Mathematicians Don’t Care About Foundations

9 06 2005

This seems to be a largely accurate statement, though of course the existence of dozens and dozens (ok, maybe even hundreds) of mathematicians actively working in set theory, model theory, recursion theory, and proof theory shows that it’s not totally true. It’s interesting that logic seems to have a slightly higher (though still quite low) status within philosophy than in mathematics. And things like modal logic and intuitionist logic seem to be far more widely known in philosophy than mathematics, even among logicians.

Anyway, an interesting post in a mathematics blog I just discovered on the list of academic blogs at Crooked Timber tries to introduce some more logic to mathematicians. Even more interestingly for me, the commenter sigfpe writes:

Mathematicians don’t care about about foundations because they don’t matter! It’s like using a well encapsulated software library. You don’t care what the inner workings are as long as the library performs as specified. If someone came along and completely rewrote the library, only the users who had been exploting the pathological cases might actually notice anything had changed. Same goes for foundations. Even if someone came along and found a contradiction in ZF, say, it would probably be in some really abstruse area, someone would come along with a ‘patch’, we’d have a new set of axioms and 99.99% of published mathematics would remain unaltered.

Something like this is a position I proposed a few months ago when I wanted to come up with a controversial position to take for a panel discussion with some of my fellow graduate students in logic at Berkeley on new axioms in mathematics. After working it out a little bit more, I actually think I believe something like this, though perhaps at a slightly lower level than this person. That is, the reason that we have any standard axiomatizations of mathematics at all is so that mathematicians don’t have to resolve all their disagreements about the philosophy of mathematics. If the platonist, nominalist, and structuralist can all agree that ZFC is a good set of axioms, then they can all return to being productive mathematicians – but if we didn’t have ZFC (or something like it), then they’d have to convince each other that their methods were valid and didn’t presuppose something about the nature of mathematical entities.

As for mathematicians not caring about the mathematical foundations (set theory, proof theory, recursion theory, model theory), that’s a different question. My account predicts that they should care about this stuff at least slightly more than the ontological and epistemological questions. But since specific results in set theory and proof theory are almost never applicable elsewhere, and only restricted amounts of proof theory and model theory seem to have applications, it seems that mathematicians outside of foundations may be able to apply my strategy at an even higher level and ignore what goes on beneath their axioms. Of course, in this case there isn’t disagreement beneath their shared axioms, unlike at the philosophical level.

Fortunately, model theorists seem to be making contact with algebraic geometers and number theorists these days, so maybe some day mathematicians will start caring about the rest of logic again.

(There’s also a connection to Penelope Maddy’s naturalism here – she suggests that mathematicians really shouldn’t care about the philosophical issues behind the axioms, for mathematical reasons. I suggest that the reasons are slightly more sociological.)




8 responses

9 06 2005
Gustavo Lacerda

Ulrich Kohlenbach applies proof theory to analysis.

From what I understand, this paper is about a technique for generating proofs by meta-reasoning: they prove constructively that there must exist a (not necessarily constructive) proof:

Please notify (I don’t see why you wouldn’t use LJ for this blog).

9 06 2005
Peter McB.

I think most mathematicians are unprincipled pragmatists (which I mean in the nicest possible way), and only care about what is useful to their work. As proof theory, model theory, etc, become useful to whatever branch of math people are working in, then they will become interested in foundations. For instance, Hrushovski’s model-theoretic proof of the Mordell-Lang Conjecture in algebraic geometry I am sure has led alg-geometers to look at model theory. Likewise, category theorists became interested in intuitionism when it was shown that the internal logic of a topos is intuitionistic. And these foundational subjects are of interest to theoretical computer scientists because they are useful there.

9 06 2005

You should enable typekey login on your blog kenny.
First of all I don’t really see why ignoring philosophical issues should have anything to do with agreeing on one set of axioms. Algebraists study both commutitive and non-commutitive groups and their lack of agreement is hardly of philosophical consequence. So why should we need to have exactly one common axiomatization of set theory to avoid philosophical issues.

Suppose we had mathematicians working in both ZFC and NF. This would not force mathematicians to bring in philosophical disagreements about which one is ‘right.’ Rather people would have theorems about groups or sets that held in NF and others that held in ZF but with most theorems in analysis and standard mathematics being unaffected. In fact this is hardly different than the situation we have in mathematics now with the axiom of choice. It is still not fully accepted and some mathematicians will try to avoid or minimize its use. Yet this causes no philosophical confusion when mathematicians who take choice for granted read their work or vice-versa. So long as the axioms which the proof depends on are clearly stated many different foundational theories can be explored at the same time with no philosophical difficulty. Heck, it is my understanding that there are some reasonable arguments that much of standard mathematics is closer to type theoretic reasoning than ZFC (though the theorems proved will be the same).

