generations of pure mathematicians. Hardy was an atheist, and makes his justification not to God but to his fellow man. Hardy initially submitted A Mathematician's Apology to Cambridge University Press with the intention of personally paying for its printing, but the Press decided to fund publication with an initial run of four thousand copies. Summary One of the main themes of the book is the beauty that mathematics possesses, which Hardy compares to painting and poetry. For Hardy, the most beautiful mathematics was that which had no practical applications in the outside world (pure mathematics) and, in particular, his own special field of number theory. Hardy contends that if useful knowledge is defined as knowledge which is likely to contribute to the material comfort of mankind in the near future (if not right now), so that mere intellectual satisfaction is irrelevant, then the great bulk of higher mathematics is useless. He justifies the pursuit of pure mathematics with the argument that its very "uselessness" on the whole meant that it could not be misused to cause harm. On the other hand, Hardy denigrates much of the applied mathematics as either being "trivial", "ugly", or "dull", and contrasts it with "real mathematics", which is how he ranks the higher, pure mathematics. Hardy expounds by commenting about a phrase attributed to Carl Friedrich Gauss that "Mathematics is the queen of the sciences and number theory is the queen of mathematics." Some people believe that it is the extreme non-applicability of number theory that led Gauss to the above statement about number theory; however, Hardy points out that this is certainly not the reason. If an application of number theory were to be found, then certainly no one would try to dethrone the "queen of mathematics" because of that. What Gauss meant, according to Hardy, is that the underlying concepts that constitute number theory are deeper and more elegant compared to those of any other branch of mathematics. Another theme is that mathematics is a "young man's game", so anyone with a talent for mathematics should develop and use that talent while they are young, before their ability to create original mathematics starts to decline in middle age. This view reflects Hardy's increasing depression at the wane of his own mathematical powers. For Hardy, real mathematics was essentially a creative activity, rather than an explanatory or expository one. Critiques Hardy's opinions were heavily