"A Mathematician's Apology" by G.H. Hardy (1940)

In mathematics, there is no political correctness, just correctness. A mistake is unlikely to offend or mislead many and for long. Hardy writes as one states mathematical conjectures: clearly, confidently, irreverently.

Mathematics is a young men's game. Most games are. But the ageing of one's mind need not entail an abrupt withdrawal from creation, and hence from life. Even if having practised mathematics has initiated a now retired mathematician into the beauty that is unmatched in other pursuits, he can still devote the remainder of his life to cultivating beauty in the non-mathematical world. Literature differs from mathematics only in degree, not substance. Flaws in the argument are easier to detect in mathematics. Literature's value resembles that of a mathematical conjecture with a proof sketch: possibly false, definitely incomplete, but inspiring. (Poetry is a conjecture without a proof sketch.)

Hardy denies to pure mathematics any utility other than its aesthetic pleasure, shared with art. Even though once-pure mathematics eventually finds practical applications, these unforeseen applications do not motivate pure mathematicians. Beauty, escapism, and competition do. A beautiful theorem is beautiful in its statement and in its proof, which link simply hitherto disparate ideas. An escape into mathematics is more credible than an escape into art. In art, one invents a world. In mathematics, one dreams of a world and then proves its existence.

The lucid minds who seek inapplicable beauty are not wasted. One would not wish to arrest the mutation of genes just because the natural selection were not forward-looking. Similarly, one should not arrest the work of a pure mathematician just because he is unable to foresee its applications. Insistence on applications will discourage him, instead of convincing him to change his topic or occupation.

"The noblest ambition is that of leaving behind one something of permanent value." Hardy believes that one can leave something of value only by associating one's name with a discovery. But name-recognition is not necessary for immortality. One can contribute to a civilization by enriching the lives of others in trite, anonymous ways, which will enable others to discover.

Hardy derides expositors and critics. Yet, a mathematical proof is the exposition of a theorem's statement. Intuition for a proof is its criticism, which can be viewed unfavourably, as the proof's corruption for the sake of the illusion of understanding, or favourably, as a puzzle explaining some steps in the proof and leaving intermediate steps to the reader. Even if criticism is the work of second-rate minds, it enables third-rate minds glimpse the beauty that would have otherwise been restricted to first-rate minds.