It's time we clarify about what infinity is.
I'm always annoyed at people who say infinity is not a number but a "concept." For one thing, that is way too broad and says absolutely nothing; I cannot think of anything in mathematics or logic that is not a concept. I say it is a number for if it is not then one divided by infinity would not equal zero anymore than one divided by a cat is a number. Here's my definition of infinity, and for simplicity I'm only referring to positive infinity: infinity is a number, but it has a characteristic that all real numbers do not possess. Namely, it is a number that is greater than any particular real number. All the rules of arithmetic applicable to real numbers do not carry over to use of infinity. Examples: infinity plus a real number is infinity: infinity divided by infinity is not equal to one: infinity subtracted from infinity is not equal to zero.
Comments (116)
One would think it not too difficult. But it seems...
And infinity + 1 = infinity is true only in the sense that if you take an infinite number and add one to it, you get an infinite number. Basically, only if infinity does not refer to a specific infinity (i.e. only if infinity - infinity =/= 0.) Otherwise, if you're working with specific infinity, infinity + 1 > infinity.
? and ?+1 are the very same. That follows from the definition given above.
Your argument about 1 divided by infinity is false.
I imagine this fits into your definition, and you'll force it.
I said in a different thread.
There has to be a great turning effort to keep anything infinite in continuum.
Infinity is always something's something, but this something dies with the user. How else are you going to conceptualize forever?
You project forever, but there must be turning forces and a specific something that is forever bound.
Otherwise it is just 'yep, forever[random, partial imagination of a great number]'.
There should be two definitions, literal infinity, and false numerical infinity.
I've always been confused about this claim that infinity is not a number but is a concept. A number is a concept too isn't it?
I guess if we look at it from a quality-quantity perspective it'll begin to make sense. Firstly, infinity arises, even if only as a concept, in the world of numbers. Examples of infinity frequently used are numerical infinities e.g. the set of natural numbers {1, 2, 3,...}. These may be be soft evidence that the true home of infinity is the world of numbers,
I'm going out on a limb here and so forgive me if I don't make sense. I know there's a mathematical notion of countable and uncountable infinity which seems bit paradoxical since infinity is, by definition, uncountable. Kindly set aside the notion of countable vs uncountable infinities for the moment and consider only that infinity basically implies uncountable. Note that the term countable is formal and only describes the situation where a given infinite set can be put in 1-to-1 correspondence with the set of natural numbers.
That aside, compare this "fact" of infinity with other concepts that are considered uncountable. How about love, courage, joy? These concepts are categorized as uncountable i.e. unquantifiable and fall under the category of quality. So, it doesn't seem wrong to say that infinity is not actually a quantity, a number, but rather a quality like love or courage, etc.
But a trillion trillion is uncountable, too, in that sense.
It's like there is a wilful disregard for the mathematics here. See the slide to:
Quoting Possibility
That's bullshit; in the technical sense - it's junk thinking indicating a lack of comprehension, wilful or otherwise.
It's just an opinion. I'm not claiming this is a mainstream view on the issue. It just seemed right to look at infinity as a quality, being uncountable as it is.
Also, did you know that our ancestors could count only upto to 2? Look below:
Cardinal - Ordinal
[b]1 - first
2 - second[/b]
3 - third
4 -fourth
.
.
.
n - nth
Notice that the names for ordinal numbers of the first two cardinals (1 & 2) are distinct viz. "first" and "second". All other ordinal numbers can be constructed from their respective numbers simply by adding "th". This is claimed to be be evidence of counting ability being limited to 2 and after that, 3, 4, 5,...,it was simply "many". So ancient counting looked like this: one (first), two (second), many. The many corresponds to the modern concept of infinity. As you can see, many and infinity represent a limit to quantification i.e. it spills over into the domain of quality - a concept and not a number.
I don't see how inifinity is used in counting, nor is it a particular quantity. It is not finite, so not a number to my way of thinking. Personally I don't like all the maths based on larger and smaller infinities. Those definitions are like saying a colour is more black or less black.
"It was 1. 1111,1111,1111,1111,1111r" - this would prove imagination is finite at most. (joke)
It is not created out of number, but mind of eternity through harmonious principles discovered in the world, or better yet, mind.
Does 1.1R imply 1.1r^i?(last joke)
That is interesting...and I have often wondered about that kind of thing. But I suspect it is a lot more complicated than that.
If an ancient had 5 hens...and one went missing, I suspect he would not just say..."I had many yesterday and I have many today, so no problem."
The problem that arises for me is...IF "infinity"...why do we only look at things for "here" out infinitely (whatever that is). IF truly infinity...then the concept should begin at "nothing"...perhaps not even the thought or the notion/concept.
Understanding the evidence about forever is like understanding how you exist out of a womb.
You get to grips with the shape, over time channeling thought, you'll project some sort of infinity.
But love, courage and joy are neither greater than nor less than any sort of number.
It is a number but also a concept.
You can have infinite numbers, but that doesn't make infinity a number, it makes the number infinity.
In what technical sense? Infinity is historically a philosophical concept, representing an unbounded limit in relation to value/potential - whether quantitative OR qualitative.
I will concede that it’s more likely both or neither - that it refers beyond the outer limit of value/potential - although in mathematics it’s more representative of this boundless outer limit itself. But to define infinity as a supposedly quantifiable concept would be inaccurate: this is usually accepted by mathematicians, at least in a conceptual sense.
Quoting Frank Apisa
Piraha is a documented and currently spoken language that has relative terms roughly translated as ‘one’, ‘two’ and ‘many’ - but no terms for more exact numbers. From the studies conducted, it’s fair to say that some would not have noticed the missing hen, while others would.
Totally agreed.
Mathematics is only about abstractions expressed in language.
Quoting Michael Lee
Cantor's work is really interesting in this regard.
Countable infinity and uncountable infinity cannot possibly be the same. Cantor's diagonal proof is simple but certainly surprising too. You may want to check. It is amazing.
If [math]\aleph_0[/math] represents countable infinity and [math]\aleph_1[/math] uncountable infinity, then the continuum hypothesis says that:
[math]\aleph_1 = 2^{\aleph_0}[/math]
There cannot possibly be one infinite cardinal. According to Cantor's proof, there are at least two. Furthermore, there is this assumption that infinity is actually an infinite sequence of infinities:
[math]\aleph_0, \aleph_1, \aleph_2, \aleph_3, \aleph_4, ...[/math]
And that the next infinite cardinal is two to the power of the previous one (generalized continuum hypothesis):
[math]\aleph_{i+1}=2^{\aleph_i}[/math]
Through the Löwenheim-Skolem theorem, this explains why first-order arithmetic (PA=Peano's Arithmetic) has more than one model (=more than one Platonic world that satisfies the theory) -- one for each infinite cardinality -- and therefore why Gödel's incompleteness theorem ends up proving that PA is "inconsistent or incomplete".
[math]\underset{x\to a}{\mathop{Lim}}\,F(x)=\infty [/math]
We may say, "F(x) goes to infinity as x goes to a", but what we mean is that for each
[math]M>0[/math] there exists a positive number [math]\varepsilon =\varepsilon (M)[/math]
such that
[math]\left| x-a \right|<\varepsilon \Rightarrow F(x)\gt M[/math]
[math]-\infty [/math] can be used in a similar manner. And there are comparable definitions for complex valued functions in the complex plane.
It doesn't go much beyond this sort of (Cauchy-Weierstrass) definition in analysis. Orders of infinity and the like normally don't appear in the literature. However, soft or modern analysis does move in the general direction of set theory and algebra. And set theory is a different story; most of the posts in these kinds of threads pertain to that subject. :cool:
:lol: If you find it difficult to believe try physicist George Gamow's book, One, Two, Three,...,Infinity which was published in 1947.
Loosely, ? is a quantity that's not a number, and one ? is the quantity of numbers.
I may be guilty of reading too much into it but there is a sense in which infinity defies quantification, at least one aspect of it and that's the difficulty or impossibility of fixing its numerical value. All I did was look at other areas where we face a similar situation and the world of quality seemed an obvious one.
