Infinity
If there are an infinite number of natural numbers, and an infinite number of fractions in between any two natural numbers, and an infinite number of fractions in between any two of those fractions, and an infinite number of fractions in between any two of those fractions, and an infinite number of fractions in between any two of those fractions, and... then that must mean that there are not only infinite infinities, but an infinite number of those infinities. and an infinite number of those infinities. and an infinite number of those infinities. and an infinite number of those infinities, and... (infinitely times. and that infinitely times. and that infinitely times. and that infinitely times. and that infinitely times. and...) continues forever. and that continues forever. and that continues forever. and that continues forever. and that continues forever. and.....(…)…
Comments (604)
Cantor showed that some infinities are larger, uncountably infinite. And then there are more than that. It's an interesting, curious area of maths. Check out Cantor's diagonal argument.
That's not true.
That's funny. Why do you think that you can line them all up? That seems like an extraordinarily irrational idea to me. You don't honestly believe it, do you?
Do you think anyone can write out all the decimal places to pi? If not, why would you think anyone can line up infinite numbers?
I gave an argument - albeit briefly. Fractions can be placed in a sequence, and so are no more than countably infinite.
Were did I go wrong?
Why not? If you can take Cantor's, you can take mine.
Quoting Banno
You didn't do that. You merely asserted that you did it.
I can do the same for finite sets. Consider A = { 0, 1, 2, 3, 4, 5 } and B = { 1, 2, 3, 4, 5 }.
Here's me using the kind of proof that you're using to prove that A and B have the same cardinality.
0 -> 1
1 -> 2
2 -> 3
...
The fact that you can't list all of the pairs when working with infinite sets is what makes it easy to fall for that trick.
Consider A = { 0, 1, 2, 3, 4, 5 } and B = { 1, 2, 3, 4, 5 }.
0??1
1??2
2??3
3??4
4??5
5???
There are not enough items in your second set to map one-to-one to the first set. Hence the cardinality of the first is larger than that of the second. Looks pretty convincing to me.
And there are not enough elements in the set A = { 1/2, 1/3, 1/4, ... } to put it into one-to-one correspondence with the set of natural numbers N = { 1, 2, 3, .. . }. It lacks exactly one element. Looks pretty convincing to me.
But that does not stop people from tricking themselves into believing that it's possible to do so by using the following "proof".
f(n) = 1/n - 1
1/2 -> 1
1/3 -> 2
1/4 -> 3
...
The fact that they can't list all of the elements is what makes it easy for them to trick themselves.
Which element is missing?
See The Enumeration of the Positive Rationals
It should be pretty clear.
Silly question. The point is that you can't put them into one-to-one correspondence. In other words, one element must be left unpaired. Which one? You can pick any one. The ellipsis allows you to hide that.
By definition, to add an element X to an existing set of elements S means to increase the size of that set. If you take a set N = { 1, 2, 3, ... } and you add 0 to it, you get a larger set. They are not the same merely because someone can pretend that they can be put into one-to-one correspondence. You can't list all of the pairs, can you? You can't. You can only list a subset. So that's not a proof you can put the two sets in one-to-one correspondence. On the other hand, the definition of the word "add" is indisputable. Nothing about infinities can change that.
Your question "Which element is left out?" is silly because I can't answer it, not because no element was left out, but because I can't tell which one you left out.
It's answerable if we're working with finite sets, e.g. A = { 1, 2, 3, 4 } and B = { 1, 2, 3 }. If you claim that A and B can be put into one-to-one correspondence and do the following:
1 -> 1
2 -> 2
3 -> 3
...
I can easily tell that you left out 4.
But with infinite sets, there's an infinite number of candidates. So how can I answer which one you left out? In fact, you didn't even CHOOSE which one to leave out. Yet, you want me to tell you which one you left out.
And you call that philosophy?
Quoting Magnus Anderson
Not for infinite sets. For obvious reasons.
? and ? ? {0} really are the same size
Take:
? = {1,2,3,…}
?? = {0,1,2,3,…}
here:
f(n) = n - 1
This is:
That is a proof of equal cardinality. Nothing is “pretended”.
The fact that this offends finite intuition is exactly what “infinite” means in modern mathematics.
You should get on well with Meta.
Not quite. Definitions are prior. Nothing can invalidate them. If "add" means "increase in size", nothing can make it change its meaning. And the word "add" means "increase in size" for all quantities -- not merely for finite ones. The rest is ad hoc rationalization.
Quoting Banno
And that is precisely what's being disputed. Your "function proof" is no proof at all. It's smoke and mirrors. The fact that f(n) = n - 1 exists merely means that you can use it on any natural number. That's all. It does not mean there's a one-to-one correspondence between N = { 1, 2, 3, ... } and N0 = { 0, 1, 2, 3, ... }. If you were to sit down and use every natural number starting at 1 as the input of that function, you won't end up producing all of the numbers from N0.
But it doesn't.
Adding four to infinity is still infinity.
And adding four to an integer is still an integer.
The resulting category is the same. If you add four to a number that is larger than every integer, you get a number that is larger than every integer.
But the resulting number isn't the same.
...is not the definition of infinity. “Larger than every integer” is a heuristic, useful for intuition, but the mathematical definitions depend on limits or cardinality. Something like:
S is countably infinite ??f:N?S that is bijective (one-to-one and onto).
A heuristic for sets is the Infinite means the set never ends; there’s no last element. That allows for sets with transfinite elements.
Quoting Magnus Anderson
Sure. Infinities are not integers.
Let A be a finite set that is { 1, 2, 3, ..., 100 }.
Let B be a finite set that is { 1, 2, 3, ..., 99 }.
Obviously, these two sets aren't of the same size.
But suppose that someone comes along and makes the claim that they ARE the same because they can be put into one-to-one correspondence.
He shows you this:
1 -> 1
2 -> 2
3 -> 3
...
It might look convincing at first, but on closer inspection, you realize that he hasn't listed all of the pairs. He has listed only a subset of them.
You inform him of this and add that B has one element less than A.
He asks you, "Which one was left out?"
But how can you tell? There are 97 possible answers. He probably hasn't even chosen which one to leave out.
But you answer anyway . . . you say, 4.
He tells you, "Ah, no! I didn't leave that one! That one is paired with 4!"
You gasp and then say . . . You left out 100.
He smiles and says, "Wrong again! 100 is paired 100!"
You keep doing this for a while, failing to prove him each time. After some time, you might get tired, give up and concede. Or you might push him till the very end -- at which point, you win.
But with infinite sets there is no point at which this process can be finished. The person can keep you playing this game for as long as they want.
So there's no point in playing this game.
The correct response is to say that you can't tell which one was left out because there is an infinite number of possibilities. Moreover, in all likelihood, the person didn't even choose which one to leave out, meaning the game is rigged from the very beginning.
Furthermore, it's useful to add that a subset is not the entire set, and that the same applies to functions. If he can't show you the entire set of pairs, he hasn't proved anything.
Read a math book or two.
If you're going to take pride in your book reading skills, even though we're on a forum that is supposedly about thinking and not reading, at least don't conflate sequences with sets.
They aren't the same size. The set of even numbers is two times smaller. It has all of the elements that N has -- except for a half of them, namely, 1, 3, 5, etc. Doesn't matter what Cantor and mathematical establishment say. They aren't reality.
Matching one to one from the left, the one left out is the 100. :meh:
With your
A = { 1/2, 1/3, 1/4, ... }
and
N = { 1, 2, 3, .. . }
There isn't last element. Nothing is left out.
Quoting Magnus Anderson
Yep, the evens only has every second number, so it must be half the size... Thanks for the giggle!
What you provided is the definition of the countable infinity. That's not the same as infinity. Furthermore, the provided definition does not contradict anything I said.
If you want to prove that my definition is false, you have to either argue that infinity is not a quantity or that it isn't larger than every natural number. You haven't done any of that.
Simply asserting that my definition is a heuristic that is useful for intuition is not an argument. Simply because your favorite books don't define it that way is not an argument. And it's not true anyways.
Quoting Banno
You're the king of missing the point. Of course they are not. But they are both categories of numbers. The only sense in which "Infinity + 1 = Infinity" is true is the same sense in which "Integer + 1 = Integer" is true. Unfortunately, that does not imply that the resulting number is equal to the one you started with.
Bravo!
Quoting Banno
Yikes. That goes against what Cantor said.
And I am pretty sure you won't be able to prove it ( asserting it isn't a proof. )
Quoting Banno
You're very clearly a non-thinker, Banno. Just a regular consumer of philosophy with ego issues.
Well, it's one infinity amongst a few others...
Quoting Magnus Anderson
Your "definition" of infinity is not a definition of infinity. It's not false, it's just an intuitive approximation.
Quoting Magnus Anderson
Yep. So I went the step further, presenting one of the standard definitions.
Quoting Magnus Anderson
It seems then that you haven't understood Cantor, either.
Quoting Magnus Anderson
A bijection exists between N and A — e.g., [math]
f(n) = \frac{1}{n+1}, \quad n \in \mathbb{N}
[/math]
You really should take 's advice and read a maths book.
And you really should take my advice and use your brain for once. Reading isn't thinking.
That's a pretty bad excuse. The dispute was over the definition of the word "infinity". You were supposed to provide a definition that is different from mine. Instead, you provided a definition of a related term that I have no problem with. So what was the point? To show us that you read books? Are you really that pathetic, Banno? Obsessed over how you look in other people's eyes? Even though it's pretty clear that your thinking skills are . . . lacking, to say the least.
Quoting Banno
You surprise me with the amount of stupidity that you can spew. "It's not a false definition of infinity, it's just not a definition of infinity."
Quoting Banno
Of a different term, you imbecile. That I don't even dispute. And that does not go against my definition. How motherfucking stupid do you have to be?
Quoting Banno
I am excused . . . I am not a fanatical book reader. But you're not allowed to make such mistakes. Cantor spoke of infinities that are not equal in size to the number of natural numbers. From that, one can conclude that he didn't believe your idiocy, "There is no last element, therefore nothing is left out."
Quoting Banno
That a bijective function exists, cretin, does not mean that the two sets can be put into a one-to-one correspondence.
Can you please stop quoting books?
This is a philosophy forum, for fuck's sake, not a reading group.
Nor is your making shit up.
Reading a maths book isn’t just passive; it’s fuel for precise thinking, especially when you’re debating infinite sets. It shows how folk have thought about these issues in the past, and the solutions they came up with that work.
Your responses are now a bit too sad to bother with. Thanks for the chat.
Quoting Magnus Anderson
:lol:
Oh, well. :roll:
"Making shit up" is what people confuse with thinking when they know nothing other than to read books and / or be sycophants.
Quoting Banno
In your case, it's obviously passive. That you can't see it is your problem. It's pretty clear that you don't know how to think.
Quoting Banno
You get what you ask for. But I'm sure you're innocent in your mind.
I’ve lived with that since 1973
For a grownup man, that's a pretty childish response.
If the word "function" is defined the way mathematicians define it, namely, as a relation between two sets where each element from the first set is paired with exactly one element from the second, then, if a bijective function that maps N onto N0 is a logical possibility, i.e. if it's not a real oxymoron, then it follows that there's a one-to-one correspondence between N and N0.
Notice the requirement that it must be a logical possibility?
f(n) = n - 1 is a bijective function that maps N onto N0. But, understood through the lens of the above definition, is it a logical possibility? How do you know that? What if it's a real oxymoron that is useful?
You don't know that. And that's why you can't use it as a premise. You can't say, "There's a one-to-one correspondence between N and N0 because there's this function f(n) = n - 1 that maps N onto NO in a bijective way." You don't know if that function is a logical possibility. You have no proof of it.
And it kind of stinks of circular reasoning, doesn't it? "There's a one-to-one correspondence between N and N0 because there is f(n) = n - 1, a one-to-one correspondence between N and N0!"
In everyday contexts, “same size” usually means something like this: if you subtract one collection from another and anything is left over, then they are not the same size. That notion is closely tied to finite counting, monotonicity, and the idea that proper subsets must be smaller than wholes. By that standard, it’s perfectly reasonable to say (for instance) that the natural numbers and the integers are not the same size.
What @Banno is appealing to, though, is a different notion that mathematicians use when working with infinite sets: sameness of size defined in terms of one-to-one correspondence. On that definition, “same size” no longer tracks what’s left over after subtraction, but whether elements can be paired without remainder. This isn’t meant to preserve ordinary quantitative intuitions; it’s meant to give a notion of comparability that still works once subtraction and counting break down.
So I don’t think the disagreement here has to be read as one side being confused or irrational. It looks more like a clash between two legitimate concepts that happen to share the same words. The intuitive notion works well for finite collections but doesn’t generalize cleanly; the mathematical notion is explicitly engineered to handle infinite cases, even at the cost of violating everyday expectations.
Once that distinction is on the table, the question isn’t really “who is right,” but what we want the concept of “same size” to do in this context. Mathematics answers that one way; ordinary language answers it another.
Sorry Magnus, but this what you say is wrong:
Quoting Magnus Anderson
A bijection does mean that sets can be put into a one-to-one correspondence.
Quoting Magnus Anderson
No. There are injections and surjections, which aren't bijections (both injection and a surjection) and they are also called functions.
Let me just remind you of this. It's looks simply, but actually it is difficult to grasp especially with infinite sets:
Dedekind Infinity, referring to the "fact" that the set of natural numbers N is equinumerous to a proper subset of itself, is an intensional concept referring only to injections of the type N --> N . This isn't the same asserting that 1,2,3,... is extensionally of the same length as 2,4,6,... since the dots "..." don't have an extensional interpretation.
A sequence S := 1,2,3,.. that is understood to be unfinished rather than complete, refers to the notion of a Dedekind-finite infinite set. This means that
1) There doesn't exist a bijection between S and and a finite set, meaning that S is unfinished.
2) Any injection S --> S is necessarily a bijection, meaning that S isn't Dedekind Infinite.
3) Any function N --> S isn't an injection, meaning that S isn't countably infinite or larger.
Unfortunately, this indispensible common-sense notion of the potentially infinite set, cannot be formulated in ZFC, because it isn't compatible with the axiom of countable choice which insists upon completing every set.
it is right for amateur philosophers to object to Dedekind Infinity being misued as an extensional concept by the media and the general public (including some physicists who ought to know better). This misuse is due to mainstream mathematics being grounded in adhoc 20th century Hilbertian foundations that assumes the existence of a completed "set" of natural numbers, to the detriment of common-sense, as well as to science and engineering.
Recall that Hilbert believed that finitary proof methods could be used to ground the notion of absolute infinity that he considered to be indispensible for mathematics, due to being under the spell of Cantor, and which the incompleteness theorems conclusively debunked - a negative result that should have been obvious from the outset - namely that an infinite amount of information is obviously not finitely compressible into finite axiom schema.
Yes, but this far too charitable. There are compelling reasons for rejecting Magnus's account. The notion of "same size" he work with is inadequate to deal with infinities coherently - using it results in inconsistencies.
Here's a formalisation of Magnus's account.
These principles are all valid for finite sets.
Let's look at a few contradictions that result.
Contradiction 1: ? vs Even Numbers
Let
[math]\mathbb{N} = {1,2,3,4,\dots}[/math]
[math]E = {2,4,6,\dots}[/math]
But define the pairing:
[math]f(n) = 2n[/math]
This is a one-to-one correspondence between [math]\mathbb{N}[/math] and [math]E[/math].
So:
Thus:
[math]E < \mathbb{N} \quad \text{and} \quad E = \mathbb{N}[/math]
This violates antisymmetry.
Contradiction 2: ? vs ?
Let
[math]\mathbb{N} = {1,2,3,\dots}[/math]
[math]\mathbb{Z} = {\dots,-2,-1,0,1,2,\dots}[/math]
But define a pairing:
[math]
0 \leftrightarrow 1,\quad
-1 \leftrightarrow 2,\quad
1 \leftrightarrow 3,\quad
-2 \leftrightarrow 4,\dots
[/math]
So:
Again:
[math]\mathbb{N} < \mathbb{Z} \quad \text{and} \quad \mathbb{N} = \mathbb{Z}[/math]
Contradiction.
Contradiction 3: Self-Subtraction
Let [math]A = \mathbb{N}[/math].
Partition [math]A[/math] into two disjoint infinite subsets:
[math]A = E \cup O[/math]
where
[math]E = {2,4,6,\dots}[/math]
[math]O = {1,3,5,\dots}[/math]
By (N1):
But:
[math]A = E \cup O[/math]
So [math]A[/math] is the union of two sets each strictly smaller than [math]A[/math].
This is impossible under the naïve size rules, which are now mutually inconsistent.
Contradiction 4: Hilbert’s Hotel
Let hotel [math]H[/math] have rooms [math]{1,2,3,\dots}[/math], all occupied.
Define:
[math]f(n) = n+1[/math]
This moves each guest up one room, freeing room 1.
Under subtraction-based size:
The governing rules of “size” break down.
Conclusion
Once infinite sets are admitted, the principles:
cannot all be maintained. The naïve notion of “same size” does not merely yield counter-intuitive results — it generates outright contradictions.
This is the sense in which the mathematical objection applies: the concept fails to define a coherent ordering on infinite collections.
Thanks to ChatGPT for help with the formatting, but even so the time taken to respond to the sort of nonsense promulgated by maths sceptics is far more than the net benefit.
Quoting Esse Quam Videri
The question is, "who is right?", and the answer is, the contradictions above show that Magnus' ideas cannot be made consistent. Formal language is nothing more than tight use of natural language - it is not unnatural. What is shown by the contradictions is not a conflict between natural and formal languages, but a lack of adequate tightness in Magnus's argument. Magnus’s argument lacks sufficient precision to handle the case he wants it to handle.
Notice also that the arguments stand alone, they are not appeals to authority.
The correct diagnosis is not conceptual pluralism, but logical failure.
I'm not sure how to proceed here.
Sounds awfully familiar and he clearly has a “book reader” ax to grind.
If someone is willing to learn something, on the contrary.
I really would hope that if I make a mistake, some fellow PF member will say that I have made a mistake and try to thoroughly explain to me what my mistake was. Not just "Read high school math 1.0".
But yes, usually the response is just an angry outburst.
Yep, at least the pattern is the same.
Cheers. I'd be interested in your take on my comments regarding formal language. I see it as a refinement of, rather than distinct from, natural language.
It's not just Dedekind Infinity, it simply is Infinity in general. Galileo Galilei noticed the puzzling aspects of infinity a long time before Dedekind or Cantor (which in my view are best explained by the example of the Hilbert Hotel).
Quoting sime
I think the term would be actual infinity that you should refer here to. Absolute Infinity is something totally else, which contradicts the Cantorian hierarchy of larger and larger infinities. Cantor simply preserved Absolute Infinity for God and as he was a deeply religious man, that shouldn't be overlooked. Yet for Absolute Infinity Cantor had no clue how to reason it.
Quoting sime
The incompleteness theorems didn't debunk actual infinity, what they debunked was Hilbert's aim to formalize mathematics and to prove its consistency and completeness by having a general answer (algorithm) to the Entscheidungsproblem. Mathematicians are usually just happy having infinity as an axiom in ZF and don't worry so much about it.
Where I’d add a small nuance is that the act of tightening isn’t always neutral. In refining a concept, we sometimes preserve certain inferential roles while deliberately abandoning others that no longer serve the new domain. In the case of “size”, the move to formal language preserves comparability and transitivity for infinite collections, but it does so by dropping the remainder-based role that functions perfectly well in the finite case.
So I don’t see formal language as distinct from natural language so much as selectively continuous with it: a refinement that’s purpose-driven rather than a mere sharpening of everything we already mean.
I have to disagree a bit with this:
Quoting Esse Quam Videri
The "remainder-based role" is not dropped; the use of bijection keeps everything that the alternative has to offer, and adds the ability to deal with infinities. The shift doesn't sacrifice the old inferential roles, it enriches them.
I see what you’re getting at, and I agree that bijection strictly extends our ability to reason about size — especially once infinities are in play. In that sense it’s an enrichment, not a rival notion.
The small point I was gesturing at is that, while the bijection criterion agrees with the remainder-based notion on all finite cases, it does so by no longer treating “having a proper remainder” as decisive for size comparison. That inferential role is preserved extensionally for finite sets, but it no longer has the same explanatory force once we move to the infinite case.
So I’m not suggesting that anything correct is lost in the finite domain — only that some intuitive cues we rely on there stop doing the work we expect of them when the concept is refined for a broader domain.
You're missing the point. What has to be shown is that the fact that one can think of f(n) = n - 1 means that there exists one-to-one correspondence, or bijection, between N and N0. To do that, you have to show that f(n) = n - 1 is not a contradiction in terms.
I can think of the concept of square-circle but that does not mean square-circles exist. You have to show the concept is not a contradiction in terms. And in the case of the concept of square-circle, it very much is ( Taxicab geometry is not a valid counter-argument, it's merely a fashionable response, peddled by people who are not particularly good at logic. )
If you're a superficial thinker -- and most people are -- you will miss the subtleties.
I can very easily show that there is NO bijection between N and N0. But of course, it's not written in the books, so sycophantic ego-driven non-thinkers dismiss it.
Quoting ssu
How exactly does that contradict anything I said?
I appreciate your response but I disagree with your conclusion, namely, that we're using two different definitions of the term "same size". I am quite confident that we're using the same definition.
When speaking of sets, the word "size" simply means "the number of elements". The word is defined the same way for both finite and infinite sets. That's how the word has been used for ages and it's the way it is used today.
With that in mind, the term "same size" simply means "equal number of elements". That's it.
The terms "bijection" and "one-to-one correspondence" refer to a relation between two sets A and B where every element from A is paired with exactly one element from B, and vice versa.
The observation is that, for every two sets A and B, if they are equal in size, they can be put into one-to-one correspondence with each other. And if they aren't, then they can't.
This means that, if we know that there's a bijection between A and B, it follows that A nd B are equal in size.
The problem they faced is that, with finite sets, one can determine whether or not they are equal in size simply by counting the elements and then comparing the resulting numbers; but with infinite sets, this isn't the case.
So they came up with the idea that, if we can show that infinite sets can be put in one-to-one correspondence with each other, we can conclude that they are equal in size.
So far so good. That is all true.
The problem lies in HOW they go about establishing whether or not any two sets can be put in one-to-one correspondence.
Their method is based on a hidden, and an erroneous, premise that, if we can think of a function that is defined as a bijection between A and B, it follows that there exists a bijection between A and B.
That's akin to saying that, if there exists a symbol that is defined as a shape that is both a square and a circle, then we can safely conclude that such shapes exist.
You've probably heard of Hilbert's Paradox. Hilbert's Paradox exposes a very serious contradiction in the way infinites are normally dealt with. Unfortunately, most pretend it's not a real contradiction, justifying themselves will sorts of silly rationalizations.
Suppose we have a hotel with a number of rooms equal to the number of natural numbers.
Suppose each room is occupied by a single guest.
That gives us a nice bijection between the set of guests and the set of hotel rooms.
R1 R2 R3 ...
G1 G2 G3 ...
Guest #1 ( G1 ) is in room #1 ( R1 ), guest #2 ( G2 ) in room #2 ( R2 ) and so on.
If there exists a bijection between N and N0, then it follows that we have a spare room for another guest. Let's call that guest G0.
---- R1 R2 R3 ...
G0 G1 G2 G3 ...
There is no longer a bijection between the two sets. G0 is not in any room. And if you try to add it to any room, you will either end up having two guests in a room ( not bijection ) or you will have to kick out one of the guests ( still not bijection. )
There's no way out of this conundrum . . . other than to pretend.
And that's what they do. They pretend.
That’s why defining a bijection counts as establishing existence in this context, and why reindexing in Hilbert’s Hotel isn’t seen as pretence. At that point the disagreement isn’t about technique, but about whether such revisions are legitimate at all.
They don't. It's called ad hoc rationalization.
Not really. Banno's argument is flawed because it is based on the erroneous premise that I already covered:
"If we can think of a function that is defined as a bijection between A and B, it follows that there exists a bijection between A and B."
Because he can think of a function that is defined as a bijection between N and E, namely, f(n) = 2n, he concludes that N and E are the same in size. He never actually proves that such a function is not a contradiction in terms. And I can easily show that it is.
The inconsistencies that he speaks of do not come from "my" use of the term "same size" ( as if anyone else is using it any differently ) but from his own logical mistakes ( which aren't really his own, he merely copied them from a textbook. )
Bullshit. You're literally copy-pasting textbook arguments. Zero thinking on your part.
Classic Banno!
Quoting Banno
Well, Magnus was very quick to pick up on your nasty habit of straw manning the other person's claims to make it appear like your own errors are the errors of the other person. I wonder why both of us come to the same very peculiar conclusion.
With finite set there's a contradiction.
With infinite set there isn't.
(In fact just look up the axiom of infinity in Zermelo-Fraenkel set theory. Or the definition of Dedekind infinity).
Sorry, but I don't think you grasp the example of Hilbert's Hotel, which above @Banno gave you. So you write:
Quoting Magnus Anderson
OK, you really don't understand the Hilbert Hotel.
How Hilbert hotel works, at first:
R1 R2 R3...
G1 G2 G3...
And then when one gest, let's say G1, leaves, it's still full (meaning there's a bijection) because:
R1 R2 R3 ...
G2 G3 G4 ...
And if another guest comes, that G0, then the hotel fills up:
R1 R2 R3 ...
G0 G1 G2 ....
Please understand when many people are saying the same thing to you. Perhaps this video would help, because it's talking exactly about the same thing, although it really shows in what circumstance there isn't any bijection:
And if you are interested in finitism, I have a great professor to listen to or watch his lectures...
This is not even an extension or enrichment of finite counting: all counting is based on bijection. In the finite case, whether you use your fingers, notches on a stick or a number system, it all boils down to the same procedure:
(Physical measurements of size and weight are also based on the same principle.)
So, to count sheep in the pen you can bend your fingers left to right, one for every sheep. Once you run out of sheep (or fingers), you can hold up your hands and say: I have that many (or at least that many). Works even if you have no notion of numbers.
Using numbers, you follow the same procedure, only you use counting numbers 1 through N instead of fingers, plus the convenient fact that the last member of your reference set is equal to its size. The size of a set is then the last member of a set of counting numbers that are in a one-to-one correspondence with that set.
Counting infinite sets works the same way, except that you have to set aside certain other assumptions that hold for finite sets but not for infinite sets. For example, you can no longer use finite subsets of natural numbers as described above. But you can use other reference sets. You can use the entire set of natural numbers as your measuring stick, or its power set if that that's not enough, or the power set of the power set, and so on.
You wouldn't expect completion from a thread titled "Infinity" would you?
Quoting SophistiCat
The problem though, is that you really cannot use the entire set of natural numbers as your measuring stick. No one can do this, because by definition, no one can get all those numbers into one's grasp, to use them that way. This renders that statement as false.
Quoting SophistiCat
The "other assumptions" which one must "set aside" are the assumptions that truth is required of a premise, to produce a sound conclusion. Once we dismiss the necessity of truth, then we might assume the premise that the entire set of natural numbers could be utilized in the prescribed way.
Yes — that’s a good way of putting it, and I agree. I didn’t mean to suggest bijection is foreign to finite counting, only that when we move to infinity the remainder-based cues we rely on in finite cases stop being reliable, even though the underlying correspondence idea remains.
You're not responding to what's in the quote. You did not prove the following: that just because you can think of a function that is defined as bijection between N and N0, e.g. f(n) = n - 1, that it follows that a bijection exists between N and N0. You conveniently ignore that. And so does Banno. It's very convenient.
The problem is not with the definition of the word "same size", the problem is with the wrong conclusion that N is the same size as E or that N is the same size as Z. They aren't equal. And once you plug that in, the contradictions are gone. But neither you nor Banno nor mathematicians want to admit that you made a mistake that you've been stubbornly preserving for decades, so instead, you blame it on the definition, as if definitions can be logically proven to be wrong ( they can't, they are prior. ) The redefinition of the term "same size" merely covers up the fact that you screwed up.
Quoting ssu
And you really don't have the ability to determine whether or not other people understand something.
Quoting ssu
It's not full. There's no longer one-to-one correspondence between the two sets.
R1 R2 R3 ...
---- G2 G3 ...
R1 is unpaired. And it cannot be paired with any other guest because they are all already paired.
Quoting ssu
Nah, that's not what you get when G0 comes.
What you get is this:
R1 R2 R3 ...
G0 G2 G3 ...
G1 left. He's not in the hotel.
Their argument is that you can create an empty room for G0 by moving all the guests one place to the right. But of course, that's impossible. They employ sleight of hand to MAKE IT LOOK like it's possible. The trick is to hide the unpaired guest in the ellipsis and to never reveal it.
And understand that I went over this many times in the past.
[math]f: \mathbb{N} \to \mathbb{N}_0, \quad f(n) = n - 1[/math]
Conclusion: The function [math]f(n) = n-1[/math] is a bijection between [math]\mathbb{N}[/math] and [math]\mathbb{N}_0[/math].
And there's a subtle difference between "You can pick any number from N and map it onto a unique number from N0" and "You can pick every number from N and map it onto a unique number from N0".
Not really, but ignoring the infinite level of irrelevance of the topic is a pretty important omission.
[math] \mathbb{N}_0[/math]?
Well, no. It is defined as f(n)=n?1 and then shown to be a bijection. That definition does not mention bijectivity at all. At this stage, the function could turn out to be injective, surjective, neither, or both. Nothing is being smuggled in.
While a square-circle is defined using incompatible properties, there is no contradiction in [math] \mathbb{N}_0[/math].
Not N0 but f(n) = n - 1. That function is a bijection by definition.
Quoting Banno
Yes. It is not explicitly stated in the definition. However, the definition implies it. And because it implies it, it is a bijection by definition.
It's like defining the symbol "S" as "a closed figure with three straight sides". It does not explicitly state that it has 3 angles but it does imply it. So it is correct to say that S has 3 angles by definition.
"By definition" does not mean "explicitly stated by the definition". It means "fixed by the definition ( either explicitly or implicitly )".
In mathematics, functions are defined as sets of input-output pairs. f( n ) = n - 1, in this view, is a set of input-output pairs where every element from N is paired with exactly one element from N0 and vice versa. That makes it a bijection by definition. But that does not mean that the bijection between N and N0 exists, i.e. that it is a logical possibility. It merely means that's what the symbol can represent. It's similar to how simply saying that the term "square-circle" means "a shape that is both a square and a circle" does not mean that square-circles exist.
The seemingly devastating consequences of accepting that no bijection exists between N and N0 is that f( n ) = n - 1 does not exist either, i.e. that it is an oxymoron. The good news is that that's merely a consequence of an incorrect definition of functions. Functions aren't sets of input-output pairs. They are sets of rules.
Here's the definition again: [math]f: \mathbb{N} \to \mathbb{N}_0, \quad f(n) = n - 1[/math]
[math]\quad f(n) = n - 1[/math] as it stands is not a definition of a bijection. It can't be, because it lacks a domain and a codomain, as provided by [math]f: \mathbb{N} \to \mathbb{N}_0[/math]
[math]\quad f(n) = n - 1[/math] could be applied to any domain, with differing results. With [math]\mathbb{Z} [/math] as the domain and codomain also [math]\mathbb{Z} [/math], it would be a bijection. If the domain were [math]\mathbb{N} [/math] and the codomain [math]\mathbb{Z} [/math], bijectivity would again depend on proof, not stipulation.
Quoting Magnus Anderson
An odd thing to say, since making that implication explicit is exactly what the proof presented above does. you treat [math] \quad f(n) = n - 1[/math] as if it secretly meant "let [math]f[/math] be a bijection defined by [math] \quad f(n) = n - 1[/math]"; but that is not what is being done. What was done, step by step, was:
1. Define a function by a rule.
2. Specify domain and codomain.
3. Prove that, given those, the function is injective and surjective.
[math] \quad f(n) = n - 1[/math] might be bijective, non-surjective, or non-injective depending on the domain and codomain.
I can't quote the entire part, the LaTeX code gets messed up for some reason.
You are right that f( n ) = n ? 1 by itself is not a complete definition of a function. A function definition requires a domain and a codomain. I conveniently left those out, assuming that they could be inferred from the context.
I was referring to the N -> N0 variant.
Quoting Banno
When I say the function is bijective by definition, I do not mean that bijectivity is explicitly stated, but that it is an unavoidable consequence of the definition. The "proof" consists solely in unpacking what is already implied by the definition, not in adding any further stipulation.
Quoting Banno
That's correct. However, I was talking about f: N -> N0, f( n ) = n - 1. That specific variant does imply bijection.
The real problem is being ignored and that is that the fact that a definition implies bijection between N and N0 does not mean that such a bijection exists in the sense of being a logical possibility.
Yep. that's what a proof does.
Why do you say the topic is irrelevant"? The concept of infinite is commonly used in mathematics, so there must be at least some relevance.
Quoting Banno
It is not "shown to be a bijection". It is stipulated to be a bijection. And, it is actually impossible to make that bijection. So what it actually is, is the affirmation of an unjustifiable, impossible, action (bijection). When what is stipulated as done or even doable, as a premise, is actually impossible, this justifies the judgement that it is a false premise.
Quoting Banno
A sound proof requires true premises. A so-called "proof" derived from a false premise, is not a proof at all. Therefore your so-called "proof" is ineptly named because it doesn't fulfil the criteria. It has been refuted. I think you actually know this already, but you tend to deny the obvious brute facts, when they are contrary to what you like to believe in.
It seems to me that this discussion keeps looping because the objection is being framed as an internal refutation of standard mathematical proofs, rather than as a foundational challenge to the notion of existence those proofs rely on.
Both of you have raised worries about the “doability” of bijection for infinite collections, which suggests a rejection of the identification of existence with formal definability and consistency. That’s a substantive philosophical position. But if that’s the objection, then it isn’t a matter of showing that the usual definitions lead to contradictions (they don’t), but of rejecting the underlying framework.
Put differently, the objection seems clearer if stated explicitly at the level of foundations, e.g.:
“I reject the identification of mathematical existence with formal definability. I require a constructive or modal account of possibility, and under that account I deny that completed infinite bijections exist.”
or
“I reject classical set theory in favor of a finitist or constructivist framework, where existence requires explicit construction.”
Framed that way, the disagreement would look less like an accusation about the failure of proof and more like a clash of foundational commitments, which is where I suspect the disagreement really belongs.
Maybe that actually allowed me to look at this from a slightly different angle.
From the comments I’ve read here, it seems to me that the discussion ultimately comes down to the question of what counts as a sound proof.
@Magnus Anderson, your reasoning reminds me of Hume. It seems to me that your requirement for a proof is to list all possible cases (one by one?) and show that a given hypothesis holds in each of them. But just as you can never know with absolute certainty that the sun will rise again (Hume), you also can’t know whether the next line in an infinite list might contradict the hypothesis.
But mathematics, as I understand, is based on definitions and sets of rules (including what counts as a proof and how to present one), not on “real objects” per se. And since the OP is talking about numbers, I think we can safely assume the topic concerns mathematical infinity.
And @Esse Quam Videri nicely laid out exactly which 'practices' of what we might call 'orthodox maths' are being rejected.
If Magus would be a constructivist or intuitionist here, then he might do well to do so explicitly. That would be a legitimate position. But what we have looks like intellectual drift rather than anything solid.
There's a few ways that a constructivist might proceed. They might reject the usual account of what it is to be a mathematical entity, ?x P(x) is true iff P(x) is derivable in a consistent formal system. They might instead insist on a constructive approach: To assert “?x P(x)” one must provide a construction (algorithm, procedure, or finite method) that yields such an x.
