A Mathematician's Apology -- notes and quotes

After listening to The Man Who Knew Infinity, a biography of Ramanujan, I was inspired to read A Mathematician's Apology by G. H. Hardy. I remember my dad mentioning it many years ago. It's only 90 short pages, and here are a few highlights, for me. It's worth noting, I think, that this was written in 1940 by a man who had spent his entire life at the most prestigious university in England and almost literally without women, except for his sister. If you can enjoy it despite his elitism and sexism, it's a fun read.

Exposition, criticism, appreciation, is work for second-rate minds.

Did I just hear someone say my name? ;)

I write about mathematics because, like any other mathematician who has passed sixty, I have no longer the freshness of mind, the energy, or the patience to carry on effectively with my proper job.

Good work is not done by 'humble' men. . . . He must shut his eyes a little and think a little more of his subject and himself than they deserve. This is not too difficult: it is harder not to make his subject and himself ridiculous by shutting his eyes too tightly.

'I do what I do because it is the one and only thing that I can do at all well.'

But:

most people can do nothing at all well. . . . perhaps five or even ten per cent of men can do something rather well. It is a tiny minority who can do anything really well, and the number of men who can do two things well is negligible. If a man has any genuine talent, he should be ready to make almost any sacrifice in order to cultivate it to the full.

Thankfully, I lack the genius to require that kind of sacrifice. (I think the gifted teacher must be the counterexample to Hardy's 'second-rate minds' barb.) Or perhaps instead of doing math because you are good at it, you do it because it came your way and you might as well:

'There is nothing that I can do particularly well. . . ' . . . most people can do nothing well . . . it matters very little what career they choose.

Approximately what Hardy is going to argue:

• Math is worthwhile even though it isn't practical (despite his recognizing practical applications of some math).
• People who are better at math than anything else should do it even if it ends up being a waste of time.
• Doing something of permanent value, even if it is small, is worthwhile and unusual. Most people don't do anything of permanent value.

The ambition to leave something permanent behind is the greatest ambition, and his target audience is those who agree with this. I once would have. Now I view it as one admirable thing, when balanced with other ethical considerations.

Three driving motivations for research:

• Intellectual curiosity
• Professional pride
• Desire for reputation, power, and/or money

I encountered a recent news article that cited research claiming that science is a reputation economy. Scientists are more interested in reputation than money or power (on average), and I can believe that. Of course, it's a sliding scale.

Math is perhaps the most permanent achievement, since languages die, but math remains. And math is pretty good about giving credit to the people who really did it.

Now for Hardy on art:

A painting may embody an 'idea', but the idea is usually commonplace and unimportant. . . . the importance of ideas in poetry is habitually exaggerated. . . . The poverty of the ideas seems hardly to affect the beauty. . . .

Math lasts longer, but

there is no permanent place in the world for ugly mathematics.

Chess problems are the hymn-tunes of mathematics. [low level of beauty]

As regards applied math:

The 'seriousness' of a mathematical theorem lies, not in its practical consequences, which are usually negligible. . .

Serious math connects many, complex mathematical ideas. Other traits that make math meaningful are depth, beauty, generality, unexpectedness, inevitablity, and economy.

Most useful math is boring, small, and mathematically unimportant. I can concur with this, since my field uses quite a lot of math, but it really is mostly boring, small, specific applications of ideas with much broader mathematical richness. Unexpected pieces of "pure" math do at times become useful, at times, but they are still small pieces.

Hardy closes equating the value of math with the value of art:

The case for my life, then, or for that of any one else who has been a mathematician in the same sense in which I have been one, is this: that I have added something to knowledge, and helped others to add more; and that these somethings have a value which differs in degree only, and not in kind, from that of the creations of the great mathematicians, or of any of the other artists, great or small, who have left some kind of memorial behind them.

I had expected to resonate with Hardy's apology more than I did, but some parts of it rang very true to me. I do aspire to leave something behind. I do hope that it will be unique and beautiful in its sphere. And I feel like I have had only one idea that even approaches the category of mathematical thought. It was the question I asked when I imagined, what if Gods evolved? It's an idea so inevitable that atheists like Richard Dawkins and Sam Harris admit the possibility, but reject it's importance and miss its implications. Yet it is an idea that ties together Mormonism and Transhumanism, religion and science, faith and the future of humanity. It accepts that Gods are limited, but goes beyond to reveal what some limits are. It answers the problem of evil--maybe not as anyone would wish, but very cleanly. It claims an empathetic middle ground between arbitrary universalism and eternal damnation. It leaves a place for God to act and to be hidden. It brings atonement into the realms of nature, and makes salvation about relationships, not satisfying unchanging justice. It creates Gods in our own image, but not falsely idealized. It may come to nothing--there's no eternity for bad philosophy--but it is surprising and beautiful to me.