It bears mentioning that the existence of the qualitative concept doesn't mean that what we view as qualities are unquantifiable per se. By that I mean quantification that will appear quite crude and unsophisticated to scientists and mathematicians but still qualify as a quantification. For instance, taking the qualities love and courage it is common practice to make statements like, "he loves me more than you" or that "x is less courageous than y". Such statements, as indicated by the words "more" and "less", directly involve quantification, the crude and unsophisticated kind I mentioned a few lines before. The whole point of quantifying is to aid us in decision-making when two or more options are available; the choice we make based on the quantitative differences between the option. For instance Mary may choose to marry John than Tom because she feels John loves her more.
Now, it seems rather easy when I put it the way I did but actually making a decision requires accurate values for what is being compared. On what basis can one ever infer, say, that x loves y more than z? Only when we can actually fix the exact numerical value of how much x loves y and how much z loves y i.e. we need actual numbers to work with: If I could determine that on the love scale, x loves y 9 and and z loves y 7 then I could rightly infer that x loves y more than z loves y. The difficulty is that this isn't possible and the word "more" or "less", in the case of love and also for other qualities, suffer from the absence of a fixed numerical value. Isn't this exactly the same problem with infinity for it too has no fixed numerical value?
The affinely extended real number system simply adds infinity as two real numbers in a Cantor-like approach:
Quoting Wikipedia on the affinely extended real number system
According to the explanations this extension approach is entirely consistent with existing results in mathematical analysis. This extension approach is actually what the Löwenheim–Skolem theorem does for Peano's arithmetic (PA) and other theories with infinite models:
Quoting Wikipedia on the Löwenheim–Skolem theorem
If you can extend the real number system with uncountable infinity then you can extend it with any other infinite cardinality (upwards), since the theory is unable to control the cardinality of its infinite model. However, this also depends on whether this real number system is still a first-order theory.
Still, the fact that an affinely extended real number system is possible, suggests that mathematical analysis may have exactly the same interpretation problem as Peano's arithmetic (PA), i.e. if one infinite cardinality satisfies the model, then all other upward infinite cardinalities also do.
Hence, mathematical analysis could suffer from the same fundamental interpretation problem surrounding infinity.
Respectively number steals the idea of infinity or you naturally think 1.
Reformatting is something that should happen often in thought.
The answer to life is a number, but it's likely not base 4.
Therefore infinity is a something, and a number of it. We make use of the infinite. Pulsation, for example, where we love doing things repetitively. Addiction, etc.
Me? Any will do, depending on context I suppose.
?[sub]0[/sub] is a quantity that's not a real number, and ?[sub]0[/sub] is the quantity of naturals/integers/rationals
?[sub]1[/sub] is a quantity that's not a real number, and ?[sub]1[/sub] is the quantity of reals
It was really just a colloquial "definition", pointing out that ? ? R, |R| is ?
Classical real or complex analysis: very doubtful. Soft analysis: no telling where that is going. :cool:
Yep. And, it is wrong.
Start with a poor definition and that's the soet of mess you get into,
See, instead, number.
It's not just that you are wrong, but that also you are not even not wrong.
Quoting Wikipedia insisting that the reals are a second-order theory
Quoting Wikipedia on real closed fields which are a first-order theory
Quoting Wikipedia: unlike natural-number logic, real-closed field logic is decidable
Real closed fields seem to be dramatically different from the standard real number system, but what is the actual difference?
Then, they say something very interesting but really complicated about the generalized continuum hypothesis in conjunction with real closed fields. I think that this is the kind of things that could shed light on the true nature of infinite cardinality in real-number theory.
(if only I understood what they are saying ...)
It means essentially that CH is equivalent to the fact that all models of the hyperreals are isomorphic. The idea is that the particular model of hyperreals you get depends on which nonprincipal ultrafilter you choose. If CH holds then all the models are isomorphic.
There's a Mathoverflow thread about this, let me see if I can find it. Ah here it is. Good luck reading. MO as you know is a site for professional mathematicians so the best one can hope for is to understand a few of the words on the page.
https://mathoverflow.net/questions/136720/why-does-ch-imply-that-there-is-a-unique-ultrapower-of-mathbbn
Also see:
https://mathoverflow.net/questions/88292/non-zfc-set-theory-and-nonuniqueness-of-the-hyperreals-problem-solved
I don't know the answers to all the good questions you raise, but I can't help thinking that you're overthinking things and letting yourself get confused by Lowenheim-Skolem.
I am confused myself over whether the completeness property is first or second order. I've seen explanations both ways. I believe it's second order. The hyperreals are a model of the first-order theory of the reals, but the hyperreals are not Cauchy-complete. That seems to imply that completeness must be second order.
Quoting Wikipedia on real closed fields which are a first-order theory
Makes perfect sense. The algebraic numbers are not Cauchy-complete but they are a real closed field, just as the real numbers (which are Cauchy-complete) are.
Quoting alcontali
This really isn't true, since the standard reals with Cauchy-completeness are second order. They provably have cardinality [math]2^{\aleph_0}[/math]. This is the part where you're confusing yourself.
Also the extended reals of analysis with [math]\pm \infty[/math] have nothing to do with any of this. The extra points don't participate in the field properties as I'm sure you know from calculus.
This is incredible. I didn't know that this was possible. It is so unlike the models of PA:
Quoting Wikipedia on nonstandard models of arithmetic
Well, yeah, I wasn't aware of the fact they behave so differently ...
Quoting fishfry
In my opinion, it is primarily a question of figuring out the foundational concepts embodied in the specialized vocabulary for the subject, i.e. real-numbers model theory. I had to do that for something a bit simpler, i.e. model theory and nonstandard models for PA. It looks like model theory related to real numbers is an entire subject in itself, even bigger than PA, with even more concepts to digest. It doesn't seem to have famous, celebrated theorems, though; unlike PA with Gödel's incompleteness theorems.
Quoting fishfry
I wasn't sure if real-number theory is first order or second order. There is a disclaimer in the Löwenheim-Skolem page that it does NOT apply to second-order theories. In fact, I was aware of that disclaimer. Real-number theory turns out to be essentially second order. That was the main source of confusion.
To tell you the truth, this is the first time I have run into documentation about real-number model theory. It is totally new to me. As far as I am concerned, it is a completely new world. Very few concepts of natural-number model theory seem to transfer unchanged ...
Quoting fishfry
Calculus was just school exam material for me consisting of endless symbol manipulation. I didn't particularly "care" about it. It is not that I have read anything about calculus ever since. It doesn't appear in computer-science subjects either. So, what am I supposed to do with it?
Field algebraic structures are more interesting to me. They reappear in cryptography. In elliptic-curve cryptography, the algebraic structure is specifically extended to contain [math]\infty[/math], which does effectively participate in the field. It is even the identity element for addition (without which the structure is not even a field):
Quoting Wikipedia on ECC
What anthropological legend is apocryphal?
If infinity is a number greater than any real number, then by definition it is not a real number. So what kind of number do you propose that infinity is? If it is a real number greater than any real number, that is contradictory. If it is some other sort of number, how would we establish a relationship between this other number system, and the real numbers?
The 'Second-order' reals (as described via second-order logic) are also 'unique' from a constructionist perspective; for if the Axiom of Choice is rejected then second-order quantification over the sets of reals is strictly interpreted as quantifying over the constructable-sets of reals. Consequently, what we then have is a first-order countable model of the reals in 'second order' disguise. The reason why the real number field is unique in this interpretation is because we are actually still working within first-order logic; and since the Ultrafilter Lemma isn't constructively acceptable, the Löwenheim–Skolem theorem for first order-logic that depends upon it fails. Therefore constructive first-order models of the reals only possess models of countable cardinality. Consequently, there cannot exist models of constructive reals that are "non-standard" thanks to Tennenbaum's theorem that denies the existence of non-standard countable models that are recursive.
From this constructive perspective , the semantic intuition behind CH is trivially correct: There are no subsets of R whose size is greater than N but less than R, simply because the real numbers are encodings of natural number elements (via Godel numbering of the underlying computable total functions) and therefore they are of the same number. But alas there only exists an effective algorithm for deciding the provably total functions, i.e the provable real numbers, and hence there is no constructive proof that the number of provably constructive real numbers equals the number of constructable real numbers.
People frequently argue that problem has been solved by the concept of a limit. But that is not the case because unlike things in nature, there is no speed limit in mathematics. All operations in mathematics, counting 1, 2, 3, etc. happen instantaneously. x doesn't tend to infinity but is infinity. In the function above, x will never reach the x axis for any real value of x no matter how large it is, the moment you say x is some real number, essentially you have stopped "tending to infinity." Here is my solution to Zeno's paradox, things that happen in reality like motion do not exactly correspond to things in mathematics. Suppose you have a rod of a finite length. You cannot measure it with a ruler to determine its mathematical length exactly and mathematics always demands things be exact or it is just an estimate.
