Tuesday, August 12, 2008

Philosophy of Mathematics

It's now 10 years since my interest in mathematics became an interest in the philosophy of mathematics, but I'm not much closer to knowing why mathematicians standardly assume an axiom of infinity (which is described in the second comment below)—not the historical or sociological reasons, but what philosophical (or amateur-scientific) reasons those applying standard mathematics have, for thinking that an axiom of infinity is true.
......That they have found few problems with that axiom in a hundred years, for example, is an answer in a different ball-park—cf. how medieval astronomers found few problems with assuming (what seems clear) that the stars circle (and the planets epicircle) a fixed Earth. Ultimately the interesting question was does the Earth turn? And while our being able to imagine a star in each cubic lightyear of some infinite space is at least a metaphysical reason (e.g. see Benardete's "Infinity: An essay in metaphysics"), mathematicians have naturally been moving away from their previous reliance upon such geometrical intuitions (towards greater rigour).
......Not that standard mathematicians need any very good metaphysical reasons for finding most interesting those models of arithmetic that contain such an axiom (e.g. there are benefits to having a common language, and there was originally the epistemic possibility to explore) but if they don't have any such reasons then physicists and metaphysicians should perhaps not presume that some such axiom is true. And note that the question arises given that numbers are objective (that proof aims at, but is not to be identified with, truth), so that it is hardly an answer to have included Constructive mathematics within academia.


Alrenous said...

Perhaps this disqualifies me from adding anything, but...what exactly is the axiom of infinity?

Enigman said...

Good question; the axiom of infinity states (more or less) that there is a set containing all the successors of the empty set - all the natural numbers, which are obtained from 0 by successively adding 1.
What that axiom is saying depends upon what a set is, upon which set-theoretic axiom system it is part of. But the basic idea is that the elementary properties - as captured (more or less) in the other set-theoretic axioms - of a simply infinite collection (e.g. 1, 2, 3, ...) are those of a finite collection.
I would say that the intuitive idea is that it is mathematically (or logically) possible for infinitely many things - a first, a second, a third and so forth - to coexist, e.g. quasi-spatially (cf. stars in an endless classical space).

The alternative is that such endlessness implies a sort of incompleteness.
The alternative is to think of all such endless sequences as existing only quasi-temporally (assume Presentism for the sake of that analogy), as a mathematical (or structural) consequence of the endlessness of the fundamental generating process (successively adding 1).
That would not mean that we were making the numbers up (which is a Constructivistic notion that is alarming close to the Naturalistic position that follows from evolution), but it would mean that we could only generalise over the natural numbers by regarding them as the indefinitely extensible products of their definite generating process.

The latter can be a Platonistic position...
If there were a God, for example, and were the axiom of infinity false, he would know that it was, and so he would not think it a failure that he did not know all about all the natural numbers.
(A rough analogy for that might be his not regarding it as a failure that he could not tell you where the spectral line between red and orange lies.)
That we intuitively expect such to be a failure (cf. Grim's arguments) could be put down to our being finite, to our confusing (subconsciously) an actually infinite simple infinity with a very big finite number.
(Primitive languages often had a word for one, a word for two, a word for three, and a word for anything bigger than three.)

Alrenous said...

I've often found it useful when people say the following kinds of things to me, so I hope that you'll in turn take it as it is meant.

Your writing style is pretty fractured. I really want you to get to the end of one idea and then add the qualifiers, rather than inserting them all over the place.

Also, I'm not going to bring this up again - it's not what I find to be a critical flaw.

Regardless, though...

I have a question about

"If there were a God, for example, and were the axiom of infinity false, he would know that it was, and so he would not think it a failure that he did not know all about all the natural numbers."

So in my understanding one can say that the numbers are infinitely extensible, but have never been infinitely extended. As such, that God does not know something analogous to the 'last' number is not a failure of omniscience.

How does this relate to what you're saying?

The basic reason as I see it is that a set with an arbitrarily large number of elements does not have different properties than some set with, for instance, three elements. It follows, then, that in the limit of infinite elements, the properties remain constant.

What do you make of that?

Enigman said...

I kindof agree about my writing style, but I think that such things are quite complex, e.g. I don't mind if readers have to slow down and reread the start of a sentence. In philosophy the meaning of a word is often fixed by its context. (And it is also helpful in philosophy to have quanlifiers embedded near what they're qualifying.) Anyway, it is not generally true that properties hold in the limit, as Hilbert's Hotel shows, for example, or indeed, that there is a limit, as shown by the necessarily proper classes that exist if infinite sets do.

