What is Mathematics?

There is a popular series of books, all with titles along the lines of "XXXXX for the Rest of Us", which I guess, never having actually read one of them, seeks to explain whatever the particular XXXXX happens to be "in layman's terms". With mathematics, this approach might run into difficulty, because what a mathematician and what the rest of us think of as mathematics are two very different things. In his entertaining miscellany of the vain efforts of amateurs, Mathematical Cranks, Underwood Dudley quotes extensively in "Crank, Case Study of a", a letter from a professor of mathematics who was acquainted with the work of a very diligent crank:

"Mr [B.] apparently thinks, or thought at the outset of his endeavors, that mathematicians spend their time doing square roots, computing logarithms, and things of that nature, and that the discovery of a new way of doing square roots or computing logarithms constitutes a major mathematical discovery."

I suspect that this is not so very far from very many other people's idea of how a mathematician might spend his or her time, and if asked to name a famous mathematician, Carol Vorderman (who appeared for many years on the UK television program "Countdown" to solve arithmetical puzzles), or her equivalent in other countries, would be high on the list. For most people, mathematics is about solving particular problems numerically or logically. Typical problems might be:

One of the oldest mathematical documents still in existence, the Rhind Papyrus (c 1650 BC), contains a list of worked problems somewhat like this. For example problem 65 shows how to divide 100 loaves among 10 men, but with double portions for the boatman, foreman, and door-keeper. In reality, many mathematicians would be uninterested in, and possibly not even very good at, performing this kind of task.† I may be judging too much by my own experience here, but books aimed at computer programmers, such as Numerical Recipes in Fortran, often point out serious pitfalls in the methods usually taught for solving such problems, and books on Computer Graphics are often the best source of information on solving practical problems in geometry. Surprisingly little attention is given to solving problems numerically in many mathematical text books.

I found the following definition of the word mathematics at the start of Wikipedia's main article on mathematics:

Mathematics is the discipline that deals with concepts such as quantity, structure, space and change. It evolved, through the use of abstraction and logical reasoning, from counting, calculation, measurement and the study of the shapes and motions of physical objects. Mathematicians explore such concepts, aiming to formulate new conjectures and establish their truth by rigorous deduction from appropriately chosen axioms and definitions.

Let us read what a very eminent mathematician has to say about being a mathematician.