So an argument might proceed by rejecting [math]f: \mathbb{N} \to \mathbb{N}_0, \quad f(n) = n - 1[/math] as a suitable account of a function, on the grounds that it relies on unrestricted quantification over a completed infinite totality, saying something like "We don't yet have a finite or algorithmic construction of the entire inverse mapping, so surjectivity is not constructively justified.”
The trouble is that for [math]f: \mathbb{N} \to \mathbb{N}_0, \quad f(n) = n - 1[/math] we do have the inverse mapping: [math]g: \mathbb{N} \to \mathbb{N}_0, \quad g(n) = n + 1[/math]. So this won't work here.
This wouldn't be a function on which a constructivist might try to stand their ground. There are, of course, other bits of maths were constructivists interestingly differ from classical maths, and there are some interesting philosophical issues here. But one needs a grounding in mathematics in order to be able to express the difficulties with clarity.
A more eccentric approach, and this is perhaps were Magnus is coming from, might reject infinities altogether. This is the most charitable interpretation Ive been able to work out. If Magnus rejects the very idea of infinite totalities, if he rejects [math]\mathbb{N}[/math], then his argument might be made consistent, but at a great cost.
On this view, computing n?1 for any given numeral would be fine, and also computing n?1 for any finite set of numerals, say {2,4,6,8}. But somewhat arbitrarily, a finitist would reject applying n?1 to any infinite set. They in effect accept [math] \quad f(n) = n - 1[/math] for any finite n, but not for [math]\mathbb{N} [/math].
Importantly, on this view the argument given above would not be invalid, or lead to contradiction, but ill-formed, because it relies on [math]\mathbb{N} [/math]. (added: It's not even true or false, since these notions do not apply to ill-formed formations )
The cost here is the rejection of succession (roughly, that for every number there is another number that is one more than it; or more accurately, that we can talk about such an infinite sequence); and consequently the rejection of the whole of Peano mathematics*. No small thing.
To be sure, this is how Magnus might have argued, but hasn't.
I, like most folk, enjoy talking about infinity, and so would reject such finitism.
* On consideration, this last isn't quite right. we might accept Peano's definition of succession and still not accept that we thereby construct an infinite set. thanks, .
Explicitly specifying a function is acceptable as a constructive proof. Constructivism shares some concerns with finitism, but it is not as [s]bonkers[/s] stringent.
I suppose so. I don't see that a constructivist would have issues with f(n)=n-1 or f(n)=n+1. Again, these are not examples with which a constructivist might take issue. They would more typically take issue with LEM, and reductio arguments, and treat infinite collections as potential rather than actual, or as given by generation rules. I've some sympathy for it, after Wittgenstein.
So I think Magnus must be basing his ideas on a finitist intuition. We'll have to see what he says.
Strange as it may sound, Peano arithmetic does not require you to accept natural numbers as a "completed" set. You can have your successor function, you can show that, given the axioms, there are "as many as you want" distinct numerals, and still that does not compel you to accept the totality of all such numerals, N.
So in both classical and constructionist maths, for any number we can construct its successor. Ok.
So constructivism will not help Magnus here. He must resort to finitism - the view that why for any number we can construct its successor, we can't thereby construct the infinite sequence [math]\mathbb{N} [/math].
PROOF
1) To say that S is larger than S' means that S' is a proper subset of S.
( A definition that applies to all sets, regardless of their size. )
2) N is a proper subset of N0.
3) Therefore, N0 is bigger than N.
This is an indisputable proof. As indisputable as 2 + 2 = 4.
However, if you're convinced by a fallacious proof, you will normally deny the validity of this one, like a cancer attacking healthy cells.
FALLACIOUS PROOF #1
The first fallacious proof they use to show that N and N0 are of the same size is the observation that, if you add 1 to infinity, you still get infinity. This is true but only in the sense that the result is also an infinite number ( i.e. larger than every integer. ) They make a mistake when they conclude that, just because "infinity" and "infinity + 1" are infinite numbers, it follows that they are equal. It's like saying that 4 equals 5 merely because 4 and 5 are integers.
FALLACIOUS PROOF #2
The second fallacious proof they use is grounded in the premise that, if you can come up with a symbol that is defined as bijection between N and N0, it follows that a bijection between N and N0 exists ( i.e. it's not a contradiction in terms. ) It's like saying that square circles exist merely because there exists a term called "square circle" that is defined as a square circle.
I wouldn't characterize this as "worries". It doesn't worry me at all. I just reject falsity for what it is, and since this matter has little if any influence on my daily life it doesn't worry me.
However I think you should reconsider what you say about contradiction. If "infinite" is defined as without limit, then it is clearly contradictory to say that the bijection could be done. It is also contradictory to say that it is doable. Further, it is also contradictory to say that the natural numbers are "countably infinite". Obviously, "without limit" means cannot be counted, so countable contradicts this.
Quoting Esse Quam Videri
I suggest we call a spade a spade. A falsity is a falsity. A conclusion derived from a false premise is unsound. An unsound argument does not constitute "a proof".
I suppose you could argue that mathematicians produce their own rules, and are not subject to the terms of logic. But what would be the point in giving mathematics such an exemption, to proceed in an illogical way. It seems like it would only defeat the purpose of the pursuit of knowledge, to allow for an illogical form of logic.
Quoting Banno
This is the reason for the distinction between "true" and "valid". Validity is concerned with the internal process. Truth is concerned with the external relations of the premises. "Proof" requires both, and this is known as soundness.
Quoting Banno
Clearly, when "infinite" is defined in the usual way, and the way that we understand the natural numbers to be, "infinite totalities" is contradictory.
In no way does this perspective make it impossible to talk about infinite succession. It only applies standard principles of logic to such talk, denying by the law of noncontradiction things like "infinite totality", and denying as false, premises such as "countably infinite". If application of these standard principles of logic expose some of current mathematics as unsound, then that is a problem, which the mathematicians ought to deal with. They ought to accept this, and not whine about having to throw away a whole lot of work.
Quoting Banno
I don't understand this objection. As mentioned above, there is no need to reject the idea of infinite sequence, nor is there a need for finitism. The problem is with the idea that an infinite sequence could be completed. That is talk which is unacceptable.
So, we can talk about tasks which will never be completed, and there is nothing wrong with this talk, it makes sense. We can even define a specific task as being impossible to complete, and this makes perfect sense. We can define counting all the natural numbers as such a task which will never be completed, and there is no problem with talking about this. The problem is when we take a task which we have defined as being impossible to complete, such as counting all the natural numbers. and then start talking about it as if it is possible to complete.
Quoting Magnus Anderson
This is false, since that definition applies only to finite sets. For infinite sets, we need something more. Consider that the even numbers form a proper subset of the integers, and yet we could count the even numbers... a bijection.
The objection that we could not actually count the even numbers because there are two many of them is trivial; we have a function f(n)=2n, that when applied to [math]\mathbb{N} [/math] gives us every even number. And we have the inverse, g(n)=n/2, which wen applied to the even numbers gives every integer. If your finitism is such that you cannot see that, I can't help you.
That's a lie you've been shamelessly pushing forward.
The definition does not apply only to finite sets. It applies to all sets. The only reason you think it does not apply to infinite sets is because it leads to contradictions when you use it in combination with your mistaken premise, "If we can define a bijection, then bijection exists." That premise is your mistake. That premise is the main point of dispute. And so far, you've been conveniently ignoring it.
The other thing you conveniently ignore, because you clearly don't understand how definitions work, is that, you cannot blame the definition in this case. When you blame the definition, what you end up doing is changing the set of relations you're talking about, while calling the new relations the same names and insisting that they are the same relations as before.
It's like saying that, for red cars, "same size as" means something different than it does for cars that are not red. For example, that for red cars, "same size as" means "same color as". Therefore, all red cars are of the same size. And whoever says there are red cars that differ in size, you will claim they are wrong, simply because you got yourself in this bunker of confusion. In reality, all that you're doing is changing the relation that you're talking about ( from "same size as" to "same color as" ) while calling it the same name ( "same size as" ) and pretending that it's the same one ( "same size as". ) Instead of talking about the equality of sizes, you're now talking about the equality of colors, while calling it the same name ( "equality of sizes" ) and pretending that you're actually talking about the equality of sizes.
It's an ugly trick. But easy to see through for people who can actually think.
Only a completely blind person can see any trace of finitism in what I'm saying.
The definition you suggest cannot be used effectively with infinite sets. But enumeration, that is a surjection, will do everything that can be done with your definition and then do more - quite a bit more - with other sets. So your definition is effectively included in yet surpassed by enumeration.
Another lie.
Quoting Banno
It's not only you. I am aware of that. Your confidence is entirely grounded in what someone else said.
Where do you think this claim appears in the proof?
The claim “infinity + 1 = infinity” does not appear anywhere in the proofs I have used.
Quoting Magnus Anderson
The proof doesn't just "define a symbol for a bijection"; it provides an explicit function:
[math]f: \mathbb{N} \to \mathbb{N}_0, \quad f(n) = n - 1[/math]
So we can read the definition as:
What is defined here is a function, not a symbol. This is a concrete mapping, not a mere linguistic construct, and it suffices to show that a bijection exists.
The symbol we're talking about is this:
f: N -> N0, f( n ) = n - 1
The definition of a symbol specifies what that symbol can be used to represent -- here, a specific bijection between N and N0.
However, that does not tell us anything about whether or not such a bijection is a contradiction in terms.
If it's a contradiction in terms, then bijection between N and N0 cannot possibly exist.
You seem to be arguing that N must be [s]bigger[/s] smaller than N U {0} because, well, 0 is left out. Is that right? (Doofus.)
But try this: instead of thinking of the numbers here as things, think of them as labels. [hide="*"](As it happens this particular case is widely known because there are programming languages that use primarily 0-based indexing and others that use 1-based indexing for arrays and lists and such.)[/hide] In one set, there is something we have labeled "0"; in the other, there isn't, but suppose there isn't not because the 0-thing isn't there, but because we've labeled it "1" instead.
The functions that have been discussed are instructions for switching from one labeling system to another. Isn't it clear that this works, and that there can be no set of objects you could label starting at 0 that you couldn't also label starting at 1? And vice versa. Or starting at 2 and using only even numbers. Or any number of other ways, so long as you are systematic about it. All these sets of labels are clearly equivalent, and in particular all equivalent to just using N. Now how can that be?
You can't do that. Logic prohibits it. There are more "labels" in N0 than there are in N.
And you're relying on a deceptive "proof". You think that, just because you can create a bijection between any proper subset of N with any equally sized subset of N0, that N and N0 are equal in size.
Oh I'm not referring to the concept of infinity, that you correctly note is important. Rather what the OP specifically referenced, which is the infinite numbers between infinitely minute numbers.
So are you saying there could be a set such that you could label every member of that set starting with 0, but you could not label every member of that set starting at 1? Is that your claim? (And I guess also that "N0" is such a set.)
That's a group of symbols... so you mean the [math]f[/math]? And your claim is that the definition
[math]f: \mathbb{N} \to \mathbb{N}_0, \quad f(n) = n - 1[/math]
"specifies what that symbol can be used to represent", but not that what it represents is not somehow contradictory? OK. Then over to you. If you think there is a contradiction here, it's up to you to show it. Exhibited it as derivations of ?.
Hence we come back to what it is for a mathematical object to exist, and the point you seem not to have accepted, that for "classical" maths ?x P(x) is true iff P(x) is derivable in a consistent formal system - as here. You appear to be rejecting that rejection, while claiming not to reject it. All very difficult to follow. Hence, the impression of an intellectual drift on your part rather than any actual argument.
I would agree with you if the object of this discussion were 'real' infinity as a 'real-world phenomenon'.
I find this 'real' infinity uncomprehensable, and so any speculation about it's properties, seems, well, at the very least, dubious. But this is not the case, as this thread concerns mathematical infinity. You're absolutely right - I argue that mathematicians set their own rules. Doesn't mean those rules can't eventualy change, of course, as paradigm shifts have occurred in other disciplines.
One of my professors used to say that pragmatism is not a philosophy at all. So perhaps a pragmatic stance on this question is not philosophical. Still, I would argue that if the 'orthodox' view of mathematical infinity solves more problems than it creates, then so be it.
It doesn't even work for finite sets. Think what it would mean if you could only compare the sizes of sets and their subsets. You couldn't say, for example, that there are more apples than oranges on the table, because neither is a subset of the other.
By contrast, the naturals are "lawful" choice sequences, which by construction are essentially dedekind-infinite functions that don't represent sequences in the flesh, and are what a type-theorist would say are purely intensional sequences that shouldn't be confused with actual sequences.
To rectify an earlier confusion, the computer-science meaning of "extension" refers to explicit data. According to this definition, the identity function on the naturals ( \lamda (n : N) => n ) is an extension in the sense of a function, whereas the graph of that function, namely the set { (n,n) | n is a Natural number} isn't an extension. But confusingly for philsophy that graph is considered an extension according to the Fregean notion of extension, since Frege defined an extension as referring to the arguments of a predicate that make it true.
In effect, Frege conflated the notion of data-at-hand with the notion of functions that can produce data on demand, as a result of thinking that functions exist independently of their domains and ranges. For Frege, and unlike the computer scientist, a function isn't a causal operation that transforms input into output, but a transcendental relation that relates a static domain to a static range. Hence Frege interpreted predicates (which he called "concepts") as being non-destructive testers of their domains, which naturally implies that concepts and Fregean extensions exist independently and in one-to-one correspondence, leading to Russell's Paradox and also led to the failure of formalists like Hilbert to predict incompleteness.
What else would this be than finitism?
You don't accept the infinite to be different from the finite and obviously treat infinite like it would finite by arguing that "infinity" and "infinity + 1" aren't equal. Just look what the axiom of infinity is, which @Magnus Anderson clearly thinks is incorrect. That n < n+1 is simply how finite numbers work.
Quoting SophistiCat
I think that @Magnus Anderson seems to think that if you take one out of an infinity set then number 1 is really missing from there.
Sorry, Magnus, but your "proof" merely begs the question. All you have done at this point is:
This is why the discussion keeps looping. If you want to move the discussion forward you need to either (1) derive (not assert) an actual contradiction [I]within[/I] the accepted mathematical framework (per ) or (2) reject the standard framework and present a coherent alternative (e.g. intuitionism, finitism, non-classical logic, etc.).
As it stands, Banno has already shown that combining your premise (1) with transitivity, antisymmetry and the existence of infinite partitions leads to contradictions. At this point there is nothing of substance left to discuss.
Yep.
And Banno, you were right.
I think this matter still has relevance. It is the issue of division. In reality, everything that we attempt to divide can only be divided according to its nature. Nature dictates the way something can be divided. We cut things up very evenly using instruments of measure, but eventually we get to molecules and then atoms, and we are greatly restricted in our capacity to divide "evenly". However, some things like space and time, we might not find the natural restrictions, and so we would be inclined to apply principles of infinite divisibility. Since the mathematical principle of divisibility (infinite) does not correspond with the real divisibility of the substance (space and time), the uncertainty principle is produced.
Quoting Zebeden
Infinity is a "real-world" phenomenon. We have examples of it as the infinite decimal extension of pi, and of the square root of two. The circle, and the square are extremely useful real world applications, yet the principles which validate their use lead us into these real-world infinities.
We might dismiss the problem by saying there is no such thing as a true square, or a true circle, in the real world, and dismiss these conceptions as ideals without real world validation, but that doesn't resolve anything. It just produces a division between conceptions and the real world, where we allow ourselves to employ false premises for the sake of usefulness, and we lose the epistemic value of "truth". Truth is no longer a requirement for knowledge, and we allow that we are not guided toward the truth.
Instead, we ought to look at these issues, where the ideal does not correspond with the real world, as demonstrations which show where our ideals have been compromised by selecting usefulness over truth. They display where our understanding of reality faulters, as reality is fundamentally different from how we represent it. If you just say "I don't care about the true nature of reality, if the principles serve the purpose that's good enough for me" this is a violation of the philosophical mindset which seeks truth. And if we're always happy with the way things are working now, knowledge never advances.
Quoting Zebeden
This is not a good standard because the comparison cannot be made. The problems which are solved can be pointed to and numbered. The problems created are associated with the unknown and cannot be counted, nor can the extent or size of the problems be determined. The resolved problems are finite, the created may be infinite and uncountable. So, for example, we created CFCs, and that resolved a whole lot of different problems which we could point to. However, at that time we didn't know what was going on with the ozone, and we couldn't compare the created problems. This is the issue then, the problems created are hidden within the unknown, and only when they start to fester do we take them seriously, and seek out their depth and roots. The example I use above, which displays the problem of unruly use of infinity is the uncertainty principle. We don't know what is hiding beneath that name.
Quoting Esse Quam Videri
1. The actual contradiction is blatant, and I've stated it.
2. Rejection of the framework because it is contradictory and false, is the task of philosophers. Presenting a coherent alternative is the task of mathematicians. Therefore you are wrong to suggest that the one who refutes the framework is obliged to present another.
Quoting Esse Quam Videri
The problem is clear. The mathematicians in this forum refuse to accept the refutation, though it is very sound. Because of this, they refuse to get on with the task of producing a coherent alternative. For the philosophers, "there is nothing of substance left to discuss", because the refutation is clear, and the mathematicians remain in denial. Until the mathematicians accept the refutation, and start again at the foundation, the philosophers will have nothing to offer, and there will be nothing of substance to discuss.
You would have to prove that. I am, however, pretty sure you can't do it. But I can show, as I already did, that YOUR reasoning is circular. So what you're doing here would be sort of like a projection.
Quoting Esse Quam Videri
Again, you're doing the very thing you accuse me of. You're asserting something without proving it.
I presented a very clear process of derivation.
Quoting Esse Quam Videri
The onus of proof is on the one making the claim. You have to show that just because you can define a symbol as a bijection between N and N0 that it follows that such a bijection exists. I can show you why that does not follow. And I kind of already did.
Quoting Esse Quam Videri
Not for sharing my standards but for not being independent thinkers ( which is fine ) while pretending that they are ( which is not fine. ) Not everyone is an independent thinker -- and does not have to be.
Quoting Esse Quam Videri
Actually showing that bijection between N and N0 exists by employing definitional logic. So far, you've been merely asserting it and relying on something that is very much like a circular argument.
Quoting Esse Quam Videri
That's not the real reason. The real reason is that people do not know how to think outside of the box. People are missing the point all over the place. The main point of dispute is never addressed.
Quoting Esse Quam Videri
I already did that. But if your argument is a non-sequitur, it's not really necessary to to do so, isn't it? All I have to do is to show that it's a non-sequitur.
Quoting Esse Quam Videri
He hasn't.
Quoting Esse Quam Videri
There is. Quite a bit. But all that is left is to think. No more room for quotations. But non-thinkers don't think.
I’m afraid it’s not, and I’ll try to clarify why.
All you’ve claimed so far is that mathematicians are working with a notion of infinity that you don’t accept, and you’ve given some philosophical reasons for rejecting it. That’s a legitimate philosophical position.
The problem is that this is a philosophical objection, not a mathematical one, and as such it doesn’t justify the claim that the mathematical notion of infinity is contradictory. The mathematical definition is perfectly sound relative to the formal system in which it is embedded.
By analogy: suppose we’re playing a game of Chess and, on your turn, you legally move your queen from d1 to a4. Suppose I respond to your move by saying: “that move doesn’t make sense because in real life kings are more powerful than queens and so only kings should be able to move like that”. That may be a fine external critique of the rules of Chess, but I haven’t thereby shown your move to be illegal. Given the established rules, it was a perfectly valid move.
Likewise, your objection to the mathematical notion of infinity is a meta-level objection. It doesn’t undermine the internal coherence of mathematics as it is standardly practiced. At most, it shows that the standard mathematical notion of infinity conflicts with your own metaphysical views.
If you wanted mathematicians to take this challenge seriously as mathematics, it would require proposing an alternative formal framework built around your accepted notion of infinity and showing that it does at least as much [I]mathematical[/I] work as the existing one. As things stand, no such reason has been given for abandoning the standard definition.
I'll leave it at that.
Functions can be malformed. They can contain internal contradictions that effectively render them as non-existent.
Consider the following example.
Let A be { 1, 2, 3, 4 }.
Let B be { 0, 1, 2 }.
Consider the function f: A -> B, f( n ) = n - 1.
Since this is a function, and since functions are relations where every element from the domain is paired with exactly one element from the codomain, this is also a contradiction. Being a function means 1) it must obey the rules, and 2) every element form A must be paired with exactly one element from B. But both are violated. The rules say that the number 4 should be paired with the number 3, but no number 3 exists in B. It being a function means that the number 4 must be paired with exactly one element; otherwise, it is not a function. But it isn't paired with any element.
So even though this function is defined as a bijection, no such bijection exists.
What this means is that you have to show that f: N -> N0, f( n ) = n - 1 is not a contradiction in terms before you can conclude that it exists.
Has anyone done that?
Of course not.
Instead, the opposite has been shown.
And is there an element n of N such that n-1 is not a member of N0?
This is a perfectly good argument, but it is not the argument you make about N and N0, which relies on the claim that if B is a proper subset of A, its cardinality must be smaller. Here no mention is made of cardinality.
To argue that f(n)=n-1 does not map every member of N to a member of N0, you must show, as you did here, that there is an n for which f(n) is undefined or not a member of N0.
Not really. And that's a commonly made mistake.
Bijection does not mean that you can take ANY element from N and uniquely pair it with an element from N0. Of course you can do that with N and N0.
It means that you can take EVERY element from N and uniquely pair it with an element from N0. And that's what you can't do.
Do you see the subtle difference?
You can't claim that bijection exists between N and N0 merely because you can take any element from N and uniquely pair it with an element from N0. That does not follow.
What is the cash value of that difference, as you see it?
The difference is that bijection means that you can take every element from N -- not merely any arbitrary subset of it -- and uniquely pair it with an element from N0.
You have to do it for all of the elements from N.
You don't know if that's possible. You just know that every element from N can be uniquely paired with an element from N0.
I see. I would say there's a difference between making a claim about "a subset" and a claim about "any subset"; many of us will treat the former as a "some" claim and the latter as an "all" claim. We similarly take "arbitrary" to imply "all" claims. Perhaps if we simply agreed on how we're using these words, there would be no dispute ...
Substantively, would you accept mathematical induction as showing that the mooted function maps every element of N to an element of N0? The proof is not hard.
Quoting Magnus Anderson
Quoting Magnus Anderson
Is there also a difference between "all" and "every"? Because you seem to be granting what you denied ...
Things would be so much easier if everyone just accepted this dictum. :wink:
Excellent use of the chess analogy.
That's not a contradiction within a function, as you diagnose, but a failure to specify a function.
That is,
In the argument we are considering, there is no such malformation. This is not just asserted, but demonstrated by the conclusion that the function is well-defined, injective and surjective.
Unlike the finite example, f: ? ? ??, f(n) = n ? 1 does satisfy the totality requirement: every natural number n has n?1 ? ??. Therefore the function exists and is indeed a bijection.
Well, yes.
In standard mathematics, we can define a function f: ? ? ??, f(n) = n ? 1, and check the definition. We saw that every n ? ? maps to exactly one element in ??.
Once the definition is satisfied, the function exists by construction. There is no need to “show it is not a contradiction.” A contradiction would arise only if the rule could not possibly assign outputs in the codomain, which is not the case here.
You suppose that before a function exists, we have to show it is not contradictory. But in mathematical thinking we define the function, check the definition, then if all requirements are satisfied, the function exists.
Again, it is up to you to show any contradiction, not up to us to show there isn't one. You have misplaced the burden of proof.
Quoting Magnus Anderson
The proof given shows that for each element in [math]\mathbb{N}[/math] there is exactly one element in [math]\mathbb{N}_0 [/math].
Take any element of [math]\mathbb{N}[/math] and there is a corresponding element in [math]\mathbb{N}_0 [/math]. Take any element in [math]\mathbb{N}_0 [/math] and there is a corresponding element in [math]\mathbb{N}[/math].
Since that works for any element, it works for every element. There is no gap. The rule at work here is Universal Generalisation.
Therefore the sets are the same cardinality.
That's not true. The definition of infinity I use is the one used in mathematics, to describe the natural numbers as unbounded, unlimited, without end. I do not reject this definition of "infinity".
Quoting Esse Quam Videri
Again this is not true. The philosophical objection is based in a fundamental logical principle, the law of noncontradiction. I demonstrated that mathematicians employ contradiction when they claim that the natural numbers are countably infinite, or a countable infinity. By the mathematicians' own definition of infinite, or infinity, it is contradictory to say that an infinity can be counted because "infinite" means that we cannot have such a count, it could never be acquired.
Quoting Esse Quam Videri
This is not analogous. I clearly show how the move of the mathematicians is 'illegal' (to use your word) within standard rules of logic, because it is contradictory. The natural numbers are defined as infinite, meaning limitless, endless, impossible to count them all. Then they say the very opposite, that the natural numbers are countable. Clearly, "countable infinity" is a contradictory concept where the first term contradicts the second. These are not my definitions which I have made up for this purpose. This contradiction is within the way that mathematicians themselves define the terms.
Quoting Esse Quam Videri
Again, this is wrong. The incoherence is internal to mathematics. The notion of "infinity" used by mathematicians themselves, is contradicted by the predication they make, when they propose a "countable" infinity. Here's an example much better than your chess proposal because the chess proposal fails to capture the situation.
Lets say we have a concept called "unintelligible" (analogous in this example to infinite). Then, we notice that there are different sorts of unintelligible things, that things are unintelligible in a number of different ways Different sorts of infinities). So, instead of studying the reason for, and the difference between, the different ways that unintelligibility appears to us, we simply name one of the forms of unintelligibility the "intelligible unintelligibility" ( analogous to countably infinite). Then we proceed to compare the other forms unintelligibility to this, under the illusion (falsity by contradiction) that we have made this type of unintelligibility intelligible by naming it so.
That is what the concept of "countably infinite" does. It creates the illusion (falsity by contradiction), that this type of infinity is actually countable. It's far better to use a concept like "transfinite", and state that the transfinite are a special type of infinite, but maintain they are not countable. This would exclude the possibility of an infinite set, or a transfinite set as this is the mistaken venture. It is the attempt to contain the boundless, limitless (infinite) into a set which is defined as an object, that requires the employment of contradiction. Putting limits to the limitless is contradictory.
Quoting Esse Quam Videri
The standard definition of "infinite" is not a problem whatsoever. so there is no need to abandon it. The proposal of "countably infinite" is a problem.
I clearly explained why it is not necessary, and actually inappropriate for me to propose an alternative framework. If mathematicians do not understand that they have incorporated contradiction within their framework, and so they are not inclined to rectify this, then I will just keep pressing this point. Maybe they never will.
Quoting Banno
The analogy is not similar. I have shown that the internal rules of the game (mathematics) are contradictory. Unless noncontradiction is not a rule in the game (mathematics), then the analogy fails. Are you and Esse Quam prepared to take that stance, to insist that the rule of noncontradiction is not a rule in the mathematician's game? if so, you might be able to make the analogy work.
You very clearly don't understand how language, definitions and oxymorons work.
Square-circles aren't squares either. They are also not circles. But they are also squares. And they are also circles. That's why they are oxymorons. They are two opposite things at the same time.
The same applies to the function that I mentioned. It is a function. But at the same time, it is not. The notation implies both. Not merely the latter. That's the mistake you're making. When you write, f: { 1, 2, 3, 4 } -> { 0, 1, 2 }, f( n ) = n - 1, that implies a function.
Symbols are capable of containing contradictions. It's not a new thing. It's the basis for the law of non-contradiction. It's not the case that symbols are necessarily X or not X. They can be both.
But either way, that's not really important, and it's nitpicking at best. The point is that the function does not exist. Which you agree with. And what that shows is that, just because you can define a function, it does not mean it exists. That was the entire point of that post.
Quoting Banno
And that's not true.
The only thing that you have shown is that you can take any element from N and uniquely pair it with an element from N0.
In other words, you have shown that, if we randomly pick an element from N, we can find its unique associate. Let's say we pick 1,345,219. Its unique associate would be 1,345,218. That holds true for every element from N. There are no exceptions.
But that does not mean we can put the two sets in one-to-one correspondence. That's a different thing. You haven't shown how many elements from N can be uniquely paired with N0. And your job is to show that you can take as many elements as there are in N0. Have you done that? Of course not.
All you have shown is that you can take an arbitrary subset of N that isn't larger than N0 and put it into a one-to-one correspondence with a subset of N0.
Do you understand the difference between the two?
The onus of proof is always on the one making the claim. If you're making the claim that bijection between N and N0 exists, you have to show it, and that means, you have to show that such a bijection is not a contradiction in terms. That's what it means to show that something exists in mathematics.
You haven't done that. But I have done otherwise ( contrary to what you say. ) But you won't that accept because you're overly attached to your fallacious proof -- essentially, a circular argument -- that there exists a bijection between N and N0. "We can take any element from N and uniquely pair it with an element from N0, therefore, there's one-to-one correspondence between N and N0."
Again, if all you're going to do is spend all of your time justifying your chosen authorities, which is precisely what you're doing, then you want see the mistake they are making.
It's a subtle point that is difficult to explain. Perhaps I should have used "any" instead of "every" in that second quote of mine.
Let me try to explain it another way.
You have a device that basically mimics the operation of f: N -> N0, f( n ) = n - 1. You can type in a natural number, starting at 1, to get a number from N0.
For one and the same input that you type in, you will get one and the same output from N0. Moreover, for every output from N0, there is exactly one number from N that can generate it.
That's what makes it a bijection in the weaker sense. And it's trivially true. We all agree about that.
But what that does not mean is that you can type in all of the numbers from N and produce all of the outputs from N0. That's a bijection in the stronger sense ( the one that matters. )
Suppose that you have all the time in the world. Suppose that means that the number of days at your disposal is larger than the number of natural numbers. Suppose that you decide to type in every number from N. On day 1, you type in 1. On day 2, you type in 2. And so on. How do you know that you will be able to produce all of the outputs? The fact that the function is bijective in the weaker sense does not tell you that.
countable
It all depends on how one defines "countable"
Infinity is a concept saying that there is no end in counting (in math), or final point (in physics or the material word or movement). If there were the end point of counting or movement is reached, then it wouldn't be infinity. Hence it is just an abstract concept, which doesn't exist in the real world.
Trying to count or prove infnity using math formulas or functions on the concept sounds silly and obtuse.
Exactly. "Countable" means something very specific within the formalism. The critique provided amounts to a [I]rejection[/I] of that notion, not a derivation of contradiction from [I]within[/I] the system.
That's right, "countable" means something very specific. But as I've demonstrated, the meaning of it, as defined, contradicts the meaning of 'the natural numbers extend endlessly'. That's where the problem lies. The natural numbers have been in use for a long time, with a very specific formulation allowing for infinite, or endless, extension. Then, "countable" was introduced as a term with a definition which contradicts the infinite extension of the natural numbers.
Please see my reply to jgill below.
Quoting jgill
As usual, I agree with you jgill. Here's the definition you provided: "capable of being put into one-to-one correspondence with the positive integers".
Nothing is capable of being put into one-to-one correspondence with all of the positive integers. We might say that the system was designed this way, to be unlimited in its capacity to measure quantitative value, 'to count'. That's why the system was formulated to extend infinitely. The positive integers derive their extraordinary usefulness from being extendable indefinitely, to be capable of counting any possible quantity. Notice, infinite possibility covers anything possible. To allow that the integers themselves may be counted. or to designate that something may be put into one-to-one correspondence with them all, is to say that there is a capacity which extends beyond them, i.e. that capacity to count them. This is to limit their usefulness as unable to measure that specific capacity. To limit the usefulness of the integers is counterproductive to the various disciplines which use mathematics.
I will attempt to clarify once more for the sake of the thread.
This statement of yours is neither a theorem, nor a definition nor a logical consequence of anything from [I]within the formal system[/I]. This is a philosophical assertion grounded in a procedural interpretation of "capable" that is [I]foreign[/I] to the mathematics. All you are saying here is that the impossibility follows from [I]your[/I] definition of "capable", and that you think your definition is the right definition. This is an external critique. At no point have you derived a contradiction from [I]within[/I] the system. Therefore, nothing you have said so far justifies the claim that the system is inconsistent.
I apologize if this comes off as rude, but this has been spelled out multiple times now from multiple different users. I think that if we still can't agree, then we have probably reached a principled stopping point that no further clarification is likely to resolve.
In other words, the problem is that you'll never finish.
Under this view, there are no functions on any infinite set. Not even f(n)=1. No functions on segments of the real line.
You could also demand that to be a set "in the stronger sense" you have to be able to finish listing its elements, and under that definition N cannot be a set.
Which, whatever. It's your sandbox, do as you like.
Quoting Metaphysician Undercover
I'm just wondering if you think somewhere in the rest of the paragraph (following the bolded sentence) you have provided an argument in its support. Is this the post you will have in mind when someone asks and you claim to have demonstrated that "Nothing is capable of being put into one-to-one correspondence with all of the positive integers"? Because it's just an assertion of incredulity followed by a lot of chitchat. (I think you have in your mind somewhere an issue of conceptual priority, but it's not an actual argument.)
Quoting Metaphysician Undercover
Sigh. You can't even pretend to be listing the reals and putting them into a one-to-one correspondence with the naturals. Rather the whole point of this kind of talk about transfinite cardinalities is that they are not all the same.
"Countable" is just a word, of course, and it doesn't bother us that it has been given a technical definition. Maybe "list-orderable" would be clearer.
Not only does none of this bother me, it has all the charm of good mathematics. Cantor's diagonal proof is simple, clear, and convincing. Even better is the zig-zag demonstration that the rationals are countable. ( (I think a more common presentation is just ordering pairs by diagonal after diagonal, but I saw it done first zig-zagging and it's stuck with me.) I think that was even more thrilling for me. In the natural ordering, in between any two, there are an infinite number -- how can they not be bigger than the naturals?! And then you see how they can be rearranged so that there is always a unique next rational. It's brilliant and convincing. People who don't ever see this, or who reject it for semantic reasons, are missing out on some lovely examples of the sort of thinking we should all aspire to.
That phrase is incoherent.
“An” refers to a discreet, limited, individuated, measured, unit.
So what does “an infinite number” point to or refer to? Certainly not some thing; certainly not some number.
We would be better off using the concept of infinity as an adverb, to describe a process. Instead of saying “there are an infinite number of natural numbers” we should say “we can count off natural numbers infinitely” or something similar.
Grammar police.
No infinite number of fractions (or infinite number of anything) exists like countable, quantifiable, distinguishable things exist. Existence sets a limit. Minds can take a whole, existing, limited thing, and then subject it to a mental process of division into fractions, infinitely. Or minds can multiply wholes infinitely, constructing bigger new wholes, infinitely. But at each step along the way, infinity is nowhere in site, and has never been reached, as it never will, infinitely.
So it is confusing to therefore assert at the get go “there is an infinite number of X.”
Saying any individual thing (like the universe, or God, or the pieces of an apple) is infinite, makes no sense, because it misunderstands where infinity exists, which is in the mind, as it constructs its descriptions and definitions of things and processes.
The notion some infinities are bigger than other infinities sounds romantic and poetic and is a curiosity - but no one can identify a discreet, whole, individuated, existing infinity to then compare it to a distinct, separately individuated whole other infinity. Infinity doesn’t quantify a single thing. It’s an adverb, tied to an existing process that theoretically is never finished processing.