All the rules of mathematics for real numbers do not transfer over to arithmetic involving infinity. For example, for real numbers except zero (not all the rules in arithmetic apply to zero either for you cannot divide by it) you cannot perform an operation on one side of the equation only and expect to obtain a true result. Whereas (infinity = infinity) and (infinity + infinity) = infinity are both true.
non-finite
You invoked the extended real numbers and claimed it has something to do with L-S, which of course it does not. Unless I misunderstood your point.
If this were true, then mathematics could not give us truth. But it's not true, because mathematics can correspond exactly with reality. Consider that I have a table with some chairs. I can count the chairs and know that there is exactly six chairs there. If I want ten chairs, I can know that I need to get exactly four more. In some instances though, the mathematics is applied in a way which doesn't correspond exactly with reality, and this creates a problem like Zeno demonstrated. "Infinite division" does not correspond to reality, so this idea is itself a problem.
When you say mathematics can correspond exactly to reality, what do you mean? Numbers consist of 0-9 symbols. I'm sure the grass is nothing to do with those symbols. I would say, grass is a number, but it's not base 4.
There is infinity between the eyes and color.
Infinity is an eternity of time; in base 4 number, it is something, but that something is base 4, and not literal infinity.
As in my example, There are six chairs at the table. I need four more to have ten chairs at the table. If this is not an exact correspondence with the reality of that situation, what more is needed? What I want is ten chairs, that is the reality of the situation. Doesn't mathematics tell me exactly and precisely that I need four more chairs?
I'd cut exactly and just put correspond.
Number can be any symbol, a C or an O, for example. It is also a 1 but that's only a property of C and O.
Yes, I could write 1 on one chair, 2 on the next, 3 on the next etc., to count them. Then I'd have a direct correspondence. After all the chairs are marked, I'd know that there is six chairs, and I could subtract six from ten to see that I still need four more to have the desired ten.
Wait, what? That's two things. The thing that exists and the number one. If a thing exists and there's no conscious entity around to comprehend it, there are no numbers. That would be my view. That numbers are an artifact of consciousness. There's a thing, but there's not the number one till someone experiences that thing; and moreover, evolves sufficient reason to count the thing. Counting's not an inherent part of the universe. It's something rational beings do. No experiencer, no numbers.
That seems unduly restrictive. By that criterion you would have rejected Riemann's non-Euclidean geometry in the 1840's because it was so obviously untrue about the world. Then when Einstein used Riemannian geometry to frame his general theory of relativity, you'd have had to change your mind. Except that by your logic, you'd have abandoned research into Riemann's work and Einstein would never have had the tool available.
Isn't it rather the job of math not to describe reality as we know it; but to provide concepts and tools that may be of use to future scientists? The existence of a mathematical object depends only on logical consistency and interestingness. Not on conformity to the limitations of contemporary knowledge of the world.
Quoting The 'Art of Solving Problems' on giving examples for the ultrafilter concept
I see. ;-)
If I understood the explanations correctly, Löwenheim-Skolem applies to the theory of real closed fields (=first order theory) but not to to the theory of real numbers (=second order theory). That last bit wasn't immediately clear to me:
Quoting Wikipedia on Löwenheim-Skolem in the context of real numbers
I had never read anything on model theory for real numbers. The materials I had run into were all about natural numbers.
To add to the Wiki quote, something I mentioned earlier: The hyperreals are not Cauchy-complete. No non-Archimedean field can be. Which leads to one of my little hobby horses. The constructive reals aren't complete because there are too few of them, only countably many. The hyperreals aren't complete because there are too many of them, the reals plus an uncountably infinite cloud of infinitesimals about each real. The standard reals are the Goldilocks model of the reals. Not too small and not too big to be Cauchy-complete. They're just right. And are therefore to be taken as the morally correct model of the reals.
Zeno's paradox is best solved by observing how you would practically explain the paradox. To practically demonstrate the paradox requires one to repeatedly move an object along the same path, but ending the motion at the half-way point of the previously travelled distance and exclaiming "the object must have earlier travelled through this point".
In other words, a demonstration of Zeno's paradox can only explain what an object position is by destroying the object's motion. In other words, this demonstration shows that the construction of a position is incompatible with the construction of a motion, and hence is an intuitive demonstration of the Heisenberg Uncertainty Principle.
In my opinion, Zeno was close to discovering this principle characteristic of Quantum Mechanics, purely from ordinary phenomenological arguments.
Too few...or too many? The subset of computable total functions that correspond to the provably convergent Cauchy sequences form a countable and complete ordered field, that is a proper subset of the provably total functions.
Quoting fishfry
Quoting fishfry
My current understanding is that there exists indeed a detailed description of the infinite model(s) for real numbers but at this point I am unable to pierce through the dense vocabulary and concepts in order to develop a correct mental picture on the matter.
Concerning the phrase "infinity is a number, but it has a characteristic that all real numbers do not possess", in my impression, it does not adequately reflect the breath and the depth of existing knowledge on real-number model(s). I personally feel that this summary is overly simplistic.
Too few, clearly. There are only countably many of them.
I do apprehend the point that the computable reals are computably uncountable, since there is no computable bijection between the computable reals and the natural numbers.
So what? The moment after Turing defined what it means to be computable, he showed that there are naturally-stated problems that are not computable. Point being that even computer scientists recognize the existence of noncomputable phenomena. See Chaitin's Omega, for example.
So yeah, there's no computable bijection. But there is a bijection, just as sure as there are only countably many Turing machines.
And no countable ordered field can be complete. It's a theorem.
I'm not sure exactly what you're looking for. To my knowledge, and I'm no specialist in these matters, the second-order theory of the real numbers is categorical, which means there is only one unique model up to isomorphism.
On the other hand set theorists do study alternate models of the reals that arise if you change the axioms of set theory. For example there's a famous example of Solovay in which, in the absence of the axiom of choice and the presence of an inaccessible cardinal, all sets of reals are Lebesgue measurable. This kind of thing may be of interest to you if you're curious about alternative models of the reals.
https://en.wikipedia.org/wiki/Solovay_model
Stephen Wolfram wrote something very relevant in that regard:
Quoting Stephen Wolfram on 'Curating the math corpus'
Being knowledgeable of say 1% of these 5 million theorems, i.e. of 50,000 theorems, is probably already overly ambitious.
Hence, that cannot possibly be what it is about.
Furthermore, it does not make sense to memorize these theorems along with their proofs, because it would turn such person into an incomplete and rather useless sub-database machine of the (curated) math corpus. Either you use the machine, or else you build the machine, because in all other cases you are just a slow, failed, useless, and sorry excuse for a machine.
People investigate what they are interested in.
There is no reason why that would necessarily include functional integration, metric spaces, or advanced calculus, none of which would give that person any understanding in other math sub-disciplines such as for example elliptic-curve cryptography or in zero-knowledge succinct arguments of knowledge.
Vitalik Buterin is a good example of what I mean:
Quoting Wikipedia on Vitalik Buterin
Vitalik does not just write about top-level mathematics, as in his medium article series:
Quoting Vitalik Buterin's article intro
Vitalik successfully implemented this mind-blowing math in the ethereum source code. As Linus Torvalds famously said:
Why would someone like Vitalik Buterin even be interested in functional integration, metric spaces, or advanced calculus? At the age of 20 he had already become a multimillionaire from his deep understanding of the math subjects that truly mattered to him, while many PhD graduates in math are associate-lecturers living off food stamps:
These people are not just on food stamps, many of them are also widely despised for their runaway arrogance. They should take an example to Vitalik, in order to improve their lives, known to suck. Vitalik, on the other hand, is a very humble person. He is friendly and pleasant. He does not try to "prove" that other people "know nothing". He is just a good human being, albeit stinking rich too. ;-)
Happy to put this into perspective.
First, this is a philosophical message board and not a mathematical or a general purpose on. It's natural that when math comes up, it's in the context of logic, mathematical logic, set theory, category theory, alternative foundations, constructivism, etc. Those are the parts of math that touch on philosophy.
