G H Hardy Quotes

The geometer offers to the physicist a whole set of maps from which to choose. One map, perhaps, will fit the facts better than others, and then the geometry which provides that particular map will be the geometry most important for applied mathematics. ~ G.H. Hardy

Exposition, criticism, appreciation, is work for second-rate minds. I ~ 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

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 art-critics, 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 second-rate minds. ~ 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 may, like poetry or music, "promote and sustain a lofty habit of mind." ~ 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 our notes of our observations. This view has been held, in one form or another, by many philosophers of high reputation from Plato onwards, and I shall use the language which is natural to a man who holds it. ~ 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

Sometimes one has to say difficult things, but one ought to say them as simply as one knows how. ~ 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

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

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

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

Twentieth-century British mathematician G.H. Hardy also believed that the human function is to "discover or observe" mathematics rather than to invent it. In other words, the abstract landscape of mathematics was there, waiting for mathematical explorers to reveal it. ~ Mario Livio

All analysts spend half their time hunting through the literature for inequalities which they want to use and cannot prove. ~ 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

Good work is not done by 'humble' men ~ G.H. Hardy

Most people can do nothing at all well ~ 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

For my part, it is difficult for me to say what I owe to Ramanujan - his originality has been a constant source of suggestion to me ever since I knew him, and his death is one of the worst blows I have ever had. ~ G.H. Hardy

A month's intelligent instruction in the theory of numbes ought to be twice as instructive, twice as useful, and at least 10 times as entertaining as the same amount of 'calculus for engineers'. ~ G.H. Hardy

We must guard against a fallacy common among apologists of science, the fallacy of supposing that the men whose work most benefits humanity are thinking much of that while they do it, that physiologists, for example, have particularly noble souls. ~ 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

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

The creative life [is] the only one for a serious man. ~ 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

I remember once going to see him [Ramanujan] when he was lying ill at Putney. I had ridden in taxi-cab 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

Asked if he believes in one G-d, a mathematician answered: "Yes, up to isomorphism". ~ G.H. Hardy

It is the dull and elementary parts of applied mathematics, as it is the dull and elementary parts of pure mathematics, that work for good or ill. Time may change all this. No one foresaw the applications of matrices and groups and other purely mathematical theories to modern physics, and it may be that some of the 'highbrow' applied mathematics will become 'useful' in as unexpected a way; but the evidence so far points to the conclusion that, in one subject as in the other, it is what is commonplace and dull that counts for practical life. ~ 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

The beauty of a mathematical theorem depends a great deal on its seriousness, as even in poetry the beauty of a line may depend to some extent on the significance of the ideas which it contains. ~ 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

Immortality is often ridiculous or cruel: few of us would have chosen to be Og or Ananias or Gallio. Even in mathematics, history sometimes plays strange tricks; Rolle figures in the textbooks of elementary calculus as if he had been a mathematician like Newton; Farey is immortal because he failed to understand a theorem which Haros had proved perfectly fourteen years before; the names of five worthy Norwegians still stand in Abel's Life, just for one act of conscientious imbecility, dutifully performed at the expense of their country's greatest man. But on the whole the history of science is fair, and this is particularly true in mathematics. No other subject has such clear-cut 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

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

Cricket is the only game where you are playing against eleven of the other side and ten of your own. ~ G.H. Hardy

The study of mathematics is, if an unprofitable, a perfectly harmless and innocent occupation. ~ 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

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 twenty-one, Abel at twenty-seven, Ramanujan at thirty-three, 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

Imaginary' universes are so much more beautiful than this stupidly constructed 'real' one; and most of the finest products of an applied mathematician's fancy must be rejected, as soon as they have been created, for the brutal but sufficient reason that they do not fit the facts. ~ 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

As history proves abundantly, mathematical achievement, whatever its intrinsic worth, is the most enduring of all. ~ G.H. Hardy

Real mathematics must be justified as art if it can be justified at all. ~ 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

If I had a statue on a column in London, would I prefer the columns to be so high that the statue was invisible, or low enough for the features to be recognizable? I would choose the first alternative, Dr Snow, presumably, the second. ~ 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

Bombs are probably more merciful than bayonets ~ G.H. Hardy

I wrote a great deal ... but very little of any importance; there are not more than four of 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

