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
The most cheerful G. H. Hardy quotes that may be undiscovered and unusual
In [great mathematics] there is a very high degree of unexpectedness, combined with inevitability and economy.
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.
I am interested in mathematics only as a creative art.
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.
It is not worth an intelligent man's time to be in the majority.
By definition, there are already enough people to do that.
Beauty is the first test: there is no permanent place in the world for ugly mathematics.
A mathematician, like a painter or a poet, is a maker of patterns.
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.
The mathematician's patterns, like the painter's or the poet's, must be beautiful.
Most people are so frightened of the name of mathematics that they are ready, quite unaffectedly, to exaggerate their own mathematical stupidity.
A science or an art may be said to be "useful" if its development increases, even indirectly, the material well-being and comfort of men, it promotes happiness, using that word in a crude and commonplace way.
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.
Mathematics is not a contemplative but a creative subject.
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.
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.
There is no scorn more profound, or on the whole more justifiable, than that of the men who make for the men who explain.
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.
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.
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.
Young men should prove theorems, old men should write books.
I was at my best at a little past forty, when I was a professor at Oxford.
Cricket is the only game where you are playing against eleven of the other side and ten of your own.
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.
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.
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.
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.
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.
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.
Exposition, criticism, appreciation, is work for second-rate minds.
Good work is not done by 'humble' men
[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.
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.
The creative life was the only one for a serious man.
Sometimes one has to say difficult things, but one ought to say them as simply as one knows how.
No mathematician should ever allow himself to forget that mathematics, more than any other art or science, is a young man's game
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.
Bombs are probably more merciful than bayonets
Most people can do nothing at all well
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.
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.
Real mathematics must be justified as art if it can be justified at all.
Asked if he believes in one G-d, a mathematician answered: "Yes, up to isomorphism".
As history proves abundantly, mathematical achievement, whatever its intrinsic worth, is the most enduring of all.
They [formulae 1.10 - 1.12 of Ramanujan] must be true because, if they were not true, no one would have had the imagination to invent them.
... Philosophy proper is a subject, on the one hand so hopelessly obscure, on the other so astonishingly elementary, that there knowledge hardly counts.
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.
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.
A person’s first duty, a young person’s at any rate, is to be ambitious, and the noblest ambition is that of leaving behind something of permanent value.
The study of mathematics is, if an unprofitable, a perfectly harmless and innocent occupation.