## What the applicability of mathematics says about its philosophy

We use mathematics to understand the world. This fact lies behind all of modern science and technology. Mathematics is the tool used by physicists, engineers, biologists, neuroscientists, chemists, astrophysicists and applied mathematicians to investigate, explain, and manipulate the world around us. The importance of mathematics to science cannot be overstated. It is the daily and ubiquitous tool of millions of scientists and engineers throughout the world and in all areas of science. The undeniable power of mathematics not only to predict but also to explain phenomena is what physics Nobel laureate Eugene Wigner dubbed the “unreasonable effectiveness of mathematics in the natural sciences” (Wigner, 1960). Yet the success of mathematics in explaining the world belies a great mystery: why is that possible? Why are our abstract thought and our manipulation of symbols able to successfully explain the workings of distant stars, the patterns of stripes on a tiger, and the weirdest behaviour of the smallest units of matter? Why is applying mathematics to the real world even possible? This is a question in the philosophy of mathematics. The traditional approach to answering it is to first decide (hopefully on rational grounds) what to believe about the nature of mathematics and its objects of study, and then to explore what this philosophical standpoint says about the applicability of mathematics to the world. In this chapter, I take a different approach. I take as given the existence of applied mathematics. On this foundational axiom, I ask the question “what does the existence of applied mathematics say about the philosophy of mathematics?” In this way, we treat the existence of applied mathematics as a lens through which to examine competing claims about the nature of mathematics. What then do we mean by the existence of applied mathematics, by the philosophy of mathematics, and what are the claims on the nature of mathematics?