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p> So Kenny over at Antimeta recently added a post arguing that the reason mathematicians accept a standard set of axioms is to avoid addressing philosophical issues. This position explains Kenny and my misunderstanding at the panel discussion as I took him to be making only the trivial claim that we use a standard set of axioms so all mathematicians are working on the same stuff. Unfortunately, I think the view he is actually advocating, while more interesting, is completely false.

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p> I take it the following passage is the key point in the post:

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.

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p> This passage can be seen as making four sub-claims:

1. If mathematicians didn’t have a common set of axioms they would have to argue philosophy.
2. These arguments would be about the methods used and nature of mathematical entities.
3. These arguments would be grounded in which philosophy of mathematics the participants believed.
4. The Platonist, nominalist and structuralist agree on ZFC.

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p> Amazingly I think all four claims are false. Even worse I think the claims interact destructively. To illustrate both these objections I will argue against each claim in turn but in case the reader doesn’t find some of my replies compelling we will assume the truth of the claim when analyzing the later claims.

1: If mathematicians didn’t have a common set of axioms they would have to argue philosophy.

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p> My first problem with this claim is that I just don’t see any evidence for the claim. Mathematicians show a remarkable resistance to allowing philosophical considerations to interfere or insert itself in real mathematics. Sure old mathematicians may write philosophy of math and certain conceptions of mathematical philosophy might inspire certain mathematical investigations but there is a remarkable ability of mathematicians to avoid philosophy qua mathematicians (my first time! should it be italics?). The only potential exception occurred when antinomies were found in mathematical practice (Russell’s paradox) and multiple competing axiom systems hardly endanger the subject in this fundamental a way.

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p> In fact those disciplines which do have multiple axiomatizations, like geometry, do not seem to suffer from this philosophical argumentation. Hyperbolic, euclidean and the other one all exist in harmony. Sure the existence of non-euclidean geometry caused a lot of philosophical hand wringing but once the axioms for those other geometries had been laid down the mathematician qua mathematician didn’t need to debate philosophy. He simply proved theorems in one of the three systems and I see no reason to believe this wouldn’t happen with foundations. In fact with the small number of people working in type theory this is how it works.

2: These arguments would be about the methods used and nature of mathematical entities.

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p> Even if the presence of competing axiom systems did lead to argumentation I don’t think it would be about the methods or nature of mathematical entities. If we are only considering other systems of axioms the methods of mathematical proof remain exactly the same. It is the type of logic used which dictates methods not the particular choice of axioms. So long as which axioms a result follows from are made clear everyone can recognize the proof as valid. This leaves the question of the nature of mathematical entities.

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p> Yet even now it seems disagreements about which axioms to use don’t have anything much to do with the nature of the objects at issue. When Shelah proves something from diamond plus or SH the individuals involved are not usually disagreeing about the nature of mathematical entities, at least not in a deep sense. They are disagreeing about what the ‘right’ mathematical entities look like not what is the fundamental nature of the entities. In other words these are differences which are settled by mathematical considerations of elegance, power and mathematical necessity.

3: These arguments would be grounded in which philosophy of mathematics the participants believed.

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p> So even supposing that mathematicians would not simply work happily in several foundational schemes I don’t think the disagreement would be about which philosophy of mathematics to which they ascribed. Except for intuitionism all of these philosophies of math could quite easily be applied to any set of axioms. In fact it is hard to see what grounds structuralists or nominalists would have to object to any system of axioms as their attitude towards mathematics seems ambivalent about the actual form of the axioms.

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p> Even the Platonist couldn’t really object to certain axioms systems on philosophy of mathematics grounds. No matter what philosophy you adopt so long as your foundational mathematics is strong enough to prove the completeness theorem you can’t deny that objects satisfying other consistent systems exist in so far as your foundational objects exist. That is different mathematicians can object to an axiom system as being not worthy of study but no philosophy of mathematics can bar a particular axiomatization based on the existence of the underlying entities. Since the different philosophies are mostly silent on what the ‘right’ sort of objects are to study (usually leaving this up to the mathematicians) it is hard to see how disagreement over axioms would boil down to differences in mathematical philosophy.

4: The Platonist, nominalist and structuralist agree on ZFC.

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p> So supposing that disagreements over the correct choice of axiom would involve some philosophical dimension it is hard to see why the choice of ZFC would settle these issues. As I pointed out above it can’t be the case that the axioms are wrong because they fail to describe objects in the correct sense. Even if mainstream mathematicians might think it is silly or useless they nevertheless recognize people doing type theory or working in NF as doing mathematics. Since any philosophy of mathematics must account for this fact at best you can allege that certain kinds of axioms aren’t the ‘right’ sort of things to study for philosophical reasons.

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p> Of course in order for this to be true there would need to be certain axiomatizations which structuralists dislike and others that Platonists dislike and so forth. That is there must be something special about the choice of ZFC which lets all the philosophies accept it. If all the philosophies would have accepted any axiomatization the argument crumbles so ZFC must have special properties that let all the philosophies agree. What are these properties?

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p> In fact the only argument about what axiomatization one should choose that I can come up with seems to point directly away from ZFC. A Platonist might try to argue that not only was it the job of mathematics to investigate the properties of arbitrary platonic objects but should explore the properties of specific objects. For instance the Platonist might think it is important to investigate the nature of ‘sets’.

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p> Yet since by the completeness theorem the Platonist must accept all these other notions of set as describing something his only objection could be, ‘That’s not what we mean by set.’ In other words the criteria for an axiomatization of set theory would be that it picks out the objects we are-theoretically identify as sets. However, our pre-theoretic usage quite freely makes use of things like the set of everything or other instances of unbounded comprehension. Conversely our pre-theoretic usage assumes no nice things about all sets being well-ordered or other nice features of ZFC. Thus to the extent a Platonist would be motivated by philosophical considerations to adopt some axiom system it seems they would be pushed to NF over ZFC.

The Real Answer

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p> I think the real reason we settle on one axiomatization is a lot simpler and more pragmatic. It is easier to collaborate and work together when everyone is working from the same axioms. Mathematicians adopt a common set of axioms for the same sociological reasons that make certain areas of research ‘hot’ or motivate intense investigation into certain problems while leaving an infinitude of other problems untouched. I think this also better reflects how the axioms were chosen. It isn’t philosophical considerations which motivate the adoption or rejection of some axiom (though some people may phrase the point this way) rather it is mathematical utility and elegance.

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p> Thus agreeing on one set of axioms is motivated by purely mathematical and social considerations. Even if everyone was an unabashed Platonist we would still choose one axiom system to work in. I do think there is something that fulfills the role Kenny talks about but that is the agreement on a deductive system (i.e. first order logic) or even the use of axiomatic systems in general not the agreement on any particular axiom system.