In short their are many pragmatic external questions which drive mathematics, will I get funding, does my advisor care about this, will this be usefull in physics. One of these pragmatic concerns is that progress is easier if we all prove things from the same axioms inside a given field. However, you have given absolutely no reason to believe their is any philosophical implication or motivation behind the pragmatic choice to all work in the same axiom system.

In fact I think the proof that this is just a pragmatic concern, rather than one motivated by philosophical agreement, is the fact that all the platonistic set theorists accept ZFC. Why do they accept ZFC? Because other set theorists work in it and it produces good mathematics. However, if they were really devoted to explicating the pre-theoretic notion of a set there would be a strong pull towards NF.

In short there is every reason to think what guides these choices is social conveince, mathematical elegance and pragmatic results. Trying to agree over philosophical issues is not a determining factor. I do however think this is the role that the axiomatic method/classical logic itself plays. It is the agreement about what should formally count as a proof which lets us push the philosophical issues aside. Once we agree on that we can take whatever axioms we like and so long as we are clear about things everyone must agree on the proof of theorems from axioms even if they don’t agree about what that means.

9 06 2005

Some nitpicks!

logicnazi: Not many people study commutative groups these days, because we have a complete understanding of them (and have done so for yonks). Noncommutative stuff is where all the action is. This is not to be confused with “commutative algebra”, which is really about rings. Generally, people work with commutative stuff because it is easier (which is not to say that it is necessarily easy :)! At any rate, the choice of working with commutative or noncommutative structures is almost always a technical decision, not a philosophical one (to be contrasted with the choice of foundations).

Kenny: Officially, I suppose that I work in proof theory these days, though I hardly ever worry about foundations! I think a lot of other people working in computational logic feel the same way. The big problem is intra-logical wars…statements such as “oh I stay away from logic X, I am not a philosopher” are prime examples of this.

10 06 2005

Well it really doesn’t matter for my example that one of the fields is not very studied and I highly doubt that we know everything about commutitive groups. Still if you want to nitpick fine then take the question of commutitive vs. non-commutative rings. These are both very important research areas.

In short kenny I don’t think you give us any reason to believe your central claim. Namely that without agreement on axioms we would have philosophical disagreements about the nature of mathematical entities.

11 06 2005

Gustavo: thanks for the link, I’ll check it out!

Peter McB: I think that’s basically what I think about why logic is ignored by most mathematicians, and why this won’t last, at least for model theory, possible recursion theory or proof theory, and (if Harvey Friedman has his way) maybe even set theory.

logicnazi: I think your second comment actually helps me clarify my argument. Once we’ve agreed that we can talk about model-theoretic structures of various sorts (because we’ve accepted ZFC, or even possibly NF or Russellian type theory), then we use different sets of axioms just to pick out just which classes of structures we’re talking about at some time. There were some foundational debates (that I should look into more) between Grothendieck and (I think) Deligne(?) about what the exact definition of a scheme should be, though they had already “proven” some results about them. Settling on the axioms in that case was a result of agreement on just what they were talking about.

However, in the most foundational cases, like sets, and model-theoretic structures, there are philosophical debates that have never been resolved, about whether the objects exist, how we know about them, and so on. But even though these debates haven’t been resolved, mathematicians have managed to agree on some axioms. So as long as they rely on these axioms for their proofs, they don’t have to worry about convincing their colleagues about any source of knowledge of the substance of these things they seem to be talking about.

I agree, it’s very odd that a platonist would agree to ZFC as an explication of the pre-theoretic concept of set, but it really seems like they do. I don’t know their reasoning behind it, but since the platonist agrees with the fictionalist on ZFC for whatever reason, they can co-operate to prove theorems. Just as the Christian Coalition and messianic Jews can co-operate to rebuild the Temple, even though both groups expect the other to be condemned or sent to hell or something.

Jon: I suppose by “foundations” I meant here just “set theory, model theory, recursion theory, and proof theory”, and I use “philosophy of mathematics” for the actual foundational underpinnings justifying the use of these axioms for studying these systems.

11 06 2005
Computational Truth

Axioms and the Axiomatic Method

Thus agreeing on one set of axioms is motivated by purely mathematical and social considerations. Even if everyone was an unabashed platonists we would still choose one axiom system to work in.

11 06 2005

Well I have a long reply now over at my blog (see the trackback). In short I’m willing to agree that the adoption of the axiomatic method avoids philosophical disagreements but I think the choice to agree on any particular axioms (or to agree at all on a set of axioms) is entierly mathematical and has little to nothing to do with philosophy. In particular I think one of the best points of evidence for this is the happy platonistic acceptance of ZFC as describing set. This shows that it is mathematical features which are driving choice of axioms and the philosophical justifications (if any) just sort of drag behind (for mathematicians).

Leave a Reply

Fill in your details below or click an icon to log in: Logo

You are commenting using your account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s

%d bloggers like this: