalization. You don't seem to have a firm grasp on this. Do you follow my point here?
Quoting Metaphysician Undercover
Well of course bowling balls are physical, and Newton was doing physics.
But there are non-physical parts of the world that we are interested in, such as quantity, order, shape, symmetry, and so forth. Those are the non-physical parts of the world that are formalized by math.
On the other hand, of course there are non-physical, non-part-of-the-world abstractions too. Chess, for instance. Chess is a formal game, it's its own little world, it has a self-consistent set of rules that correspond to nothing at all in the real world. Knights don't "really" move that way. Right? Say you agree. How can anyone possibly disagree?
Perhaps that's why math is special. It's a formal game, but it's a formal game that seeks to model certain aspects of the world that are themselves not quite physical. Order, quantity, shape, symmetry.
Quoting Metaphysician Undercover
But you are the one that insists that physical collections of things have an inherent order. And that's what the mathematical concept of order is intended to formalize. Things in the world have order, and we have a mathematical theory of order that seeks to formalize the idea.
Right? When mathematicians formalize numbers, they're abstracting and formalizing familiar counting and ordering. When they create abstract sets, they are formalizing the commonplace idea of collections. A bag of groceries becomes, in the formalization, a set of groceries. Surely you can see that. Why would you claim math is not based on everyday, common-sense notions of the world?
Quoting Metaphysician Undercover
How can you say that? Some of them obviously are. Most of them are. All of math is ultimately inspired by the world, just as the fictional story of Moby Dick was inspired by a real-world incident in which a ship was sunk by a whale. All fiction is inspired by the real world in one way or another, surely you know this.
Quoting Metaphysician Undercover
Like what? Can you name some of these? Sets correspond to collections. Bags of groceries, baseball teams, solar systems. Cardinal numbers correspond to quantity, ordinal numbers to order. Group theory is the study of symmetry. Crystallographers study group theory.
What mathematical ideas don't have any correspondence or at least ultimate inspiration from some aspect of the real world?
Quoting Metaphysician Undercover
Your notion of induction is wrong. "All bowling balls fall down," is an inductive conclusion. F = ma is a formalization.
But again I ask you, exactly WHICH mathematical ideas are not based on or inspired by the natural world? You must have something in mind, but I am not sure what.
Quoting Metaphysician Undercover
That's a useful mindset to have, so that we don't allow our everyday intuitions interfere with our understanding of the formalism. But of course historically, math is inspired by the real world. Even though the formalisms can indeed get way out there.
Quoting Metaphysician Undercover
Yes exactly. You say that like it's a bad thing! That's what formalization is. We have some aspect of the world, and we invent a mathematical formalization of it that captures its important features but that is distinct from the thing itself. So that we can use math and logic to draw mathematical conclusions, and use those conclusions to get insight about the original thing were were interested in.
Quoting Metaphysician Undercover
You aren't making your case. Surely you don't reject all of science because science builds mathematical models of certain aspects of reality, and that those models are not identical with the aspects of reality that they model.
Quoting Metaphysician Undercover
Yes. As opposed to chess, say, which is a pure abstraction not intended to help us understand the world, but rather intended as an entertainment and pastime in and of itself.
Quoting Metaphysician Undercover
Well as you know, math consists of logical implications. IF we assume this, THEN we may conclude that. We don't necessarily assert the truth of the antecedent. I think Bertrand Russell pointed this out. Because F = ma is not "literally" true of the world, it only formally represents certain aspects of the world. You have to be willing to make that conceptual split, between what is, on the one hand, and our abstract formalization, on the other. The formalization can never be true, because it's distinct from the thing it represents. The truth is in the thing. The formalization can't be true or false, it's not a thing in the world.
Quoting Metaphysician Undercover
They pay a lot of attention to the suitability of the axioms for a given purpose. But in the end, the axioms must be lies, because they are not, and CAN NOT BE, identical with the things they represent.
If I want to study the planets I put little circles on paper and draw arrows representing their motion. The truth is in the planets, not the circles and arrows. I hope you can see this and I don't know why you act like you can't.
Quoting Metaphysician Undercover
First, sets are intended to model our everyday notion of a collection. And in order to do a nice formalization, we like to separate ideas. So we have orderless sets, then we add in order, then we add in other stuff. If I want to put up a building, you can't complain that a brick doesn't include a staircase. First we use the bricks to build the house, then we put in the staircase. It's a process of layering.
Quoting Metaphysician Undercover
The objects themselves that are in the world may well have inherent order. Our formalization begins with pure sets. It's just how this particular formalization works. I don't know why it troubles you. If I represent a planet as a circle, you don't complain that my circle doesn't have rocks and and atmosphere and little green men. I'll add those in later. Right? Do you reject representing planets as little circles on paper, devoid of features, even though the planets they represent have features? How on earth can we get science off the ground without the process of abstraction, in which we begin with only certain aspects of things, leaving other aspects out.
You act like all this is new to you. Why?
Quoting Metaphysician Undercover
I refer you to Galileo's sketch of Jupiter's moons. With this picture he started a scientific and philosophical revolution. Yet anyone can see that these little circles are not planets! There are no rocks, no craters, no gaseous Jovian atmosphere. Why do you pretend to be mystified by this obvious point?
If Galileo showed you this diagram would you complain that it's a lie because it doesn't show the features of Jupiter? It is "true" insofar as it faithfully represents the small aspect of reality that it's trying to model. The fact that Jupiter has moons was a huge, massive, world-changing discovery. That's what was important here. You reject this line of thinking entirely? When you go to the planetarium do you complain that those aren't the real planets, that the models are made of plastic and are too small and are therefore lies?
You can't be this obtuse. Are you trolling?
Quoting Metaphysician Undercover
We are usually perfectly clear about these demarcations. When we look at a globe of the earth, we see the oceans but the oceans are not wet. We say to ourselves, "The real ocean is wet, but the ocean on the globe is made of hard plastic. It's a lie, but it's a lie in the service of the truth." Nobody in the world is confused by this but you!
Quoting Metaphysician Undercover
Do you feel the same way about maps? Ever use a map? The map is not the territory. Yet the map shows us true things about certain aspects of the territory, like the street names and where the freeway is.
Tell me this, @Meta. When you see a map, do you raise all these issues? "The rivers aren't wet. The streets aren't filled with cars. It's made of paper." Well ok I can't remember the last time I saw a paper map. But you get the idea. A map is a representation of some aspects of the world that we find of interest. Maps are lies, of course. In fact maps ARE lies, since maps are flat and the earth is a sphere. The projection's all off. You know this, right?
Do you rail at maps, at planetariums. at Galileo's crude drawings that changed the world? Darwin draws a finch, and you say, "It doesn't cheep. It doesn't lay eggs. It doesn't eat worms. It's only a pencil sketch. It's a lie, it's a lie I tell you!" Do you do this? Frankly I doubt it. You only act this way to play a character on this site.
Bottom line: Abstraction is a process of capturing the essence of some aspects of a thing of interest, by leaving out all other aspects. Abstractions are necessarily lies because they must leave important things out. Yet from them, we discern truth.
I assume there are many different senses to the word "see". The word is used sometimes to refer strictly to what is sensed, and other times to what is apprehended by the mind.
Quoting Luke
I really don't know how, it's just the reality of the situation. We sense things without apprehending what it is that is being sensed, as in my example of hearing a foreign language. There is a matter of distinguishing the individual elements, one from another, which the sense organ does not necessarily do, despite sensing the elements together as a composite.
Quoting fishfry
Don't worry about that, the conversations are completely different. Luke is on a completely different plane.
Quoting fishfry
I don't see the distinction you're trying to make here, between an inductive conclusion, and "an abstraction intended to formalize an aspect of nature". What do you mean by "formalize" other than to state an inductive conclusion.
I see the majority of definitions as inductive conclusions. Either they are like the dictionary, giving us a formalization (inductive conclusion) of how the word is commonly used, or they are intended to say something inductive (state a formalization) about some aspect of nature.
Quoting fishfry
I think it's you who is missing the point. I do not have a firm grasp on the distinction you are trying to make, because there are no principles, or evidence to back up your claim of a difference between these two.
F=ma says something about a much broader array of things than just bowling balls. So one could not produce that generalization just from watching bowling balls, you'd have to have some information telling you that other things behave in a similar way to bowling balls. Mass is a property assigned to all things, and the statement "f=ma" indicates that a force is required to move mass. How can you not see this as an inductive conclusion? It's not just a principle dreamed up with no empirical evidence. In all cases where an object starts to move, a force is required to cause that motion. It might have been the case that "force" was a word created, thought up, or taken from some other context and handed that position, as being what is required to produce motion (acceleration), but this does not change the inductive nature of the statement.
Quoting fishfry
As I said, I really do not understand how a "formalization" as used here, is anything other than an inductive conclusion. So I do not understand how you think my notion of induction is wrong. Perhaps you should look into what inductive reasoning is, and explain to me how you think a "formalization" is something different. I think induction is usually defined as the reasoning process whereby general principles are derived from our experiences of circumstances which are particular.
Quoting fishfry
That such things are non-physical is what I dispute. How could there be a quantity which is not physical? "Quantity" implies an amount of something, and if that something were not physical it would be nothing. "Order" implies something which is ordered, and if there was no physical things which are ordered, there would be no order. And so on, for your other terms. It makes no sense to say that properties which only exist as properties of physical things are themselves non-physical.
Quoting fishfry
When you say "formalize" here, do you mean to express in a formal manner, to state in formal terms? If it is physical things in the world which have order, and mathematics seeks to express this order in a formal way, then how is this not making a generalization about the order which exists in the phyiscal world, i.e. making an inductive conclusion?
Quoting fishfry
How can I agree with this? Chess is a game of physical pieces, and a physical board, with rules as to how one may move those physical pieces, and the results of the movements. The physical board and pieces are not "nothing at all in the real world", they are all part of the world.
What's with your motive here? Why do you insist on taking rules like those of mathematics, which clearly refer to parts of the real world, and remove them from that context, insisting that they do not refer to any part of the real world? Your analogy clearly does not work for you. The chess game is obviously a part of the world and so its rules refer to a part of the real world, just like quantity, order, shape, and symmetry are all parts of the real world, and so the rules (or formalities) of these also refer to parts of the real world.
Quoting fishfry
Yes, I agree with this here. Now the issue is how can you say that there is a collection of things which has no inherent order. If things in the world have order, and mathematicians seek to formalize that order, then where does the idea of "no inherent order" come from? That notion of "no inherent order" is obviously not derived from any instance of order, and if mathematicians are seeking to formalize the idea of order, the idea of "no order" has no place here. It is in no way a part of the order which things have, and therefore ought not enter into the formalized idea of "order".
Quoting fishfry
Have you lost track of our conversation? The idea of "no inherent order" is what we are talking about, and this is what I say does not correspond with our observations of the world. We observe order everywhere in the world. Sets do not correspond to collections, because any collection has an inherent order, existing as the group of particular things which it is, in that particular way, therefore having that order, yet as a "set" you claim to remove that order.
Quoting fishfry
I'll repeat. It's what we've been discussing, your idea of "a set", as a collection of things with no inherent order. Something having no inherent order is not based in, nor inspired by the real world, we don't see this anywhere in the world. We can also look at the idea of the infinite. It is not inspired by anything in the natural world. It is derived completely from the imagination.
Quoting fishfry
Let's try this. We'll say that a "formalism" relates to the real world in one way or another, and then we can avoid the issue of whether it is an inductive conclusion. We'll just say that it relates to the world. Now, can we make a category of ideas which do not relate to the real world? Then can we place things like "infinity", and "no order" into this category of ideas? But rules about quantifying things, and rules about chess games do relate to the real world, as formalisms.
Can you see that these ideas are not formalisms, nor formalizations in any way? Because they are purely imaginary, and not grounded in any real aspects of the natural world, there is no real principles whereby we can say that they are true or false, correct or incorrect. They cannot be classed as formalizations because they do not formalize anything, they are just whimsical imaginary principles. To use your game analogy, they are rules for a game which does not exist. People can just make up rules, and claim these are the rules to X game, but there is no such thing as X game, just a hodgepodge of rules which some people might choose to follow sometimes, and not follow other times, because they are not ever really playing game X, just choosing from a vast array of rules which people have put out there. Therefore there is nothing formal, so we cannot call these ideas formalisms or formalizations.
Quoting fishfry
I disagree with your notion of truth. I think truth is correspondence, therefore not in the thing itself, but attributable to the accuracy of the representation of the thing. Identity is in the thing, as per the law of identity, but "true" and "false" refer to what we say about the thing.
Quoting fishfry
I think this is a completely unreasonable representation of "truth", one which in no way represents how the term is commonly used. We say that a proposition is true or false, and that is a judgement we pass on the interpreted meaning of the proposition. We never say that truth is within the thing we are talking about, we say that it is a property of the talk. or a relation between the talk and the thing.
Quoting fishfry
Take a look at your example. The bricks are never "orderless". They come from the factory on skids, very well ordered. Your idea of "orderless sets" in no way models our everyday notion of a collection.
Quoting fishfry
The point is that orderlessness is in no way a formalization. A formalization is fundamentally, and essentially, a structure of order. Therefore you cannot start with a formalization of "no order". This is self-contradictory. As I proposed above, the idea of orderlessness, just like the idea of infinite, must be removed from the category of formalizations because it can in no way be something formal. To make it something formal is to introduce contradiction into your formalism.
Quoting fishfry
What I'm complaining about is your attempt to represent nothing, and say that it is something. You have an idea, "no inherent order", which represents nothing real, It's not a planet, a star, or any part of the universe, it's fundamentally not real. Then you say that this nothing exists as something, a set. So this nothing idea "no inherent order" as a set. Now you have represented nothing (no inherent order), as if it is the property of something, a set.
Quoting fishfry
The idea of contradictory formalisms is not at all new to me. I am very well acquainted with an abundance of them. That's why I work hard to point them out, and argue against them.
Quoting fishfry
I don't see how this is analogous. Galileo represented something real, existing in the world, the motions of Jupiter's moons. What I object to is representing something which is not real, i.e. having no existence in the world, things like "no inherent order". This is not a representation, it is a fundamental assumption which does not represent anything. If a formalism is a representation, then the fundamental assumption, "no inherent order" cannot be a part of the formalism.
Quoting fishfry
Consider this analogy. The idea of "no inherent order" describes nothing real, anywhere. So why is it part of the map? Obviously it's a misleading part of the map because there is nowhere out there where there is no inherent order, therefore I would not want it as part of my map.
Quoting fishfry
Yes, I get very frustrated when the map shows something which is not there. I look for that thing as a marker or indicator of where I am, and when i can't find it I start to feel lost. Then I realize that it was really the maker of the map who was lost.
I'm not familiar with any sense of the word "see" which means "not see".
Quoting Metaphysician Undercover
Can you not hear foreign languages? This is a terrible analogy. This is something which we can perceive but cannot apprehend. Your analogy with molecules is equally bad, since it is something we can apprehend but cannot perceive. It is (or very recently was) your position that we can neither perceive nor apprehend the inherent order. Remember? You said that order is "not visible".
I don't think the analogies are bad. That there is order inherent within the thing seen is something inferred, just like that there is meaning in the foreign language which is heard, is something inferred, and that there are molecules in the object seen is something inferred. We neither perceive nor apprehend the inherent order but we infer that it is there, just like we infer that there is meaning in the foreign language, and that there are molecules within the thing seen. But we neither perceive nor apprehend the meaning in the foreign language, nor do we perceive or apprehend the actual molecules in the object seen. We apprehend a representation of the molecules, just like we apprehend a representation of the inherent order. And, when we come to understand the language we apprehend a representation of the meaning intended (what is meant) by the author.
Earlier, you said that we sense or perceive a foreign language without apprehending it:
Quoting Metaphysician Undercover
Now you say that we neither sense nor perceive the meaning of a foreign language:
Quoting Metaphysician Undercover
Do you have an attention deficit or memory disorder? You seem incapable of maintaining your own position. A moment ago you were talking about "the reality of the situation" that we see the inherent order without apprehending it, and now you've switched back to saying that we can neither see nor apprehend the inherent order. If you think there are two different senses of the word "see" involved here, then define them both. Because one of them seems to refer to what we cannot see, which is not a familiar definition of the word "see".
I don't see the problem. Do you not grasp a difference between hearing people talking, and apprehending the meaning? Meaning as analogous with order, was the example.
Quoting Luke
You seem to be going through great effort to create problems where there are none. Oh well, it's what I've come to expect from you.
The problem is that you weren't talking about meaning before. You said that we sense a foreign language without apprehending it. Now you're talking about something else: that we can neither sense nor apprehend the meaning of a foreign language. What happened to your position from a day ago about being able to sense the inherent order?
Quoting Luke
According to your new position, we cannot see the meaning of a language. But you only introduced this analogy to support your claims regarding our ability to see the inherent order. I could now counter your new position, explaining how the analogy still does not work because we are able to apprehend a foreign language (by learning it), but we can never apprehend the inherent order. But why I should I bother? I'm tired of your constantly moving target. It's intellectually dishonest.
To apprehend the language being spoken, is to understand the meaning. You work very hard to make understanding difficult for yourself.
Quoting Metaphysician Undercover
Keep spinning your bullshit.
This is what we were discussing before you changed the subject to the meaning of a foreign language.
You didn't get the molecule analogy so I went back to the language one, I believe I used it earlier. Now you claim not to get the language one, but that appeared to me like an intentional misinterpretation. You pretended as if you didn't understand that apprehending language is understanding meaning.
Each is an example of sensing something without apprehending what is being shown by the thing being sensed.
You think it must be "hidden", if we sense something without understanding it, but I think that idea is what's misleading you. It's not at all hidden, the mind is just lacking in the capacity to understand what is being sensed. Thinking it is "hidden" is a feature of your accusative nature. When you can't understand a person you blame the other, instead of introspecting your own capacity. And if you cannot see what is right in front of you, you think it must be "hidden" from you, instead of considering the possibility that your eyes are actually sensing it, but your mind is just not apprehending it.,
1. I note your recent change from talking about "seeing" to talking about "sensing". Have you rejected your claim that we can see the inherent order?
2. How do you reconcile this with your statements that order is not visible?
Quoting Metaphysician Undercover
Quoting Metaphysician Undercover
No, I think we see it in exactly the way that I explained.
Quoting Luke
I explained that. We see the object. The object exists as an instance of ordered parts, inherent order. Therefore we must be seeing the inherent order even though strictly speaking the order is not visible to the person who is seeing it. The "not visible" property is due to a deficiency in the capacity of the person who is seeing the order.
I used the molecule example. Molecules are not visible to the naked eye. But we see the object, and the object is composed of molecules, therefore we must be seeing the molecules. That the molecules are not visible to the person seeing them is due to a deficiency in that person's capacities.
It's the same principle as when someone is pointing something out to you, and you're looking right at it, so you're definitely seeing it, because it's right there in your field of vision, yet you don't see the particular thing that the person is pointing out. Have you ever looked at stars, and had someone try to point out specific constellations to you? You can be looking right at the stars, and see them all, therefore you are seeing the mentioned constellation, yet you still might not be able see that specific constellation.
See the different senses of "see", and how "visible" might be determined based on the capacity of the observer, or the capacity of the thing to be observed? The inherent order is not visible to us, due to our deficient capacities, yet we do see it, because it exists as what we are seeing. Go figure.
Maybe I should start reading the rest of this thread so we can have a free-for-all instead of a tag team.
Quoting Metaphysician Undercover
Good question, let me see if I can sharpen my explanation.
If I see 100 bowling balls fall down, "bowling balls always fall down" is an inductive conclusion. But F = ma and the law of Newtonian gravity are mathematical models from which you can derive the fact that bowling balls fall down. It's a physical law, meaning that if you assume it, you can explain (within the limits of observational technology) the thing you observe.
But this is not an important point in the overall discussion.
Quoting Metaphysician Undercover
Ok. I don't think the definition of induction versus a formal model is super important here. But "bowling balls always fall down" is simply a generalization of an inductive observation, whereas the law of gravity lets you derive the fact that bowling balls fall down; and that in fact on the Moon, they'd weigh less. The latter is not evident from "bowling balls always fall down," but it is evident from the equation for gravitational force.
Quoting Metaphysician Undercover
It's not an important point, but for what it's worth, I think you are missing a major point as to the nature of science.
Quoting Metaphysician Undercover
It's not important to the larger conversation.
Quoting Metaphysician Undercover
And yet it lets us derive the falling of bowling balls. After all this you DO understand the difference.
Quoting Metaphysician Undercover
This point is not central to the main point, which is that models must necessarily omit key aspects of the thing being modeled.
Quoting Metaphysician Undercover
Not central, let's move on.
Quoting Metaphysician Undercover
Ok fine, then order is physical and the mathematical theory of order is an abstraction or model that necessarily misses many important real-world aspects of order yet still allows us to get some insight. That's the point of abstraction, which I already beat to death in my last post.
Quoting Metaphysician Undercover
One, to abstract, and two, to build a mathematical model of the abstraction. Or maybe those two are the aspects of the same process. I'm making a larger point, not splitting hairs.
Quoting Metaphysician Undercover
Ok they are. What is your problem with this?
Quoting Metaphysician Undercover
That's pure sophistry. There is no physical law that requires the piece to move the way they do.
Quoting Metaphysician Undercover
I'm explaining to you that whatever your concept of physical order is, mathematical order is an abstraction of it, which is necessarily a lie by virtue of being an abstraction or model, yet has value just as a map is not the territory yet lets us figure out how to get from here to there.
Quoting Metaphysician Undercover
I don't say that. Now that I understand what you mean by order, I'm happy to agree that every collection of physical things has an inherent order, namely "where every item is in time and space."
Quoting Metaphysician Undercover
Now that I understand what you mean by inherent order, I no longer need to argue this point. Physical collections have inherent order, if by order you mean "where everything is, or how everything is arranged, in time and space."
Quoting Metaphysician Undercover
I've conceded your point, now that I understand what you mean by inherent order.
Quoting Metaphysician Undercover
Once I concede that, what else have you got?
Quoting Metaphysician Undercover
Now that I understand what you mean by order, I see what you are talking about and I no longer oppose your point. Simple matter of understanding what you mean by inherent order, which you could have, but inexplicably chose not to, explain many posts ago
Quoting Metaphysician Undercover
It's an abstraction that necessarily includes SOME aspects of the thing being modeled and excluces OTHER aspects. Just as a street map includes the orientation of the roads but ignores the traffic lights.
Quoting Metaphysician Undercover
How do we make maps without drawing in the cars?
Quoting Metaphysician Undercover
That's right. A map is correct about some aspects of the world and incorrect about others. It's an abstraction. It's a representation of SOME ASPECTS of the thing being modeled but by necessity not ALL aspects otherwise the map would have to be an exact copy of your entire city or state. Globes would have to be as big as the earth. That they are smaller means that they are incorrect regarding size. That the oceans on a globe are not wet means they are incorrect about the wetness of the oceans.
Quoting Metaphysician Undercover
Maps are imaginary principles and don't formalize anything? Do you see why I think you're trolling?
Quoting Metaphysician Undercover
You're just playing games now, not seriously engaging with me.
Quoting Metaphysician Undercover
What would a true map be, in your opinion? A map of your city or town, say. Would it have to be the same size? Would the rivers and lakes have to be wet? Would the cars have to be on it?
Quoting Metaphysician Undercover
Map map map map map. Engage with the point, please.
Quoting Metaphysician Undercover
Map map map map map. Engage with the point, please.
Quoting Metaphysician Undercover
Why are the oceans dry on globes? Engage with the point, please.
Quoting Metaphysician Undercover
If you would engage with my examples of maps and globes, I would find that helpful.
Quoting Metaphysician Undercover
If you would engage with my examples of maps and globes, I would find that helpful.
Quoting Metaphysician Undercover
And sets represent aspects of collections, which exist in the world. And they omit "inherent order," which for sake of argument I'll agree collections in the world have.
Quoting Metaphysician Undercover
I don't think I have anything left to say. Perhaps we're done. This isn't fun and it isn't educational.
Quoting Metaphysician Undercover
Well that has nothing to do with anything. Maps don't show things that aren't there. The question is, how do you feel when a map omits things that ARE there, like wet lakes and rivers, cars, and the size and scale of the actual territory being modeled.
This is like saying that we can see infrared or ultraviolet light with the naked eye. We can't; not according to any common usage of the word "see".
Quoting Metaphysician Undercover
Then you don't see it.
Quoting Metaphysician Undercover
No, we simply don't see it. And you claimed earlier that we could not possibly see it, in principle:
Quoting Metaphysician Undercover
The law of gravity is the more general statement, saying all things with mass will fall down. The statement that bowling balls will fall down is more specific. Inductive reasoning is to produce a general statement from empirical observations of particular instances. So, the law of gravity as a general statement, is an inductive conclusion. And, bowling balls may or may not have been observed in producing that inductive law, but the law extends to cover things not observed, due to the nature of inductive reasoning, and the generality of what is produced. This is why inductive reasoning gives us predictive capacity. That mathematics is used to enhance the predictive capacity of inductive reasoning is not relevant to this point.
Quoting fishfry
It is important, because induction, by its nature, requires observation of particular instances. And you seem to be arguing that there is a type of abstraction, pure abstraction, which does not require any inductive principles. So it is important that you understand exactly what induction is, and how it brings principles derived from observations of particular instances, into abstract formulae. Do you see that the Pythagorean theorem for example, as something produced from practice, is derived from induction?
Quoting fishfry
It is the inductive conclusion, which allows for the derivation, the prediction, which you refer to. As it is a general statement, it can be applied to things not yet observed. It is not the mathematics which provides the capacity for prediction. mathematics enhances the capacity
Quoting fishfry
The central point is the difference between the inductive conclusion, which states something general, and the modeling of a "thing", which is a particular instance. At this point, we take the generalization, and apply it to the more specific. It must be determined how well the generalization is suited, or applicable to the situation. This requires a judgement of the thing, according to some criteria.
Quoting fishfry
I think your description of abstraction as missing things, is a bit off the mark. What abstraction must do is derive what is essential (what is true in all cases of the named type), dismissing what is accidental (what may or may not be true of the thing). Now, if order is essential to being a thing, then we cannot abstract the order out of the thing, to have a thing without order, because it would no longer be a thing.
Quoting fishfry
This is not true in a number of ways. First, good abstractions, inductive conclusions, or generalizations, do not lie because they stipulate what is essential to the named type. They speak the truth because every instance of that named type will have the determined property.
Second, your proposed "mathematical order" is not an abstraction, inductive conclusion, or generalization. You started with the principle that there is a unity of things with no inherent order. So you have separated yourself from all abstraction, induction, or generalization, to produce a purely imaginary, and fictitious starting point. You cannot claim that the imaginary, and purely fictitious starting point, of "no inherent order" is a generalization, or an inductive conclusion, or in any way an abstraction of the physical order. You are removing what is essential to "order", by claiming "no order", therefore you have no justification in claiming that this is an abstraction of physical order.
Do you recognize the difference between abstracted and imaginary? Imagination has no stipulation for laws of intelligibility, while abstraction does.
