Top 15 G.H. Hardy Quotes

It is never worth a first-class man's time to express a majority opinion. By definition, there are plenty of others to do that. (pg 46)

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. (pg 28)

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.

I still say to myself when I am depressed and find myself forced to listen to pompous and tiresome people "Well, I have done one thing you could never have done, and that is to have collaborated with Littlewood and Ramanujan on something like equal terms."

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.

We have still one more question to consider. We have concluded that the trivial mathematics is, on the whole, useful, and that the real mathematics, on the whole, is not

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.

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, because mathematical reality is built that way.

...there is probably less difference between the positions of a mathematician and of a physicist than is generally supposed, [...] the mathematician is in much more direct contact with reality. This may seem a paradox, since it is the physicist who deals with the subject-matter usually described as 'real', but [...] [a physicist] is trying to correlate the incoherent body of crude fact confronting him with some definite and orderly scheme of abstract relations, the kind of scheme he can borrow only from mathematics.

[T]here is no mathematician so pure that he feels no interest at all in the physical world; but, in so far as he succumbs to this temptation, he will be abandoning his purely mathematical position.

The play is independent of the pages on which it is printed, and 'pure geometries' are independent of lecture rooms, [rough blackboard drawings] or of any other detail of the physical world.
This is the point of view of a pure mathematician. Applied mathematicians, mathematical physicists... take a different view... preoccupied with the physical world itself, which also has its structure or pattern. ...We may be able to trace a ...resemblance between the two sets of relations, and then the pure geometry will become interesting to physicists; it will give us ...a map which 'fits the facts' ...The geometer offers ...a whole set of maps from which to choose.

There is the science of pure geometry, in which there are many geometries, , , non-Euclidean geometry... [etc.]. Each... is a , a pattern of ideas... judged by the interest and beauty of... pattern. It is a map or picture, the... product of many hands, a partial and imperfect copy (yet exact so far as it extends) of a section of mathematical reality. But... there is one thing... of which pure geometries are not pictures, and that is the spatio-temporal reality of the physical world. ...[T]hey cannot be, since earthquakes and eclipses are not mathematical concepts.

[M]athematical reality lies outside us ...our function is to discover or observe it, and ...the theorems ...we prove, and ...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 [...]

A man who could give a convincing account of mathematical reality would have solved very many of the most difficult problems of metaphysics. If he could include physical reality in his account, he would have solved them all.

It is... natural to suppose that there is a great difference in utility between 'pure' and 'applied' mathematics. This is a delusion...