★★★★☆ "The best account of what it was like to be a creative artist"

A Mathematician's Apology

G.H. Hardy

First, a word about the title. "Apology" is not here used in the most common modern sense of "an admission of error or discourtesy accompanied by an expression of regret", bur rather in the older sense of "a formal explanation or defense of a belief or system, especially one that is unpopular". G.H. Hardy is in no way expressing regret or admitting error for being a mathematician. In fact, he makes it clear that it is the thing in his life that he is most proud of. His purpose is to explain for non-mathematicians what a mathematician does and why it is valuable. Briefly, this is what he has to say

A mathematician, like a painter or a poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas...
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.

If you take that and nothing else away from Apology, you will have learned the main lesson. Graham Greene described A Mathematician's Apology as one of "the best accounts of what it is like to be a creative artist."

Mathematics is, to my knowledge, the only profession whose first concerns are explicitly truth and beauty. (I do not except the arts. I have never met an artist who would agree that beauty was a goal of his or her work.)

Hardy knew, of course, that it was not adequate to simply assert the importance of beauty. He attempts therefore to demonstrate it, and it is here, I fear, that Apology fails. To this purpose he presents two classical theorems, Euclid's proof that the number of primes is infinite, and the proof that there is no rational number whose square is two. Hardy believes that the capacity to perceive and appreciate mathematical beauty is widespread, and that these two theorems will be perceived as beautiful by his readers, allowing him to demonstrate its nature.

I fear he is too optimistic. I suspect the vast majority of people will not put in the effort to understand these proofs, simple though they are, and even if they do, will not perceive any beauty therein. Apology is not, in my opinion, a book you can give to non-mathematicians to explain math to them, except for those who are already more than halfway there.

Hardy also spends some time on the "usefulness" of math. The money quote is "very little of mathematics is useful practically, and that that little is comparatively dull." This argument has not aged well. For instance, he writes

If useful knowledge is, as we agreed provisionally to say, knowledge which is likely, now or in the comparatively near future, to contribute to the material comfort of mankind, so that mere intellectual satisfaction is irrelevant, then the great bulk of higher mathematics is useless. Modern geometry and algebra, the theory of numbers, the theory of aggregates and functions, relativity, quantum mechanics—no one of them stands the test much better than another.

In fact, in the 84 years since he published those sentences, quantum mechanics gave us light-emitting-diodes and transistors, transistors gave us high-speed computers, and number theory gave us public-key encryption standards. Reading this, you are using results of these mathematical disciplines.

Still, Hardy's main point, that math is not to be justified by its practical usefulness, but by its beauty, holds.

This edition of Apology begins with a biographical introduction by C.P. Snow, a good friend of Hardy's and a successful writer. This introduction does perhaps a better job of presenting Hardy's arguments than Hardy himself does.

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