Quoting fishfry
OK, now lets proceed to look at your imaginary "mathematical order". Do you concede as well, that by removing the necessity of order from your "set", we can no longer look at the set as any type of real thing. Nor is it a generalization, an inductive conclusion, or an abstraction of physical order. It is purely a product of the imagination, "no order", and as such it has no relationship with any real physical order, no bearing, therefore no modeling purposefulness. It ought to be disposed, dismissed, so that we can start with a new premise, a proper inductive conclusion which describes the necessity of order.
Quoting fishfry
The idea of something with "no inherent order" is not an abstraction, as explained above. It is a product of fantasy, imaginary fiction.
Quoting fishfry
A map is not an abstraction, it is a representation. I see that we need to distinguish between abstraction, which involves the process of induction, producing generalizations, and as different, the art of applying these generalizations toward making representations, models, or maps. Do you see, and accept the difference between these two? We cannot conflate these because they are fundamentally different. The process of abstraction, induction, seeks what is similar in all sorts of different thing, for the sake of producing generalizations. The art of making models, or maps, involves naming the differences between particulars. These are very distinct activities, one looking at similarities, the other at differences, and for this reason abstraction cannot be described as map making.
Quoting fishfry
I was referring to the principles of "no inherent order", and "infinity", with the claim that these do not formalize anything. I wasn't talking about maps.
Quoting fishfry
The map analogy is not very useful, for the reason explained above, it doesn't properly account for the nature of inductive principles, abstraction. Generalizations may be employed in map making, but they are not necessarily created for the purpose of making maps. Now the map maker takes the generalizations for granted, and proceeds from there, but must choose one's principles. In making a map, what do you think is better, to start with a true inductive abstraction like "all things have order", or start with a fictitious imaginary principle like "there is something without order"? Wouldn't the latter be extremely counterproductive to the art of map making, because it assumes something which cannot be mapped?
Quoting fishfry
So, for the sake of argument, we can make the inductive conclusion, all collections which exist in the world have an inherent order. This is a valid abstraction, based in empirical observation, and it states that what is essential to, or what is a necessary property of, a collection, is that it has an inherent order. Do you agree then, that if we posit something without inherent order, this cannot be a collection? It doesn't have the essential property of a collection, i.e. order; therefore it is not a collection.
Quoting fishfry
Each map maker, based on the needs of that map maker's intentions, chooses what to include in the map. Abstraction, inductive reasoning, is very distinct from this, because we are forced by the necessities of the world to make generalizations which are consistent with everything. That's what makes them generalizations
Quoting Luke
Perhaps, but I disagree. It's a matter of opinion I suppose. You desire to put a restriction on the use of "see", such that we cannot be sensing things which we do not apprehend with the mind. I seem to apprehend a wider usage of "see" than you do, allowing that we sense things which are not apprehended. So in my mind, when one scans the horizon with the eyes, one "sees" all sorts of things which are not "forgotten" when the person looks away, because the person never acknowledged them in the first place, so they didn't even register in the memory to be forgotten, yet the person did see them.
Quoting Luke
No, I think you misunderstood. Perhaps it was the use of "perceive" which is like "apprehend". I said we could not apprehend it with the mind, the mind being deficient. This does not mean that we cannot sense, or "see" it at all. But your limiting of "see", to only that which is apprehended by the mind, instead of allowing (what in my opinion is the reality of the situation) that we are sensing things which are not being apprehended by the mind, not "perceived", is making you think that just because we cannot apprehend it with the mind, therefore we are not sensing it at all.
I know it's a difficult issue and it appears as incoherency, as ontological issues often do, because they are difficult to understand, but I think we need to establish a separation between what is sensed, and the apprehension of it, to account for the differences between how different people apprehend very similar sensations.
The point is that the process of abstraction necessarily, by its very nature, must omit many important aspects of the thing it's intended to model. You prefer not to engage with this point.
Quoting Metaphysician Undercover
That's fantastic. The Pythagorean theorem is a beautiful, gorgeous, striking, brilliant, dazzling, elegant, sumptuous, and opulent example of an abstraction that is not based on ANYTHING in the real world and that has NO INDUCTIVE CORRELATE WHATSOEVER. I am thrilled that you brought up such an example that so thoroughly refutes your own point.
The Pythagorean theorem posits and contemplates a purely abstract, hypothetical, mathematical right triangle such that the sum of the squares on the legs is equal to the square on the hypotenuse. [i]No such right triangle has ever, nor will ever, exist in the real world.
@Meta this is such a great example. I wish I had thought of it myself. In soccer they call this an own goal, where you kick the ball into your own net and score a point against your own team.
Oh this is good. Just perfect. You made my day.
To make this clear: The exactitude of the Pythagorean theorem is FALSE for every actual right triangle that's ever existed. It's only in pure abstract mathematical space that it's true. So we go from a fact that is NEVER true in the real world, to one that is ALWAYS true in the abstract mathematical world. This is the complete opposite of induction. It's deduction. It perfectly shows the power of pure abstraction to reveal things about the real world while being based on nothing at all of the real world.
I drop a thousand bowling balls, they all fall down. "Bowling balls fall down." That's induction. I observe a thousand, a million, a gazillion, right triangles, and I note that the sums of the squares on the legs is NEVER equal to the square on the hypotenuse, but only sort of close. From that I DEDUCE -- not "induce," I DEDUCE -- that for a perfect, abstract, Platonic right triangle, the theorem is exact.
Meta you are secretly on my side. I knew it all along! Like a double agent I dispatched into the world long ago and forgot was secretly working on my behalf. I welcome you back to the world of pure, abstract mathematics, in which things can be deductively proven true that are NEVER inductively true in the real world.
Quoting Metaphysician Undercover
Your own example falsifies this. I can never confirm the Pythagorean theorem in by observation of the world. I can only prove it deductively and never inductively.
Quoting Metaphysician Undercover
No middle 'e' in judgment. I can't take anyone seriously who can't spell.
Quoting Metaphysician Undercover
I'm sorry, I can't focus. You so thoroughly demolished your own argument with the Pythagorean example that I can't focus on what you're saying.
But getting back to the larger point: A map is a formalization of certain aspects of reality that necessarily falsifies many other things. Just as a set is a formalization of the idea of a collection, that necessarily leaves out many other things. I have a bag of groceries. I formalize it as a set. The set doesn't have order, the grocery bag does. The set doesn't have milk, eggs, bowling balls, and rutabagas, the grocery bag does.
Quoting Metaphysician Undercover
I think what you are saying is that an abstraction faithfully represents some aspects of the thing, and leaves out others.
So the essence of your argument is that "inherent order" is so tightly bound to the thing, that it can not be separated by an abstraction. I believe that's the core of our difference. Do I have that right?
Let me say that again, because these posts are getting too long and I believe I've found the essence.
[b]The essence of your argument is that "inherent order" is so tightly bound to the thing, that it can not be separated by an abstraction. I believe that's the core of our difference. Do I have that right?
[/b]
In other words I could not separate, "Fat bearded guy in a red suit who flies around at Christmas time and climbs down chimneys," from the concept of Santa Claus, because the two notions are so tightly bound that to omit one is to forever de-faithfulize the representation.
Am I now understanding your point?
Quoting Metaphysician Undercover
Not necessarily in the world, only in the formal model. Which is no problem.
Quoting Metaphysician Undercover
For purposes of the conversation, yes, I'll stipulate that. So what? It's how mathematical abstraction works. Just like the Pythagorean theorem does the same thing. There is no right triangle in the world that obeys Pythagoras. Only fake, idealized, imaginary, formal, completely-made-up mathematical right triangles do. Euclid would have been glad to explain this point to you. There are no points, lines, and planes. They're pure mathematical abstractions inspired by, but very unlike, certain things in the real world.
Quoting Metaphysician Undercover
Well now you're just arguing about my semantics. If I don't use the word generalization (which by the way I have not -- please note); nor have I used the phrase inductive conclusion. I have not used those phrases, you are putting words in my mouth. I said mathematical models are formalized abstractions, or formal abstractions. Or formal representations. That's what they are. The need not and generally don't conform to the particulars of the thing they are intended to represent. Just as an idealized right triangle satisfies the Pythagorean theorem, and no actual triangle ever has or ever will. Oh what a great example, I wish I had thought of it. Thank you!
Quoting Metaphysician Undercover
Well, abstraction is inspired by things in the real world, and imaginary isn't. But both are instances of formal systems. For example a mathematical right triangle is an abstraction, and chess is imaginary.
Quoting Metaphysician Undercover
As I just defined it, mathematical order is abstract and not imaginary, since it's inspired by the order found in nature.
Quoting Metaphysician Undercover
My gosh, @Meta, have I ever in all the times we've been conversating EVER referred to sets as real things? They're abstract mathematical objects, hence "real" if by real you mean objects of human thought; as opposed to things in nature like rocks. Of course sets are not "real things." In fact unlike most mathematical objects , sets don't even have a definition. Nobody knows what a set is. A set is anything that satisfies the axioms of some set theory; and there are many distinct axiomatic set theories.
I would never call a set real. But I have never TRIED to call a set real. Why on earth do you think you're challenging me with such a silly question? "No longer" look at a set as real? I never did.
Quoting Metaphysician Undercover
There are alternative foundations. I don't see how the choice of foundation is troubling you so much. If you don't like sets, try type theory. I'd say try category theory, but you can do set theory within category theory so that's no escape.
But of course that's not what you're saying. You are objecting to the mathematical concept of set. Well a lot of mathematicians have done the same. On far more sophisticated grounds, which is why it would help you to learn some math if you want to throw rocks at it.
But we conceive of sets as abstractions of collections; and for purposes of getting the formalization off the ground, we conceive of sets as having no order; and then we add the order back in via order theory. I truly don't see why you find this troubling, but I'll accept that you do.
Quoting Metaphysician Undercover
You say that like it's a bad thing! Ok, imaginary fiction then. But a useful one! Functional analysis and differential geometry are based on set theory, and quantum physics and general relativity are based on FA and DG, respectively. So you can't deny the utility of set theory, even as you rail against its unreality. On the contrary, the unreality is the whole point of abstraction. But you deny it's abstraction. Ok then, fiction. Ok fine, here's SEP on mathematical fictionalism. There's a philosophical school of thought that completely accepts your premise that math is fiction, nevertheless an interesting and a handy one. That's pretty much the philosophy I'm expressing in my posts to you. Though to be fair, some days I'm a Platonist. Both points of view are useful.
So: Yes math is a fiction. A complete lie. Stuff someone dreamed up one day. What of it? It's still useful. Remember the great essay with the perfect title: "The Unreasonable Effectiveness of Mathematics in the Physical Sciences.' Doesn't that just say it all? Math is so fictional, so clearly NOT based on reality, that it's UNREASONABLE that it's so effective. Yet is is.
So nobody's disagreeing with your point. You need to get beyond your point that math is a fiction, to try to come to terms with why it's so useful.
Quoting Metaphysician Undercover
You know, I see that I am no longer even trying to argue that math is based on reality or represents reality. I could, but then you'll just tangle me up in semantics and fine points. A stronger argument is for me to simply agree with you, completely and wholeheartedly, that math is fiction. And useful. So if you have a problem, it's your problem and not mine, and not math's.
Quoting Metaphysician Undercover
Well one is hard-pressed to do physics these days without mathematical infinity, even though the world as far as we know is finite. And I take your point about order, that you think order is so tightly bound to "collections of things" that the two concepts can't be separated by any abstraction. But set theory falsifies that claim, since set theory DOES separate collection from ordered collection.
Quoting Metaphysician Undercover
Well set theory isn't map making, of course. and so map makers should start by trying to capture the inherent order of the layout of the streets in a city. But set theorists don't have to do that. So the hell with the map analogy then.
Like I say you have now helped me to clarify my thinking. I have a much stronger position. Math is fiction, and it's useful, so what of it?
Quoting Metaphysician Undercover
Yes. I'll stipulate that. And all right triangles in the world violate the Pythagorean theorem. Yet the mathematical version of collection, a set, need not and does not have inherent order; and mathematical versions of right triangles necessarily satisfy the Pythagorean theorem.
Quoting Metaphysician Undercover
It can't be a real-world collection, accepting your definition that the molecules in the ocean are "inherently ordered" by virtue of where each and ever one is at any particular moment. Likewise real world right triangles violate Pythagoras. Oh what a great example!
Quoting Metaphysician Undercover
Well a set isn't actually a collection. A set is an undefined term whose meaning is derived from the way it behaves under a given axiom system. And there is more than one axiom system. So set is a very fuzzy term. Of course sets are inspired by collections, but sets are not collections. Only in high school math are sets collections. In actual math, sets are no longer collections, and it's not clear what sets are. Many mathematicians have expressed the idea that sets are not a coherent idea. It would be great to have a more sophisticated discussion of this point. I'm not even disagreeing with you about this. Sets are very murky.
Quoting Metaphysician Undercover
I never use the word generalizations. I say abstractions. But if you won't let me do that, then I'll retreat to, "Fiction, and so what?"
Quoting Metaphysician Undercover
Was this for me? Oh I see that was for @Luke. LOL.
Well. I hope we can shorten this going forward. I think there are some key points.
* You think that inherent order is so tightly bound with the idea of collection, that the two notions can not be separated by any abstraction. Like Santa Claus and the fat bearded guy in the red suit. That's an interesting point.
* You think math is utter fiction. To which I say, Ok, I'm a mathematical fictionalist myself, and what of it? And Wigner makes the same point. Math is so clearly untrue, that it is unreasonable that it should be so effective. This should be a starting point for your thinking, not an end point. Yes math is fictional. I not only don't argue that point, I have been trying for years to get you to see that. You are the one who wants to reify it.
And that whether or not math is "really" a fiction, which frankly is doubtful, it is nonetheless highly useful to adopt that stance when trying to understand it, so as to take math on its own terms. If you try to figure out whether it's "real" you can drive yourself nuts, because the abstractions get piled on pretty high. So it's better just to take it as fiction and learn the rules. as you do when learning chess.
* So therefore perhaps I should stop wasting my time trying to argue that math is inspired by reality, or that math is an abstraction of reality, and simply concede your point that it's utter fiction, and put the onus on you to deal with that. I could in fact argue the abstraction and inspired-by route, but you bog me down in semantics when I do that and it's tedious.
I think these are the key points here.
What kind of petty bullshit is this? Fuck you fishfry, I thought we were trying to be civil with one another. I see you've gone off the deep end already, and it's only Monday.
Quoting fishfry
I've engage with this point, explaining that I think it is wrong. If it's an important aspect, an essential feature, then if the abstraction processes "misses" it, the abstraction is wrong. If it is something which can be left out of the abstraction, it is in Aristotelian terms "accidental" or "an accident", and is not an important aspect. Abstraction separates the important from the unimportant, and if it omits important aspects it is faulty.
Quoting fishfry
That's amazingly wrong, to think that the Pythagorean theorem is not based in anything from the real world. It's based in the method used to produced parallel lines for marking out plots of land. Check into the history of "the right angle", and you will learn this. Clearly this is something in the real world.
Quoting fishfry
Huh? Construction workers prove the Pythagorean theorem in the real world, many times every day. Make a 3,4,5 triangle, tt never fails to produce the desired angle. How is this not proof? Try it yourself. Mark two points to produce a line. Use the Pythagorean theorem to make a right angle at each of the two points, and make two new points on those right angles, at equal distances from the original points. Measure the distance between the two new points, and you will see that it is the same as the distance between the two original points, and you have proven the Pythagorean theorem because you have used it to produce right angles, and have proven that the angles produced are in fact right angles by producing two more equivalent angles.
Quoting fishfry
What you seem to not grasp, is that people were producing right angles long before the Pythagorean theorem was formalized. The Pythagorean theorem came into existence as a formalized description of what those people were doing. Therefore it is a generalization of what people were doing when they succeeded in producing the right angle, so it is an inductive conclusion. Try and see if you can apprehend pi as an inductive conclusion? It is a generalization, what all circles have in common, just like the Pythagorean theorem is a generalization, what all instances of "the right angle" have in common. If you produce an angle which is not consistent with what the Pythagorean theorem says, you have not produced the right angle.
Quoting fishfry
As explained above, if an abstraction, or formalization, leaves out important aspects, then it is faulty. And if you insist on using the map analogy after I've explained why it is unacceptable, I will insist that if a map leaves out important things, then it is obviously a faulty map.
One reason why the map analogy is faulty, is because the map maker can decide, based on the purpose for which the map is being made, which aspects are important, and which are not. In the case of abstraction, formalizing, or generalizing, we have no choice but to adhere to the facts of reality, or else the formalizations will be incorrect.
Quoting fishfry
An abstraction is a generalization. It does not represent "the thing" in any way, nor does it represent aspects of the thing. It represents a multitude of things, by creating a category or type, by which we can classify things. Again, another reason why the map analogy is misleading. It appears to make you think that an abstraction represents a thing, like a map does. That is incorrect, the abstraction is a generalization, a universal, which represents a multitude of things.
Quoting fishfry
This is not really a good representation of my argument, because you don't seem to understand what abstraction is in anyway near to the way that I do. It's a good start anyway. But let me put it in another way. Let's suppose a category, or type called "thing". The abstraction, generalization, or formalization, would be a statement of definition, what it means to be a thing. This would be a statement as to what all things have in common, which makes it correct to call each of them a "thing". To be an acceptable definition, would be to be a good inductive conclusion. My argument is that the good inductive conclusion is that all things have inherent order therefore it would be a bad formalization, generalization, or abstraction, to posit a thing without inherent order because this is contrary to good inductive reasoning. Furthermore, I've argued that since such a principle, is not based in any inductive reasoning, it cannot truthfully be called an abstraction, generalization, or formalization, it is simply an imaginary fiction.
Quoting fishfry
As I've explained, it is false to call this an abstraction. To make up a purely imaginary, fictitious principle, is not abstraction. And, the Pythagorean theorem is not at all like this. Creating the Pythagorean theorem was a matter of taking what people had been doing on the ground, producing the right angle and parallel lines, and using inductive reasoning to determine what all these cases of producing the right angle had in common. Therefore it is not a purely imaginary and fictitious principle, it is a truthful inductive statement about what all instances of the producing the right angle have in common.
Quoting fishfry
Remember, you claimed a difference between a formalization, and an inductive conclusion. I did not accept such a difference, and asked you to validate this claim. You have not yet done so, but continue to speak as if your proposed distinction is a true distinction, while I have demonstrated that it is not. Therefore I suggest that you give up, as false, this claim to a difference.
Quoting fishfry
Yes, fictions are useful. The principal use of these is to mislead and deceive. A secondary use is entertainment, but this requires consent to the fact that what is presented is fiction.
Quoting fishfry
Of course, deception is a problem for the one being deceived, not the deceiver. Or maybe I'm just not entertained by your proposed entertainment. Again, still my problem, but perhaps you have made a poor presentation.
Quoting fishfry
It was you who called a set a collection, and referred to some sort of mystical process of collecting, which allows for your proposed "no inherent order".
Quoting fishfry
I dealt with this. Most math is not fiction, as evident in the example of the Pythagorean theorem. I differentiated the types of mathematical principles which are imaginary fictions though, things like "no inherent order", and "infinity".
LOL. Should I read the rest? Second time today someone took one of my little jokes too seriously, I'll practice up on my smileys. :smile: :yikes: :cool: :rofl:
Say, did you know that the Pythagorean theorem is false in the real world? What do you make of that?
ps -- Ok I see you did respond to the rest. By claiming the Pythagorean theorem is literally true. I'll respond later to the entire post. But you know, there are no right angles in the real world. That you think there are is a problem. Must I really walk you through the basics of the philosophy of physical measurement? Sigh. And remember: Only one 'e' in judgment. I trust you won't make that error again. Smiley smiley smiley smiley smiley. Jeez.
You did not address my argument. Do you think that we can see infrared and ultraviolet light just because it exists in the world? This is your argument regarding molecules. Infrared and ultraviolet light are defined as frequencies of the non-visible spectrum. We cannot see them with the naked eye, by definition. This is not a matter of opinion over the definition of the word "see", unless you think that the dictionary, or the way that most people use the word "see", is an "opinion".
If you insist that we can "see" ultraviolet light and molecules then I propose that you use a synonym which has the meaning "to see what is not visible", in order to avoid any confusion. However, I don't know of any words with that meaning. Maybe you could use the notation "see(not see)" or "see(MU)" instead.
Quoting Metaphysician Undercover
Not "forgotten"? LOL. What are you talking about?
Quoting Metaphysician Undercover
I have not misunderstood. Your recent talk about "seeing" molecules and "scanning the horizon" is not about "apprehending" order; you are talking about perceiving order with the senses. This is also what your contradictory quote is referring to:
Quoting Metaphysician Undercover
Did you even read the quote? Your latest argument regarding molecules is that we can somehow "see" invisible things because we "sense" them. However, you already contradicted this earlier, as demonstrated by the above quote: "1) We do not perceive order with the senses". - You are now arguing that we do perceive (i.e. see) order with the senses, correct?
Quoting Metaphysician Undercover
Quoting Metaphysician Undercover
But then again:
Quoting Metaphysician Undercover
Quoting Metaphysician Undercover
Quoting Metaphysician Undercover
No, you've got that wrong. The Pythagorean theorem is true in the real world, because it works well and has been proven. Where it is false is in your imaginary world. It works very well for me. I use it regularly. That you think my right angle is a wrong angle is a bit of a problem though. We know induction is not perfect, it just describes what is experienced or practised. (Am I spelling practise wrong?) That the Pythagorean theorem is false in your imaginary world which you call "abstraction", is just more evidence that what you call "abstraction" is not abstraction at all, but fiction.
Quoting Luke
Yes, I think the eyes most likely do sense infrared and ultraviolet in some way: https://www.sciencedaily.com/releases/2014/12/141201161116.htm
Quoting Luke
Yes, you are very clearly misunderstanding, and I'm tired of trying to explain. You don't seem to have a mind which is inclined toward trying to understand complicated ontological problems, instead thinking that everything can be described simply by is or is not, because otherwise would be contradiction.
The article does not mention ultraviolet light. I'm sure you understand my point. The same point applies to molecules.
Quoting Metaphysician Undercover
It's not complicated; you contradicted yourself. I think you see that now, which is why you have given up.
So, your failure to recognize the distinct ways that I used "see", which I explained over and over again, constitutes contradiction on my part. OK, I must have contradicted myself then, according to the way that you use "contradicted", therefore I give up.
This is an important point for you to recognize. It's not in the real world, (where truth and falsity is determined by correspondence), where the Pythagorean theorem is false, it's tried and tested in the real world, and very true. It's only in you imaginary world, of so-called pure abstraction, where the only test for truth is logical consistency, or coherency, that it appears to be false. All this indicates is that your imaginary world is not to be trusted, as it does not give us coherency between even the most simple mathematical principles. On the other hand the Pythagorean theorem alone, can be trusted, because it does give us the right angle. So the quest for logical consistency, or coherency, is not a quest for truth..
What failure to recognise? I suggested that you use another word or notation to mark the distinction. One meaning is to see what is visible, the other is to "see" what is not visible.
You have stated both that we do and do not perceive (i.e. see) order with the senses. This is not a failing on my part.
It suddenly occurred to me today, why you are having so much trouble understanding. It's not so much the ambiguous use of "see" which is throwing you off, but I now see that you are not respecting the distinction between the two completely different referents for "order", which I thoroughly explained to you.
In the case of "inherent order" the order is within the thing sensed. It is sensed (in the manner I described), but not apprehended by the mind due to the deficient capacity of the sensing being. I've also used "order" to refer to orders created by the mind, within the mind, sometimes intended to represent the inherent order, as a model does. This order is apprehended by the mind, being created within the mind, but it is in no way sensed, because it is created within the mind and is therefore not part of the thing sensed.
You can see that in one context the referent of the word "order" is sensed but not apprehended by the mind, while in the other context the referent order is apprehended by the mind, but not sensed. Without adhering to the particulars of the context, and maintaining the differentiation between the two very distinct things referred to with the word "order", it would appear like "order" is used in a contradictory way; both sensed and not sensed, apprehended by the mind and not apprehended by the mind This is what you have been doing, taking my statements concerning "order" out of their context, failing to respect the described difference between the two distinct types of order, and claiming that I have contradicted myself
I will get to your earlier longer post when I get a chance, a little busy this week.
That you don't understand that all physical measurement is approximate, and that math deals in idealized exactness that does not correlate or hold true in the real world, is an issue I would have no patience to argue with you. You are simply wrong. Physical measurements are limited by the imprecision of our instruments. This is not up for debate. But I do see a relation between your misunderstanding of this point, and your general failure to comprehend mathematical equality.
Complete misunderstanding of the nature of physical science and the inexactness of all physical measurement. You are living in your own world of delusion.
To be clear, your argument is now that:
1. We do not perceive (i.e. see) order with the senses; but that
2. We do perceive (i.e. see) inherent order with the senses.
Inherent order is only one type of order (you also allow for other types such as best-to-worst). How is it that we do not perceive order with the senses in general, but that we do perceive inherent order with the senses specifically?
Quoting Metaphysician Undercover
Let's take a look at the context, then. It was not until recently that you began arguing that we do perceive inherent order with the senses and can "see" or otherwise "sense" invisible physical entities such as molecules, ultraviolet light, and the inherent order. This contradicts what you said earlier, that: "1) We do not perceive order with the senses." This was said in response to a fragment of mine that you quoted:
Quoting Luke
You will note I maintain the distinction here between order and inherent order. You must have been aware of this distinction in your own response when you contradicted your latest argument and affirmed that: "1) We do not perceive order with the senses". It is therefore a complete fabrication to attribute your own contradiction to my misunderstanding or lack of awareness of the distinction between order and inherent order.
Furthermore, as quoted at the top of this page from your earlier remarks:
Quoting Metaphysician Undercover
In other words, you explicitly state here that we do not sense the inherent order specifically.
Anyway, I look forward to you once again trying to attribute this to my misunderstanding instead of your own blatant contradiction.
The fact that you believe that mathematics deals with "idealized exactness", is the real problem. Look at the role of things like irrational numbers and infinities in conventional mathematics, these are very clear evidence that the dream of "idealized exactness" for mathematics is just that, a dream, and not reality at all, it's an illusion only. Idealized exactness never has been there, and probably never will be there.
You seem to deny this brute fact concerning mathematics, to insist on a separation between real world measurement (deficient in exactness) and ideal mathematics (consisting of perfect exactitude). You hide behind this denial, to completely ignore the reality that the principles of mathematics have been created from the acts of, and for the purpose of, real world measurements.
These are two facts you need to recognize, 1) Mathematical principles have been derived from acts of measurement, and 2) Mathematical principles are created for the purpose of measurement. Since this separation which you espouse cannot be accomplished, due to the fact that the principles of mathematics have been derived from the practise of measurement (1), as I explained with the example of the Pythagorean theorem, you ought to dismiss that intent to separate, altogether. And, because measurement is the purpose of mathematics in its practise (2), it is itself an instrument of measurement. So your observations that physical measurements are limited by "the imprecision of our instruments" ought to inspire you to a recognition of the imprecision of our mathematics.
Quoting Luke
I can't answer the how, but I have answered the why. The two types of order are completely distinct and different.
Quoting Luke
Go way back, to when I said "see" the inherent order in the dots on the plain in the diagram.
Quoting Luke
Right, that's a good quote, showing context. I think I generally indicated inherent order with the word "inherent", or "inheres within", indicating order within the object itself, (noumenal if that helps). If I just said "order", I likely was referring to the type of order created within the mind.
