A Mathematician's Apology
I finally read A Mathematician’s Apology, G.H. Hardy’s classic defense of a lifetime dedicated to the study of pure (“impractical”) mathematics. It’s a remarkably sad book, in which Hardy, near the end of his life, famously describes mathematics as a “young man’s pursuit”^{1} in which the elderly have little to contribute. However, it also contains some really wellcomposed thoughts:
A man who is always asking, “Is what I do worth while?” and “Am I the right person to do it?” will always be ineffective himself and a discouragement to others. He must shut his eyes a little and think a little more of his subject and himself than they deserve.
The mathematician’s patterns, like the painter’s or the poet’s, must be beautiful; the ideas, like the colours or the words, must fit together in a harmonious way. Beauty is the first test: there is no permanent place in the world for ugly mathematics. … It may be very hard to define mathematical beauty, but that is just as true of beauty of any kind—we may not know quite what we mean by a beautiful poem, but that does not prevent us from recognizing one when we read it.

The usual depressing formulation of Hardy’s rule is that, “if a mathematician’s going to do any significant work, it’ll be done before they’re thirty.” This is true so long as we ignore the later work of Archimedes, Cauchy, Descartes, Euler, Fermat, Frege, Gauss, Hilbert, Newton, Peano, Poincare, Russell, von Neumann, Weierstrass, Erdős, and most recently Andrew Wiles. I would guess that Hardy’s opinion on the matter was influenced by his relationship with the mathematical prodigy Ramanujan, who died at 33. ↩