Introduction

            Philosophy of math can seem like a juggling or balancing act. One wants to explain a number of curious facts, most notably the following two: that the kinds of things which mathematicians do can give us mathematical knowledge, and that this knowledge has or allows the kinds of empirical applications which it does. But the views of mathematics which allow us to account for one of these facts often make the other into a formidable mystery. In this work I aim to do two things: advocate a particular theory of math which I think is capable of satisfying the relevant requirements, and provide a sort of toolkit for constructing such an account by considering the nature of these requirements in detail and generality which may be accepted by those who reject my particular theory.

            My first chapter will orient us by considering what the philosophical problem of accounting for mathematics amounts to. Following Wittgenstein’s comment at the beginning of his Lectures on the Foundations of Mathematics that, “One might think that I am going to give you, not new calculations but new interpretations of those calculations. But I am not going to do that…Mathematicians tend to think interpretations of mathematical of mathematical symbols are a lot of jaw – a sort of gas that surrounds the real process, the essential mathematical kernel.” I will put forward a view of philosophy of mathematics on which it does not aim to provide truth conditions for mathematical statements in terms of something else, but rather to clarify certain things about mathematical practice.[1] And I will then show how this understanding of the project of philosophy of mathematics two classic arguments: Burgess’ dilemma for the nominalist and a problem about the law of the excluded middle raised by Dummet.

            In the second chapter I will present my own view of mathematical practice. This is a hybrid of three more well known views, namely: fictionalism, deductivism and practice projection. Since logic plays a vital role I start with the matter of gaining knowledge of logical possibility and necessity, where these notions are understood to have to be linguistic and substance-less in the manner of the Tractatus. This gives us two kinds of logical facts: facts which can be proved in logic and facts about logical possibility i.e. consistency (on my view the latter kind are prior to the former). I will then turn to the sense in which math is a matter of training and ‘carrying on in the same way’ and show how this kind of a practice can become moored to some preferred physical or (more importantly in this case) logical application. Mathematics generally, I propose is a combination of one fragment which is, ultimately, fixed by Tractarian logical facts and another which involves the logical and informal working out of fictional kinds of objects which are originally suggested by mathematical training.

            In the third chapter I will turn to the normativity associated with math and logic. I will argue against some Dummetian criticisms of practice projection views and provide a naturalistic picture of this normativity of mathematics and logic (both the sense in which problems have right and wrong answers, and (as time permits also) the sense in which a person can be justified in/reprehensible for not adding certain logical consequences of their beliefs. As part of the nessicary ground work for this (it’s hard to talk about why people shouldn’t believe logical contradictions if you can’t make sense of how they could do so) I will address the problem of logical omniscience.

            Finally, in the fourth chapter I will turn to the empirical applications of mathematics. I first consider the apparently nessicary consequences which math has for our empirical observations, discussing counting procedures and criticizing another Dummetian objection to Wittgenstein which arises from these applications. I then turn to the use of math in phrasing contingent physical laws, arguing against the Quinean view of mathematical objects and knowledge as a part of empirical science. I then conclude by considering the smaller questions of why so many confirmed physical theories can be stated simply in the language of mathematics (i.e. the ‘god is nice’ question) and describing how, once certain background conditions are set up, we can use mathematics to learn physical facts and physical experiments to learn mathematical facts.