Formal Tools and the Philosophy of Mathematics

Abstract

In this chapter, I won’t try to defend a particular philosophical view about mathematics, but, in the spirit of New Waves, I would instead like to think about how we can hope to make real progress in the field. And to do that it seems best to think about what in the field as it is now is holding up progress. What in the field should change so that we can hope to do better than what has been done before? Having considered the question, I propose the following answer: The single biggest obstacle to real progress in the philosophy of mathematics is a lack of reflection on which questions should be addressed with what methods. And this is particularly apparent in the role that formal tools have in the discipline as it is today. Formal tools encompass paradigmatically formal, artificial languages, formal logic expressed with such languages, and mathematical proofs about such languages. These tools were developed during the rise of logic over 100 years ago, and they are ubiquitous in the philosophy of mathematics today. But even though formal tools have been used with great success in other parts of inquiry, in the philosophy of mathematics they have done a lot of harm, besides quite a bit of good. In my contribution to this volume, I would thus like to think a bit about what role formal tools should have in the philosophy of mathematics. I will argue that they should have merely a secondary role, unless one holds certain substantial views in the philosophy of mathematics, ones that very few people hold.