You need to recognize the complete separation between what is referred to with "inherent order" and what is referred to with "order". Inherent order, as inhering within the object, is not a type of order, as created by the mind, like the description indicates, this is impossible. The complete separation is required by their contradictory natures. However, there may be similarities by which we could place both, order and inherent order, into one category, but we haven't approached that yet.
In the quoted passage you seem to be looking at what is referred to by "inherent order", as a type of what is referred to as "order". This would constitute a misunderstanding, they are completely distinct and one is not a type of the other.
Quoting Luke
Sorry, that was a mistaken statement, instead of "sense" I should have used a better expression, like "perceive" or "apprehend". I was flustered by your ridiculous claim that I had earlier implied that sense was not involved at all. This is the complete context:
Quoting Metaphysician Undercover
I should have said "my point was that we do not receive, from the senses into the mind (apprehend), the order which inheres within the thing, we produce an order within the mind". This would allow clearly that the inherent order is present to the senses (is that a better way to say it?), as I had been describing. The intent was to establish the complete separation between the order constructed, and the order inherent in the object, described above. To clarify, the inherent order is present to the senses, but not present to the mind, when the mind produces a representation. "Present to the senses" I have been arguing qualifies as being sensed, but in the quoted passage I mistakenly said that this is not a case of being sensed
Again, I apologize, that was a sloppy post. I was a little rushed. and extremely put off by your claim that I was saying sense was not at all involved in the act of showing, so my reply was a reflex, consisting of a poorly chosen word, rather than clearly thought out. If you understand what I have presented in this post, you'll see that the senses are the medium which separate the order produced in the mind, from the inherent order which exists within the object. And this is why the order produced by the mind is completely distinct from the order which inheres within the object, though it is very true that the senses, and sensation have a relation to both of these distinct things.
@Meta, I'm going to withdraw from this phase of our ongoing conversation. Perhaps we'll pick it up at some time in the future. If you can't agree that real world measurement is necessarily imprecise and that mathematical abstraction deals in idealized exactness, we are not using words the same way and there is no conversation to be had. I don't think you would be able to cite another thinker anywhere ever who would claim that physical measurement is exact. That's just factually wrong.
I did not claim that physical measurement is exact. We agree that real world measurement is necessarily imprecise. Where I disagreed is with your claim that mathematics has obtained ideal exactness. That is what is factually wrong. Some mathematicians might strive for such perfection, and I would not deny that, but they have not obtained it, for the reasons I described.
Principally, mathematics has a relationship of dependency on physical world measurements which I described. This has ensured that the imprecision of physical world measurements has been accepted into the principles of mathematics. The lofty goal of idealized exactness has always been, and will continue to be, compromised by the need for principles to practise physical measurement, where idealized exactness is not a requirement. Therefore mathematics will never obtain idealized exactness. Look at the role of infinity in modern mathematics for a clear example of straying from that goal of idealized exactness.
Someone using your keyboard, perhaps your cat, wrote
Quoting Metaphysician Undercover
The Pythagorean theorem in the real world is literally false. It's close but no cigar. It's approximately true, that's the best you can say. But the point here is that you are on record claiming the Pythagorean theorem is "very true." So you are not in a position to deny saying that.
Quoting Metaphysician Undercover
Idealized mathematics (as opposed to say, numerical methods or engineering math, etc.) is perfectly exact. That's its supreme virtue.
Now it seems to me that the starting point for an interesting discussion is to note that the Pythagorean theorem is literally false in the world, and perfectly exactly true in idealized math; and from there, to meditate on the nature of mathematical abstraction. How we can literally tell a lie about the world, that the theorem is true, and yet that lie is so valuable and comes to represent or model an idealized form or representation of the world.
But if you deny both these premises, one, that the P theorem is false in the world (close though it may be) and perfectly true in idealized math, then there is no conversation to be had. And for what it's worth, your opinions on these two statements are dramatically at odds with the overwhelming majority of informed opinion.
Quoting Metaphysician Undercover
Of course. "Inspired by." Just as the great work of fiction Moby Dick was inspired by the true story of the Essex, a whaling ship sunk by a whale.
Nonetheless, Moby Dick is a work of fiction. A valuable one, I might add. Fiction is often valuable. Moby Dick teaches us not to follow our obsessions to our doom. That contradicts a point you made earlier that I didn't get a chance to comment on. I believe you said that fiction is always bad, that lies about the world are always bad. Math consists of lies about the world. Nothing in pure math is literally true about the world. Yet fiction, and fictional representations, tell a greater truth by their lies.
THAT is an interesting topic of conversation. Not claiming that the Pythagorean theorem is true in the world and false in idealized math, both claims contrary to fact.
Quoting Metaphysician Undercover
Of course, statisticians have a highly developed theory of measurement error. What of it? Idealized math is inspired by the world.
Quoting Metaphysician Undercover
The fact that Moby Dick changed the name of the ship from the Essex to the Pequod, changed the names of the characters, and invented episodes and stories that never really happened, does not detract from the novel in the least. A representation or abstraction stands alone. We do not denigrate the Pythagorean theorem for the "crime" of being exact, when the real-world approximations that inspired it are not. But you so deeply disagree with this point of view that there's little point in continuing. We're just repeating our mutually incompatible premises at this point.
Quoting Metaphysician Undercover
It obtains it every day of the week. It obtained idealized exactness in the time of Euclid. Euclid perfectly well understood that his lines and planes and angles were idealized versions of things that did not actually exist in the real world. How you fail to agree to this point of view I don't know. What's important about Euclid is first, the idea of deriving mathematical truths from premises, or axioms; and two, the process of abstraction, meaning that those premises are, strictly speaking, absolute falsehoods about the world. There are no dimensionless points, lines made up of points, and planes made up of lines in the world. Euclid showed how to start with abstract falsehoods (inspired by the world but not literally true about the world); derive logical consequences from them; and thereby obtain insight into the world. Perhaps you should consider that. I can't argue with someone who denies the power of mathematical abstraction.
Quoting Metaphysician Undercover
I have spent a fair amount of time in my life doing exactly that.
Quoting Metaphysician Undercover
The mathematical theory of infinity is a classic example of an abstraction that has nothing at all to do with the real world. And yet, without the mathematical theory of infinity we can't get calculus off the ground, and then there's no physics, no biology, no probability theory, no economics. So THAT is the start of an interesting philosophical conversation. How does such a massive fiction as transfinite set theory turn out to be so darn useful in the physical sciences? Where's Eugene Wigner now that we need him?
But you don't want to have that conversation because you want to utterly reject transfinite set theory simply because it's not literally true about the world. But that's such a boring and trite point of view. Of course it's literally false about the world. The more interesting conversation is to ask how it can nonetheless be so supremely useful in the world. It's the same question as how Euclid's idealized points, lines, and planes can be so useful.
How can lies, in the form of idealized abstractions, lead to truth? That's a good question. Stopping your thought process because the abstractions aren't literally true is not very interesting.
While I've got you here, I wanted to mention that in another thread someone pointed me to Quine's great essay On What There Is (pdf link]. There is a passage that jumped out at me:
[quote=Quine]
If I have been seeming to minimize the degree to which in our philosophical and unphilosophical discourse we involve ourselves in ontological commitments, let me then emphasize that classical mathematics, as the example of primes between 1000 and 1010 clearly illustrates, is up to its neck in commitments to an ontology of abstract entities. Thus it is that the great mediaeval controversy over universals has flared up anew in the modern philosophy of mathematics.
[/quote]
[My emphasis]
Is this a reference to what you've been trying to talk to me about from time to time? Universals, and how they bear on mathematical abstraction? What does it mean, exactly? After all I frequently point out to you that mathematical ontology posits the existence of certain abstract entities, and this is exactly what you deny. If I understood this point about universals better (or at all, actually) I'd better understand where you're coming from.
This was before you let anyone know that the inherent order was noumenal and invisible, which is right around the time I believe you changed your position. You started from this position:
Quoting Metaphysician Undercover
That is, you started out telling us that the actual/inherent order can be perceived with the senses and apprehended, then you changed your position to say that the inherent order cannot be perceived with the senses or apprehended, and now you're saying that the inherent order is invisible but it can (again) be perceived with the senses. At least, that's your latest position.
Quoting Metaphysician Undercover
Inherent order is not a type of order? Then what have you been talking about this whole time?
Quoting Metaphysician Undercover
If inherent order is not a type of order, then I don't understand what you have been arguing about regarding mathematical order. Why did you previously allow for other types of order, such as best-to-worst?
Quoting Metaphysician Undercover
Oh come on. You previously spoke of "perceive" and "apprehend" as opposing concepts, but now you consider them synonymous? For a long stretch of the discussion, you repeated in various forms that we perceive with the senses, as distinct from apprehending with the mind:
Quoting Metaphysician Undercover
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Clearly, this is hardly a once-off error said only because you were flustered, so I do not find this reason for contradicting yourself to be credible.
Apart from your attempt to re-write history regarding your use of "perceive", these quotes are further evidence that you are contradicting your earlier comments by now saying that we "see" or somehow "sense" the inherent order and other invisibles. -- And we can "see" or "sense" the inherent order but not perceive it?
"Idealized exactness" is not "truth". The Pythagorean theorem is very true in the real world. Where we disagree is on what constitutes being true. That has been obvious all along, you allow that fictions like "no inherent order", may be a part of your idealized exactness, thereby compromising your supposed truth with falsity.
Quoting fishfry
No fishfry, "infinity" is in no way perfectly exact. You did not address this, and there are a multitude of other examples of the imprecision of mathematics, such as the mathematician's ability to choose between incompatible axioms, and the various different roles which "zero" plays, as evidenced by imaginary numbers.
Quoting fishfry
You keep insisting that the Pythagorean theorem is false in the world, but it is used many times every day, and every time it is used it proves itself. Where's your evidence that it's false? And if you produce competing mathematical principles as your evidence, you are only proving the inexactness of mathematics, not the falsity of that particular theorem..
Quoting fishfry
Of course I deny those premises. They are both false. But we have different ideas of what constitute truth. I think truth is correspondence with reality, you think truth is some form of idealized exactness. But "idealized exactness" does not even describe mathematics in a true way. How can justify your claim that the Pythagorean theorem has idealized exactness when the square root of two is irrational? What defines an irrational ratio is that it is lacking in perfect exactitude. So both you premises are false. The Pythagorean theorem is not false in the real world, it gives us the right angle every single time, and there is no perfection to its truth in idealized math, because it gives us irrational ratios. See both your premises need to be rejected.
Quoting fishfry
You still do not seem to have any respect for the difference between a creative work of the imagination, and a representation. A "representation" cannot stand alone, because it necessarily represents something, that's why it's called a representation. If it stood alone it could not be called a representation. An imaginary fiction, like Moby Dick, stands alone as a creative piece of art, not meant to represent anything. It is not a representation.
This problem appears to deeply affect the way that you think about truth. You seem to think that a representation can be true without any rules of correspondence, simply by standing alone. Of course this is not true because it is the rules of correspondence which provide for the truth or falsity of any representation. Consider your map analogy, the key, or legend, tells you what the symbols stand for, allowing for the truth or falsity of the map.
Your map analogy fails because of your desire to extend it into the artistic world of fictitious creations which are not meant to represent anything, and therefore not similar to maps. Here, we have works of art, created by imaginative power, which are enjoyed for aesthetic beauty, This is where you place your "idealized exactness" striven for by mathematicians in their acts of imaginative creation, as a high form of beauty. If mathematicians could obtain to that highest level, ideal exactness, they might create the highest form of beauty, "truth". However, idealized exactness is not a part of the real world, just like "no inherent order" is not a part of the real world, nor is infinity part of the world, while mathematicians and mathematics are parts of the real world. So these beautiful works of art produced by the mathematicians, which have great aesthetic beauty, but do not represent anything, are simply beautiful works of art, which, as any other part of the world, contain imperfections.
Now, you present these works of art to me as "representations", and claim that there is truth within them, as "idealized exactness". However, they very clearly do not obtain to that level of "ideal exactness" so if ideal exactness is supposed to be truth for you, then these works of the mathematicians are obviously not true.
Quoting fishfry
I think Euclid's parallel postulate is somewhat questionable in some modern geometry. You've just given me more proof that idealized exactness has not been obtained. If it had been obtained, there would be no need for new forms of geometry which cast doubt on the old. Geometry works in the field, in real world situations it gives us truth, but it clearly does not give us the ideal (absolutely perfect) exactness, which you seem to believe it does.
Quoting fishfry
Hmm, an infinity of dimensionless points could not produce a line with dimension, more evidence that ideal exactness has not been obtained.
Quoting fishfry
I had no doubt that you'd have good things to say about infinity in mathematics, but you didn't address the point. The use of infinity in mathematics is clear evidence that mathematics does not not consist of idealized exactness. And now that you mention it, calculus itself is based in principles of allowing less than perfect exactness, with notions like infinitesimals.
Quoting fishfry
Yes, well maybe we'll continue this discussion for a few more years.
Quoting Luke
I didn't change my position. We've been through this already. You misunderstood my use of words. I went back and explained how the position was consistent, but the choice of words was difficult.
Quoting Luke
This is false, I never said inherent order is apprehended. I've remained consistent and I've clarified this already.
Quoting Luke
If you recall fishfry introduced "inherent order" by claiming that a set has no inherent order. I haven't been using "mathematical order" so I don't even know what you're talking about here. Near the beginning of the thread there was no consensus between the participants in the thread as to what "order" referred to. I developed the distinction between inherent order, and the order created by the mind as the thread moved on.
Quoting Luke
OK, so I should have used "apprehend" then, and "perceive" was not a good option. As I said the choice of words is difficult, that is the nature of ontology. Just one little mistake after weeks or months of trying to explain the same thing to you over and over again, in as many different ways as possible, day after day, I think that's pretty good. You know, trying to explain the same thing in many different ways, so that a person who is having trouble understanding might have a better chance to understand, requires saying the same thing with different words. The appearance of contradiction is inevitable, to the person who refuses to look beyond the appearance, and try to understand what the other person is trying to say.
Your response to my last post makes it overwhelmingly clear that you are trying to see contradiction in my words, and not trying to understand. What a surprise!
This is why it's not productive to continue this convo. I've spent the last couple of posts saying that math is a lie, math is fiction, math is untruth in the service of higher truth, and you put words in my mouth. It's not fun and there's no point.
Quoting Metaphysician Undercover
Ok whatever. I have some stuff I'm dealing with in meatspace and maybe I'll get back to this tomorrow or the day after. But there's no point to this. I hope you will accept that. You claim I said things I've been saying the exact opposite of, and you take positions that I don't find sufficiently reasonable to interact with. There is no point to my replying. You do need to understand the concept of the necessary approximateness of all physical measurement. I can't imagine why you would take a stance so fundamentally wrong. You cannot draw a line of length 1 in the real world nor an angle of exactly 90 degrees. And you're the one who's convinced the square root of 2 doesn't exist, and now you say it not only exists, but you can draw it as the diagonal of an exact unit square. In order to have a conversation there has to be some small sliver of shared reality, and I find none here.
Your previous comments suggest otherwise:
Quoting Metaphysician Undercover
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You strongly imply that the inherent order is able to be apprehended in these quotes. We must be able to apprehend the inherent order if it is "describable" and we are able to see it.
Quoting Metaphysician Undercover
That's right, and then you forced upon the conversation your idiosyncratic idea of "the inherent order" that is unrelated to sets or ordering in mathematics. Fishfry and Tones tried telling you this, but you weren't interested.
Quoting Metaphysician Undercover
"Order" in relation to sets and ordering as it is understood in mathematics. I don't claim to be an expert, but I know you aren't talking about the same thing.
Quoting Metaphysician Undercover
Yes, because of you. Your concept of "the inherent order" was the main obstacle to a consensus. Did you think you were helping to build consensus?
Quoting Metaphysician Undercover
Thanks? I guess. But this does not answer the question of how your concept of "the inherent order" relates to "order" more generally. You could start with your own ideas of "order" and "the inherent order" and explain how these relate to each other. Why is "the inherent order" not a type of "order"?
Quoting Metaphysician Undercover
I'm not responsible for your contradictory statements or your inability to account for them. If you say that we can and cannot see the inherent order, or that we do and do not perceive the inherent order with the senses, then that is a contradiction. Have you considered that what you say might appear to be contradictory because it is contradictory, and that the problem is with your metaphysical edifice rather than with my understanding?
Quoting Metaphysician Undercover
The principle of charity is a two-way street.
I didn't see any mention of "higher truth". And I really don't think you've provided any explanation as to how lies, fictions, and untruths could be "in the service of higher truth". That sounds similar to Plato's "noble lie", where the rulers of the State lie to the common people for the sake of their own good. The proposal that such lies and deception are for the sake of a "higher truth" is extremely doubtful.
The problem we have here is that you seem to believe that mathematics give us idealized exactness, when really it fails at this. So this is a self-deception on your part. It all seems to stem from your idea that you can separate a "pure mathematics" as pure abstraction, away from applied mathematics, so that this pure mathematics deals with idealized exactness, while applied mathematics deals with the imprecision of real world measurement. You do not seem to understand that those who engage in so called "pure mathematics" are actually working on ways to solve the problems of imprecision in real world measurement, not trying to create pure abstractions. The problems of real world imprecision are not solved by axioms of idealized exactness, because of the fundamental incompatibility between the two.
Here's a proposal for you to consider. Imagine that human beings, when they first came out with mathematics many thousands of years ago, believed that mathematics provided them with ideal exactness. So we go way back, before Pythagoreans, Egyptians, or even Babylonians, and see that the people knew things from math, such as the example that they could derive a perfect right angle all the time, and they believed that mathematics gave to them idealized exactness. However, there were slight problems in applying numbers to spatial projections, such as the irrational nature of the square root of two, and pi.
Spatial projections are a creation based in real world appearances. I do not think you would deny this. So if we say that numbers are based in this idealized exactness, then when they are applied to real world problems, spatial projects, an incompatibility appears. The mathematician is faced with the problem of solving this incompatibility. But the mathematician is incapable of altering the reality of our spatial temporal existence, so there is no choice but to alter that lofty "idealized exactness" of the fundamental mathematical principles, numbers, if that incompatibility is to be resolved. The mathematician therefore is attempting to produce axioms which will bring numbers away from idealized exactness.
What I propose for you to consider then, is that mathematics may have been based in the idea of idealized exactness, many thousands of years ago, just like religion might have been based in the idea of an eternal immortal soul, but the progress which is made in mathematics, by mathematicians, is to bring us down from this idea, bring us away from it, to make mathematics something more compatible with the real world. For mathematicians to be working in some lofty world of ideal exactness, while this is completely incompatible with the real world, is pointless. So what the mathematicians are really doing is finding ways to bring the principles down out of that pointless realm of ideal exactness, compromise them in a way which makes them applicable, while maintaining as much exactness as possible.
Quoting fishfry
You see, my stance is directionally opposed to yours. Which one is fundamentally wrong is debatable. You seem to think that the way to "higher truth" is for the mathematician to work with "pure abstractions", of ideal exactness, which have no correlation to anything in the real world. I think the way to "higher truth" is to rid mathematics of such "pure abstractions", (because they are based in nothing but imagination), and to find "higher truth" we need to replace these principles with principles that correlate with the real world, even if this means to forfeit "ideal exactness". We both know that ideal exactness is impossible in the real world, therefore it cannot be a good principle upon which to judge mathematical principles which have the sole purpose of real world measurements.
Quoting Luke
You are not respecting the difference I described between what "order" refers to, and what "inherent order" refers to. The former we can apprehend, the latter we cannot, though they are both fundamentally intelligible. That's why I said, ultimately they can both be placed in the same category, I'll call it "intelligible".
I explained those differences already, and you are simply taking things out of context. The "inherent order" is fundamentally intelligible, but not by us, due to deficient capacity. Earlier I used the analogy of the way theologians speak of God. God is fundamentally intelligible, but not intelligible to use. Also, as Kant describes, noumena is fundamentally intelligible, but not by us.
Quoting Luke
In this entire thread, no one but jgill has offered any sort of alternative explanation as to what "inherent order" could refer to. Clearly it refers to an order which inheres within something. Jgill proposed that it is the order which inheres within a biological organism. But I see no reason to restrict this term to living things, as inanimate things also display reason to believe they have an inherent order. Until you bring forth another idea for "inherent order", it appears like you have no reason to say that mine is idiosyncratic, it really seems to be the only coherent understanding of "inherent order" possible.
You seem to be missing the point of my argument. By removing "inherent order" from the things called sets, as fishfry did, with the assumption of "no inherent order", these things (sets) can be assigned any possible order (in the sense of humanly created order), with absolutely no regard for truth or falsity, because it is stipulated that the set has no inherent order. My point of contention is that there is no such thing as something with no inherent order, it is an impossibility as self-contradictory, a unity of parts without any order to those parts. So this concept of a set, as a thing with no inherent order, is fundamentally wrong, and ought to be rejected.
Quoting Luke
What is referred to by "inherent order" is not a type of what is referred to as "order" because of the separation between them. This is described by Kant as noumenal and phenomenal. Inherent order is within the thing, as essential to the existence of the thing being the precise thing that it is. This is associated with the law of identity, which refers to the uniqueness of a thing. It has a unique inherent order, which makes it fundamentally intelligible as the unique thing which it is, and discernible from other unique things. However, the human mind does not grasp and understand the uniqueness of the thing, it grasps the thing relative to others, by similarities of universals, abstractions.
So the "order" understood by the human mind, created by the human mind as universal, is fundamentally different from the order inhering within the particular individual. If we said inherent order is a type of order (as conceive by humans), we'd dissolve the distinction between universal and particular, in a category mistake, making a particular into a universal. The law of identity prevents us from doing this.
How what is referred to as "inherent order" (within the particular) is related to what is referred to as "order" (a universal created by the human mind), is that they are both fundamentally intelligible. The former is not intelligible to the human being though. We could switch to the Aristotelian term "form" here. Aristotle distinguished two principal senses of the word "form", the form which inheres within the particular object, making it the unique object which it is, and the "form" which we assign, in describing the object, which involves universal abstractions. The latter, the form we assign to the object, neglects, or leaves out, the object's matter. "Matter" is assigned to the particular object to account for its accidents, the parts which are not grasped by the human mind, and this accounts for the object's uniqueness. Under this structure, the form of the particular, its inherent order, complete with accidents, is fundamentally intelligible, but not intelligible to the human intellect which understands through universal forms (orders), leaving the particular, inherent order, incomprehensible.
Quoting Luke
You haven't given me any real evidence that this might be the case, so no. I'll continue to wait for you to produce some substance, and indication that you understand, rather than demonstrating that you can search keywords throughout a lengthy thread, and take quotes out of context to produce the appearance of contradiction.
Possibility has "no regard for truth or falsity"? What does that mean?
Quoting Metaphysician Undercover
You can't have impossibility without possibility.
Quoting Metaphysician Undercover
And I'll continue to wait for you to produce some support for your accusation that I have taken any of your quotes out of context, or that the clear contradictions I have quoted are merely apparent. Your contradictions are a result of your constantly changing position. Since you are unable to clear them up, you can only accuse me of misunderstanding. Or else complain of the difficulty in choosing your words and say you were flustered and didn't mean to say that.
That's not true. Possibilities are limited by the actual state of the world. Anything claimed to be possible, which is not allowed for by the present state, is actually impossible.
Quoting Luke
And this is not even true. If determinism is the true description of reality, then true possibility is actually impossible, such that we would have the impossibility of changing the eternalist block universe, without any real possibility.
What's not true? You said: "(sets) can be assigned any possible order (in the sense of humanly created order), with absolutely no regard for truth or falsity." I asked what it means for the possibility (of the order) to have "absolutely no regard for truth or falsity".
Quoting Metaphysician Undercover
Right, so if I presently have a set (or bag) of three balls coloured red, white and blue, then there are six possible orderings in which I can draw out those three balls: (RWB, RBW, WRB, WBR, BRW, BWR). What's wrong with that? And what is their order before they are drawn from the bag?
Quoting Metaphysician Undercover
Has that been the basis of your argument from the start? Funny, since I've seen you argue against the eternalist block universe in other threads. You really are a troll.
I said, the assignment of possibility is done without regard for order.
Sorry Luke, your interpretation is so bad (no wonder you see contradiction in everything I say) , I have extreme difficulty communicating with you. I don't see how any reasonable mind could interpret the way you do, therefore I can only conclude that you make these unreasonable interpretations intentionally.
A set is not a bag of items, it's an abstraction, that's the point fishfry has been stressing. A bag of items has an inherent order, as I've spent months describing to you. If we want to represent that bag of items as a thing called "a set" we cannot truthfully predicate of that subject, the property of "no inherent order" because the thing being represented necessarily does have an inherent order. No inherent order is a false representation.
Fishfry claims that a pure abstraction is an imaginary fiction, so it doesn't matter that it's not a true representation, and claimed that the imaginary fiction is useful toward a "higher truth". However, fishfry insisted on using a map analogy for explaining abstractions, and a map is a representation, so there is inconsistency in what fishfry was presenting. Furthermore, fishfry could not explain how an imaginary fiction could be useful toward obtain a higher truth.
Quoting Luke
Going through this thread, and taking statements out of context isn't enough for you, so now you have to refer to other threads. You bring "taking things out of context" to a whole new level. I really am a troll but you're just an ass hole.
Do you think I misquoted you? Here:
Quoting Metaphysician Undercover
My question, again, what do you mean possibility has "no regard for truth or falsity"?
Quoting Metaphysician Undercover
What interpretation? I asked you a question.
Quoting Metaphysician Undercover
So tell me what is the order of the three coloured balls before they are drawn from the bag.
Quoting Metaphysician Undercover
Can you tell us how the imperceptible, unapprehendable inherent order could be useful to anyone?
I replied to this, in the last post go back and read it. That is such an unreasonably bad interpretation of what I said, that I think any reasonable person could only have presented me with such a thing intentionally.
Quoting Luke
We've gone through this before. I can't tell you the inherent order.. Luke, you've got an extremely bad habit of getting me to spend endless time explaining something to you, then you start right back at square one, as if we've never talked about it before.
So, to assist you in understanding, I'll use different words than the last time. Then, you turn around to the last time, and say 'look, these words are different from the last time, therefore you contradict yourself'. And you say I'm a troll!
Quoting Luke
It's useful to recognize the reality of it, to understand the deficiencies of mathematics.
You said:
Quoting Metaphysician Undercover
So you are saying that possibility has no regard for truth or falsity, i.e. no regard for the inherent order. I still have no idea what this means. But what regard should the inherent order be given if it cannot be perceived or known?
Quoting Metaphysician Undercover
Yeah, that's why I asked. It's a bullshit assumption that can't be known.
Quoting Metaphysician Undercover
The "reality of it" is nothing more than your useless assumption.
No, I am saying that the person who assigns possibility, in that situation, has no regard for truth or falsity, in that act. How could possibility be the type of thing which might have a regard for truth or falsity? Your interpretation is simply ridiculous, and I can't see any reason for such a ridiculous interpretation other than that you intentionally make an unreasonable interpretation in an attempt to make what I say appear to be unreasonable.
Quoting Luke
You are ignoring the fact that I repeatedly said that we see the inherent order without apprehending it with the mind. You just can't seem to grasp this fact of reality, that we see things without understanding what is seen. To you, this is pure contradiction, but until you grasp it, you will never understand what I've been saying.