Yes!
The diagonal argument and its friends are amongst the most beautiful and impressive intellectual presentations. I pity those who do not see this. The exercise here is to show folk something extraordinary; but it seems that there are a small but vocal minority who for whatever reason cannot see.
Why not work with the definition unless some contradiction is shown?
And in the cases of infinite sets, you have not shown a contradiction.
Quoting Magnus Anderson
This is perverse. That is exactly what has been shown. That each element of ? can be paired with an element of ??, and that each element of ?? can be paired with an element of ?. The bijection is fully established.
And a circle contains an uncountably infinite number of points. Oh well, no more analytic geometry.
The very first line of the proof does exactly what you ask for here. A function maps a each individual in one domain with an individual in the other. Hence:
If there is some other contradiction, then that is your claim, and up to you to demonstrate.
Indeed. And not just that. Much of modern maths would be unavailable or need reworking, with no apparent gain.
Magnus's position appears incoherent, in that he makes use of ? and other infinities while disavowing the relations between them. Meta is perhaps more consistent in apparently simply rejecting any infinities - or something like that.
Esse, please read what is written. I took the definition from a mathematics site, provided by a mathematician, jgill. The definition was "capable of being put into one-to-one correspondence with the positive integers". Please, for the sake of an honest discussion, recognize the word "capable" in that definition. And please recognize that your diatribe about my use of the concept "capable" is completely wrong, and out of place.
"Capable" is not a concept foreign to mathematics. Mathematicians employ the concept of "capable" with the concept of "countable", and surprise, there it is in that definition. You have no argument unless you define "countable" in a way other than capable of being counted. Are you prepared to argue that "countable" means something other than capable of being counted for a mathematician.
Or, are you proposing that mathematicians have their own special definition of "capable", designed so as to avoid this contradiction. Are you proposing that they have a meaning of "capable" which applies to things which are impossible, allowing that mathematicians are "capable" of doing something which they understand to be impossible? If so, then let's see this definition of "capable" which allows them to be capable of doing what they know is impossible to do.
Quoting Srap Tasmaner
Sorry Srap, it seems you haven't been following the discussion. I suggest you start at the beginning.
There is a "point at infinity" in complex analysis that arises when the complex plane is mapped onto the Riemann sphere. But it is simply the north pole of the sphere.
I wonder if and when physics will find uses for transfinite objects. Perhaps it already has.
Here's an example to consider Esse. Would you say that someone is "capable" of producing the entire decimal extension of pi? If not, then why would you say that something is "capable" of being put into one-to-one correspondence with all of the positive integers? Or do you equivocate on your meaning of "capable"?
God forbid you repeat yourself ...
Quoting Metaphysician Undercover
The key word in all this seems to be "all". You might as well bold it each time you use it.
Now, it's a known fact that you can line up all the rationals, in the sense of "fact", "can", "all", and even "you" that matters to mathematics. You disagree, and so far as I can tell only because anyone who tried to do this would never finish. Which --
Okay but when you said
Quoting Metaphysician Undercover
what are you referring to with this phrase, "all the positive integers"? I know what I would mean by that phrase; I genuinely do not know what you mean.
I don't see anything special about that word. Why do you think I should embolden it?
Quoting Srap Tasmaner
That's right, we know, by the defining features of the system, that no one could ever finish this task. It is impossible, by definition.
So, tell me how it is that you claim "it's a known fact that you can line up all the rationals"? Has someone produced this line of all the rationals, to prove this fact? Of course not, because it is also a known fact that this is impossible to do, because no one could ever finish. What's with the contradiction?
Quoting Srap Tasmaner
I probably mean the very same thing as you're thinking. Jgill raised the the issue of the meaning of "countable", and provided a reference. The definition from that referred page was: "capable of being put into one-to-one correspondence with the positive integers". So, think of what a "positive integer" is, a whole number greater than zero, and imagine all of them. Now do you know what I mean?
How on earth do you imagine all the natural numbers?
If you re-read my reply carefully you will see that I did not say that mathematicians do not use the word "capable", but that they use it in a different way.
"A is countable" means "?f such that f is a bijection between A and ?". That's it. There is nothing procedural in this definition. That was the point I was trying to make.
This is just one example of the way in which, when you change one feature of a language-game (conceptual structure), you often have to change the meaning of other terms within that structure.
So, "countable" in the context of infinity cannot possibly mean the same as "countable" in normal contexts. In the context of infinity, it means that you can start counting the terms and count as many as you like, and there is no term that cannot be included in a count; the requirement that it be possible to complete the count is vacuous, since there is no last term. It's not a problem.
Quoting Metaphysician Undercover
Well, perhaps it needs putting in a slightly different way. For example, how about "there is no rational that you cannot place on the number line"?
When you define the successor function - Successor(n)=(n+1) - you can see that there will never be a last number, and you don't have to try to write all the numbers down to do it. You can also see that each and every number is defined - or perhaps better, there is no number that is not defined. So you do not need to complete the task in order to see that the conclusion is true. In the relevant mathematical system, that is a proof that they all exist and can be located on the number line.
Is there something going on in the background here about actual and potential infinities?
I suppose that while transfinite numbers are not much used in physics, continuum cardinality and so on are present as background commitments. So if the space-time manifold in General Relativity is continuous, then I suppose transfinite cardinals are included by default in that formalisation; or so I believe. Of course, quantum theories would involve granularity, but this is entering into speculative physics, a can of worms.
There are oddities. In particular, I've had discussions previously with "finitist" folk who denied limits and such, and so were unable to make sense of differential calculus, and so in turn were led to denying corresponding physical entities such as instantaneous velocity. @Metaphysician Undercover has been known to do something along these lines.
I've thought about that, but always assumed that someone would then demand how I explain "unbounded but finite", which, I'm led to believe is also possible. I've sometimes used "there is no last term". Is there any problem with that? (Mathematically, I'm sympathetic layman.)
Quoting jgill
I certainly woudn't bet against that. I'm only deterred from betting in favour by the fact that it could take a long, long time before it happened.
the existence and meaningfulness of transfinite cardinals rests upon the Axiom of Choice, but that principle also implies unfettered resource duplication via the Banach Tarski Paradox, which goes way what is needed to define mathematical continuity, and well beyond the physical requirements and assumptions of general relativity, not to mentioning violating energy conservation.
By definition, the computational content of physics (i.e. physical inferential semantics) cannot rest upon choice axioms, because they represent what cannot be computed. In practice, physical continuity doesn't refer to an idealised continuum in the sky, but only to the ability to construct or measure vanishingly small changes in output in response to vanishingly small changes in input, for which multiple alternative languages are available, that don't carry the metaphysical baggage.
Moreover, the standard dogma of the transfinite cardinals, harms theoretical physics, by denying the ability speak of potentially infinite sets that naturally describe the content of a physical process better than Dedekind-infinite completed set balony.
Once AOC is relinquished, the unreal "beauty" of the ideal cardinals is replaced with the ugly and uncertain truth of equivalence classes of set bijections that are generally undecidable, and only potentially infinite, such they cannot hide their complexity behind a veil of cardinal representatives obeying a simple cardinal arithmetic.
In addition, it cannot even be proven, without begging the question, that transfinite induction up to ?0 is sound, since ?0 might not be well-founded. Hence a theoretical physics claim cannot rest upon transfinite induction. In practice, a "proof" that PA is consistent essentially amounts to beating the skeptic into submission with pseudo-religious dogma about the metaphysical "truth" of transfinite induction as decided by classical mathematicians who don't ultimately care about the physical truth and use-value of such theories.
Only the boring transfinite ordinals up to ?0 are empirically and computationally meaningful.
I can't, neither can you. Get the point?
Quoting Esse Quam Videri
I know you said this, but I do not believe you . The concept of "capable" is very straight forward with very little ambiguity. It means having the ability for. So, if you read through to the end of my post, I requested that you provide this special definition of "capable", which you claim mathematicians are using.
Quoting Esse Quam Videri
You are wrong again Esse. "Countable" is defined as a form of "capable" which is defined as "ability for". Therefore it is very clear that something procedural is referred to by "countable". Producing a bijection is a procedure. That is the point Magnus took up with Banno. You might obscure this fact with reference to 'function", and insist on a separation between "function" and "procedure" or employ a variety of other terms to veil this reality, but all this amounts to is a dishonest attempt to obscure the facts, deception.
Why do you keep insisting on things which you really ought to know are wrong? That is the problem. Instead of acknowledging, 'oh yeah, there are some problems with mathematical principles, and this is one of them', you go off and try to hide the problem. You see, in philosophy we meet these sorts of problems all the time, everywhere, in metaphysics, theology, free will, mathematics, physics, biology, etc.. Philosophers are critical, and look for these issues, that is critical thinking. Those things always pop up, because knowledge evolves, and what was once cutting edge becomes old, a then the problems get exposed. The faster knowledge progresses the more these issue get overlooked, and they multiply.
Now, philosophical criticism seems to be expected in some fields, relative to ancient ideas like metaphysics, theology, etc.. When a philosopher demonstrates problems in an ancient concept of God for example, this does not surprise anyone. However, in my experience on this forum, there are certain fields, mathematics and physics, for example, where criticism is regarded as unacceptable. It's like the dogma takes hold of the people, and is adhered to in such a religious manner, that criticism (heresy) must not be allowed. Those who faithfully uphold these principles seem to be programmed to disallow criticism. When problems are pointed out, they deny that their chosen dogma and ideology could even have such issues, and use whatever means possible to hide those features.
The critical point here is that these issues, which we as philosophers point out (inconsistencies and contradictions), are not unusual in human knowledge. They are common, widespread, extending throughout all the fields of knowledge. They are nothing to be ashamed of. We all make mistakes, and the human species in general is a growing and learning culture. The real problem arises from failure to recognize mistakes as mistakes, when they are exposed and the ensuing denial. That ought to elicit shame.
Quoting Ludwig V
Let's say that any language game is always evolving. Someone will dream up a new idea, or a new rule, in one's own private mind, and propose it to the others. They start using it, and if the others accept it, it becomes integrated into the game. If the new rule is not consistent with what's already existing then the others ought to notice this, point it out, and rectify the situation. Adopting it for use, would appear to justify it, and if it is inconsistent with some existing rule, that would be a faulty justification. It's analogous to someone offering you a proposal, and instead of thinking about it, to determine if you really agree, you just accept it, and carry on.
Obviously there is a problem in the concept of "countable". I submit that your proposal would not solve the problem. You are suggesting that when it becomes evident that the recently accepted rule is really contradictory to a previously existing rule, and ought not have been accepted in the first place, that we ought to just alter the definition of the offending word in one of the rules. But this is still not acceptable within a logical system because it amounts to equivocation. What this would do is simply obscure the obvious problem, contradiction, with a less obvious problem equivocation. Then all the problems created by what is really a contradiction would be obscured, hidden and more difficult to determine. This would amount to intentional deception, to recognize a problem of contradiction, then try to hide it behind equivocation. That's like taking a shotgun to your problem, blowing it to smithereens, so that you're left with a multitude of little problems instead of one big one.
Quoting Ludwig V
How does this make sense to you? To "place on the number line" is a procedural expression, to use Esse's word. We know that it is impossible to make the procedure of placing all the rationals on the number line. Therefore the proper conclusion and procedural statement is exactly opposite to what you propose: "there will always be rationals which you cannot place on the number line".
It is not my intention to obscure the facts. I am engaging honestly with you - and in good faith - even if it may not seem like it to you.
Here are the facts as I understand them:
The formal definition I provided to you (or similar variation) is the one you will find in many of the standard textbooks on Real Analysis, Set Theory and Discrete Mathematics that discuss countably infinite sets. This is why it confuses me when you say that you don't believe that this is the standard formal definition of "countably infinite".
Likewise, and for the same reason, I am also confused by your insistence that the definitional existence of a bijection requires that the bijection be temporally or procedurally executable. Within the global mathematics community it is commonly understood and accepted that procedural execution is not a requirement for definitional existence. This is why you will not find such a requirement listed in the aforementioned textbooks. This is also why I previously stated that adding this requirement would amount to something like an external constructivist critique of the dominant paradigm.
I hope that this helps clarify my perspective on this. I understand that you may not agree with the criticisms that I have offered, but they are based on sincere and honest confusion regarding your claims, given my current understanding of academic mathematics. I am certainly open to being mistaken on these points, but it's currently hard to see how given that these are fairly basic observations about how mathematics is currently done. Thanks.
You don't seem to understand the problem. "Countably" implies a procedure which you continue to deny. When we looked at the definition of "countable" it is defined by "capable", which implies "able to" perform a specified procedure. Then you claimed that mathematicians use a different definition of "capable" which doesn't imply the ability to perform a procedure. That's when I accused you of intentionally trying to obscure the issue, instead of facing the reality of it.
Quoting Esse Quam Videri
Well, it appears like "the global mathematics community" is mistaken then. When something is defined in terms of the capability to perform a procedure, and then it's understood that actually being able to perform that procedure is "not a requirement" for fulfilling the criteria of that definition, then this is obviously a mistaken understanding. Don't you agree? And please, live up to your claim of "open to being mistaken on these points".
I am very much open to be mistaken. I have had numerous discussions with mathematicians on this forum, and have learned a lot, altering my perspective on many things. This issue though, as I see it, is so simple, clear, and obvious, that it would require a substantial argument to prove that I am mistaken here. But that substantial argument has not been forthcoming. People simply assert that I am mistaken, and ridicule me for arguing against "the global mathematics community", as a form of appealing to authority, rather than actually addressing the matter with clear principles.
Allow me to apologize if my previous replies came off as an attempt to ridicule you. That was not my intention.
I see that what I've said so far has not convinced you. That's understandable. That said, I'm not sure I have the ability to express my critique any more clearly than I already have. I say that not in an attempt to blame you for misunderstanding me, but more as an acknowledgement of my own limitations in that regard. I still stand by my arguments, but I'm not sure how to productively move the discussion forward from here. Thanks.
No, my apology too, I didn't intend to imply that you have done this, in particular. But I might mention@Banno, and a few other members in the past.
Quoting Esse Quam Videri
As I said, I've learned a lot in my past discussions, so I'll offer you a perspective which you may be able to make sense of. Let's suppose that bijections simply exist without needing to be carried out as a procedural thing. This might be what's intended with the term "function".
For example, imagine that there is forty chairs in a room somewhere. There is simply an existing bijection between the chairs and the integers, so that the count is already made without having to be counted. It's just a brute fact that there is forty chairs there, without anyone counting them. This is a form of realism known as Platonic realism. The numbers simply exist, and have those relations, which we would put them into through our methods, but it is not required that we put them into those relations for the relations to exist.
I discussed this before, with @Banno I believe, in a discussion about the nature of measurement. The example was a jar of marbles. Our common intuition is that when there is a jar of marbles, or something like this, there is a measurement, a count, associated with it, the number of marbles which are in the jar. "Truth", or the correct count, would be to produce a count which corresponds with this already existing relation. You can see that this is completely different from a procedural "correct count". The procedural correctness is produced by performing the procedure correctly according to the rules, and the answer then is the correct answer without any necessary assumption of an independent measuring system (Platonic Ideals) already related, "truth".
The reason this issue came up, is because of the so-called measurement problem in quantum physics. In quantum physics it has been demonstrated that there cannot be an already existing independent measurement. So measurement is not a case of producing the result which corresponds with the already existing relation, it must be a matter of correctly carrying out the procedure.
Therefore, I argue that this is actually the true nature of "measurement" in all cases, that the correct answer is always a matter of carrying out the prescribed procedure correctly. Consequently I also argue that the Platonic realism which supports the other, intuitive notion of measurement, that there are independent numbers, which are already associated with things, as the true measurement, is misleading. This issue becomes very evident in the notion of infinity.
1 is a number, and every number has a successor. That's enough to show that the natural numbers exist.
That was me.
Now, of course, it's true there are issues with counterfactual definiteness in quantum mechanics, and "experiments which are not performed have no results." Sure.
It is also well-known that those issues do not arise in the same way at the macro scale.
Which is not to claim that acquiring knowledge at the macro scale is easy-peasy and there are no challenges.
Logic and mathematics are mental tools or technologies, habits of mind, that we have developed for dealing with things at the macro scale. And it is of the essence of these forms of thought that we use them to acquire knowledge without messing about with things in the real world.
This is unsurprising since our mental lives consist, to a quite considerable degree, of making predictions. Logic and mathematics enable us to figure out ahead of time whether the bridge we're building can support six trucks at once or only four.
The real world is not always cooperative, of course, and some of our predictions fail. But the link between logic and mathematics, on the one hand, and prediction, on the other, is so strong that it is not implausible that mathematics developed precisely as a refinement and systematization of our pre-existing efforts at prediction.
Which leads, at last, to my point, such as it is: there is something perverse, right out of the gate, about the insistence on "actually carrying it out". It misses an important point about the value of logic and mathematics, that we can check first, using our minds, before committing to an action, and we can calculate instead of risking a perhaps quite expensive or dangerous "experiment". ("If there is no handrail, people are more likely to fall and be injured or killed" -- and therefore handrail, without waiting for someone to fall.)
What's even more perverse is to take the difficulties we find in making good predictions about the natural world, even using logic and mathematics, and conclude not only that there is no way to have knowledge of the natural world ahead of time (which may be true, absolutely, but all we really need are reasonably well-calibrated expectations) but also that we have a similarly absolute inability to deal with our own minds, our own mental tools, that even logic and mathematics are not within our control, as if every time we multiply 5 and 7 the answer might turn out something other than 35.
The natural numbers turn out to go on forever, and we can prove this without somehow conclusively failing to write them all down. We now know that a certain sort of axiomatization of mathematics is not possible, though we once thought it was, and we know this without trying every conceivable way first.
What you get when you turn logic and mathematics upon themselves has a very different flavor than what you get when you try to tame the natural world with them. Mathematics has almost the character of pure thought, like its cousin music.
To see the demonstration that the rational numbers are equinumerous with the natural numbers and complain that it is not conclusive because no one can "actually do them all" is worse than obtuse, it is an affront to human thought.
There's an ontology which presumes that numbers exist, it's called Platonism. It's been demonstrated to be a very problematic ontology, and many philosophers claim that it was successfully refuted by Aristotle, as inconsistent with reality.
Quoting Srap Tasmaner
That's the problem with this type of issue. The supposed universal principles work extremely well in the midrange of the physical domain. Since the midrange is our worldly presence, and that is the vast majority of applications, we tend to get the impression that the principles are infallible, and "true". However, application at the extremes evidently produces problems. Therefore we must take the skeptic's eye to address the real possibility of faults within the supposed ideals.
Quoting Srap Tasmaner
What you call "the macro scale" is really the midrange, the realm of human dealings. Other than the micro scale and the macro scale, we need a third category which might be called the cosmological scale.
Quoting Srap Tasmaner
It is true, that this midrange scale, what you call the macro scale envelopes pretty much the entirety of our day to day lives. However, as philosophers with the desire to know, we want to extend our principles far beyond the extent of the macro scale. And this is where the issue of incorrectly representing infinity may become a problem.
For example, let's say that the macro scale is in the range of 45-55 in a scale of 0-100. So we might hypothesize and speculate about that part of reality beyond our mundane 45-55 range. If the application of mathematics, to the physical hypotheses leads to infinity in both directions at what is really only 35 and 65, then we have a problem because we place the majority of reality beyond infinity. And, if we close infinity by making it countable, then there is no way for us to know that there is even anything beyond 35 and 65. It appears from our physical hypotheses that we have reached infinity, therefore the extreme boundaries. And, if the mathematics has closed infinity, in the way that it does, then by that principle we actually have reached infinity. Therefore, by that faulty closure of infinity, 35 and 65 are conclude as the true ends of the universe, the true limits to reality, when reality actually extends much further on each side.
Quoting Srap Tasmaner
I don't see how this is relevant. The issue is not properly with "actually carrying it out", the problem is with the assumption that it is possible to carry it out. The defining feature of "infinite" renders it impossible to carry it out. So when we say that it is possible to carry out something which is defined as impossible to carry out, this is a problem regardless of "actually carrying it out".
This denigrates the status of "impossible". Now, "impossible" is a very important concept because it is the most reliable source of "necessity". When something is determined to be impossible, this produces a necessity which is much stronger and more reliable than the necessity of inducive reason. So the necessity of what is impossible forms the foundation for the most rigorous logic. For example, the law of noncontradiction, it is impossible for the same thing, at the same time, to both have and have not, a specified property. this impossibility is a very strong necessity. In mathematics, the impossible, and therefore the guiding necessity, is that we could have a count which could include all the natural numbers. if we stipulate that this is actually possible, then we lose that foundational necessity.
Quoting Srap Tasmaner
So this exposes the problem. We know that the natural numbers go on for ever. Therefore it is impossible to count them, or that there is a bijection of them. They could not have all come into existence therefore it is impossible that there is a bijection of them. This impossibility is a very useful necessity in mathematics. So if we stipulate axiomatically, that it is possible to count them, or have a bijection, then we compromise that very useful necessity, by rendering the impossible as possible.
Quoting Srap Tasmaner
This is a misrepresentation of what I am arguing. My claim is that it is definitively impossible to count the numbers. Therefore to represent this as possible is a contradiction. This has nothing to do with whether a human being, computer, or even some sort of god, could "actually do them all". The system is designed so that they cannot be counted. Nothing can do them all, and this is definitional as a fundamental axiom. So, whether or not anything can actually do them all is irrelevant because we are talking about a definition. Therefore, to introduce another axiom which states that it is possible to do them all, is contradictory.
This is to spectacularly miss the point.
Because we can prove what the result would be, we do not have to actually carry out the pairing of every rational number with a natural number. Proof is a further refinement of prediction, beyond even calculation. Of course it's impossible to count the elements of an infinite set as you would the elements of a finite set. But for the results we're interested in here, you do not need to. That is the point. We already know what the result would be if it were in fact possible.
(In my old computability textbook, this was described by having Zeus count all the natural numbers: he could finish, by using half as much time to count each successor. But even Zeus could not count the real numbers, no matter how fast he went.)
Infinite sets obviously present a barrier to calculation. So we deduce. Having deduced, we label our results, and then calculation becomes available again. We continually cycle between logic and mathematics, not just here but everywhere.
We don't need much ontology. Quantification will suffice.
How do you know that the natural numbers go on for ever? Have you tried to count them and failed? That doesn't prove that they go on for ever.
But perhaps you have counted some of them. That's easy enough to do.
When we are counting numbers, it is natural to start at the beginning. But we could start at any point in the sequence, and go on as long as we like from there. So we cannot find any numbers that cannot be counted.
So they are countable in the sense that some of them can be counted and we cannot find any numbers in the sequence that cannot be counted.
Admlttedly, the step from there to saying that they go on for ever is an induction.
But it is not an induction like the conclusion that the sun will rise tomorrow morning. We can see from the first few steps that anything that emerges from the function will be a number (because it is the successor to a number) and this is nothing to stop the next number emerging.
So the induction is secure. I know that there is some debate about this, but that is the debate we should be taking up.
Quoting Metaphysician Undercover
Ah, so this is about actual and potential infinities. My problem with that is that I don't see how the idea of a possible abstract object can work. In an Aristotelian system, as I understand it, the concepts of matter and potentiality are linked. But that only applies to physical or material objects. Since abstract objects are not material, I don't see how they can have any potential for anything.
I have to admit that much of the talk about functions, suggests that they produce their results only when we feed in values for the variables. This is misleading. The answers are "always already" defined, before we start calculating. Nothing more is needed for a number to exist.
The philosophical parameters for the debate what it means for a mathematical (abstract) object to exist are well enough defined, so that's the debate we are really involved in.
As the popularity of this post shows, we do need clarity on the mathematical object called infinity.
In my view the question comes down to simply just what does it really mean when Cantor showed us that the natural numbers cannot be put into 1-to-1 correspondence with the reals. The standard answer, that the infinity is simply larger, and thus we have larger infinities etc. doesn't really answer everything. It simply lacks the rigorous logic that is so ever present in mathematics. The problem with the Continuum Hypothesis shouldn't come as a surprise.
I was surprised, on enlisting in these fora, to find that there are folk who don't get to the stage of understanding that every natural number has a successor, that "...and one" works for any natural number. (not ordinals... another bit of the puzzle.)
And that infinity and one is still infinity. This hazy number play sets up the kid's intuitions. Especially where it doesn't work. Infinity is not part of the structure that lets us play the number game. It needs new rules.
Yes, the purely constructive meaning of the diagonal argument, is that any constructable injection from the naturals to the Reals defines the construction of a new real. And all that this constructively implies, is that it is impossible to define a surjection from the natural numbers to the reals. Hence it says nothing about whether or not an injection exists from the reals to the natural numbers, and hence the diagonal argument does not rule out the possibility that the real numbers might in fact be a subset of the natural numbers.
As far as the computable Reals are concerned, all that the diagonalization argument implies is that the computable reals are a subset of the Naturals that cannot be "detached" from the rest of the Naturals by an algorithm. For we know by definition that there aren't more computable reals than naturals, since a computable real refers to a computer program of some sort that has a godel number.
The hypothesis that every real number can be listed by an algorithm, is equivalent to knowing the limiting behaviour of every computer program. So what Cantor actually showed, is an indirect proof that the halting problem cannot be solved, and not that there are "more" real numbers than natural numbers.
I demonstrated already, it's not a proper "proof" because it relies on a false premise. This produces your incoherent, unsound conclusion "we can prove what the result would be". The incoherency of proving that the result of an impossible task is anything other than incompletion, is obvious.
Quoting Srap Tasmaner
Deduction from false premises produces absurdities. That's what Zeno is famous for having demonstrated.
Quoting Banno
Maybe not much, but some. Claiming that numbers "exist" is ontology. If you avoid the ontology, then what are you quantifying?
Quoting Ludwig V
Mathematical ideals are produced by definition. People decided that this would be really good, and so the system was designed and maintained that way.
Quoting Ludwig V
But the issue is whether an infinite quantity is countable. Any finite quantity is, in principle countable. But, since "infinite" is defined as endless, any supposed infinite quantity is not countable.
This problem is strictly confined to Platonism, which treats a number as an object which can be counted. So it inheres within the principal axioms of set theory which premises mathematical objects. Numbering is the means by which we measure a quantity of distinct, individual things. When we assume that a number is a distinct individual thing (Platonism), then we might be inclined to measure the quantity of numbers (cardinality).
The problem which jumps out, is that now we are trying to measure the measurement system with itself. And the ontological issue is that it is fundamentally false to represent a number as a distinct individual thing which can be counted. In reality "a number" is a concept which has its meaning in relation to other numbers (ordinality). Therefore we cannot isolate "a number" to be a distinct object, it would lose its meaning and no longer be able to serve its purpose as the concept it was meant to be. Therefore Platonism, which treats ideas as distinct objects which can be counted is ontologically unacceptable.
Quoting Ludwig V
I'll give you a brief description why abstract "ideas" are classed as potential by Aristotle. This forms the basis of his claimed refutation of Platonism, and provides the primary premise for his so-called cosmological argument which demonstrates that anything eternal must be actual.
The Pythagorean Platonists, as distinct from Aristotelian Platonists, insisted that ideas, specifically mathematical ideas, had actual existence as eternal objects, eternal truths in themselves. Aristotle premised that it is the geometer's mind which gives actual existence to the ideas. The actual existence of the idea is within the mind. Therefore if we premise that the idea had existence prior to being "discovered" by the mind (as the assumed eternal), we must conclude that it existed potentially. (The cosmological argument then goes on to show that anything eternal must be actual.) So the refutation of Pythagorean Platonism sets up a distinction between an idea within a mind, and the supposed independent idea. Within the mind it is actual, and however it exists in the medium outside of minds, is as potential. So when we assume numbers to be independent objects rather than thoughts within a mind, then they exist potentially, not as actual objects.
Quoting Ludwig V
This is probably the heart of the issue. "Exist" is a term which is properly defined by ontology rather than mathematics. Therefore the discipline of ontology is the one which ought to determine whether numbers exist. Notice, @Banno makes some seemingly random claims about the existence of numbers. Since the distinction between what exists and what does not exist forms the basis for our judgements of true and false, we can't simply make an arbitrary, or completely subjective stipulation, or axiom, which defines "exist". That would imply total disregard for truth.
Exactly, and we aren't understanding those rules yet. What we see are paradoxes and we simply want to avoid them or assume there's something wrong. There isn't anything wrong, it's that we start from the wrong axioms.
Indeed it's very interesting. I have just later understood when grandfather, a math teacher himself, just shook his head and said it was too difficult, when at first grade at school they started teaching math with set theory. But so progressive and courageous were math-teaching in the 1970's in Finland.
Yet I think mathematics will suprise us some day... once we really understand infinity, it likely is something that can be taught at school for kids. Mathematics is just so beautiful.
I think the real breakthrough will be in when we understand that there is the mathematics that is not computable and thus doesn't start with addition, but is inherently important to mathematics. Yet we (and I think other animals too) have started using mathematics from the practical side of it. "No lions, one lion, two lions, many lions." could be the "mathematical system" for a zebra in the African Savannah, which is very useful for it's survival. It doesn't need calculus, it doesn't ponder infinity. Yet for us when we want to make mathematics a logical system. For the grazer on the Savannah it might be enough, but for us the "0,1,2,many" system isn't enough at all. We need calculus, infinity is obvious in mathematics. Yet if we assume that everything in mathematics starts from counting, from finite natural numbers, we are making a mistake.
It's just extending the way we talk about numbers. What started with the Biggest Number game gets extended into infinity, both ? and ?, the difference being that while ?+1=?, ?+1>?; The first reflecting the teacher's answer "infinity plus one is still infinity", the second, the player's answer "infinity plus one is bigger than infinity". What we have is a division in how we proceeded, in the rules of the game, not in what "exists" in any firm ontological sense. It's chess against checkers, not cats against dogs. Neither set of rules is "true" while the other is "false".
And the great thing about these games is that they are extensible, in that we add more rules as we go, keeping the game coherent, while being able to talk about more and different stuff.
Part of where Meta and Magnus have difficulty is in their insistence that one way of talking is right, the other, they call variously incoherent or inconsistent, both without providing an argument and in the face of demonstrations to the opposition effect. To establish incoherence, they would need to show a violated rule internal to the system, or an explicit contradiction derivable from its axioms.
Mere discomfort with plural rule-sets doesn’t suffice.
Within cardinal arithmetic, ?+1=? is true; within ordinal arithmetic, ?+1>? is true. Cross-applying the rules is what generates the illusion of contradiction.
I'd also relate this back to my essay Two ways to philosophise, and to the arguments in Logical Nihilism. It's better to have an incomplete theory that is coherent than a complete theory that is inconsistent or artificially restricted. And better to have many differing, incomplete logics than one, monolithic yet restricted logic. These allow for growth. Advocating for new rules, new distinctions, new domains of discourse gives us a normative standard that is neither realist nor relativist.
Critics may conflate pluralism with anything-goes relativism. But only because coherence is doing real work; incoherent extensions are still excluded. Others will insist that without a privileged logic, critique collapses. But critique is local, rules are criticised from within practices or at their interfaces, not from a mythical God’s-eye view.
To be is to be the value of a bound variable. ? and ? are cases in point. In maths, Quine's rule fits: existence is not discovered by metaphysical intuition but incurred by theory choice. Quantification, ?(x)f(x), sets out what we can and can't discuss.
It's really good that now people are more and more noticing the simple link with Cantor and undecidability resuls of Turing and Gödel. Negative self reference is a very powerful tool in logic.
Quoting Banno
Indeed. And this is why it's actually very informative and interesting to listen to actual finitists as they can make valid criticism of ordinary mathematics. Just like every school in philosophy or economics or whatever, also in mathematics various schools make interesting viewpoints that shouldn't be categorized as being either right or wrong.
Quoting Banno
The first uncountable ordinal is the interesting question. What is it, what does it mean and what is the logic then?
I think the main problem is that proving something is inherently close to computation, thus no wonder that we have the undecidability results lurking with the uncountable.
Tough nut to crack, but actually it's great that some large basic questions are still open in math and not everything has been done before us. Because that's the error many do: Russell and Whitehead thought that as everything is already there, they just had to write everything out then in a "small" book.
As I said, Platonism, which is an unacceptable ontology.
Platonism is indeed unacceptable, but quantification is not platonic. Sad you can't see that.
Quantification does not require Platonic commitment; it merely specifies the domain of discourse and what statements about it are true. This is consistent with nominalist or structuralist interpretations.
Do you recognize that set theory is based in Platonism?
No, Meta. Quantification or assigning a value does not require Platonic commitment. A value can ‘have being’ within a formal system, a constructive framework, or a model, without existing independently as Plato would claim.
Quoting Metaphysician Undercover
Sad. Formally, set theory is just a system of rules. Treating its sets as independently real is a Platonic interpretation, not a necessity.
Guess it's back to ignoring your posts.
You were claiming that numbers "exist", and how to be, is to be a value. Now you've totally changed the subject to "assigning a value".
Quoting Banno
Sure, and those rules are axioms about "mathematical objects". When you were in grade school, were you taught that "1", "2", and "3" are numerals, which represent numbers? Notice, "2" is not a symbol with meaning like the word "notice" is. It's a symbol which represents an object known as a number. In case you haven't been formally educated in metaphysics, that's known as Platonism.
Quoting Banno
And I'll opt to believe that you willfully deny the truth, rather than simply misunderstand.
Same thing. Again, not my problem that you don't understand this.
Quoting Metaphysician Undercover
Very sloppy work. Platonism is not the claim that symbols refer to something, but that mathematical objects exist independently of any theory, language, practice, or mind, and are discovered, not constituted, by mathematics. Nothing here commits to that. You are equivocating between reference and ontological independence.
Quoting Metaphysician Undercover
You are looking for a rhetorical dodge to get out of the mess you find yourself in.
As I said, denial!
When numbers are assumed to be mathematical objects, these objects simply exist independently of any human mind. The supposed object is not in my mind, nor your mind, because it would be in many different places at the same time. Under the assumption of Platonism, a bijection does not need to be carried out, it can be represented, because it is assumed to already exist independently of any minds, so we just need to reference it. Likewise, the natural numbers can be represented with "{1, 2, 3, ...}" but only if they exist independently of any minds.
Do you understand the difference between the representation of a set of objects, and a formula for the procedure which you called "assigning a value"? Could you read 1, 2, 3, ... as a formula for assigning a succession of values?
Quoting Banno
What you stated her is blatant Platonism.
If however, "1, 2, 3..." signifies to you, a formula for the process of "assigning a value" in a specified sequence, rather than representing an infinity of numbers, this is not Platonism. Can you apprehend the difference? I think you can.
And I think, that's why you switched form "to be is to be the value of a bound variable", to, "assigning a value". right after you stated "Platonism is indeed unacceptable". You know the mathematical principles you argue are thoroughly Platonist, and you feel ashamed of this. So you tried to cover this up. Why the dishonesty? Accept mathematics for what it is, and get on with it. Shameful deception and attempts to disguise your ontology get us nowhere.
Every time that you say 1, 2, 3... represents an infinity of numbers, that is blatant Platonism. It is absolutely necessary that the referenced infinity of numbers must have independent existence because it is absolutely impossible that they could exist within any minds.
Your view is called finitism. It's from Aristotle.