So if you are saying that the default position should be that such sets exist, with properties like those of the finite sets, then my reply is that since all sets have the set-theoretic properties (i.e. they satisfy all the axioms), and yet their limit (their proper class) cannot, so the default position should only be that such sets might exist. The assumption that they exist (the axiom of infinity) is probably formally consistent, but we cannot even justify that "probably" fully.

Alrenous said...

In that case, I think you're exactly right.

There is no philosophical reason for the axiom of infinity.

It's empirically valid, but that's all we've got.

Enigman said...

But what makes you claim that the axiom of infinity is empirically valid? (Incidentally I would say that - depending upon what you mean by "valid" there - that could be a philosophical reason.)

I can think of a few possible reasons, but also counters for all of them - e.g. supertasks (e.g. this one that I talked about at Aberdeen last month) claim that it is not empirically valid.
The standard response to supertasks is usually to point out that since the axiom of infinity is true hence we should instead deduce the falsity of the supertask's weakest physical assumption (for mine, that quantum-mechanical probabilities are single-case propensities).
So if that's all we've got (but thirdly why is it?) then it's not so much empirically valid as empirically challenged...

Alrenous said...

Sadly I'm clearly not qualified to understand the analysis of the supertask.

However, the thing about empirical proofs is that they tolerate the odd falsification here or there. If it works, we just assume we haven't thought of something and continue blithely on.

Enigman said...

Oh dear, I was hoping that people like you at least could easily understand that supertask. Would you mention where you got stuck?

Still, there's that difference between finite arithmetic and transfinite arithmetic - the former is hardly falsifiable. And the thing about a (possible) falsification of the latter is that is shows that the latter (possibly) fails to work at such levels of application as we can actually reach.

As Hamming noted, when you're studying equations to see if the device you're building is likely to destroy our ecosphere, or not, you want an answer, and you want to be as sure you can be that the mathematics you're using, to get you from the empirical data to the answer, is not going to introduce any falsities. You also want thinking more logically about a problem to be a better way towards the truth, not a way of increasing the errors (and standard set theory is the foundation of standard logic as well as standard mathematics).

As philosophers put it, we want our mathematics at least to be true in all possible worlds. Philosophers are in the business of understanding what is going on, not just continuing blithely on. After all, by ignoring such (possible) falsifications we would be ignoring opportunities to get ourselves a better theory (and our history clearly indicates that there are likely to be empirical benefits from getting such).

Alrenous said...

The first problem is that I'm bad with terminology. A full explanation is really long, but in short I'm very often familiar with a concept I cannot write down.

In this case, I'm having to guess at the meaning of way too many words to be certain I'm getting the right things out of it. Now, I do seem to be good at guessing, but this one is stretching it.

But actually, having slept on it, it seems to be clearer. We'll see; if my comment/question is ridiculous, it's because I made a bad guess somewhere. :-j.

I guess the first problem I'm going to have is with the tendencies.

For instance, when you say:

Note that Integers may not have numerical probabilities; but if they do then, since they are equally likely, those probabilities must (via T2) be less than any positive real number—e.g. they might all be 0 or some nonzero infinitesimal.

I kind of have to infer what the tendency actually means.

Now, I have no idea what a multiplicavity is. But it seems that one consequence of T2 is that when you multiply an infinite series of probabilities of less than one, you get an infinitesimal or zero, unless there's no sensible way to construct a numerical probability at all.

That's just one example.

The thing about the link seems to be that the probabilities of the integers is inconsistent across observers.

However, it's still not clear to me what a supertask is, let alone a bifurcated supertask.

I'm not sure how this relates to set theory. (I've never studied set theory, though I am interested.) Though certainly I can see that infinite series of probabilities lead to much more interesting places than arbitrarily long but finite series.

Anyway, that's the data. Hopefully you can make your own estimation of what I'm not understanding.

Your remarks about empiricism vs philosophy;

Well, yes, which is why I say of the reasons, 'but that all we've got.' It is a reason but not a good reason.

And your question was not, intially, "Is the axiom of infinity valid?" But "Why do mathematicians assume an axiom of infinity, not including historical reasons?"

The answer to that question is easy; no reason at all.

This comment thread constitutes novel research on the matter at this point.

Do not forget, however, that I certainly don't have anything better to do with my spare time.

Enigman said...