So, when the mind produces an order, which is supposed to be a representation of the inherent order, within the thing, the order which is being sensed must be regarded in order that the representation be a good one.
Quoting Luke
If you are convinced that the assumption of an inherent order is a "bullshit assumption", then why didn't you just say this two months ago, and we could have avoided all of your nonsense bad interpretation, and out of context quotes, in your effort to make it look like what I am saying is contradictory?
But no, this you MO, to produce nonsense interpretations, and out of context quotes, with the intention of making it look like the author is inconsistent. So of course you couldn't have been up front with your difference of opinion. You had to carry on and on, in the pretense that you were trying to understand, but couldn't, to get more words, more phrases, sentences, and statements, as ammunition in your pointless attack, without any intention of trying to understand. And then you call me a troll!
Maybe it took you this long to figure it out, that the assumption of an inherent order is a bullshit assumption, but if this is what you believe, then there's no point in going any further with this discussion, because I have no desire to try to convince you otherwise. Because of this belief which you have, that inherent order is a bullshit assumption, there is no point in discussing the relation of inherent order to a set, because you do not believe there is any such thing as inherent order in the first place.
It concerns what is possible in reality; what may come true or happen.
Quoting Metaphysician Undercover
And for several pages prior to this position, you said that we could not perceive the inherent order. You still have not defined the two different meanings of "see" that you claim is at work in this apparent contradiction. More recently, you said that we do not perceive the inherent order with the senses, only that it is "present to" the senses. That does not mean "see". Ultraviolet light might be considered "present to" the senses in the same way, but we cannot see it with the naked eye, either. When I put this argument to you, you argued a minor point from an article regarding infrared light which we might be able to see in some cases, but you did not address any of the other forms of electromagnetic radiation that we cannot see. How can we both "see" and not see these other forms of radiation?
Quoting Metaphysician Undercover
You would need to apprehend the inherent order in able to compare and judge the representation as good or bad. You claim we cannot apprehend the inherent order. Unless you can justify or explain your different meaning of the word "see", then neither can we see the inherent order.
Quoting Metaphysician Undercover
Because I thought you were changing your position and I wanted to prove that you were. Alternatively, if you were not changing your position, then you were just espousing obvious contradictions, and I thought you might come to realise that it was a bullshit assumption after your several glaring contradictions were shown to you. Alas, you are not prepared to even view your plainly contradictory statements as contradictory.
Quoting Metaphysician Undercover
This is not true, we use empirical evidence, and deductive logic. It's called science. The hypothesis is the representation, and it is judged according to the evidence and logic.
You are assuming the existence of an inherent order that lies beyond conscious recognition. Is there another aspect of mind that might register this phenomena? Is the fact we can discuss IO due to this possibility?
I don't think I'd call it an aspect of mind, rather an aspect of life. It is evident, that at the most fundamental level, living beings make use of inherent order, by creating extremely complex molecules, etc.. I don't know how the inherent order is recognized at this level, but it must be, in some way, in order for this organizational activity to occur.
I wouldn't say it is an aspect of "mind" that recognizes it because then we get into panpsychism, or something like that, and I think that there is a clear separation, or difference between mind, along with consciousness, and what happens at this fundamental level. So I think it is good to keep them separate, and understand that this is not a part of mind though mind may be a part of this. But the part only performs its particular function, without understanding its relation to the whole.
I brought this up earlier, the notion that "ordering" and "order" were not the same, saying that order had to do with biological and other systems, rather than listing. How order relates to ordering the elements of a set is unclear.
Yes, I think that's the distinction I've been describing to Luke, the difference between inherent order, and order as created by the human mind, what you call "ordering". The two are completely different, and how they relate is in some respects "unclear".
The issue I brought up with fishfry is the distinction between representation and imagination. Fishfry allows that "abstraction" might encompass both of these, such that imaginary ideas could be a useful part of a representative model. But this was not borne out by fishfry's map analogy. The purely imaginary in this case, is the thing with no inherent order, as a "set". Another example of the purely imaginary is infinity.
The point that I am making, is that the current order which a thing has, the inherent order, limits the possibility for future order. Therefore the possibilities for ordering (by the mind), in any true sense, must be limited by an assumed inherent order that a thing has. To remove the necessity of assuming an inherent order, for the sake of allowing infinite possibilities of ordering, is a fiction of the imagination, which has no place in a representative model.
So here's an example. Suppose there's a set which contains the "numbers" 3,4,5. The "numbers" are assumed to be abstract objects. As abstract objects, they necessarily have meaning. The meaning which the abstract objects have, necessarily gives them an order, an order which inheres within that group of objects, the set. You cannot have five without first having four, and you cannot have four without first having three. Or even if we define purely by quantity, there inheres within the meaning of "quantity", more and less. Therefore there is an order which inheres within that set of abstract objects, necessitated by the meaning of the objects. To assume to be able to remove that order, which is essential to the existence of those abstract objects, and therefore inheres with the set, for the sake of claiming that any ordering of the set is possible, is a mistaken adventure.
There is one thing about the relation between inherent order, and ordering, which is very clear. This is that to represent the possible orderings of any group of objects, one must have respect for the limitations imposed by the order that they already have. To deny that they have any inherent order is to produce a false representation based in imaginary fiction. And to give abstract objects a special status as fishfry does, to allow exemption from this rule, is to render them free from the restrictions of definition which enforces logical relations (order) between them, thus allowing the imagination to run free.
But as simple symbols, rather than meaningful symbols, they may have no IO or a different IO. If I make up three random symbols from finite lines, say, would you then state the order in which I created them gives IO to the set?
You know it's an oxymoron to talk of a symbol without meaning. If it doesn't symbolize something we can't call it a symbol. Anyway, a physical symbol is not much different from any other physical thing, except it's produced with intention. Each symbol as a distinct entity has order inherent within as the relations of its parts, and as a group of three, there is order inherent in the relations between them, As symbols, with meaning, there is a reason for the order in which you write them. if they are supposed to be devoid of meaning, then why would you be making them in the first place? It's an intentional act, so there must be reasons, therefore order.
True enough. I should have said random configuration of line segments or something like that.
Quoting Metaphysician Undercover
Are all acts founded on reason?
Is there an axiom in set theory that requires the display of elements of sets to be done in inherent order?
No, not all acts are based in reason. Reason is a property of the conscious mind. But that they are not based in reason does not mean that there is not order to them, as there is order in the actions of the inanimate world. That's the difference I referred to. If all order is derived from consciousness, or "reason", we are lead toward panpsychism.
Quoting jgill
No I really do not believe there is such an axiom, as fishfry stated, a set has no inherent order. And regardless of the displayed order, one could express a statement like 'these elements without any order'. The issue is whether it is possible to conceive of something like that, 'elements without any order'. If not, it would be an incoherent concept. It is possible to state things which are contradictory, or something like that, making what is stated impossible to conceive.
Ironically, it is the the platonists who insist that every set must be "well-ordered" which is an assumption equivalent to the axiom of choice. But for those who deny the axiom of choice, it is nevertheless meaningful to compare the "sizes" of different sets even if the determined sizes are not synonymous to counting elements.
Then there is the little matter of potential infinity. Mathematically, it might well be the case that the number of grains of sand in a heap is neither finite nor actually infinite, but indefinitely large. To argue differently is to argue the religion of physics rather than maths.
Suppose that a heap of sand is indefinitely large, in that every time a grain of sand is extracted from the heap it might be possible to remove from the heap yet another grain of sand. Even though the heap of sand is indefinitely large, it is nevertheless meaningful to speak of the original heap of sand as being larger than the heap with a grain of sand removed, and yet in this case it is only possible to count the grains removed from the heap.
As a mathematician who has dabbled for years in complex analysis - a branch of math that involves the limit process, calculus, etc. in the complex plane - I have rarely if ever interpreted infinity as a kind of "number" but as shorthand for "unbounded", which correlates to you always finding another grain of sand.
As for inherent order, I admit that writing {c,a,b}, although permissible, just doesn't seem right. I would always prefer {a,b,c}. :cool:
I foresee big trouble with this definition or my name's not Gottlob Frege. Well my name's not Gottlob Frege, but Frege proposed the same definition. Then Russell came along and said, "Ok smart guy, take the predicate to be [math]x \notin x[/math] and see what happens. You get a contradiction. Busted!
https://en.wikipedia.org/wiki/Russell%27s_paradox
In fact a set is ... well nobody knows what a set is. A set is whatever satisfies the axioms of some particular set theory, of which there are many. A thing could be a set in one set theory and not in another.
What you gave is the definition used in "high school set theory," sometimes called naive set theory. But that definition fails as soon as you examine it closely.
Quoting sime
News to me. You claim that being a Platonist ixs equivalent to believing in the axiom of choice? I'd take those two things to be totally independent of one another. You could be a Platonist or not, and pro-choice or not. I don't see the connection.
Quoting sime
The problem is that absent the axiom of choice, there are infinite sets that are not comparable to each other. That is, there are cardinals [math]x[/math] and [math]y[/math] such that none of [math]x = y[/math] nor [math]x < y[/math] nor [math]x > y[/math] are true.
That's actually the purpose of the axiom of choice, to make infinite sets behave. Without choice there's an infinite set that's Dedekind-finite; that is, a set that's not bijectively equivalent to any natural number 0, 1, 2, 3, ..., but that is not bijectively equivalent to any of its proper subsets.
So in fact your statement is inaccurate. If you deny the axiom of choice, it is sometimes MEANINGLESS to compare the sizes of sets.
Quoting fishfry
Don't feel badly. As a professor of math I rarely thought of sets that did not satisfy this definition. In passing I thought of Russell's paradox as quirky, requiring set theory be a little more sophisticated. Living in a state of "high school" naivety all those years has been devastating. How can I recover, now I am near the end of my days? :cry:
I was just reading about the Frege-Hilbert dispute. As I understand it, Hilbert was saying that axioms are formal things and it doesn't matter what they stand for as long as we can talk about their logical relationships such as consistency. Frege thought that the axioms are supposed to represent real things. I'm not sure if I'm summarizing this correctly but this is the feeling I got when I was reading it. That I'm taking Hilbert's side, saying that the axioms don't mean anything at all; and you are with Frege, saying that the axioms must mean whatever they are intended to mean and nothing else.
https://plato.stanford.edu/entries/frege-hilbert/
Is this what you're getting at?
You then complained about my map analogy, but I hope you will agree that you could object to my analogy without necessarily refuting my thesis. If you don't like maps, forget the maps.
ps -- I found another nice article.
https://academic.oup.com/philmat/article/13/1/61/1569375
You can skip the category theory stuff, scroll down to here:
I take this to be a reference to the fact that geometers studied various models of geometry, such as Euclidean and non-Euclidean, and were no longer concerned with which was "true," but rather only that each model was individually consistent. And philosophers, who said math was supposed to be about truth, were not happy.
And:
(my emphasis).
I think that last bolded part is the heart of our discussion. I'm saying that we can study aspects of the world by creating formal abstractions that, by design, have nothing much to do with the world; and that are studied formally, by manipulation of symbols. And that we use this process to then get insight about the world.
And you say, how can these meaningless abstractions possibly tell us anything about the world? They're meaningless, they're not true. You are taking the Fregean position, that if the axioms are not about the world, they're nonsense.
Here's Paul Bernays explaining Hilbert's point of view:
(my emphasis again)
I do believe this is what our conversation is about. You're a Fregean and I'm with Hilbert.
What do you think? For my part these articles have given me some insight into your point of view. If collections in the real world have inherent order, what sense does it make to postulate sets that have no inherent order? The answer is Hilbert's side of the debate; which, for better or worse, is the prevalent view in modern math.
There's more. This is a great article.
(my emphasis)
This is EXACTLY what you are lecturing me about! But as SEP notes, Hilbert stopped replying to Frege in 1900. Just as I ultimately had to stop replying to you. If you and Frege don't get the method of abstraction, Hilbert and I can only spend so much time listening to your complaints. I found a great sense of familiarity in reading about the Frege-Hilbert debate.
I'll join you for a hot dog and a beer.
Yes, I'm already aware of all of that, and was only speaking approximately on set theory. My point was only attacking the idea that quantity is reducible to ordering.
The connection is the fact that the axiom of choice is equivalent to the law of excluded middle, which for infinite objects dissociates truth from derivation. This in itself wouldn't imply platonism if it wasn't for the fact that most proponents of classical logic and ZFC make no attempt to justify the formalisms pragmatically with respect to real world application.
"And of course, we know that LEM does not imply AC, since we know that ZF is consistent with ¬AC while LEM holds." (MathStackExchange) :chin:
That is not correct. It is the case that Z (even without the law of excluded middle (LEM)) and the axiom of choice (AC) together imply LEM. But it is not the case that Z (which includes LEM) implies AC.
Sorry for the confusion. Yes that is true for ZF, since it is built upon classical logic. In set theory, controversial instances of the excluded middle are the result of both the underlying logic if it is classical as well as the set theoretic axioms of choice and regularity.
What i had in mind wasn't ZF, but intuitionistic set theory, in which choice principles and LEM are approximately equivalent as documented in the SEP article on the axiom of choice.
https://plato.stanford.edu/entries/axiom-choice/
(One reason math has become so abstract is that classical areas of investigation have been "mined out". Professors need suggestions for research topics for their PhD students. So, create new definitions and/or generalize.)
That is not the reason. The reason is that LEM does not imply AC, whether with intuitionistic or classical logic.
Quoting sime
I looked at that article briefly. I did not see mention of an "approximate" equivalence.
Whether classically or intuitionistically, there is not an equivalence. Rather, there is only the one direction: AC implies LEM. But the other direction that is needed for equivalence - LEM implies AC - is not the case.
Explicitly constructive mathematics goes back at least a hundred years, and with roots in the 19th century too. It has great importance toward understanding foundations. I think interest in it goes well beyond any need for assigning research topics.
I do not believe you have correctly stated Diaconescu's theorem. (Didn't think I knew that one, did you!) I quote from Wiki:
So AT BEST, AC implies LEM. And NOT the converse. So your claim of equivalence is not supported by Wikipedia. I have to trust Wiki on this point because I haven't studied this much, just remember hearing about it.
Secondly, the qualifier, "in constructive set theory." Does that mean that the implication ONLY holds in constructive set theory? Or is that an ambiguous statement on the part of some Wiki author. I don't know. But you've gone way too far in your claim of equivalence.
Moreover, you have not even remotely justified your claim that AC is equivalent to Platonism. You wrote a sentence that I could neither parse nor understand, and I don't think it's true. I'm talking about, "This in itself wouldn't imply platonism if it wasn't for the fact that most proponents of classical logic and ZFC make no attempt to justify the formalisms pragmatically with respect to real world application." Can you rephrase that so I can understand what you're saying?
(edit) I'm beginning to unpack a little of that. "Most proponents of classical logic and ZFC make no attempt to justify the formalisms pragmatically." Why on earth should they? That's not their job. And that's not even Platonism. Platonism doesn't say that the abstract math thingies refer to the real world. It says that the abstract math thingies exist in some nonphysical Platonic world. So even here I don't think you're using the technical terms correctly.
Also, "... which for infinite objects dissociates truth from derivation. " What? if you reject LEM then truth = derivation? That's false. Derivation is syntactic, truth is what's true in some model. That doesn't change just because you reject LEM.
Quoting sime
My most humble apologies, I do not know what this is in reference to. Quantity and order are surely different, that's why there are cardinals and ordinals.
Quoting sime
"Approximately equivalent?" What does that mean? Are 12 and 14 approximately equivalent? Apples and rutabagas? I don't think you are making your point. And to be fair I don't know much about constructive math (and not for lack of trying), so I assume you have something interesting to say here, but you're not saying it clearly.
Quoting TonesInDeepFreeze
Glad you said that, I thought I was being too snarky jumping on that phrase.
Quoting jgill
@jgill This place is corrupting you! You're supposed to say that as a complex variables guy you don't think much about foundations! This was most impressive.
Quoting jgill
Constructive math is making a comeback these days because of the influence of computers. There are computerized proof assistants, and they tend to operate on some form of constructive logic or intuitionistic type theory. Brouwer was the great 1930's intuitionist, so I thin of all this as "Brouwer's revenge."
Quoting GrandMinnow
Would that others show so much circumspection. Myself included from time to time.
Bear in mind
1) All of the non-constructive content of classical logic is discarded by jettisoning LEM.
2) The axiom of choice holds trivially as a tautology in sets constructed in higher-order constructive logic, because in this logic existence is synonymous with construction.
So one could even say that absence of LEM implies AC (or perhaps rather, that AC is an admissible tautology in absence of LEM).
But this statement isn't enlightening, because it conflates the difference in meaning that AC has in the two different systems, for AC holds trivially and non-controversially in constructive logic as a tautology, where it doesn't imply anything above and beyond construction.
In the constructive sense, i think it is fair to say that LEM implies AC, when speaking of AC not in the sense of an isolated axiom, but in the commonly used informal vernacular when speaking of choice principles in their structural and implicational senses
Of course, semantics for intuitionistic systems are different from semantics for classical systems. But the question of equivalence is that of derivability.
Quoting sime
Reference please.
Quoting sime
That makes no sense. 'Implies' means proves in this context. And one cannot prove what is otherwise not provable by weakening the proof logic. (But I can't opine on 'admissible tautology' since I don't know your definition.)
Quoting sime
AC is an exact formulation. It is not expressed as "commonly used informal vernacular when speaking of choice principles in their structural and implicational senses", whatever you might exactly mean by that.
There are different formulations that may have equivalences, and there are complications throughout, but I know of no proof nor mention in the article you cited that shows the equivalence of AC with LEM in intuitionistic set theory. The SEP article does say "each of a number of intuitionistically invalid logical principles, including the law of excluded middle, is equivalent (in intuitionistic set theory) to a suitably weakened [italics in Bell's earlier article] version of the axiom of choice. Accordingly these logical principles may be viewed as choice principles." But the question was not that of various choice principles but of AC itself, and we have not been shown a proof that AC and LEM are equivalent in intuitionistic set theory.
My comment about abstraction wasn't really in reference to foundations, just a general reflection on the profession. I mean no disrespect to constructive mathematics. I recall my advisor fifty years ago talking about classical analysis being somewhat mined out and mathematicians moving toward generalities and abstraction to find new territory. He thought interest in the classical might come back at some point.
(Actually, I'm a fan of Brouwer, more so of Banach, whose fixed point theorem I extended to suit my special interests.)
Quoting fishfry
Makes my head spin. Thanks for opening my eyes a bit to formalized set theory and for your patience!
I disrespect constructive math all the time. Actually over the past couple of years a handful of constructivists showed up here and I had interesting conversations with them, but always ended up baffled and they left. I hope it wasn't something I said, I'm just trying to learn. These were generally people coming to the subject from the computer science side and never the math side, so they had no appreciation for standard math. Made for frustrating conversations both ways.
Quoting jgill
Glad to help.
Thanks for the reading material fishfry, I've read through the SEP article a couple times, and the other partially, and finally have time to get back to you.
I think Frege brings up similar issues to me. The main problem, relevant to what I'm arguing, mentioned in the referred SEP article, is the matter of content.
[quote=SEP]The difference of opinion over the success of Hilbert’s consistency and independence proofs is, as detailed below, the result of significant differences of opinion over such fundamental issues as: how to understand the content of a mathematical theory, what a successful axiomatization consists in, what the “truths” of a mathematical theory really are, and finally, what one is really asking when one asks about the consistency of a set of axioms or the independence of a given mathematical statement from others.[/quote]
In critical analysis, we have the classical distinction of form and content. You can find very good examples of this usage in early Marx. Content is the various ideas themselves, which make up the piece, and form is the way the author relates the ideas to create an overall structured unity.
Hilbert appears to be claiming to remove content from logic, to create a formal structure without content. In my opinion, this is a misguided adventure, because it is actually not possible to pull it off, in reality. This is because of the nature of human thought, logic, and reality. Traditionally, content was the individual ideas, signified by words, which are brought together related to each other, through a formal structure. Under Hilbert's proposal, the only remaining idea is an ideal, the goal of a unified formal structure. So the "idea' has been moved from the bottom, as content, to the top, as goal, or end. This does not rid us of content though, as the content is now the relations between the words, and the form is now a final cause, as the ideal, the goal of a unified formal structure. The structure still has content, the described relations.
Following the Aristotelian principles of matter/form, content is a sort of matter, subject-matter, hence for Marx, ideas, as content, are the material aspect of any logical work. This underlies Marxist materialism
However, in the Aristotelian system, matter is fundamentally indeterminate, making it in some sense unintelligible, producing uncertainty. Matter is given the position of violating the LEM, by Aristotle, as potential is is what may or may not be. Some modern materialists, dialectical materialists, following Marx's interpretation of Hegel prefer a violation of the law of non-contradiction.
So the move toward formalism by Frege and Hilbert can be seen as an effort to deal with the uncertainty of content, Uncertainty is how the human being approaches content, as a sort of matter, there is a fundamental unintelligibility to it. Hilbert appears to be claiming to remove content from logic, to create a formal structure without content, thus improving certainty. In my opinion, what he has actually done is made content an inherent part of the formalized structure, thus bringing the indeterminacy and unintelligibility, which is fundamental to content, into the formal structure. The result is a formalism with inherent uncertainty.
I believe that this is the inevitable result of such an attempt. The reality is that there is a degree of uncertainty in any human expression. Traditionally, the effort was made to maintain a high degree of certainty within the formal aspects of logic, and relegate the uncertain aspects to a special category, as content.
Think of the classical distinction between the truth of premises, and the validity of the logic. We can know the validity of the logic with a high degree of certainty, that is the formal aspect. But the premises (or definitions, as argued by Frege) contain the content, the material element where indeterminateness, unintelligibility and incoherency may lurk underneath. We haven't got the same type of criteria to judge truth or falsity of premises, that we have to judge the validity of the logic. There is a much higher degree of uncertainty in our judgement of truth of premises, than there is of the validity of logic. So we separate the premises to be judged in a different way, a different system of criteria, knowing that uncertainty and unsoundness creeps into the logical procedures from this source.
Now, imagine that we remove this separation, between the truth of the premises, and the validity of the logic, because we want every part of the logical procedure to have the higher degree of certainty as valid logic has. However, the reality of the world is such that we cannot remove the uncertainty which lurks within human ideas, and thought. All we can do is create a formalism which lowers itself, to allow within it, the uncertainties which were formerly excluded, and relegated to content. Therefore we do not get rid of the uncertainty, we just incapacitate our ability to know where it lies, by allowing it to be scattered throughout the formal structure, hiding in various places, rather than being restricted to a particular aspect, the content.
I will not address directly, Hilbert's technique, described in the SEP article as his conceptualization of independence and consistency, unless I read primary sources from both Hilbert and Frege.
Quoting fishfry
Geometers, and mathematicians have taken a turn away from accepted philosophical principles. This I tried to describe to you in relation to the law of identity. So there is no doubt, that there is a division between the two. Take a look at the Wikipedia entry on "axiom" for example. Unlike mathematics, in philosophy an axiom is a self-evident truth. Principles in philosophy are grounded in ontology, but mathematics has turned away from this. One might try to argue that it's just a different ontology, but this is not true. There is simply a lack of ontology in mathematics, as evidenced by a lack of coherent and consistent ontological principles.
You might think that this is all good, that mathematics goes off in all sorts of different directions none of which is grounded in a solid ontology, but I don't see how that could be the case.
Quoting fishfry
I know you keep saying this, but you've provided no evidence, or proof. Suppose we want to say something insightful about the world. So we start with what you call a "formal abstraction", something produced from imagination, which has absolutely nothing to do with the world. Imagine the nature of such a statement, something which has nothing to do with the world. How do you propose that we can use this to say something about the world. It doesn't make any sense. Logic cannot proceed that way, there must be something which relates the abstraction to the world. But then we cannot say that the abstraction says nothing about the world. If the abstraction is in some way related to the world, it says something about the world. If it doesn't say anything about the world, then it's completely independent from any descriptions of the world, so how would we bring it into a system which is saying something about the world?
Quoting fishfry
Actually, I think that often when one stops replying to the other it is because they get an inkling that the other is right. So there's a matter of pride, where the person stops replying, and sticks to one's principles rather than going down the road of dismantling what one has already put a lot of work into, being too proud to face that prospect. You, it appears, do not suffer from this issue of pride so much, because you keep coming back, and looking further and further into the issues.
it is not the case that Frege and I do not get the method of abstraction. Being philosophers, we get abstraction very well, it is the subject matter of our discipline. You do not seem to have respect for this. This is the division of the upper realm of knowledge Plato described in The Republic. Mathematicians work with abstractions that is the lower part of the upper division, philosophers study and seek to understand the nature of abstractions, that is the upper half of the upper division. Really, it is people like you, who want to predicate to abstraction, some sort of idealized perfection, where it is free from the deprivations of the world in which us human beings, and our abstractions exist, who don't get abstraction.
That is evidence that mathematics, and what fishfry calls "formal abstractions", are not separate, or independent from the world, as fishfry argues. They are not ideal perfections, separate Forms, but they must share in the imperfections of the material world, as only being useful in that world, if they are part of that world. The proposition that they could have independence from the material world is a false premise. So to create a formalism and present it as free from the negative influence of content, is to present a smoke and mirrors illusion, because the most one can do in this respect, is hide that negative influence.
As a working math person I rarely thought about these philosophical issues, But "formal abstractions" in foundations appears to be a very sophisticated game. Whether this game is entirely separated from the material world might depend on how one interprets "separate".
I have a number of issues with this passage. Any concept which originates in real world concerns cannot be said to have been produced from an appeal to aesthetic beauty. And if we suppose that there are some of each, wouldn't it be the ones which deal with real world concerns which get accepted into the community. So, as much as I see the claim that "pure mathematicians" are motivated by aesthetic beauty, as opposed to real world concerns, I don't see that any such concepts as being produced purely for aesthetic beauty actually exist in mathematics.
The next is with the phrase " the intellectual challenge and aesthetic beauty of working out the logical consequences of basic principles". What we are discussing is the production of the basic principles themselves. If "pure mathematics" simply involves working out the logical consequences of already established principles, then it is not really relevant to what we were discussing, which is the derivation of those basic principles. The question is whether those principles ought to be derived from pure imagination, or ontology.
I can attest this (my) mathematical concept was produced purely for aesthetic beauty. The fact you don't see a string of symbols is immaterial. It is found on a mathematics page in Wikipedia.
Quoting Metaphysician Undercover
Is there such a thing as "pure imagination" that does not arise ultimately from observations and experiences in the physical world?
That's a good question. But if all such principles can be said to have empirical causes, then how can you say that you have a mathematical concept which was produced purely for aesthetic beauty? If there are experiential concerns which enter into your conception then how can you say that your intention of aesthetic beauty is pure?
This is essentially the freewill vs determinism question. To make your conception pure, you'd need the capacity to mentally make a clean break between past and future. It is the assumed continuity between past and future, which forces real world concerns into our thinking. We cannot escape the reality of what we experience as what has just happened, and how this bears on what is about to happen. But if we make a break between past and future, then past experience has no necessary bearing on what we produce for the future because what has just happened will not influence our thinking about what is about to happen. Then your goal for the future, a creation of pure aesthetic beauty, could be completely free from the notion that past occurrences put a necessary constraint on your future production, and you could draw from your past experiences, in complete freedom. The creation of your aesthetic beauty could be done purely without any real world concerns, i.e. knowing that the past has no necessary causal relation with the future, allowing you complete freedom from real world pressure.
