Epistemology of Math

Introduction

In this section I’m going to assume the picture of the nature of mathematical practice given in previous sections and describe how it can be used to account for our knowledge of mathematics, and the sense in which it is a priori. If mathematics really is a matter of following out the logical consequences of hypotheses chosen for philosophical, scientific, practical aesthetic or other extra-mathematical reasons, then the question of giving an epistemology for math really comes down to that of giving an epistemology for logic. We know what to say about the fiction in which certain axioms hold or a certain intuitive picture is realized because we know what the logical consequences of these axioms, or some axioms which largely characterize this picture, are.

So, how do we acquire logical knowledge? The task of giving an epistemology for logic sounds hubristic, and it probably is if we understand it to involve giving logic a foundation which is more certain than itself. I, at least, don’t mean to be giving a justification for logic. Rather I want to assume logic, induction and a good deal of other science and psychology besides and try to describe how the process of people coming to know logical truths and consequences could work. I think this kind of project is sometimes called ‘naturalized epistemology’.

Epistemology of logic

I want to say, following Wittgenstein in the Tractatus that logical knowledge is knowledge about certain patterns in the way we use words. So, for example, A implies B if no situation could count as making A true which wouldn’t also count as making B true. Insofar as this view of logic takes logical truths out of the realm of external metaphysical necessity and puts makes them out to be, instead, artifacts of human dispositions and practices, one might hope that it makes the gruesome issue of providing an epistemology for logic easier. But if the question is easer, it certainly isn’t trivial.

We can put it like this. Suppose someone knows how to use all the words of a language which doesn’t include logical terms (like contradiction and logical truth or consequence) correctly. So, for example, they will accept ‘it’s raining or it’s not raining’ and reject ‘it’s raining and it’s not raining’ and if you tell them ‘it’s raining and grass is green’ they will come to believe that it’s raining. But suppose they don’t yet know that it’s a logical truth that ‘it’s raining or it’s not raining’, or what’s a contradiction or a logical consequence. Their use of the language does infact determine that they will agree to all sentences of the form ‘A or ~A’, and the do agree, and remember agreeing to all such sentences which have occurred to them. But how do they go from this state to learning the higher order fact about their use of language that all sentences of the form ‘A or ~A’ come out true?

I want to say that we do it by induction. Psychologically speaking this seems pretty accurate. When you read about some purported logical law, or rule of inference the main way you (or at least I) try to evaluate it is by plugging in a bunch of different instances and seeing if you can come up with a way to make the law false or the inference rule lead from truths to falsehoods. So maybe we learn these logical facts about the relationships between the ways we apply different words by considering our judgments about particular cases and then applying induction. This is not to say that all logical knowledge springs directly from induction on actual cases. Once we have some ‘seed’ knowledge of logical possibility and necessity there are lots of things we can use to increase this knowledge, including conceivability arguments, truth tables and the construction of models. I’ll get into talking about these, and the reasons why they can’t be the origins of our logical knowledge but they can increase it another section. But first I want to consider a serious problem.
The problem arises from the similarity between our problem (how to go from actual judgments to logical/linguistic laws) and Hume’s empirical problem of induction (how do you go from knowledge of what actually happens to knowledge of the laws of nature). I am giving, in a sense, the same answer to both: look at what has actually happened and then perform induction. And that seems like a big problem since we distinguish between merely physical necessity on the one hand and logical necessity on the other, while applying the same procedure (induction) to the same body of cases our actual judgments) should yield the same answer in both cases. To put the problem another way, induction tells us both that we’ll never assent to ‘this is a cup and it isn’t a cup’ and that we’ll never assent to ‘this cup is floating’, but how do we distinguish the linguistic necessity of the former from the merely physical necessity of the latter?

I’m pretty sure that I do know that ‘cups don’t float’ is a contingent physical necessity rather than a matter of language, but when I try to say how, I get into a circle of roughly the kind that Quine describes his ‘Two Dogmas’.

Me: I know that the absence of floating cups from my experience is a matter of contingent physical law, rather than linguistic necessity, because there is something which I would count as a cup floating. Namely, if things were arranged like such and such (maybe I draw a picture here) then that would count as a cup floating.

The Quinean: The mere fact that you can give a description of something that would incline you to say that a cup was floating doesn’t suffice. I could say ‘If a person was married and they were a bachelor then this would be a situation that I would count as containing a married bachelor. Thus the lack of married bachelors is only a contingent physical law.’ The description you gave only helps if we first know that it is linguistically possible.

M: Well, maybe I know that the situation is linguistically possible, because I know that the description I’ve given is consistent, and I know that it’s consistant because I can keep describing it in detail without arriving at a contradiction.

