(source) 
G. H. Hardy
(7 Feb 1877  1 Dec 1947)

Science Quotes by G. H. Hardy (64 quotes)
>> Click for G. H. Hardy Quotes on  Mathematician  Mathematics  Ramanujan_Srinivasa  Theorem  Work 
>> Click for G. H. Hardy Quotes on  Mathematician  Mathematics  Ramanujan_Srinivasa  Theorem  Work 
[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
[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
[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 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
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
A man’s first duty, a young man’s at any rate, is to be ambitious … the noblest ambition is that of leaving behind one something of permanent value.
— G. H. Hardy
A mathematical proof should resemble a simple and clearcut constellation, not a scattered cluster in the Milky Way.
— 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
A mathematician, like a painter or a 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
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. … The poverty of ideas seems hardly to affect the beauty of the verbal pattern. A mathematician, on the other hand, 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
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
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
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
As history proves abundantly, mathematical achievement, whatever its intrinsic worth, is the most enduring of all.
— 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
Beauty is the first test: there is no permanent place in the world for ugly mathematics.
— G. H. Hardy
Chess problems are the hymntunes of mathematics.
— G. H. Hardy
Everything is what it is, and not another thing.
— G. H. Hardy
For any serious purpose, intelligence is a very minor gift.
— 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
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
He adhered, with a severity most unusual in Indians resident in England, to the religious observances of his caste; but his religion was a matter of observance and not of intellectual conviction, and I remember well his telling me (much to my surprise) that all religions seemed to him more or less equally true.
— G. H. Hardy
I am interested in mathematics only as a creative art.
— G. H. Hardy
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 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
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
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
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. [The things I have added to knowledge do not differ from] the creations of the other artists, great or small, who have left some kind of memorial beind them.
— 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
I remember once going to see him when he was lying ill at Putney. I had ridden in taxi cab number 1729 and remarked that the number seemed to me rather a dull one, and that I hoped it was not an unfavorable omen. “No,” he replied, “it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways.”
— G. H. Hardy
I was at my best at a little past forty, when I was a professor at Oxford.
— 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
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
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
In [great mathematics] there is a very high degree of unexpectedness, combined with inevitability and economy.
— 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
It is a melancholy experience for a professional mathematician to find him 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 men who make for the men who explain. Exposition, criticism, appreciation, is work for secondrate minds.
— 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
It is not worth a first class man’s time to express a majority opinion. By definition, there are already enough people to do that.
— 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
Mathematics is not a contemplative but a creative subject.
— G. H. Hardy
Mathematics may, like poetry or music, “promote and sustain a lofty habit of mind.”
— 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
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
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
No one should ever be bored. … One can be horrified, or disgusted, but one can’t be bored.
— G. H. Hardy
No other subject has such clearcut 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.
— 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
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
Sometimes one has got to say difficult things, but one ought to say them as simply as one knows how.
— 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
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 creative life [is] the only one for a serious man.
— G. H. Hardy
The fact is that there are few more “popular” subjects than mathematics. 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 may 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
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
The mathematician's patterns … must be beautiful … Beauty is the first test; there is no permanent place in the world for ugly mathematics.
— 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.
— 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
The public does not need to be convinced that there is something in mathematics.
— G. H. Hardy
The study of mathematics is, if an unprofitable, a perfectly harmless and innocent occupation.
— 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
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
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
Young men should prove theorems, old men should write books.
— G. H. Hardy
Quotes by others about G. H. Hardy (7)
[Godfrey H. Hardy] personified the popular idea of the absentminded professor. But those who formed the idea that he was merely an absentminded professor would receive a shock in conversation, where he displayed amazing vitality on every subject under the sun. ... He was interested in the game of chess, but was frankly puzzled by something in its nature which seemed to come into conflict with his mathematical principles.
Littlewood, on Hardy’s own estimate, is the finest mathematician he has ever known. He was the man most likely to storm and smash a really deep and formidable problem; there was no one else who could command such a combination of insight, technique and power.
I do not think that G. H. Hardy was talking nonsense when he insisted that the mathematician was discovering rather than creating, nor was it wholly nonsense for Kepler to exult that he was thinking God's thoughts after him. The world for me is a necessary system, and in the degree to which the thinker can surrender his thought to that system and follow it, he is in a sense participating in that which is timeless or eternal.
Replying to G. H. Hardy’s suggestion that the number of a taxi (1729) was “dull”: No, it is a very interesting number; it is the smallest number expressible as a sum of two cubes in two different ways, the two ways being 1³ + 12³ and 9³ + 10³.
Plenty of mathematicians, Hardy knew, could follow a stepbystep discursus unflaggingly—yet counted for nothing beside Ramanujan. Years later, he would contrive an informal scale of natural mathematical ability on which he assigned himself a 25 and Littlewood a 30. To David Hilbert, the most eminent mathematician of the day, he assigned an 80. To Ramanujan he gave 100.
One day at Fenner's (the university cricket ground at Cambridge), just before the last war, G. H. Hardy and I were talking about Einstein. Hardy had met him several times, and I had recently returned from visiting him. Hardy was saying that in his lifetime there had only been two men in the world, in all the fields of human achievement, science, literature, politics, anything you like, who qualified for the Bradman class. For those not familiar with cricket, or with Hardy's personal idiom, I ought to mention that “the Bradman class” denoted the highest kind of excellence: it would include Shakespeare, Tolstoi, Newton, Archimedes, and maybe a dozen others. Well, said Hardy, there had only been two additions in his lifetime. One was Lenin and the other Einstein.
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 proofsheets I read (what now stands), “It was Littlewood who said…”. What had happened was that Hardy had received the remark in silence and with poker face, and I wrote it off as a dud.
See also:
 7 Feb  short biography, births, deaths and events on date of Hardy's birth.
 Godfrey Harold Hardy  context of quote Languages die and mathematical ideas do not  Medium image (500 x 350 px)
 Godfrey Harold Hardy  context of quote Languages die and mathematical ideas do not  Large image (800 x 600 px)
 Godfrey Harold Hardy  context of quote Young men should prove theorems, old men should write books.  Medium image (500 x 350 px)
 Godfrey Harold Hardy  context of quote Young men should prove theorems, old men should write books.  Large image (800 x 600 px)
 A Mathematician's Apology, by G. H. Hardy.  book suggestion.