This is the issue of inherent order, which we've discussed for months now, in a nutshell. If there is order which inheres within a thing, then that order puts a necessary constraint on future possibilities of order, due to the continuity (causal relation) which we assume to exist between past and future. Logically, we want to start with the assumption of unlimited possibility, to give us the capacity to understand any possible ordering. So, we start with the premise fishfry suggested of "no inherent order". But this is not a real representation of the necessity imposed by inherent order. To remove the necessity of inherent order in a more realistic way, I think, requires that we make a clean break between past and future, annihilating the supposition of continuity, thus allowing that the order which inheres within a thing has no real bearing on the thing's future order. This would allow for the true possibility of any order, it doesn't start with the premise of no inherent order, but it rejects the order which is imposed by the supposition of continuity.
Interesting ideas, but a tad too ethereal for me. I prefer the solidity of pure mathematics. :cool:
Glad you found some of this helpful. You talked about a lot of things here I'm not qualified to comment on, but I wanted to go to the end and respond to this first.
Quoting Metaphysician Undercover
Another reason is that the conversation has gone past the point where there's any light being generated relative to the word count. And the fact that you think I'm arguing right or wrong is telling. I'm not arguing my point of view is right, I'm not even arguing a point of view. I'm telling you how modern math works. It's like this, if you don't mind a Galilean dialog.
I hope you see the parallel. I am NOT saying I'm "right" that math should be the way it is. I'm reporting to you from the front lines of math, about how things are. It's pointless for you to tell me that I'm wrong about how things are, because my report is objective. And it's pointless for you to tell me math "should" be other than it is, because I'm not the Lord High Commissioner of mathematics. I'm just telling you how the twentieth century went. It's the tediosity of holding up my end of this theater of the absurd that leads me to withdraw from the field of play.
Quoting Metaphysician Undercover
That doesn't even make sense. I can take no pride in how math is, I was simply trained in universities in the modern style of the subject. Hilbert could take pride, he was one of the major instigators of the movement to abstraction. I'm only a very humble student. I'm putting my knowledge at your disposal, I'm not claiming modern math is right or wrong. I'm describing, not advocating.
I am not claiming the modern approach is right. I am only telling you how they do it. I'm telling you how the knight moves, I'm not claiming the rules of chess are "right" or "wrong." The fact that you don't see this emphasizes the futility of any time I spend typing here.
Quoting Metaphysician Undercover
I keep thinking I might get through to you. I'm not trying to convince you the modern math approach is right. Why do you think I am?
Quoting Metaphysician Undercover
Right. You are taking the Fregean point of view, and modern math the Hilbertian. But I don't believe I'm arguing the rightness of the modern abstract way; only trying to describe it to you.
Quoting Metaphysician Undercover
All this is stipulated. I can't continue the convo since I haven't said anything new in weeks if not months.
Quoting Metaphysician Undercover
Karl or Groucho? You seem to still want to argue that I'm "wrong" somehow, or explain your point of view. I now understand your point of view. There's nothing I can do about it, I can't imagine doing math the way you suggest, not because one couldn't, but because Frege lost the 20th century and I know no other way.
Quoting Metaphysician Undercover
From math, not from logic. Surely you don't claim there's content in logic "P implies Q" and "Q" imply P. There's no content there but that's one of the most ancient logical forms there is.
Quoting Metaphysician Undercover
You say modern math is misguided as if you want me to defend the opposite proposition. I'm not defending anything. I'm just describing to you how modern math is. I'm not defending it. I'm reporting to you on what I know about it. And for what it's worth, if it's misguided, the abstract point of view not only won the 20th century, math got even more abstract in the second half of the 20th century and into the present.
Quoting Metaphysician Undercover
I'm not arguing the point.
Quoting Metaphysician Undercover
Over my head, way out of my bailiwick. Can't respond.
Quoting Metaphysician Undercover
This is all very impressive-sounding but is an alien language to me. I can't respond, I have no stake in the matter and no understanding of what you're talking about.
Quoting Metaphysician Undercover
Ok. I'm sure you have the core of a nice essay there, but why are you telling it to me? And again, what do you mean "remove content from logic?" When was logic EVER about content? P and Q, remember?
Quoting Metaphysician Undercover
You're just going on without me. I don't relate any of this to anything we've been discussing.
Quoting Metaphysician Undercover
My point was that, having learned about the Frege-Hilbert dispute, I see that you have been arguing Frege's view and I Hilbert's. And in math, Hilbert won the 20th century. This is a matter of fact. There is nothing to argue and no right or wrong. But your discourse in this present post is alien to me, I have no idea what it's about. I am sure you are making interesting points, but they're lost on me.
Quoting Metaphysician Undercover
Modern math is what it is, and nothing you say changes that, nor am I defending it, only reporting on it. You think it's bad and wrong, ok, I'm no longer arguing the point with you if I ever was.
Quoting Metaphysician Undercover
Sprichst du Deutsch?
Quoting Metaphysician Undercover
In math, axioms used to be self-evident truths, and now they're more or less arbitrary assumptions to get a given theory off the ground. No truth is claimed. I had no idea philosophers were still clinging to the old concept of axioms. No wonder they're so far behind in understanding modern math.
Quoting Metaphysician Undercover
It was forced on math by the discovery of non-Euclidean geometry. Once mathematicians discovered the existence of multiple internally consistent but mutually inconsistent geometries, what else could they do but give up on truth and focus on consistency?
I'm curious to hear your response to this point. What were they supposed to do with non-Euclidean geometry? Especially when 70 years later it turned out to be of vital importance in physics?
Quoting Metaphysician Undercover
You say that like it's a bad thing! It was forced on math by non-Euclidean geometry. Physics is about ontology now. But of course even contemporary physics has abandoned ontology, and if you say that's a bad thing, I'd be inclined to agree with you.
Quoting Metaphysician Undercover
It's not good or bad, it is simple inevitable. What should math do? Abolish Eucidean or non-Euclidean geometry? On what basis?
Quoting Metaphysician Undercover
As evidence I give you "The unreasonable effectiveness of math etc."
Quoting Metaphysician Undercover
You've given me not the slightest evidence that you have any idea how math works. And a lot of evidence to the contrary.
Quoting Metaphysician Undercover
To the extent that philosophers can't deal with mathematical practice as it is, they have no claim on such exaltation.
Quoting Metaphysician Undercover
I wish. I predicated nothing. I only struggled to learn what I was taught, and I'm reporting back to you how the subject works. "People like me." Jeez man what are you going on about?
The knight moves the way it does. Or as Galileo would have said: "Yet it moves."
/
In mathematical logic, of course, any set of formal sentences is an axiomatization of a formal theory. But many (maybe the preponderance of) mathematicians regard the axioms for particular fields of study not to be merely arbitrary, but rather as meaningful and true.
True. The degree of meaningfulness in direct proportion to the number of hours spent working in that discipline. In real and complex analysis one takes (as non-arbitrary?) and axiomatic the definitions of convergence , continuity, etc. given by several mathematicians, including Cauchy and Weierstrass (my math genealogy ancestor - along with 36K other descendants). :cool:
That's pretty cool!
This is not true. You've been making arguments about "pure math", and "pure abstractions". So it is you who is making a division between the application of mathematics, "how modern math works", and pure mathematics, and you've been arguing that pure mathematics deals with pure abstractions. You've argued philosophical speculation concerning the derivation of mathematical axioms through some claimed process of pure abstraction, totally removed from any real world concerns, rather than the need for mathematics to work. So your chess game analogy is way off the mark, because what we've been discussing here, is the creation of the rules for the game, not the play of the game. And, in creating the rules we must rely on some criteria.
Nowhere do I dispute the obvious, that this is "how modern math works". That is not our discussion at all. What I dispute is the truth or validity of some fundamental principles (axioms) which mathematicians work with. This is why the game analogy fails, because applying mathematics in the real world, is by that very description, a real world enterprise, it is not playing a game which is totally unrelated to the world. So the same principle which makes playing the game something separate from a real world adventure, also makes it different from mathematics, therefore not analogous in that way
I have a proposal, a way to make your analogy more relevant. Let's assume that playing a game is a real world thing, *as it truly is something we do in the world, just like scientists, engineers and architects do real world things with mathematics. Then let's say that there are people who work on the rules of the game, creating the game and adjusting the rules whenever problems become evident, like too many stalemates or something like that. Do you agree that "pure mathematicians" are analogous to these people, fixing the rules? Clearly, the people fixing the rules are not in a bubble, completely isolated from the people involved in the real world play. Of course not, they are working on problems involved with the real world play, just like the "pure mathematicians" are working on problems involved with the application of math in science and engineering, etc.. Plato described this well, speaking about how tools are designed. A tool is actually a much better analogy for math than a game. The crafts people who use the tool must have input into the design of the tool because they know what is needed from the tool.
In conclusion, your claim that "pure mathematicians" are completely removed from the real world use of mathematics is not consistent with the game analogy nor the tool analogy. Those who create the rules of a game obviously have the real world play of the game in mind when creating the rules, so they have a purpose and those who design tools obviously have the real world use of the tool in mind when designing it, and the tool has a purpose. So if mathematics is analogous, then the pure mathematicians have the real world use of mathematics in mind when creating axioms, such that the axioms have a purpose.
Quoting fishfry
The problem is that you have been "reporting" falsely. You consistently claimed, over and over again, that "pure mathematicians" work in a realm of pure abstraction, completely separated, and removed from the real world application of mathematics, and real world problems. That is the substance of our disagreement in this thread. My observations of things like the Hilbert-Frege discussion show me very clearly that this is a real world problem, a problem of application, not abstraction, which Hilbert was working on. And, the fact that Hilbert's principles were accepted and are now applied, demonstrates further evidence that Hilbert delivered a resolution to a problem of application, not a principle of pure abstraction.
Quoting fishfry
I can't say that I see what I'm supposed to comment on. The geometry used is the one developed to suit the application, it's produced for a purpose. With the conflation of time and space, into the concept of an active changing space-time, Euclidean geometry which give principles for a static unchanging space, is inadequate. Hence the need for non-Euclidean geometry in modern physics.
Quoting fishfry
This is why there is a need for solid ontological principles, an understanding of the real nature of time, the real nature of space. Only through such an understanding will the proper geometry be developed.
This is why it makes no sense to place the "pure mathematician" in a completely separate realm of "pure abstraction". The "pure mathematician" could dream up all sorts of different geometries, and have none of them any good for any real purpose, if the "pure mathematician" had absolutely no respect for the real nature of space.
Quoting fishfry
Sorry fishfry, but this is evidence for my side of the argument. "The unreasonable effectiveness of math" is clear evidence that the mathematicians who dream up the axioms really do take notice, and have respect for real world problems. That's obviously why math is so effective. If the mathematicians were working in some realm of pure abstraction, with total disregard for any real world issues, then it would be unreasonable to think that they would produce principles which are extremely effective in the real world. Which do you think is the case, that mathematics just happens to be extremely effective in the real world, or that the mathematicians who have created the axioms have been trying to make it extremely effective?
Quoting fishfry
Our discussion, throughout this thread has never been about "how math works". We have been discussing fundamental axioms, and not the application of mathematics at all. You are now changing the subject, and trying to claim that all you've been talking about is "how math works", but clearly what you've been talking about has been pure math, and pure abstraction, not application.
Ok, so the conversation is shifting now to arguing about what we've been arguing about. Slow day at Chez fishfry, so I'll play. But full disclosure, my heart's not in it.
Quoting Metaphysician Undercover
Arguing in the sense of describing to you how modern math sees certain things, such as sets and order relations.
Quoting Metaphysician Undercover
Me personally? You give me too much credit. Those divisions were there long before I was born. I am just talking about them. But actually we've never been talking about pure versus applied math at all, I don't know where you're getting this. Applied math is the use of math is fields like physics, economics, biology, and so forth. We haven't been talking about that at all. We've only ever been talking about pure math. The meaning of 2 + 2, the nature of sets, the nature of order relationships, how mathematicians formalize things.
Quoting Metaphysician Undercover
LOL. I'd be glad to argue that any day. Pure math deals with pure abstractions? What's your counter proposition? That's like saying barbering deals with cutting hair. There is no sensible negation to the proposition.
Quoting Metaphysician Undercover
On the contrary, I have argued that the choice of mathematical axioms is pragmatic. Possibly not that much in our convos but in general. The axioms are chosen because they let you build up good theories above them. I've always argued that. But "real world" concerns are not involved, that I agree with.
Quoting Metaphysician Undercover
My chess analogy is perfectly apt. By what criterion is the rule for how the knight moves chosen? Why is the lower right-hand square always white ("white on the right") and never black as is often erroneously portrayed by careless prop artists in movies?
Quoting Metaphysician Undercover
Now THAT is funny. You do nothing but, starting from "2 + 2 does not denote the same mathematical object as 4," several years ago, right up to the present moment. You constantly dispute the obvious.
Quoting Metaphysician Undercover
For the past several weeks I've been explaining to you that mathematical sets have no inherent order and you've been arguing that this is somehow "wrong." So we have definitely been discussing "how modern math works."
Quoting Metaphysician Undercover
How can you argue with the truth of things that are not claimed to be true? Nobody claims that the axiom of replacement or the axiom of powersets is true. NOBODY says that. This is your own personal strawman. And it's a tremendous misunderstanding on your part that anyone is claiming the axioms of math are "true." The axioms are strings of formal symbols, true in some models and false in others. Your failure to comprehend this is a great failing of yours.
Quoting Metaphysician Undercover
I have never had any interest in applying math to the real world. I wonder why you think I do, or should? I'm with the great British mathematician G. H. Hardy, who argued in his great essay, A Mathematician's Apology, that the beauty of an area of math is measured by how utterly useless it is; and that by this criterion his own field, that of number theory, is the most supremely beautiful area of math. How ironic, then, that number theory, which was supremely useless for 2000 years, has in only the past few decades become the core technology behind Internet security and cryptocurrencies. Hardy would spin in his grave. Hardy was played by Jeremy Irons in The Man Who Knew Infinity, highly recommended. A very rare math film that gets the math right and tells a great human story too, the tragic story of Ramanujan. A must-see for all readers of this site.
Pure math is not about the real world. Now you may not like that, and you math think it "should" be otherwise, but I am only telling you how it is. You can't argue with me about that. I don't know why you persist in trying.
Quoting Metaphysician Undercover
It's entirely analogous. Chess is a formal game, there's no "reason" why the knight moves as it does other than the pragmatics of what's been proven by experience to make for an interesting game. And there are equally valid variations of the game in common use as well.
Quoting Metaphysician Undercover
Perhaps you're thinking of engineers.
Quoting Metaphysician Undercover
My God man, pure mathematicians are not concerned with the problems of the world. And when they are, they are doing applied math, not pure.
Quoting Metaphysician Undercover
Math is justified only by itself. That the physicists find it useful is good for them. It's not what drives math. You truly don't understand this. Now you are perfectly entitled to argue that things SHOULD be different. But you can't credibly argue that they ARE different, because they are not. Just ask a physicist about math, they'll tell you the mathematicians are off in the clouds totally untethered from the real world. As if that's a bad thing!
Quoting Metaphysician Undercover
You didn't move me with such a weak and fallacious argument. Your argument that math is concerned with the real world was true a thousand years ago, but has not been true for a long time.
Quoting Metaphysician Undercover
By your ignorant measure. As measured by reality, I've been reporting accurately.
Quoting Metaphysician Undercover
Yes.
Quoting Metaphysician Undercover
Hilbert's famous quote is that ""One must be able to say at all times--instead of points, straight lines, and planes--tables, chairs, and beer mugs"" I suggest you meditate on this point.
Of course Hilbert did some applied math too. He famously offered to help Einstein finish general relativity, only for Einstein to discover, almost too late, that Hilbert was trying to finish first and gain credit. Fortunately Hilbert had an error in his calculations and in the end, graciously conceded priority to Einstein. Otherwise we'd say to this day, "He's no Hilbert!" instead of "He's no Einstein."
Quoting Metaphysician Undercover
Yet both theories are internally consistent. So math alone can't determine truth. I believe you've conceded my point.
Quoting Metaphysician Undercover
Of course math is inspired by the world. It's just not bound by it. A point I've made to you a dozen times by now.
Quoting Metaphysician Undercover
News to me.
Quoting Metaphysician Undercover
I agree with that. I've never had the slightest interest in applied math. I'm with Hardy. Math is worthwhile to the extent that it's useless. Of course he was being a bit facetious, I suppose. And in the end even his belovedly useless number theory came to be indispensable to the world.
Quoting Metaphysician Undercover
LOL. Now that's funny. As if I've ever been talking about anything else.
@Meta surely this convo has run its course, don't you agree?
Wow! I am really impressed to realize Omar Khayyam (1048-1131) had the perspicacity to realize his efforts at Non-Euclidean geometry involved notions of space-time. Thanks, MU. I would not have guessed. :chin:
Many mathematicians and philosophers of mathematics regard certain axioms and theorems to be true not just relative to models. It might even be the dominant view.
My sense of the matter is as follows. The overwhelming majority of working mathematicians are not set theorists or involved in foundations. They pay no attention to set theory and would be hard-pressed to even name the axioms. It's not like your average anabelian geometer ever gives explicit though to the truth or falsity of the axiom of replacement. The question doesn't come up.
Among those who study foundations, it must be abundantly clear that the axioms are arbitrary and not literally true, since it's consistent to accept or deny Foundation, Powerset, and other axioms that are never questioned in standard math. Powerset negation is its own cottage industry these days, even though it's an extremely niche interest from a mainstream point of view.
It's hard for me to believe that anyone thinks the axioms of set theory are literally true about the world; or even about the abstract world of mathematics. There may be a few.
So at best I would say that "a few" mathematicians claim the axioms are literally true in some sense.
The foremost philosopher of set theory, Penelope Maddy, argues persuasively that the axioms are chosen pragmatically, on a variety of practical grounds. See https://www.cs.umd.edu/~gasarch/BLOGPAPERS/belaxioms1.pdf for example.
Among philosophers, who could seriously argue that the axioms of set theory are "true" in any meaningful sense; or even meaningful in any meaningful sense! I am less familiar philosophy than math, but my sense is that just as with mathematicians, most philosophers aren't concerned with the axioms of set theory at all, let alone their truth.
That's my sense of the matter. I'd go further and say that it's perfectly clear that some of the axioms, such as the axiom of infinity, are literally false. That strengthens my point to @Meta, which is that the axioms are chosen pragmatically for their utility in developing math, and not for any real-world reasons.
I would add to all that the growing importance of neo-intuitionist, constructivist, and category-theoretic approaches, in which set theory is not only false, but irrelevant. (Category theorists pay lip service to set theory by defining "small" or "locally small" categories where certain collections are required to be sets, and some of the category theorists do worry about such things, but in the mainstream of category theory, they'd don't worry too much about set theory).
But of course you said, "certain axioms," and I suppose if you want to say that high school notions like unions and intersections are true or instantiable in the real world, you'd have a point. At least for finite sets. But finite sets are not of much interest to set theorists. Finite sets belong to combinatorics.
I would expect that there is a wide range of interest in foundational axioms among mathematicians - from no interest to intense interest. But even among mathematicians with only little interest in foundations, there are those who regard certain axioms as true without having to quality by saying "only relative to models". For example, there are mathematicians who regard the Peano axioms as true, without saying "but only relative to certain models".
Moreover, as a special case, the logical axioms are true in every model, so those are axioms that would be said to be true without qualification as to models.
Quoting fishfry
The question whether mathematical truth is merely model-theoretic doesn't usually come up in studies outside foundations. But the question whether a given mathematical statement is true or false comes up all the time. Indeed, the fact that a great many mathematicians don't even care about foundations leads to them saying about certain axioms that they are true (while they don't qualify "but only in certain models"). The axioms adopted in the field of study are often regarded as true, even without bothering about model-theoretic truth. Indeed, for a good number of mathematicians, it is repugnant to view mathematics as just symbol study with the formulations not expressing mathematical truths unqualified by models.
Quoting fishfry
What you say must be is just not the case. There are indeed mathematicians in foundations and philosophers who regard certain axioms as true without having to add "but only in certain models". This is not being ignorant of model-theoretic truth, but rather to view that there are other senses of truth - mathematical realism (Godel being the most famous), operational, et. al, or even just a naive sense in which mathematicians regard certain axioms and theorems to express truths in their fields of study, ranging from concrete finitary truths to greater abstractions.
Quoting fishfry
Then that is at lease some movement from 'NOBODY' [all caps original]. Of course for an accurate quantification, we would need an accurate poll. But from my readings in mathematics, from conversations with mathematicians, and from reading posts of mathematicians, I have found that there are many who regard certain axioms and theorems to be true without having to qualify to models. And my impression, though not derived by polling, is that that is the case for most mathematicians. Ask some mathematicians "Is the fundamental theorem of arithmetic true?" Then when the answer is 'yes', ask "Do you mean it's true only in certain models of the language of arithmetic, or do you mean it's just true"? I bet you dollars to donuts that most would answer 'just true' to the second question.
Also, adding "literally" narrows your first claim. There are notions of truth including, model-theoretic, literal, realism, operational, true-to-a-concept, et. al.
Quoting fishfry
You may think it not wise to think that they are true or meaningful in a sense other than model-theoretic, but that does not entail that no mathematicians or philosophers (or even only a few) have that view. Indeed, for example, mathematical realism, broadly put, is the view that there are mathematical objects and truth about them independent of consciousness about them. Ordinarily, mathematical realism would regard that there are true axioms that are true even without having to qualify "but only relative to certain models". For a good number of mathematicians and philosophers, they recognize that sentences that are not validities are not true in every model, but they still regard certain axioms to be true in a sense other merely model-theoretic..
Quoting fishfry
I don't know what your definition of 'literal' is when it comes to mathematics, but you are of course entitled to your own view about certain axioms, but that does not entail that no (or even only few) mathematicians share your view.
Quoting fishfry
Yet people do work in constructivist and intuitionist set theory. I don't know a lot about category theory, but it can be axiomatized by ZFC+Grothendieck-universe.
Quoting fishfry
I mean only that there may be axioms some mathematicians don't believe to be true. For example, some mathematicians might regard the axioms of Peano arithmetic to be true but not, the axiom of choice, or whatever. I don't mean to say that those axioms that mathematicians do believe to be true don't include set theory axioms such as power set, schema of replacement, infinity, choice, and even some who believe that the continuum hypothesis is true and some who believe the negation of the continuum hypothesis is true, or certain large cardinal axioms.
Your claim was that 'NOBODY' [all caps original] believes axioms are true without model-theoretic qualification. Now it is that at best only a few believe axioms are "literally" true. 'literal' is not defined yet in this conversation as to mathematical truth, and we should expect that different mathematicians would have different definitions if you forced them to respond to it, but it is not the case that only a few mathematicians and philosophers hold that there is no mathematical truth other than model theoretic, and, it is at least my impression that most mathematicians and a fair number of philosophers do hold that there is mathematical truth other than the model-theoretic sense.
Which requires the existence of an inaccessible cardinal, the existence of which is not even provable in ZFC.
We're arguing over what other people think, we can't ever get to the bottom of that. @jgill posted a while back about the tiny percentage of overall math papers that are devoted to set theory. Few working mathematicians give any of these matters the slightest thought. In what sense could the Peano axioms be true in the real world? There are only [math]10^{78}[/math] hydrogen atoms in the observable universe.
Yes, which makes it even more curious what one would mean by saying the axioms of ZFC are false, while proposing a theory that is equivalent to ZFC PLUS another axiom.
Quoting fishfry
That doesn't entail that a lot mathematicians aren't aware of axioms, including those not of set theory and those of set theory. And, again, probably most mathematicians don't get hung up on mathematical logic and its model theoretic sense of truth, yet mathematicians speak of the truth of mathematical statements.
And it's not even a given that only a few mathematicians who do understand models in mathematical logic hold that there are other senses of truth, including realism, instrumental, true-to-concept, et. al. Indeed, we know that there are mathematicians who well understand mathematical logic but still regard a sense of truth no restricted to that of "true in a model".
Quoting fishfry
You made a clam about it. We don't have a scientific polling, but we can see that there are many people who don't think that mathematical truth is confined only to the model-theoretic sense.
Quoting fishfry
So 'real world' is now added to the question.
Again, that you view certain notions about mathematics to be untenable doesn't entail that there are not plenty of people who don't share your view
The axioms aren't false, either, any more than the way the knight moves in chess may be said to be true or false. It's just a rule that's been found by experience to make the game interesting.
Quoting TonesInDeepFreeze
I truly can't argue about what the majority or substantial plurality or "some" or "a few" or whatever mathematicians believe. I have no data or evidence, neither do you. But the subject matter that most mathematicians work on, as evidenced by the number of papers published, is so far removed from foundations that I can't imagine that many mathematicians spend five minutes thinking about the subject in a year or in a career.
Quoting TonesInDeepFreeze
We can talk math, or we can talk philosophy of math, but arguing popular opinion is not fruitful. What outcome are you looking for? Would you like me to go from "a few" to "a whole bunch?" I'm not sure what outcome would satisfy you. This is not a meaningful conversation.
Quoting TonesInDeepFreeze
That's what true and false typically mean. The axiom of infinity is manifestly false about the real world. At the very least it's inconsistent with contemporary physics. But it's an essential axiom of standard mathematics. Perhaps you can put your concept of truth into context for me such that the axiom of infinity could be regarded as even having a meaningful truth value other than it being generally accepted as an axiom of modern set theory.
Quoting TonesInDeepFreeze
"Plenty." Ok I can live with that. If, given that there must be 100,000 or so math professors in the world, I concede that "plenty" of them believe whatever you say they believe, would that satisfy you? Respected mathematician Alexander Abian wanted to blow up the moon; and prolific author of high-level math texts Serge Lange was an AIDs denier. The Unabomber had a doctorate in math, as did the guy who swindled the CIA during the Iraq war, Ahmed Chalabi. Mathematicians are human, they believe all sorts of things.
I'll go with "plenty" if this will mollify your sense of right and wrong here.
A progression of views (not necessarily your own):
(1) "Hilbert said that mathematics is only a meaningless game of manipulating symbols."
False. Hilbert was very much concerned with the contentual aspect of mathematics.
(2) "Al mathematicians view mathematics as only a meaningless game of manipulating symbols."
Clearly false.
(3) Formalism in mathematics is the view that mathematics is only a meaningless game of manipulating symbols.
False.
(4) There is a form of extreme formalism that views mathematics as only a meaningless game of manipulating symbols.
True.
(5) All mathematicians and philosophers hold that truth in mathematics pertains only to truth per models in mathematical logic.
Clearly false.
(6) All mathematicians and philosophers who understand truth per models hold that there are no viable senses of mathematical truth other than that of models.
Clearly false.
(7) Most mathematicians and philosophers who understand truth per models hold that there are no viable senses of mathematical truth other than that of models.
Not known. My impression is that it is false, but would deserve a poll.
(8) Among mathematicians who know nothing, or very little, about models in mathematical logic, all (with possibly only few exceptions) regard axioms (incuding Peano, set theory) as true only with regard to models.