Pure 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

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

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

What is the proper justification of a mathematician's life? My answers will be, for the most part, such as are expected from a mathematician: I think that it is worthwhile, that there is ample justification. But I should say at once that my defense of mathematics will be a defense of myself, and that my apology is bound to be to some extent egotistical. I should not think it worth while to apologize for my subject if I regarded myself as one of its failures. Some egotism of this sort is inevitable, and I do not feel that it really needs justification. 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

[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

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. We may say, roughly, that a mathematical idea is 'significant' if it can be connected, in a natural and illuminating way, with a large complex of other mathematical ideas. Thus a serious mathematical theorem, a theorem which connects significant ideas, is likely to lead to important advances in mathematics itself and even in other sciences. ~ G.H. Hardy

{Replying to G. H. Hardy's suggestion that the number of a taxi (1729) was 'dull', showing off his spontaneous mathematical genius}

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. ~ Srinivasa Ramanujan

A mathematical proof should resemble a simple and clear-cut constellation, not a scattered cluster in the Milky Way. ~ G.H. Hardy

Poetry is more valuable than cricket, but Bradman would be a fool if he sacrificed his cricket in order to write second-rate minor poetry (and I suppose that it is unlikely that he could do better). ~ 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

I am interested in mathematics only as a creative art. ~ G.H. Hardy

A chess problem is an exercise in pure 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

I was at my best at a little past forty, when I was a professor at Oxford. ~ G.H. Hardy

The theory of numbers, more than any other branch of mathematics, began by being an experimental science. Its most famous theorems have all been conjectured, sometimes a hundred years or more before they were proved; and they have been suggested by the evidence of a mass of computations. ~ G.H. Hardy

Chess problems are the hymn-tunes of mathematics. ~ 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

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; 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

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

Emotions, moods, impulses, ebb and flow with the tide of my life. Tidal waves, at times, in a bipolar mind. ~ H.g.

Let the Angels lead me to you, if only they fly in this land. ~ John G.H. Dickinson

There is no treasure but Truth, there is no Truth but Wisdom. There is no Wisdom, but from Learning, and Learning is won by the devotion of hours, years, days and nights to the works of Nature and the Treasures of Truth that others have gathered. ~ John G.H. Dickinson

When authors write best, or at least, when they write most fluently, an influence seems to waken in them which becomes their master, which will have its own way, putting out of view all behests but its own, dictating certain words, and insisting on their being used, whether vehement or measured in their nature; new moulding characters, giving unthought-of turns to incidents, rejecting carefully elaborated old ideas, and suddenly creating and adopting new ones. Is it not so? And should we try to counteract this influence? Can we indeed counteract it?
from a letter to G.H. Lewes, 12 January 1848 ~ Charlotte Bronte

I suppose you could say my father's world was Thomas Hardy and my mother's D.H. Lawrence. ~ Seamus Heaney

Organisation - key to success. ~ Tom G.H. Adams

The occultist is brought into intelligent communication with the spirits of the air, and can receive any knowledge which they possess, or any false impression they choose to impart ... the demons seem permitted to do various wonders at their request. ~ G. H. Pember

Oh, but to reach silence, what a huge effort of voice. My voice is the way I go seek reality; reality prior to my language exists as an unthinkable thought, but I was and am fatefully impelled to have to know what thought thinks. Reality precedes the voice that seeks it, but like the earth precedes the tree, but like the world precedes the man, but like the sea precedes the view of the sea, life precedes love, bodily matter precedes the body, and one day in its turn language shall have preceded possession of silence. - Clarice Lispector, The Passion According to G.H. ~ Clarice Lispector

History covers the Texas landscape like a multi-covered quilt, each square particular to different cultures and traditions. ~ G.H. Lambert

We only see what interests us, and we have only insight in proportion to our sympathy. ~ G.H. Lewes

Writing is easy, good writing is hard and excellent writing is hard work. ~ G.H. Guzik

One of the things he liked about playwriting as to any other kind of writing is that a playwright is a w-r-i-g-h-t, not a w-r-i-t-e; in other words, that a playwright is more of a craftsman than an artist of the big novel. ~ Simon McBurney

A b c d e f g h I j k l m n o p q r s t v w x y z

Did I miss something?
Yes! I missed u :( ~ Yazan Ammar

Freedom is what you can do, Liberty is what they cannot do to you. ~ G.H. Thomas

For not the revelation of God, but the expounders of that revelation, are responsible for the diversities of Christendom: the fault rests with the fallen and corrupt nature of man, which so affects him that he cannot clearly discern truth even when it is set before his eyes. ~ G. H. Pember

the time is always right to do what is
R I G H T ~ Nobody!

