The really short version of my attitude to the problem of mathematical knowledge is “What Problem?” I mean obviously mathematical knowledge is subject to the same skeptical doubts that other forms of knowledge are but I’m unconvinced that there is any particular problem unique to mathematical knowledge. More specifically I would say that mathematical knowledge is nothing but a limiting case of other sorts of knowledge so it poses no problem over and above the problem of understanding the meaning and our knowledge of other sorts of statements. Of course explaining meaning is a notoriously difficult problem in it’s own right but I’m tempted to think that it’s a hopeless problem. Ultimately one must merely take meaning to be a primitive concept but that’s another discussion.

I need to get back to working on my thesis so I won’t give more than a very very quick sketch of my thoughts here but roughly I take it there are two primary reasons one might think that mathematical knowledge requires special explanation.

1. The Benacerraf problem: How could we come to know anything about numbers if they don’t have causal powers, we don’t interact with them and so forth.
2. How could it be that our mathematical theories turn out to be useful in the way they are.

Platonism and Reference

So if one accepts a platonic theory of mathematical meaning then there may indeed be special problems about mathematical knowledge. That is if the meaning of a statement like 2+2 =4 is really that some special 2 object out there bears a certain relation to itself and the four object one might wonder how it is that we come to know about these platonic objects. However, I’m inclined to simply turn the question around and ask whether the platonic theory in question provides any reason to think that “2″ refers to something we would ‘recognize’ as an integer or whether it could (logically not metaphysically) be that 2 refers to the concept of bunny rabbits and all our statements about arithmetic are really nonsensical. If the platonic interpretation of mathematics tells us that the reference of two must really behave like 2 to qualify as the correct reference then we know exactly how we come to have true beliefs about the numbers — because if our beliefs weren’t largely true we would be talking about something else[^enough]. On the other hand if we don’t have any restrictions about what sort of platonic object 2 might refer to then we aren’t justified in adopting this kind of theory in the first place.

Unfortunately the debate about Platonism and competing philosophies of mathematical largely distracts from what I think are the important issues. As I’ve argued previously Platonism in and of itself says very little about mathematics. What the last paragraph as well as my previous post on the issue emphasize is that it isn’t really Platonism that is doing the work it is your theory of reference. Really on it’s own Platonism says nothing very significant¹, it’s the means by which our talk maps to particular platonic objects that really does the work in the theory. This raises the obvious question of what we even mean when we say that the reference of a certain term is such and such. Are we merely making a claim about dispositions and talk or are we invoking some real metaphysical relation. While Platonism provides a good motivation to consider the issue I think a proper examination of this question of what sort of thing the meaning relation is in the first place illustrates the non-problem of philosophy of mathematics in general.

Platonic Realism About Reference

There are two ways one could understand claims about meaning and reference One could think that the relation of meaning is a truly objective notion with metaphysical substance. That is that the relation between words/mental states/speaking contexts is some and references/meanings is something like a platonic entity in it’s own right. On such a theory it is presumably logically possible (but not metaphysically) that when I say “2″ it really (by virtue of this objectively existing meaning relation) refers to rabbit. In other words the meaning of word is a notion much like the moral status of an action under on a realist moral theory.

Just as with moral realism I think the appropriate response to this notion of meaning is to challenge that it counts as meaning at all. Ultimately there is just this relation out there mapping situations/worlds/utterances/mental states/whatever to references/intentions but why should we think this picks out what we talk about when we use the term meaning? Additionally on this sort of platonic realism about meaning we don’t have any reason to actually believe that we really do have knowledge. After all maybe the objective meaning relation isn’t what we think it is at all and what we take to be true statements aren’t true at all.

One might still be tempted to insist that obviously we have knowledge thus the fact that this theory can’t explain this fact is a puzzle requiring explanation. However, this simply gets things backwards and implicitly rejects the very assumptions of the theory itself. If we accept this sort of theory we need to just bite the bullet and say we don’t know if we really know anything and thus how we know things doesn’t require explanation. Personally I think our intuition that our usage determines meaning is a good reason to reject this sort of theory but in either case this leaves no special problems for mathematics. Of course you might try and say that the mapping between statements and meanings/references must obey certain restrctions but this does no good at all since of course any actual map will have some facts that are true of it but this does nothing to offer us reason to think we have any knowledge of what they are.

