By G. H. Hardy, 1940, 153 pages, partial text, amazon
I propose to put forward an apology for mathematics; and I may be told that it needs none, since there are now few studies more generally recognized, for good reason or bad, as profitable or priceworthy… A mathematician does need not now consider himself on the defensive… The public does not need to be convinced that there is something in mathematics. All this is in its way very comforting to mathematicians, but it is hardly possible to be content with it. Any genuine mathematician must feel that it is not on … crude achievements that the real case for mathematics rests, that the popular reputation of mathematics is based largely on ignorance and confusion, and that there is room for more rational defence. At any rate, I am disposed to try to make one.
And that is what he gives in this tiny book. More than a third of it is a foreword by C. P. Snow, which introduces Hardy to the reader, introduces him as a mathematician proud of what he’s doing, person with very delicate and unorthodox thinking who places high expectations on everyone and highest on himself.
Then you start reading the essay itself and whether you are a mathematician or not, you’ll be absorbed. For non-mathematicians this is an amazing opportunity to peek into mathematician’s way of thinking, reasoning, motivation and notion of beauty. For mathematician (in addition to all of the above) this is a unique verbalization (sparse, but very much to the point) of what they understand on an intuitive level, but would probably have a hard time expressing so clearly.
Even though you might not agree with all of Hardy’s ideas (I don’t, especially with the those concerning applied mathematics… which is probably due to the fact that applied mathematics is today something very different from what Hardy considered it to be), you cannot help yoursel but admire the way he presents his points.
And what is he actually apologizing for? You need to read that for yourself. Highly recommended!
PS On a similar note (but not as beautifully written), from much younger (but at least as talented) mathematician Terry Tao comes an article What is good mathematics?