By G.H. Hardy. Find the book.
At his best, he said, he was for a short time the fifth best pure mathematician in the world.
— from the Forward, by C.P. Snow
Together they [Littlewood and Hardy] produced nearly a hundred papers, a good many of them “in the Bradman class”. Mathematicians not intimate with Hardy in his later years, nor with cricket, kept repeating that his highest term of praise was “in the Hobbs class”. It wasn’t: very reluctantly, because Hobbs was one of his pets, he had to alter the order of merit. I once had a postcard from him, probably in 1938, saying “Bradman is a whole class above any batsman who ever lived: if Archimedes, Newton and Gauss remain in the Hobbs class, I have to admit the possibility of a class above them, which I find difficult to imagine. They had better be moved from now on into the Bradman class.”
— from the Forward, by C.P. Snow
Good work is not done by “humble” men. It is one of the first duties of a professor, for example, in any subject, to exaggerate a little both the importance of his subject and his own importance within it. A man who is always asking “Is what I do worth while?” and “Am I the right person to do it” will always be ineffective himself and a discouragement to others. 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.
— Part 2, p. 66
No mathematician should ever allow himself to forget that mathematics, more than any other art or science, is a young man’s game. To take a simple illustration at a comparatively humble level, the average age of election to the Royal Society is lowest in mathematics.
We can naturally find much more striking illustrations. We may consider, for example, the career of a man who was certainly one of the world’s three greatest mathematicians. Newton gave up mathematics at fifty, and had lost his enthusiasm long before; he had recognized no doubt by the time that he was forty his great creative days were over. His greatest ideas of all, fluxions and the law of gravitation, came to him about 1666, when he was twenty-four–‘in those days I was in the prime of my age for invention, and minded mathematics and philosophy more than at any time since’. He made big discoveries until he was nearly forty (the ‘elliptic orbit’ at thirty-seven, but after that he did little but polish and perfect.
Galois died at twenty-one, Abel at twenty-seven, Ramanujan at thirty-three, Riemann at forty. There have been men who have done great work a good deal later; Gauss’s great memoir on differential geometry was published when he was fifty (though he had had the fundamental ideas ten years before). I do not know an instance of a major mathematical advance initiated by a man past fifty.
— Part 4, pp. 70-72
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 Atilla or Napoleon: but the noblest ambition is that of leaving behind one something of permanent value–
Here, on the level sand, Between the sea and land, What shall I build or write, Against the fall of night?
The me of runes to grave That hold the bursting wave, Or bastions to design For longer date than mane.
— Part 7, p. 77
… on the whole the history of science is fair, and this is particularly true of mathematics. No other subject has such clear-cut or unanimously accepted standards, and the men who are remembered are almost always the men who merit it. Mathematical fame, if you have the cash to pay for it, is one of the soundest and steadiest of investments.
— Part 8, p. 82
A mathematician, like a painter or a poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because theirs are made with ideas. A painter makes patterns with shapes and colours, a poet with words. A painting may embody an “idea”, but the idea is usually commonplace and unimportant. In poetry, ideas count for a good deal more; but, as Housman insisted, the importance of ideas in poetry is habitually exaggerated: “I cannot satisfy myself that there are any such things as poetical ideas… Poetry is not the thing said but a way of saying it.”
Not all the water in the rough rude sea Can wash the balm from an anointed King
Could any lines be better, and could ideas be at once more trite and more false?
— Part 10, p. 84
It is undeniable that a good deal of elementary mathematics–and I use the word “elementary” in the sense in which professional mathematicians use it, in which it includes, for example, a fair working knowledge of the differential and integral calculus–has considerable practical utility. These parts of mathematics are, on the whole, rather dull; they are just the parts which have least aesthetic value. The “real” mathematics of the “real” mathematicians, the mathematics of Fermat and Euler and Gauss and Abel and Riemann, is almost wholly “useless” (and this is as true of “applied” as or “pure” mathematics). It is not possible to justify the life of any genuine professional mathematician on the ground of the “utility” of his work.
— Part 21, pp. 119-120
It may also be urged … that the equalization of risks which science was expected to bring would be in the long run salutary; that a civilian’s life is not worth more than a soldier’s, nor a woman’s than a man’s; that anything is better than the concentration of savagery on one particular class; and that, in short, the sooner war comes “all out” the better.
— Part 28, p. 142