False, essentially a contradictory claim.
(9) Among mathematicians who know nothing, or very little, about models in mathematical logic, only a few are familiar with the set theory axioms.
Not clear. My guess is that it is false.
(10) Of those mathematicians who are familiar with the axioms of set theory, all (with possibly only a few exceptions) view the axioms as false.
Almost surely false.
(11) Of those mathematicians who are familiar with the axioms of set theory, all (with possibly only a few exceptions) view the axioms as meaningful only as syntactic objects for syntactically proving other syntactic objects.
Clearly false.
/
In any case, Quoting fishfry is false.
Well then I don't feel bound to justify them. I'll let you have the last word on almost all of this. The one thing I'd like you to explain to me is that if you deny that the axiom of infinity is "manifestly false about the real world," which is a statement I actually DID make, in what sense to you regard it as physically true? Or if not physically true, how is it meaningful to say it's either true or false in some other sense, Platonic, formal, or otherwise? How is the axiom of infinity different than the way the knight moves?
Here is Maddy (linked above) quoting Hallet about infinity:
This is a pragmatic argument. Ontologically we could do without the axiom of infinity. We adopt it on purely pragmatic grounds, in order to get a decent theory of the real numbers. That makes it neither true nor false, words that are not meaningful in this context; but rather useful, which is my position on the matter.
That's your view. My point is not nor has been to convince you otherwise. Rather, my point is that no matter that it may be your view, it is not true that nobody (or only a few) people disagree with it.
Quoting fishfry
Fine. And so there's not basis to claim that nobody (or merely a few) views axioms as true in a sense other than relative to models.
Quoting fishfry
I have evidence from writings, conversations, and posts. From those, it is manifestly clear that it is false that "Nobody claims that the axiom of replacement or the axiom of powersets is true. NOBODY [all caps in original] says that. [,,,] [no one] is claiming the axioms of math are "true."" Then, as to what the majority of mathematicians believe, I've stated my impression based on what I have read and heard from mathematicians, while I've said that of course that impression is not scientific.
Quoting fishfry
Again, that is the wrong road of argument for your position. I don't doubt that the vast majority of mathematicians don't care about foundations, in particular the model-theoretic notion of truth. But that only adds to my argument, not yours. Clearly, commonly mathematicians speak of the truth of mathematical statements, and even many mathematicians not occupied with foundations understand axioms in their field of study and often enough even the set theory axioms. So when such mathematicians say things like "the fundamental theorem of arithmetic" is true, then they don't mean it as "the fundamental theorem of arithmetic is true only in the sense that it is derivable in a consistent formal theory so that it is true in some models".
Quoting fishfry
It's become a point of contention only because I responded to your claim about it, and not just in popular opinion, but your claim of totality of opinion.
Quoting fishfry
I don't care what you go to. I am making my own point that it is not the case that NOBODY (or even only a few) people regard axioms as true other than model-theoretically.
Quoting fishfry
If it was meaningful for you to make the claim, then it is meaningful for me to reply to it, and to reply to your replies.
Quoting fishfry
Views of mathematical truth don't have to be limited to what is typical otherwise. Whether or not departures from "typical" are justified, my main point was that it is not the case that all (or nearly all) mathematicians regard truth as merely model-theoretic.
Moreover, some mathematicians do regard certain mathematical statements in what is arguably a typical sense of finitary combinatory statements being concrete and true. And, as I mentioned, validities are true no matter what the models.
Quoting fishfry
That's your view. But it doesn't refute my point that it is not the case that all (or nearly all) mathematicians and philosophers regard axioms as true only as pertains to models.
Quoting fishfry
It doesn't matter toward my point. I have not claimed nor disagreed with any notion of truth. I don't have to just to point out that it is not the case that nobody regards axioms as true except relative to models. This reminds me of an article I read today. The writer claimed that nobody finds Colbert funny. I don't have to opine whether Colbert is funny to point out that it is false that nobody laughs at his jokes.
Quoting fishfry
That opens another question. Whether one agrees with notions of mathematical truth other than model-theoretic, I'd be inclined not to claim that thinking philosophically or heuristically of mathematical truth as rather than model theoretic is among the ilk of proposing detonation of the moon or claiming that AIDS doesn't exist.
I neither denied it nor affirmed it.
Two different things: (1) "P is the case" and (2) "Nobody claims that ~P is the case".
Today when I read "Nobody thinks Colbert is funny", my first thought was not "But Colbert is funny" nor "I agree that Colbert is not funny", but rather how ludicrous it is to start an opinion article about American society with such a manifestly false claim as "Nobody thinks Colbert is funny."
If 'to make interesting' includes 'to provide an axiomatization of the mathematics for sciences'.
I have conceded the word "plenty." I can't continue to argue with you about what (a few, some, many, a strong plurality, a majority, an overwhelming flood) of people think. I won't respond any more to that subject. If you want to talk about whether the axiom of infinity may be meaningfully said to be true or false, that's a good conversation. If you want to argue about what people think, I can't engage on that anymore. Having conceded the word "plenty" already, I would think you would be happy, and that's as far as I'll go.
Quoting TonesInDeepFreeze
99% of professional mathematicians are not involved in foundations (more or less objective number, I didn't look it up but recall @jgill's post regarding the percentage of recently published papers) and therefore have no professional opinion on the subject at all.
Quoting TonesInDeepFreeze
Enough. No more of this for me.
Quoting TonesInDeepFreeze
No más, por favor
Quoting TonesInDeepFreeze
You made your point then got tedious and are now beyond that.
Quoting TonesInDeepFreeze
I think you've expressed yourself with sufficient conviction on the matter.
Quoting TonesInDeepFreeze
Perhaps you are taking things a bit too literally.
Quoting TonesInDeepFreeze
As you've said.
Quoting TonesInDeepFreeze
Enough. Please.
Quoting TonesInDeepFreeze
You have pointed it out.
Quoting TonesInDeepFreeze
Ok!! I'm glad to change the subject.
Quoting TonesInDeepFreeze
The question of foundations is as far from the practice of most mainstream mathematicians as blowing up the moon or AIDs denialism. If someone is classifying the finite simple groups, they are not thinking about the axiom of replacement.
Quoting TonesInDeepFreeze
It would be fun if you did, then we could have a conversation.
Quoting TonesInDeepFreeze
The former being interesting, the latter tedious beyond belief.
Quoting TonesInDeepFreeze
So you didn't change the subject after all.
Quoting TonesInDeepFreeze
This could never be true. Physics has not been axiomatized at all. They can't even reconcile quantum mechanics and relativity. And the idea that set theory could ever be a foundation for physics seems to me to be an unlikely stretch. But at least that is an interesting and substantive topic in the philosophy of math and science.
I would say that if someone asks, "Is it meaningful to ask if the axiom of infinity is true or false; and if so, which?" I would be willing to argue any side of that. That the axiom of infinity both is and isn't meaningfully true or false; or that if it is, it's true; or that if it is, it's false. I could whip up a good argument for each of those three propositions meaningless, meaningful/true, and meaningful false.
I don't see why this is a problem for you. You hand me a proposition, and I refuse to accept it, claiming that it is false. You say, 'but I am not claiming that it is true'. So I move to demonstrate to you why I believe it to be false. You still insist that you are not claiming it to be true, and further, that the truth or falsity of it is irrelevant to you. Well, the truth or falsity of it is not irrelevant to me, and that's why I argue it's falsity hoping that you would reply with a demonstration of its truth to back up your support of it.. If the truth or falsity of it is really irrelevant to you, then why does it bother you that I argue its falsity? And why do you claim that I cannot argue the falsity of something which has not been claimed to be true? Whether or not you claim something to be true, in no way dictates whether or not I can argue its falsity.
Quoting fishfry
Either you are not getting the point, or you are simply in denial. Playing chess, is a real world activity, as is any activity. Your effort to describe an activity, like the game of chess, or pure math, as independent from the real world, as if it exists in it's own separate bubble which is not part of the world, is simply a misrepresentation.
Now, you admit that there actually is a pragmatic reason why the knight moves as it does, and this is to make an interesting game. And of course playing a game is a real world activity. so there is a real world reason for that rule. Now if you could hold true to your analogy, and admit the same thing about pure mathematics, then we'd have a starting point, of common agreement. However, the reason for mathematical principles beings as they are, such as our example of the Pythagorean theorem, is not to make an interesting game. It is for the sake of some other real world activity. Do you agree?
Quoting fishfry
You keep on insisting on such falsities, and I have to repeatedly point out to you that they are falsities. But you seem to have no respect for truth or falsity, as if truth and falsity doesn't matter to you. Mathematics has been created by human beings, with physical bodies, physical brains, living in the world. It has no means to escape the restrictions imposed upon it by the physical conditions of the physical body. Therefore it very truly is bound by the world. Your idea that mathematics can somehow escape the limitations imposed upon it by the world, to retreat into some imaginary world of infinite infinities, is not a case of actually escaping the bounds of the world at all, it's just imaginary. We all know that imagination cannot give us any real escape from the bounds of the world. Imagining that mathematics is not bound by the world does not make it so. Such a freedom from the bounds of the world is just an illusion. Mathematics is truly bound by the world. And when the imagination strays beyond these boundaries, it produces imaginary fictions, not mathematics. But you do not even recognize a difference between imaginary fictions, and mathematics.
Quoting fishfry
The reason why I can truthfully say that our discussion has never been about how math works, is that you have never given me any indication as to how it works. You keep insisting that mathematical principles are the product of some sort of imaginary pure abstraction, completely separated from the real world, like eternal Platonic Forms, then you give no indication as to how such products of pure fiction become useful in the world, i.e., how math works.
Quoting jgill
As I said, one can produce any sort of geometry depending on the particular purpose. My reference to space-time was in reply to fishfry's talk of a specific incidence, the use of non-Euclidian geometry in modern physics
I don't seek to be assuaged. You don't need to assent to 'plenty' on my account. Rather, one can assent to it merely on the grounds that it is obvious.
Quoting fishfry
Yet you write:
Quoting fishfry
You continue to miss the point. That a vast number of mathematicians don't care about foundations doesn't imply that the vast number of mathematicians don't think axioms are true except model-theoretically, as indeed the fewer who care about foundations then reasonably we would expect the fewer who think truth is merely model-theoretic.
Quoting fishfry
It may have been tedious, but my entries have been responses to your continued posting on the subject. It seems odd to me that one would continue to reply, and repeat some points that are essentially the same, but then complain that responses on point to your own replies are thereby tediousness.
Quoting fishfry
Whether that is the case, my point is that having a foundational view that there is mathematical truth other than model-theoretic is not remotely outlandish in the class of advocating that we destroy the moon or that AIDS does not exist.
Quoting fishfry
You claimed the latter, so it is reasonable to reply to it whether you find that tedious or not.
Quoting fishfry
I addressed the additional matter of outlandishness; I didn't thereby declare that I am forever changing any subject or not changing it, especially as you continued to post as if I had not already made clear that the question of what people think is distinct from whether they are wise to think it.
Quoting fishfry
I didn't say "axiomatization of physics". I said "axiomatization of the MATHEMATICS for the sciences" [all-caps added]. Of course, though there is some consensus that set theory does axiomatize the branches of mathematics needed for the sciences, one may question whether indeed all of the needed mathematics is captured. But even a negative answer would not refute my point that among the salient reasons for adopting the axiom of infinity, at least we may say those reasons include an intent to lend support to axiomatizing the mathematics for the sciences, which is far beyond merely adding it to make things interesting. Also, I don't know that physics has not been axiomatized "AT ALL" [all caps added].
I used a figure of speech called hyperbole. You pointed out repeatedly that what I said was not literally correct. I conceded that "plenty" of mathematicians disagree with what I said. Yet this is still not enough for you. What more do you want?
Quoting TonesInDeepFreeze
I used a figure of speech called hyperbole. You pointed out repeatedly that what I said was not literally correct. I conceded that "plenty" of mathematicians disagree with what I said. Yet this is still not enough for you. What more do you want?
Quoting TonesInDeepFreeze
I used a figure of speech called hyperbole. You pointed out repeatedly that what I said was not literally correct. I conceded that "plenty" of mathematicians disagree with what I said. Yet this is still not enough for you. What more do you want?
Quoting TonesInDeepFreeze
I'm sure the standard axiomatization of math is an overkill for that.
Quoting TonesInDeepFreeze
You don't know at all if it's been axiomatized ? That's something that can be looked up. Or you don't know if even small parts of it have been axiomatized? Your sentence was a little ambiguous. I'm sure there are axiomatizations of parts of science. Newtonian gravity has a nice axiomatization in Newton's three laws.
I don't know what hyperbole you have in mind. Maybe 'nobody'. Because you seemed adamant with all-caps, and, as I recall, three variations of 'no', I didn't know it was hyperbole. So I merely replied to it at face value. Of course I would not have begrudged you then declaring it was only hyperbole. But still, I don't think what was hyperbolized was correct, even if given non-hyperbolized restatement.
Anyway, your response again misses my point. My point that you just quoted is not to take issue with your hyperbole, but rather to point out how your more recent argument goes wrong.
Quoting fishfry
Okay, but my point quoted above was not about that.
Quoting fishfry
Do you mean the hyperboles "blow up the moon" and "AIDS denier"? If so, that's fine that you say now it was hyperbole. But I did take your comments at least to be a claim that a view that mathematical truth is not confined to model-theoretical is on its face preposterous even outlandish. I said that a lot of mathematicians don't view truth as merely model-theoretic, and you replied to the effect that there are intellectually talented people who believe a number of crazy things. It is reasonable for me to say that believing that truth is not merely model theoretic is not that kind of crazy, if it is even crazy at all.
Quoting fishfry
Some set theorists have pointed how we can reduce some axiomatic assumptions and still get the mathematics for the sciences. And even if ZFC is too productive, that doesn't refute that a good part of the interest in the axiom of infinity is to axiomatize (even if too productively) the mathematics for the sciences.
Quoting fishfry
No, not that I don't at all know. Rather, I don't know that it hasn't been axiomatized at all (i.e. hasn't been axiomatized to any extent whatsoever or with no progress toward axiomatizing it). That was in response to what you wrote, "Physics has not been axiomatized at all."
I am on record as holding that the axioms of set theory are neither true nor false, as they are syntactic entities whose truth or falsity can only be determined after an interpretation, or model, is provided. This is perfectly in keeping with standard practice in mathematical logic.
After all the "Abelian axiom" that xy = yx is true in the real numbers, and false in the set of 2x2 matrices whose entries are real numbers. "It is snowing" is true in Alaska in the winter, but never in San Diego. It's not possible for the axiom of replacement to be true OR false in isolation from an interpretation.
Quoting Metaphysician Undercover
Because it shows that you misunderstand the distinction between syntax and semantics, between a formal axiomatic system and its models. So if you say an axiom is true you're wrong, and if you say it's false you're wrong! An axiom isn't true or false. Now if you would supply a model, I can tell you whether it's true in that model.
Quoting Metaphysician Undercover
I couldn't talk you out of arguing that 2 + 2 and 4 represent distinct mathematical objects. I suppose I shouldn't bother with the axiom of replacement, which actually is a bit of a subtle and powerful axiom schema.
Quoting Metaphysician Undercover
Or, logically, you are in one of those two states.
Quoting Metaphysician Undercover
Doing set theory is a real world activity too, done by set theorists and undergrads the world over. Even high school students get a watered-down version of it. So my analogy holds. You're trying to say chess is "real world" because you can sit at a board and move the pieces. But it's still a formal game. You're being very disingenuous here. Sure the pieces are made of atoms, but there is no fundamental physical reason why the knight moves that way. And sure, set theorists are made of atoms too, but there is no fundamental physical reason to adopt or reject the laws of set theory.
Quoting Metaphysician Undercover
Yes yes yes yes. That is correct. We are in agreement. People play chess for years and find that some rules give more interesting versions of the game than others. Who invented en passant pawn captures, or castling? These are obviously historically contingent developments, introduced for purely pragmatic reasons.
As are, thinking ahead here, each and every one of the axioms of set theory.
Quoting Metaphysician Undercover
But I do. If you drop a mathematician from a height, he or she will fall in accordance with gravitational acceleration. But the axioms of set theory are historically contingent, pragmatically derived, matters of agreement. Like traffic lights. Red and green wavelengths are laws of nature. Which means go and which means stop is a social agreement. One which, if you violated it, can be fatal; but a social agreement nonetheless.
Quoting Metaphysician Undercover
I agree that math is different from chess in that math is inspired by the real world (ancient bookkeeping and surveying), and has vast applications in the real world. I certainly agree that math is subtly different and that generations of philosophers have tried hard to put their finger on exactly what that means. Ideas like indispensability and so forth. Of course I agree with this point.
Quoting Metaphysician Undercover
That math is inspired by the world and not bound by it? To me this is a banality, not a falsehood. It's true, but so trivial as to be beneath mention to anyone who's studied mathematics or mathematical philosophy.
Quoting Metaphysician Undercover
Since you're wrong on this point, repetition doesn't help. If you were right, you'd only have to say it once. That echoes Einstein's remark on being told that a hundred physicists disagreed with him. "If I'm wrong, one would be enough."
You actually disagree with the statement that "Math is inspired by but not bound by the world?" I propose to drill down on this because it's a clear point that we could discuss and perhaps shed some light. You disagree that math is inspired by the world? Or that it's not bound by it? I suspect you disagree with the latter. In which case I whip out non-Euclidean geometry as the classic example in support of my point.
Quoting Metaphysician Undercover
Not in axiomatic systems, no. Absent a model there is no truth or falsity.
Quoting Metaphysician Undercover
If you give me an axiomatic system plus a model, or interpretation, then truth or falsity can be determined, and matters to me.
Quoting Metaphysician Undercover
That's an interesting point. Yet you can see the difference between representational art, which strives to be "true," and abstract art, which is inspired by but not bound by the real world. Or as they told us when I took a film class once, "Film frees us from the limitations of time and space." A movie is inspired by but not bound by reality. Star Wars isn't real, but the celluloid film stock (or whatever they use these days) is made of atoms. Right? Right.
Quoting Metaphysician Undercover
Science fiction, abstract art, novels. Moby Dick is based on a true story of the Essex, a whaling ship sunk by a whale. But it's not bound by the story of the Essex. The characters and events are wholly made up. The point holds even more strongly for abstract art, science fiction, surrealism, and all other creative works of people.
Quoting Metaphysician Undercover
Because my pencil is made of atoms? Are you now taking the cranky anti-Cantorian position? Everything since 1870 is bullshit? Is this your stance? Are you like this at the art museum too?
Quoting Metaphysician Undercover
Maybe you just don't have enough imagination. You seem to be taking a radical realist position of some kind whereby science fiction and abstract art and Star Wars either don't exist or aren't real or are lies that should be banned. What exactly is your position here? How far will you take this stance that imagination has no place in the world?
Quoting Metaphysician Undercover
I recognize the difference between pure and applied mathematics. And you seem to reject fiction, science fiction, surrealist poetry, modern art, and unicorns. Me I like unicorns. They are inspired by the world but not bound by it. I like infinitary mathematics, for exactly the same reason. Perhaps you should read my recent essay here on the transfinite ordinals. It will give you much fuel for righteous rage. But I didn't invent any of it, Cantor did, and mathematicians have been pursuing the theory ever since then right up to the present moment. Perhaps you could take it up with them.
Quoting Metaphysician Undercover
Well that's Wigner's point in the Unreasonable Effectiveness paper. I don't claim to have the words or the philosophical background to give a good account of how math, which is perfectly obviously a massive fiction, can be so darn useful in the world. A lot of people have taken a shot at the question. Surely you know this.
You would not have begrudged me then, but you will begrudge the living bejeebus out of me now? LOL. Does the phrase, "Give it a rest," have any resonance with you?
Quoting TonesInDeepFreeze
I don't remember making any recent argument with you other than that it's pointless to argue about how many mathematicians believe this or that, and that in any event I have happily conceded that "plenty" of them disagree with what I wrote.
Quoting TonesInDeepFreeze
I may be lost by now. I have no idea what we're discussing.
Quoting TonesInDeepFreeze
No, those are literal facts of record. I supplied the relevant Wiki links.
Quoting TonesInDeepFreeze
I will stipulate that "plenty" of people don't think mathematical truth is merely model-theoretic. I have been stipulating this for some time, but evidently it's just not enough for you. I hereby stipulate that my use of "nobody" in that context was hyperbole, that I didn't mean for it to be taken literally; and that in retrospect I would have been better off saying that the "complement of plenty" of mathematicians have never given the matter five minutes of thought in their entire lives.
Do you find this satisfactory, or are further mea culpas necessary?
Quoting TonesInDeepFreeze
In other words its adoption is pragmatic. But ok your point was to include axiomatizing science in the discussion and I'm happy to agree. But of course that is the interesting point. The axiom of infinity is in contradiction with known physics; yet physics is based on the real numbers, and the theory of the real numbers requires the axiom of infinity. Truly it's a mystery. There is, by the way, a field of study in its very early stages called constructive physics, which attempts to build physics within the framework of constructive mathematics. Exactly to try to get around some of these issues. Not around infinity, necessarily, but around nonconstructive mathematics.
The field is new enough that there are only papers, and no Wiki entry. But at least a handful of people are thinking about the problem. Mathematics is way too much and way too false about the real world for it to serve as a suitable foundation for physics. That's my interpretation, not necessarily anyone else's.. But again this is Wigner's point, the UNREASONABLE effectiveness of math in the physical sciences. (There I go with the all caps again, a bad typographical habit for sure). Math is so "out there" that it's unreasonable that it finds such indispensable (Putnam & Quine's word) application in the world. It's a mystery.
Quoting TonesInDeepFreeze
LOL. You said, "I don't know that it hasn't been axiomatized AT ALL," your caps. Which could mean:
a) You don't know AT ALL if it hasn't been axiomatized; or
b) You don't know if it's been axiomatized AT ALL, as opposed to in its entirety.
I pointed out that parts of science have been very nicely axiomatized, such as Newtonian gravity. That would be an agreement with (b). Whereas (a) refers to the state of your knowledge.
But surely we can agree that ALL of science is very far from being axiomatized. At least I hope we can agree on that.
The concept of infinite infinities is already part of mathematics today. Therefore, in your dubious distinction between mathematics and “imaginary fictions”, your placement of infinite infinities on the side of "imaginary fictions" makes no sense; infinite infinities is already on the side of mathematics. Your attempted stipulations to the contrary are pointless.
No. I did not begrudge you hyperbole. Rather, (1) I explained why previously it was not unreasonable for me not to infer that you were writing hyperbolically, and (2) That even factoring for hyperbole, I disagreed with the non-hyperbolic claim behind the hyperbole.
Quoting fishfry
I don't see that when you continue to reply, with both repeated points and arguments and new points and arguments that I should then not also reply.
Quoting fishfry
The arguments are in your posts recently made.
Quoting fishfry
You quoted me:
Quoting fishfry
And to that you replied that you were being hyperbolic (presumably when you said that nobody takes as axioms as true other than model-theoretically). So I replied that your hyperbole is not connected to what you quoted by me.
Quoting fishfry
Quoting fishfry
Since the context was outlandishness, when you replied that you were being hyperbolic, my reply was wondering whether you meant that you were being hyperbolic when you compared "blow up the moon" and "no AIDS exists" to a view that mathematical truth is not limited to the model-theoretic. So, since that is not what you meant, now again, I don't know what connection you intended between my point about outlandishness and your having been hyperbolic in saying that nobody regards axioms as true except in the model-theoretic sense.
Quoting fishfry
I just explained very clearly that I meant that I don't know that physics has not been axiomatized to any extent. I'll say it yet another way: I don't know that there is no extent to which physics has been axiomatized.
(1) YOU said, "Physics has not been axiomatized at all". So I replied "I don't know that physics has not been axiomatized at all".
Obviously I'm replying to YOUR OWN sense of YOUR OWN statement "Physics has not been axiomatized at all". Especially in context, I didn't take YOUR OWN statement as ambiguous.
Quoting fishfry
You didn't mention Newtonian gravity in the post to which I first replied.
Here is the exchange:
Quoting fishfry
Especially in context, I took you to mean that there has been no progress in axiomatizing physics.
If someone says, "The house has not been built at all", then one would understand that to mean that there has been no progress in building the house, that no one has completed even the first phase in construction of the house.
So when you said, "Physics has not been axiomatized at all", I took you to mean the obvious sense that there has been no progress in axiomatizing physics.
If you don't mean that sense, then what sense did you mean when you said "Physics has not been axiomatized at all"?
I don't know that. The axiom of infinity says there is an inductive set and, with other axioms, entails that there is an infinite set. Set theory doesn't say that there is an infinite set of particles or that physical space extends infinitely outward or whatever. Also, is it definitively established that there are not infinitely many particles or that space does not extend infinitely outward?
I'm glad to know there's still hope for Wikipedia!
I want to bounce something off of you.
Scenario A (This universe):
1. Speed limit: 186000 miles per second (light speed)
2. If infinite energy is applied on an object, that object will attain light speed.
Scenario B (Another universe)
3. Speed limit: None! Go as fast as you can.
4. If infinite energy is poured into an object, that object will attain infinite speed
Basically, in terms of energy (infinite energy in both cases), there's no difference between light speed (this universe/scenario A) and infinite speed (another universe/scenario B).
Light speed then is an actual infinity. It's completed (1860000 miles per second) and it's equivalent to infinity (another universe/scenario B)
[s]As a side note a speed limit (light speed, 186000 miles per second in this universe) violates the law of conservation of energy. What's happening to the energy I'm exerting on an object if its speed doesn't change proportionately?[/s]
I don't know anything about physics. Nothing I've said here pertains to the physical universe.
2. If infinite energy is applied on an object, that object will attain light speed.[/quote]
There's no such thing as "infinite energy" as far as contemporary physics is concerned, nor is it sensible that an object could attain light speed by any means at all.
Sorry can't be of any assistance here, this is speculative physics and seems to contradict known physics.
No one I knew in my corner of the mathematical community had anything much to say about set theory or foundations, other than, perhaps, something along the lines of "Oh yeah, neat how Peano Axioms do that."
In my Intro to Grad Math course in 1962 when we got into the second half of Halmos Naive Set Theory most of us lost interest. I remember thinking, "chains, towers, etc.- sounds medieval".
If an inductive set that's not physical "exists," what does that mean to you? I've stated repeatedly to @Meta that it exists "mathematically," and he has correctly challenged me on this. I can play formalist and say it's just a formal game, like chess. I concede that this is not entirely satisfactory. If you play Platonist, I can ask you where your inductive set lives, and what else lives there with it? The Baby Jesus? The Flying Spaghetti Monster? Captain Ahab? Platonism is full of conceptual problems. As is formalism. But you're the one claiming that an inductive set "exists," so I would ask you what you mean by that.
Quoting TonesInDeepFreeze
It's unknown, but disallowed by all physical theories except the highly speculative ones like Eternal Inflation, positing infinite time and an actual infinity of universes. Even that theory's inventor no longer believes in it, but the papers keep getting published anyway. Physics is in a heck of a state these days. Einstein and the other early 20th century giants cared about ontology. Modern physicists lose their careers for talking about ontology. Sad state of affairs.
Quoting TonesInDeepFreeze
"I explained why previously it was not unreasonable for me not to infer that you were writing hyperbolically" broke my parser. And that hurts!
