G. H. Hardy famous quotes
Last updated: Sep 5, 2024

I believe that mathematical reality lies outside us, that our function is to discover or observe it, and that the theorems which we prove, and which we describe grandiloquently as our "creations," are simply the notes of our observations.
 G. H. Hardy 
I am obliged to interpolate some remarks on a very difficult subject: proof and its importance in mathematics. All physicists, and a good many quite respectable mathematicians, are contemptuous about proof. I have heard Professor Eddington, for example, maintain that proof, as pure mathematicians understand it, is really quite uninteresting and unimportant, and that no one who is really certain that he has found something good should waste his time looking for proof.
 G. H. Hardy 
Good work is no 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 in 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.
 G. H. Hardy 
I am interested in mathematics only as a creative art.
 G. H. Hardy 
No mathematician should ever allow him to forget that mathematics, more than any other art or science, is a young man's game. ... Galois died at twentyone, Abel at twentyseven, Ramanujan at thirtythree, Riemann at forty. There have been men who have done great work later; ... [but] I do not know of a single instance of a major mathematical advance initiated by a man past fifty. ... A mathematician may still be competent enough at sixty, but it is useless to expect him to have original ideas.
 G. H. Hardy 
[P]ure mathematics is on the whole distinctly more useful than applied. For what is useful above all is technique, and mathematical technique is taught mainly through pure mathematics.
 G. H. Hardy 
It is not worth an intelligent man's time to be in the majority. By definition, there are already enough people to do that.
 G. H. Hardy 
Beauty is the first test: there is no permanent place in the world for ugly mathematics.
 G. H. Hardy 
A mathematician, like a painter or poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas.
 G. H. Hardy 
Archimedes will be remembered when Aeschylus is forgotten, because languages die and mathematical ideas do not. "Immortality" may be a silly word, but probably a mathematician has the best chance of whatever it may mean.
 G. H. Hardy 
Real mathematics must be justified as art if it can be justified at all.
 G. H. Hardy 
317 is a prime, not because we think so, or because our minds are shaped in one way rather than another, but because it is so, because mathematical reality is built that way.
 G. H. Hardy 
As Littlewood said to me once [of the ancient Greeks], they are not clever school boys or "scholarship candidates," but "Fellows of another college."
 G. H. Hardy 
Most people have some appreciation of mathematics, just as most people can enjoy a pleasant tune; and there are probably more people really interested in mathematics than in music. Appearances suggest the contrary, but there are easy explanations. Music can be used to stimulate mass emotion, while mathematics cannot; and musical incapacity is recognized (no doubt rightly) as mildly discreditable, whereas most people are so frightened of the name of mathematics that they are ready, quite unaffectedly, to exaggerate their own mathematical stupidity
 G. H. Hardy 
There is no scorn more profound, or on the whole more justifiable, than that of the men who make for the men who explain.
 G. H. Hardy 
A personâ€™s first duty, a young personâ€™s at any rate, is to be ambitious, and the noblest ambition is that of leaving behind something of permanent value.
 G. H. Hardy 
In these days of conflict between ancient and modern studies, there must surely be something to be said for a study which did not begin with Pythagoras, and will not end with Einstein, but is the oldest and the youngest of all.
 G. H. Hardy 
A science or an art may be said to be "useful" if its development increases, even indirectly, the material wellbeing and comfort of men, it promotes happiness, using that word in a crude and commonplace way.
 G. H. Hardy 
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).
 G. H. Hardy 
The creative life [is] the only one for a serious man.
 G. H. Hardy 
I was at my best at a little past forty, when I was a professor at Oxford.
 G. H. Hardy 
The study of mathematics is, if an unprofitable, a perfectly harmless and innocent occupation.
 G. H. Hardy 
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.
 G. H. Hardy 
It is hardly possible to maintain seriously that the evil done by science is not altogether outweighed by the good. For example, if ten million lives were lost in every war, the net effect of science would still have been to increase the average length of life.
 G. H. Hardy 
Young men should prove theorems, old men should write books.