Naturalist Theories of Meaning

I think a much more promising approach² is to jettison all the metaphysical baggage and start from the assumption that meanings, ultimately must be defined in terms of sounds, dispositions, actions and other arrangements of matter. That is nothing special or magical goes on with meanings. They are just a concept introduced to organize very complicated descriptions of human behavior in terms of atoms and physical laws. Thus the ultimate standard against which we judge a theory of meaning is it’s predictive accuracy and theoretical utility (how well does it work with other models we wish to use). In some sense already this approach should suggest that there shouldn’t be any deep paradoxes in terms of meaning. After all we are confident that the description of human behavior at the level of atoms is consistant thus any apparent difficulty at the level of meanings either reflects a confusion on our part or a poor choice of definition.

To put the point a bit differently we should think about a theory of meaning much the way we think about thermodynamics as derived from statistical mechanics. Yes, it can be a powerful theory with useful concepts and important impacts but ultimately just as debates about whether entropy is the log of the number of possible states holding X,Y and Z fixed or just X and Y doesn’t reflect any fundamental fact about the universe but a definitional choice we make that is judged on it’s utility. Thus theories like fictionalism or formalism shouldn’t be understood as making different philosophical claims but rather judged simply on their utility in predicting how people actually use words. Indeed one might very well conclude that different models are most appropriate in different circumstances.

Ultimately then the question about how we can come to have mathematical knowledge is largely a non-question. I can point to the actual ways that mathematicians prove theorems and reach conclusions and that right there shows how we come to have mathematical knowledge. Still one might ask but why are the results of our proofs actually true? However, this has a totally trivial answer. The reason that proofs give us true mathematical results is that every step of the proof is truth preserving. Indeed we can go through this and using the fact (in the meta-language) that A and B is true if and only if A is true or B is true show that the methods mathematicans use to reach theorems really do produce true theorems. Asking for anything more is a demand to know why logic is true. Obviously at a very basic level we have to just assume that logic is true (see Quine’s arguments about this point in his discussions of radial translation) so it’s unclear what is left to be explained at all.

To put the point slightly differently it’s contradictory to worry about how we get mathematics correct. Either the question tells us how we have reason to believe we do get mathematics right, in which case it tells us the answer or it offers no such explanation and we have no need to explain a phenomena that we don’t have reason to accept as true.

Usefullness of Mathematics

This finally brings us to the question of why mathematics turns out to be useful. One might think that it’s surprising that the results of mathematics tells us useful things about the world. Certainly in one sense it is surprising, but that’s the sense in which the understandability of the world is surprsing, i.e., that induction works. While it may appear that mathematics directly makes predictions about the world (if I have two apples in my bag and place another two apples into my bag I have four apples in my bag) in fact it’s only the combination of mathematical theorems with contingent bridge laws that makes these predictions (apples don’t appear or disappear when I place more of them together). One might try and minimize the significance of these bridge laws by saying something like “So long as apples don’t appear or disappear the number of apples in my bag is the number of apples I added minus the number I removed.” However, this merely begs the question by working in our expectation that the plus operation on the natural numbers describes how objects behave into the definition of appear or disappear. I could equally well claim that apples were disappearing and reappearing all the time but if they didn’t do so we could see that adding n apples to a bag with m apples in it results in a bag with n x m apples in it.

In fact the usefulness of something like mathematics is an easy consequence of a well known theorem in recursion theory. Supposing we have a language complex enough to express arbitrary procedures³ then that language will contain infinitely many different ways to state the same procedure, some subset of which will be possible to construct a verification that they are equivalent. In other words no matter how weird your language is you can’t get around the fact that some things will turn out to be non-obviously equivalent which suggests that it will be useful to have a means to identify at least some of them.

Usefullness and Knowledge

The final worry is that one might try and link the two concepts and ask how it is that we come to have mathematical knowledge that yields useful results. Thus even if we don’t have abstract reasons to believe that the syntactic manipulations of mathematicians meet some independent standard of being true we do notice that they let us build rockets and cure disease and the like. Thus one might think the utility of mathematics requires some explanation.