Quoting TonesInDeepFreeze
You're right. If I say I am not replying, that would constitute a reply. I shall henceforth simply not reply to this inane self-referential conversation. When you something substantive, as opposed to looping back on the syntax of whatever I may have said, I'll reply.
(several paragraphs later)
Well that didn't leave much. I think there's a a potentially interesting conversation about the axiom of infinity.
I'm not claiming any particular sense of existence. Nor am I disputing any particular sense of existence. In context of the question whether set theory is inconsistent with physics, I am interested in the context of formal axiomatization (I'll just say 'axiomatization'). In Z we have the theorem: Ex x is infinite. I would think that that would provide an inconsistent axiomatization T of mathematics/physics only if T has a theorem: ~Ex x is infinite.
Quoting fishfry
I don't.
Quoting fishfry
I claim that set theory has a theorem: Ex x is an inductive set. I don't opine as to what particular sense we should say that provides. I do tend to think that whatever that sense is, it is at least some abstract mathematical sense. And I appreciate that there are variations held by different people. I can "picture" in my mind certain notions such as "the least inductive set is an abstract mathematical object that I can hold in my mind as "picked out" by the predicate of being a least inductive set". I find it to be a coherent thought for myself. But I don't have any need to convince anyone else that such a view of mathematical existence should be be generally adopted or even considered coherent by others.
Quoting fishfry
Then your parser is weak handling double negation. I chose double negation because it best suits the flow of how I think about the proposition. With less negation: I explained why previously that it was reasonable for me not to infer that you were writing hyperbolically.
Quoting fishfry
Just to be clear, my replies were not merely to you saying that you are not replying.
Quoting fishfry
My part is not inane. And whether or not you think that conversation about conversation should be eschewed, I don't think that way.
Quoting fishfry
I did not merely "loop back on the syntax of whatever you may have said". It's interesting that you want an end to posting about the conversational roles themselves, but you want to do that while still getting in your own digs such as "inane" and dismissive mischaracterization such as "looping back on the syntax".
And I have not the least interest in the subject of corporate financing. Go figure. But I do make it a point to go into discussion threads about business and economics and make my boredom with the subject well known.
Yes there is a physical reason for this. The pieces cannot be floating in air, nor can they randomly disappear and reappear in other places, nor be in two places at once. There are real physical restrictions which had to be respected when the game was created. So a board and pieces, with specific moves which are physically possible, was a convenient format considering those restrictions.
This is the problem with your claim that mathematics is not bound by real world restrictions. You can assert that it is not, and you can create completely imaginary axioms, such as a thing with no inherent order, but when it comes to real world play (use of such mathematics) if these axioms contain physical impossibilities, it's likely to create problems in application. The creator of chess could have made a rule which allowed that the knight be on two squares at once. or that it might hover around the board. But then how could the game be played when the designated moves of the pieces is inconsistent with what is physically possible for those pieces?
I have full respect for the notion that mathematical axioms might be completely imaginary, like works of art, even fictional, but my argument is that such axioms would be inherently problematic when applied in real world play. You seem to think that it doesn't matter if mathematical axioms go beyond what is physically possible, and it's even okay to assume what is physically impossible like "no inherent order". And you support this claim with evidence that mathematics provides great effectiveness in real world applications. But you refuse to consider the real problems in real world applications (though you accept that modern physics has real problems), and you refuse to separate the problems from the effectiveness, to see that effectiveness is provided for by principles which are consistent with physical reality, and problems are provided for by principles which are inconsistent.
Quoting fishfry
I've explained to you very clearly why it is false to say that mathematics is not bound by the real world. Perhaps if you would drop the idea that it is a banality, you would look seriously at what I have said, and come to the realization that what you have taken for a banal truth, and therefore have never given it any thought, is actually a falsity.
Quoting fishfry
I've gone through this subject of formalism already. No formalism, or "formal game", of human creation can escape from content to be pure Form. You seem to be having a very hard time to grasp this, and this is why you keep on insisting that there's such a thing as "pure abstraction". Content, or in the Aristotelian term "matter" is what is forcing real world restrictions onto any formal system. So when a formal system is created with the intent of giving us as much certainty as possible, we cannot escape the uncertainty produced by the presence of content, which is the real world restriction on certainty, that inheres within any formal system.
Quoting fishfry
Let's take this analogy then. Will you oblige me please to see it through to the conclusion? Let's say that abstract art is analogous to pure, abstract mathematics, and representational art is analogous to practical math. Would you agree that if someone went to a piece of abstract art, and started talking about what was represented by that art, the person would necessarily be mistaken? Likewise, if someone took a piece of pure abstract mathematics, and tried to put it to practice, this would be a mistake.
Bear in mind, that I am not arguing that what we commonly call pure math, ought not be put to practice, I am arguing that pure math as you characterize it, as pure abstraction, is a false description. In other words, your analogy fails, just like the game analogy failed, the distinction between pure math and practical math is not like the distinction between abstract art and representational art.
Quoting fishfry
I do not reject fiction, I accept it for what it is, fiction. I do reject your claim that pure mathematics is analogous to fiction. Here is the difference. In fiction, the mind is free to cross all boundaries of all disciplines and fields of education. In pure mathematics, the mathematician is bound by fundamental principles, which are the criteria for "mathematics", and if these boundaries are broken it is not mathematics which the person is doing. And, these boundaries are not dreamt up and imposed by the imagination of the mathematician who is doing the pure mathematics, they are imposed by the real world, (what other people say about what the person is doing), which is external to the pure mathematician's mind. This is why it is false to say that pure mathematics is not bound by the real world. If the person engaged in such abstraction, allows one's mind to wander too far, the creation will not be judged by others (the real world) as "mathematics". Therefore if the person wanders outside the boundaries which the real world places on pure mathematics, the person is no longer doing mathematics.
edit: This again is the issue of content. If the content is not consistent with what is judged as the content of mathematics, then the person working with so-called "pure" abstractions cannot be judged as doing mathematics. Therefore the abstractions cannot be "pure" as there are restrictions of content, as to what qualifies as mathematics.
Quoting Luke
My argument is that such things are wrongly called mathematics, due to faulty conventions which allow imaginary fictions, cleverly disguised to appear as mathematical principles, to seep into mathematics, taking the place of mathematical principles. And obviously, it's not a stipulation but an argument, as I've spent months arguing through examples.
Too bad. Thanks for replying though.
Among the two or three participants? I would think it might provide viewer relief. But to each his own. :roll:
Quoting fishfry
Just thought I'd supply recent information regarding.
Not sure. Einstein did joke that the moon doesn't need to exist when we aren't looking towards where it should be. Would the idea of infinity be like a reality simulation in which things are only added to the information data base when it is needed for observation?
Of course that is vacuously true, since there is no axiomatic formulation of physics. The axiom of infinity is inconsistent with known physics since there is no principle of modern physics that stipulates the existence of any infinite set, let alone an inductive set of natural numbers. Since it's known that there are only [math]10^{78}[/math] hydrogen atoms in the observable universe, I'll take that as evidence that contemporary physics can not accept the axiom of infinity as a physical principle. You can't equivocate your way out of this. Perhaps you could simply acknowledge that the axiom of infinity is plainly at odds with known physics, yet a cornerstone of modern mathematics.
Quoting TonesInDeepFreeze
Why not? What prevents you from dipping a toe in the water and taking a stand? It's clear that there aren't infinitely many physical objects, except in the most speculative physical theories having no experimental support. So why not say something like, "The axiom of infinity is a formal statement that, as far as we know, is false about the world, yet taken as a fundamental truth in mathematics. And I account for that philosophically as follows: _______." Ducking the question doesn't help.
Quoting TonesInDeepFreeze
Yes. Very good. The axiom of infinity is taken as true in "some abstract mathematical sense." My point exactly, on which we are now in agreement. There are models of set theory in which it's true; at least if there are any models at all.
Quoting TonesInDeepFreeze
You've come to be in agreement with me. The only way the axiom of infinity can possibly be accepted as true or meaningful is in the context of purely abstract math, and NOT physics. Hence the axiom of infinity is a statement that is clearly false about the world, yet taken as a basic truth in math. My point exactly.
Quoting TonesInDeepFreeze
No longer responding to this line of discourse (Nlrttlod).
Quoting TonesInDeepFreeze
Nlrttlod.
Quoting TonesInDeepFreeze
Eschewed and espit out. Masticated to death. Munched and crunched.
Quoting TonesInDeepFreeze
Nlrttlod.
Quoting TonesInDeepFreeze
Nlrttlod.
Quoting TonesInDeepFreeze
Nlrttlod to this and to the rest of it.
What "disguise"?
Of course it's an attempted stipulation, since you are attempting to stipulate what the mathematical conventions should be (because you consider them "faulty"). Regardless, does your claim that mathematics should conform to "real world boundaries" (whatever that could mean) apply not only to infinite infinities, but also to infinity, zero, negative numbers, imaginary numbers and the like?
Right. Just like mathematicians can't fly by flapping their arms. The rules of chess are arbitrary and constrained by physical law, as are the axioms of set theory.
Quoting Metaphysician Undercover
Well this is Wigner's point. Some aspects of mathematics is so obviously fictional that it is UNREASONABLE that math should be so effective in the physical sciences. I don't expect to be able to personally explain how this works, but I hope you would agree that this is a mystery that more than one clever person has tried to sort out. It is UNREASONABLE that math is so effective in the physical sciences. You can't hang that problem around my neck as if it's mine personally. Everyone knows about the problem.
Quoting Metaphysician Undercover
In that respect math is even more free to be fictional, since sets and other mathematical entities are not bound by the laws of physics. If you drop a set near the earth, it doesn't fall down, unlike a bowling ball or a chess piece. You are making my point for me. Math is MORE free to be utterly fictional even than chess.
But actually your point is wrong. Physical chess pieces are not required. If I say "e4" to a chess player they know exactly what that means, and it is not necessary to physically push the white King's pawn forward two squares. Indeed, people play chess blindfolded, keeping the position entirely in their mind. Physical pieces are not necessary to play chess.
Quoting Metaphysician Undercover
All right! Took a long time but I appreciate that you have acknowledge this.
Quoting Metaphysician Undercover
Take it up with Wigner. I can not personally take responsibility for this mystery, which is indeed problematic. Some aspects of math are patently false, yet so amazingly useful in the physical sciences. Your argument? I think Wigner beat you to it. I hope you take this point. You are only repeating a well-known problem in the philosophy of math.
Quoting Metaphysician Undercover
It's not only ok, it's one of the cornerstones of modern math. And again, this isn't me advocating for such a position. This is me describing the facts of modern math. Pick up any book on set theory. Look at the axiom of extensionality. I am not advocating for its rightness or wrongness; only reporting to you that extensionality is the first rule of set theory. Sets aren't affected by gravity or electric charge either. You have a problem with that?
Quoting Metaphysician Undercover
I hardly need to supply evidence. I hope you don't deny it. Do you deny it?
Quoting Metaphysician Undercover
On the contrary. I agree with Wigner than parts of math are so obviously fictional and divorced from reality that it is UNREASONABLE, Wigner's famous word, that math should be so effective in the physical sciences. You act like nobody ever thought of this before but it's a famous essay and a famous mystery. There are no sets, at least not as conceived by set theory. Yet set theory founds the real numbers which underly most of physics. I "refuse to consider" this mystery? Of course not. I point you to Wigner's essay time after time after time, and you come back and act like this is some great mystery that you yourself have personally uncovered and that I deny.
Quoting Metaphysician Undercover
If you drop a set near the earth, it doesn't fall down. Sets have no gravitational or inertial mass. They have no electric charge. They have no temperature, velocity, momentum, or orientation. In what sense are sets bound by the real world?
Quoting Metaphysician Undercover
I've given the matter quite a bit of thought and expressed my thoughts in this conversation. It's a banality that mathematical objects are not bound by the laws of nature, except that -- stretching a point -- mathematical objects are products of the human mind and the human mind is bound by the laws of nature. So perhaps ultimately there's a physical reason why we think the thoughts we do. I'd agree with that possibility, if that's the point you're making.
Quoting Metaphysician Undercover
I gather that "pure Form" is a term of art in philosophy with which I'm not familiar.
Quoting Metaphysician Undercover
As opposed to what? Impure abstraction? There are abstract things in the world. SEP has an article on the subject.
Quoting Metaphysician Undercover
Matter has a very specific technical meaning in physics. Matter has gravitational mass. Mathematical objects don't. You can't weigh a topological space. I can't speak to what Aristotle might have thought, but he also believed that bowling balls fall down because they're like the earth and thing go to their natural place. We no longer take him seriously on that. I don't know why you think I should take him seriously on whether mathematical sets, which Aristotle didn't know about, have mass. You don't think sets have mass, do you? In this post you are pursuing a line of argument I find nonsensical. Of course mathematical objects are not bound by the laws of nature.
Quoting Metaphysician Undercover
That was a little word salad-y. If you mean that mathematical sets are sort of like bags of groceries but definitely not completely like bags of groceries, of course I agree. For one thing, mathematical sets can contain infinitely many elements.
Quoting Metaphysician Undercover
Ok.
Quoting Metaphysician Undercover
The analogies are only good to a point, but ok I'll play along. For example abstract art is not indispensable to the formulation of modern physics, whereas abstract math is.
Quoting Metaphysician Undercover
Of course not. Art critics do it all the time. I may not agree or even understand, but people who care about such things see meaning in abstract art, even real world meaning. After all Moby Dick is a work of fiction yet cautions us to not follow our obsessions to our doom.
Quoting Metaphysician Undercover
What???? Are you joking? Physicists do it every day of the week and twice at grant proposal time. Group theory, part of the subject matter of an undergrad math major course called Abstract Algebra, is the heart and soul of particle physics these days. Functional analysis and differential geometry, two highly abstract areas of math, are the foundation of quantum mechanics and general relativity, respectively.
Your questions are wildly off the mark. Physical scientist apply the most abstruse and abstract areas of mathematics constantly, as part of their daily work.
Quoting Metaphysician Undercover
The physicists will be relieved.
Quoting Metaphysician Undercover
I quite agree. Let me say that again I quite agree. Therefore it is a MYSTERY that it is so UNREASONABLY effective. Wigner Wigner Wigner. These thoughts have already been thought, this problem is well known. Much of abstract math is false as false can be, as regards the physical world; and yet, those same parts of math are indispensable -- Quine and Putnam's word -- in the physical sciences. It's a puzzler alright. But a very well-known puzzler.
Quoting Metaphysician Undercover
I don't think you made your case. You didn't make your case at all.
Quoting Metaphysician Undercover
Mathematical principles are historically contingent, and the greatest advances have been made when someone transcends and violates the established principles of their time. Negative numbers, irrational numbers, non-Euclidean geometry, transfinite numbers and set theory. Cantor caused a revolution. His radical ideas on transfinite quantities was received with great skepticism bordering on horror. Today his ideas are taught to high school students. Cantor was told that what he was doing was not mathematics. His great opponent, Kronecker, who had actually been Cantor's teacher, said, "I don't know what predominates in Cantor's theory – philosophy or theology, but I am sure that there is no mathematics there." https://en.wikipedia.org/wiki/Controversy_over_Cantor%27s_theory
In math, violating the "fundamental principles" is how progress is made.
Quoting Metaphysician Undercover
I've just shown that some of the greatest advances in math have been made by blowing up the opinions of the world. What happens is that the opinions of the mathematical world change. Or as Planck said, scientific progress proceeds one funeral at a time. Meaning that the old guard die off and the young Turks readily adopt the radical new ideas.
Quoting Metaphysician Undercover
They're creating radical new mathematics. As has happened innumerable times throughout history.
Your insufficient knowledge of mathematics and its history causes you to have such mistaken ideas, that there are eternal principles that never change. On the contrary, each generation blows up the ideas of the past. Standards of rigor, what counts as a proof, what counts as a number, what counts as mathematics, is constantly changing and is a matter of historical contingency.
Much appreciated.
I thought it was obvious that since we don't have in front of us an axiomatization, then my question is hypothetical regarding whatever proposed axiomatization might be presented. Such an axiomatization would have mathematical axioms and also extra primitives and axioms for physics. I don't see that such a theory would have to be inconsistent. For example, "Ex x is infinite" is not inconsistent with "~Ex x has infinitely many particles" (I'm just using 'particles' as a placeholder for whatever technical rubric would actually be used).
Quoting fishfry
That doesn't entail inconsistency. Just because a theory doesn't have a certain principle doesn't entail that adding that principle causes inconsistency. But if physics had a principle that it is not the case that there exists an infinite set, then yes, there would be inconsistency. But even if physics had a principle that there are not infinitely many particles, that is not itself inconsistent with the existence of infinite sets, such as infinite sets of numbers if numbers are not axiomatized to be particles.
Quoting fishfry
I never said that it would be a physical principle. It would be a mathematical theorem to which are added primitives and axioms for theorems of physics.
Quoting fishfry
If I wanted to take a normative stance on the subject then I would.
Quoting fishfry
Because I am not motivated to do that. Moreover, I am not inclined to accept that my thoughts (including lack of conclusions) about a subject needs to conform to a Procrustean format "this and that, and I account for that philosophically ______."
Quoting fishfry
I haven't made claims that I've ducked supporting.
Quoting fishfry
Of course, a consistent theory has models. But nothing I've said then commits me to adopting a position for or against the proposition that mathematical statements are true only relative to models.
Quoting fishfry
Nothing I've said commits me to such a claim. I said that I have a sense of the abstract meaning of mathematical statements (also, I can add, that certain axioms fit my intuitive concept of what sets are as abstractions). That doesn't entail that I also must go on to say that my sense previously described must be the ONLY correct, meaningful or useful sense, and especially it does not require that I take a stance that the notion of model-theoretic truth is the ONLY correct, meaningful or useful sense.
Quoting fishfry
Again, I haven't claimed that the axiom of infinity "says" anything about physical objects or has anything to do with physics other than it provides the mathematics used for the sciences such as physics.
The real numbers however, are nonsensical with respect to experimental physics and engineering, where their literal definition is at odds with respect to how the formalism is actually used. There, real numbers aren't used literally in the sense of referring to infinitely precise quanitities, but are used non-rigorously or "non-standardly" to refer to indefinite and imprecise quantities and taken together with noise and error terms. For this reason, in conjunction with the rapid ascent of automated theorem proving and functional programming that are based on type theory, the awkward, misleading and practically false language of real analysis can only die fast.
Quoting TonesInDeepFreeze
yes, originally I was speaking roughly in relation to that article while making what i considered to be a tangential point in relation to the thread topic. As an axiom, LEM when interpreted in the Set category by the usual Tarskian approach, is an axiom of "finite choice" in the sense of asserting 'by divine fiat' the existence of a choice function for every relation into a finite set, i.e. that every finite set is 'choice '. Stronger choice principles additionally assert the existence of choice-sets that are the non-constructive infinite unions of the finite choice sets.
I don't know your personal use of the terminology. Of course even without AC every finite set of nonempty sets has a choice function. I don't know whether LEM is needed for that (at least on first glance I don't see that it is).
Well, I'm glad I won't be around to see that.
Has an actual, real live physicist posted on this thread? There have been a lot of assumptions about physics, interesting opinions, but I wonder what people in the profession have to say about the number systems they employ. fishfry provided a link to a novel paper on constructivism in physics that shows there is some degree of interest in the subject in the physics community. Kenosha Kid is a quantum researcher. :chin:
http://tph.tuwien.ac.at/~svozil/publ/1995-set.pdf
https://www.sciencedirect.com/science/article/abs/pii/S0960077996000550
and from a set theorist:
http://logic.harvard.edu/EFI_Magidor.pdf
Hello. Was, not is. Sold my soul for a bag of gold and sick guitar skills.
I haven't followed the thread, sorry, and responding to the OP 19 pages in seems weird. But I'll chuck a thought in and you can just pretend I wasn't here if I'm off-track.
I feel loathe to claim that physicists "believe" much. Physics is grounded on curiosity; if you're chock full of beliefs, why go and ruin it by learning things? I'm probably particularly minimalist when it comes to beliefs though. I don't "believe" in quarks and Higgs bosons and even quantum theory in the way I "believe" in electrons, protons, evolution and the special theory of relativity.
When we endorse a theory, what we're saying is "whatever is actually going on, it's got to be something like _this_". Sometimes the theory is compelling enough to base an actual belief upon, but really it's all a work in progress.
In terms of the boundary of the universe, I'm intrigued but uncertain. My intuition is that the universe is temporally bounded and from relativistic considerations, that being so, we should expect space to be bounded also. But that's unjustified and falls short of a belief.
I'd be a bit more confident about saying that the universe is either infinite or has periodic boundary conditions, i.e. some surface of a finite hyper-object. I don't think when you get to the end there's a wall, like in The Truman Show. :)
Infinity crops up in mathematical physics all the time but it's usually a mathematical artefact not a physical one. It could be a trick, like in integration, where you integrate over infinity knowing that the thing you're integrating attenuates to zero. Or it could be a consequence of representing something difficult as a power series or some other kind of expansion, like Feynman diagrams. Speaking of, quantum field theories a rife with infinities, requiring a renormalization procedure to get rid of them. Feynman invented renormalization and refered to it as a trick. As the fourth greatest physicist of all time, we should take him seriously, not renormalization.
The most overt infinity is probably the singularity: a point of finite energy but infinite energy density. It's perfectly feasible there's no such thing, that black holes are tiny but finite, but there'd be an intriguing question of what's holding it up since, by that point, even the strongest known pseudo-force in existence -- statistics (Pauli exclusion) has given up the ghost! But there's no shortage of theories in which black holes do all manner of crazy things (like birthing new universes).
Thing is, if there's infinity out there, we're not likely to encounter it. Infinity is either a limit we can't reach or a sign that something's broken. The visible universe is large but finite. The effects of black holes that we can observe are large but finite. If it is out there, either it or old age will kill you before you find it.
Just for a minimal start, it is clearly evident that it is not in principle inconsistent merely to add non-mathematical axioms to ZFC, even if the non-mathematical axioms include a declaration that the non-mathematical domain is finite.
For the most simple example:
To ZFC, add a 1-place predicate symbol: P (intuitively, Px means x is a particle). Add an axiom E!x Px. Define: p = the unique x such Px. So the set of particles is finite.
Or, instead of having the axiom E!xPx, we could have: ExPx & ~Ey(y is infinite & Axey Px). Then there are an indeterminately many particles but not infinitely many of them.
[i][EDIT: The above two paragraphs are not what I meant:
Add a 1-place predicate symbol: R (intuitively, Rx means x is a particle). Add an axiom E!xAy(yex <-> Ry). Define P = {y | Ry}. Add axiom: P is finite.
And delete the second paragraph.][/i]
Those are not inconsistent theories. [EDIT: That is not an inconsistent theory.]
In Z set theories, we may define:
x is a class <-> (x = the-empty-set or Ey yex)
x is a set <-> (x is a class & Ey xey)
x is a proper class <-> (x is a class & ~ x is a set)
x is an urelement <-> ~ x is a class
Then we have theorems:
Ax x is a class
Ax x is a set
~Ex x is a proper class
~Ex x is an urelement
That entails that every particle is a set. And we might not like that. But we could say "Oh well, that is an artifact of abstraction that won't hurt the physical theorems we'll prove, similar to the fact that the set theoretic definition of numbers burdens numbers with abstract set theoretical artifacts that however don't interfere with the mathematics we will do with those numbers."
Or we could reformulate as follows:
Delete the axiom of extensionality.
Add primitive: S (Sx intuitively meaning "x is a set").
Add axiom: E!x(Sx & Ay ~yex) (there is a unique empty set).
Add axiom Axy((Sx & Sy) -> (Az(zex <-> zey) -> x=y)) (revised axiom of extensionality).
Add axiom: Px -> x is an urelement. [EDIT: Rx -> x is an urelement]
Adopt all the other axioms of ZFC.
It seems a safe bet that that is a consistent theory. Granted, it is not ZFC and not strictly speaking set or even class theory (such as NBG) that both are characterized by perhaps the most crucial property of classes and sets - extensionality. But set theory with urelements may be recognized as a reasonable variant.
And one may add additional primitives such as a function for mass, primitives about spacetime, primitives and definitions about subparticles, physical objects made from particles, and axioms about particles, subparticles, and physical objects, their masses, interactions among them in space and time, etc... And with axioms with mathematics about the physical objects, etc, hopefully deriving theorems of physics.
I don't see that there would be a correct argument that merely in principle such a theory must be inconsistent.
/
See Suppes's 'Introduction To Logic' pgs 291-305 for another example: axioms for Particle Mechanics.
The vast majority of mathematical principles, like pi, and the Pythagorean theorem which we discussed, are not fictional, and that is why mathematics is so effective. So there is nothing UNREASONABLE in the effectiveness of mathematics. You can argue from ridiculous premises, as to what constitutes "true", and assume that the Pythagorean theorem is not true, as you did, but then it's just the person making that argument, who is being unreasonable.
Quoting fishfry
The word "set" is a physical thing, which signifies something. And it only has meaning to a human being in the world, with the senses to perceive it. Therefore sets, as what is signified by that word, are bound by the real world.
Quoting fishfry
Now you're catching on. Consider though, that the physical forces of the real world are not the "reason why" we think what we do, as we have freedom of choice, to think whatever we want, within the boundaries of our physical capacities. The physical forces are the boundaries. So we really are bounded by the real world, in our thinking. We do not apprehend the boundaries as boundaries though, because we cannot get beyond them to the other side, to see them as boundaries, they are just where thinking can't go. Therefore it appears to us, like we are free to think whatever we want, because our thinking doesn't go where it can't go.
You will not understand the boundaries unless you accept that they are there, and are real, and inquire as to the nature of them. The boundaries appear to me, as the activity where thinking slips away andis replaced by other mental activities such as dreaming, and can no longer be called "thinking". We have a similar, but artificial boundary we can call the boundary between rational thinking and irrational thinking, reasonable and unreasonable. This is not the boundary of thinking, but it serves as a model of how this type of boundary is vague and not well defined.
Further, we have a boundary between conscious and subconscious, and this is closer to being a real representation of the boundary of thinking. The subconscious activity of the mind is not called "thinking", So dreaming is not thinking. You can see that the descriptions of these two types of boundaries are somewhat similar, being modeled on some form of coherency. A form of coherency marks the difference between reasonable and unreasonable, and a different type of coherency marks the difference between waking mental activity (thinking), and dreaming (mental activity outside the bounds of thinking). In the latter division, you ought to see clearly that it is the physical condition of the body (being asleep), which provides the boundary that we model with a description as to what qualifies as thinking, and what does not. However, there is a coherency or lack of it, within the mental activity, which corresponds with the physical boundary which we model as the difference between awake and asleep. In the case of reasonable/unreasonable, we have cases of physical illness, and intoxication, which demonstrate that the boundary, which is a boundary of coherency, has a corresponding physical condition.
So here's the point. We have mental activity which is thinking, and mental activity which cannot be classified as thinking. Therefore there must be a boundary to thinking which separates it from that other activity. The difference is described as a difference in coherency But, since there are real world physical differences which correspond with the described boundaries of coherency, I propose that it is the real world physical boundaries which impose upon our mental activity, the inclination to create corresponding mental boundaries of coherency. So for instance, the law of non-contradiction is a boundary of coherency. To violate the law is to put oneself outside a boundary of coherency. But the law of non-contraction is a statement of what we believe to be impossible, in the real physical world. So it is the acknowledgement of this physical impossibility, as being impossible, which substantiates the assumed mental boundary of coherent/incoherent.
