A Mathematician’s Apology

Key Takeaways

1. As I had plenty of opportunities to realize in the future, Hardy had no faith in intuitions or impressions, his own or anyone else’s. The only way to assess someone’s knowledge, in Hardy’s view, was to examine him.
2. I owed my friendship with Hardy to having wasted a disproportionate amount of my youth on cricket. I don’t know what the moral is. But it was a major piece of luck for me. This was intellectually the most valuable friendship of my life. His mind, as I have just mentioned, was brilliant and concentrated: so much so that by his side anyone else’s seemed a little muddy, a little pedestrian and confused. He wasn’t a great genius, as Einstein and Rutherford were. He said, with his usual clarity, that if the word meant anything he was not a genius at all. At his best, he said, he was for a short time the fifth best pure mathematician in the world.
3. There is something else, though, at which he was clearly superior to Einstein or Rutherford or any other great genius: and that is at turning any work of the intellect, major or minor or sheer play, into a work of art. It was that gift above all, I think, which made him, almost without realizing it, purvey such intellectual delight.
4. He was the classical anti-narcissist. He could not endure having his photograph taken: so far as I know, there are only five snapshots in existence.
5. His life remained the life of a brilliant young man until he was old: so did his spirit: his games, his interests, kept the lightness of a young don’s. And, like many men who keep a young man’s interests into their sixties, his last years were the darker for it.
6. His inner life was his own, and very rich. The sadness came at the end. Apart from his devoted sister, he was left with no one close to him.
7. Four hours creative work a day is about the limit for a mathematician, he used to say.
8. Hardy was, in Newton’s phrase, ‘in the prime of his age for invention’, and this came in his early forties, unusually late for a mathematician.
9. Underneath his shyness, he just didn’t give a damn.
10. He would have been the first to disclaim that he had any special psychological insight. But he was the most intelligent of men, he had lived with his eyes open and read a lot, and he had obtained a good generalized sense of human nature—robust, indulgent, satirical, and utterly free from moral vanity. He was spiritually candid as few men are (I doubt if anyone could be more candid), and he had a mocking horror of pretentiousness, self-righteous indignation, and the whole stately pantechnicon of the hypocritical virtues.
11. ‘It is never worth a first class man’s time to express a majority opinion. By definition, there are plenty of others to do that.’
12. Sometimes one has to say difficult things, but one ought to say them as simply as one knows how.
13. That is why A Mathematician’s Apology is, if read with the textual attention it deserves, a book of haunting sadness. Yes, it is witty and sharp with intellectual high spirits: yes, the crystalline clarity and candour are still there: yes, it is the testament of a creative artist.
14. There is no scorn more profound, or on the whole more justifiable, than that of the men who make for the men who explain. Exposition, criticism, appreciation, is work for second-rate minds.
15. Good work is not done by ‘humble’ men.
16. A man who sets out to justify his existence and his activities has to distinguish two different questions. The first is whether the work which he does is worth doing; and the second is why he does it, whatever its value may be. Their answers, if they are honest, will usually take one or other of two forms; and the second form is merely a humbler variation of the first, which is the only answer which we need consider seriously. (I) ‘I do what I do because it is the one and only thing that I can do at all well.
17. If a man has any genuine talent, he should be ready to make almost any sacrifice in order to cultivate it to the full.
18. No mathematician should ever allow himself to forget that mathematics, more than any other art or science, is a young man’s game.
19. It is quite true that most people can do nothing well. If so, it matters very little what career they choose, and there is really nothing more to say about it. It
20. What we do may be small, but it has a certain character of permanence; and to have produced anything of the slightest permanent interest, whether it be a copy of verses or a geometrical theorem, is to have done something utterly beyond the powers of the vast majority of men. And—
21. A man’s first duty, a young man’s at any rate, is to be ambitious. Ambition is a noble passion which may legitimately take many forms; there was something noble in the ambition of Attila or Napoleon: but the noblest ambition is that of leaving behind one something of permanent value
22. Here, on the level sand, Between the sea and land, What shall I build or write Against the fall of night?   Tell me of runes to grave That hold the bursting wave, Or bastions to design For longer date than mine.
23. Ambition has been the driving force behind nearly all the best work of the world. In particular, practically all substantial contributions to human happiness have been made by ambitious men.
24. We must guard against a fallacy common among apologists of science, the fallacy of supposing that the men whose work most benefits humanity are thinking much of that while they do it,