No, that's not quite right. I reject the assumption of any "mathematical objects" finite or infinite, as Platonism, and unacceptable. It just happens that the absurdity of assuming such objects becomes very evident with supposed infinite objects. In other words, the reality of "infinite", and "infinity" serves very well to demonstrate the falsity of Platonism.
Quoting Banno
In case you think I did not address this claim, 'having being within a system' is fiction. It can be said that Frodo Baggins has being within a system, but this type of being is well known as "fictional".
Being within a mathematical system, is fictional being. Therefore it is a false premise rendering any supposed "proof" as unsound.
Ok.
So you do know that the series is infinite without completing the count of them all.
Quoting Metaphysician Undercover
Zeno, if I remember right, thought that he had developed a proof that change is impossible. It is other people who treat his proof as a reduction. The catch is that they have not yet discovered which of his premises is false.
Quoting Metaphysician Undercover
That's just playing with words. We agree on the facts.
Quoting Metaphysician Undercover
I thought following paragraph very interesting. It made sense of your arguments. Not that I agree with it.
Quoting Metaphysician Undercover
I was going to put that argument to you. But I see it is not necessary.
Quoting Metaphysician Undercover
And yet, Frodo Baggins exists - in the way that fictional characters exist. They can even be counted. Similarly, numbers exist - in their way.
Quoting Metaphysician Undercover
I'm not quite sure that I understand you. I think that it is not necessary for the infinite number of numbers to exist in my mind. All I need to have in my mind is S(n) = n+1.
Quoting Banno
Quite so. The difference is, however, that while we can work quite happily with both chess and checkers, it seems pretty clear that this game is not just an addition to the menu of possible ways we might amuse ourselves. @Metaphysician Undercover believes it is illegitimate in some way. It turns out that the disagreement turns on a metaphysical disagreement. Tackling that needs a different approach.
Quoting Banno
A consummation devoutly to be wished for. But can you explain a bit what that standard is?
This is exactly right, and it is the sort of move I have been trying to hold up as a triumph of human thought. We cannot list them all, but we can give a rule, and a rule we can hold in our heads and work with. (In a similar spirit, Ramsey suggested that universal quantification is actually an inference rule: to say that all F are G is to say, if something is F then it's G.)
I didn't know that came from Ramsey. But it is in the same spirit. Instead of having to remember each specific F and that it is G, and that my list of F's is complete, we just have to remember a rule.
If I recall correctly, he specifically said "habit" rather than "rule", which suggests naturalizing logic, and indeed I think that's where he was headed.
We do not find numbers in nature, not in the same way we find trees and rocks and clouds. Did we then make it all up, our mathematics? Is it just a game we play with arbitrary rules?
I don't think so. I think naturalizing mathematics means understanding it as behavior, mental behavior. We can interact with rocks carryingly or stackingly or countingly. The concept "rock" is already an abstraction, a mental skill we can apply to objects found in nature. Mathematics, on the other hand, is more like abstraction turned inward, applied to our own mental behavior. The cardinality of a set of rocks is an abstraction of an abstraction of an abstraction. (In many contexts, "idealization" is an even better word than "abstraction". I think Plato was desperate to understand how this works, and if you turn the tools you have, such as idealization, upon thought itself, something like the theory of forms is all but inevitable.)
We never encounter in nature infinitely long lines, endless planes, perfect circles, and so on. But that's fine: those 'objects' are not exactly abstracted from objects we encounter in nature, but from how we think about nature. And it turns out this way of thinking has many properties that are convenient to its smooth functioning that will never be found in nature. And that's fine, because mathematics is not a picture of nature but a tool we use in dealing with it.
If you think Meta has convincingly shown that numbers do not exist, then I suppose that's an end to this discussion. And to mathematics.
But I hope you see the incoherence of his position.
This is Aristotle's finitism. Finitism is like this: if we put you in a spaceship that has an odometer, you will never see any but a finite number on it. You'll never see an infinity symbol, though you never stop moving forever and ever.
Per Aristotle, infinity exists in potential. The actual is always finite. Set theory, by handling infinity as a set, appears to be defying finitism. This is an unresolved issue in phil of math. Someday it may result in a shift in thinking about set theory.
There's a lot that could be said about that. It is tempting to classify some rules as habits. Habits neither have, nor require, any kind of justification. They are what they are and that's all that can be said. Rules, on the other hand, can provide justifications and, just for that reason, are, mostly subject to justifications. Some rules do not have, and do not require, any further justification - especially if they set the standard for right and wrong. So these are like habits in that they do not require justification and unlike habits in that they can provide justification. (I'm sketching here.)
Quoting Srap Tasmaner
That's true. But as soon as we recognize a tree, and recognise this is a different tree and this is the same tree again, we have have sown the seeds of counting. Not so much in the case of rocks and even less to in the case of clouds.
I don't think we make mathematics up like a story or even like a game (although the comparison can be useful). But it is a mistake to move from saying that we make mathematics up to saying that it (or its objects) don't exist. They don't exist in the way that trees and rocks do, but that shouldn't be a problem.
I don't like saying we discover or recognize mathematics either. It's not wrong, exactly, so long as we don't compare such discoveries to what explorers like Columbus do, thus positing a world of mathematics comparable to the physical world we live in.
Mathematics shares features with other practices in our lives and so comparisons can be useful. But I am unable to adopt any one comparison as the whole truth. (Comparison is always partial and involves differences as well as similarities.)
I'm in favour of naturalizing, if that means locating things in our lives and practices. But one has to acknowledge that there is a somewhat different project, at least in relation to mathematics, which is the search for foundations for a structure (as opposed to a set of practices).
Quoting Banno
No, I don't think that @Meta has shown that numbers don't exist. I'm inclined to think that he doesn't believe that, either. He has been explicit that he rejects what he calls Platonism, but I don't think it follows that he thinks that numbers do not exist. I'm not sure he even rejects the idea that there are an infinite number of them - since he realizes that we can't complete a count of the natural numbers. I do think that we can't get to the bottom of what he thinks without taking on board the metaphysical theory that he has articulated.
Quoting frank
I didn't intend it to be. Surely, it works like this - Aristotle thinks that infinity cannot be real if we cannot complete the count. I intended to say that infinity is real even if we cannot complete the count, because the successor function tells us so.
Quoting frank
Yes, I'm aware of that - and of the startling results that followed when his view was set aside and infinity was treated as real, thus enabling the invention/discover/development of the calculus.
Come to think of it, one might argue that Aristotle does not actually say that infinity doesn't exist, just that it exists ("in potential"). But everyone thinks that something that potentially exists doesn't exist and it would be perverse to suggest otherwise, I suppose.
Aristotle is not set aside by calculus because it does not deal with actual infinity. Set theory is a different matter.
Of course, why would say that? it's defined as infinite. That's the whole point. It is infinite and infinite is defined as boundless, endless, therefore not possible to count. So any axiom which states that it is countable contradicts this.
Quoting Ludwig V
Yes, a finite number of fictional thigs is countable. But infinite is defined as endless, therefore it is impossible to count an infinite number of fictional things.
Quoting Ludwig V
I'll explain again. If numbers are assumed to be independent Platonic objects, we can assume that bijections simply exist, without needing to be produced by a human being. However, the infinite bijection is a matter of contradiction, even if bijections simply exist. Therefore it ought to be rejected as incoherent. If, on the other hand, numbers are assumed to be fictional objects, created by human minds, the same contradiction still remains. It is the idea that numbers are infinite, yet countable as objects, which is incoherent. So it does not matter how you validate the existence of numbers as objects which can be counted, the incoherency cannot be avoided.
Quoting Ludwig V
I don't understand why, so many people on this thread seem to think that if they can make symbols which represent something incoherent, this somehow makes it coherent. When we speak contradictions, that's what we do, use symbols to represent something incoherent. Why would you think that writing it somehow makes it coherent? I can say RNR stands for the thig which is both red and not red at the same time, but how does symbolizing it make it coherent? How does "S(n) = n+1" make you believe that an uncountable number of objects is countable? I truly cannot understand this.
Quoting Ludwig V
It's not a metaphysical problem directly. As a matter of contradiction between basic axioms, it is an epistemic problem within the mathematical system (set theory). As explained above, it doesn't matter which metaphysics you use to validate the existence of numbers as countable objects, the problem remains.
I do propose that it could be resolved with a metaphysical solution though. The solution is to reject the ontology which supports the idea that numbers are countable things, along with the mathematics which follows (set theory). An idea is not a thing which can be counted, and that is a basic flaw in the ontology which supports set theory.
Notice, it's not a metaphysical problem in itself. We can assume that numbers and all sorts of ideas are objects, and maintain that ontology. The problem is epistemic. We think that since numbers are objects then they ought to be countable. that's what produces this problem. To resolve the problem we might say that numbers are a type of object which is for some reason not countable, but that creates a problem with the concept of "object". Therefore it's better, and actually provides a better foundation for understanding what concepts are, if we deny that numbers are objects.
Quoting Banno
The point is that a number is not a thing which can be counted, it is something in the mind, mental. I think you understand the difference between physical, sensible things which can be counted, and mental thoughts which are not individual things that might be counted. You did read Wittgenstein's Philosophical Investigations didn't you? Did you learn anything from it?
There is a very significant error in the idea that a measuring system could measure itself.
Quoting Ludwig V
The point is that "numbers" do not exist as individual countable things. This is a misrepresentation of what a number is, and the problem becomes evident when we allow the infinite capacity of numbering, and then try to count those numbers. So it doesn't matter if you represent the number as an independent Platonic object, or an object of human construct, either way is faulty. A supposed individual number is really an idea, which is dependent on other ideas for its meaning, and cannot be accurately represented as an individual object.
Maybe for you. For me, that's a theorem.
Quoting Metaphysician Undercover
Quoting Metaphysician Undercover
Then this is nothing to do with infinite sequences, infinite sets, or infinity.
Your position is that you can't count how many numbers there are between 1 and 10.
Now many integers are there between zero and five?
"Infinite" means "not finite", not "not countable".
And to be countable is to have an injection into ?; or equivalently, to have a bijection with a subset of ?. For infinite items, that subset is ? itself.
? is both infinite and countable. ? is infinite and uncountable.
You have yet to show a contradiction in bijection; indeed, you have yet to show what that might even mean. Mathematics on the other hand takes a bijection between two sets A and
B to mean there is a rule f such that each element of A is paired with exactly one element of B, and each element of B is paired with exactly one element of A. Here is a proof that ? has a bijection with ?, and so is countable:
Take
[math]f : \mathbb{N} \to \mathbb{N}, \quad f(n) = n[/math]
Conclusion:
The function [math]f(n) = n[/math] is a bijection from [math]\mathbb{N}[/math] to [math]\mathbb{N}[/math].
Therefore [math]\mathbb{N}[/math] is countable.
The bijection is not assumed, it is demonstrated.
I can't really follow anything you are saying.
Quoting Banno
Again, "integer" is a faulty concept, because it assumes that "a number" is a countable object. That's exactly the problem I explained to you. We ought not treat an idea as an individual object. Providing more examples of the same problem will not prove that the problem does not exist. The problem of Platonism is everywhere in western society, even outside of mathematics, so the examples of it are endless.
Quoting Banno
This does not address the point. A rule can contradict another rule within the same system. Saying that there is a rule which allows a specific bijection doesn't necessarily mean that there is not another rule which disallows such.
Quoting Banno
That's false, it's not a demonstration, at best, it's begging the question. You have no definition of "countable", so your conclusion, "Therefore N is countable" does not follow. It would follow, if you provide a definition of "countable" which begs the question. The proper conclusion is N is countable if N is countable. Then a definition of "countable" could be provided which contradicts the infinite nature of the natural numbers, making "N" "countable". Voila, begging the question with contradiction. I think Magnus already explained this to you, so you're just continuing to demonstrate your dishonest denial. I don't see any point to further discussion, you'll only continue to refuse to look at what is shown to you, and rehash the same faulty arguments.
So apples are countable, but numbers aren't. :grin:
You got it bro!
We measure the object not the measurement tool. The standard metre cannot be measured. Numbers are the tool, not the thing to be measured.
:joke:
Oh oh, the set {1,2,3} has 3 numbers. :gasp:
It's called nominalism. I would ask one favor though. Stop capitalizing the P in Platonism. The phil of math view of platonism. Plato pitted opposing ideas against each other, so for instance, in Parmenides, he outlines a lethal argument against the Forms. That's why they use a little p: platonism.
Quoting SEP
Quoting jgill
A nominalist entirely rejects set theory because it's a mountain of abstract objects.
I agree. but my spell check doesn't like little p platonism. And, I count the distinction as unimportant because there really would be no such thing as big P Platonism if we maintained that distinction. Plato pitted ideas against each other so there's no real ontological position which could qualify as big P Platonism. So they end up being the same meaning anyway.
I think you're discounting the importance of community. If it's not stretching your spine out of shape, you can go along with the rest of the phil of math and write it as platonism. It's a little nod to the deep bonds that hold us together over the millennia as our brothers and sisters try to take freakin' Greenland and what not.
We can make it simpler for you: How many whole numbers are there between one and three?
I say one. You say, they can't be counted.
Quoting Metaphysician Undercover
Then show us that other rule. Pretty simple. Set the supposed contradiction out.
Quoting Metaphysician Undercover
“Countable” is defined as “there exists a bijection with ? (or a subset of ?).” I bolded it for you. Again, if you think there is a contradiction in that, it is up to you to show it.
Begging the question would look like this:
But what actually happens is:
- Definition: “countable” = “there exists a bijection with N”.
- Construction: Exhibit such a bijection (the identity).
- Conclusion: Therefore N is countable
.That is definition + witness, not circularity.
You of course do not have to respond to my posts. Keep in mind that everything I have set out here is standard ZFC, and has been examined by countless mathematicians, yet remains solid. It is your account that is eccentric.
This issue is more complicated though. The Neo-Platonists took Plato's name and claimed to continue Plato's school, but their ontology is consistent with what you call platonist. Aristotle's school claimed to be the true Platonists but the Neo-Platonists took the name. So you have to take on the Neo-Platonists, and tell them that they should call themselves Neo-platonists, as not true Platonists. But this problem has been around for millennia, and they do not like being accused of misrepresenting Plato, they like to claim the true continuation of Plato's teaching.
Quoting Banno
i say it's a loaded question, like "have you stopped beating your wife?". If we give up on the idea that there are numbers in between numbers, we get rid of an infinity of problems from infinitely trying to put more numbers between numbers. This supposition that you have, that there are numbers between numbers is very problematic.
Quoting Banno
I did it all ready in this thread, numerous times. If you're truly interested go back and reread my posts. But I'm tired of it. And I know you, you'll just deny anyway so what's the point?
Quoting Banno
Right, begging the question. "There exists a bijection with N" is explicitly saying "N is countable". Are you kidding me in pretending that you don't see this?
I am a Neoplatonist, and I don't care whether you capitalize the P or not! :grin:
So your argument is that 2 is not between 1 and three.
Righto.
Quoting Metaphysician Undercover
Well, no. You claimed there is a contradiction, repeatedly, but never showed what it was. So go ahead and quote yourself.
Quoting Metaphysician Undercover
"There exists a bijection of N" is the conclusion, not an assumption.
He's saying that 2 isn't a thing. It's a modifier like pink. You can't count pinks because it's not a thing you count. It's nominalism.
Yes, indeed he is.
And the counterpoint is that "being a thing", especially in mathematics, consists in being the value of a bound variable. As in "for any whole number, if it is between one and three then it is two"
This is not saying numbers are spooky abstract objects. It's saying that whatever our theory quantifies over, exists according to that theory. The nominalist’s complaint about “thinghood” simply misses the target.
Quoting frank
“Pink” is not something that can be the value of a bound variable in arithmetic. “2” is. The analogy fails.
The other error here is to think that If something is countable, it must be a concrete or abstract thing. it ain't so. And note the equivocation of "thing".
Ok. Here's some stuff that won't work if we accept Meta's ideas.
We could go on.
Yes. Nominalists believe we don't need to posit abstract objects to make sense of math. It's generally considered that they have the burden of proof, and they take that seriously.
No mainstream nominalism does this, because it destroys the grammar of mathematics.
And so on. It's not nominalism as usually understood. Even predicativist or fictionalist views preserve quantificational structure.
Ok. I'm probably wrong then.
Yes absolutely, if we interpret "refuting cardinal analysis" as ditching ZF/ZFC for being computationally inadmissible due to the infinite hierarchy of cardinals being computationally meaningless and poorly motivated, given the fact that ZF/ZFC are set theories that are descendents of Cantor's theological prejudices that aren't true by correspondence to anything of relevance to science and engineering.
More specifically, the Cantor-Schröder-Bernstein Theorem that is the foundation of infinite cardinal analysis, is abjectly false in any constructive intepretation of the diagonalization argument that is conscious of undecidability.
In reverse mathematics, where we start by analysing a theorem without first assuming a particular axiomatic foundation, then the CSB theorem becomes the assumption that if f : A --> B is an injection (written |A| <= |B|) and g : B --> A is an injection (written |B| <= |A|) , then there must exist a bijection between A and B (written |A| = |B|).
So let's take P(N) to be the decidable subsets of the natural numbers. Then is CSB true or false?
1. We know that we can construct an injection P(N) --> N via Turing machine encoding of decidable sets. (|P(N)| <= N)
2. We can build 'any old' injection f : N -> P(N) to show that |N| <= |P(N)|.
3. Hence according to CSB, the set of decidable sets of P(N) has the same size as N.
And yet f cannot be a surjection: For diagonalising over f must produce a new member of P(N), but this isn't possible if f is surjective. Hence f cannot be a surjection, and this is the reason why diagonalization can produce new members of P(N), without P(N) ever being greater than N.
This particular case turns CSB against it's originator Cantor, for CSB insists that P(N) and N must necessarily have the size, in spite of the fact a surjection N --> P(N) cannot exist.
So this is where Cantor specifically went wrong: he should have interpreted diagonalization as showing that a surjection cannot always exist between countable sets. But instead, Cantor started with the premise that a surjection A --> B must always exist when A and B are countable, which forces the conclusion that diagonalisation implies "even bigger" uncountable sets, which is a conclusion that Cantor accepted because it resonated with his theology.
That's exactly right. To say that 2 is between 1 and 3 is to say that it serves as a medium. However, in the true conception and use of numbers, 1, 2, 3, is conceived as a unified, continuous idea. This unity is what allows for the simple succession representation which you like to bring up. No number is between any other number, they are conceived as a continuous succession. To say that 2 comes between 1 and 3 is a statement of division, rather than the true representation of 1, 2, 3, as a unity, in the way that the unified numbering system is conceived and applied.
Quoting Banno
"Infinite" means limitless, boundless. The natural numbers are defined as infinite, endless. limitless. All measurement is base on boundaries. To say a specific parameter is infinite, means that it cannot be measured. Counting is a form of measurement. Therefore the natural numbers cannot be counted. To propose that they are countable, is contradictory, because to count them requires a boundary which is lacking, by definition.
This is why "open sets" are used to justify unmeasurable spaces, resulting in an incoherent concept of continuity. Incoherent "continuity" is the result of the false opinion that there exists numbers "between" numbers, instead of representing the numbers as a unified concept.
Look, to say that something is infinitely heavy or light means its weight cannot be measured. To say that it is infinitely long or short means that its length cannot be measured. To say that it's infinitely hot or cold means that its temperature cannot be measured. To say that it is an infinitely large or small quantity means that its quantity cannot be measured.
Why do you think the proposition that the natural numbers is countable does not contradict the proposition that the natural numbers are infinite, in the way I explained?
Quoting Banno
Correction. There is no difference between a number and a numeral, the number is the symbol. What the symbol represents is an abstract value. It's a category mistake to say that a value is an "object" unless we define "object" in the sense of a goal.
Regardless of what you assert, to say that the value represented is an object called "a number" is platonism. Calling it an imaginary, or fictional object, doesn't fulfill the ontological criteria of "object". Therefore we'd have to treat it as an idea because treating it as an object would be a false premise. We cannot truthfully treat a fictitious object as an object, because it is an idea and the existence of ideas is categorically distinct from the existence of objects.
:heart:
As a slogan, that looks almost right.
To say that a parameter is infinite, means that it cannot be measured relative to a given basis of description. Hence the distinction between an ordinary task and a hypertask depends on how the task is described, and this distinction can be regarded as the logically correct solution to Zeno's Dichotomy paradox.
(The finitude of an object's exact position in position space, becomes infinite when described in momentum space, and vice versa. Zeno's paradox is dissolved by giving up the assumption that either position space or momentum space is primal)
Are you suggesting that is a reason for rejecting his conclusions? Either way, I would suggest that we leave Cantor's theology as a matter between Cantor and his God.
Quoting Metaphysician Undercover
It depends, as I explained earlier, how you define "countable". I don't say that it's just all just a matter of definitions, but it's probably a good idea to get those agreed so that we can be sure we are talking about the real issues. As it is, we don't agree and so we never get to identify and discuss the real issues.
Quoting Metaphysician Undercover
I'm not sure what you mean by "serves as a medium". I accept you are right to observe that the numbers are defined as a succession. (I don't know why you call the successor function a representation of something, but let that pass...) But the point of a succession is that every step (apart, perhaps, from 0) has a predecessor and a successor. That is what it means to say that n is between n-1 and n+1. It is not wrong to say that 2 unites 1 and 3 and it is not wrong to say that 2 divides 1 and 3. But it is wrong not to say both.
Quoting Metaphysician Undercover
This just turns on your definition of what it is to count something.
Using a ruler to measure a (limited) distance means counting the units. Obviously, we need enough numbers to count any distance we measure. So having an infinite number of numbers is not a bug, but a feature. It guarantees that we can measure (or count) anything we want to measure or count.
I maintain that if you can start to count some things, they are countable. You maintain that things are countable only if you can finish counting them., It's a rather trivial disagreement about definitions. But I do wonder how it is possible to start counting if I can only start if I can finish.
Quoting Metaphysician Undercover
I see what you are saying here. I was coming at this from a slightly different angle.
I take it that you are aware that there are several different axiomatizations of set theory. Some examples are: ZFC, ZF, Z, CZF, IZF and various Finitistic and even Ultrafinitistic systems.
The argument about measurement that you provided in your reply is interesting, and I can see how it is relevant to question of whether (or in what sense) a countably infinite set can be said to "exist". But the word "exists" can have different meanings depending on the context. Within the context of ZFC set theory, to say that a countably infinite set "exists" doesn't imply that it exists in some Platonic heaven. That's not to say that you couldn't interpret it in a Platonic way, just that nothing in ZFC itself forces this interpretation.
Now, I see that a few others on the thread have raised a similar point and that you have not been convinced. That's fair. I doubt that I will be able to convince you either, but I will try to explain how I see it and then you can let me know what you think.
The way I (and many others) interpret the word "exists" with respect to ZFC set theory is something like "there is a derivation from the axioms of ZFC using the inference rules of classical first-order logic, of the formula ?x P(x)". Or, more compactly, ZFC ? ?x P(x).
So to say that "a countably infinite set exists" is just to say "ZFC ? ?x CountablyInfinite(x)". The actual derivation follows very simply from the axiom of infinity in combination with the definition of "countably infinite".
In my view, accepting this does not mean that you have to believe that countably infinite sets "exist" in any other sense, whether that be in a Platonic heaven, in the mind of God, or as an actual collection of objects somewhere within the physical universe.
What are your thoughts on this?
i don't consider this to be a solution, because the result is the uncertainty principle. What you indicate is two distinct concepts of space which are incompatible, "position space", and "momentum space".
Quoting Ludwig V
We went through the common definition of "countable" provided by jgill, and the contradiction remained. So I really don't know what type of definition of "countable" you might be thinking of.
Quoting Ludwig V
"Medium is commonly defined as "something in a middle position". If something is between two things, it is distinct from each of the two as in the middle.
Quoting Ludwig V
Yes, every step has a successor, but the succession is described as a continuous process. No individual step can serve as a division between the prior step and the posterior, as each is continuity, not a division. To say that one step is a division would produce two distinct successions, one prior one posterior. then the one which served as the divisor would have no place in either of the two successions.
So I dont't understand what you are saying here, especially what you mean by "2 divides 1 and 3". One divided by two produces a half, and three divided two produces one and a half. But it doesn't make sense to say that two acts as a division between one and three in the way that you propose.
Quoting Ludwig V
So it looks like you disagree with my premise that counting is a form of measurement. Since you claim that starting to count something is sufficient to claim that it is countable, then if we maintain consistency for other forms of measurement, puling out the tape measure would be sufficient to claim that the item is measurable. Since this is obviously not true, it seems you are claiming that counting is not a form of measurement at all. How would you define "countable"?
I'm saying that in the presence of an inconsistency between ZFC and computable notions of mathematics, coupled with the obvious uselessness of of non-constructive cardinal analysis, the theological origin of ZFC becomes conspicuous.
Quoting Metaphysician Undercover
yes, there are two incompatible bases for describing dynamics, and in line with your proposal, what looks like a hypertask iwhen measured in one basis, is a mere task in the other (where measurement is understood as destructive interference).
I presently suspect that the structure of the uncertainty principle, that concerns non-commutative measurements, is a logical principle derivable from Zeno's arguments, without needing to appeal to Physics.
I'm sorry. I should have said "separates", not "divides".
Quoting Metaphysician Undercover
Can you think of a form of measurement that is not counting - apart from guessing or "judging"?
Quoting Metaphysician Undercover
I disagree. Since this is not an argument, it seems inappropriate to reply.
Quoting sime
OK. Obviously I'm not in a position to comment.
The statement “We can construct an injection P(N)?N via Turing machine encoding of decidable sets”
would mean every subset of N can be uniquely encoded by a natural number. But that is equivalent to saying ?P(N)???N?, which directly contradicts Cantor’s theorem. So if the statement were true, Cantor’s theorem would already be false.
There are undecidable subsets of N. We cannot construct an injection P(N)?N via Turing machine encoding of decidable sets
I'll stop there. I can't see that your account works.
There is a point at which one's interlocutor's commitments collapse the subject matter under discussion.
That's where we are at with Meta.
As Frank points out,
Quoting frank
And Meta's view undermines most of mathematics, despite what we do with it.
Meta treats the ? of quantification, a logical move within the game of maths that understands there is a symbol n in the domain of discourse that satisfies P according to the rules of the theory, as if it implies n exists as an abstract object independent of language, symbols, or human conventions. That's just a muddle. At the core he perhaps does not understand the difference between syntax, semantics and ontology.
Given that Meta asserts that 2 is not between 1 and 3, I think I'm done here. I don't see any gain in showing further absurdities in his position.
A nominalist would provide an argument for why we can use math without committing to abstract objects. I guess Meta is a math skeptic.
I'm glad you're back.
Quoting Esse Quam Videri
The issue of platonism is more about the existence of any bijection in general, and the question of whether a measurement exists without requiring someone to measure it. It's a form of naive realism, which in our conventions and educational habits, we tend to take for granted. We look at an object, a tree, a mountain, etc., and we assume that it has a corresponding measurement, without requiring that someone measures it. Then, when someone goes to measure it, the correct measurement is assumed to be the one which presumably corresponds with the supposed independent measurement. This type of realism requires platonism because there must be independent numbers and measuring standards which exist independently from any mind, in order that the object has a measurement before being measured.
But if we understand that numbering conceptions, and measuring conceptions are products of the human mind, then it's impossible that an independent object could have a measurement prior to the measurement being made by a human being. This rules out the possibility that the natural numbers could have a measurement, or be countable because we know that human beings could not count them all.
Quoting Esse Quam Videri
So we could say, that numbers "exist" in a way other than platonic realism, but we must consider what would be meant by this. We need to ask, what is the criteria for existence. Consider the difference between the following two statements. 1. "the set of natural numbers between 0 and 5". 2. "{1,2,3,4.}". We might say at first glance that they both say the very same thing, and they both necessitate the existence of those four integers, but this would not be correct. That is because the first is a formula, and the existence of the named integers requires that the formula be carried out correctly. So we need to respect this difference, the existence of the integers in the first example is conditional, or contingent, on "correctness", and in the existence is necessary. And when we say 1, 2, 3, ..., or use the successor function, the existence of those numbers is conditional, contingent on correctness. Now we have the problem mentioned above, the natural numbers cannot have a measurement, because the procedure cannot be carried out to the end, "correctly".
The inclination might be to deny that distinction between necessary existence and conditional existence, which I provided. But we cannot do this because we need to account for the reality of human error. A formula does not necessitate the existence of numbers because error may arise in a number of different ways. The formula might be carried out incorrectly, or it might be a mistaken formula in the first place. So we might add, "the designated numbers exist if the the formula is properly formulated, and if it is carried out correctly. But that makes it conditional.
Quoting Esse Quam Videri
So you are talking about a conditional existence. The supposed existence of the natural numbers, is dependent on the correct procedure. The issue with the definition of "countably infinite" is that the procedure cannot be carried out. The formula states something which is impossible to correctly finish, therefore the numbers cannot exist.
Furthermore, platonism doesn't solve the problem because the infinite is defined as being impossible, so the numbers cannot even exist in a platonic realm. That would be like saying that the full extension of pi exists in a platonic realm, when this has been demonstrated to be impossible.
Quoting sime
I agree. Many people conclude that calculus solved Zeno's paradoxes. I've argued elsewhere, that all calculus provided was a workaround, which was sufficient for a while, until the problem reappeared with the Fourier transform.
Quoting Ludwig V
I don't think this makes any difference.
Quoting Ludwig V
Sure, I believe measuring is fundamentally a form of ordering. So most comparisons which are intended to produce an order are instances of measuring. Get a bunch of people, compare their heights, and order them accordingly. That's a form of measuring.
Quoting Banno
You mention "what we do with math", but are neglecting something very important, "what we can't do with math". This is the limitations, like the uncertainty principle mention above. We do a lot with math, sure, but there is a lot more we would be able to do if we work out some of the bugs. Then there's the even worse problem of the many things that people believe we do with mathematics, which we really don't. Some people think that calculus has solved Zeno's paradoxes. It has not. Some people think that mathematics allows us to determine the velocity of an object at a single instant in time. It doesn't. Some people think that mathematics has provided a way to make infinite numbers countable. It has not. That's what I'm talking about. To have the attitude that math is perfect, ideal, therefore it is wrong to subject it to philosophical skepticism is the real problem.
Quoting frank
I like to apply a healthy dose of skepticism to any so-called knowledge. Nothing escapes the skeptic's doubt.
I suppose so, but the GPS in your phone was designed using math invented by Descartes. It's so weird that your GPS works even though math does not exist. :confused:
I'm not saying that the powerset of N is defined as only referring to the decidable sets (apologies if that is how it looked). Rather, I was overloading the notation of P(N) to refer exclusively to the decidable subsets of the natural numbers (i.e. to what is sometimes written Pdec(N))), in order to inspect what CSB implies in that special case, because the results are illuminating.
A decidable set A is a set whose members can be enumerated, and whose complement ~A can also be enumerated. Applying diagonalization to A, as per Cantor's theorem, must produce a novel decidable set, at least if we assume that an injection N --> Pdec(N) represents an effective procedure (a point that I ought to have stressed earlier). Thus to improve upon the above, diagonalization shows that:
if N --> Pdec(N) is a computable injection, then N --> Pdec(N) cannot be a computable surjection. Hence in this case, diagonalisation is a proof of the undecidability of the halting problem, rather than a proof of "more numbers".
Furthemore, Pdec(N) --> N exists as a computable injection. Hence according to CSB, N --> Pdec(N) must necessarily exist as a surjection, which is false if by surjection we mean a computable surjection.
The biggest failure of CSB in this context, is its insistence that if A --> B is a surjection, then a surjection B --> A must also exist. This is constructively false as discussed above, and the reason for why cardinal arithmetic is pointless, misleading, and false from a computational perspective.
Formally, it is might be said that CSB isn't refuted by the above, due to an "apples versus oranges" argument: Classical set theory makes no reference to decidability, meaning that N --> Pdec(N) is allowed to exist in ZFC 'rent free' as a non-computable surjection in the strictly syntactical sense of a quantiified predicate that cannot be converted into data.
In the constructive setting however, CSB is usually said to be false rather than inapplicable.
Personally, I am of the opinion that CSB along with infinite cardinal analysis, should be called "correct" in relation to the language of ZFC, but false when no background set theory is specified, due to the fact 1) that exclusively classical theorems aren't relatable to reality, and 2) they are often appealed to without anyone remembering that their correctness is relative to a classical axiomatization of set theory.
This seems to be the crux of the issue for you, and I can appreciate the tension that you are raising, but personally I don't see this as an issue. I see the logical proof of the bijection as adequate to accept its existence, but scoped only to within the "game" of ZFC set theory.I'll try to explain my reasoning as clearly as I can.
For many on this thread, to say "the bijection exists", is literally to say nothing more than:
(1) the bijection is formally derivable from the axioms of ZFC in combination with the inference rules of classical first-order logic.
That's it. So when we say "the bijection exists" we are saying something more like "the bijection exists within ZFC".
For many of us on the thread, (1) is straight-forwardly true. So when someone denies "the bijection exists", we hear it as a denial of (1), since that is all we mean, whereas I think the people making the denial are (perhaps?) not intending it in this way. Hence all of the confusion.
What are your thoughts on this?
It's not even an issue of does math, or does math not exist. We are talking about whether we have an accurate description of it. To assign a name "exists" without any principles for understand what that word means, does nothing toward the purpose of describing.
If set theory starts with the premise that numbers are countable objects, and this is false, then it is a theory based on a false premise. We can ignore the question, and say that a number is "an object" in a different sense of the word, from how we commonly use "object", but then we have to ask, in what sense is it "countable" then. Counting is how we measure ordinary objects, and now we are not dealing with ordinary objects. It's incoherent to say that a number is countable in the sense of an ordinary object, but the number is not an object in the sense of an ordinary object. That's mixing apples and oranges. So this type of object, the mathematical object, must be "countable" in a different way from the way that ordinary objects are.
And the matter of infinity confirms this. If we assume the real possibility of an infinity of ordinary objects, they would not be countable. But then we say that an infinity of the type of object, which numbers are supposed to be, is countable. This confirms the problem. We count ordinary things with a bijection. Numbers are not counted in the same way as ordinary things. How are they counted? Or is the act of counting just a pretense? It appears to me to be the latter. Numbers are not counted at all, there is a stipulated formula for determining cardinality.
The matter of how does math work if it does not exist, is a completely different issue. That math works, implies that it is the means to an end. As such it is a technique, a "way of acting". Even though a "way of acting" cannot be said to exist, it can reliably bring about the desired end. A way of acting is perhaps best described as a sort of conditional. If the conditions are X, then the response is Y. So it might be described as a prescriptive rule. It's a way of applying a general principle to the particular situation.
The issue of whether a general principle, a Form "exists" is difficult. When you ask a bunch of philosophers what type of existence a rule has, you get as many different answers as people. But now we go to a further extent, and question the relationship between the general rule and the particular, in application, as a way of acting. It's like asking about the existence of a habit. It's best to avoid "exists" altogether in this context. Plato exposed this difficulty with "the good", as something which does not fit into the realm of existence. So we just judge habits as good or bad, without regard to whether they have "existence".
Quoting Esse Quam Videri
That's fair, it appears like most people do not see it as an issue.
Quoting Esse Quam Videri
How I interpret this, is that you believe it exists by stipulation. if something is stipulated to exist, then it does. I have a problem with this, because it circumvents the judgement of truth, allowing you to employ premises (axioms) without the requirement of truth. Ultimately the conclusions are unsound.