 G. H. Hardy 
If intellectual curiosity, professional pride, and ambition are the dominant incentives to research, then assuredly no one has a fairer chance of gratifying them than a mathematician.
 G. H. Hardy 
[Regarding mathematics,] there are now few studies more generally recognized, for good reasons or bad, as profitable and praiseworthy. This may be true; indeed it is probable, since the sensational triumphs of Einstein, that stellar astronomy and atomic physics are the only sciences which stand higher in popular estimation.
 G. H. Hardy 
A science is said to be useful if its development tends to accentuate the existing inequalities in the distribution of wealth, or more directly promotes the destruction of human life.
 G. H. Hardy 
No discovery of mine has made, or is likely to make, directly or indirectly, for good or ill, the least difference to the amenity of the world.
 G. H. Hardy 
A chess problem is genuine mathematics, but it is in some way "trivial" mathematics. However, ingenious and intricate, however original and surprising the moves, there is something essential lacking. Chess problems are unimportant. 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.
 G. H. Hardy 
As history proves abundantly, mathematical achievement, whatever its intrinsic worth, is the most enduring of all.
 G. H. Hardy 
Reductio ad absurdum, which Euclid loved so much, is one of a mathematician's finest weapons. It is a far finer gambit than any chess play: a chess player may offer the sacrifice of a pawn or even a piece, but a mathematician offers the game.
 G. H. Hardy 
In [great mathematics] there is a very high degree of unexpectedness, combined with inevitability and economy.
 G. H. Hardy 
I have never done anything 'useful'. No discovery of mine has made, or is likely to make, directly or indirectly, for good or ill, the least difference to the amenity of the world... Judged by all practical standards, the value of my mathematical life is nil; and outside mathematics it is trivial anyhow. I have just one chance of escaping a verdict of complete triviality, that I may be judged to have created something worth creating. And that I have created something is undeniable: the question is about its value.
 G. H. Hardy 
Greek mathematics is the real thing. The Greeks first spoke a language which modern mathematicians can understand... So Greek mathematics is 'permanent', more permanent even than Greek literature.
 G. H. Hardy 
Chess problems are the hymntunes of mathematics.
 G. H. Hardy 
A mathematician ... has no material to work with but ideas, and so his patterns are likely to last longer, since ideas wear less with time than words.
 G. H. Hardy 
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.
 G. H. Hardy 
The mathematician is in much more direct contact with reality. ... [Whereas] the physicist's reality, whatever it may be, has few or none of the attributes which common sense ascribes instinctively to reality. A chair may be a collection of whirling electrons.
 G. H. Hardy 
[I was advised] to read Jordan's 'Cours d'analyse'; and I shall never forget the astonishment with which I read that remarkable work, the first inspiration for so many mathematicians of my generation, and learnt for the first time as I read it what mathematics really meant.
 G. H. Hardy 
I propose to put forward an apology for mathematics; and I may be told that it needs none, since there are now few studies more generally recognized, for good reasons or bad, as profitable and praiseworthy.
 G. H. Hardy 
All analysts spend half their time hunting through the literature for inequalities which they want to use and cannot prove.
 G. H. Hardy 
Mathematics may, like poetry or music, "promote and sustain a lofty habit of mind."
 G. H. Hardy 
The Babylonian and Assyrian civilizations have perished; Hammurabi, Sargon and Nebuchadnezzar are empty names; yet Babylonian mathematics is still interesting, and the Babylonian scale of 60 is still used in Astronomy.
 G. H. Hardy 
The public does not need to be convinced that there is something in mathematics.
 G. H. Hardy 
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.
 G. H. Hardy 
When the world is mad, a mathematician may find in mathematics an incomparable anodyne. For mathematics is, of all the arts and sciences, the most austere and the most remote, and a mathematician should be of all men the one who can most easily take refuge where, as Bertrand Russell says, "one at least of our nobler impulses can best escape from the dreary exile of the actual world."
 G. H. Hardy 
Mathematics is not a contemplative but a creative subject.
 G. H. Hardy 
It is 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.