Now, you are proposing a type of thinking "pure mathematics", which is not at all bounded by the real physical world. How could there possibly be such a thing? As I explained, the mental activity which is called thinking, is already bounded in order that it be separate from the mental activity which is not thinking, and there is a corresponding physical condition, being conscious, or awake, which provides the capacity for thinking. Since this physical condition is required for, as providing the capacity for, thinking, then thinking is necessarily bounded by the real physical world. A person cannot go in thought, beyond the capacity given to that person by one's physical body.
Suppose we allow that thinking might move past the boundaries of coherency, (which I admit we have created), to be not at all bounded by coherency. The problem is that there is a corresponding real world boundary which is responsible for the creation of the boundary of coherence. Do you allow that subconscious mental activity, and dreaming, are thinking? See, it makes no sense to say that thinking can go beyond the bounds imposed upon it by the real physical body which capacitates it.
Now suppose we say that there is a special type of thinking, "pure mathematics", which we give that privilege to. How can we even call this thinking? We create boundaries of coherency to define what "thinking" is, keeping "thinking" within the range of conscious mental activity, but now you want to allow a special type of thinking which is not bound by this rule. In reality, "pure mathematics" is a special type of thinking, so it has stricter binding of coherency than just thinking in general does. And, corresponding with those binds of coherency are features of the real physical world.
Quoting fishfry
I agree with this fully. But the need to violate fundamental principles just means that what was once considered to be a boundary of coherency can no longer be consider such. It does not negate the real physical boundaries, which the boundary of coherency was meant to represent. The boundary of coherency did not properly correspond, with the real world boundary, and therefore it needed to be replaced. The need to replace fundamental principles is evidence of faulty correspondence.
Quoting fishfry
Yes, yes, I think you truly are catching on. The need to change mathematical principles is a feature of poor correspondence. It is not the case that pure mathematics is thinking which is not bounded, because it truly is bounded, as described. But it is thinking which does not adequately understand the real world conditions which are its boundaries. It does not understand its boundaries, that's why it might even think as you do, that it is not bounded. Therefore it often poorly represents these boundaries, and when the boundaries become better understood, the representations need to be replaced. Understanding the true real world boundaries is what produces certainty.
Thanks for chiming in. Thought-provoking comments! :cool:
And contrariwise in other articles. I'm sure you wouldn't want to cherry pick just gainsaying quotes.
And the article describes Bridgman's notions as stemming from Bridgman's philosophical framework. That does not preclude other frameworks, and especially doesn't entail that any form of ZFC+physics must be inconsistent. (Though the author of the article does give other vigorous arguments against set theory as a foundation for physics.)
I skimmed the first article and this caught my eye. I only know renormalization from afar.
[math]\sum\limits_{1}^{\infty }{n}=-\frac{1}{12}[/math]
:scream:
It doesn't work that way.
It's true that physics "uses" the real numbers. And it's true that the real numbers are formalized using infinite sets. It does NOT follow that physics uses or is formalized by infinite sets.
I don't have the philosophy vocabulary to name this phenomenon, where A is a part of B and B is a part of C but A is in no way even remotely a part of C. Perhaps others can suggest the right concept.
Regardless, it's just not true and it's not valid thinking to say that "physics uses math and math uses infinite sets therefore ..." It's wrong, I just can't verbalize why.
ps -- To add a little. If you want to use the real numbers, you can just take their properties as given, namely that they are a complete totally ordered field. That uniquely characterizes them, there's only one model. (The completeness is second-order, that's why this works. I'm a bit fuzzy on that aspect of the logic but I think that's what's going on).
Now if you are doing math and you want to show that you can cook up a complete totally ordered field within set theory, you use one of the standard constructions involving the use of infinite sets. But the ontological commitments, if that's the right phrase, are of a different type. Physics uses real numbers and mathematicians formalize real numbers using infinite sets, but there's no ontological commitment in physics to infinite sets.
1+2+3+ ...
I didn't say it does. And I am not saying that it would be consistent if it did. I am saying that I haven't seen an argument that it would be inconsistent if we added to set theory, primitives and axioms for physics, and that it seems plausible that we might be able to do so.
Did you think I argued that physics uses or is formalized by infinite sets? If you did, then that would be yet another example, from different threads, in which you read into what I posted claims that I did not make in those posts.
Quoting fishfry
And using set theory doesn't entail that physics would take on an ontological commitment. Again, I am asking about a possible axiomatization. That is syntactical. Consistency is syntactical.
Yes, I deleted my reply because immediately after posting it, I read about the Euler and Riemann sums.
So that I can understand what you mean, can you give an example? What kind of axiom would we add to set theory that would be an axiom for physics?
My main interest is this claim:
Quoting fishfry
If there is not some formalization of physics in mind, then it is not clear what it means for a formal axiom to be in contradiction with a set of unformalized statements. It might mean that any conceptualization of the meaning of the axiom of infinity is incompatible with the concepts of physics, or something along those lines. I don't make a claim pro or con about such informal senses of 'contradiction', but I am interested in the question whether any possible reasonably sufficient formal axiomatization of physics would entail the negation of the axiom of infinity. If such a theory can't formulate the axiom of infinity in the language of the theory, then, a fortiori, there is not a contradiction with the axiom of infinity, so that would settle the question. On the other hand, if the theory includes set theory or any variant of set theory (such as set theory with urelements) that includes the theorem "there exists an infinite set", then the question is whether it is possible to have a consistent system that combines set theory or such variant with a reasonably sufficient set of physics axioms. (I'll leave tacit henceforth that we might need a variant such as set theory with urelements. But moving to a variant would not vitiate my point, since the variant would still include the "there exists an infinite set", which is the supposed source of inconsistency. Also, I'll take as tacit "reasonably sufficient".)
There are two questions: (1) Can there be a consistent set of axioms for physics? I don't opine, especially since I have no expertise in physics. (2) If there can be an axiomatization for physics that combines with set theory, then would any such axiomatization be inconsistent? My point is that I have not seen an argument that it is not at least plausible that there might be a consistent axiomatization that combines physics with set theory, and that I do think it might be plausible.
Some earlier points bear on the question and need to be kept in mind:
Quoting TonesInDeepFreeze
Quoting TonesInDeepFreeze
Quoting fishfry
I don't know that there are not axiomatizations of any part of physics or even of a large part of it. At the very least, Suppes provides an axiomatization of particle mechanics. Granted, that's not an axiomatization of modern physics. But at least the question of infinity is addressed, as Suppes combines infinitistic set theory with added physics primitives and a definition of a system of particle mechanics such that the set of particles is finite. I was along those lines when I said that the existence of infinite sets is not inconsistent with having a finite set of particles. Also, not claiming an axiomatization of physics, but arguing for the plausibility that certain questions in physics might be affected by set theory: http://logic.harvard.edu/EFI_Magidor.pdf at page 10.
/
Quoting fishfry
I will not suggest any particular axioms, as I am not expert in physics. It is better anyway that anyone may nominate, at their own will, any postulates of physics (or any formulas of physics deemed fundamental and productive enough) for axioms. These can be, for example, the postulates of special relativity. (Nota bene: Again, I am not claiming that this project would be successful. I am saying only that it might be plausible.) Examples from Suppes for particle mechanics:
The set of particles is finite.
The mass of a particle is a positive real number.
For particles p and q, and elapsed time t, the force of (p q t) = - the force of (q p t).
I won't defend any particular formulations from possible criticism. The particulars are not my point, on an assumption that any details that raise objection could be adjusted to suit whatever formulation the physicist more prefers.
Contradicts axiom of infinity.
Quoting TonesInDeepFreeze
What's a particle? What's mass? These are axioms, remember? Everything's defined in terms of a single primitive, [math]\in[/math]. You've claimed you can add physics axioms to the axioms for set theory, so the burden on you is to challenge yourself to see if your idea holds water.
No it does not. If you can't see that, then discussion with you is hopeless.
That some sets are finite does not contradict that some sets are infinite.
Quoting fishfry
They are primitives.*
Quoting fishfry
No, we add primitives for physics. I covered that very clearly in previous posts. You are just skipping the explanations given you.
I did not say that physics can be formulated using only the axioms of set theory. I explicitly said that we take the axioms of set theory and add primitives
* Technical note: Strictly speaking, Suppes doesn't add primitives, but instead he defines a 'system for particle mechanics' as a tuple with certain properties. The tuple is a structure or sometimes called a 'system', in the same way as an algebraic structure or other structures in mathematics and science. Anyway, this is not an essential difference from adding primitives. His definition of a certain kind of structure can be easily transformed into adding primitives. Moreover, defining a certain a kind of tuple adds even less to set theory. Moreover, the physics axioms also could be conveyed instead as properties of the structures.
You have ignored and outrageously misconstrued what I wrote, yet again. I didn't want to comment on the discussion itself again, but your reading confusions, as seen in this thread and other threads, are quite remarkable.
You haven't convinced me of your point in the least.
Quoting TonesInDeepFreeze
Perhaps we can mutually agree on at least this. I'm always for achieving agreement.
What possibly could refute that it is consistent that some sets are finite and other sets are infinite?
Quoting fishfry
What I wrote is:
Quoting TonesInDeepFreeze
You continually ignore and terribly misconstrue what I write, and now you can't even see that the finititue of a particular set does not contradict the axiom of infinity. So if you persist that way, then discussion with you is pointless.
We're quite in agreement. We tend to talk past each other and I'm content to leave it at that.
Quoting TonesInDeepFreeze
I'll concede that point. But if you adopt as an axiom claims that are subject to experiment and investigation, your science won't get you very far.
No, you regularly ignore and misconstrue, sometimes even to the point of posting as if I said the bald negation of what I actually said. Meanwhile, I respond on point to you, and make every reasonable effort not to misconstrue you or mischaracterize your remarks, and I'm happy to correct myself if I did.
That is not an equivalence.
Quoting fishfry
Meanwhile, your original point that the axiom of infinity combined with physics is inconsistent has not been sustained. I don't know whether you recognize that now.
Next, as to your new point, perhaps I don't understand what you're saying. Axioms can be interpreted and then the interpretations subjected to experiment, so that either the experiments support the axiom as interpreted or refute the axiom as interpreted, in which case the theory would need to be reformulated, if possible, with different axioms. I don't see a problem with that.
In any case, again (since so much effort was spent to get to this juncture): It has not been shown here that the axiom of infinity is inconsistent with possible axiomatizations of physics.
I am truly curious why you even disputed it to begin with, and then persisted in yet another post. Especially as this is typical with you. You weren't reading correctly? Your weren't reading correctly because you mostly only skim? A mental lapse? A mental lapse because you have a continual preconception that when I disagree with you or question whether your claim is supported that I am bound to be wrong about it?
Is it possible that you're not always as clear in your meaning as you think you are?
Quoting TonesInDeepFreeze
Can you see that it's possible that this is not my perception?
Quoting TonesInDeepFreeze
I humbly apologize for whatever grave offense I may have caused. Peace be with you, my friend.
ps -- But ok, you asked a fair question and you deserve an answer. You said "The number of particles is finite." Now you proposed that as an axiom to be added to the standard axioms of set theory. Being familiar with the latter, and not knowing what a "particle" is, I assumed you meant mathematical sets, or mathematical points. In which case your formulation would indeed be in contradiction with the axiom of infinity.
So I asked. And you THEN -- after I challenged you on this point -- admitted that "particle" is a primitive, something you had not said before. After you said that, it was clear to me that there could indeed be only finitely many of them without creating a contradiction.
Can you see that I had to ask you twice in order to dig out your hidden assumption that "particle" is a brand new primitive in set/physics theory? Without that information, your statement that there are only finitely many particles makes no sense.
Therefore can you see that my asking twice was necessary in order to smoke out the hidden information you didn't bother to say up front? And that this would be a perfectly sensible explanation for my having to ask you twice about your claim?
You see you are not always as clear as you think you are. If you want to add new primitives to the theory, and you don't bother to tell me that, then it's perfectly understandable that I would have confusion about your meaning.
(ps, a couple of hours later) Hilbert's sixth problem is to axiomatize physics. It's still an open problem. So if you think you have an idea, or if you even claim it's logically possible, the burden is on you to be crystal clear in your thoughts; because nobody in 120 years has axiomatized physics, let alone unified it with set theory, which seems logically contradictory on its face (to me at least).
Pi and the Pythagorean theorem are not mathematical "principles."
Quoting Metaphysician Undercover
If you knew or understood more math, you'd understand the point. That you come up with "pi" as an example of a mathematical principle exemplifies the problem. Pi is a particular real number, known to the ancients. Hardly a principle.
Quoting Metaphysician Undercover
If you're not willing to agree that set is a term of art in math that designates a purely abstract thing, having nothing to do with the physical world, we just can't have a conversation.
Quoting Metaphysician Undercover
Are you saying that because humans are physical and sets are a product of the human mind, that sets are therefore physical? By that definition everything is physical, yes, but you are ignoring the distinction between physical and abstract things. Makes for a pointless conversation.
Quoting Metaphysician Undercover
Well then your point is trivial and pointless. Everything is physical if we can imagine it. The Baby Jesus, the Flying Spaghetti Monster, the three-headed hydra, all physical because the mind is physical. Whatever man. Pointless to conversate further then if you hide behind such a nihilistic and unproductive point.
Quoting Metaphysician Undercover
What a bullshit argument. I'm not going to play. Can we please stop now?
Quoting Metaphysician Undercover
So everything is physical because some mind thought it up. What a childish talking point to deny abstract objects.
I skipped the rest, this is too childish. All the best. You've talked yourself into an unsupportable position. Everything is physical because a human thought of it. Therefore there are no abstract objects. I can't continue. I see no continuation. You won't acknowledge the existence of abstract or fictional objects not bound by the world. What basis is there for me to continue?
Not only do I think it is possible, but I bet it's true. I am keenly aware that (1) It is difficult to write about these topics on-the-fly and in the confines of posts, and therefore, no matter how hard I try, I'm bound to sometimes falter, and (2) Looking back at some of my posts, I see that what I wrote could have been clearer or needs certain corrections.
But I don't think that is the main problem with you. With you, even things that really are quite clear get misconstrued. There was even an incident in which you continued to insist that I made a certain claim, but I actually wrote the explicit negation of that claim. You had overlooked the word 'not' in my post. Sure, such a mistake can happen to anyone, but it was remarkable that you persisted even after it had been pointed out to you and even claimed the word 'not' did not appear! And then, after another poster also alerted you that the word appeared, as I recall, you still did not post that you recognized it finally.
And, time after time, even when I state clear and simple things, chronically, you read into them things not there nor implied.
And there have been even a couple of bizarre incidents (even lately) where you conflated what you said with what I said. Again, it can happen to anyone - but you persist in such incidents even after your error has been pointed out to you.
Quoting fishfry
Indeed I do. But I don't recall an instance in which I unintentionally misconstrued you, then refused to recognize it when pointed out, let alone went on and on doing it over the same point, as you do with me. And, though I can't recall the specific incidents, I think there were one or perhaps two times when I misunderstood you and posted that I recognized that when it was pointed out to me.
Quoting fishfry
Just to be clear, I wrote it as a report of what an author wrote. I made clear that I don't personally propose any particular axioms for physics.
Quoting fishfry
No! You're doing it again! You're ignoring what I posted. I wrote explicitly about the alternatives (1) particles are sets or (2) particles are not sets but instead are urelements.
https://thephilosophyforum.com/discussion/comment/563405
[Note: I just now marked some edits to that post. But even with the pre-edited version, it was explicit that that there are two approaches, and in one approach particles are urelements not sets.]
And even if particles were sets, having the set of them being finite still would not violate the axiom of infinity. You're repeating the mistake on the point you conceded just a few posts ago!
Even if particles were sets (which, as I mentioned, we can avoid anyway) it is not inconsistent with the axiom of infinity to have the set of particles as a finite set.
So you blew right past my earlier post. I can see at least four possibilities (1) You skip reading a pretty good amount of the posts or (2) You read them but have a comprehension or retention problem or (3) You are nuts or (4) You willfully mangle the conversation for some kind of trolling effect. My guess is that it's a combination of (1) and (2).
Quoting fishfry
I duly note what I take to be your sarcasm. But there's no grave offense. Hardly even offense. More like a feeling that it's too bad that someone who is not altogether uninformed sometimes reads so poorly, reasons so abysmally, and is so characteristically recalcitrant about it.
Quoting fishfry
Quoting TonesInDeepFreeze
When we add a symbol without definition, then it is clear that we are adding a primitive.
And I said over and over in various posts, that we add primitives and axioms for physics.
Quoting TonesInDeepFreeze
And even if 'particle' weren't meant as a primitive, but were defined, it would not detract from my point of giving you examples of axioms.
A reasonable conversation would be:
F: Is 'particle' primitive or defined?
T: Primitive.
F: Okay, now it's clear.
or
F: Is 'particle' primitive or defined:
T. Defined.
F. Then what is its definition?
T. There is a chain of definitions leading up to it. I can't practically type it all out here. But my point is not to convince you of the Suppes's cogency, but rather just to give an example of an axiom.
Even if I hadn't earlier mentioned 'particle' as added, that would not have been misleading you, since any lack of details can still be supplied on request. There is nothing I posted or didn''t post that was my fault of you misconstruing me.
Alice: C is the case.
Betty: That's wrong [or, That's problematic, or whatever criticism].
Alice: No, it's right, because [fill in S, which is support for C, here].
Betty: You withheld the information S.
Alice: I didn't include S when I said C, but that doesn't justify claiming that C is wrong, especially as It is not the case that C is wrong. Instead, you could have said, "You claimed C, but have not supported it", to which I could respond (1) "Even if I don't personally support it, it is still the case that P" or (2) "Here is the support S".
What you are arguing essentially is that there is something wrong with my arriving at (2). That's illogical.
And anyway, I did mention previously that 'particle' is primitive.
Quoting fishfry
Wrong. You make no sense. Even if I didn't give any information about the set of particles other than it is finite, that doesn't entail that saying that it is finite makes no sense. You are again abysmally illogical.
Quoting fishfry
Again, the consistency doesn't depend on the predicate being a new primitive. Your notion is ridiculous. Even if the set where made without a new primitive, it still is not inconsistent to have a finite set while other sets are infinite.
Your illogic is stunning.
Quoting fishfry
Nothing was "hidden". (1) Even if I didn't mention it at a particular juncture, that doesn't entail that I am "hiding" it. (2) I did mention in previous posts that we add primitives and axioms. (3) In an earlier post, I did mention that we add 'particle' as a predicate, and without giving it a definition; so of course it would be primitive. (4) I did mention that Suppes himself adds primitives and axioms (except with an inessential technical qualification that I explained). So one may allow that it is primitive in his axiom, or if one doesn't want to take that as granted, then one can ask. But there was no misleading you about it.
And you failed to count to to even the number one. When you FIRST asked me about primitives, answered you IMMEDIATLY in the next post:
Quoting TonesInDeepFreeze
And you make the false claim that you had to ask me twice, not as a causal matter of fact, but rather while claiming that I was "hiding" and you had to "smoke it out". Yeah, you had to "smoke it out" by asking and then receiving my immediate reply. And even IF you had to ask twice that's not so bad really. Meanwhile, so many questions and points I've made to you that you have ignored; you continually make arguments that I rebut and then you ignore the rebuttal but still go on repeating your more basic misconstruing and strawmen. You are bizarre.
Quoting fishfry
It is perfectly reasonable to say I didn't mention it or to ask about it. It is very unreasonable to claim that I was "hiding" or that you had to "smoke it out" (especially as I answered you about it immediately). And I did mention 'particle' as an added to the language in an earlier post, and I did mention at least a few times that we add primitives and axioms, and one might take from context, in Suppes's formulation that it is primitive or, if not taking it from context, ask before falsely claiming that without the information you are justified to claim there is a contradiction. And, even more basically, even if it were not primitive, then it is still ludicrously illogical to claim there is a contradiction between stating a given set is finite and having the axiom of infinity.
Quoting fishfry
I have been clear. (I noticed today that I botched a formulation in an earlier post, but it is not material to the particular argument we are now having.) Also, what I claimed to be consistent is merely the initial setup of adding a 1-place predicate, and adding that there is a unique set all and only those objects that have the predicate and adding that that set is finite. And it is consistent with ZFC. It's trivial that is consistent. Anyone can trivially see it for themself.
Meanwhile, the first claim on this subject was YOUR claim that the axiom of infinity contradicts physics. You have not supported that claim. And, again, here's what I said about that:
Quoting TonesInDeepFreeze
I read this far and gave up. You started out by agreeing that your exposition was unclear and that I was asking clarifying questions. I became hopeful that a productive and interesting conversation could ensue. Then you go back to the personal insults. Have a nice day.
ps -- Also my name's not Betty.
This is clearly wrong. The ancients did not have real numbers, so they could not have known pi as a real number. They knew pi as the ratio of a circle's circumference to it's diameter. Further, they discovered that this ratio is irrational. You really amaze me with the nonsense you come up with sometimes fishfry.
Quoting fishfry
No, I was saying that human actions are limited by the physical world, and mathematical thinking is a human action therefore it is limited by the physical world.
Quoting fishfry
Obviously, I wasn't saying "everything is physical". Metaphysically, I believe in the immaterial, or non-physical. But human thoughts, as properties of physical beings, do not obtain this status.
Quoting fishfry
Sure, you seem to have run out of intelligent things to say.
It's clearly wrong that pi is a particular real number? @Meta, please understand that you give me no basis to continue this conversation. Maybe we'll chat about something else in some other thread sometime. For what it's worth, and for your mathematical education, pi is a particular real number.
It's clearly wrong that the ancients knew pi as a real number.
What the ancients knew is you changing the subject. It has no bearing on the nature of pi or whether pi may be called a "principle of mathematics," which was yet another error on your part.
Quoting Metaphysician Undercover
I got tired of you trolling me and have put a stop to it the only way I can.
Pi is the ratio of the circumference of a circle to it's diameter. Why is that not a "principle of mathematics"?
I don't want to be rude but at some point I have to stop responding and I'm at that point. The definition of a particular number is not sufficiently general or broad or foundational to be called a principle.
But Pi obviously is a principle, the ratio of the circumference of a circle to it's diameter. So, clearly it's you who is wrong, to say that pi is "the definition of a particular number". Again, you really amaze me with the nonsense you come up with sometimes fishfry.
I didn't mean in this particular argument. Granted, I could have been made it explicit that I recognize that I can be clearer sometimes but that I am not saying that I was unclear in this argument.
Quoting fishfry
I did not say that. You are making that up out of thin air. Again, bizarre.
And, let's take this one point again:
You said that you had to ask me twice. But you asked me and then I IMMEDIATLY answered.
Sure, one can make a mistake like that innocently, but with you it's a dominant pattern and you won't recognize such instances when they are pointed out to you.
Quoting fishfry
When your posting is so bizarrely off-base and patently illogical, through so many different conversations, then it's reasonable to point that out and to wonder aloud what is the source of your problem.
Then surely it's no loss to you to stop talking to me.
But pi is not a particular real number? How can I have a conversation with you? It's like a trained brain surgeon arguing about medicine with someone who just learned how to apply a band-aid. You lack the knowledge to be an interesting conversational partner. And then you get insulting about your ignorance.
Pi is not a particular real number? Wow.
Quoting TonesInDeepFreeze
Like the girls in junior high school used to say: Let's not and say we did.
Of course not with you. You don't wish to respond to the point that you made a false claim about me: So clearly false that all one has to do is look at the two posts.
When someone makes a false claim about someone, then waves off even responding to being shown that it is false, then that is included in the rubric of 'lying'.
I mention it because fishfry was so bizarre; it's an object lesson.
Starting here, looking at the my posts (I'm GrandMinnow) and fishfry's posts::
and ending with the third party here:
At the risk of further encouraging your fixation by replying to you: For the record, I reread the thread in question and I agree with myself. I stand by every word I wrote. Moreover, your use of onto was regarding a "claim," and not a mapping. You write imprecisely then complain when you're misunderstood. I will let you have the last word after this.
Here is your direct quote:
Quoting GrandMinnow
How would anyone take from that, that you are referring to an onto mapping rather than meaning a "claim unto itself" and simply misusing the locution? That's how I took it then, that's how I took it tonight, and that's how any native speaker of English would read it. That you meant to write "unto" but wrote "onto" by mistake; because unto makes sense in context, and onto does not. Re-reading your sentence over and over with "onto," i can't figure out what you are trying to say. "The following claim onto itself is not contradictory?" I mentally swapped in "unto" and thought nothing of it. No other interpretation is possible. What is a "claim onto itself?"
It's your own muddled writing that leads people to have no idea what you're talking about.
And now you can have the last word. I will not be replying to you anymore even if you post easily-refuted examples as you just did. You don't express yourself well and you blame other people for misunderstanding you. You should have the self-awareness to remedy your own imprecise writing.
You continued to claim that I didn't write 'onto itself'. You even quoted the post where I wrote 'onto itself' and said you do not see it, and yet it was there right in the quote. And you went on. Then another poster referred to one of your posts about it and quoted me yet again showing that indeed I did write 'onto itself'. Then nothing from you.
And you say you stand by your part in the discussion! But, even more crazy, now you are talking about my having written 'onto itself'! You are contradicting yourself right here!
Quoting fishfry
No, it was about a formula.
Quoting fishfry
I wasn't referring to a mapping. The statement I made about the consistency of the formula had nothing to do with a mapping.
The FORMULA onto itself is not inconsistent. I meant that the formula taken alone is not inconsistent. And I even followed up to you to explain exactly that the formula alone is not inconsistent but is inconsistent with one of the axioms. And you even argued about THAT with me.
Note: Reflecting on this, I think it is possible that 'onto itself' in the way I used it is not standard English, and if that is the case, then my use could be unclear. But, there are two prongs here (1) That I did write it and you continued to say that I didn't write it, even though you quoted it yourself, yet now you do recognize that I wrote it, which makes it bizarre to say you still stand by what you wrote in the thread. (2) Even if my use is not standard, I still did illustrate my point in the thread: The formula is not inconsistent but rather it is inconsistent with the other axiom. Moreover even if I had not qualified with 'onto itself' or that 'onto itself' is not clear or standard, then my original statement is still correct. The formula is not inconsistent, though, of course, it is inconsistent with the other axiom.
/
Meanwhile, still you can't even recognize that you asked me about 'particle' and I immediately replied that it is primitive, though you falsely claim that you had to ask me twice.
/
I'm wondering whether what is going on with you is that you are so determined to think that you are right that you are willing to make the most preposterously false statements to do it.
You are not a crank in the sense of cranks who argue ignorantly and illogically about Cantor, Godel, et. al, but you share some traits: (1) terrible confusions,(2) blatant illogic, (3) skipping rebuttals to you, (4) persistently misconstruing posts and then posting as if something was claimed that was not claimed - so effectively ongoing strawman. Of all the people I've met on on the Internet, you're among the worst.
You must know by now, that I do not accept "real number" as a valid concept. Your insistence that I must accept real numbers as a premise for discussion with you, is simply an act of begging the question.