Once again though I think a careful examination of the question reveals it to be a non-worry. If by mathematics you merely mean the sort of thing that mathematicians do then it’s undeniable that what counts as mathematics is partially determined by what is useful. While many types of mathematics are very abstract the subject in the large is influenced by what has solved problems presented by the world. This point is made even more forcefully if you try to define mathematics as any abstract rule based manipulation of symbols. After all under such a definition certain types of astrology would qualify which most assuredly is not useful. Similarly any other means by which you tried to formally define the problem is likely to either reduce to triviality or not call our for any explanation at all.

This was a pretty hurried and scattered explanation of my thouhts so hopefully people ould follow it. If you are confused but curious about what I’m trying to say anyway feel free to post a comment or ask me via email

It seems indisputable that we do have inductive evidence in mathematics. Clearly our confidence in something like Fermat’s last theorem increased as we tested many possible counterexamples but couldn’t falsify the hypothesis. This may seem like a perfectly standard use of induction but mathematical truths are supposed to be necessary and it is mathematical truths which form both our evidence and out conclusion. Hence every possible world has the same set of inductive evidence (in this case facts like 222+444 != 666) and the same result (Fermat’s last theorem is true in every world) therefore if induction about mathematics works in our world it works in ever world. Hence if the statement ‘Induction is a valid principle of inference in mathematics’ is true it is necessarily true.

At least in the case of mathematics then it would appear that Hume was wrong. However, it is pretty weird to think that there are completely different reasons why induction on mathematical truth works and why induction in science works but if the former is a necessary truth and the later is only a contingent truth they can’t be true for the same reason. One interesting solution is to revive the Carnap approach and argue that even scientific induction has a necessary justification. As readers of this blog will be aware I am somewhat sympathetic to the project of using recursion theory to show something of this kind. However, this isn’t something I expect most philosophers to believe at the moment.

What are the other options for dealing with this apparent paradox? One could just bite the bullet and just say it’s happenstance that the same principle of inference works in both domains but that is highly unsatisfying. Alternatively one could try to reject the idea that mathematical induction is necessary though this would seem to involve denying it’s obvious truth. One potential avenue would be to deny that in general observing a finite number of instances gives confidence in a universal truth but it happens to be true in the type of mathematics humans find interesting. Then perhaps one might argue that induction works in mathematics because we are interested in mathematics which describes the physical world. Perhaps, but even if investigating the integers was inspired by the physical world induction seems to work even on abstract questions that don’t seem to be so inspired. Alternatively one might try and argue that there is something about the type of question that humans find interesting and mathematical induction works on that class. This offers the intriguing possibility that by formalizing what sort of questions people ask (questions about generalizations of computable phenomenon or something) we could prove induction would work on these even in the scientific sense.

Overall I’m sympathetic to the idea that induction will turn out to have a necessary justification. However, I think induction will turn out not to have real ‘content’, i.e., saying induction works just gives theories which don’t disagree with experiment not which reflect ‘truth’. In any case it raises the interesting question about how one could formalize and prove meta-mathematically that induction works.

However, either Brays misses a subtlety of Putnam’s argument or the issue is easily fixed. Thus it appears that this detail of Putnam’s argument does go through. Brays other objections, however, to the philosophical significance of Putnam’s proof are much more compelling though I won’t go into them here. Even if Putnam’s argument goes through, however, I still don’t think it adds much force to the argument against realism. Brays argues that Putnam makes a mistake in his argument. In particular the failure of any recursive axiom system to prove its own consistency means that if A is determined to be the right set of axioms then Putnam cannot (in A) prove the existence of a non-standard model of A. This is in fact correct. However, it is sufficient for Putnam to prove that assuming realism realism is semantically under determined. With this in mind the trick I used in my assault on platonic determination of mathematical axioms.

Assuming realism there is not only are their real sets, their are real sets which get the integers right. That is there is an actually well founded collection of sets. Now just consider the actual set which contains all the sentences of whatever axiom system you want (recursive or not) and since that set is actually consistent it has a model. Once you have a model you can use whatever model-theoretic tricks you like. The fact that Putnam went out of his way to point out he was really using a Skolem Hull construction rather than the Lowenheim-Skolem theorem suggests he might have been aware of this trick.

I think Putnam’s argument is wrong for other reasons. These I will address in another post but even assuming that Putnam is correct I don’t see what all the fuss is about. Physical realism is completely compatible with the idea that mathematical ideas don’t have a determinate referent. The difficulty with physical realism comes only because the same argument can be extended to show that there are multiple models that are compatible with our ideal theory of physical reality and hence can’t be sure our terms refer correctly. Yet unless one doubts that it is possible for us to be brains in a vat and unable to discover this even with our ideal theory I don’t see what contribution the model theoretic argument makes.