Q: Well, you can’t describe it without describing a physical impossibility (that there is a floating piece of clay) and our question is exactly how you know that this is merely a physical impossibility and not a contradiction.

M: OK, maybe I’ve been taking the wrong tack so far, and important difference is that there’s a sequence of possible experience –even physically possible if you like- which I haven’t had, and won’t have, but such that if I did have this experience I would be convinced that cups sometimes did float, while there is no such sequence of experience which would convince me that some bachelors were married.

Q: I have two objections to that. First off, this criterion for contingency is too weak. Take some long sum in artihmatic. There’s certainly a physically possible sequence of expereinces you could have which would convince you that you were wrong about the sum e.g. remembering working it and getting a different answer, having different calculators reliably give you a different answer etc. And secondly, you probably don’t want to require that there be a physically possible sequence of experience which would convince you of every linguistic possibility. Isn’t it linguistically possible that your visual field could reach around 360 degrees or that you could see more than three primary colors?

These difficulties lead me to adopt for the present moment an extremely radical, almost insane, view of the foundations of the difference between our ideas of physical and logical possibility. It’s almost the kind of thing which would have been at home in the 1980’s culture wars. Namely, it might be that the original seed for our distinct ideas of analytic/logical/metaphysical necessity is a matter of story telling. The idea would be that part of our learning to use certain terms was not only learning to apply them correctly but also as fictional or possible but not actual. This training in certain fictional examples would mean that we would have two distinct bodies of judgments to apply induction to. First there would be the facts about what you had actually judged as true, and applying induction to these would give you an idea of what is physically necessary. Then there would be the body of things you have actually judged to be true together with the things you have judged to be true of certain stories which you had been trained to consider as possible. The inclusion of these additional data points then makes a huge difference to which inductive generalizations are supported, and this different, weaker set of generalizations becomes your idea of what is linguistically/analytically nessicary.

To appreciate the full extremeness of this solution, note that if this is right a) the extent to which we extend the realm of metaphysical possibility beyond that of physical possibility is arbitrary and culturally determined and b) people who didn’t tell discuss the non-actual wouldn’t have a concept of logical/analytic/metaphysical truth or consequence.

Koellner’s problem

I’ve sketched a view on which mathematical practice is, roughly , a matter of following out the logical conseqences of certain axioms, informal descriptions or pictures which have been chosen for arbitrary practical or aesthetic reasons. I now need to give an account of logic and logical consequence, and not just for the sake of making the description above more complete. The issue is more urgent than that because there is some reason to worry that this kind of anti-realist view can’t account for a certain partly logical kind of knowledge. Peter Koellner told me about the problem which he put like this: I know that a certain axiom system S is consistant because mathematics provides us with a model of it. So I know the following thing about the world: that no one will ever come up with a proof of contradiction from S. Thus mathematics seems (contra to the picture I have advocated) to be making substantive and correct empirical predictions on its own.

There are a few points to make here. First off, I think this problem does show that the practice of logical talk about a certain language can’t be understood as progressing separately from the practices which give meaning to each of the terms of this language. If we separate mathematical talk from logical talk about consistency then we create a mystery about why we never end up accepting mathematical proofs of contradiction for systems which we have dubbed concistant. But this kind of independence is very far from the Tractarian view of logic or my view of mathematics. For, once you have fixed how certain terms apply, you have already fixed the internal relations between these uses which give us logical facts: you have already fixed that anything would count as an A would also count as a B hence that A(x) implies B(x) or that there is nothing we would count as both a C and D hence that C^D is inconsistent. So I accept and propose to explain the fact that the kind of reasoning just mentioned does states something true about the possibilities for future mathematical proofs.

But I want to deny that the mathematical reasoning above provides us with substantive or empirical knowledge. We don’t know, from the above reasoning, that people won’t ever believe something to be a proof of contradiction from S –human nature is fallible. We don’t know that the uses and meanings of the axioms in S won’t radically change over the course of human history so that the words which express S come to express a contradiction and this can be proved. So there’s a kind of sociological prediction (if you bet that no one would find a proof of contradiction you would win money) which this reasoning might seem to justify but it doesn’t. Rather, what it shows is that there is nothing which our actual present standards would count as a proof of contradiction from S, and hence as a trivial consequence that no one will create such a thing. And this ‘there is nothing which we would count as an X’ is the paradigm of the non-substantive logical/analytic necessity of the Tractatus.

