John Edensor Littlewood Quotes

I had been struggling for two months to prove a result I was pretty sure was true. When I was walking up a Swiss mountain, fully occupied by the effort, a very odd device emerged - so odd that, though it worked, I could not grasp the resulting proof as a whole. But not only so; I had a sense that my subconscious was saying, 'Are you never going to do it, confound you; try this.'

The difference when you do know (when, for example, we are looking for a new proof) is enormous. Like a Bridge problem 'if the thing is possible it must be necessary to lead the Ace and trump with the Ace in Dummy.'

My research began, naturally, in the Long Vacation of my 3rd year, 1906. My director of studies (and tutor) E. W. Barnes suggested the subject of integral functions of order 0... [After success,] Barnes was now encouraged to suggest a new problem: 'Prove the Riemann Hypothesis'.

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.

The Astronomer's fallacy. It is very hard to make a random selection of stars. If, for example, you see a star (with the naked eye) it is probably bright (as stars go). A lecturer was once making the point that middle class families were smaller than lower class ones. As a test he asked everyone to write down the number of children in his family. The average was larger than the lower class average. The obvious point he overlooked were that zero families were unrepresented in the audience. But further, families of n have a probability of being represented proportional to n; with all this, the result is to be expected.

There is much to be said for being a mathematician. To begin with, he has to be completely honest in his work, not from any superior morality, but because he simply cannot get away with a fake. ... A mathematician's normal day contains hours of close concentration, and leaves him jaded in the evening. ... This is why we tend to relax either on mild nonfiction like biographies, or - to be crude, and to the derision of arts people - on trash. There is, of course, good trash and bad trash. ... Minor depressions will occur, and most of a mathematician's life is spent in frustration, punctuated with rare inspirations. A beginner can't expect quick results; if they are quick they are pretty sure to be poor. ... When one has finished a substantial paper there is commonly a mood in which it seems that there is really nothing in it. Do not worry, later on you will be thinking 'At least I could do something good then.' At the end of a particularly long and exacting work there can be a strange melancholy. This, however, is romantic, and mildly pleasant, like some other melancholies.

I began on a question on elementary theory of numbers, in which I felt safe in my school days. It did not come out, nor did it on a later attack. I had occasion to fetch more paper; when passing a desk my eye lit on a heavy mark against the question. The candidate was not one of the leading people, and I half unconsciously inferred that I was making unnecessarily heavy weather; the question then came out fairly easily. The perfectly high-minded man would no doubt have abstained from further attack; I wish I had done so, but the offence does not lie very heavily on my conscience.

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...' (What had happened was that Hardy had received the remark in silence and with a poker face, and I wrote it off as a dud....)

G. H. Hardy said he thought on paper ('with my pen'). He wrote everything out (in his invariably admirable handwriting), scrapping and copying whenever a page got into a mess. When I am thinking about a difficult problem everything goes onto a single page - all over the place with odd equations, diagrams, rings. However appalling the mess, I feel that to scrap this page would somehow break threads in the unconscious.

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. ... There must exist a collection of well-authenticated coincidences, and I regret that I am not better acquainted with them. ... I sometimes ask the question: what is the most remarkable coincidence you have experienced, and is it, for the most remarkable one, remarkable? (With a lifetime to choose from, 10⁶ : 1 is a mere trifle.) ... Eddington once told me that information about a new (newly visible, not necessarily unknown) comet was received by an Observatory in misprinted form; they looked at the place indicated (no doubt sweeping a square degree or so), and saw a new comet. ...

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 possibly more fruitful, approach. (So a rock climber might force a dreadful crack, instead of finding a subtle and delicate route.)

It was Mr. Littlewood (I believe) who remarked that "every positive integer was one of his personal friends."

Besicovitch and Harry Williams asked me what God was doing before the Creation. I said: 'Millions of words must have been written on this; but he was doing Pure Mathematics and thought it would be a pleasant change to do some Applied.'

(A. S. Besicovitch) A mathematician's reputation rests on the number of bad proofs he has given. (Pioneer work is clumsy.)

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.