Before creation, God did just pure mathematics. Then He thought it would be a pleasant change to do some applied.Collection: Would Be

Try a hard problem. You may not solve it, but you will prove something else.Collection: Trying

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..."Collection: Friendship

The infinitely competent can be uncreative.Collection: Math

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.Collection: Teaching

I listen only to Bach, Beethoven or Mozart. Life is too short to waste on other composers.Collection: Life Is Too Short

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.Collection: Change

The first test of potential in mathematics is whether you can get anything out of geometry.Collection: Tests

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 1:10. Mathematics is a dangerous profession; an appreciable proportion of us go mad, and then this particular event would be quite likely.Collection: Learning

Mathematics is a dangerous profession; an appreciable proportion of us go mad.Collection: Mad

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: 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.Collection: Two

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".Collection: Learning

A good mathematical joke is better, and better mathematics, than a dozen mediocre papers.Collection: Math

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.Collection: Thinking

A precisian professor had the habit of saying: "... quartic polynomial ax^4+bx^3+cx^2+dx+e , where e need not be the base of the natural logarithms."Collection: Math

The surprising thing about this paper is that a man who could write it would.Collection: Writing

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.Collection: Math

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.Collection: People

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.Collection: Math

A Miscellany is a collection without a natual ordering relation.Collection: Mathematics

A heavy warning used to be given that pictures are not rigorous; this has never had its bluff called and has permanently frightened its victims into playing for safety.Collection: Safety

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.)Collection: Strong

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".Collection: Running

The first lecture of each new year renews for most people a light stage fright.Collection: New Year

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.Collection: Book