Without birds to feed on them, the insects would multiply catastrophically ... The insects, not man or other proud species, are really the only ones fitted for survival in the nuclear age ... The cockroach, a venerable and hardy species, will take over the habitats of the foolish humans, and compete only with other insects or bacteria. ~ H. Bentley Glass

She was working as a waitress in a cocktail bar when she decided to summon the devil and sell her soul for fame. As you do. Hell, we've all been there… Of course, things didn't go according to plan. Never have, never will… ~ G.H. Finn

My heart is hardy, for I have suffered much on the seas and the battlefield: this will be only something more. But a ravenous belly cannot be hid, damn the thing. It gives a world of trouble to men, makes them fit out fleets of ships and scour the barren sea, to bring misery on their enemies. ~ W.H.D. Rouse

The chances of anyone colonizing Mars are a million to one!" Or so the newspapers said. But still we came. ~ G.H. Finn

I once lay in a white hospital for the dying and the dying self, where some god pissed a rain of reason to make things grow only to die, where on my knees I prayed for LIGHT, I prayed for l*i*g*h*t, and praying crawled like a blind slug into the web where threads of wind stuck against my mind and I died of pity for Man, for myself, on a cross without nails, watching in fear as the pig belches in his sty, farts, blinks and eats. ~ Charles Bukowski

Angel is right,"said Dr. G-H quickly. "This is my clumsy way of demonstrating."
"Demonstrating what?" I was barely able to keep a snarl out of my voice. "How to get yourself beat up in one easy step? ~ James Patterson

I spend roughly $80 per year watching bananas go brown. ~ G.H. Eckel

It was still early, and though the sun's lower limb was just free of the hill, his rays, ungenial and peering, addressed the eye rather than the touch as yet. There was not a human soul near. Sad October and her sadder self seemed the only two existences haunting that lane. ~ Thomas Hardy

Always wanting another man than your own. ~ Thomas Hardy

The clash of discord between mood and matter here was forced painfully home to the heart; and, as in laughter there are more dreadful phases than in tears, so was there in the steadiness of this agonized man an expression deeper than a cry. ~ Thomas Hardy

Once victim, always victim-that's the law. ~ Thomas Hardy

The season developed and matured. Another year's instalment of flowers, leaves, nightingales, thrushes, finches, and such ephemeral creatures, took up their positions where only a year ago others had stood in their place when these were nothing more than germs and inorganic particles. Rays from the sunrise drew forth the buds and stretched them into long stalks, lifted up sap in noiseless streams, opened petals, and sucked out scents in invisible jets and breathings. ~ Thomas Hardy

Unsuccessful people carry their goals around in their head like marbles rattling around in a can, and we say a goal that is not in writing is merely a fantasy. ~ Darren Hardy

Ideas cannot be limited to the confines of a silo. They need space to run around and occasionally bump into strangers. ~ Steve Hardy

And then her cooing voice, plaintive in expostulation, disturbed the darkness, the velvet touch of her lips passed over his brow, and he could distinguish in the air the warmth of her breath. ~ Thomas Hardy

Glaring, Kai leaned back against the headrest. "I'm already uncomfortable with you piloting this ship and being in control of my life. Try not to make it worse."
"Why does everyone think I'm such a bad pilot?"
"Cinder told me as much."
"Well, tell Cinder I'm perfectly capable of flying a blasted podship without killing anyone. My flight instructor at the Andromeda - which is a very prestigious military academy in the Republic, I will have you know-"
"I know what Andromeda Academy is."
"Yeah, well, my flight instructor said I was a natural."
"Right," Kai drawled. "Was that the same flight instructor who wrote in you official report about your inattentiveness, refusal to take safety precautions seriously, and overconfident attitude that often bordered on ... what was the word she used>? 'Fool-hardy', I think?"
"Oh, yeah. Commander Reid. She had a thing for me." The radar blinked, picking up a cruiser in the far distance, and Thorne deftly changed directions to keep them out of its course. "I didn't realize I had a royal stalker. I'm flattered, Your Majesty."
"Even better - you had an entire government team assigned to digging up information on you. They reported twice daily for over a week. You did run off with the most-wanted criminal in the world, after all. ~ Marissa Meyer