I told a guest at a recent party that I use mathematics to try to understand migraines. She thought that I ask migraine sufferers to do mental arithmetic to alleviate their symptoms. Of course, what I really do is use mathematics to understand the biological causes of migraines.
My work is possible because of a stunning fact we often overlook: the world can be understood mathematically. The party goer's misconception reminds us that this fact is not obvious. In this article I want to discuss a big question: “why can maths be used to describe the world?”, or to extend it more provocatively, “why is applied maths even possible?” To do so we need to review the long history of the general philosophy of mathematics — what I will loosely call metamaths.
Before we go any further, we should be clear on what we mean by applied mathematics. I will borrow a definition given by an important applied mathematician of the 20th and 21st centuries, Tim Pedley, the GI Taylor Professor of Fluid Mechanics at the University of Cambridge. In his Presidential Address to the Institute of Mathematics and its Applications in 2004, he said “Applying mathematics means using a mathematical technique to derive an answer to a question posed from outside mathematics.” This definition is deliberately broad — including everything from counting change to climate change — and the very possibility of such a broad definition is part of the mystery we are discussing.
The question of why mathematics is so applicable is arguably more important than any other question you might ask about the nature of mathematics. Firstly, because applied mathematics is mathematics, it raises all the same issues as those traditionally arising in metamaths. Secondly, being applied, it raises some of the issues addressed in the philosophy of science. I suspect that the case could be made for our big question being in fact the big question in the philosophy of science and mathematics. However, let us now turn to the history of metamaths: what has been said about mathematics, its nature and its applicability?