But I've discussed this with others, and it appears like "pure mathematics" likes to provide itself with the ultimate freedom of being unfettered by empirical judgements. This is advantageous to the mathematician for a number of reasons, but significantly it allows judgement based on results, rather than prior constraints. Therefore "unsound" is not necessarily bad. Forging ahead with unsound principles often produces what is good.
So in the reply to frank above, I described mathematics as "a way of acting". The way of acting is judged relative to the end, the consequences, rather judging the premises. This is pragmaticism, if it produces good results then the way of acting is itself good. We don't need to address the truth or falsity of the premises (axioms), and if you analyze the numerous different ways of acting you'll find that often the basic rules being followed cannot even be identified. At the base level, we have habits, and when a person acts by habit one cannot say that there is a rule which is being followed. So the pragmatist perspective renders the exact nature of the rules, and principles which are followed in a procedure, as fundamentally unimportant, because success is what is desired and therefore the focus.
This means that my strategy of attacking the premises, axioms, is rather pointless, as you and others will say "personally I don't see this as an issue". Even if there are blatant contradictions, it wouldn't be an issue, because success is what is important, and that's what frank pointed to. Contradiction at the base level is unimportant if the system reliably produces success. This means that the true test, the real judgement requires a focus soley on "success". In ethics this is consequentialism. However, it requires that we clearly define the intended goal, the end, in order to determine whether there really is success or not.
As a philosopher, I look beyond all the worldly goods, GPS mentioned by frank, computers, internet, AI, all these things, to say that the ultimate goal is to understand the true nature of reality. If this is the case, then as sime pointed out, mathematics delivers us a problem known as the uncertainty principle. This is a roadblock which stands in the way of success, under that description. Since this problem has been revealed, and is demonstrably the result of the mathematics, it is incumbent on us philosophers to analyze the axioms and premise of the mathematics, to determine where we are misguided. Since success is lacking, we need to take issue with the rules.
I think we are on the same page now. I personally don't think that the axioms of ZFC are "true" in any metaphysical, transcendental or empirical sense. However, I accept that existence claims derived from those axioms are nonetheless valid within the formal system. This is formal/heuristic truth, rather than metaphysical or empirical truth.
Is this a form of pragmatism? Yes, I think it probably is. I am not adopting the axioms because they are "true" in any robust sense, but because they enable so much interesting, beautiful and indispensably useful mathematics.
That said, I don't deny that your critique may have real bite against those who would take the axioms to be true in a more robust sense.
It is possible, and has been the case at times in this discussion, that both sides of a debate are thinking in terms of "straightforwardly true". But there is a case for saying that, in this instance, "straightforward truth" just isn't available. To put it one way, there is truth in the orthodox account of infinity, and the "Aristotelian" and nominalist accounts. Much of this debate has circled round this, without producing much in the way of mutual understanding. Classic philosophy.
Wittgenstein's approach seems to me much more likely to be fruitful - which is not to say that everything that he argues for is equally convincing - ref; "infinity". The aim of dismantling philosophical theories by showing that some term or other had not been effectively defined is an excellent test. But nonetheless, the comparison of a philosophical view with an interpretation of a picture suggest that a more laid-back approach is more likely to enable us to understand the issue and its difficulties, at least. Then the option of admitting that the problem is insoluble or has more than one solution may be open to us or that the issue
Quoting frank
Indeed it was. And it is important to see and to notice that mathematical ideas apply to the physical universe, at least sometimes. Whether this is unreasonable or not, is not obvious to me. But reasonable or not, it is so. Pragmatic approaches often get short shrift around here and I don't equate "works" with "true". But where some idea or technique does work, that seems an important fact about it - just as whether it is verifiable or not is not the whole story, but is an important part of it.
Quoting Esse Quam Videri
Yes. That's why I prefer the classical approach here. But I don't rule out that perhaps there is something about the alternative approaches that has not yet come to light. There is a connection to theology, which might explain why those approaches survive, though I confess it would not recommend them to me.
OK, this might be difficult to understand as I don't have a clear way to say this, but I'll try to be as clear and as simple as possible:
Cantor's diagonalization, just as the example with Turing's machine, is basically about negative self reference and this negative self-reference and it's effects are the key issue. With Cantor's diagonal argument, one way to say is that it's about showing that there's a real number which isn't on the list of real numbers where every real number ought to be. With Turing, it's about another Turing Machine that does the opposite of the first Turing Machine, where the first Turing Machine should be capable of computing and giving an answer on everything. Both cases, the limitation is, de facto, negative self reference. Negative self refence simply means that it's not possible for me or anybody to do something, that I or they, don't do.
I myself have used many times the following example of how negative self reference, how easily this diagonalization works:
Forecast what number, 1 or 2, I will write in my next response and make the forecast before I respond (in a day, at least). I will follow exactly these lines: If you say I will write 1, I will write 2. If you say I will write 2, I will write 1. If you don't answer anything, just copy this or answer something else or disregard this, I will write 1.
In game theory, the answer is obvious:
_You say_/_I say_
__1__/__2__
__2__/__1__
__"something else"__/__1__
Is it easy for anyone else than you to forecast correctly what I will say. Yes, very likely it's going to be 1, because usually people won't even bother to participate in this simple forecasting game of mine. But you cannot write the correct forecast because the correct forecast will be what you don't write. Hence the negative self-reference. Is there a correct number to be forecasted? Definitely, yes, but it depends on your action.
Let's put this back into the context of what we have been talking about:
Is Cantor-Schröder-Bernstein theorem correct? Yes, it is.
Just as Gödel's completeness theorem goes perfectly with his incompleteness theorems, the real question is what changes in the more complex systems?
You argue it this way:
Quoting sime
OK, perhaps I don't get 100% of this, but I assume your on the correct track.
Yet does diagonalization really "produce a new member"? I don't think so. Diagonalization shows us the limits of computability or in the case of Cantor the futility of trying of treating an uncomputable set as a computable set. Remember that the diagonal proof is a Reductio ad absurdum proof. A list of all real numbers cannot simply done. That it cannot be done means that it's uncomputable. Yet is there the set of real numbers? Yes. And obviously there's a bijection from the set of real numbers to the set of real numbers.
I would argue that the limits of computability and provability just show where the line between objectivity and subjectivity go. The part of mathematics that is uncomputable tells actually a lot about subjectivity and being part of the universe, which creates problems for objectivity.
P(N) and Dec(N) are different sets. Pdec(N) is an odd notation; I presume you mean it as the decidable subsets of P(N). I'll use Dec(N) there, to avoid any ambiguity.
We can inject Dec(N) into P(N) but not P(N) into Dec(N).
That is, there are undecidable subsets of N.
This is true computationally as well as classically.
No, we don't. Encoding Turing machines only enumerates decidable subsets of N, not all of P(N).
The Cantor–Schröder–Bernstein theorem states:
If there exists an injection f:A?B and an injection g:B?A, then there exists a bijection h:A?B.
That's all.
CSB does not claim:
We're still not on the same page, because I claim contradiction within the system, therefore I cannot conclude that the existence claims are valid. That would require that existing things could violate the law of noncontradiction. I say that there is no coherent way to make whatever is said to be infinite also countable, regardless of how you describe "existence" (except by violation of noncontradiction).
The problem is that "infinite" puts severe restrictions on what type of "existence" the supposed infinite thing can have. Some definitions of "existence" might not even allow for anything infinite. Therefore, if "exist" is to be a relevant term it needs to be defined in a way which would allow for an existing infinity. The specified "infinite" parameter must be limitless, boundless. This makes that parameter impossible to measure. Therefore if the infinite parameter is quantity, it is impossible to count.
So it's not a question of whether or not the said bijection "exists", it's a matter of whether it is possible for it to exist. I argue that the meaning of the terms, "infinite" and "bijection" render it impossible for that specified bijection to exist, no matter how you define "exist", unless you allow contradiction.
Quoting Ludwig V
The connection to theology which I get a glimpse of, is the ontological argument, which became unacceptable even in theology. The principal problem which I see with that argument is that it attempted to but a boundary, a limit to the limitless. The "infinite" God was described as "That than which nothing greater can be conceived". Notice how it is a deceptive way of describing what is supposed to be limitless, infinite, (God), as an actual limit, the limit to thought. (The greatest possible conception is a limit to thought.) So it's a sort of self-contradicting premise, because we can conceive of the limitless, "infinity", but then the premise presents this as a limit to thought, 'the greatest possible' conception.
So set theory clearly demonstrates how the ontological argument is faulty. If we represent the limitless, i.e. "infinite" as a limit to our thinking (as the ontological argument does), then our thinking will just think up a way to surpass that boundary. That is because the thinking mind will not allow itself to be limited by such self-deception. So when "infinite" is proposed as a limit, as is the case in calculus, then the mind which refuses such a constraint, will just say 'that's a countable infinity', and surpass it with a new, 'uncountable infinity'. Therefore, when the limitless is proposed, and accepted, as a limit to the thinking mind, the true reality of the thinking mind is that it will not allow itself to be limited in that way, so it simply surpasses that limit, and creates a new limitlessness for itself.
How I see this as a problem is that this method is complex and convoluted, so it needs to be simplified. And, the complexity obscures inherent contradiction, which I've argued for. This is because the way of set theory doesn't actually address the problem, it builds on top, working to cover up. The true problem is with the principles of calculus, where the limitless (infinite) is proposed as a limit. That's where the base of the contradiction lies. Then, instead of addressing this as fundamentally flawed, the mathematicians simply forge ahead, thinking that if the limitless has become a limit to us, we're just going to create a new limitlessness. That's the way the human mind works, if you constrain its freedom it will find itself a way around that constraint.
I understand. I do not see a formal contradiction. It sounds like you do. I think this is where we must part ways.
I wouldn't call the contradiction "formal", because it's not explicit as a statement of X and not-X. However, i see it as implicit, and unavoidable from the meaning given to those terms within the system. "Countable" implies capable of being put into a bijection, and "infinite" implies not capable of producing a bijection. The closest we can get to resolving this is with platonism, by saying that the bijection simply exists. But then "countable" would need to be replaced with "counted", and that implies that the limitless (infinite) is not limitless. So in my mind, the contradiction is unavoidable.
I see what you are saying, but I would gently push back here. In my last post I distinguished between formal, empirical and metaphysical truth. I see the claim "there exists a bijection..." as being straight-forwardly true in a formal sense—given the axioms and the inference rules, it follows as a matter of course. As far as I am able to tell, this is precisely what is being denied by some others on the thread.
If we were talking at the level of metaphysical or empirical truth I would agree with you. But at the level of formal truth, either ZFC ? ?f (bijection) is true, or it isn't. I have a hard time making sense of the claim that it isn't.
I realize that you see the contradiction as implicit and unavoidable. But you are not recognizing the meaning given to the terms within the system.
"countable" within the system means only that some of them can be counted and we cannot find any numbers in the sequence that cannot be counted. Actually, since we had that discussion, I've come across the term "countably infinite" which I think is much less misleading.
And I think that you are not aware of how the term "limit" is used within the system. A limit, in this context, is a value that the series gets closer to, but never reaches. It is not a value derived from the function. It is not the last term in the series.
It does not constrain the series at all. So, in Zeno's paradox, Achilles gets closer and closer to the tortoise but never reaches it. (Forgive my inexpert account.)
From my perspective, the adjusted meaning of terms within the system is one of the biggest differences between us.
Quoting Metaphysician Undercover
Thanks for that. I hadn't thought about it. But that wasn't what got Cantor into trouble.
I think that a dissection of the ontological argument here might be thought off-topic. So I won't go any further into territory that I don't understand anyway.
Quoting Esse Quam Videri
Yes. I was thinking in terms of a truth that could be recognized unconditionally, as it were. But then, "either ZFC ? ?f (bijection) is true, or it isn't" is just that. So I missed that truth. The difference is, I suppose, is that I doubt if one could defend similar claims beginning "In metaphysics..." or "Empirically..."
The definition of "number" (in bold, in the quoted statement), is relevant.
The number of natural numbers is not a natural number. A natural number is a number that can be counted to. The same thing applies to every instance in your post where you refer to an "infinite number".
The only meaningful measure of infinities are transfinite numbers, and it's based on a mapping relation. The "number" of natural numbers (positive integers) equals the number of integers because a 1:1 mapping can be identified between the sets. Similarly, the (transfinite) number of real numbers between 1 and 2 is the same (transfinite) number of real numbers.
Transfinite numbers should not be confused with quantities, in the everyday sense. When one mistakenly treats them that way, it leads to paradox.
I think it is important to underline that the mapping between the sets is identified between the first few steps in the series (I may have the exact terms wrong). But the written identification runs out at some point and is conventionally followed by .... The conclusion that it applies right through the sequence follows from the fact that there is nothing to stop it continuing on the principle that every step in the sequence is structurally identical. (You would need to give a reason why things should be different at some point.)
I've found that this is not obvious to everyone.
So we picture {2, 4, 6...} and carry on in this way.
But we know that there are innumerable ways to continue this sequence.
The mapping is not based on "carry on in this way" but on the function
[math]f: \mathbb{N} \to \mathbb{E}, \quad f(n) = 2n[/math]
And this is what shows the bijection. At every n?N, f assigns exactly one even number, and every even number is assigned to exactly one n.
We know exactly how to carry on.
I don't agree with that interpretation, because along with "countably infinite" there is "uncountably infinite". Furthermore, Banno produced the proof that the set of natural numbers can be in a bijection with itself. And there can be a bijection between the naturals and the even numbers, and so on, so that the cardinality of these types of sets is the same. To assign cardinality is not a case of "some of them can be counted".
Quoting Ludwig V
That is exactly the problem. It is treated as, and called "the limit" but it is not the limit. Since it is not the true limit, you don't actually ever get closer to it. There is an infinite number of places between here and the designated "limit", and no matter how many places you proceed through, there is still always an infinite number of places. Therefore you are never actually getting any closer to the supposed limit through that means. It is a faulty means. It's supposed to be the limit, and it's supposed that the series gets closer to the limit, but it does not.
That is why this representation of "limit" is faulty. It is a false representation, because there is always a gap of infinity between where you are and the supposed limit. Therefore it's not the true limit, you're never closer to it, or further from it, by that method. So you don't actually keep getting closer, and it's nothing but an arbitrary, and false designation of "limit".
To say that you actually keep getting closer would be like saying as you count the natural numbers you keep getting closer to the end. Or, as you work out pi to more and more decimal places you keep getting closer to the end. That's false representation. You do not. Likewise, to say that the series gets closer to, but never reaches the limit is equally a false representation, because there is always an infinite gap between the series and the limit. You are not getting any closer. This shows it to be a faulty representation, and a faulty method. Zeno's paradox is unresolved.
.
No one ever says either of those things. You're arguing with someone in your head who knows no more about mathematics than you do.
*
Zeno's paradox comes down to this: the rational numbers in their natural order do not form a sequence, unlike the natural numbers.
As it happens, they can made to form a sequence; and as it happens, the real numbers cannot. But I don't think either of those things really matter.
Zeno quite reasonably approaches the problem of moving by attempting to break it into a sequence of "steps" as we call them, for obvious reasons. (A very powerful technique that underlies much of what we do.) But that sequence of actions cannot be mapped onto the rationals in their natural order because that's not a sequence.
Zeno's paradox is a convergent series, dude. It doesn't matter what order you sum it in.
Thanks, Banno. I knew I would not get it quite right.
My apologies for my curtness. I'v'e in mind heading off a divergence into discussions of rules.
Yep.
[math]\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \cdots[/math]
is
[math]\sum_{n=1}^{\infty} \frac{1}{2^n}
[/math]
that is, 1.
there are an infinite number of steps in this description of the distance between 0 and 1, but that simply does not stop it being traversed in a finite time.
Zeno mistook an infinite description of motion for an infinite obstacle to motion.
Zeno saw himself as proving that all motion is an illusion. You're saying that he's wrong, but you aren't providing an argument. That's fine.
Of course. The ghost at the feast, perhaps.
Quoting frank
Well, one could simply argue that the argument is not a proof, but a reductio of a certain approach to space, time and infinity.
We can compute when Achilles will achieve his goal as soon as we know how fast he is running and how large the distance is. That figure does not change as the race progresses. Unless Zeno can find a fault in that calculation, it proves that the issue is in the approach to the question, not in the situation as described.
In a fixed period of time, Achilles passes an increasing number of distances, culminating, no doubt, in his traversing infinitely many distances in an infinitely small amount of time. Zeno seems to think that he takes a non-infinitesimal amount of time to traverse an infinitesimally small distance.
E.g, starting from 0, what is the next rational number to count? since this doesn't have an answer (when counting in order), this means that the topology of the rational numbers cannot represent dynamics. Sure, the rationals can represent displacement, including an infinite sum of displacements, but not the process of displacement, namely motion.
To represent motion in a way that avoids the paradox, requires a smooth and differentiable continuous topology that doesn't contain points that are in need of traversal, but only open sets that can finitely intersect to create spots, but not infinitely often so as to create points. Yet on the other hand, to represent positions requires a discrete point-based topology of infinitely thin spikes that doesn't blur position information. Hence motion and position require incompatible topologies.
Fourier Transforms are the best way to visually understand the solution to zeno's paradox, but on the proviso that the FT is understood as creating an output topology (e.g motion) from a different input topology (e.g position).
see .
No need to overcomplicate things.
That's not an argument either. Some people are just emotionally averse to paying attention to things like Zeno's paradox. You can lead a horse to water, but you can't make him drink, that sort of thing.
What does math have to do with the structure of space and time? Read the SEP article on Zeno's paradox.
:grin:
I just did. Did you? Which paradox would you like explained?
Maybe tomorrow.
What I need is for you to explain why you think calculus tells us something about space and time. It's in the article.
Huh? Someone in my head knows more about mathematics than I do? Isn't that contradictory?
Quoting Banno
Obviously then, the description is wrong.
Quoting Ludwig V
The approach to the situation, is logically prior to the description. What appears to be the case, is that the description, and consequently the approach, are both wrong.
Quoting sime
Or, we can represent motion as discontinuous, which is the way that quantum physics seems to demonstrate is the real way. The particle has a position, then it has another position, without traversing the intermediary. I believe, that what happens in between cannot completely be represented as "a smooth and differentiable continuous topology". Issues with the wavefunction demonstrate that this is not quite right. So what happens in between ought to be represented as truly unknown, though it is actually represented in a not very accurate way, as a continuous topology of superpositions.
Quoting Banno
Reality is complex, this is philosophy, and the common mistake is to oversimplify. Sometimes Ockham's blade just doesn't cut it.
Yes, there is no traversing anything unless a particle is in a smooth motion-state as a result of applying a motion operator to it, which cannot be the case if the particle is in a spiky position-state as a resulting of applying a position operator to it. The question is, at what level of explanation should this incompatibility be situated? at the physical level, as physics usually assumes, or at the level of the rules of mathematics?
I think we should consider the fact that Newton and Leibniz didn't invent calculus for the purpose of solving Zeno's paradox, but for describing trajectories under gravity. Hence the mathematical definition of differentiation that we inherited from them and use today, isn't defined as a resource-transforming operation that takes a mutable function and mutates it into its derivative; rather our classical differentiation is merely defined as a mapping between two stateless and immutable functions.
But if Zeno's paradox is to be exorcised from calculus, such that calculus has a dynamical model, then I can't see an alternative than to treat abstract functions like pieces of plasticine, that can be sliced into bits or rolled into a smooth curve, but not at the same time.
Or it was a critique of Plato and other mainstream philosopher's idea of the potential infinite.
We should remember that we unfortunately have lost Plato's original book, where likely the Eleatic school would have made their own viewpoint. Now we have just the texts of those who were against the Eleatic school, the "mainstream" Socratic-Platonic school.
Boy, would that book be nice to resurface. It's said that Zeno had even more paradoxes. Loved to have known what they were.
No, but the issue in the core of Zeno's paradoxes. And we should note that calculus had problems with the infinitesimals, like the famous critique from bishop Berkeley.
And basically logism and the set theoretic approach hoped to find some rigorous ground for calculus, but the paradox resisted to die with Russell's paradox. And with Cantor's hierarchial system, there's still questions...
I really think that there's more to it than we know now. Math is just so beautiful and so awesome.
I'm not claiming calculus tells us what space and time are; I'm denying that this is a coherent question.
If we want calculus to solve Zeno's paradox, we have to assume that the math is telling us something about space and time.
It doesn't solve, it dissolves.
The paradoxes only appear to work because they slide between the mathematical and ontological games.
Pick one, and set it out, and we can see how this happens in the detail.
One can argue that calculus doesn't solve Zeno's paradoxes as we don't have yet a clear understanding of infinity.
The point is that the paradox isn't fundamentally a math problem. It's a series of questions that point to a contradiction.
I guess you could put it that way.
See how explicit the admixture of two differing language games is here?
Quoting ssu
What we have are ways of talking, language games, a grammar, or a paradigm - whatever you want to call it. Infinity is a mathematical notion that we can use to calculate physical results. It is not an ontology.
I suppose so, yes.
(And all of this makes sense only if we agree that there is a whole number between one and three.)
Zeno wasn't arguing that we can't plot satellite orbits with acceptable precision.
I do admire your devotion to the practical. Detaching yourself from it and purely following the contours of the mind will set you out in front of contradictions.
Well, he was, from what we know, arguing that motion was not real.
Paradoxes occur when we say things incorrectly. The world cannot be wrong, but what we say about it can be.
Yes, but we can get along just fine in an illusion. Contradictions are just little sign posts that things aren't exactly as we're imagining them. They can't be.
Quoting Banno
Thank you, John Locke. Your faith is commendable.
Have you some alternative? :wink:
No, it's faith.
Quoting Banno
Everybody grows the psychological structures they need to deal with the life they have. I can't tell you how you need to think in order to successfully be you. If deep suspicion about mental stuff, coupled with strong faith in the world is the outlook your psyche thrives with, then God bless it.
But this is about method rather than belief. What is suggested is that if there is an inconsistency we reconsider what it is we are saying about how things are, rather than deciding that the world must be inconsistent.
There presumably is a point at which the world is so confusing that our reconsidering of what we say is insufficient to explain what is going on. But I hope we're not there yet.
The risks are that we are hiding behind grammar, using it as a shield against metaphysical inquiry rather than engaging it, or that we miss phenomena that actually resist conceptual capture. I acknowledge that.
That would indeed be of great interest. I wonder if we could construct a reply that they might have made?
Quoting frank
It may well do so. It may also set you in front of outright fantasies that have no connection with any kind of truth. The theoretical stance needs a grounding in ordinary life, if only because there is no escaping ordinary life. Not even philosophers can really escape from it.
Quoting Banno
There used to be a story that aerodynamics showed that bumble bees cannot fly. Did anyone doubt that bumble bees can fly? I don't think so. I understand that aerodynamics is now clear that bumble bees can fly. But in that case, it was clear how the world is, as opposed to how we thought about it, or described it. Why is it that we don't just point out that the arrow will leave the bow, and that Achilles will catch up with the tortoise? It seems that we cannot simply correct infinity, but have to learn to live with it. Calculus fits in to that project.
I'm not convinced that all paradoxes can be resolved. Some of them, like Zeno's, may be inherent in the project of saying things about the world. Self-reference is another part of our language that we struggle to escape from. Are we sure that we cannot just live with at least some of them?
Quoting frank
Each to their own, I suppose. But is that how you think about your own views, as well? If that's what's going on, why do we bother arguing with each other?
It is not uncommon for people to believe that the ordinary world is not really real, but is some kind of dream or fantasy or shadow. I doubt they would welcome a pat on the head and permission to believe whatever they need to believe.
But you are right that 's response is just his way of excusing his own views from critique... :wink:
Actually I just assumed my views would bore you.
And we do use it. It is, well, essential.
What we don't have is a proof. Or how it fits everything else.
What we have is threads like this constantly coming up. That itself tells something.
Now, am I crazy to argue that there might be something more to be said here? Perhaps I'm annoying in repeating myself, but I think this is a topic worth wile to talk about.
And ontology? Well, what is the relationship of infinity with numbers?
If we define numbers being arithmetic values that represent a certain quantity over all other quantities, what if we skip away the "arithmetic" part? What if we say that infinity represents a certain unique quantity over all quantities also?
Let me try to be a bit clearer. I cited the bumble bee just because it was a case where there isn't much, if any, doubt about how the world is as opposed to how we think about it. I wanted to contrast that with the issues about infinity. There are two ways of approaching Achilles & co. One is Zeno's way, the other is simple arithmetic, which one might think is how the world is. But that's not how we respond. I'm not sure I understand why, exactly, except that both are methods of calculation, so both come from the same stable. (Contrast the bumble bee). Possibly, we could choose to stick with simple arithmetic in the Zeno case. So perhaps the reason is that we need it for other calculations, such as the orbits of planets and other issues in geometry. In which case we need both. In other words, this choice cannot really be posed as between how the world is and how we talk about it.
This is an interesting remark. Many would say that "space" is conceptual only. And if it is, how could it be anything other than the way we represent it, as infinitely divisible?
"Space" might be the distance between two objects, but space is not what is measured, distance is. Furthermore, we commonly assume a medium between two objects, air or something. And space is not the air. Clearly, if we were talking about air, we wouldn't represent it as infinitely divisible.
So this is why there is a problem, when we get down to the basics, the medium between the nucleus of the atom and the electrons of the atom for an example, we really don't know what the medium is. The proposal of aether has been dismissed, so we just produce an artificial (imaginary) medium, some fields or something like that. Since the concept of "space", and its accompanying mathematics provide for infinite divisibility, and the proposed medium is simply conceptual, how could the medium be modeled in any way other than a way which is consistent with the concept "space", and the related mathematics, i.e. as infinitely divisible. Without proposing a real medium with real restrictions to divisibility, to propose that the fundamental medium between things is not infinitely divisible, according to how it is conceptualized as "space", is somewhat incoherent.
Space is an aspect of gravity. Mass tells space how to curve, space tells mass how to move.
When people say, “there are infinitely many fractions between 1 and 2,” they mean you can always find another fraction in between, no matter how close two numbers are. For instance, between 1 and 2 you can pick 1.5, and between 1 and 1.5 you can pick 1.25, and you can keep doing that forever. It never ends.
But that doesn’t automatically mean there are “different levels” of infinity stacked on top of each other. It’s the same idea repeating you can keep splitting the space and still find more numbers.
If you want a real case where infinities truly differ in size, you have to switch from “fractions” to “all decimal numbers,” like numbers with endless digits. There are way more of those than there are whole numbers or fractions. That’s where mathematicians say one infinity is bigger than another.
And if I put on a “Wittgenstein hat” for a second, he migft say: don’t let the word “infinite” hypnotize you. Most of the time it just means “this process can continue without end,” not “we’ve discovered a weird tower of endless infinities.”
I think it must be both. The theoretical mathematicians who practise what they like to call pure mathematics want to be free from the constraints of the physical world. So, they may produce axioms independently of the requirements of physics, and other sciences. However, the axioms which get accepted and become conventionalized are the ones which are applicable. Then by the time the use of any axioms become standard practise, they have been selected for, by the needs of the scientists.
Therefore we can separate the two in principle only. We put the hypotheses of science in one category and the hypotheses of mathematics in another category. But if we maintain the supposed separation into a description of actual practise, science must have logical priority. Ultimately then, conventional and standard mathematics has been shaped to meet the descriptions of scientists. So the incompatibility is within the descriptions provided by physicists. We must maintain a different, real separation though, and that is between descriptions and the real world. It is not necessarily a feature of the real world, which causes incompatible mathematics to be accepted, but perhaps a mistaken description.
So the conventional mathematics is shaped by demand, and the demand is the sciences. The incompatibility manifests in the mathematics which has been conformed to the descriptions of the sciences. Therefore the descriptions provided by the sciences must be faulty, they require incompatible mathematical principles. Like @Banno says, we cannot conclude that the real world is faulty. So what Zeno demonstrated is that our descriptions of motion are faulty, and the mathematics as applied to these descriptions, reveals this by leading to paradox.
Quoting sime
What you say here about Newton and Leibniz demonstrates how modern mathematics was fundamentally subservient to physics. Since then, the study of pure mathematics, and number theory have become more distinct and separate from that foundation. This actually provides an advantage toward solving these issues, because it allows us to look directly at the incompatibilities within the mathematics, without being influenced by empirical prejudice. Plato's principle "the senses deceive us" is very important.
I believe this is how the heliocentric model of the solar system was figured out. When we remove the mathematics from the influence of our observations of the empirical world, which the mathematics is formed around, then we can extend it in all directions to see where the infinites appear. Each appearance of infinity represents a problem within the empirical description. (In the case of the solar system eternal (infinite) circular motion was the fundamental problem demonstrated by Aristotle.) Then we can make a map of just the problems themselves, and attempt to correlate them and determine a unified underlying cause. A huge number of problems can actually have one simple cause.
Quoting sime
I think the key issue is that we improperly represent time. Modeling time as the fourth dimension of space implies that time emerges from space. So at the Big Bang, there is something spatial, and time emerges. But that's fundamental incoherent because all activity, including "emergence" requires time. So this representation implies that time existed before time. To rectify this we need to model time as the zeroth dimension, and allow that space emerges from time. This requirement is also indicated through an analysis of the pure mathematics, removed from empirical prejudice. The non-spatial, non-dimensional, "point", is very real and necessary to mathematics. Therefore it must be accommodated for in our modeling of activity in the empirical world. Currently, empirical descriptions do not allow for the reality of non spatial activity (time passage without spatial change). This is a very big problem, which makes the modeling of activity at a non-spatial point completely speculative, and somewhat incoherent.
That is part of what @sime was talking about, the incompatibility between representing objects as having position relative to each other, and as being in motion relative to each other. So when we attempt to unite the concepts of space and time, we must alter them. "Space" gets altered by being "curved", so that standard Euclidian geometry doesn't serve. And "time" gets altered by the relativity of simultaneity, so that there is no objective present.
Never!
(If the ever did, I'd just not respond.)
A proof of what, and to what ends? We know it's consistent and we do have rigorous axiomatisations...
I'm not following this, since Quoting ssu
seems to be saying that it does fit in with everything else...
What would count as success here? hat doubt would be removed, what practice would change if the "proof" were given?
Quoting Ludwig V
Limits, as against calculating velocities? Let's be clear, these two descriptions are quite consistent with each other. If you are pointing out that Zeno's description is incomplete because he doesn't include the bit where Achilles passes the tortoise, I think we agree.
Yep; and bringing Wittgenstein in explicitly is interesting. There's a fine line between quantifying over infinities and hypostatising them.
It just assumes first order logic and extensional equivalence over a domain of the reals. We could go into the difference between a limit and a least upper bound, between ? and ?, and how infinity never appears in the axiomatisation - except as shorthand for a process.
Might be novel to consider this stuff in detail. I have only a hazy memory of it from first year pure maths.
Sounds interesting. Let's do.
:wink:
Stealing from ChatGPT,
[*]B. Order axioms
There is a relation < on ? satisfying:
[*]C. Completeness (least upper bound property)
Every non-empty subset of ? that is bounded above has a least upper bound (supremum) in ?. Formally:
This is what allows limits, convergent series, and calculus to exist without introducing actual ?.
[/list]
This last says, informally, that every non-empty subset of R that is less than some number has a smallest number that is bigger than it... and is what distinguishes the reals from the rationals. SO it's perhaps where our attention might dwel.
:up:
Axioms of Real Number System: an explanation of field structure, order and completeness.
An Introduction to Real Analysis: a University of California text, as a PDF. We're looking at getting to 6.1 and perhaps 8.1.
I suppose the thought here is to show that the limit is not so much made up or defined, but sitting there waiting to be found within ?. We construct ? then find these interesting results.
It seems to me that the question of a medium in space is secondary. The first move is to set up a co-ordinates and rules for plotting the position of objects on those. (In other words, the concept is defined by the practice.) Once we have co-ordinate and objects, the question of a medium makes some sense. How non-mathematicians develop the concept is another question. But we can be pretty sure it is by interacting with the ordinary world. Mathematics, in my book, is a development of that.
Quoting Banno
I never intended to suggest that they were in some way inconsistent. On the contrary, the point is that they are both in order. So the question is, why do we prefer to use one rather than the other. Your suggestion is plausible - narrow focus in an analysis can be very helpful, but also very misldeading. The paradox of Zeno's paradox, for me, is that Achilles is precluded from reaching a point that defines the system - the limit. The first step is to divided the distance from the start to the goal, limit, by 2, and so on. The limit is not an optional add-on, (as it seems to be in the case the natural numbers).
Quoting Sam26
I'm sure he would. But it is not so easy to rest content with "this process can continue without end". On one hand, we think that the result of the function for each value is "always already" true. On the other hand, we feel that the result is not available until the function has been applied to each value. What makes this game even more puzzling, is that it seems we can know things about the whole sequence without working out the results of the whole sequence. The first example of this is that we can know that the process can continue without end.
Quoting Banno
We are not comfortable with the fact that rules have consequences when they are surprising or not what we want.
Ya, I agree it’s hard to rest content with “the process can continue without end,” i.e. we feel a real pull in two directions. On the one hand, once the rule is fixed, we want to say the value at each input is “already settled.” On the other hand, we want to say the value is not actually there for us until we run the rule at that input. This is exactly where the philosophical itch lives.
Wittgenstein, as I read him, is to separate “already settled” from “already computed.” The rule determines what counts as the correct next step, and in this sense the sequence is fixed, but it doesn't follow that the whole infinite list exists as a finished object waiting to be inspected. The “always already” feeling comes from the grammar of the rule, not from possession of an infinite completed totality.
The point about knowing things about the whole sequence without grinding through each case is just more of the same. We can know global facts because they are proved from the rule, for example, “this can go on without end” isn't discovered by checking every term, it’s a consequence of how the procedure is defined. The puzzle is real, but the solution isn't to posit a hidden, completed infinity in the background. It’s noticing what proofs actually license us to say about a rule-governed practice.
A rule can fix the standards for correctness without implying that the entire infinite list exists as a finished thing. We often feel “it’s already there” because the rule is firm, but what’s “already there” is the method, not a completed infinite inventory.
Well, I can't say I understand exactly what you are proposing, but it seems like you are saying the question of the medium is secondary, but then you explain why it must be primary.
The nature of the medium, in relation to the nature of the substance which is moving, determines the possible positions. So without determining the medium and the substance first, one could set up a co-ordinate system with infinite possible positions, but it would be false if the medium doesn't allow for it. That is also the case with divisibility. The mathematical system could allow infinite divisibility, but in reality divisibility must be determined according to the substance to be divided, and the means of division. So we might start with the co-ordinates and rules for plotting, as you say, but then it would just be trial and error, in application.
So you start out by saying that mathematics ought to be prior, "The first move is to set up a co-ordinates and rules", but then you end with the statement that mathematics is a development from our interacting with the world, which would place it as posterior.
Quoting Ludwig V
The problem in this paradox of Zeno's, is the issue which is explained above, as starting with the designation of rules and limits, instead of determining the true limits of the medium and substance first. The rules allow for infinite divisibility, but this does not correspond with the true medium.
Here's a way of looking at it. Suppose the measurement is on the ground, a long tape measure on the ground. Each time Achilles takes a step, the foot is at a new position on the tape measure. And, the section of the tape measure between there and the last step, is never traversed by Achilles. he steps from one position to the next, with a gap in between. So the false premise which Zeno makes is that all the area has to be covered. It doesn't Achilles steps from one spot to the next. Achilles could give the tortoise a short head start, then take one step and be past the tortoise, without ever properly catching up. This is why the nature of the movement and the medium is so important.