Second, on a site like this each person brings their own knowledge and experience to the table. Many people these days come to mathematical topics through computer science or related disciplines. So they may know quite a bit about constructive math or category theory as applied to logic, or things like Boolean algebras and so forth, without necessarily having a traditional math major background in analysis, abstract algebra, and so forth. There's no reason to "shudder" at the fact that you know things others don't. Perhaps others know things you don't. I daresay your own mathematical orientation includes advanced knowledge of some things, and maybe not so much on others. You don't want us to shudder at what you know and what you don't, right? People who live in glass houses should live and let live, I say. Don't you agree?
Quoting jgill
You're free to start any kind of thread you like; and if the moderators, of which I'm not one, see fit to let it stand, then it was good. Else not.
But as much fun as it would be, this isn't really the place to talk math. There are some math-oriented sites, one of my favorites is https://mymathforum.com/. If you go over there are start talking about philosophy you'll be off-topic; but your advanced math comments and questions will be welcome. Likewise there is the famous https://www.physicsforums.com/, which has some pretty decent mathematicians. They're strictly anti-philosophy. They delete anything even remotely philosophical. And being a physics forum, they're much stronger on differential geometry than abstract algebra. It all depends on the orientation of the board.
To use an analogy, say we're talking about life. On a philosophy forum we might ask, what is the meaning of life. On a biology forum we'd ask about the role of osmosis in the Krebs cycle. Likewise when we talk about the brain here we talk about the mind and consciousness; and not so much about the electrochemical mechanisms underlying neurotransmitter reuptake.
Make sense?
But still. In an online forum the only true rule is: Post whatever you want and let the mods take it down.
The computable total functions are sub-countable. An enumeration of all and only the constructively convergent cauchy sequences isn't possible as this is equivalent to deciding every mathematics proposition. Nevertheless we can construct a countable enumeration of a proper subset of the computable total functions, namely the provably convergent cauchy sequences with locateable limits, which collectively constitute a complete and ordered field, where by "complete" we mean with respect to a constructive least upper-bound principle.
The computable numbers are countable. That's because the set of Turing machines is countable. Over a countable alphabet there are countably many TMs of length 1, countably many of length 2, etc.; and the union of countable sets is countable. QE Freaking D.
"where by "complete" we mean with respect to a constructive least upper-bound principle."
Well sure, if you supply your own definition of complete then you can make anything you like conform to your made up definition.
Turing recognized the importance of non-computability. Too many Wiki pages, not enough math, that's my diagnosis of your posts.
The sequence of n-th truncations of the binary expansion of Chaitin's number is a Cauchy sequence that does not converge to a computable real. End of story. Then you say, "Oh but that sequence isn't computable," and I say, "So freaking what?" and this goes on till I get tired of talking to yet another disingenuous faux-constructivist.
Thanks for your post regarding mine, fishfry. Your quote above to sime is germane.
I’ve ruffled some feathers with my post, for which I apologize. I got a bit irritated last night and didn’t express my thoughts well.
First, I’m not coming from a feeling of superiority regarding math. As a retired prof my interests are in a sliver so small it’s barely visible, one low-interest page among 40,000 on Wikipedia. There are sophisticated discussions on this forum about math, computer science, and logic that I can only stand aside and watch. And most conversations about foundations are beyond me.
But sometimes posters will make statements about mathematics in general that are erroneous, but said with conviction. Such as claiming that math proofs are computer programs, or that there are no more geometrical proofs. Or saying that fiddling with axioms makes the entire body of mathematics flawed, when, in fact, most mathematicians wouldn’t even notice. Claiming that irrational numbers are a mistake and that this undercuts the entire structure of mathematics. Stating that calculus is largely manipulating symbols and that formal education is detrimental. That adding a symbol, a “number”, for infinity will undermine current mathematics. For misusing the expression “chaos theory” when discussing randomness. For claiming that much of what we know of math now was derived or discovered two thousand years ago. On and on. I've probably misinterpreted some of this. If so, apologies.
It’s this moving away from what one knows to speculative territory, but being convinced one is correct – that’s a little annoying to me. But this is a philosophy forum, so no harm done.
As for physics, well all is not well in that discipline. For example, there is an argument about the aether that seemingly goes as follows: The premise is that every wave must travel through a physical substance, and that the aether exists. Electromagnetic pulses are waves, therefore must be propagated through the aether. Hence, electromagnetic waves travel through a physical substance. Makes sense if the premise is true. It's conjecture stated as fact.
I took a year of physics in college, and as a math prof used some physics in my classes. But I would feel incompetent to engage in a discussion about anything beyond the simplest ideas. But here we have string theory, differentiable manifolds, general relativity, entanglement, Bell’s theorem, and on and on – all as if the poster is sure of what he is talking about and not merely parroting Wikipedia. Maybe it’s no more than a lack of modesty. If I have offended anyone, sorry.
?? Perhaps I should have been clearer from the beginning, but i took everyone's understanding for granted that a computable number refers (in some way) to a computable total function. Apologies if that is the case. For surely you appreciate that the computable total functions aren't countable?
The computable total functions are a proper subset of the computable functions that also contain partial functions. i.e. that do not halt on a given input.
It is true to say that the whole set of computable functions is countable, for reasons you'e sketched. It is not true to say that the set of computable total functions are countable, for we cannot solve the halting problem. Hence the reason why we say the computable numbers are sub-countable: the only way we could 'effectively' enumerate the computable numbers is to simulate every Turing machine and wait forever, meaning that any 'candidate enumeration' we construct of our computable numbers after waiting a finite time is also going to contain computable functions that aren't total and hence are not numbers.
For the constructivist, this "subcountability" is all 'that 'uncountability' means. It is simply means that we can never construct a total surjective function from the natural numbers onto the computable numbers. It doesn't mean in any literal sense that we have more computable real numbers than natural numbers.
Quoting fishfry
We have to be careful there. We can run every Turing Machine and at any given time create a bar-chart of the ones which have halted, and this histogram comprises a sequence of computable functions whose limit isn't a computable function. To my understanding this sequence of functions isn't cauchy convergent, for we cannot construct a bound on the distance between successive histograms. Let's not forget that there are an infinite number of computer programs of every size.
Compare this situation to a computable total function f(n) representing the "values" of the Goldbach's Conjecture; Let's say that f(n) = 0 if every even number less than n is the sum of two primes, otherwise f(n)=1. Here we can also compute the individual digits in finite time. If GC is decidable, i.e. GC OR ~GC, then f(n) is Cauchy convergent to either 0 or 1. But if GC isn't decidable, then as with Chaitin's constant f(n) doesn't have a cauchy convergent limit, even though f(n) is a computable total function.
Therefore, in order to know that one has constructed a complete and ordered field of computable numbers, one must only use a set of provably Cauchy-convergent computable total functions, for which every cauchy-convergent sequence of these functions is also provably cauchy-convergent.
You were already clear. I reviewed the wiki article on subcountability and nothing you said caused me to change anything I wrote.
https://en.wikipedia.org/wiki/Subcountability
There seems to be an ambiguity between two definitions of completeness. If Dedekind completeness is understood to be an axiom of construction then it is trivially satisfiable in the sense that the axiom itself can be used to assist in the generation of a real from an existing list of real numbers. After all, if there wasn't a countable model of the Axioms of the reals, then they would be inconsistent, since Second-order quantification can always be interpreted as referring only to the sets constructively definable in first-order logic.
On the other hand, if completeness is understood to refer to a finished list of PCCTFs, our list is not complete in that sense.
So it seems to me that countable model of reals, both first and second order, are especially useful ( not to mention the only models we use in practice),for clarifying the relationship between Dedekind completion, Cantor's theorem and ordered fields.
If one abandon's the second-order completeness axiom, and possibly cauchy convergence, then there are less constraints in the construction process, allowing one to define a potentially larger field of computable numbers that includes infinitesimals as is done with the (constructive) Hyperreals, and one can even include computable 'numbers' that are aren't provably total. In which case ones countable list is now finished, but now there are no more numbers to be added, because now the diagonal argument cannot be used to construct a new numbers in virtue of one's list including non-numbers that aren't guaranteed to halt on their inputs.
So i hope this had lead to a satisfactory conclusion.
Now that's something I've never run across. Both too big and too small at the same time. But it takes a weak form of the axiom of choice to have a nonprincipal ultrafilter, which is needed to construct the hyperreals. Do constructivists allow that?
