Proofs, applications, and other mathematical activities

Why is there a whole field of inquiry, a discipline if you like, called the philosophy of mathematics? This unusual question, the very title of this book, will not begin to be examined with care until Chapter 3, but two summary answers can be stated at once.

First, because of the experience of some demonstrative proofs, the experience of proving to one’s complete satisfaction some new and often unlikely fact. Or simply experiencing the power and conviction conveyed by a good proof that one is taught, that one reads, or has explained to one. How can mere words, mere ideas, sometimes mere pictures, have those effects?

Second, because of the richness of applications of mathematics, often derived by thinking at a desk and toying with a pencil. Or more poetically, in the words of the historian of science A. C. Crombie (1994 I, ix), ‘the enigmatic matching of nature with mathematics and of mathematics by nature’.

Thus this book is a series of philosophical thoughts about proofs, applications, and other mathematical activities.