Quoting Sam26
What bothers me is that we seem driven to talk about processes in connection with infinity, as you do in the first sentence. But does such a concept make sense in the context of mathematics? Or does it mean that constructivism must be true, at least in the context of infinity?
No, it is simpler than that. We are using "medium" is different ways. I think. For me, empty space is not a mediium. A medium is substance that fills a space. Space is a co-ordinate system, which defines the possibilities where certain kinds of object may be. Objects are distinct from mediums because the latter are found everywhere, but objects have a locating within space.
In math, process doesn’t have to mean a thing happening in time. It may just mean a rule, a precise recipe that tells you how to get the next step, or how to compute the nth term. Infinity shows up because the rule has no final step.
That doesn't itself prove constructivism. Math is just comfortable saying this exists even when you don’t have a method to build it. Constructivism demands the method, at least that's my take.
Wittgenstein’s point is to be careful not to treat the infinite as a finished object sitting out there. What we really have is a rule and the proofs we proceed with. That leans constructive in spirit, but it isn’t a knockdown argument that constructivism must be true.
Of course it's not empty space, or else it wouldn't qualify as a medium. That's the point I was making. There is no such thing as empty space between objects. So to make a co-ordinate system which shows the positions which an object could have requires knowing the type of object and the type of medium.
Quoting Ludwig V
So "space" here is completely conceptual. And the point I was making is that it needs to be conceptualized according to the objects which are to be mapped and the medium between the objects. If we make a co-ordinate system which allows any objects to be anywhere (infinite possibility) that produces Zeno paradoxes. It's the faulty conception of space which allows for infinite possibility that creates Zeno type paradoxes.
How could "the next step" not imply "a thing happening in time"?
Good point. Does a typical mathematical sequence imply motion in time?
I made it far enough in mathematics, before getting too ornery, to know that you have to do multiplication and division before you do subtraction and addition.
Because next can mean two different things.
1) Next in the definition (logical next).
In mathematics, next often just means “the item with the next label in the sequence.” It’s part of how the rule is set up, so if you tell me where you are, the rule tells you what counts as the next one. That doesn’t require anything to be happening in time.
2) Next in our activity (temporal next).
When you or I actually work it out on paper, then there really is a next moment: first this line, then the next line, etc.
So, the word next doesn’t automatically imply time. It can be about the structure of the sequence, or it can be about our act of calculating it.
Because it doesn't mean that.
"Next" here implies a relation, and mathematics is the study of the relations between its "objects," which it is happy to treat as effectively undefined. That's why "What is a number really, and do numbers actually exist?" is not a question mathematicians are much interested in, though non-mathematicians of all sorts, even philosophers, are.
Quoting jgill
You know it doesn't, unless you mean something pretty subtle by "imply".
I wouldn't argue Wittgenstein's point, though doesn't that point us firmly in the direction of the Aristotelian distinction between actual and potential infinity? Which itself leans heavily on our actions in relation to infinity. The second sentence is true if we are talking about our activity in relation to mathematical formulae.
Quoting Sam26
Fair comment. I used to think that constructivism was the way to go. No longer. Now, I'm seriously bewildered and working things out. I have noticed how time and process show up so often in talk about infinity and am wondering how deeply rooted it is.
Perhaps it is a metaphor. Perhaps it is an application of terms in a new, stretched, language game. Notice, though that your talk of the infinite as unfinished implies a process.
Quoting Sam26
Aren't you leaning here on an idea of what exists and/or is real? Isn't it that idea that leads us into difficulties about the status of the sequence. In one way, you are right. In another, you seem to be saying that there are natural numbers that don't exist or aren't real (non-mathematical sense of real). Aristotelian talk of potential numbers tries to find a half-way house, though I think it is a most unhelpful concept.
Quoting Srap Tasmaner
Are you happy to defend an interpretation which regard S(n)=n+1 as a remark about the relations between numbers? It must be that, unless you are thinking of the number line, which is a spatial metaphor. But if is just a remark about the relations between numbers, it seems more like a generalization that a rule.
Quoting Metaphysician Undercover
Empirically, that may be true - especially if you regard a field (gravity, magnetism) as a medium. But setting up a set of co-ordinates does not require a medium in addition, so far as I can see.
On your Aristotelian comment, Wittgenstein might ask what “actual” and “potential” are doing in our language, and whether they clarify the use of symbols or just swap one picture for another.
And on existence, I am not denying that numbers exist. I’m blocking a slide in what “exist” means here. In mathematics, “exists” is governed by proof and use, not by the idea of a completed infinite inventory sitting somewhere. So, the rule can be firm without that extra picture.
The philosophical problem isn’t infinity; it’s the pictures our words seem to imply when we remove them from the practice that gives them sense. When we keep the use fixed, the mystery largely disappears.
OK.
Quoting Sam26
As I said, I don't think the Aristotelian account clarifies anything much. If anything, it deepens the mystery.
Quoting Sam26
Perish the thought of denying that numbers exist!
Quoting Sam26
Yes, but here, we need to deal with the adaptation of terms that already have a use in some contexts, but need adaptation for this specific context.
What are you blocking it with? Sentiment?
The "logical next" is next in time in this context. The only other option is "beside" in space, and this is clearly not the case. "The item with the next label in the sequence" is the one which comes after the other. Therefore the sequence is temporal. Without the separation of before and after, there is no sequence. The rule tells you "what counts as the next one", but unless you follow the rule, and produce "the next one", then the next one never comes. And following that rule is a temporal process. Therefore the sequence is a temporal process.
One might argue, that the order of such mathematical things simply exists, as eternal platonic objects, and that "the rule" is a description of that platonicly existing order. Then we'd have the nontemporal order, without having to fulfil the process of following the rule. But platonism is clearly wrong here. the rule is clearly not descriptive, because the proposed platonic objects cannot be observed to be described. They have no spatial/temporal existence. Therefore the rule is a prescriptive rule, and the sequence only comes into existence by following the process temporally.
Quoting Srap Tasmaner
Yes, "next" implies a relation, as you say. It implies a temporal relation. "Next" has two distinct meanings, a spatial relation, or a temporal relation. In this case it is not a spatial relation, therefore it must be a temporal relation.
You may insist that mathematics keeps "objects" as undefined, But mathematics would be useless if it cannot define its relations. And this is a serious consequence of having "object" as undefined. If we cannot identify an object, how can we formulate relations? In other words, we cannot unequivocally understand the proposed relations between objects if we do not know what an object is.
So, you are asserting that "next" implies a relation. Do you think you could explain what "next" means in the context of a mathematical sequence, without describing it as either temporal or spatial? Otherwise you are simply making an unjustifiable claim.
Quoting Ludwig V
I know we can do that, and that's the point I was making. We can, and do set up sets of co-ordinates without reference to the medium. That is a universal conception of "space", which allows in principle, for infinite positioning. But it is conceptual only. And if one sets up such a universal set of co-ordinates, with infinite possibility, and applies it to a real medium, it is a false representation as the primary premise in the representation which will follow. That false premise is what creates Zeno's paradoxes.
The point being that we can, and do set up such co-ordinate systems, I'm not arguing against that. What I am saying is that when we apply them they are applied as false premises, As such, they produce unsound conclusions as demonstrated by Zeno. Zeno concluded that motion cannot be real.
With regard to the number line, I'll say first that the intuitions most of us have, formed in school days, can be a bit misleading, because we are on the far side of a great many developments in mathematics, which bring together the numerical and spatial through measure. The "purely spatial" without any sense of measure gives you not geometry, not the number line, but topology. In short, I wouldn't agree that the number line is purely spatial.
But I think I understand what you had in mind. You can talk about one number coming later or earlier than another in a temporal sequence, or you can talk about a number being to the left or to the right of another, as they are laid out in space. And I'm saying those are much more the same thing than you might think at first, because a 1-manifold of 0 curvature doesn't have any numbers on it at all.
The number line of grade school is neither Euclid's "breadthless length" nor the 1-manifold of topology. It's an axis ripped out of a Cartesian coordinate system because they intend eventually to teach you analytic geometry and calculus.
Now there is a question about whether our mathematics is built upon one sort of fundamental intuition or two: is it all numbers (and collections and so on) or is it also shape and space? There's a pretty strong case for saying that the spatial intuition is distinct, and that much of mathematics has been occupied with somehow bringing together the two sorts of intuition (as in the number line).
But if they can be brought together, what enables that? Doesn't that indicate these are two different ways of looking at the same thing? Maybe. It's at least clear that the ways of doing things with our numerical and our spatial intuitions are closely related, so we can generalize at least enough to say something about that, and that's why we say mathematics is the study of systematic relations among things, be those things numbers or shapes, integers or angles of polygons, or what have you. (The proof of Fermat's last theorem, the statement of which looks like the barest number theory, takes a very long detour through algebraic geometry, if I recall correctly, and falls out as a special case. Part of the interest of that series of results, as I remember it, was how many fields were brought together in those proofs.)
Finally, you ask whether we're talking about a generalization or a rule, which sounds quite a bit like asking me if mathematics is discovered or invented. It's an unavoidable issue, and I've suggested before where my intuitions lie, which of course involves answering "neither". I'll only add that I think too often we think we can fruitfully approach this issue by staring really hard at the natural numbers or at triangles and circles to figure out what they really are and where they came from, when we would do better to look at the practice of mathematics to see what's going on there. It is empirically false that mathematics is all working out the consequences of arbitrarily chosen rules.
I can give a small example, not very good, but maybe it'll indicate what I have in mind. I was recently asked to look at a bit of statistical analysis someone had done of sales in several stores. There were all these numbers and percentages calculated, the usual stuff, but it didn't actually mean what they were saying it meant. There were no errors in the calculations, but the numbers they were comparing just shouldn't be compared, and certainly not in the way they were doing it. Why not? I couldn't really explain why, except to say that I had never seen it done, it had never occurred to me to do it, and I knew in my bones that it shouldn't be. I suggested that someone smarter than me and higher up the pay scale might be able to explain why we don't do this, but I could only give hunches. Still, I knew intuitively that it was gibberish.
I think you can see the same sort of thing among mathematicians. There are certain ways of developing the field that feel like mathematics. If you're doing something quite odd like inventing non-Euclidean geometry, you might get some pushback, but the way you'll win over the naysayers is by getting them to dig into it enough that they get a feel for it and can see that it is not arbitrary, not chaotic or random or meaningless, but still recognizably mathematics. There are other things you might try that just feel off, or feel wrong, that just aren't mathematics.
(You can see exactly the same thing in chess: there are legal moves that are, in effect, meaningless, because they don't address the position; there are also the obvious moves, but sometimes there are moves that don't make sense at first but once you understand them, they address the position even more deeply than the obvious moves, which come off looking superficial. Really playing chess is something different from just following the rules.)
Is any of this in the neighborhood of what you were asking?
One additional thought. We've alluded to the spatial and temporal metaphors we often use talking about mathematics, but another very common metaphor in mathematics (and in mathematics-adjacent discourse) is the tree. Trees are interesting because the main thing we want out of them is the parent-child relation, which suggests numerical change over time, but that relation is also naturally related to thoughts of growth, or spatial change over time.
Do you mean the premiss that space can be infinitely divided, not merely conceptually, but also physically?
I think most people would accept some version of that. But a physical limit to the process of division doesn't undermine the conceptual description. The physical limit will allows the conceptual division to continue.
Zeno produces an paradoxical analysis of the race. We can brush it aside and stick with the conventional analysis. There is an alternative, which is not paradoxical. Simple arithmetic and the definition of speed and (distance/time) tells us when Achilles will overtake the tortoise. So it is only a question of how you look at it. But still, people get hung up on the paradox. However, I think the real problems emerge in the analysis, for example, of circles and ellipses, which are not so easily dealt with in that way.
That's fair enough. Thank you for your comments.
Quoting Banno
Now the field structure and the order axioms are the rules that @Sam26 and @Ludwig V have been discussing, that set up the sequence of numbers in order.
We've already left Meta behind, since he has claimed numbers are not ordered...
The completeness axiom is a second-order statement (because of the quantification over subsets S), and it expresses completeness of ?.
? S ? ?, (S ? ? ? ? M ? ?: ? s ? S, s ? M) ? ? L ? ?: (? s ? S, s ? L) ? (? L' < L, ? s ? S, L' < s)
? S ? ? says the axiom quantifies over subsets of ?, and does so without specifying which subsets.
(S ? ? ? ? M ? ?: ? s ? S, s ? M) is the antecedent in the axiom. "S ? ?" discounts an empty domain. ? M ? ?: ? s ? S, s ? M specifies that there be numbers bigger than or equal to those in S; mathematically, S is bounded above.
So the antecedent is "if there is a non-empty set of real numbers with some upper bound..."
? L ? ?: (? s ? S, s ? L) says that there is some number that is larger than or equal to every number in S.
and
(? L' < L, ? s ? S, L' < s) says that there is also always some number that is less than that larger number, but still a part of S. That is, we have an L such that we can't lower L even slightly and still have the upper bound, and yet anything smaller than L fails to give us that upper bound.
Putting it together, For every non-empty set of real numbers that is bounded above, there exists a real number which is the smallest number greater than or equal to every element of the set.
By way of an example, If S were {.9, 0.99, 0.999...}. And S is not empty; S is bounded above by 1; and so by the completeness axiom, there is a real number which is the smallest number greater than or equal to every element of the set. Again, int his case, 1.
With S: {.9, 0.99, 0.999...}, we would have a process, "keep adding another 9", that would be acceptable to a Wittgensteinian - a rule that allows arbitrarily many interruptions. But the true-blue Wittgensteinian would deny that we thereby have a whole set; we have a rule for constructing an arbitrarily long string, and nothing more. To get to the compete set we need S={x?R?x=1?10?n for some n?N}, which presumes ? and so presumes already that we can talk about infinite sets.
From a modal perspective, S is a subset of ? with 1 as its supremum. From a Wittgensteinian perspective the rule “add another 9” never produces 1, “approaching 1” is not “being bounded by an element”, and the talk of a completed S is a projection of grammar. So the statement "The supremum of S is 1” is treated as a useful way of talking, not a statement reporting a fact about a completed domain.
Now what I would maintain is that the two are for all intents and purposes the same. That is, the ellipsis as it stands does not tell us how to continue on, and so falls to the sort of view expressed by Kripke; but we dissolve this by insisting that there is a correct way to carry on, given by the model theoretical account.
And I would add that this amounts to no more than following more rules. We have the definiendum ? on the left, and on the right we have a rule setting out what counts as ?.
And all this by way of showing that some rules are not procedural at all; they are constitutive norms.
______________
My apologies for that post, it's sloppy, and under argued, and I moved from limits to ? as I went through the argument. The whole needs reworking, but I'll let it stand because it sets out the direction of my thinking in response to the last page or so from @Sam26, @Ludwig V, @Srap Tasmaner and @frank; that we can legitimately reify a procedure with a "...counts as..." constitutive rule. In this case the axioms count as setting up ?.
Okay, I guess I do have something more to say. I can see why @Banno would connect it to PI 201, at least as an analogy. PI 201 is about the gap between a rule and its application; the worry is that any finite formulation can be made to fit different continuations unless our practice fixes what counts as going on correctly, which is close in spirit to the worry about ellipses and “…” in the infinity case.
But I'd be careful about saying Wittgenstein had the real numbers or completeness axioms in mind there. In 201 he is not doing foundations of analysis; he is diagnosing a philosophical temptation: the idea that the rule must contain its own application as rails laid to infinity. His answer is that correctness is in the shared practice of following a rule, not in a ghostly extra fact.
If the link is: “reifying a procedure with a counts as norm is one way of making our practice explicit,” then yes, that resonates with 201. But if the link is: “201 is really about completeness or model theory,” I think that is a stretch. The Wittgensteinian moral is still that a rule does not interpret itself, and neither does an ellipsis. What settles the continuation is the rule plus the practice that gives it grip.
Yep, and that diagnosis applies to the foundations of maths - the area in which he thought he had made the greatest contribution.
A rule does not interpret itself. Yet we have rules that set up novel interpretations. Following a rule can involve treating something as if it were something more. The move is essentially to build a new language game on the back of another. And something like this seems implicit in a form of life. The whole remains embedded in human activity, in a form of life.
I was thinking some days ago that, though I'm not sure what the favored way to do this is, if pressed to define the natural numbers I would just construct them: 1 is a natural number, and if n is a natural number then so is n+1. I would define them in exactly the same way we set up mathematical induction. (Which is why I commented to @Metaphysician Undercover that the natural numbers "being infinite" is not part of their definition, as I see it, but a dead easy theorem.)
And this will be handy later when we want to prove things by mathematical induction because our definition of the natural numbers is ideally suited to just that use.
Is this the sort of thing you're getting at? I have a procedure for producing one natural number from another, but more to the point is that the natural numbers just are what you get when you do that. It's the definition. It doesn't "turn out" that adding 1 to a natural number gives you another. That's not something we discover. It's part of what we mean by "the natural numbers".
On the other hand, it seems you could easily prove that adding 1 to an integer must produce an integer. The question is, what would you be doing in that proof? I think it would amount to showing that the definition you started with is good enough, that is, not self-contradictory in some sneaky way, and that it's all you need to generate the objects you want.
I guess that last sentence points to the fact that even here, we're talking about coming up with rules that give a complete account of a pre-theoretical practice of counting. So there's something a little disingenuous about saying I'm "defining" the natural numbers. (Famously, the Big Guy did that.) But I think we can still say that such a definition is an adequate account of our practice, so in that sense it's not quite the norm itself, but a usable form of it -- because having a definition in hand allows us to do all sorts of clever things.
I agree with most of that.
Wittgenstein does think his approach bears on the foundations of mathematics: of course, the temptation is to imagine that the rule, or the proof, carries its own application and its own interpretation independent of what we actually do. A rule doesn't interpret itself, it's not an aside, it's aimed at that picture.
At the same time, youa'e right that we can introduce further rules that effectively stabilize new ways of speaking. We can take an earlier practice and add a counts as norm that extends it. In this sense, following a rule can include treating a construction as if it were something more, because we have adopted criteria that make that treatment correct within the extended game.
But the Wittgensteinian idea is that this isn't a metaphysical ascent to a realm of completed entities. It's a reworking of our practice (what we do), still embedded in human activity and a form of life. The novelty comes from what we now allow as a correct move, not from discovering a new kind of object behind the calculus.
Pretty much. So we have "Any number has a subsequent number", a procedure - if something is a number, then there is a subsequent number. But we need another step - "1 counts as a number" - to get the procedure moving.
Calling on procedure alone is insufficient. We need there to be stuff to perform the procedure on.
And I just don't suppose that Wittgenstein, a clever chap, had missed this point as was saying that all we need in maths is procedures.
I keep thinking about how we teach basic arithmetic with applications, and it's a very subtle thing. We say, "If I hold up 1 finger, and then 2 more, I'm holding up 3 fingers" and the important thing is getting the child to say that this is because 1 + 2 = 3. That "because" is very interesting.
Yep.
It's not platonic.
So we get "One counts as a number" and "every number has a subsequent number" and discover that the pattern does not end, and then learn to talk of the whole as being unbounded and that infinite counts as being unbounded... iterating the "...counts as..." to invoke more language games.
So abstraction isn't a philosophical folly. It's the result of an astounding innovation.
I might have said property - this counts as being mine. Basic idea is right.
That existed before money. They bartered. The problem was that corruption in bartering was rampant. They would put the good dates on the top of the caravan, and it was just mud-balls below that. It was so bad that it inhibited trade.
Money set trade free from corruption because it was these little pieces of gold which were stamped to assure a specific weight and purity.
Next came banking, which was mainly invented by the Italians. Now we have virtual money, which allows economies to grow past their present means. The human world as we know it today is a result of money and banking.
I think the tricky bit is that philosophers hear "1 finger and 2 fingers make 3 fingers because 1 + 2 = 3," or even "1 finger and 2 fingers must make 3 fingers, because ..." and this sounds to them like the natural world obeying the "laws of mathematics" or some such. As if the fingers might try to add up some other way, but they would always fail, because there's a law.
But it's actually more like this: if I'm already committed to saying 1 and 2 make 3, then I'm also committed to saying 1 finger and 2 fingers make 3 fingers; if I didn't, I'd be inconsistent. Similarly, I can't say it works with fingers but not with train cars.
Children do have to learn, through trial and error, how much they're supposed to generalize. (Calling cows "doggies" and all that. And learning the difference between count nouns and mass nouns.) And of course what counts as success or failure is determined not by nature alone but also by the adults that mediate a child's understanding of nature.
What's difficult for us, in talking about mathematics, or about language, or about concepts, is that we want to pass over the generation upon generation of practice and refinement, to recreate the primordial scene in which someone, however far back, came up with a way of doing this sort of thing that worked, and we want to identify the features of the environment that enabled it to work, very much as if we expect there would only be one way. Some aspects of our thinking we find relatively easy to change, but some are so deeply embedded that we cannot quite imagine an alternative, so we think this way uniquely fits how the world is.
But it's not just a question of whether other ways of thinking were adequate to "our" needs, but recognizing that there was already adaptive behavior and already learning before there was any conception at all, and even our first conceptual steps were built on that.
Quoting frank
Are you saying there could have been a period when people had money, but didn't have amounts of money?
I agree with the spirit of your history lesson, that abstraction was a practical, observable, behavioral thing, but I don't understand the idea that money is the basis of math.
No, I've repeated this numerous times now, "space" is purely conceptual. it doesn't make sense to talk about dividing space physically. Physically there is substance, and that's what is divided. And representing that substance as "space" which is infinitely divisible is what I called the false premise which produces Zeno's paradoxes.
Quoting Ludwig V
It means that the conceptual description is false. And, this falsity, because it is a falsity, produces the absurd conclusions which Zeno demonstrates.
Quoting Banno
As usual, a completely false and utterly ridiculous representation. I said it doesn't make sense to use "next" in a way which is not either spatial or temporal. If we switch the term to "order" rather than "next", this allows all types of hierarchy such as good/bad, big/small, etc.. But the principle of the hierarchy, and the order of things within the category still needs to be defined. There is no such thing as simply "order" in the general sense. And to have a next implies a direction, which implies either a temporal or spatial ordering.
Therefore we cannot avoid expressing the order itself in spatial or temporal terms. If the scale is big and small for example, then for there to be an order one of the two extremes must be prior to the other, and this turns out to be a temporal order. If there was a supposed order which was infinite in all ways it could not be an order, because infinite possibility is disorder.
Quoting Srap Tasmaner
You just show that it is limitless which is how "infinite" is defined, so there is no difference and you are not getting away from it being so, by definition.
Quoting Banno
The prerequisite platonist premise.
Quoting Banno
The usual denial. That "1 counts for a number" rather than signifying a quantitative value, is platonic. That's what platonism does, it makes values which are inherently subjective mental features, into countable independent objects. This is a faulty attempt to portray what is fundamentally subjective (of the subject) as something objective (of the object)
Quoting Banno
Your statement "every number has a subsequent number" is a stipulation. Therefore it is something produced by design, definition, it is not something that we "discover". So you continue in your misguided attempt to justify mathematical platonism.
Are you a cartoon character? Do you know SpongeBob?
People trapped in a perpetual vortex are those likened to Plato's famously quoted analogy of 'The Cave:.
The only way that "1" can refer to an object called "a number", instead of referring to distinct ideas in the minds of individual subjects is platonism. Platonism is the only way that "1" can refer to the same thing (a number, an object) for multiple people. Otherwise "1" refers, for you, to the idea you have in your head, for me, to the idea I have in my head, and so on. This is the way that values such as mathematical values are presumed to be objective rather than being subjective like many other values. It's known as platonism.
You are right of course. Like you, I am disinclined to back either option. But I prefer to treat each claim as a comparison or analogy and to note similarities and differences between the language-games. This may appear to be a cop-out, but I think it is more judicious than drawing up battle-lines. The same goes for intuitions, and you give a good example. There is, I think, a similar phenomenon wherever people acquire in-depth expertise; it's not something we are born with, but something that is born of long and intimate acquaintance with the relevant skills.
Quoting Srap Tasmaner
Wittgenstein is very good on this, as I'm sure you know. It is important. I'm fond of the adage that a rich diet of examples is very helpful. That is also part of Wittgenstein's practice.
Quoting Banno
It seems clear to me that Wittgenstein would agree with you:- [quote "PI ]201"] That there is a misunderstanding here is shown by the mere fact that in this chain of reasoning we place one interpretation behind another, as if each one contented us at least for a moment, until we thought of yet another lying behind it. For what we thereby show is that there is a way of grasping a rule which is not an interpretation, but which, from case to case of application, is exhibited in what we call “following the rule” and “going against it”.[/quote]
You can't follow a rule or go against it until you start applying it. Kripke's mistake was to demand that everything is settled in advance. There's a lot of discussion of similar ideas in the Blue Book (see p.34, 36 etc.)
Quoting Sam26
Paper money is a good example.
Quoting Banno
You are right that not all rules are of the same kind. In addition to procedural, there are constitutive rules.
Quoting frank
Yes, I think that may well be fair. But I can't help observing the ancient Egyptians had ordinary arithmetic, which, it would seem, was primarily aimed at the logistics of huge work forces - rations, supplies, etc. Ancient Sumer, China and Lombardy all contributed. There's plenty of people to share in the credit and the blame.
Quoting Srap Tasmaner
Yes. I do like bits of history as a way of understanding something about our present practices. But I wouldn't want to treat history as sacrosanct in some way. There's nothing wrong with inventing language games to bring out one point or another. Wittgenstein does it all the time, so it can't be wrong, can it?
Quoting Metaphysician Undercover
The problem with Plato's ideas is that he tries to apply the model of 3D physical objects to abstract objects. Both exist and can be referred to, but they are not the same kind of objects. Your idea that the only kind of object that is not a 3D physical object is an idea in the mind. Numbers are not just ideas in the mind, but are rooted as objects in our shared practices.
Let's be clear, numerals are objects in our shared practices. Numbers if they are assumed to be objects are nothing other than platonic objects.
The question is, what do you think a numeral like "1" refers to. If you think it refers to an object, in the type of "number", or the "mathematical" type, that is a platonic object. If you think it refers to an idea of quantitative value, or order, in your mind, that is meaning, not an object. If you think it refers to an object of shared practise in your mind, there is no such thing. Numerals are objects of shared practice in your minds, not numbers.
Before we even get to the question of what a numeral refers to, you face an issue of what makes any given numeral count as a 1 (or as a numeral, or as a symbol). If each individual 1 is a token of the type <1>, you have to say what sort of thing the type is. That's not going to work out. A natural move to avoid types as abstract objects is to claim that the various numerals 1 belong to an equivalence class, but that's not so much an explanation as a restatement of our starting point, that each numeral 1 counts as a numeral 1, and it gives you no help actually defining the equivalence class.
Numbers are not like rocks, nor are they like sensations.
That's part of the reason that he can't make sense of logical precedence, restricting himself to temporal or spatial precedence. His metaphysical picture cannot represent logical priority at all, since it's neither purely mental or purely of the world. And along with that go other things that rely on public standards for correctness, such as normative dependence, and rule-dependence.
The following makes his error particularly clear:
Quoting Metaphysician Undercover
Notice that this odd position is blandly asserted, not supported by any argument.
He relies on presuming that all reference must be object-reference, that object-reference must be either mental or Platonic, and that public sameness requires numerical identity of a referent. Meta relies on an unargued slide: “same object” ? “same referent” ? “same use” He treats these as equivalent, but they are not. What is required for reference to function is not that we talk about the same object but that we have a public criteria for correctness. It's learning that public criteria that so clearly portrays; learning to count is learning to participate in public activities involving fingers and toy cars and slices of pizza. Numerals get their identity from roles in activities, not from reference to entities.
I don't think I can add anything to your replies. I would likely just confuse the issue.
Frege had a pretty persuasive argument for it.
And then we have Benacerraf's identification problem. There are multiple equally valid set-theoretic constructions of the natural numbers. If numbers are “objects” in the Fregean or set-theoretic sense, which objects are they? There is no fact of the matter that uniquely picks one construction over another. In contrast, public reference and logical precedence do not require objecthood at all. Benacerraf's argument shows objecthood is doing no work, the Wittgenstinian account offers an alternative.
Why should I care?
Because you're procrastinating from whatever it is you're supposed to be doing right now?
There was a lot of strenuous protesting in this thread to the effect that infinity is a thing. Turns out you actually agree with Meta. Numbers aren't things. They're just elements of language games.
I don't understand you. In each instance where 1 is taken to be a token, the type is a symbol. And the type of symbol is mathematical. And the type of mathematical symbol is a numeral. How is there a problem with this?
Quoting Banno
I have no problem with that story. we are human beings with minds and free will, and we figure things out and do things. Don't you think that's the case?
Quoting Banno
It appears like I didn't make the argument clear enough for you, when I stated it earlier. So, here it is.
If a numeral refers to an object, which is within a human mind, it is a different object for me as it is for you, due to the nature of subjectivity. My thoughts are not the same as your thoughts, so we'd have distinct objects being referred to because we have distinct minds. Therefore, since a numeral is supposed to refer "an object", not a bunch of different objects, and also to the same object for you, as it refers to for Srap, that object must be independent from both of you. The referenced "number", as "an object", must be an independent object This is known as a platonic object. Hence, assuming that a numeral refers to an object called a number, is platonism.
To state it simply, without assuming that the object referred to is an independent, platonic object, it is impossible that the numeral refers to the same object for distinct people, because we each have distinct minds with distinct thoughts. Then the numeral would refer to a bunch of different objects in different minds, instead of "an object", the specified "number". Therefore the assumption that a numeral refers to an object called a number is platonism.
Quoting Banno
That's another one of your very absurd misrepresentations. I explicitly stated, in the passage you quoted, that the symbol might refer to an idea in a mind. Never did I imply that I believe all reference must be object-reference.
What I said, is that if a numeral is taken to refer to an object, a thing called a number, that object must be a platonic object. This is supported by the argument above. However, I do not believe that a numeral refers to an object called a number. I believe that it refers to an idea called a value. I believe that values are not objects, yet they are referred to. Therefore, in no way do I believe that all reference is "object-reference".
Wow! Being productive. :up:
So math is just language games, right?
Elements in a language game can be things - because we quantify over them... all these numbers are even, all those numbers are prime.
Quoting frank
Drop "just" and you might be getting there.
Quoting frank
Despite my obvious addiction, I am still functional. But it's going to be 36? today, 42? tomorrow, so productivity occurs inside or in the early morning.
You're trying to have your cake and eat it too.
Quoting Banno
-15C here. :confused:
That's permitted, under the rules...
You have the minority view, so you must be following the minority rules. :razz:
https://survey2020.philpeople.org/survey/results/5030
Dude. According to that survey, a small minority think the foundation of math is set theory. There aren't a lot of experts in phil of math, but they would all roll their eyes at that. :grin:
You could say that. The point though is that if a numeral refers to a number which is an object, and that object is said to be an idea in someone's mind, then it would be a different object in each mind. We are all distinct individuals with different bodies, different minds, and different ideas. It could not be the case that the idea in my mind (if we call it an object) is the same object as the idea in your mind even if we each refer to our ideas with the same word. We might use the same name "1", and even be trained to describe it with the same words, but it's still not the same idea.
You might consider the beetle in the box analogy. We use the same word, "beetle", and we might even describe it in the same way, but we still have distinct objects. The only way to assume that the numeral refers to the same object for distinct individuals, is to assume that the object is independent. That's Platonism. For whatever reasons, I do not know, @Banno insistently denies the obvious, to say that a numeral refers to a number, which is an object, is classic platonism. No one ought to be surprised by this. Western ideology is firmly based in idealism.
Quoting frank
Compare and contrast... https://survey2020.philpeople.org/survey/results/5030?aos=47 ...note change in AOS.
We could go in to a discussion about whether the view expressed here is structuralism or constructivism, if you like. But none is a majority opinion, even amongst those who study in the area.
So you've just completely changed your mind here. You were quoting ZFC as if it were scripture a few pages back. Then you were a deflationary nominalist. Now.. I have no idea what you are. I think you might be constructing your view as you go along.
An object in your mind is called a mental object. An object in your hand is a physical object. An abstract object is something that isn't physical, but it's not simply mental either.
Quoting Metaphysician Undercover
That is correct.
Could be.
:meh: What of quantification?
What about it?
We've shown how quantification can be handled without invoking abstract objects at all — it’s rule-based, normatively grounded, and socially coordinated.
Ok. I don't object to that. I doubt you can do the same thing for ZFC, though. So are you now suspicious that ZFC might be bullshit?
This is platonism. The abstract object is independent from minds, but accessed by them.
Quoting Banno
Quantification doesn't require platonism. The proposition that a numeral represents a thing which is a number is platonism. But we can quantify without that premise. For example, we can do a bijection between the numerals and the things to be quantified. The presumption of "numbers" is superfluous in this case.
Yes. I know. You pretty much came up with Frege's argument all by yourself. That's pretty cool.
:roll:
Is that a no?
I've just been observing the different stances people are taking. The only book on phil of math I've read is Mary Tiles' book. After reading it, I realized the ways that set theory is conceptually objectionable, which might not be surprising since Cantor was a mystic, and his mathematical views were directly related to mysticism. Tiles doesn't campaign against set theory by any means, but she does leave the reader with the thought that we may one day rethink the whole thing. It may be that Aristotle was right after all.
Much appreciated, thank you.
Yes, you read these threads as people and their interactions rathe than as about ideas.
You said this:
Quoting Banno
Now apply that strategy to the empty set. You'll find that you can't. Set theory is fundamentally platonic. Eject platonism, and you've ejected set theory.
Why not? I have nothing in my pocket, therefore I have nothing. :meh:
It doesn't sound like you know what a set is.
I was referring to the previous two posts. Beyond that, there's much that I agree with, but I still have puzzles (questions), which is not quite the same as disagreement. Partly, they centre on the questions about what it is for a mathematical object, such as a number, to exist. Partly, they centre on what the timeless present means in this context.
Quoting Banno
I agree with that. I don't have a problem about the timeless present in the case of constitutive norms. But in relation to procedures, I do. For the obvious reason, that a procedure takes place in time.
Quoting Banno
Of course. You may care to know that, as I understand it, the reason the Pythagoreans did not count 1 as a number was, at least partly, because they saw it as the source of all the other numbers. But don't we also need 0, as the starting-point?
Quoting Srap Tasmaner
That's reassuring! But I'm not quite clear what it means to "produce" a number. It's not as if we say to ourselves "I need another number here" and so instigate the procedure. Does your procedure create the numbers it produces from scratch or does it just produce another copy of the number????
Quoting Banno
You are not wrong. But now we are getting into trouble with the difference between numerals and numbers. I have a feeling, however, that we may need numbers in order to identify correspondences between numeral systems and perhaps even number systems with different bases.
I'm also getting puzzled about "to be is to be the value of a variable", or, more expansively, the idea that existence is defined within language games and the rejection of single (absolute?) criterion of existence across language games. I think that approach has a great deal to be said for it.
Quoting Metaphysician Undercover
I think many people believe that if something is referred to, it counts as an object.
It is true that we equate numbers with values, in the mathematical sense. That's to do with the uses that we put numbers to. So you are right to foreground what we do with numbers - or numerals if you prefer. But I think you slip up when you say that the numeral refers to an idea. That just resuscitates that argument you gave about numbers as ideas. The assignation of value in this context is public and shared, so it cannot be about ideas in our individual minds.