 G. H. Hardy 
Exposition, criticism, appreciation, is work for secondrate minds.
 G. H. Hardy 
I count Maxwell and Einstein, Eddington and Dirac, among "real" mathematicians. The great modern achievements of applied mathematics have been in relativity and quantum mechanics, and these subjects are at present at any rate, almost as "useless" as the theory of numbers.
 G. H. Hardy 
No mathematician should ever allow himself to forget that mathematics, more than any other art or science, is a young man's game
 G. H. Hardy 
Bombs are probably more merciful than bayonets
 G. H. Hardy 
Mathematics is not a contemplative but a creative subject; no one can draw much consolation from it when he has lost the power or the desire to create; and that is apt to happen to a mathematician rather soon. It is a pity, but in that case he does not matter a great deal anyhow, and it would be silly to bother about him.
 G. H. Hardy 
I remember once going to see him [Ramanujan] when he was lying ill at Putney. I had ridden in taxicab No. 1729, and remarked that the number seemed to me rather a dull one, and that I hoped it was not an unfavourable omen. "No," he replied, "it is a very interesting number; it is the smallest number expressible as a sum of two cubes in two different ways."
 G. H. Hardy 
If I could prove by logic that you would die in five minutes, I should be sorry you were going to die, but my sorrow would be very much mitigated by pleasure in the proof.
 G. H. Hardy 
A chess problem is an exercise in pure mathematics.
 G. H. Hardy 
Sometimes one has to say difficult things, but one ought to say them as simply as one knows how.
 G. H. Hardy 
If a man is in any sense a real mathematician, then it is a hundred to one that his mathematics will be far better than anything else he can do, and that it would be silly if he surrendered any decent opportunity of exercising his one talent in order to do undistinguished work in other fields. Such a sacrifice could be justified only by economic necessity of age.
 G. H. Hardy 
I wrote a great deal during the next ten [early] years,but very little of any importance; there are not more than four or five papers which I can still remember with some satisfaction.
 G. H. Hardy 
The primes are the raw material out of which we have to build arithmetic, and Euclid's theorem assures us that we have plenty of material for the task.
 G. H. Hardy 
No one has yet discovered any warlike purpose to be served by the theory of numbers or relativity, and it seems unlikely that anyone will do so for many years.
 G. H. Hardy 
There is always more in one of Ramanujan's formulae than meets the eye, as anyone who sets to work to verify those which look the easiest will soon discover. In some the interest lies very deep, in others comparatively near the surface; but there is not one which is not curious and entertaining.
 G. H. Hardy 
Perhaps five or even ten per cent of men can do something rather well. It is a tiny minority who can do anything really well, and the number of men who can do two things well is negligible. If a man has any genuine talent, he should be ready to make almost any sacrifice in order to cultivate it to the full.
 G. H. Hardy 
Bradman is a whole class above any batsman who has 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.
 G. H. Hardy 
For any serious purpose, intelligence is a very minor gift.
 G. H. Hardy 
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.
 G. H. Hardy 
Most people are so frightened of the name of mathematics that they are ready, quite unaffectedly, to exaggerate their own mathematical stupidity.
 G. H. Hardy 
It is a melancholy experience for a professional mathematician to find himself writing about mathematics. The function of a mathematician is to do something, to prove new theorems, to add to mathematics, and not to talk about what he or other mathematicians have done. Statesmen despise publicists, painters despise artcritics, and physiologists, physicists, or mathematicians have usually similar feelings: 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 secondrate minds.
 G. H. Hardy 
The fact is there are few more popular subjects than mathematics. Most people have some appreciation of mathematics, just as most people can enjoy a pleasant tune.
 G. H. Hardy 
Cricket is the only game where you are playing against eleven of the other side and ten of your own.
 G. H. Hardy 
The case for my life... is this: that I have added something to knowledge, and helped others to add more
 G. H. Hardy 
I do not know an instance of a major mathematical advance initiated by a man past fifty
 G. H. Hardy 
The mathematician's patterns, like the painter's or the poet's, must be beautiful.
 G. H. Hardy
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