First, your own logic results in the fair conclusion that infinity is a concept. If infinity is a number and all things mathematical are concepts, then infinity is a concept.
Second, even if all things within the realms of mathematics or logic are a concept, that does not mean that all concepts are within the realm of mathematics or logic.
Consequently, it you are correct and infinity is a number, then it is okay to call infinity a concept. On the other hand, if you are incorrect and infinity is not a number, then it is still okay to call infinity a concept.
Perhaps you should have settled for infinity as a mathematical concept and argued for a mathematical representation of the that mathematical concept.
Just saying.
emmm......... Nope :) for the reason you've just mentioned. For where is the algorithm of construction? Of course , the trivial principle ultrafilter is permitted, which then produces a countable model..
By "constructive hyperreal" i was merely colloquially referring to using functions such as f(n)=1/n as numbers according to some constructive term-oriented method that didn't involve assuming or using cauchy limits.
From what I've seen, constructivists typically allow weak forms of choice, for the reason that otherwise you can't get satisfactory math. So it wouldn't surprise me if some constructivists allow nonprinciple ultrafilters. The trivial ultrafilter of course doesn't give you the hyperreals but I can see how it might produce something that might be called constructive hyperreals.
Quoting sime
Wait so you just made that up? It's not a real thing? You had me convinced. Why not mod out the reals by the trivial ultrafilter and see what you get? What do you get?
Why are there so many die-hard constructivists on this forum? If you go to any serious math forum, the subject never comes up, unless one is specifically discussing constructive math. You never see constructivists claiming that their alternative definitions are right and standard math is wrong. Only here. It's a puzzler.
Just a guess, but I would imagine that one typically becomes a constructivist in the first place for primarily philosophical reasons--e.g., dissatisfaction with the philosophical basis of standard math, hence the desire for and advocacy of alternative definitions. Since this is a philosophical forum, rather than a mathematical forum, it is a natural place for committed constructivists to make their case.
Well obviously from a pure mathematics perspective, every proof in ZFC is considered construction, in contrast to Computer Science that has traditionally had more natural affinity with ZF for obvious reasons, and there is a long historical precedent for using classical logic and mathematics. As a language, there is nothing of course that classical logic cannot express in virtue of being a "superset" of intuitionistic logic, but classical mathematics founded upon classical set theory IS a problem, because it is less useful, is intuitively confusing, false or contradictory, lacks clarity and encourages software bugs.
In my opinion, Constructive mathematics founded upon intuitionistic logic is going to become mainstream, thanks to it's relatively recent exposition by Errett Bishop and the Russian school of recursive mathematics. Constructive mathematics is practically more useful and less confusing for students in the long term. Consider the fact that the standard 'fiction' of classical real analysis doesn't prepare an engineering student for working in industry where he must work with numerical computing and deal with numerical underflow.
The original programme of Intuitionism on the other hand (which considers choice-sequences created by the free-willed subject to be the foundation of logic, rather than vice versa) doesn't seem to have developed at the same rate as the constructive programme it inspired. However, it's philosophically interesting imo, and might eventually find an applied niche somewhere, perhaps in communication theory or game theory.
BTW, i'm not actually a constructivist in the philosophical sense, since the constructive notion of a logical quantifier is too restrictive. In a real computer program, the witness to a logical quantifier isn't always an internally constructed object, but an external event the program receives on a port that it is listening. What's really needed is a logic with game semantics. Linear logic, which subsumes intuitionistic and classical logic is the clearest system i know of for expressing their distinction and their relation to games.
As for a trivial ultrafilter, its an interesting question. Perhaps a natural equivalence class of Turing Machine 'numbers' is in terms of their relative halting times. Although we already know that whatever reals we construct, they will be countable from "outside" the model, and will appear uncountable from "inside" the model.
Don't worry, nobody noticed or cares. You only ruffled a feather or two of mine, and I'm easily ruffled.
Quoting jgill
Very few working mathematicians care about foundations. A lot of philosophers and pseudo-philosophers imagine that all mathematicians sit around writing proofs directly from the axioms of ZFC. As has been often noted, the average mathematicians couldn't write down the axioms of ZFC if challenged.
Quoting jgill
Yes. A lot of that around in the public discourse as well, wouldn't you agree?
FWIW there are the strictly moderated forums that are no fun, and the more loosely moderated forums that allow a bit of give and take, but are thereby welcoming to people with varying degrees of knowledge and sanity. Among all the loosely moderated philosophy forums, this place is by far the best. It gets a lot worse on some other similar forums. It's just part of the fun of being online. This is not the proceedings of the Royal Society.
Quoting jgill
This is in fact true. It's the famous Curry-Howard correspondence.
If you think about it, it's quite sensible. Say we prove that some wildly non-constructive object has mathematical existence. Vitali's nonmeasurable set for example.
Nevertheless the proof of existence is a constructive object. It's a sequence of syntactic moves starting from a set of axioms, which are well-formed formulas of some formal language; and a set of inference rules. Given the axioms and the inference rules, a computer could calculate whether a given derivation is legal.
In effect we're ALL constructivists. We construct proofs, even if those proofs claim the existence of nonconstructive objects. Our proofs literally are translatable to computer programs in an abstract sense.
One of our resident constructivists got me to understand this a while back. I got some insight into constructivism from that.
Quoting jgill
Haven't seen that one. But this is not a technical forum. It's a freewheeling discussion forum that is nevertheless far more intelligent in general than most other online discussion forums out there. And of course Wikipedia is partially to blame. A lot of people think they know things these days that they really don't know.
Quoting jgill
Hard to explain to philosophers how little working mathematicians care about foundations. If ZFC were discovered inconsistent tomorrow morning, hardly anyone would care besides the specialists. Nobody ever heard of set theory before Cantor but a lot of great math was done. Attitudes towards foundations come and go as a matter of historical contingency.
Quoting jgill
Our friend @Metaphysician Undercover is a special case. He is so sure of himself and he writes well; so the challenge on a forum like this is to try to engage him rationally and see how well one understands and can advance their own point of view. I've always found that when I'm debating someone online who has an unorthodox/alternative/cranky/crazy opinion, the real challenge is to see if I can be transcendentally clear and persuasive myself. Either that or just ignore what you don't like. That's what free speech is about IMO, and discussion forums are about community-moderated free speech.
Quoting jgill
To be fair, that's exactly how we teach it. "Bring down the exponent and subtract one." Calculus is a service course for the benefit of the engineering, physics, economics, pre-med, and other departments. It's got very little to do with math. You can't blame the kids for being confused. As my grad advisor put it to me once, when I was about to embark on being a calculus TA: "Freshman calculus is a futile exercise in mind fucking." Truer words were never spoken.
Quoting jgill
Also to be fair, many of the high and mighty in the land say the same. Isn't Elon Musk one of those masters of the universe telling kids to drop out of school and just get to work doing what they care about?
And from what I've heard about higher education these days, education's not what it used to be. I'm not sure any of us are in a good position to defend what passes for formal education these days.
Quoting jgill
Lot of confusion about the extended reals and their relation (which is none whatsoever) to the transfinite numbers of set theory.
Quoting jgill
Now this is a malady common to science journalism in general. A lot of your concerns are better addressed to the miserable state of science journalism in general. AI hype, quantum computing hype, hype in general.
Quoting jgill
Most people don't know much about math, even educated people. Might as well yell at the tides. The challenge of a venue like this is to state your case as clearly as you can and see if anyone's convinced. If all you want is technical questions and authoritative answers, that's what Stackexchange is for.
Quoting jgill
None needed, really. You do give the impression of not having been on the Internet much. The world has had this problem since Gutenberg. Once you give the public a voice, no telling what they'll say. Reminds me of something Churchill said. "The best argument against democracy is a five minute conversation with the average voter."
Quoting jgill
It's the nature of online discourse. And public discourse too. If you have no idea what you're talking about, say it real loud and with a sense of self-righteousness. Again, there's always Stackexchange. This place ain't that. And it's a good thing in general. Think of it more like the corner bar. Takes all kinds.
Quoting jgill
This was resolved in the 1900s. There is no luminiferous aether. I did not know this is still an issue. I don't think it is.
Quoting jgill
Well of course it's the Wikipedia factor. Someone reads a Wiki page and they feel emboldened to vociferously promote their own mistaken understandings; even if they are talking to someone who actually knows what they're talking about. Nature of modern society.
