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Background and motivation

The following is copied from my blog since someone thought it was the clearest statement I had made regarding a problem I recently posed. On their advice, it is a community wiki. I would encourage, however, I discussion somewhere else (meta.MO perhaps?) on what seems to be a serious disconnect between pure mathematicians and physicists (note: I straddle the line but my PhD is in Math). I should add that nowhere on this site does it say that questions are to be restricted to pure mathematics (as opposed to applied mathematics). Some of the greatest discoveries in the history of mathematics came out of the consideration of physical problems. And I can't emphasis enough how important it is that the communities not drift apart (and I see it happening even within my own sub-discipline). On to the question, then.

Background: One particularly contentious question in the foundations of both mathematics and physics is whether mathematics is discovered or invented. Related to this is whether mathematics is the way the world actually is or if it is simply a way in which we can model the world.

This is a particularly difficult question to answer since it is quite clear that there are physically impossible situations that can be spoken about in mathematics.

Example: taking $n$ copies of a quantum channel should start to approach something that can be approximated by unitaries as n approaches infinity, i.e. in the asymptotic limit, $n$ copies of a channel have a unitary representation (roughly speaking and if the theorem is correct). But, from a physical point of view, this is ridiculous since it is literally impossible for an infinite number of channels (or anything else, for that matter) to physically exist. Yet whole branches of mathematics are devoted to working with infinity (infinite sets, for example).

Clearly some mathematical results have been originally thought to be entirely abstract only to much later find some application in the physical world. But it could be argued that concrete, mathematical analyses of infinite things could never, by their very definition, find an application in the physical world.

Main questions

What are the different notions of infinity as used by mathematicians? To what extent does the use of infinity inform our understanding of the relationships between mathematics and the real world? Specifically:

  • To what extent is mathematics invented and/or discovered? Is there variance that correlates with how infinitary the area of mathematics is? How does "infinity" inform this question in other ways?

  • Should the use of infinity within mathematics inform our understanding of the (meta-)physical world? One popular interpretation of the quantum-field-theory formalism is in terms of "many worlds", but there's a lack of consensus on exactly how seriously to take this interpretation. Should the use of infinity be taken as “evidence” for the Many-Worlds Interpretation (MWI), e.g. because MWI includes infinitely many universes, and therefore gives physical meaning to "infinity".

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There are many distinct notions of infinity in mathematics. In your question, you are not distinguishing between them, which makes it hard to make sense of your question, other than than it is based on false assumptions. –  Douglas Zare Feb 5 '10 at 13:40
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I certainly think that this is a good question, but it's on the border of too discussion-y for MO. Which is to say: because of the way MathOverflow is implemented, answers change orders depending on the number of votes, so it is almost impossible for there to be a real discussion in the answers (and comments are intentionally limited in length to prohibit technical discussions there). Personally, I wish MO had a "discussion forum" section. –  Theo Johnson-Freyd Feb 5 '10 at 16:50
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A remark: your question is titled "the problem of infinity", and yet it seems that you actually want to ask the orthogonal question of to what extent mathematics is invented/discovered. (Aside: the "or" in your post is way oversimplifying the situation, and will lead to less-developed answers. Certainly it can be both: eg modern QFT as done by physicists, cf Kaiser 2005 "Drawing Theories Apart".) I think the "infinity" question is more mathematical, and the answers have responded to it. As this is CW, I may edit the post to separate the questions. Feel free to revert. –  Theo Johnson-Freyd Feb 5 '10 at 17:03
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I would like to point out that Neel Krishnaswami took the time to craft an informative answer and you responded with an insult. Please make an effort to use your "adult voice" in the future. –  S. Carnahan Feb 5 '10 at 19:35
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I wouldn't call this a "borderline discussion" question. It's at least two pure discussion questions. While discussion/philosophical questions can be very interesting, MO is designed for questions what have more concrete answers, and as a side effect, is bad for such questions (as Theo suggested). Consider reposting them to reddit.com/r/math, which seems to host lots of mathematical discussion. If you want to get into a discussion about whether MO should have discussions, go to tea.mathoverflow.net/discussion/21/philosophical-questions –  Anton Geraschenko Feb 6 '10 at 3:20
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closed as not constructive by Andrew Stacey, Charles Siegel, Anton Geraschenko Feb 6 '10 at 3:20

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3 Answers

up vote 3 down vote accepted

I agree with Mike that even the example of $\mathbb{R}^3$ shows that talking about infinite mathematical objects can still make finite physical sense. For example, in $\mathbb{R}^3$ we can show that there are only so many regular polyhedra, and then we invent the microscope and find out that these polyhedra occur all over the natural world - in crystals, in the shape of viruses, and so forth - and no other ones do. That's a pretty good indication that $\mathbb{R}^3$ is a good model of at least some parts of the physical world. The way I think this result should be interpreted is that, even if you believe that there are only a finite number of possible locations in the physical universe (or whatever), the "mesh size" of the universe is small enough that we can take a limit and work in the infinitary setting, which boils down to discarding some negligible error terms.

Terence Tao's post on hard and soft analysis may also be of interest. I also want to quote somebody who, on MO, once said something like "do we ever talk about the infinite, or do we only talk about symbols that talk about the infinite?"

