Improbabilities are apt to be overestimated. It is true that I should have been surprised in the past to learn that Professor Hardy had joined the Oxford Group. But one could not say the adverse chance was 106 : 1. Mathematics is a dangerous profession; an appreciable proportion of us go mad, and then this particular event would be quite likely.
A good mathematical joke is better, and better mathematics, than a dozen mediocre papers.
I recall once saying that when I had given the same lecture several times I couldn't help feeling that they really ought to know it by now.
In passing, I firmly believe that research should be offset by a certain amount of teaching, if only as a change from the agony of research. The trouble, however, I freely admit, is that in practice you get either no teaching, or else far too much.
It is possible for a mathematician to be "too strong" for a given occasion. he forces through, where another might be driven to a different, and possible more fruitful, approach. (So a rock climber might force a dreadful crack, instead of finding a subtle and delicate route.)
I constantly meet people who are doubtful, generally without due reason, about their potential capacity [as mathematicians]. The first test is whether you got anything out of geometry. To have disliked or failed to get on with other [mathematical] subjects need mean nothing; much drill and drudgery is unavoidable before they can get started, and bad teaching can make them unintelligible even to a born mathematician.
The infinitely competent can be uncreative.
In presenting a mathematical argument the great thing is to give the educated reader the chance to catch on at once to the momentary point and take details for granted: his successive mouthfuls should be such as can be swallowed at sight; in case of accidents, or in case he wishes for once to check in detail, he should have only a clearly circumscribed little problem to solve (e.g. to check an identity: two trivialities omitted can add up to an impasse). The unpractised writer, even after the dawn of a conscience, gives him no such chance; before he can spot the point he has to tease his way through a maze of symbols of which not the tiniest suffix can be skipped.
A linguist would be shocked to learn that if a set is not closed this does not mean that it is open, or again that "E is dense in E" does not mean the same thing as "E is dense in itself".
The surprising thing about this paper is that a man who could write it would.
A precisian professor had the habit of saying: "... quartic polynomial ax4 + bx3 + cx2 + dx + e, where e need not be the base of the natural logarithms."
I read in the proof sheets of Hardy on Ramanujan: "As someone said, each of the positive integers was one of his personal friends." My reaction was, "I wonder who said that; I wish I had." In the next proof-sheets I read (what now stands), "It was Littlewood who said..."
We come finally, however, to the relation of the ideal theory to real world, or "real" probability. If he is consistent a man of the mathematical school washes his hands of applications. To someone who wants them he would say that the ideal system runs parallel to the usual theory: "If this is what you want, try it: it is not my business to justify application of the system; that can only be done by philosophizing; I am a mathematician". In practice he is apt to say: "try this; if it works that will justify it". But now he is not merely philosophizing; he is committing the characteristic fallacy. Inductive experience that the system works is not evidence.
The theory of numbers is particularly liable to the accusation that some of its problems are the wrong sort of questions to ask. I do not myself think the danger is serious; either a reasonable amount of concentration leads to new ideas or methods of obvious interest, or else one just leaves the problem alone. "Perfect numbers" certainly never did any good, but then they never did any particular harm.
The paper is very heavy going, and I should never have read it had I had not written it myself.
The first lecture of each new year renews for most people a light stage fright.
Before creation God did just pure mathematics. Then He thought it would be a pleasant change to do some applied.
I've been giving this lecture to first-year classes for over twenty-five years. You'd think they would begin to understand it by now.
The referee said it was not acceptable, but the Press considered they could not refuse to publish a book by a professor of the university.
I listen only to Bach, Beethoven or Mozart. Life is too short to waste on other composers.
The higher mental activities are pretty tough and resilient, but it is a devastating experience if the drive does stop. Some people lose it in their forties and can only stop. In England they are a source of Vice-Chancellors.
Try a hard problem. You may not solve it, but you will prove something else.
Mathematics is a dangerous profession; an appreciable proportion of us goes mad.
The first test of potential in mathematics is whether you can get anything out of geometry.