Thanks for the compliments. The biggest stumbling block between us is your concept of "mathematical existence". The proof that something has mathematical existence is really meaningless unless we have a rigorous definition, or convention, concerning what "mathematical existence" means. If it simply means to be consistent with some set of axioms, and we have no standard as to how an axiom might be justified, then all sorts of fictions may be proven to have mathematical existence. And, if mathematical existence is not consistent with "existence" in the more general, philosophical sense, then it's not even a type of existence at all, and use of the term "existence" is misleading. Then we'd be better to replace "mathematical existence" with "pseudo-existence", or "crackpot existence", so as to be less misleading with our terms.
Does the knight's move have chess existence? The other day you said you reject chess because it doesn't refer to anything in the real world. That's extreme nihilism. You can't get out of bed in the morning with a philosophy like that. How do you know it's your own bed? Property's an abstraction.
This is certainly valid regarding the structure of a mathematical argument. But by itself it leaves the impression that mathematics is merely symbol manipulation and not what it really is: exercises in imagination and creativity. On the other hand, it may be that sometime in the future AI will explore and develop mathematics so convoluted and complicated that the results will be on the edge of human understanding, or beyond. Perhaps issues like the nature of time will be resolved, but humans will not be able to comprehend the results. Who knows? :chin:
Quoting fishfry
Oh, I am well aware of that! :nerd: I taught calculus at all levels for 29 years. But my introduction to the subject was unusual: after taking an excellent course in analytic geometry as a freshman at Georgia Tech in 1955, I was recruited, along with about fourteen other students, into an experimental first quarter calculus course taught by two professors. Epsilons and deltas on the first day and no text book. Mostly we were bewildered at first, the exception being two brainiacs who caught on instantly. There was no attempt to continue the experiment into the second quarter, so we all migrated back to the standard curriculum for engineers, physicists, etc. What a huge difference!
At the University of Chicago in the fall of 1958, I was surprised to learn that the physics department was no longer allowing its students to enroll in courses from the math department and was teaching its own mathematics.
Long ago, what attracted me to mathematics was the same thing that attracted me to my avocation as a rock climber: exploration, discovery and creativity. The course work in grad school was someth9ing I needed to plow through, and doing assigned problems that had been solved by generations of students was a chore. Like repeating well established climbs. The original research at the end was a delight, however.
Quoting fishfry
Yes, a few spectacular success do just that. I disagree. I've watched students of mine graduate and move on into successful careers.
Quoting fishfry
:smile: Wrong impression, Dude! I was on an outstanding climbers' forum for years until it folded last May. Along with a great deal of climbing discussion, there were threads about other subjects. The one I particularly enjoyed delved into the nature of mind and consciousness.
By the time the forum ended, this thread had well over 20,000 posts. Among those participating were several mathematicians, a well-known physicist, a neuroscientist, an academic anthropologist, a retired management prof, several academic philosophers, a well-known author who has practiced Zen for decades, and many others at all educational levels who chimed in from time to time. In particular, the debates between the physicist and the Zen person were stimulating. Also, even though avatars were used we all knew the identities of the primary contributors. So you see where I'm coming from. :cool:
Thanks for your comments!
What I said, or at least meant, is that I refuse to play chess because I find it irrelevant to my endeavours, so it's a waste of time. What could you possibly mean by "chess existence"? Let's say that the game consists of some physical pieces, and some stated "rules". What you have referenced is "a rule". How do you think that a rule exists? Does it exist as the symbols on the paper as the stated "rules of the game"? If so then these rules require being read, and interpreted, understood, in order for someone to actually play the game. Then the person's play is dependent on the person's interpretation. If the rules exist as the interpretation, within someone's head, then who's interpretation is correctly called "the rule". Consider President Trump's, impeachment trial. Who's interpretation of what is required for impeachment constitute the actual existing "rules"? And if it's what's written on paper, that is the existing "rule", how does it have any meaning as a "rule" without being read and interpreted?
Because I have not seen any resolution to these questions, I would not say that a "rule" has any existence at all. I think it is a simplification of something we do not understand. There is a subject of human behaviour, habituation, etc., which is not well understood, and some people like to represent it as understood, so they say there are "rules" which human beings are following. The use of "rules" creates the illusion that human behaviour is understood. The human being follows rules, just like matter follows rules of physics. Use of the term "rule" is just a convenient fiction, used to hide the fact that this subject is not well understood. It's a fiction because it doesn't represent any real, existent thing, it just creates an illusion. So when I get out of bed in the morning (hopefully it's the morning), I am not following rules, I am acting on my own terms. That's what it means to be a free willing human being.
What do you think "rule" signifies?
I take your point to heart.
The actual meaning of mathematical existence is that it's whatever working professional mathematicians say it is. You don't accept that, but that is how it works.
I do take the point that this is not sufficient for you; and that if ALL I mean by mathematical existence is something I can prove from arbitrary axioms, that's not much of a criterion for existence. I could posit the existence of purple flying elephants but that wouldn't mean I've proven their existence.
I would be willing to stipulate that although the criterion I gave: that mathematical existence is whatever professional mathematicians say it is; I do owe you a better explanation. I haven't got one at the moment that would be satisfactory to you. But I do want to say that I take your point and I'm mulling the question over in my mind.
You have the same objection to football, baseball, Chinese checkers, and whist? You reject playing poker because the only Queen you know is Elizabeth? Nihilism. Childish rejection of the very concept of abstraction.
I see this a lot among those who have seen a little category theory in the context of computer science, and think they understand the deeper meaning of math. I also see this among novices who find out for the first time that math is based on axioms. They immediately leap to the conclusion that math is about writing down the consequences of the axioms. On the contrary, the math itself precedes the axioms. We know what's true and then we try to formalize it. The formalization is distinctly secondary to the math.
My sense is that professional philosophers of math (Maddy et. al.) perfectly well understand this point. it's the amateurs on the online forums who don't.
Quoting jgill
When I studied math at UC Berkeley I called up the Physics dept one day and asked them if they had a fast track intro to physics for math majors. The person I spoke to said no and was attitudinal about it. The only way to learn physics is from the official physics courses! At that time physicists had a genuine dislike of the math curriculum. I believe things aren't quite as bad as that today.
Quoting jgill
Bad sample space. Anonymous forums are entirely different.
On the other hand I entirely agree with you about certain aspects of this forum. I think I've just finally gotten used to it. And like I say, this forum is the best of the philosophy forums out there.
Yes I take that point. But note that it's a theoretical result about abstract, idealized proofs. In actual every day professional mathematics, proofs are not only not programs -- they're not even proofs as a logician would recognize them. They're mostly informal arguments, as much prose text as symbology.
In other words if a proof is a sequence of statements, each one following from an axiom or a result of previous statements, then no working mathematician has ever seen a proof.
I wonder if that's part of the disconnect between philosophers and mathematicians. Working mathematicians don't write proofs the way philosophers and (some) computer scientists conceive proofs. @jgill has made this point.
Quoting sime
Less than ZF in fact. A Turing machine is an unbounded tape and not an infinite tape. The tape is as long as it needs to be but at any step is always finite. It's a potential infinity and not a completed one if you like. Computer science does not require the axiom of infinity. The Peano axioms will do. Except for those parts of CS that do require infinite sets.
Quoting sime
You and @Mephist are in agreement but again, the question isn't that one framework's better than another. They're all tools in the service of discovering higher truth. The mathematics that's being talked about is the same mathematics whether you represent it in type theory or category theory or intuitionist logic or classical logic. They're all tools to be used as appropriate. It is not a cage match to the death as some seem to believe.
As far as one approach or another being better for programming, there's a long history of one false panacea after another. "Common business-oriented language," or COBOL, was going to make it possible for business analysts to write code. Didn't work. Procedural programming would make software more reliable. Didn't work. Structured programming was the answer. Didn't work. Object-oriented programming, everyone is dumping on it these days. Inheritance is a lie, nobody ever ended up building useful industry-wide libraries of base classes. Now functional programming's the thing. It will solve all our problems.
I've seen a lot of this history first-hand and I'm not likely to be impressed by the latest proposed solution to the eternal software crisis. That to me seems like a very different discussion than the role of intuitionism in math.
Quoting sime
Well sure, it's all a matter of historical contingency and intellectual fashion. I've argued that point myself. I may have my own doubts about constructivism, but I don't deny that it's inevitably gaining mindshare in our age of computation.