25. There are many highly respectable motives which may lead men to prosecute research, but three which are much more important than the rest. The first (without which the rest must come to nothing) is intellectual curiosity, desire to know the truth. Then, professional pride, anxiety to be satisfied with one’s performance, the shame that overcomes any self-respecting craftsman when his work is unworthy of his talent. Finally, ambition, desire for reputation, and the position, even the power or the money, which it brings. It may be fine to feel, when you have done your work, that you have added to the happiness or alleviated the sufferings of others, but that will not be why you did
26. 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.
27. The best mathematics is serious as well as beautiful—‘important’ if you like, but the word is very ambiguous, and ‘serious’ expresses what I mean much better. I am not thinking of the ‘practical’ consequences of mathematics. I have to return to that point later: at present I will say only that if a chess problem is, in the crude sense, ‘useless’, then that is equally true of most of the best mathematics; that very little of mathematics is useful practically, and that that little is comparatively dull. The ‘seriousness’ of a mathematical theorem lies, not in its practical consequences, which are usually negligible, but in the significance of the mathematical ideas which it connects. We may say, roughly, that a mathematical idea is ‘significant’ if it can be connected, in a natural and illuminating way, with a large complex of other mathematical ideas. Thus a serious mathematical theorem, a theorem which connects significant ideas, is likely to lead to important advances in mathematics itself and even in other sciences. No chess problem has ever affected the general development of scientific thought; Pythagoras, Newton, Einstein have in their times changed its whole direction.
28. I said that a mathematician was a maker of patterns of ideas, and that beauty and seriousness were the criteria by which his patterns should be judged.
29. There are two things at any rate which seem essential, a certain generality and a certain depth; but neither quality is easy to define at all precisely. A significant mathematical idea, a serious mathematical theorem, should be ‘general’ in some such sense as this. The idea should be one which is a constituent in many mathematical constructs, which is used in the proof of theorems of many different kinds. The theorem should be one which, even if stated originally (like Pythagoras’s theorem) in a quite special form, is capable of considerable extension and is typical of a whole class of theorems of its kind. The relations revealed by the proof should be such as connect many different mathematical ideas. All this is very vague, and subject to many reservations. But it is easy enough to see that a theorem is unlikely to be serious when it lacks these qualities conspicuously;
30. The second quality which I demanded in a significant idea was depth, and this is still more difficult to define. It has something to do with difficulty; the ‘deeper’ ideas are usually the harder to grasp: but it is not at all the same. The ideas underlying Pythagoras’s theorem and its generalizations are quite deep, but no mathematician now would find them difficult. On the other hand a theorem may be essentially superficial and yet quite difficult to prove (as are many ‘Diophantine’ theorems, i.e. theorems about the solution of equations in integers). It seems that mathematical ideas are arranged somehow in strata, the ideas in each stratum being linked by a complex of relations both among themselves and with those above and below. The lower the stratum, the deeper (and in general the more difficult) the idea. Thus the idea of an ‘irrational’ is deeper than that of an integer; and Pythagoras’s theorem is, for that reason, deeper than Euclid’s.
31. In both theorems (and in the theorems, of course, I include the proofs) there is a very high degree of unexpectedness, combined with inevitability and economy.
32. For my own part I have never once found myself in a position where such scientific knowledge as I possess, outside pure mathematics, has brought me the slightest advantage. It is indeed rather astonishing how little practical value scientific knowledge has for ordinary men, how dull and commonplace such of it as has value is, and how its value seems almost to vary inversely to its reputed utility. It is useful to be tolerably quick at common arithmetic (and that, of course, is pure mathematics). It is useful to know a little French or German, a little history and geography, perhaps even a little economics. But a little chemistry, physics, or physiology has no value at all in ordinary life.
33. High thinking of one kind is always likely to affect high thinking of another
34. I cannot remember ever having wanted to be anything but a mathematician. I suppose that it was always clear that my specific abilities lay that way, and it never occurred to me to question the verdict of my elders. I do not remember having felt, as a boy, any passion for mathematics, and such notions as I may have had of the career of a mathematician were far from noble. I thought of mathematics in terms of examinations and scholarships: I wanted to beat other boys, and this seemed to be the way in which I could do so most decisively.
35. A mathematician may still be competent enough at sixty, but it is useless to expect him to have original ideas.
36. 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.

What I got out of it

1. This book shows the power and pleasure that can come from mathematics. His focus on ambition, doing important work, cultivating that craft through deep work, what makes for good mathematics (being serious and beautiful, general and deep), mathematicians are makers of patterns. Really beautiful language and a fascinating story I knew nothing about.