Quoting Metaphysician Undercover
I'm getting the impression that your objection is simply to the concept of an abstract object, which you call platonism. Would that be fair?
Quoting frank
Yes. Though there are lots of different kinds of physical object, not all of which can be held in your hand. Shadows, reflections, clouds, lightning, colours, sounds, surfaces, centres of gravity and on and on. Similarly with mental objects. Abstract objects also come in lots of different kinds.
Quoting Metaphysician Undercover
In the Roman number system "V" counts as five. The Chinese system has ? (w?) for the same number. The ancient greeks used the letters of their alphabet as numerals, so five was the letter epsilon. If you just talk about numerals, you lose the equivalences across different systems.
I think that would be an odd use of language, if every word referred to an object. Definitely not suitable for a rigorous logic. For example, we distinguish noun and verb, object from subject, subject from predicate. To disregard these distinctions would incapacitate logical procedures.
Quoting Ludwig V
I actually don't mind when people refer to numbers as objects, that's the way I learned in school. But when people do this they need to respect the ontological consequences.
When you count something publicly, you share your assignment of value. Other people can observe and correct you if they think you make a mistake. This clearly is about ideas in our minds.
Quoting Ludwig V
Banno was denying that the principles he asserted were platonist, and so I was trying to get him to acknowledge that they are. My objection was to the hypocrisy of publicly rejecting platonism then employing platonist principles.
Quoting Ludwig V
That's exactly the reality of translation. In most cases there is no true equivalence "across different systems". The different language games come into being and evolve under different social contexts. The assumption of platonism, produces the idea of eternal unchanging objects which words refer to, and disables us from being able to understand the reality of the nonequivalent aspects.
However, in the case of symbols used in calculation, an equivalence can be established.
Quoting Metaphysician Undercover
So you think that "to be is to be the value of a variable" is a platonist principle? I know you sometimes use words in ways I find hard to understand. This seems to be another case.
Quoting Metaphysician Undercover
Very true. Except that ordinal numbers don't assign a value; that assigns a place in an order. Assigning a value in mathematics just means what you do when you substitute a specific number (or word or sentence) to a place in a formula that is designated for such "values".
Quoting Metaphysician Undercover
No, it isn't. It is about whatever I am assigning a value to.
Quoting Metaphysician Undercover
In the context of traditional grammar, an object can be almost any noun, limited only by the specific subject and verb that you are talking about.
Quoting Metaphysician Undercover
Not all words refer to anything. That's why there's such a fuss about dragons and the present king of France.
if being is reduced to value, that's idealism, not necessarily platonist though, but most cases yes. That's classical Pythagorean idealism, the cosmos is made up of mathematical objects.
Quoting Ludwig V
A place in an order, or hierarchy is a value.
Quoting Ludwig V
What we were discussing was the act of assigning value, counting. That was the subject. Now you are changing the subject to claim that we were not talking about this act, but that we were talking about the thing which you assign the value to. Clearly we were not, as whatever it is assumed to be was not even mentioned.
Quoting Ludwig V
Why do you allow that sometimes when words refer to ideas (two, three, for example), they refer to things, but sometimes when words refer to ideas (dragons, present king of France), they do not refer to things? Why not just maintain consistency and recognize that these are all cases of words referring to ideas?
Eh. A procedure, as I'm using the term here, accepts some input and yields some output. You show me a natural number, and I can show you another.
What I was suggesting was that we can replace our pre-theoretical understanding of counting with this system, consisting of exactly two rules (that 1 is a natural number, and every natural number has a successor), and we will (a) lose nothing, and (b) gain considerably in convenience for doing things that build on counting.
I consider (a) and (b) more or less facts, but there's nothing wrong with examining them closely. Philosophy spends a lot of time doing exactly this sort of thing, but not only philosophy. Linguistics is an easy example quite nearby, where people want to describe a great mass of complex behavior in terms of a smallish set of rules that could account for it. A more or less universal scientific impulse.
So the "axiomatization" of counting here is open to criticism, and I believe it will withstand it.
But it doesn't necessarily tell you what counting actually is.
It might. In a sense, when you come up with a little set of rules like this, if it works pretty well, then what you definitely have is a model of the behavior you want to understand. Whether that model reflects the underlying mechanisms of the behavior, or only simulates the behavior itself, relying on different mechanisms, that's not always perfectly clear. (In one formulation of Chomsky's program, it was of the utmost importance that you have a finite system of rules that can, through recursion, generate an infinite number of sentences, because the system has to be instantiated in a human brain.)
I've been thinking a little, as we've gone along, about the most famous "primitive" counting systems, the "1, 2, 3, many" type. Is "many" a number there? Not exactly. Is it open-ended enough that it might even apply to endless or unbounded sequences? Maybe, maybe not. What I'm trying to say is it might not be quite the same thing as us saying "1, 2, 3, 4 or more" or "1, 2, 3, more than 3", because in our system of counting numbers there is definitely no upper bound.
I suppose I'm bringing that up because we might ask whether people using one counting system are doing something psychologically different from people using another, but we might also ask if there is some difference that philosophy ought to be interested in. The latter, I suppose, would be something about the system itself, and the thoughts that it enables or doesn't, and therefore what would be available as truth, given such a system.
I was using "procedure" as a generalisation of "function". Where a function will have exactly one result for each input, a procedure need not. I hadn't considered that someone would suppose that logical procedures are somehow temporal. I find that idea quite odd.
Again, it hadn't occurred to me that this wasn't obvious... do we want natural numbers or counting numbers? It's not needed, as such, unless you have nothing in your pocket.
That is, which game are you playing?
Again...
Well, what I was pointing to is the difference between a numeral and a number is in the use to which it is put; one counts with numbers, not numerals. "Numerals get their identity from roles in activities, not from reference to entities" is intended to point this out. The difference between numerals and numbers is not ontological, it is grammatical.
Herein lies much confusion, that can be sorted by looking at quantification.
Again, it hadn't occurred to me that this would be problematic. It's quite legitimate to move from "7 > 5" to "Something is greater than five", or "There is a thing greater than five". That doesn't commit us to bumping in to fives and sevens along with chairs and tables. Quantification tells us what a grammar ranges over, not what exists as a spatiotemporal object. The confusion here is between differing language games; to think that "object" only means tables and chairs and not 7 or fully incorporated companies.
Being an object is a role in a language game, not an ontological status.
Failing to recognise this is what sits behind the confusion of calling things "platonism" hereabouts.
Who said anything about reducing being to value?
Quoting Metaphysician Undercover
Hierarchy, yes. Order not necessarily. Alphabetical order doesn't imply value.
Quoting Metaphysician Undercover
Oh, dear. How can one assign a value without assigning it to something? In any case, counting chickens, for example, answers the question "How many" and assigns a value to the brood, if you like. But it doesn't assign any particular value to any of the chickens.
Quoting Metaphysician Undercover
When I say that the President is bold, I am talking about the President, not the idea of the President. When I say that the President has executive power, I'm talking about the idea of the President. The idea of something is a different entity (if it is an entity at all) from the something that it is an idea of.
OK. In that case, you carry out the procedure. What bothers me is the idea that a formula like S(n)=n+1 is not a set of instructions about how to do something, but actually does it. So someone might say that formula generates the infinity of numbers. That's not at all the same thing.
Quoting Srap Tasmaner
I don't have a problem with that. Something like regularizing, tidying up, making explicit - even get a whole new perspective on something entirely familiar. I can see a point to that.
Quoting Srap Tasmaner
Yes. One would need a demonstration of the written instructions as well. It's the gesture of adding one to the total, letting one sheep through the gate, and one more, let through the next one and so on.
Quoting Srap Tasmaner
Yes. I do like half-way houses. They can be very instructive.
Quoting Srap Tasmaner
It would depend on the details.
So the same thing will work for "abstract" and "platonism.". They're parts of a language game. You can't reject them without special pleading.
Godel said we perceive abstract objects. He would know.
Thanks for that distinction. I wasn't aware.
Quoting Banno
I'm glad you agree with me. I had noticed that people often speak as if the procedure (or function) somehow executed itself. Obviously a procedure or function only achieves the result if someone follows the instructions. In that case, talk of a function yielding a result is short-hand, omitting the proviso "when someone follows the instructions. Would that be right? The problem is the idea that the rule executes itself in advance of our following it.
Quoting Banno
OK. It depends on what you are doing. I was thinking of the point of origin on a graph, but that's not quite the same as counting numbers.
Quoting Banno
So the numeral is the number in the way that lump of wood is the king in chess? Yes, that's much neater.
Ockham would be pleased.
Quoting Banno
Oh dear. I obviously made my point very badly. I was trying to get at the point that there are different kinds of object, that's all.
I think this is important - see how what we are up to changes what number system we are using?
I had to double-check but I never posted this! A couple times I wrote a post which contained exactly this point. (This post is what was left.) It would have gone something like this:
You can count sheep in a field just by looking but there are a number of challenges. A better way is to force them through a chute into another field or paddock or something, and then counting them as they come through is easy. It's interesting that you needn't care what order they come through in; you have your helper — the dog — start a number of fleeing-toward-the-chute processes that run concurrently, and you count them as they terminate. It doesn't even matter that they interfere with each other.
Zeno insists that we count the sheep — that is, the rational numbers — as we find them, in their natural order. But Cantor showed that there is a way to force them through a chute so that you can count them one-at-a-time. It's interesting that it turns out you cannot do this with the real numbers. (And I'll note again that we might take from Zeno not what we're usually told to, but a clever illustration that the rationals in their natural order do not form a sequence, or as an illustration simply of the reason: Zeno shows us that there is no smallest rational number greater than 0, and so there's no "first step". That was worth learning.)
Quoting Ludwig V
I was thinking more of (a) how we individuate objects in our environment, (b) how we consider some of them countable and some not, and especially (c) the idea of associating one list with another. There's quite a little leap in (c), because you have to recognize that two collections have structures that can be treated "isomorphically". In our case, the word "collection" seems a bit out of place, but it's not, because we know what kind of structure the natural numbers have without collecting them all. The rational numbers with that same order (that is, "<") do not have the same structure as the natural numbers, but you can order them differently so that they do. That's (d), the cattle chute, re-ordering a collection (even an open-ended one) so that you can map it onto another, or vice versa. Between (c) and (d) it's hard to say which is the bigger leap in imagination. I lean toward (c). When did shepherds start using notched sticks or knotted strings to count cattle? How on earth did they come up with such an idea? Extraordinary.
I think it's just a coincidence. I used this example because it occurred to me at the time, not because I had read it before.
Quoting Srap Tasmaner
I imagine that there was a problem on the second day that someone took someone else's sheep out and came back with fewer. There has to be an agreed record of how many sheep went out.
Quoting Srap Tasmaner
You are making me very curious about the rationals, reals, etc. But I think I'll leave them for another occasion. Thank you for your help. .. and you for yours.
That's what I meant. I was very pleased you had the same thought.
Cheers. Interesting chat.
That's a step in the right direction. You have to switch from a count noun to a mass noun. Water from a fire hose. But even that's not good enough, because with an election microscope you can count individual molecules of water. Maybe the real numbers are closer to something like an electromagnetic field, something where the idea of counting instead of measuring is not just impractical but unthinkable.
Maybe there's no joy there. Still, forcing the unwieldy mass of rational numbers to line up single file to be counted was a master stroke.
It's just that the extension of the idea of the real numbers seems to be somehow bigger than the extension of the idea of the natural numbers. We could express that by saying it appears the set of natural numbers is a subset of the set of reals.
Neither set is countable, but that sense that one is bigger than the other was expressed in terms of cardinality.
The natural numbers are also a proper subset of the rationals, but they're the same size.
The natural numbers are countable.
Quoting Srap Tasmaner
As is, there is a bijection between them.
You couldn't finish counting them.
You mean they have the same cardinality. Neither one really has a size.
Denumerable, yes. Let's not mistake that for countable in the common sense of the term. I think that's where some of the confusion in this thread is coming from.
Which some authors prefer, but it means what other authors mean by "countable". So long as we know what we mean, "The natural numbers are violet" would do just fine.
Absolutely. Let's keep in mind that it does not mean the same thing as countable as the word is commonly understood.
I don't think anyone in this thread had forgotten, or that anyone was confused. Some people reject talking about infinite collections, I think, or reject talking about performing operations on them. We who accept and they who reject disagree, but we all agree on what we're talking about.
I think I could find cases of it in this thread. I'm not going to mine it to find them though
Quoting Srap Tasmaner
And since you bring that up, let's look at the difference between a collection, an extensional definition, and a set. Just because I think we need to stuff that difference down this thread's throat. :blush:
The extension of an idea need not be thought of as an abstract object. A set has to be thought of that way. There's no choice. The people who invented set theory knew that.
You left out classes, often in this context called "proper classes," I believe (since the word "class" has many uses), collections that are too big to be a set, for example.
I used to know a lot more of the technical side of this stuff than I do now, but I don't think where people have disagreed it was primarily about technical issues anyway. It looks to me like even our differences regarding mathematics are not primary, but result from broad differences in outlook.
Supertask. It's the reason Zeno's paradox stands.
Maybe. I'm still mulling it over.
If, as I've suggested earlier, you think of mathematics as the long working out of how to join two sorts of intuitions into a single enterprise (number or count, on the one hand, and something like shape or space, on the other), then Zeno's paradox is a kind of speed bump there, and indicates that this will not be so simple as we might have hoped.
I think that's one reason so many of us grew up thinking calculus somehow "solves" the paradox, or overcomes it, because calculus does represent some kind of completion of a very long process of drawing together various fragments of mathematics.
But something else we might say comes oddly out of the discussion above, about how the reals cannot be counted, and the standard alternative, if we're casting about for a different metaphor, in English anyway, would be that they must be measured. (If you're not a count noun, you're a mass noun.) Somewhere I suggested that "measure" is the first step in joining the two strands of intuition, number and shape, and that's obviously true. But it's also true that a distinction persists. So — to get to it — we don't count distances; we measure them. But the whole structure of Zeno's analysis, despite relying on dividing by 2 and all that, tends toward counting, as if it's an attempt to force distance into the procrustean bed of counting. The whole procedure seems intended to undermine the idea of measuring at all through a perverse insistence on the model of counting. (If that's not clear, I can take another swing at it.)
As for the supertask business, it's the framework that interests me. Zeno insists, apparently reasonably, that to carry out the task of traveling from A to B, you must perform a series of actions — indeed you could say this about anything. It's hard to imagine an alternative, but it's quite an odd thing really. Everything is done by carrying out certain steps one at a time, in order? That's demonstrably false for a great number of things we do. The universe is a concurrent place, and we are concurrent beings. In order to walk, you don't move a leg, then an arm, then the other leg, then the other arm. If you tried to walk by performing a number of actions sequentially, you'd fail.
The most interesting case is probably thinking itself, because we know for a fact that the brain is a massively parallel system, and yet we have put enormous effort down through the generations into shaping that mass of simultaneous activity into something linear and sequential. We get logic that way, and human speech, which is one damned word after another, linearly. We are very proud of our linear triumphs, but it is, so far as I can tell, impossible to say whether that linearity is an illusion.
In short, what interests me about the paradox has less to do with "infinity" and more to do with "sequentiality".
I read you as basically saying there's a higher truth missed by Zeno. Per tradition, his point was exactly that: that the way we picture the world, the way we commonly think, is missing something.
If you go back and look at one of the paradoxes, they're pretty simple. Looking at in terms of truth it starts here:
Isn't it true that in order to get from point A to point B, you have to travel half the distance between them?
Who would say no to that? How could you get from A to B without arriving at a point that's halfway between? If you say no to that, you've already ejected yourself from common sense. If you say yes to it, you're on your way to being ejected from common sense because there's a convergent series of points between. Either way: common sense has a problem.
There are two kinds of people: ones who can tolerate a threat to common sense, and those who can't. I think the first category is usually non-linear thinkers.
When the measuring stick needs to be measured, it's time to throw away the measuring system completely, and devise anew. Otherwise paradoxes are produced, like Russell's.
Quoting Srap Tasmaner
Of course, an infinite collection by any standard definition of "collection" is nonsense. A collection consists of things which have been collected, not things designated as collectible. And that's problems arise in set theory, "collection" becomes a designated collectible type, rather than the collection itself.
This is how the concept of "the empty set" creates paradoxes like Russel's. A "collection" with no items is not a collection at all. It is only a criterion for collection, therefore an abstract 'type" distinct from an item. Allowing for an empty set means that "the set" itself is not the collection of things (or else an empty set would not be a set), but "the set" is the abstract type, which describes the things to be collected. The things themselves, therefore, the elements of the set, must be categorically distinct from the sets, or else the empty set is the contradictory notion of a collection of nothing. Failing to follow this categorical distinction, which is necessitated by "the empty set", and allowing that a set might itself be an element of a set, produces problems.
But if the collection consist of things designated as collectible, and there is none of them, then it makes sense to talk about an empty set. However, this leaves cardinality as completely unjustified because the elements are just possibly collected, and therefore not counted.
Quoting frank
I think that what Srap is saying is that we cannot reduce motion to a succession of truths. That's what Aristotle demonstrated as the incompatibility between being and becoming. If change is represented as a succession of different states of being, one after the other, then there will always be the need to posit a further distinct state, in between any two. Then we have an infinite regress, without ever accounting for what happens between two states, as the change, or "becoming" which occurs as the transition from one to the next.
So if motion is described as getting from A to B, A and B are the two points of being, you are at A, then you are at B. Since they are not the same, there is distance between, and we can posit a middle point. You are at C. Then we posit a point of being in between A and C. You are at D. Notice, we've reduced motion to "being at a point which is different from the previous point". But this produces an infinite regress without ever addressing the real issue of how you get from one point to the other, what happens in between. This is the real nature of motion, what happens in between, and it cannot be represented as being at a designated point.
Ok. All I know is that it's common sense that if you're driving from Washington DC to Alaska, you will, at some point, be in British Columbia. Those who claim this view is wrong should at least acknowledge that what they're saying sounds bizarre.
I would say the opposite is the case, what you say sounds bizarre. You are representing driving through British Columbia, as being in British Columbia at some point. What does "at some point" even mean in this context? You use it because it's an acceptable figure of speech, but taken literally, it doesn't fit. So what does it really mean?
Being in British Columbia usually entails waking up in your car with a Canadian citizen tapping on your window to see if you're ok. You roll your window down and try to do a Canadian accent so they don't know you're American, at which point they just stare at you. Does that explain it?
No, you described a long process, and the problem is with the use of "at some point". How does a process occur at a point?
I think you just need some more of whatever mind altering substance you have available. Then you'll get it.
@frank A Great quip for a reader lost in the woods. I think I was losing the grasp of the argument presented, due to the multiple citations of the author in the text, some of which I have none or only cursory experience.
What, if anything, in the supposed paradoxes of motions from Zeno, is not answered by limits, infinitesimals and calculus?
I suggest that what does remain is not a problem about motion, space, or time, but about conceptual confusion over infinity, divisibility, and description.
The key is that an infinite sequence may have a finite sum: ½ + ¼ + ? ... = 1
Velocity is not defined at an instant by a finite displacement, but as: [math]v = \lim_{\Delta t \to 0} \frac{\Delta x}{\Delta t}[/math]
I think it’s time to play Tom Waites; The Piano has been Drinking.
Joking aside, Zeno’s paradox is an anomaly, just like infinity, they’re both anomalies thrown up by thinking. They don’t apply to the real world.
Maybe there are two kinds of worlds, one where nothing happens (I can’t describe that world), or a world where everything is in motion, everything is happening (moving), such that there isn’t anything that isn’t happening (moving). Now we’ve solved the problem of how to get from A to B, but we can’t stop now, we’re just going to have to keep moving (happening) now, add infinitum.
For God’s sake, who the hell decided to set everything in motion.
You mean the key is to put an end to the infinite sequence by rounding off. That's what we've done with pi for thousands of years. But if you think that this puts an end to the infinite sequence, and solves Zeno's paradoxes, you misunderstand.
This obviously works in practise. But the paradoxes are theoretical, they always have been, and they've always been irrelevant to practise. "Limits, infinitesimals and calculus" change the practise, but have no affect toward answering the paradoxes, which remain unchanged despite the changes in practise.
No. Nothing to do with rounding.
Your failure to understand mathematics is not our problem.
I'll repeat, since you did not address the issue.
It is a difference between theory and practise. In theory, the sum approaches the limit. In practice the sum is the limit. The latter can be understood as "rounding off". Failure to recognize this is to misunderstand.
It's not an ordinary sort of rounding off, though. The difference between the limit and the sum is an infinitely small number. We could say that this solves Zeno's paradox as along as space and time actually conform to the calculus framework. I think the average scientist would agree that they do conform, but there is still room to reject the calculus angle.
The infinite sum of (1,1,1,1,0,0,0,0....) = 4, as is its limit.
The infinite sum of the geometric series (1,0.5,0.25,...) is technically undefined, for in this case, every partial sum S(n) is non zero, since S(n) = 2 - 0.5^(n-1).
Sure, we can call 2 the "infinite sum", but it isn't an infinite sum in any literal sense of the word, rather 2 is the limit of the geometric series.
A limit isn't defined as a position on a sequence, but is defined as a finite winning strategy in a finite game, that involves cutting off the tail of an infinite sequence at a position ?, such that the height of the tail is within a prespecified distance ? to a prespecified value called "the limit", as per Cauchy's (?, ?) definition of a limit.
E.g we have a sequence game S. Eloise first chooses a value v, then Abelard chooses a positive rational ?, then Eloise chooses a natural number ? in response. Abelard is now tasked with choosing an n greater than ? such that |S(n) - v| >= ?, otherwise Abelard loses. If Eloise's choice of v can guarantee her victory over Abelard, then we say that the limit of S is v.
There is no approaching the limit when proving a limit, for a proof of a winning strategy for Eloise doesn't involve multiple rounds of Abelard choosing ?1 then Eloise choosing ?1, then Abelard backtracking to choose ?2 < ?1 then Alice choosing ?2 > ?1 etc. Rather, a proof of a limit is just an inductively defined function ? -> ? established in two steps, for mapping Abelard's possible choice of ? to Eloise's choice of ?.
This is really cool, but I'm not convinced.
The gist of it is that there is a dominant strategy iff the sequence has a limit. If you countenance classical mathematics, you do an existence proof; if you don't, you do a constructive one. And then you have an answer about the game-theoretic application.
I guess what I don't get is that if you want to go the other way—to actually define the limit as a strategy—then you still have to start with an account of how to construct a ? given an ?. You seem to be doing the same thing but saying it's for a different reason.
The limit as strategy is interesting, and it's cool that it can be presented that way, but it looks like you still end up doing exactly the same math (of your preference) you would if you just presented it as a bare question, does this sequence converge? What am I missing?
"Infinitely small number" really has no meaning in this context. If the formula is applied to spatial distance, as in the Zeno paradox, it means infinitely short distance, not infinitely small number.
Quoting frank
I don't agree. I think the average scientist would say that it doesn't make sense to talk about infinitely short distances. So if they round something off to zero it wouldn't be an infinitely short distance which is being rounded off, because the limitation of practise would require rounding off before infinitely short distance (whatever that actually means) is reached.
For example, when I use pi I round off to 3.14. Some scientific applications might request something more precise, but really the precision of the outcome is relative to the precision of the actual measurement. But, it's never an infinitely short amount which is being rounded of. So in the other example, are the measurements such that you are rounding 1/2+1/4+1/8 +1/16 to 1, or are you rounding 1/2+1/4+1/8+1/16+1/32+1/64 to 1? In the first case, 1/16 would be lost, as rounded to zero. In the second case 1/32 would be lost. The smaller the size becomes, the more difficult it becomes to measure it, and the required precision is application dependent.
The fact that no partial sum equals 0 does not imply anything about whether the limit exists, or what it is. Limits routinely exist even when no term (or partial sum) ever equals the limiting value.
??The infinite sum of a series is defined as the limit of its partial sums (when that limit exists):
[math]
\sum_{k=1}^{\infty} a_k := \lim_{n \to \infty} S_n
[/math]
For the geometric series
[math]
1 + \tfrac12 + \tfrac14 + \cdots
[/math]
the partial sums are
[math]
S_n = 2 - \left(\tfrac12\right)^{n-1}.
[/math]
Since
[math]
\left(\tfrac12\right)^{n-1} \to 0 \quad \text{as } n \to \infty,
[/math]
it follows that
[math]
\lim_{n \to \infty} S_n = 2.
[/math]
The fact that every partial sum is non-zero is irrelevant; convergence depends on the existence of the limit, not on whether any partial sum equals the limiting value.
Even on this “game” interpretation, the geometric series trivially has a winning strategy for every
?. So by your own account, the limit exists.
No.
There is no principled theory/practice gap here. “Approaches the limit” and “equals the limit” are not in tension. Introducing “rounding off” does not correct or deepen the mathematics—it changes the subject.
Maybe this is won't help, but "rounding" is something you do when all you need is an approximation.
It's not that the adjacent members of a sequence become "infinitely close": they become "arbitrarily close", and so the series (in this case, the sum of the members of the sequence) becomes arbitrarily close to — well, that's the thing, to what? And that's your limit.
We are indeed talking about approximation, and therefore approximation of some value. It turns out we can imaginatively construct a "perfect approximation" which just is the value we are approximating. If you can get arbitrarily close to it, you can figure out what you're getting close to.
For a convergent series the sum is defined as the limit. There is no residual “infinitely small difference” between the sum and the limit. The sum is the limit. Partial sums are less than the limit, but their difference goes to zero in the standard real number system.
There is a branch of mathematics called Numerical Analysis which, among other tasks, attempts to predict how far out one has to go in an infinite expansion to achieve an approximation of the limit to a specified degree of accuracy. I wrote some papers about this topic concerning continued fraction expansions. For example:
https://www.researchgate.net/publication/303490331_An_error_estimate_for_continued_fractions
There are various infinite expansions beyond sums: Infinite series, infinite products, infinite compositions, infinite continued fractions, etc. as well as infinite sequences arising from other algorithms.
That's exactly when rounding off is employed, when things are designated as "arbitrarily close". How have you done anything other than described a case of rounding off?
I'll defer to your experience. My understanding is that what I said holds for classical convergence in Real Analysis.
Keep up the good work!
It's the difference between saying (1) here is an approximation of the value that is within some tolerance you have specified (or precision, or significant digits, whatever), and (2) here is a value that is within any tolerance you might specify, however small. For (2) to be possible, I must be offering you the actual value.
(Decades, it's been decades since I did this in school. I could look everything up on wiki, but it's more fun to see if I can piece back together stuff I used to actually know.)
Back a few pages I began a bit on the definition of a limit. I got as far as completeness and the least upper bound. Every nonempty set of real numbers that is bounded above has a least upper bound in ?, the smallest real number that is greater than or equal to every element of the set. It's the existence of this number that guarantees the existence of a limit when one uses the sequence in a calculation... if it's a monotone increasing sequence that is bounded above...
But as you found, the interesting stuff is the variations on these themes. The thread is focused on a small, very specific region of maths, and mostly failing to get a good handle on even that.
But an electron is conceived as a point. It doesn't have any length or height. Isn't that the same as the idea of an infinitesimal in math?
Ok.
This is all from proofs by Cauchy that I don't understand. Do you understand it?
Not just Cauchy.
Tell me where I’m wrong if you can.
According to Zvi Rosen, the sum and the limit are not equal (according to Cauchy). They're just as close as we "want" them to be. This is Cauchy's definition of a limit:
Quoting Cauchy
So it's true that the idea of an infinitesimal was removed, but the idea of infinitely small remained, and we added the idea of "as small as we want."
We'll see what the other's said.
To which we might add, as a corollary, The limit is not “almost” the value.
That's incorrect.
Sorry Srap, I can't see how you make this conclusion. 'Within a specified tolerance' does not indicate "the actual value" has been given. It just indicates that the value is within a specified tolerance. In neither case is the value which is being rounded off, actually specified. If "the actual value" was specified the procedure would be unnecessary. So I don't see any significant difference between the two, just two different forms of rounding off.
Quoting frank
I don't think so, electrons are conceived as a cloud of probability, with a variable density.
Quoting frank
An electron could not be infinitely small, because this would reduce the probability of them having any location to practically zero. And that is contrary to what is observed and verified by the cloud of probability conception.
The issue is actually quite complex, because "point particle" is really just a conception of convenience. It's not meant to actually indicate the physical properties of the supposed particle. Rather it's a convenient way to conceptualize interactions. Compare this to the concept of "centre of gravity" for example. This is meant to represent a point which indicates where a body's weight or mass is centred around. But it's just a conception of convenience which helps to model interactions, it doesn't indicate a real point that the body is centring itself around. Nor does the "point particle" concept indicate a real point where an electron is located. They are both conceptions of convenience, intended to facilitate the representation of interactions.
Quoting frank
It's just a matter of definition. Notice what you say, that they are as close as we want them to be. Banno wants them to be equal, and so he defines them that way. But in the context of this discussion such a stipulation is really meaningless.
Quoting Banno
Then why did you say to@jgill, "a more intricate form of 'rounding off'"? That really looks like "rounding off" to me. The point being, that applying a limit to that which is limitless (infinite), is nothing other than a form of rounding off. it's really no different from saying that pi is 3.14, or that it is 3.14159, or however you want to round it off. You are apply a limit to what is limitless, and that is a form of rounding off.
Ok. Details?
Simple example of a limit with an exact value
Consider the sequence
[math]a_n = \frac{1}{n}[/math]
Claim
[math]\lim_{n \to \infty} \frac{1}{n} = 0[/math]
Proof (?–N)
Let [math]\varepsilon > 0[/math] be arbitrary.
Choose [math]N > \frac{1}{\varepsilon}[/math].
Then for all [math]n \ge N[/math],
[math]\left| \frac{1}{n} - 0 \right| = \frac{1}{n} < \varepsilon[/math]
Since this holds for every positive [math]\varepsilon[/math], the difference between [math]\frac{1}{n}[/math] and [math]0[/math] can be made smaller than any positive real number.
Therefore,
[math]\lim_{n \to \infty} \frac{1}{n} = 0[/math]
Conclusion
There is no “infinitely small but non-zero” remainder. In ?, being smaller than every positive real number forces equality with zero.
With the help of ChatGPT. Let me know if it's wrong. Looks OK to me.
Because he was looking at Numerical Analysis not Real Analysis.
It was a short post, making a single point, which answers exactly this question.
Quoting frank
It's also an answer to this, I think.
Yea, you're wrong.
Quoting Srap Tasmaner
How so?
It is. An electron is a point particle.
This is a counter instance to your insistence. The "=" is not an approximation.
So if you would keep your credibility, show your working.
Damn keyboard keeps sticking.
But the point I made is that "point particle" is a conception of convenience, designed for the purpose of representing interactions. It does not represent how the electron is actually conceived as existing. The electron is modeled as a "point particle", but it does not exist that way, the probability cloud is a better representation (though still very inadequate) of how electrons exist.
You're right that per Cauchy, the sum of the series is the limit. However, the devil is in the details. The sum and the limit are never equal. see here.
I'm not concerned about credibility or showing that I'm working. :grin:
I actually took a deep dive on this at one point. The electron is, in fact, conceived by scientists as a point. It's startling, but true.
Banno's proofs continue to be a matter of begging the question. Stipulate that the limit is the value, then use that as a premise in proving an instance of this.
Quoting frank
That's half true, because the electron is also conceived as a probability cloud. Hence the wave/particle duality.
Were it says
The limit will be called the sum of the series.
:meh:
You misread.
What is stipulated is what is meant by a limit:
Definition (limit of a sequence)
[math]L[/math] is the limit of the sequence [math](a_n)[/math] iff
for every [math]\varepsilon > 0[/math]
there exists [math]N[/math] such that for all [math]n \ge N[/math],
[math]\lvert a_n - L \rvert < \varepsilon[/math]
If ?x?0, then x=0 is not a stipulation about limits; it is a theorem about the real numbers, derived from the order structure of ?.
The structure of the argument is:
We are nto doing numerical analysis.
Yes. I don't think either of us is interested in arguing about what that actually means. Let's leave it.
The meaning of of this was just given.
[math]L[/math] is the limit of the sequence [math](a_n)[/math] iff
for every [math]\varepsilon > 0[/math]
there exists [math]N[/math] such that for all [math]n \ge N[/math],
[math]\lvert a_n - L \rvert < \varepsilon[/math]
it says: The terms of the sequence can be made as close as you like to L by going far enough out in the sequence.
Importantly, there is no little bit left over because in the real numbers there is no positive number smaller than every positive number. So if the difference between [math]a_n[/math] and L can be made smaller than any positive number you choose, the difference must actually be zero.
But yes, I am getting a bit sick of working on the tags... especially since folk seem to ignore them.
The conclusion "x=0" is not valid without a further stipulation that there can be nothing between the least ? and zero. But we know there is no least ? and there will always be another lesser ? . Therefore x has no place in that number system, and is wrongly inserted as a category mistake. What is x? And how is it allowed to fit into the number line in this way, when it is not itself a number?.
Since what constitutes "the real numbers" is a matter of stipulation, you are wrong to say it's not a stipulation. You have inserted, through a category mistake, something called x which is not a number, but somehow you claim that it is equal to a number, zero in this case. That is a stipulation.
That stipulation is what ? is. It is not an extra, and it does not make the argument that there is a limit circular.
It is not a stipulation about limits.
Added: the pedagogic problem - it's not a mathematical problem - is how to dissipate the notion that the limit is "a little bit more" than the sequence? Notice that the limit is set out in terms of the sequence - the limit is provided by the sequence alone! so the limit results form the sequence. But it need not be one of the elements of the sequence. It's not something the sequence reaches toward — it is a property of the sequence itself. The limit isn't something the sequence is trying to get to; it's a concise description of how the sequence behaves. The sequence doesn't "know about" or "aim for" its limit - the limit is simply our label for a pattern in the sequence's terms.
Quoting Banno
Obviously, there is always "a little but more" in terms of how close we can get to the limit. that is implied by your definition of "limit". If we'd like to get closer to the limit, than any previously proposed closeness, we can do that, and get closer to that limit. This is what your definition indicates. Therefore, to "dissipate the notion" that there is always more, would be a big mistake, contrary to the definition. Why would you aspire to do this?.
Quoting Banno
So here is where your mistake lies. The limit is the condition for the sequence, the sequence is derived from it, as a formula, a repetition of "half the value between this point and the specified limit". Therefore your mistake is in saying "the limit results form the sequence". The limit is necessarily specified prior to producing the sequence. Then, the sequence is produced from the way that "limit" was defined. We can always get closer to the limit, if that is what we want to do.
Because the limit is prior to the sequence it is not "a property of the sequence itself". The limit preexists the sequence as a necessary condition for it. So if one is to be said to be a property of the other, the sequence is a property of the limit. By this mistake, what you say which follows, is all wrong.
Quoting Banno
The sequence is designed, and produced from the limit. Therefore knowing the limit, and aiming for it, in this way of getting ever closer to it, is an essential aspect of the sequence. the sequence is derived from the nature of "the limit". And this is clear from the way you define "limit". We know "some number", and we also know that there is no limit to how close we can get (we can get as close as we like) to it. The sequence is derived from the specified number.
You are making this sound more esoteric than it is. This is freshman calculus that has been studied by generations upon generations of students. In our class you were expected to understand the theorems and their proofs and to be able to sketch some of them from memory. (Of course, most of this has been thoroughly forgotten decades ago...)
