Towards a Philosophy of Applied Mathematics

Abstract

If there is any trend in contemporary philosophy of mathematics worthy of the label “New Wave”, it is surely the call to turn our attention to the practices, priorities and developments that are prized by working mathematicians. Most actual mathematicians, unsurprisingly, have little interest in the questions that have dominated the philosophy of mathematics since the 1960s. Few mathematicians, for example, are likely to be troubled by Benacerraf’s argument in “What Numbers Could Not Be” that numbers are not objects for the simple reason that they typically do their mathematics without worrying about what numbers might be. Similarly, the extended debates between platonists and nominalists, and the associated epistemo-logical worries about our knowledge of ordinary mathematics, have little impact on what is usually called “mathematical practice”.

Most contemporary philosophy of mathematics focuses on a small segment of mathematics, mainly the natural numbers and foundational disciplines like set theory. Although there are good reasons for this approach, in this chapter, I will examine the philosophical problems associated with the area of mathematics known as applied mathematics. Here mathematicians pursue mathematical theories that are closely connected to the use of mathematics in the sciences and engineering. This area of mathematics seems to proceed using different methods and standards when compared to much of mathematics. I argue that applied mathematics can contribute to the philosophy of mathematics and our understanding of mathematics as a whole.