What is Mathematics?

“Richard Courant and Herbert Robbins wrote an entire book attempting to answer this question. When I was a student or even a junior faculty member, I surely would have responded differently than I do now. But now, however, I think of Mathematics as that area where, through curiosity and observations, one looks for, hopes to find and then verifies the existence of patterns that appear in nature or in the abstract.” —Gary Chartrand, author of Graphs and Digraphs, Sixth Edition

G. H. Hardy was one of the world's leading mathematicians in the first half of the 20th century. In his book "A Mathematician's Apology" he elaborates at length on differences between pure and applied mathematics. He discusses two examples of (beautiful) pure math problems. These are problems that some middle school and high school students might well solve, but are quite different than the types of mathematics addressed in our current K-12 curriculum. Both of these problems were solved more than 2,000 years ago and are representative of what mathematicians do.

After the fall of Rome, the development of mathematics was taken on by the Arabs, then the Europeans. Fibonacci was one of the first European mathematicians and was famous for his theories on arithmetic, algebra, and geometry. The Renaissance led to advances that included decimal fractions, logarithms, and projective geometry. Number theory was greatly expanded upon, and theories of probability and analytic geometry ushered in a new age of mathematics, with calculus at the forefront.

One famous philosopher who finds mathematical necessity an inconvenience is Peter Singer. In one of his best-selling books on ethics, he argues that we cannot rely on intuiting ethical truths, since the most convincing case of intuition, in mathematics, is not correct. ‘The self-evidence of the basic truths of mathematics,’ he says, ‘could be explained… by seeing mathematics as a system of tautologies… true by virtue of the meanings of the terms used.’ Singer is wrong to claim that this philosophy of mathematics, called logicism, is ‘widely, if not universally accepted’. It has not been accepted by any serious philosopher of mathematics for 100 years. But it is clear why anyone who, like Singer, wishes to explain away the strange power of human intuition might want a deflationary philosophy of mathematics to be true.