The Values of Mathematical Proofs

Proofs are central, and unique, to mathematics. They establish the truth of theorems and provide us with the most secure knowledge we can possess. It is thus perhaps unsurprising that philosophers once thought that the only value proofs have lies in establishing the truth of theorems. However, such a view is inconsistent with mathematical practice. If a proof’s only value is to show a theorem is true, then mathematicians would have no reason to reprove the same theorem in different ways, yet this is a practice they frequently engage in. This suggests that proofs can have a wide variety of values. My purpose in this chapter is to provide a survey of some of them. I will discuss how proofs can have practical value, for example, by helping mathematicians to make new discoveries or learn new mathematical tools, and how they can have abstract value, for example, by being beautiful or explanatory. I will also explore the relationships between different proof values.