I’m kinda worried though that I might be missing something significant about Putnam’s arguments.

While I have never been really convinced about the argument for theoretical entities in physics I think the argument is even weaker for mathematical objects. At the very best I think one could use it to support the idea that ‘conditional’ mathematical statements have truth. Where by conditional mean ones of the form “If axioms A1..AN are true then so is blah.” It does not support either platonic existence or the idea that there is a ‘correct’ answer in cases of axiomatic independence as it is commonly deployed to do. The most commonsense objection to the indispensability argument is the fact that those who actually use the theories don’t put the mathematical background on the same level as the other ontological claims of the theory. Physicists are genuinely concerned about finding the ‘correct’ description of physical reality and are not content to shrug their shoulders in the face of observational equivalent theories. On the other hand they are perfectly happy to accept differing mathematical presentations of the theories as simply being restatements of the same facts. If asked most physicists would swear to the real existence of electrons without question but the same could not be said for platonic mathematical objects. Until we have a good theory about why usefulness in theory is a good epistemic indicator it seems wise not to exceed the intuitions of those who create the theory.

It isn’t like physicists are just arbitrarily discriminating against mathematical entities. There are very real differences in the roles they play in the theories. Physical entities play casual roles. They literally interact with us to create our experiences of them, or at least the physical realist would have us believe. Mathematical entities do not. As some famous philosopher said assuming the story given us by the physical realist we can explain our knowledge of entities like electrons. Our best physical theory is likely to have much in common with the true one if we assume induction. Even assuming the story of the mathematical realist we don’t have such an explanation and physical theories do us no good. It is the very lack of casual power that the argument from indispensability is supposed to save us from and without any casual power it is totally unclear why the best theory should tell us anything about mathematical entities.

In fact it seems the indispensability argument can establish nothing unless one has already rejected every non-platonic theory of mathematics. Whatever this other theory is it will have it’s own explanation for the observation of mathematical truths in the physical world, i.e. counting beans, using abacuses. This explanation will likely easily extend from beans to all physical objects. For instance if you favor some sort of linguistic explanation of mathematical truths theorems merely represent different ways of naming the same thing. This easily explains the usefulness of mathematics in physical theories, it is just appropriate use of abbreviations and restatements to emphasize important aspects. Or to put it another way once we know some statement is true in some fashion why should it’s use in some physical theory surprise us and force us to say it is not only true but ontologically substantial.

The argument that physical theories give us cause to decide on axioms for our theories is even more fallacious. As we have seen in a previous post mere platonic reality is not enough to make questions about the ‘right’ axioms substantially meaningful. Besides as we have already seen use in a physical theory gives us little cause to believe in Platonic existence anyway. The only other option seems to be that the physical theory directly gives us an indication about what axioms are ‘correct’ by it’s choice of axioms.

Problematically it isn’t even clear what axioms a theory is really using. For instance is a theory which describes physical quantities as real numbers giveing evidence for the axioms of the real numbers or evidence for the axioms for the particular subfield which physical measurements end up being taken from? In particular, to echo the objection of one of Kenny’s commentators, does our current theory give us evidence for the rationals or the reals? If one responds that the simplest or most elegant choice (likely the considerations driving the choice of the physicist) is the right one you have just short circuited the entire indispensability argument and might as well do without it. If we already believe that elegance is grounds for believing a mathematical theory is literally true we don’t need the indespensability argument. Trying to make use of some natural kind of mathematical object which serves the purpose just makes the argument circular. After all the whole point is to show that mathematical objects are actually natural kinds with real existance and not something else.

Or to put the objection differently we naturally expect the facts about our universe constrain what is actual but we are reluctant to let it constrain what is necessary. The discovery that every electron has exactly the same charge makes it physically impossible for us to have two differently charged electrons. We would not, however, suspect it made it metaphysically impossible for such a result to occur. So how is it that the particular laws about our universe tell us what is mathematically valid and hence necessary in all worlds? Suppose our physical theories made no use of imaginary numbers. Would this tell us they didn’t really exist or that our world just happened not to have any particles with imaginary charges?