So what there remains to explain is just this: how the present process of reasoning just mentioning, finding a mathematical model for S, can tell us about constraints on our possible future logical practice. One part of the story is relatively easy: something counts as a proof if it’s composed of inferences such that it would never be right to accept the premises without accepting the conclusion, and some statement is a contradiction if no circumstances would suffice to make it true. And something counts as a model of S if it shows how the domain and extentions of the terms in S could be related so as to make S true. Thus if one can so much as imagine a model of S, there are possible circumstances under which S could be asserted so nothing could count as a proof with had S as premise and contradiction as its conclusion. But this rehearsing of the Tractarian reading of the familiar relations between proof, consistency and possessing a model leaves an important epistemic issue out. Given that we will only call something a valid inference if whenever the premises apply the conclusion does as well the story works, but how do we ever come to have this kind of higher order knowledge of our standards of correct use. How do I know, on this view, that I will always assent to each of the infinitely many possible instances of A or ~A? If there is no meaning of ~ which I have in my head which runs ahead and decides all future uses, how can I know that I will always assent to sentences of this form?

One thing to notice is that this kind of knowledge isn’t immediate, and neither are our judgments about our future use infallible. Sometimes we think that every instance of a certain form must be true, until a counter example occurs to us or we’re uncertain about the validity of some proposed basic inference until we discover it’s equivalence to other things we are sure of, or run through enough examples. The psychology of coming to accept some logical truth is more like that of being convinced of a physical law by seeing diverse instances of it, or being convinced to reject a proposed physical law by making an experiment in which it is violated, then like immediate introspective awareness.

I have proposed an extreme view on which logical laws and physical laws are both known by induction, as part of an overall scientific project for understanding our actual observations by proposing structural theories of how our language applies as well as physical constraints on what we will ever actually observe. The difference between the laws which are counted as physical and those which are counted as linguistic arises from the tradition of countenancing certain scenarios as coherent but non-actual (e.g. in fiction) which forms part of our linguistic training. Since the laws of the linguistically necessary need to hold for these fictional scenarios as well, they end up being different and less restrictive than the physical laws. This strategy accounts for knowledge of logical laws, at the cost of making the degree to which logical possibility extends beyond physical possibility out to be a sort of cultural artifact.

Where initially we had wanted to make out the sense in which there could be a floating teacup but there couldn’t be a teacup which was heavier than itself, or not self-identical, we are contenting ourselves with saying that there is a practice of telling stories about one and not the other. Here I want to address two concerns. First there is the question of whether this strange epistemology for logic can be plugged into Koelner’s question. Secondly, the apparent explanatory gap in this account: the boundaries of logical/analytic possibility seem to limit not just what we do suppose but what we could possibly coherently suppose, and invoking the application of induction to a tradition of story telling doesn’t explain why these two boundaries (the sort of thing that we do suppose, and the sort of thing which we coherently could) fall in the same place.

I think the issue with Koellner’s problem is pretty simple. All we need is access to some, single, kind of possibility – it might even be physical possibility. Then it’s a contstraint on a model that it schematically indicates a (e.g. physically) possible scenario in which S would be true, a constraint on proofs that it is (physically) impossible for their premises to be true while their conclusions false, a constraint on contradictions that they be (physically) impossible so that it comes out to be impossible for anything to count as a ‘proof’ of contradiction from S given that a ‘model’ has been found. All we need for this explanation to work is a single notion of possibility which gives these other concepts their meaning and which we have original epistemic access to.

Now, what about the mysterious fact that the boundaries of what we are inclined to actually suppose seem to happen to lie just in the same place as the boundaries of what we can coherently suppose. Here I have two answers, which I have not yet chosen between, and you can choose the one which you prefer. The first would be to take a Quinean line, and, inspired by the impossibility of indicating the boundaries of the metaphysically possible, hold that there are no such boundaries. In this case the facts about which scenarios our linguistic training leads us to countenance and which it leads us to reject are all there is, so there is no further agreement to explain.

The second follows the strong intuition that inspite of the difficulty of explaining what it takes for something to be metaphysically possible there really is a sense in which the cup could be floating over the table but it couldn’t be heavier than itself and invokes externalism and interpretative charity to explain the match between the coherent possibilities and those which we actually do countenance. It involves a familiar Kantian reverse. We started off wondering how we learned that it was metaphysically possible for the cup to float so that we then could talk about this scenario would be like, but not about other metaphysically impossible ones. But the relationship between what possibilities we countenance and which are metaphysically possible might run the other way around. Maybe our practices of talking about certain possibilities and not others are (all that) determines the essential properties and hence metaphysical possibilities for our terms. So, for example our willingness to talk about a cup floating would be what determines that the essence of being a cup isn’t something that essentially requires being in an earth-like atmosphere. And this theory would also explain how the essential properties of certain objects can be vaguely or ill defined (could electrons have had a different charge?) for in these cases the background of fictional talk is absent.