Mathematics as an academic discipline is more like science and less like philosophy, in that the focus is not so much on authors and their original texts. The formulations and proofs that you find in modern textbooks are often not the same as those given by their original discoverers, even when they bear the names of Cauchy, d'Alembert, Weierstrass, etc., because clearer, simpler, more robust or more general versions have been developed. The history of mathematics is a worthy subject in itself, but that is not the topic here.
Being obvious to Meta is not a proof.
Always keep in mind that Meta argues that there are no numbers between 1 and 3.
Quoting Metaphysician Undercover
This is exactly arse about. The limit is a result of the sequence. Those who care to look can see exactly that in the proofs offered earlier.
Cauchy, Weierstrass, and Riemann saved calculus from mounting criticism that it doesn't make sense. So, that did become the topic. We've finished talking about it now, though. :grin:
Take a look at the quoted sequence:
Quoting Banno
Quite clearly, the limit must be assumed prior to taking half of it.
One might set the limit to one then look for a sequence, but of course there is more than one such sequence... quite a few more.
Your posts make less and less sense as we proceed.
Absolutely. Never to come up again. :grin:
My first calculus teacher was awesome. He told us this story about when he was young and he dropped some mercury on the floor. They tried to sweep it up with a broom and it turned into a "blue bloody million" little balls of mercury.
So...?
I read about it, yes. Cauchy's original solution was eventually rejected in favor of Weierstrass's solution. As has been mentioned, this is history that usually isn't taught. I know this irritates you, but what's most interesting to me is the way people defend it when they don't actually understand it.
Would it be better to attack it without understanding it?
Your denial never ceases to amaze me.
No. Can you explain the difference between point-wise and uniform convergence?
Oh dear. It seems you can't.
As it stands, you're just being a bit of an arse hole, not wanting to address the content here but to play with personalities instead.
:yawn:
Notice that @jgill, our resident mathematician, shows only passing interest here. Maths doesn't much care, and part of getting the conceptual work right might well be explaining why it doesn't matter. Nothing essential to his mathematical work turns on the choice.
At the core the difference might be seen as between an approach the closes of mathematical possibilities by saying "you can't do that" and an approach the encourages trying stuff out. One stance says: only methods that fit a preferred ontology count as legitimate; the other says try it and see whether it can be made rigorous...
Which in turn comes back to two different ways of doing philosophy.
Why? As in, where does it fit?
Point-wise convergence is considered to be a weak explanation for how a series converges in a way that allows us to say the limit is the sum.
Uniform convergence is considered to be the stronger explanation.
It's become the custom to express the two ideas in mathematical terminology, which doesn't do much for me. I need a "verbal" explanation. It appears you can't get that without whole biographies of everyone involved.
Gee, I don't know, frank. Isn't that mostly what people do here, no matter what the topic? Or: isn't that the claim of their opponents, should there be an actual debate? @Banno claims not to be a platonist, and @Metaphysician Undercover claims he is anyway—that Banno either doesn't understand his own position or that he doesn't actually hold the position he thinks he does.
And so far as that goes, this is par for the course among real philosophers, not just amateurs like us.
Much like @SophistiCat, there was a time in my life when I could have demonstrated Dedekind cuts for you and proved the Mean Value Theorem on demand. Nowadays, no. Much of the little knowledge of mathematics I once had is gone, along with my undergraduate expertise, but my appreciation of mathematics, the love of mathematics I've had since I was a kid, that remains. Sometimes I like these math threads because it's a chance for me to brush up, blow away some of the cobwebs, and it's a chance to look at math.
I was probably never all that good at math, much as I loved it, but even though I no longer have at my fingertips even the fingertips of the body of mathematical knowledge, I have never stopped looking at the world mathematically. So I enjoy these chances to exercise my math muscles a bit more directly than usual, and I take deep offense at @Metaphysician Undercover's repeated dismissal of mathematics as a tissue of lies, half-truths, and obfuscations.
Yes, we don't always understand everything we're talking about. What else is new? But it's a challenge. I like trying to understand things, and the best way I know of determining whether I do is trying to explain it myself. If I can't, I have some work to do. What else is new? I always have work to do.
Too many participants in too many discussions here evince no such desire to understand. I can take it on faith that they're participating in good faith, but I could not prove from their posts that they are not simply trolling. Maybe some people think the same of me, but I hope not, and if I thought so it would bother me, and I'd rethink how I write. (This is not hypothetical. I have had an analogous experience on the forum.)
By the way, if there's something mathematical you want to know and wikipedia doesn't work for you—some of its mathematics articles are not exactly for the general reader, in my experience—and you can't find another website with a nice explanation, you don't want TPF, you want Stack Exchange. There will be material there that's over your head, sure, but there are also people that know what they're talking about and put a surprising amount of effort into explaining it.
and here we are dealing with real analysis and uniform convergence, so this is stuff is peripheral..?
I wasn't being critical. Defending an idea without understanding it is a sign of a conservative spirit. Nothing wrong with that.
Ok. We can drop it.
Failure to commit. Again.
What? I thought you were telling me to drop it. :lol:
Yes! What I'm finding interesting here are the links to set theory and first order logic, but it's a strain to recall the little undergrad calculus I did study.
I've tried to present my working as explicitly as possible - and ChatGPT is invaluable here, for both checking arguments and formatting Mathjax. I'd have hoped that if there were real objections, the objector would at least take the trouble to set them out formally.
You're talking about dogma, I get that, but I think you're missing another possibility.
The other possibility is the sort of thing suggested by Mercier and Sperber in The Enigma of Reason. If you think of reason not primarily as a system a solitary individual would use to deduce one truth from another, that sort of thing, but instead as a tool for critiquing the views of others and supporting your own view against objections raised by others—if, in short, you see it primarily in its social function, then the sort of thing we do around here makes a little more sense.
It's very late in the day, of course, and some people, the sort of people who have devoted some time to systematic thought (logic, mathematics, law, and so on), have been able to internalize the process, and we think of the usage we see there as the norm.
But in its origin, the important thing is the process of communal decision-making and communal understanding. Seen in that light, it's no surprise that we are pretty good at spotting the flaws in the ideas of others and not so good at spotting the flaws in our own ideas. And it also makes sense that logic and argument tend toward dichotomy, black and white, true and false, right and wrong.
Why? Because in the group discussion, each individual is not responsible for figuring it all out on their own; they are responsible for bringing a view to the group and advocating for it, and everyone else does the same. You give reasons to support your view not as an explanation for how you came to hold that view—you probably don't really know that—but to build support among others.
If you start with a view that doesn't hold up, you'll discover that as others critique it, and you begin to see its weakness. But you won't have that experience if you don't bring your idea forward. In hindsight, it might very well look like you were advocating a position you didn't fully understand, but so what? The whole point was to put it to the test. If it failed, so be it, and you're the better for it.
So, no, I don't think it's always just a matter of defending that old time religion, or a conservative mindset. In some cases, it's just playing your part.
I understand. In my approach to any topic, I need a skeleton, and then I put flesh on it. As it grows, my comprehension grows. So with any philosophical topic, I need to know what the skeleton is: what is the bone of contention? What are the different arguments? What are their strengths and weaknesses? That way I can take something a person says and see where it goes on the skeleton. I'm aware that other people start with the flesh and sometimes never really coordinate it all with a skeleton. I just can't do that. My brain doesn't work that way.
Quoting Banno
I'm really not. I learned calculus in an engineering setting. It never really occurred to me that anyone thought the sum of a convergent series actually equals the limit. At face value, that really makes no sense. It turns out Newton would agree with me. Leibniz would not. So this conflict is at the beginning of this kind of math. Since people have been struggling with it for like 300 years, you should cut me some slack for trying to get it. :razz:
Meh. You seem more interested in the drama than the maths.
I feel your pain, but it is the result of applying a philosophical approach to mathematics. Philosophers love to dwell in the past and compare one master with another, one ancient idea with another. Mathematics is a social agreement and looks forward, not backward. I've taught engineering calculus, although its been quite a while ago, and the notion of the convergence of, say a power series, is fundamental. An analytic function is defined by convergent power series.
I notice numerous posters have the same attitude: that math is somehow immune from philosophical inquiry, and that if it's all built on nonsense, that's ok. I think it's really unfortunate that people got that impression. It's arrogant ignorance.
It depends on whether you are referring to a recursive sequence or to a choice-sequence.
A recursive-sequence is an algorithm for generating a sequence prefix of any finite length, where a limit refers to a convergence property of the algorithm, as opposed to referring to a property of any prefix that is generated by using the algorithm.
On the other hand, a choice-sequence S is an unfinishable sequence of choices that is both
- Dedekind finite - meaning we don't have an injection N --> S.
- Of unbounded length - meaning we don't have an injection S --> {0,1,2,..n} for any finite n.
Such potentially infinite sequences do not possess a limit unless the choices are made in accordance with an epsilon-delta strategy that obeys the definition of "limit". So in this case, we can speak of approaching a limit, because Eloise and Abelard are endlessly cooperating to produce a strategy for continuing a live sequence that literally approaches their desired limit, as opposed to the previous case of Eloise having a one-move winning-strategy when competing against Abelard for proving a convergence property of a dead algorithm.
Unfortunately, ZFC grounded classical mathematics cannot formally recognize potentially infinite (live) sequences due to the axiom of Choice that "finishes" them. Hence there is a clash between common-sense mathematical intuition (i.e. intuitionism) on the one hand, that correctly thinks of infinite sequences as referring to either unfinishable processes or algorithms, versus the formal straight-jacket imposed by the timeless world of ZFC, that cannot express the notion of a live process approaching a limit.
By default, classical mathematics is implied when talking about calculus, and even though ZFC isn't explictly assumed in textbook discussions of calculus, the logic they appeal to when discussing logical concepts such as limits, is classical in which calculus proofs are inductive proofs, which aren't applicable when reasoning about choice sequences, whose proof-theory is coinductive.
(I've never read a textbook definition of a limit as a two-player game - but they nevertheless informally appeal to such games when encouraging students to rote learn - a short term pedegogical payoff leading to long-term confusion after the students forget the game-theoretic reasoning behind the proofs)
I don't understand this feeling of offense. This is philosophy, and what we do is critical thinking, and therefore criticize. What I don't get, is that many people think it's acceptable, even warranted and expected, that we criticize metaphysical principles, yet some of the same people believe it's for some reason unacceptable, and offensive to criticize mathematical principles. Where is the consistency in this type of attitude?
What I apprehend here is that some people take mathematics as a sort of religion. So in the same way that some people get seriously offended when their "God" is criticized, some others get seriously offended when their "mathematics" is criticized.
Quoting sime
This is what @Banno seems to be in denial of. The intent behind creating the infinite sequence, is to create an infinite sequence. This implies that the so-called "limit", as defined by Banno, is prior to the sequence, as a requirement for the creation of the sequence.
On the other hand, we could look at the infinite extension of pi, as an unintentional infinite sequence. Notice, that now there is no "limit". This exposes the nature of "the limit", it is a concept which serves the purpose of creating an infinite sequence. When an infinite sequence is created unintentionally, there is no "limit".
This leads to a question about the intentionality of the infinity which is the natural numbers. If this is an unintentional infinite sequence, we ought to assume that there is no limit. But if it is intentional, then there ought to be some sort of limit, as the source of its creation.
Infinity is a word, that presents the concept of "indefinite continuation" in terms of a beings and unity. Just like we think of the numbers 1 and 2 in terms of being and unity, these things in themselves... so obviously that there is infinitely many things in themselves between these two things in themselves... as grammatical objects these things in themselves are seen as limits. People will see "infinity" as the thing in itself (only 1 infinity), or they will see that infinite meta regress between two things in themselves (Zeno's paradox [infinite infinities]).
Infinity isn't a known truth in terms of indefinite continuation in reality.
It's only possible in meta.
Care to remedy the confusion, per the unspoken goal and purpose of most philosophies, or merely take pot-shots from a place whose elevation and understanding I'd frankly question. :wink:
Was editing as you were typing.
But there ya go.
Grammar Psychology trick fuckin yall...
Quoting Metaphysician Undercover
Yes, I attach value to mathematics, but that's like saying I attach value to logic or to language or, you know, to thinking. The basis of mathematics is woven into the way we think, and mathematics itself is primarily a matter of doing that more systematically, more self-consciously, more carefully, more reflectively. The way many on this forum say you can't escape philosophy or metaphysics, I believe you can't escape mathematics, or at least that primordial mathematics of apprehending structure and relation.
When you say you are critiquing mathematical principles, here's what I imagine: you open your math book to page 1; there's a definition there, maybe it strikes you as questionable in some way; you announce that mathematics is built on a faulty foundation and close the book. "It's all rubbish!" You never make it past what you describe as the "principles" which you reject.
So, on the one hand, I think you're simply making a mistake to think that the definition you read on page 1 is the foundation of anything. We are the foundation of mathematics. The definitions and all that, they come later. And, on the other hand, even if mathematics did have the structure you think it does, so that attacking some definition did amount to attacking the entire edifice of mathematics in one blow, I would still disapprove of your failure to engage in the material past page 1. It's childish. Maybe what the adults are doing is foolish, but the evidence for that is not a child, who doesn't understand what they're doing, announcing that it's "dumb."
Recently, one of my supervisors was explaining something to a bunch of us, and she insisted that what she was talking about was true "not theoretically, but mathematically." Put that in your pipe and smoke it.
OK, so you believe that mathematics is very much comparable to metaphysics, as I suggested. Do you also believe that to maintain consistency, if a philosopher believes that there is a need to be critical of metaphysical principles, that same philosopher ought to also believe that there is a need to be critical of mathematical principles?
Quoting Srap Tasmaner
Your imagination misleads you then.
But this does not invalidate ZFC nor the axiom of choice, nor need we conclude that a limit is something the sequence approaches dynamically rather than a property of the sequence as a completed object.
And the larger point: At issue is whether there is one basic ontology for mathematics. Sime is seeking to replace one ontology with another, to insist that we should think of infinite sequences as processes or algorithms, not completed totalities.
This in contrast to the Wittgensteinian approach, ontological questions dissolve into grammar and use.
What has not been shown is that something goes wrong, concretely, in classical practice if sequences are treated as completed totalities.
Not at all. But philosophically examining mathematics requires knowing something of the subject. Otherwise it becomes a babble of word definitions. Philosophy of mathematics as an academic subject is certainly alive and well, practiced by those familiar with foundations and at least something of the branches of math.
That is not to say philosophical discussions of math here on TPF is inappropriate, but merely speculative and more concerned with how words are interpreted. That's fine. Actually, I am curious about "choice sequences" - an example perhaps?
Indeed. Neither has it been shown that something goes wrong in practice if we treat a convergent series as unequal to the limit by an infinitesimal amount.
Quoting jgill
:up:
Quoting SEP
Quoting SEP
Can you set this out clearly, so we can see what you are claiming?
Sure. Prior to the 19th Century, a convergent series would have been treated as if it reaches the limit, though it would have been ok to say it's actually just approaching it. In the 19th Century, they decided that it doesn't just approach it, it actually gets there because the function is continuous. This doesn't really make a lot of sense to me, but I haven't finished reading about it.
I'll tell you a story to illustrate how it used to be. A student was studying electronics and was confused to find that on a test, the resistance across two points was specified as infinite. The student was asked to state what the voltage across this span would be.
The student tried to apply Ohm's law, voltage = current x resistance. So the voltage would be zero (the current) times infinity (the resistance). Except, looking again, that would mean that the voltage divided by zero = infinity. Which makes no sense.
The student went to the professor after the test and asked what had gone wrong with Ohm's Law, and he was told: "Oh, the resistance isn't really infinite. It's infinite for all practical purposes. It's just really big. We multiply the really big resistance by the really small current, and we get 12. We know it's 12 because the power source is a 12V battery."
The student walked away re-committed to paying attention to practical purposes. If you get too entranced by the philosophy, you'll realize there's no way anything is actually the way we say it is.
I agree. Wittgenstein understood set theory is platonism, and rejected it as an inadequate representation of thinking. Thinking is the private property of subjects and is therefore inherently subjective. Platonism presents us with the products of thinking as something independent from the act of thinking, these are what we call "thoughts".
But this neglects a very important feature of thinking which is communication. We present our thoughts to each other through communication. When we allow that communication must be represented as a necessary aspect of "thoughts", then the true "objects" produced by thinking are the spoken and written symbols (leading toward nominalism), rather than some ideas which are called "thoughts".
The issue is that when we accept the reality, that communication is a necessary aspect of what is commonly called "thought", then it becomes very clear that the other representation of "thought" as some sort of ideas which are produced by thinking, has no grounding accept in platonism. The notion that thinking produces some sort of objects, ideas, is a misrepresentation, because what is actually produced is a system of symbols. And the existence of those supposed objects (ideas) have no grounding accept in platonism. So platonism is a false representation because it does not account for the role of symbols in the act of thinking.
Banno clearly takes the platonist perspective, which ignores the role of symbols, and we can see this from the following.
Quoting Banno
We need to consider the role of symbols in representation, to understand the thinking which is being represented in these situations. Consider the following two ways to represent the natural numbers, "1, 2, 3, ...", and "N". Would you agree that these two symbolizations each signify something different? The latter represents a complete object which we know as 'the natural numbers". The former represents an endless sequence, which by that understanding, could never be complete.
With respect to those two distinct ways of representing "the natural numbers", would you agree that it is possible, even acceptable, and conventional, to represent what we know as "the natural numbers", in two contradictory ways? One symbolization means something, the other means something else, and the two contradict each other. This implies that there is two contradictory ways to understand what "the natural numbers" means, depending on the symbolization employed in usage.
@Srap Tasmaner.
This proposition, that what is meant by "the natural numbers" has contradictory meaning depending on the application, ought not be taken as an offence. You ought to accept it as a proposed description of the reality of mathematics, and judge honestly whether it is a true description or not.
And, the idea that an "object" within a highly specialized field of study like mathematics has contradictory definitions ought not be surprising to you. Take a look for example at the difference between rest mass, and relativistic mass in physics for example. The concept of "mass" has contradictory meaning depending on the application. This is just a description of the reality of human knowledge.
Yeah, because the student doesn't understand basic math. If resistance is infinite then you can't tell what voltage is being applied - unless, of course, you have another piece of information available, such as a place on the diagram where it clearly tells you what it is!
Anyway, I don't know what point this wooly analogy is supposed to illustrate, other than wooly thinking. Which, I suppose, is appropriate when it follows this:
Quoting frank
It sounds like you're saying that zero times infinity equals 12. :lol:
That is correct.
Now some students see the ?-? definition and think: "For any ? > 0, we can get within ? of L. So we can get arbitrarily close to L. But we never actually reach L. Therefore the limit only approximates L"
Here the student misses the quantification: For All ? > 0, the value is within ? of L.
This isn't saying we can get close. It's saying we can get closer than any specified distance. The universal quantifier ?? is doing the heavy lifting. And bear in mind that ?x F(x) is ~?x ~f(x). This is an existential statement: there is a limit; it has a specific value, and is not an approximation.
This is a pedagogic point, not a mathematical one - and one that I learn is brushed over in the schools of engineering, perhaps because their use of rounding is so routine.
And to that history we can add the return of infinitesimals, this time with a firm foundation, in the development of non-standard analysis and hyperreal numbers, *?, whcih includes positive numbers smaller than any standard real, but not zero.
For two centuries, students were told: "Leibniz and Euler were being sloppy. Infinitesimals don't exist. Here's the rigorous way (?-?)." Then Robinson showed "Actually, infinitesimals exist just fine. The old guys were on to something."
[math]0.\dot{9}[/math] really does equal 1.
What nonsense. Platonism treats mathematical propositions as descriptions of independently existing objects; psychologism treats them as reports of mental acts. Both misunderstand mathematics, which consists in public techniques governed by rules.
Really? 0.999... = 1 ?
Ask ChatGPT about the popularity of NSA. It is on target.
Quoting Metaphysician Undercover
Depends on whether the first symbolism is time dependent. Does counting actually require temporal steps. Can you think of 1,2,3 as instantaneous? Just speculating.
:smile:
And so maths is a game that never ends...?
So far as I can see, [math]0.\dot{9}[/math] is not an infinitesimal, but a real.
And again we must avoid mixing up a completed infinite definition with a process imagined as still going on.
I'm tempted to use a constitutive definition here, that 0.999... counts as 1.
I don't think that leads to any contradictions, and cleans things up nicely. But...?
"1, 2, 3..." isn't rigorous, of course - there are many different ways to continue the sequence. ? is rigorous - well, at least more rigourous. So there is a sense in which they are the same only if the sequence is continued in a certain way... the rule is shown in the doing, as Wittgenstein put it.
Yet it is also stated in the definition of ?
However, this is not time-dependent. We understand what it would be to continue the sequence correctly or incorrectly, without doing so.
[math]\dot{9}[/math] = 999999...
I'll go back and edit.
Edit: Ah - it's an English/USA thing? - do you use [math]\overline{9}[/math] ? Interesting. Down another rabbit hole. This is the sort of of thing I was taught, and taught to others: Recurring Decimals.
What babble, but entertaining. :cool:
Wittgenstein would agree with this view, and it's why he rejected set theory. I posted a couple of quotes above that show that.
Platonism treats numbers as independently existing; psychologism treats them as things in the mind; Wittgenstein showed how they are a public, social practice.
W. didn't reject set theory What he rejected was both the platonic and psychological interpretations of set theory, together with the false antipathy that thinks we must choose one or the other.
According to the SEP he was a finitist. You're imagining that there is some finitist approach to set theory. There isn't. Look it up. :brow:
Well, again, that needs some finesse:
Quoting Stanford
This is well worth working through, as well as was he right?
My contention - and I haven't put it together into a PhD yet, so it is incomplete - is that he lacked, or missed, the mechanism that allows us to move from a definition to a quantification, the "counts as" of the constitutive definition.
That's the direction taken by Austin, and then Searle, and a large part of why their work is worth considering alongside that finitism. We bring things into existence by with we do with words, in a way that Wittgenstein might not have recognised.
If you keep reading, the SEP explains that the arguments that he wasn't a finitist are weak.
Quoting Banno
He would say there's no fact of the matter regarding who is right. As you mentioned before, there is no change in practice if we accept or reject finitism.
My contention - and I haven't put it together into a PhD yet, so it is incomplete - is that he lacked, or missed, the mechanism that allows us to move from a rule to a quantification, the "counts as" of the constitutive definition.
That's the direction taken by Austin, and then Searle, and a large part of why their work is worth considering alongside that finitism. We bring things into existence by with we do with words, in a way that Wittgenstein might not have recognised.
It's a rejection of set theory. We wouldn't even talk about the extension of the real numbers. There is no extension.
Well, finitism doesn't automatically reject set theory. Arguing in terms of 'isms' will not get us as far as setting out the detail. some might see ZFC or other set theories from a finitist perspective, treating infinite sets as symbolic devices or potentialities, without committing to their actual existence. Finitism rejects the Platonist reading of infinite sets, but I think I've shown that there is at least one alternative.
It kind of looks like your alternative involves people walking into a fictional world and pretending it's all real, drawing conclusions based on it's reality, when they know good and well it's all a lie.
Is that how you see math?
That's were you live.
Don't bait me into giving my sociological report on this thread. :cool:
Ok. I'm going fishing.
I'm actually Socrates. I forgot to tell you.
Everyone here uses that excuse.
That's insignificant drivel. We could say it about any discipline, they all consist of techniques governed by rules. That's education, learning the rules. The issue here however, is what do the rules say. If the rule says that "the natural numbers" refers to a completed object, that's platonism. If the rule says that "the natural numbers" refers to a count which can never be completed, then this refers to a mental act. The problem is that we cannot have both rules in the same system without contradiction within the system.
Quoting jgill
This is the issue of platonism which Banno intentional avoids. The only way to believe that "N" could refer to a non-temporal, eternal object, is platonism.
Better, education is learning to use the rules. And the issue is, what can we do with the rules.
Opening up, instead of closing off.
If the rules of a single system contradict each other, as in the example, then "learning to use the rules" has a nuanced meaning, which includes choosing which of the opposing rules best suits one's purpose. Providing for an individual to choose from contradictory rules, according to one's purpose, allows subjectivity to contaminate the discipline which is supposed to provide for objective knowledge.
Which system? What contradiction?
Go back and finish reading my post, instead of just replying to the second sentence.
The proffered alternative is that mathematical statements are true, and we can talk about mathematical objects existing, but this doesn't require positing some separate realm outside space and time where numbers "live." Instead, mathematical language works the way it does - we can truly say "there is a prime number between 7 and 11" - without needing to tell some grand metaphysical story about what makes this true. The truth of mathematical statements is connected to their role in our practices, proofs, and language games rather than correspondence to abstract objects in a Platonic heaven.
This view preserves mathematical realism (mathematical statements have objective truth values) while avoiding the metaphysical commitments of Platonism (no need for causally inert, spatiotemporally transcendent entities).
This is what we call trying to have our cake and eat it too.
Quine Putman Indispensability Argument
If you were willing to set this out as an argument, rather than just wave at it, we might have an interesting discussion.
Failure to...
I don't really have the burden of proof here.
I'll give you time to read the article.
Just to be clear, the indispensability argument gives us reason to commit to the existence of mathematical entities. The proffered account does just that.
So, where's the issue?
So in what specific ways are you different from a platonist?
:brow:
Quoting Banno
Platonism is not just "numbers exist", as Meta supposes.
Why are you changing the topic back away from indispensability...?
So let's take Wittgenstein's objection to set theory.
He says we can talk about what goes on in the first 10,000 decimal places of pi, but it makes no sense to talk about the full extension.
If you claim the full extension exists in space and time, where? and when? It's not part of any social practice, so where is it?
Where?
“The decimal expansion of ? is not a completed object. It is an instruction for producing digits.” RFM I §32
“It is not as if all the digits were already there and we merely hadn’t yet discovered them.” RFM I §35
Wittgenstein is certainly not saying that talk of the value of ? does not make sense. It does make sense to talk of the value of ?. We do so all over mathematics. Consider: which digit are we not able in principle to determine? There is no digit that is in principle undeterminable; but there is also no completed totality of digits waiting to be surveyed.
The response is not to reify the procedure that produces each digit; yet ? is a quantified value within mathematics. It figures under quantifiers, enters inequalities, is bounded, approximated, compared, integrated over, etc. None of that is in dispute, and none of it commits us to Platonism. ? is quantified intensionally, via its defining rules and inferential role — not extensionally, as a completed set of digits.
? is not 3.1415926... but it is the ratio of the circumference of a circle to its diameter.
How is your catch of the day? Indispensability not such good bait?
Quoting Banno
Compare your interpretation of quus. There are multiple ways for us to continue the sequence 3.1415926... but only one is ?. This is were Kripke starts to slip.
Quus: scepticism arises if meaning is tied to finite behaviour alone.
?: determinacy is secured by publicly available rules and standards.
Do you?
Well then, tell me. Say something. Commit.
You said your constructivism was compatible with realism, which would imply that the reality of numbers is a byproduct of social practices. Social practices are objective, and numbers are an aspect of them, so in that sense numbers are objective.
I handed you the problem with that theory, which is that a constructivist is forgetting about the existence of things like the extension of decimal pi.
You would notice that Quine struggled with the same issue and reluctantly agreed with platonism based on the indispensability argument. And if you think about it, Frege, Godel, Quine, and Putnam had time to sit around pondering it full time. You can't really half-ass your way to rejecting their arguments. :grin:
Quoting frank
This?
Quoting Banno
Tell me what you think realism is - how you are here using it... Ontological realism (Platonism), Semantic realism, Quantificational or something else/combined? I've been pretty explicit that the 'reality' of numbers is little more than our ability to quantify over them.
...?
Banno, the assumption that mathematical objects exist requires justification or else you're just talking through your hat. When anyone tries to justify their existence, Platonism is exposed in that attempt.
Quoting Banno
If your practise is to start with the premise that numerals refer to abstract objects, then the truth of this premise requires a platonic realm where these abstract objects exist. Otherwise any logic which follows is unsound, based in a false premise.
Here is the problem. For convenience sake, and common vernacular, we talk about numbers as if they are objects, and this in principle has no effect on mathematics, as mathematics is used. There is a clear separation between the talk about mathematics, people talking about numbers as objects, etc., and how the mathematicians are actually using the language of mathematics.
Describing mathematics in that way is just done to facilitate talk about mathematics. The talk about mathematics is in that way false, but it's a falsity of convenience, it facilitates our talk about mathematics. However, if the assumption that numbers are abstract objects makes its way into the axioms of mathematics (set theory), and this assumption is false, then we have a false premise within that logical system.
Quoting Banno
If it is the case, that within the axioms of mathematics, abstract objects are assumed, then "this view" which you present is a false description of mathematics. Clearly, set theory assumes within its axioms, abstract mathematical objects. Therefore the "objective truth" of mathematical set theory requires platonism.
You want to have it both ways (your cake and eat it, as frank says). You say that we can talk about numbers as abstract mathematical objects, though we know they really are not, and when we do mathematics the objective truth of mathematics is not dependent on this. That is fine in principle, if it is true. However, the truth about mathematics is that set theory assumes the existence of platonist objects, and the logical system is dependent on this assumption. This means that when we do mathematics using set theory, "abstract mathematical objects" is assumed, and the objective truth of mathematics is dependent on the "abstract mathematical objects".
So it is not just a matter of talking about numbers as mathematical objects, it is a matter of premising that numbers are platonic objects, and constructing a structure of mathematical logic with this premise as the foundation. That is set theory
Therefore, this talk about numbers as abstract objects, which we might recognize as false, yet still use, for simplicity sake, has been allowed to infiltrate and contaminate the system itself. We say that we recognize this assumption as not really a truth, but do we recognize the consequences? A vast logical structure, set theory, is based on what we recognize as a false assumption.
Quoting Banno
Platonism is "numbers are objects". "Object" implies existing. When you propose that "X" stands for an object, or "2" stands for an object, the existence of that object must be justified. That's what Wittgenstein showed with the private language analogy. One can point to a chair, and say that is the object I'm talking about. But we can't point to a number this way. If I say that there is an object which is a number, this object must be independent from my mind, for its existence to be publicly justifiable, and that is platonism.
Otherwise the beetle in the box analogy applies. I have an object in my mind which I call "2", and you do too. We call them the same name, maybe even describe them in a very similar way, but your object is not the same as mine. therefore we do not have a proper "object" referred to with "2". The only way to justify 2 as an independent object is to assign platonic existence to it.
Quoting Banno
But you do not apply this principle infinite sequences. You do not say that each of these "is not a completed object. It is an instruction for producing digits". You insist on the very opposite, that these are completed objects That requires platonism to justify.
What do you take Quine to have said about ontological commitment with regard to mathematical entities? It'd be helpful to understand how you think it differs from the view I expressed, which makes use of his "To be is to be the value of a bound variable."
Following Quine,
Unfortunately, you do not agree, having said:
Quoting Metaphysician Undercover
Quine's approach has a distinct advantage over your own, in that it allows us to do basic arithmetic.
Nothing in the above commits us to numbers existing independently, in the way of chairs or mountains. Nothing commits us to a hard platonic world of floating numbers. It is open for mathematical entities to be more akin to property, money or countries, a convenient way of talking about how things are.
Quine's approach does not commit us to Platonism in any robust or traditional sense.
What you post shows is your failure to follow the argument.
Would it be better to start a thread on ontological commitment? If so, I can do that.
Dude. If you were committed to the topic, you would at least know what mathematical realism is. :razz:
But you are fishing again. What happened to indispensability?
You don't have the foundation to understand the argument. :confused:
You know, Quine's dictum is a funny thing.
On the one hand, it seems to treat "there exists" as univocal, when discussions like this seem strongly to suggest different sorts of things exist in different ways.
But on the other hand, Quine's dictum does, in its own way, recognize that "is" is "substantive hungry". (Austin's phrase? It's the point that "Alfred is" strikes us as incomplete -- "Alfred is what?") Variables don't float around on their own in classical logic; even when not bound by a quantifier, they only show up governed by predicates.
("What about the domain of discourse? Surely that's just a collection of objects we have assigned names to." But Quine was also inclined to do away with names and use only predicates.)
I think we could follow Quine in saying that, so far as logic is concerned, "there exists" is univocal, while recognizing -- perhaps against his wishes -- that because bound variables are always governed by predicates, there is room for allowing that dogs exist the way dogs exist, numbers the way numbers exist, quarks the way quarks exist, and so on.
(I have complained on several occasions that our logic does not distinguish between predicates and sortals, and this looks like another one of those occasions. But we can similarly recognize that truth functions don't care about that distinction, even if sometimes we do.)
So we might think that it moves the “question of existence” back a step, back to asking what it is to be part of the domain. And the domain is a construct; this or that counts as an item within the domain.
Yes, I think that's right.
In a sense, what the formalism of FOL identifies is that being a member of a domain, or not, and satisfying a predicate, or not, are the same operations for all domains and for all predicates.
In that sense, it is a just a further step along the path Aristotle discovered when he noted the structural similarity of classes of arguments, setting aside the specific contents of the premises and conclusions.
The same will be true if one quantifies over physical objects. Doing so is a statement about what ontological commitments are required by the theory.
Sorry, I don't see the relevance. We can do arithmetic without assuming that there are numbers which are positioned in between other numbers. All we need is symbols which represent values. Care to explain what you are trying to say?
While Frank continues to say very little.
But if you want to reject platonism, you can't do semantic realism because platonism is the status quo (among people who know what they're talking about).
You pretty much have to pick one if you're going to avoid contradiction.
We did this already.
https://survey2020.philpeople.org/survey/results/5030?aos=47
That's the data from philosophers of mathematics. 43 respondents. Structuralism was ahead, with 18 agreeing. Platonism is in the alternatives, with 15 respondents.
Not perfect data, but far from a consensus for platonism.
What's the supposed contradiction?
Come on Banno, get with it, and quit your ridiculous straw manning. I'm saying there's no such thing as "numbers".
Unless you are referring to the symbols, what we call numerals, the assumption of "numbers" is blatant platonism, which we have no need for. We can still do orders, when 1, 2, 3, refer to first, second, third, etc..
Notice how this is much more realistic than the platonism you espouse. We get a clear distinction between "2" referring to a quantity, and "2nd" referring to an order position. This is simply a difference in the usage of the symbol. It's much clearer, easier to understand, and a more realistic representation, than the ambiguity of 'the number 2' being somehow both an order position, and a quantity.
Quoting Banno
The problem here has become obvious. there are probably many platonists, like Banno, who either don't realize it, or simply deny it. So response to a survey asking "what are you", would provide very misleading information. Better information would be obtained with some sort of questionnaire.
Perhaps the difficulty is to do with how a model-theoretical account relaters to intuitionist mathematics. On the on hand we have a clear idea of truth as satisfaction, and considerable progress in math. On the other, we have truth as relative to proof. It'd make for a good topic. But not here, with so many clowns.
I don't know who responded to that survey. I don't really care.
I would say yes to the first question. There is clearly room for improvement on mathematical principles, therefore new fields waiting to be discovered. The second question is not quite so straight forward. But I would say, that unless you are one of the individuals who is going to forge those new fields, there is no immediate impact on your daily life.
Seems to be the case.
This was in my YouTube feed, thought you would appreciate it. I guess Mary Tiles was right: some people are starting to rethink actual infinity in favor of potential Infinity.
Interesting, thanks frank.