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.

]]>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.

]]>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.

]]>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.

]]>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.

]]>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:

http://www.mathematik.tu-darmstadt.de/~kohlenbach/matilde.ps.gz

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

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