Quoting sime
I made an honest, good-faith attempt to understand free choice sequences once. I simply could not get past the idea of a "subject" that makes choices. Too woo-woo for me. And you're right, modern intuitionism became important when the computer became important in the world. Brouwer's revenge.
Quoting sime
I spent much of my professional life working with networked applications, but never thought much about the abstract semantics. What does it mean when Turing machines get external input. I gather it can't make too much of a fundamental theoretical difference otherwise I'd have heard about it, but I could be wrong.
But it sounds like your approaching programs from a proof-of-correctness point of view rather than a day-to-day software engineering perspective. Am I getting that right?
Quoting sime
Needed for what, exactly? You seem to be relating the math to programming theory. Surely little or none of this relates to the building of actual computer systems except at a theoretical level.
Quoting sime
I have no idea. I remember about three years ago I spent some time coming up to speed on the technical aspects of the hypperreals and ultrafilters but I've forgotten most of it.
I hope you see the problem with this. You're saying, if we (mathematicians) agree that it exists then it exists, without any definition of what it means to exist. In any other field, no one would agree that such and such "exists", unless there was a definition of "exists" and some evidence to show that the thing actually exists. For example, would some biologist come in with a fictitious life form and ask the other biologists, can we agree that this life form exists, so that it can be a real existent life form? Or would a physicist propose the existence of a fictitious particle?
What I think is that "existence" is just a facilitator. The mathematicians realize that if they posit the existence of these things, they can treat them logically as we would treat objects, and this makes things much easier. A symbol represents an object, nice and simple. The problem though is that we can't really treat these things like objects. So the mathematicians have created a wall of illusion which separates them from reality. And now, they do not even know how to properly deal with these things which they have assumed to be objects, because they have spent so long wrongly assuming that they are objects, that they have no understanding of what they really are any more.
Quoting fishfry
What I reject, is not the concept of abstraction, but the childish notion that an abstraction is an existing object
Trying to change the subject to distract from the fact that you're wrong? If so, you're unsuccessful because you're still wrong. The critics determine the value and meaning of the piece of art, not the artist.
Quoting tim wood
It seems like you misunderstand "instantiation". From Wikipedia: "The instantiation principle, the idea that in order for a property to exist, it must be had by some object or substance; the instance being a specific object rather than the idea of it".
"I have a chair" is not incoherent, but we need to respect the fact that it may be false. You might not really have a chair when you say this, and then you would be deceiving us. What fishfry has finally started to realize above, this principle: "if mathematicians say it exists then it exists", is a faulty principle.
I would say that the laws of Mathematics and Logic are normative principles pertaining to conduct regulation so as to make the world easier to describe and manipulate.
These normative principles cannot be given a logical justification on pain of circularity, rather their justification stands or falls with their general overall usefulness.
Think of mathematicians sitting around a table and creating a game, discussing the pieces that are played, the environment in which they are played, and the rules that are agreed upon. Once done, would you then say, "The game does not exist."? You fail to recognize that math is a social endeavor, frequently deriving from observations of the physical world, but just as frequently not.
From this perspective, would you say the rules are the axioms? I would say no, there are ill-defined patterns of thought that precede the establishment of the rules, and that might be the subject of study and formalization at a later time - as is the case of the foundations of mathematics.
Just a thought. :smile:
But yes and yes.
Physicists thought one day there must be atoms. Then they discovered the atoms are made of protons and electrons and neutrons. Then they discovered the protons are made of quarks. Now they think the quarks are made of strings. Do any of these abstractions exist? Yes they do, in the sense that they are part of an abstract mathematical theory that explains the experiments we're capable of doing at any moment in history.
Physicist invent new existing things all the time. And de-exist things to. The luminiferous aether was once regarded as existing, till Michelson and Morley couldn't find it and Einstein did away with its necessity.
A scientific entity has existence when it's a necessary ingredient of a successful physical theory. Nobody can say whether a quark or an electron "really" exists; only that positing their existence gives a good theory. That is the definition of scientific existence. And mathematical existence too. I'll go with that, since I challenged myself to define mathematical existence for you.
Biology? Once, disease was caused by ill humours in the blood. Then they came up with the germ theory of disease. Germs are an abstract thing that gives a good theory of disease. Now we can study germs under a microscope, but really, what are they? Bundles of biological material. More abstractions. In the end, they're all quarks and the properties that emerge from various organizations of quarks. But now we treat infections with antibiotics and not leeches, so there is slow progress towards the good. Our abstractions become real because they work. In the future some of the things we think are real will turn out not to be (like the force of gravity) and other things we didn't think were real will turn out to be (electrons, quarks, strings, loops ...)
I gather you call "real" only what is "really out there." But if the 20th century taught us anything, it's that the existence of such a thing as "real things out there" is an assumption and not a fact. I believe if I'm not mistaken this is called scientific realism. It's only an idea. We could kick it around. But you have no logical basis for claiming it's true and everybody else is wrong. The days of Euclidean geometry and Newtonian physics are gone. Now we know the world consists of probability waves that are everywhere at once till we measure them. What can that mean? We don't know. But you claiming that you personally know what things are real, is a delusion on your part. Since you called me delusional the other day, which I can live without.
The problem is, as I demonstrated, the concept of "empty set" is self-contradicting. Sure, a contradictory concept is functional and purposeful, but that purpose is nothing other than deception.
Quoting tim wood
I told you already, abstractions do not exist as objects. That is the oversimplification of platonic realism which Plato himself demonstrated as false. Abstraction is an activity of individual human minds, and the proposition that there is "an abstraction" which is created by numerous human minds, is dependent on both a category mistake and a composition fallacy.
First, abstraction is a process of the human mind, there is no evidence that it produces an object, called "an abstraction". There is recollection of the image, representation with symbols, and application, but no evidence of an abstracted object. The proposition, that abstraction does produce an object, "an abstraction" is a falsity intended to simplify reasoning. It is a convenient falsity, accepted because it produces efficiency, but false because it is based in the category error that mental activity, "abstraction", can be represented as an object, an abstraction.
And, even if we are fooled by the category mistake, and accept that mental activity produces an object, "an abstraction", we have to get past the composition fallacy involved with the proposition that there is "an abstraction" common to numerous human minds.
So, the idea of "an abstraction" is supported by a double falsity. Some human beings might argue that two wrongs make a right, because a double negation is a positive, but that assumption as well, is based in faulty principles.
Quoting sime
At least someone here has a reasonable perspective. Still, there is a problem basing justification in "overall usefulness", because deception is a valid intention. So unless we allow that "useful for the purpose of deception" is valid justification, we need further principles to judge "overall usefulness". I think it is necessary to exclude "useful for the purpose of deception" as a valid justification.
Quoting jgill
Saying "the game exists" has the same problem as saying "the rules exist", or saying "the concepts exist". It is an over-simplification made to facilitate communication. Each of these terms, "game", "rules", "concepts", refers to a complexity of physical objects, symbols, and mental interpretations of the symbols. To make communication smooth and swift we refer to those complexities with simple words. The problem is, that common language use which is an habitual activity, clouds our minds as to what is really behind those terms. Because we use the language as if there is an existent thing referred to by "game", "rule", or "concept", we fall under the illusion that there is such existent things.
But a careful, clear, and rigorous analysis of what is actually referred to by these words reveals that there is no such existent things. Each of these words is used to refer to a massive complexity of social interactions which we do not properly understand how to represent. So, we have a word, we assume that the word represents an existent object, and we go about our business ignoring the fact that the word does not represent an existent object, it really represents a massive complexity of social interactions which is not understood. In philosophy though, we seek to unravel these mysteries of the misunderstood, and that is why I insist on recognizing the reality that there is no existent objects referred to by these names.
Quoting jgill
If I understand you correctly, you are suggesting that the axioms are like proposals for rules. Each axiom is presented by a mathematician as a proposition to be accepted, or rejected, by the others. The various mathematicians will then take these proposals and try them out, relate them to each other, combine them with each other, etc., in a sort of trial and error fashion, and after some time of doing this some axioms will emerge as "the rules". I accept this representation, it's similar to the way we do science, hypotheses are presented, they are related to each other, tried and tested with experimentation, until certain theories emerge as "the rules".
Here's something we ought to respect though. The proposals don't ever change their inherent nature as "proposals", despite the fact that they get accepted into the fold as "rules". Therefore we ought not change our attitude toward any proposal just because it has obtained the status of "rule". In reality, things change, human beings and their social structures evolve, so the "rules" change accordingly. Therefore we cannot allow that a proposal, long ago accepted into the status of "rule", is beyond reproach.
Quoting fishfry
The problem here is that you assume "they are part of an abstract mathematical theory", when there is no such unified "theory". There is a multiplicity of theories, related or connected to each other in various different ways, dependent on an individual's interpretation. So this assumption is misleading. It is the belief that the abstractions are all part of one overriding theory, which gives credence to the claim that they exist. Once we recognize the falsity of this assumption, the assigned unity is lost, and the entire structure which depends on the reality of this unity, falls apart into individual ideas in individual human minds. The assumed reality of these ideas, as existing "abstractions" is dependent on this unity of "an abstract mathematical theory", which is not supported or justified.
Quoting fishfry
Again, to call these things "existing" is an over-simplification designed to facilitate communication. That we say at one time X exists, and later x does not exist is an indication of this. At one time we are comfortable using these theories, that is the convention, so we talk as if they exist, at another time we are not comfortable using them, they have become unconventional, so we talk as if they do not exist. Using the word "exist" is just a convenient fiction to refer to what is and isn't conventional. But if we take that fiction literally, and assume that because a conventional idea "exists", it is therefore an object, and we try to treat the idea as an object, we have been misled down the path of misunderstanding.
Quoting fishfry
Again, you are over-simplifying, referring to a "physical theory" as a unified object. Every theory is interpreted in numerous different ways, by numerous different people, and one interpretation may be demonstrated as unacceptable while another one is acceptable, so there emerges a conventional interpretation.
Quoting fishfry
Sorry, there was no ill intent with the word delusional, and it wasn't meant to single you out. I think we are all delusional, it's a function of where our institutions and conventions have misled us. You might think that science has guided us to the ontology of model-dependent realism, meaning that there are no "real things out there", but this necessitates also that there are no "real things in there". So this form of "realism" is not a realism at all because it cannot validate anything as real. That's why we're deluded, we base reality on usefulness and eloquence, having dismissed truth as unreal. But without truth, eloquence is useful for, and justified through, deception.
Interesting metaphysical limb upon which you are perched. Nothing really exists because there are no entities of sufficient purity that they are not compositions of things, many of which fail to exist themselves. Believe me, mathematics does indeed exist, as do the peculiar thoughts that bubble up into your consciousness, having complex pedigrees which apparently do not exist as well.
Seek help, my friend. You limb is but a twig.
Right, how could a thing which is composed of parts which do not exist, itself exist? This doesn't mean nothing exists, only those things composed of fictional parts do not exist.
quote="jgill;380008"]Seek help, my friend. You limb is but a twig.[/quote]
Size is irrelevant. What matters is that it provides support. Non-existent things provide no support.
You're made up of quarks whose position is a probability wave smeared across the universe until somebody looks, at which time you end up in the place you're most likely to be found. If that ain't fiction I don't know what is. But it works. And you exist.
It doesn't work though, because it doesn't explain how I'm here when no one's looking. So my real, true existence, is not supported by that fiction. That it is, is a delusion.
Perhaps you could publish your interpretation of quantum physics in a reputable journal. Or just explicate it here. A Nobel awaits.
You and the moon don't need the attention.
Notice what you're doing: you're defining an extension to real numbers and real arithmetic. This doesn't magically transform infinity into something it's not (it's not a real number) it just means that the concept of infinity within your extended system, is coherent.
In ordinary arithmetic, infinity is not a number to which arithmetic operations can be applied. It is false to claim that (1/infinity) = 0. Rather, one can abstractly consider where this series leads:
1/n, for n=1,2,3...
the series never ends, but it gets increasingly closer to zero. This leads to the (extended) concept of "limit". The misconception that (1/infinity)=0 is an inexact way of saying that the limit of (1/n), as n approaches infinity, =0. It's useful math, it's fundamental to calculus, but that doesn't "prove" infinity exists in the real world - not in the sense that "7" exists in any collection of 7 objects in the real world.
You would deem the mathematics "supporting" the moon landing and Mars' vehicles non-existent. You would also label the very thoughts you post here non-existent. :roll:
It seems you didn't read my earlier posts
Quoting jgill
Right, the symbols on the paper, and on this page have existence. The thoughts which were used to produce those symbols are events which are in the past, and no longer have existence.
But those who read and interpret those symbols revive those thoughts and give them renewed existence. Thus, like monks reading and reciting scripture, were an order to so illuminate and pronounce mathematical works with unflagging resolve those thoughts would exist forever.
Hemingway's thoughts exist unendingly, for someone, somewhere is reading them now.
Not quite, the readers produce new thoughts, within a new context. So if thoughts are existent things, the old thoughts of the author are different things from the new thoughts of the reader. There is no continuity of existence between a thought at one time and aa thought at a later time, so the two are not the same thing. The moon landing, along with all the thoughts involved, is a distant memory. It may be recreated with new thoughts, but the new thoughts are not the same thoughts as the old thoughts, as is evident from conspiracy theories. The thoughts which supported the moon landing are not existing.
Quoting jgill
This is a myth which science has dispelled. Symbols, which represent thoughts, might exist indefinitely, but not the thoughts themselves. And, as time passes, the context within which the symbols exist, changes. Since the meaning of any symbols is context dependent, the interpretation of the symbols changes with the passing of time.
Nonsense. Take the Pythagorean Theorem: [math]{{a}^{2}}+{{b}^{2}}={{c}^{2}}[/math]
The original thought occurred millennia ago, and it has been transmitted through the intervening years both by a variety of symbols and word of mouth. It remains essentially the same in Euclidean geometry, which by and large is the world in which we live, even though there are other forms of geometry.
Quoting Metaphysician Undercover
Probably true if you qualify your statement. But even then there is indeed an underlying structure of thought which might be analogous to a step function. The function, though discontinuous at points of time, exists throughout a long period and in effect provides a continuity of existence across time in that interpretations exist at each instant, although they may differ, changing abruptly from time to time.
Symbols, and word of mouth do not transmit thoughts from one person to another. It's you who is speaking nonsense. When I write this symbol "A", do you think that there is a thought inside there, which is coming from my mind to go into your mind?
Are you familiar with the law of identity? When I read the symbols you wrote, I do not have the same thoughts as you had when you wrote those symbols. When two things are similar, like my thoughts and your thoughts, they are of the same type, they are not the same thing.
The PT is an idea transmitted down through the ages. Thought = Idea. Your definition of "thought" is far too narrow. You clearly want to keep all your thoughts to yourself. :roll:
There is a reason why we have the law of identity. It was established to prevent the faulty arguments of sophistry. If you think my definition of "thought" is far too narrow, because I adhere to the law of identity, and you'd like to allow that thoughts in different people's minds might be "one and the same" thought, then I see no reason why you would propose this, unless you are trying to argue some trick of sophistry.
I have no problem saying that a thought is "a single idea", where I have the problem is in saying that my ideas are your ideas. As evidence of the difference, there are laws of intellectual property, based in the real separation between the ideas of distinct people. But in this case, it's not that I "want" to keep my thoughts to myself (otherwise I would not be here), but I have a healthy respect for the constraints of reality, which make my thoughts uniquely my own.
This is more primitive than the concept of number: Two children can confirm that they each have the same number of pieces of candy by pairing one child's pieces with the other's — which does not require knowing how to count.
In math, an infinite set is often defined by the condition that it has the same cardinality as some proper subset of itself (a subset that doesn't include all the members of the original set). And sure enough, the positive integers {1, 2, 3, ...} can be put in one-to-one correspondence with the set of even positive integers {2, 4, 6, ...} just by the formula n <—> 2n. (As everyone knows, you can't do this with a finite set.)
There is a very simple proof that shows that for any set X, the "set of all subsets of X" can never be in one-to-one correspondence with the original set X, no matter how big or small X may be. So the set of all subsets of the integers is strictly larger than the set of integers. And doing this again, we find that the set of all subsets of *that* set is larger than that set is. There is no limit to how many times this can be done.
But hold on to your hat! Because the operation of "taking the set of all subsets of a set" can be done infinitely many times! And the union of all the resulting sets is larger than any previous one. So there are actually infinitely many distinct infinities.
This suggests why there is no largest infinite set — because the set of its subsets is strictly larger.