This issue is of particular difficulty in set theory. Given any physical theory it seems perfectly plausible that it could just be some special case of some theory which referred to every possible collection in the simple theory as a set. In other words unless we are confident that the lack of application in our physical theory is evidence of lack of existence we have no reason to disbelieve that the set theory we are finding useful is just a set model inside the real set theory. If we are confident enough to conclude non-existence from non-application we can dismiss almost all of modern set theory, certainly everything beyond V_(w+w).

Unfortunately, the belief that if true Platonism would regard certain axiomitizations as true and others as false is simple incorrect. At least without significant additions to the theory. In fact as I shall show Platonism, just like the other philosophies of mathematics, in fact regards all consistent axiomitizations of set theory as equal with only our preferences to choose between them. As Platonism is essentially the extension of our normal notion of truth to mathematical objects it behooves us to first consider when this standard notion. Importantly, even for normal objects it is not just the external world but also our choice of language which determines what sentences we accept as true or false. For instance the fact that tables are flat surfaces one sets plates upon rather than objects designed for sitting upon has nothing to do with the external world and everything to do with our choice to use the symbol ‘table’ to mean the former and ‘chair’ to mean the latter.

This point has little import for most statements about the physical world as one of the implied components of meaning for most words is that they refer. Thus even though we might only stipulate that an electron means an indivisible object of negative electric charge found in hydrogen atoms this leaves no linguistic ambiguity as to whether electrons interact with the strong force. Since it is understood that we mean electron to refer to all existent objects satisfying the definition, all we need to do is go out and test whether those objects in the world meeting our definition of electron are affected by the strong force to determine the truth of this claim. In other words the use of the word electron is inessential in the claim “Electrons are affected by the strong force.” We can remove the word electron and translate the claim into, “There exists an indivisible object found in hydrogen atoms and affected by the strong force.”

The same tactic will not work for set theory. According to Godel’s completeness theorem, which is provable from ZFC, any consistent theory has a model. Thus if it is not possible to deduce a contradiction from some set of sentences there is an actual set of objects which satisfy those sentences. Since all Platonists about ZFC agree about what (recursive) sets of sentences are consistent they all agree about what sets of sentences have a model. In particular the both agree that the axioms of ZFC+~CH and ZFC+CH are consistent since ZFC proves CH is independent of ZFC. The upshot of this is that both Platonists who believes sets actually satisfy ZFC+CH (Continuum Hypothesis) and those who believe sets actually satisfy ZFC+~CH both agree that the sentences, “Objects exist which satisfy ZFC but not CH” and “Objects exist which satisfy ZFC and CH” are true.

Thus even assuming Platonism there is still no external matter of fact about what the right axioms are for set theory. The question is still entirely linguistic: which objects that all Platonists agree exist should we call sets.? While this result can probably be interpreted several ways at the very least everyone must agree on the following. If there is a sense in which one of “Sets satisfy CH” or “Sets don’t satisfy CH” than we must mean more by set than merely “an object which satisfies ZFC.” This means that any Platonist about sets who wants to argue that there is a determinate answer to CH has to meet the challenge of explaining what else he believes we mean by set beyond ZFC which entails his result.

Some (Kenny) might object that this shows no more particular difficulty for a realist about mathematics than does the existence of observationally indistinguishable theories about the physical world. For instance a theory asserting the reality of a electrons and another theory only asserting certain patterns in sense data. However, the difference here is significant. On a empiricist theory of meaning the choice between these two theories might be only a matter of meaning but once you accept ‘physically real’ as a real valid predicate one of the statements becomes true and the other false. In this mathematical case even granting the Platonist their ontological categories the difficulty still remains.

Moreover, Quine’s theory of indeterminacy of translation does nothing to diminish this result either. It only drives home the stake that the choice between sets which satisfy CH and those that do not is purely linguistic. Whether or not other statements occupy this category as well is unimportant. If you try to claim that set couldn’t just mean object that satisfies ZFC on Quine’s theory since this would be an unacceptable analytic truth you are just biting the bullet and admitting you mean more by set. Moreover, retreating to Quine seems self-defeating for a mathematical Platonist who wants to believe there is an object matter of fact about statements like CH.