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Tao's hard-versus-soft article is very good, and I was going to link to it too. The point is that "the limit as n\to \infty" makes very good sense without any supposition of an infinite: if there's a theorem that says "in the limit, it's unitary", then there's a theorem that says "we can approximate unitarity this well with that many channels." –  Theo Johnson-Freyd Feb 5 '10 at 16:57
    
@QY: Thank you for your answer. I don't agree with it, but it is a "hard" answer to a "soft" question which is what I was looking for. –  Ian Durham Feb 7 '10 at 21:02
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This question does not have a mathematical answer.

However, different approaches to formalizing the concept of infinity have been considered in philosophy of mathematics for a very long time (since Aristotle, at least), and have resulted in quite a bit of interesting mathematics proper. Most people here are likely familiar with the modern Cantorian concept of infinity (which is what undergirds modern set theory), and so I will describe a somewhat different mathematical conception infinity.

So, one way of interpreting constructive mathematics is by means of "realizability interpretations". Here, we take the view that a proposition is true when it is possible to give evidence for its truth, and then inductively for each proposition we give conditions for evidence:

  • $\top$ has the string $()$ for its justifying evidence
  • $A \land B$ is justified when we can give a string $(p, q)$, where $p$ is evidence of $A$ and $q$ is evidence of $B$
  • $\bot$ has no evidence
  • $A \vee B$ is justified when we can give a string $(i, p)$, where $i$ is either 0 or 1. If $i$ is 0, then $p$ is evidence of $A$ and if $i$ is 1 it is evidence of $B$
  • $A \implies B$ is justified by a computer program $c$, if $c$ computes evidence for $B$ as an output whenever it is given evidence for $A$ as an input.
  • $\forall x. A(x)$ is justified by a computer program $c$ for evidence, if $c$ computes evidence for $A(n)$ as an output whenever it is given a numeral $n$ as an input.

Now, note that in the cases of implication and quantification, we ask for a computer program which is total on its inputs. So, the question arises, how can we tell whether or not a given program is a legitimate realizer for a proposition or not? The Halting Theorem ensures that we cannot accept arbitrary programs and decide after-the-fact whether or not they are evidence. So we must circumscribe what we will accept to some class of total functions.

So we can decide that certain patterns of recursive program definition (for example, primitive recursion) are acceptable forms of looping, which we believe will always lead to halting programs. These patterns correspond to induction principles. Proof-theoretically stronger theories correspond to logics whose realizability interpretation allows more generous recursion schemes as realizers. In this setup, large infinite sets correspond to very strong induction principles. So this gives a way of understanding Cantorian infinities without having to posit the actual existence of sets with huge numbers of elements.

This should illustrate that how we read mathematical statements shapes what ontological commitments we make regarding them, and so you can't answer physical questions only through pure mathematics. That is to say: physicists can't get out of doing experiments. But funny readings of mathematics may help you interpret those experiments!

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No, it doesn't have a purely mathematical answer, but it is a legitimate question about the nature of mathematics. Several people who are regular contributors to this site said it would make a good community wiki question. I cannot believe how myopic the users of this site are. See mathoverflow.net/questions/14162/… –  Ian Durham Feb 5 '10 at 14:24
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Don't get angry at someone who attempted to answer your question just because they happened to not have read the discussion about your previous question. –  Zev Chonoles Feb 5 '10 at 14:43
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@Ian: I think you misread Neel's answer. It is obvious to me that Neel thinks your question is a legitimate math philosophy question. Neel is pointing out how realizability suggests a different view of the infinite than Cantorian/Platonic infinity, a view which is more compatible with empiricism. –  François G. Dorais Feb 5 '10 at 14:44
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@Ian: remember that there are many users here. And reasonable disagreements about what do and don't make good questions. –  Theo Johnson-Freyd Feb 5 '10 at 16:52
    
Apparently my view of 'reasonable disagreement' appears to be very different from many people on this site. What I don't understand is a) if this question has been closed, how people can still answer and comment, and b) if 'reasonable disagreement' is allowed, why questions can be closed in the first place. This question was posted after Qiaochu Yuan posted a link to my blog post saying it was a clear explanation of what I was asking with my previous question. I can't believe people don't realize how contradictory this seems. –  Ian Durham Feb 7 '10 at 22:38
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You say, "There are physically impossible things that can be spoken about in mathematics." When mathematics is used to model a physical system, it is always an approximation. So, the fact that the model is not a perfect match for the system (that is, it describes some "physically impossible things") is not a surprise. There's no need to get fancy to show examples: our space is not plain old Euclidean 3 dimensional space, but that model has been very useful. Or even something as basic as possible: using natural numbers to count stuff. Well, it's physically possible that half of one of the things you were counting could burn up, and it turns out the mathematical model you were using is inadequate to describe the system (because you can't talk about half an object).

The fact that some mathematics is abstract but finds applications seems like a different issue. Some number theory is such an example, but I don't think anyone ever denied that integers were highly useful in modeling real systems (and thus, in some sense I guess, "real"); it's just that the questions people were asking about them seemed to not have practical uses.

So I actually think there is less here than meets the eye. It's not really a big deal that we talk about infinite sets of things even though maybe there is no infinite collection of objects in reality, just like it's not really a big deal that we treat our space like it's continuous even though maybe it's actually discrete.

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It is a bigger deal than one would think when taken in the context of something like the Quine-Putnam indispensability argument which suggests that mathematical objects are as 'real' as the scientific theories they help explain. It gets at the heart of whether mathematics is invented or discovered. It seems that quite a few people on this board seem to be of the opinion that math is invented (and there's nothing wrong with that view, but it is wrong to assume the question is not worth debating). –  Ian Durham Feb 7 '10 at 21:00
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