My apologies - I think you're right about the obscurity of my talk. This unpublished paper on the same supertask is a bit less obscure (the words you don't know should be easier to google), although I shall have to rewrite even that paper to make it more accessible (when I get round to it).

Basically, if we interpret quantum mechanics in terms of indeterministic collapses in this world, then the most realistic way to think of such probabilities (as given by the uncollapsed wavefunctions) is as single-case propensities (introduced by Popper), and 'tendencies' is just another word for them.

My supertask shows that such a view of chance is false, if the axiom of infinity is true. A supertask is an endless sequence of tasks, e.g. tossing a coin forever. They were originally used to try to show that the axiom of infinity is false, but more recently they are used with the assumption of standard mathematics to refine physical theories (classical ones usually). The best introduction to them is this entry to SEP.

Multiplicativity of probabilities is just that Prob (A and B) = Prob (A) times Prob (B), when A and B are independent. That this property should be a property of tendencies can be seen by thinking of collapse as the selection of one possibility from a lot of them, as akin to blindly picking a ball from a box of balls. If there are 2 balls, a red one and a blue one, then Prob (red) = 1/2. If you take balls from two such boxes, then Prob (2 reds) = 1/4.

Steve Gimbel said...

The reason is that mathematics becomes either too boring or too cumbersome without it.

Let me be Hilbert here and argue that it's an axiom and we can take or leave it and see what we can and can't prove. Brouwer and the intuitionists (sounds like a name for a really geeky bar band) did their level best to reconstruct as much math as possible without it, but it ended up disallowing potent and exciting areas and making others unnecessarily difficult.

Like all axioms, we could consider a mathematical universe without it or we could consider the mathematical universe with it. With it is more fun (in the technical sense, of course), so we posit it conventionally.

Enigman said...

Hmm... but surely that's not a reason for taking it to be axiomatic for the maths taught in our schools (to future scientists and such), as true so to speak?
It being 'fun' (I think it is) was a reason for mathematicians to go for it a hundred years ago. But I was really hoping (10 years ago, before I knew better:) for better from scientists and philosophers (who pass this buck to the mathematicians, who respond by being formalistic, which is to duck the question)...

Incidentally, the alternative is not Brouwer but a set of axioms based on Realistic intuitions and that lacks the axiom of infinity - ideally it would have an axiom of anti-simple-infinity. I'd guess that something like that is what philosophers who look at proper classes want, although they are hampered by having the axiom of (simple) infinity.
Interestingly, something more constructivistic (more like Brouwer and Wittgenstein) is being slowly forced upon the standard view by its Naturalism, but that's a horrendously interweaved story...

abo said...

Well, the axiom of infinity is a set-theoretic axiom - so I think it misses the point in any case.

I think the proper question is whether the Successor Axiom (that every natural number has a successor) is true.

" Ideally it would have an axiom of anti-simple-infinity." Why ideally? It would seem that the best approach is to be agnostic about the Successor Axiom (or if you prefer, the Axiom of Infinity). Second-order Peano Arithmetic without the Successor Axiom can prove quite a few things.

Enigman said...

Hi abo... I'd've thought that if the successor axiom was false then there'd be some natural number to which one could not add 1 to get the next natural number, which seems absurd. I've talked to a strict finitist about such things, but it still seems impossible to me. Still, it must be a serious proposition, if work has been done on Second-order Peano Arithmetic without the Successor Axiom.

I totally agree with you about being agnostic though. I don't know why standard mathematics is not agnostic about such things. But the alternatives we were being agnostic about would still need investigating, e.g. standard set theory (until it is shown to be inconsistent) and the obvious alternatives.

Standard set theory without the axiom of (simple) infinity would be OK for arithmetic, but would have to come down one way or another on the question of the structural possibility of aleph-null stars within an infinite space, if more advanced mathematics was being covered.

abo said...

Let Sx,y mean "y is the successor of x." And let Nx mean "x is a natural number." Mathematical axioms would be:

1/ S is a function
2/ S is one-to-one
3/ 0 is not in the image of S
4/ Induction, i.e. if phi(0) and (n)(m)(Nn & phi(n) & Sn,m => phi(m)), then (n)(Nn => phi(n)).

The axioms are consistent, since every finite segment (such as {0,1,2,3}) is a model. Indeed - which is a nice feature IMHO - the system can prove its own consistency.

Olga Lednichenko said...

Since this is an axiom: It cant be challenged. Unless of course you change the reference frame itself.

Olga Lednichenko

who said...

what's "the" reference frame though?