Besides the entire excursion into Quine is really quite pointless as his arguments are just wrong. The inability to coordinate language through purely experimental methods is simply irrelevant to whether or not creatures with a similar physical basis might have more coordination of meaning. The entire idea that we can talk about meaning as if it was just some fact purely determined by an arrangement of atoms is just downright absurd. There simply have to be brute level facts about meaning somewhere if you want to admit anyALL meaning facts or answer the famous argument that this is a necessary consequence of indeterminacy in translation.

The motivation of this idea is kinda an attempt to combine the theory that the meaning of logical connectives is given by the deductions they support (‘and’ just means that connective which allows the derivation of A, B from ‘A and B’) and if-then-ism. While it seems there is some difficulty in just using the same technique from the logical connectives with mathematical objects themselves, because for instance we don’t in fact have words referring to every integer even if we have a procedure to produce them, it seems we might be able to make conditional statements about what would happen if we did have such words.

Note: I DON’T think this provides a good solution for philosophy of math. Rather I think it shows most of the worries about philosophy of math infect our talk about philosophy of language so an answer to these difficulties in philosophy of language might provide some help addressing them in phil of math.

Once again this is hardly a well thought out theory but here goes. Alright so one simple approach to understanding mathematical statements is if-then-ism. That is to say that the Goldbach conjecture is true (provable) in the integers is just to say that if some structure satisfies the axioms of the integers then that structure will also satisfy Goldbach’s conjecture. There are a fair number of complications I am glossing over but the appeal of the approach is obvious. It seems to automatically give an answer as to why mathematics provides useful results in counting. However, it also has several serious problems. For instance what sort of conditional are we to take the ‘if’ to be? If it is only material and the universe is finite all statements about the integers should come out trivially true. Even if it is some form of counterfactual it is unclear why we should believe there are any structures in any possible worlds that model set theory. If the existence of some structure modeling set theory is needed for the theory to work why not just take that structure and be a Platonist about it?

So it seems many of the problems in if-then-ism stem from what we mean by ‘if something satisfies the axioms’. However, it seems the same problems can be easily replicated in meta-analytic statements. For instance consider the statement (substituting some specific number for N), “If X1…XN are different words meaning the same thing as hot then very X1…very XN are different words meaning the same thing as very hot.” Should we consider this statement as trivially true if there are only N-1 different words meaning the same thing as hot? That doesn’t match with my intuitions about such statements. It seems to me we expect such statements to be true independent of the actual existence of words we can substitute in for X1…XN.

So how can this vague analogy be pressed into an interpretation of mathematical truth? Well the idea is that instead of using statements of the form ‘if blah satisfies the axioms’ which has the problems discussed above take statements of the form ‘If blah satisfies the axioms as a matter of meaning.’ Or in other words the statement of the Goldbach conjecture could be rephrased as something like, ‘If X0…XN… satisfy the axioms of arithmetic by virtue of meaning then they also satisfy the goldbach conjecture by virtue of meaning.”

Admittedly this is still mostly imprecise analogy. All I’m hoping to convey is the sense that hey statements about analytic truth seem to have alot of the same difficulties as statements about mathematical truth. The interpretation of mathematical statements in terms of meta-analytic statements is likely to suffer from serious problems. However, I’m hoping just explaining why it is wrong in a precise way would go a long way to answering questions about both problems, which I am convinced are not as far apart as most people think.

However, it does seem there is a much closer analogy between mathematical statements and meta statements about analytic truths. For instance consider the statement, “If X means the same thing as Y and the word big applies to both X and Y similarly then a ‘big X’ means the same thing as a ‘big Y.’” Such a statement seems to be a sort of necessary truth and also incorporate some kind of quantification but yet is a perfectly acceptable sentence in philosophy of language. Moreover, like mathematical statements this sort of meta statement about analytic truth seems to resist any easy interpretation in terms of meanings or as a straightforward analytic truth.

I don’t mean to suggest that philosophy of math is actually really easy. Far from it. Rather I mean to suggest perhaps we don’t really understand the sorts of statements we make in philosophy of language as well as we think we do. Hopefully a good theory could be developed to handle both cases. Still it is equally possible that I am making a silly error and missing something important. Anyone have any thoughts?

UPDATE: I only meant this post as a vague note of an analogy between mathematical statements and meta-analytic ones. Namely they both have quantification, presumably reflect necessery truths etc.. Thus I want to leave this post just as a question about what we mean with these meta-analytic statements and provide a longer post (probably tomorrow) detailing what I think can be made of this analogy.

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.