So my fiance Sharon Berry posted a very early (like 2 years early) draft of her thesis on a wiki here. The broad question she is addressing is how we can come to have accurate mathematical knowledge which I figured might be of interest to some people who check out the philosophy part of my blog. I also figured I’d take this chance to share my own thoughts on the subject. However, to give credit where credit is due I would never have really thought through these issues if Sharon hadn’t brought up the subject and many of the ideas are really hers. However, I take them in a very different direction than she does.

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

Sorry it’s been awhile since I last posted. I got a little distracted posting about the Spying scandal over on my other blog [Now part of this blog] and then I was visiting my parents for Christmas without internet access. Sometime before school starts again hopefully I get around to finally explaining my views on axiomatic set theory. Now though I wanted to post about a little puzzle which has been bouncing around in my head for some time. How can we have inductive knowledge in mathematics? Since I will be talking about number theory a bit let me stress that induction in this post only refers to the scientific kind and never the formal mathematical kind.

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.

In response to our recent discussion about philosophy of math Kenny sent me a related paper by Timothy Bays. The paper takes issue with Putnam’s model theoretic argument against realism. Those who are unfamiliar with the argument can read about it at the link above but the part in dispute is whether Putnam can prove the existence of a model of ZFC +V=L containing any given real.

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. (more…)

Reading a recent post over at antimeta.org/blog I was reminded again about my problems with the indispensability argument for the existence of mathematical reality. You can read the original post for a better description of this argument but the basic idea is that we have the same reason to believe in mathematical objects as we have to believe in theoretical objects like electrons. They are both indispensable to our best empirical theories.

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. (more…)

Among actual set theorists Platonism is quite popular as it gives them the feeling that they are discovering some kind of pre-existing eternal truth. Of course many other philosophies of mathematics would support the idea that mathematicians were discovering eternal truths about what are logical consequences of a particular axiom system, i.e., ‘ZFC entails Ramsey’s theorem.’ However, the special affection mathematicians have for Platonism seems to stem from it’s apparent ability to make the choice of axioms themselves more than a matter of convention or convenience. Most Platonists believe that the axioms are either true or false of actual sets in the same way the earth is either round or flat. Thus in this Platonistic framework math is no longer merely discovering conditional truths but truly investigating the nature of sets.

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. (more…)

So my last entry mentioned a surface analogy between meta-analytic statements (statements about analytic statements) and philosophy of mathematics. What I was aiming at in that post was merely an exploration of how we might deal with these meta-analytic statements in philosophy of language and a suggestion that such explanation could throw light on philosophy of math. However, much of the interest in that post focused on what sort of specific application I had in mind so I will try to explain one possible interpretation of mathematical statements as meta-analytic statements in this post.

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. (more…)

A while ago an entry over at antimeta.org caused me to think again about what makes philosophy of math so hard. In particular what prevents us from just taking mathematical statements as plain old analytic truths. That is 2+2=4 is true for the same reason a bachelor is an unmarried man is true, they are just different ways to say the same thing. Now of course there are difficulties with such a simple approach. For instance what is it that 4 means? If both 2+2 and 4 are supposed to have the same meaning what is that meaning? Also it seems more sophisticated attempts along these lines tend to push toward if-then-ism.

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.

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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.