# G. H. Hardy

Appearance

**Godfrey Harold Hardy** FRS (7 February 1877 – 1 December 1947) was an English mathematician, known for his achievements in number theory and mathematical analysis. In biology, he is known for the Hardy–Weinberg principle, a basic principle of population genetics. Hardy is usually known by those outside the field of mathematics for his 1940 essay *A Mathematician's Apology*, often considered one of the best insights into the mind of a working mathematician written for the layperson. He had a long collaboration with John Edensor Littlewood, and he discovered and supported the work of Srinivasa Ramanujan.

## Quotes

[edit]- Mathematicians have constructed a very large number of different systems of geometry, Euclidean or non-Euclidean, of one, two, three, or any number of dimensions. All these systems are of complete and equal validity. They embody the results of mathematicians' observations of
*their*reality, a reality far more intense and far more rigid than the dubious and elusive reality of physics. The old-fashioned geometry of Euclid, the entertaining seven-point geometry of Veblen, the space-times of Minkowski and Einstein, are all absolutely and equally real. ...There may be three dimensions in this room and five next door. As a professional mathematician, I have no idea; I can only ask some competent physicist to instruct me in the facts.

The function of a mathematician, then, is simply to observe the facts about his own intricate system of reality, that astonishingly beautiful complex of logical relations which forms the subject-matter of his science, as if he were an explorer looking at a distant range of mountains, and to record the results of his observations in a series of maps, each of which is a branch of pure mathematics. ...Among them there perhaps none quite so fascinating, with quite the astonishing contrasts of sharp outline and shade, as that which constitutes the theory of numbers.- "The Theory of Numbers,"
*Nature*(Sep 16, 1922) Vol. 110 p. 381

- "The Theory of Numbers,"

- If I knew I was going to die today, I think I should still want to hear the cricket scores.
- Quoted in
*The Joy of Cricket*(ed. John Bright-Holmes, 1984)

- Quoted in

- 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.
- Quoted by Bertrand Russell,
*Logical and Philosophical Papers, 1909–13*, Routledge, 1992, p. xxix.

- Quoted by Bertrand Russell,

*Ramanujan* (1940)

[edit]*Ramanujan : Twelve Lectures on Subjects Suggested by His Life and Work*

- He could remember the idiosyncrasies of numbers in an almost uncanny way. It was Littlewood who said that every positive integer was one of Ramanujan's personal friends. I remember once going to see him when he was 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."
- Ch. I : The Indian mathematician Ramanujan.

- 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.- Ch. I : The Indian mathematician Ramanujan.

*A Mathematician's Apology* (1941)

[edit]- … 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.

- If a man has any genuine talent, he should be ready to make almost any sacrifice in order to cultivate it to the full.

- 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 a good deal later; Gauss's great memoir on differential geometry was published when he was fifty (though he had had the fundamental ideas ten years before). I do not know an instance of a major mathematical advance initiated by a man past fifty. If a man of mature age loses interest in and abandons mathematics, the loss is not likely to be very serious either for mathematics or for himself.

- 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.

- 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*.

- 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, [...] the importance of ideas in poetry is habitually exaggerated: '... Poetry is not the thing said but a way of saying it.' [In poetry,] the poverty of the ideas seems hardly to affect the beauty of the verbal pattern.

- 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.

*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*.

- Chess problems are the hymn-tunes of mathematics.

- 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.

- I am interested in mathematics only as a creative art.

- 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 [...]

- ...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.

- 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.

- 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.

- 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

- 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.

- 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 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.

#### Quoted by C. P. Snow

[edit]*Quoted by C. P. Snow in his foreword to the 1967 Cambridge edition*

- 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)
- 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)

## Quotes about G. H. Hardy

[edit]- Hardy was a great internationalist who worked with foreign mathematicians, visiting them, encouraging them to visit him and settling some, including Besicovitch, in England. There were some major probability figures in Hardy’s network: George Pólya (1887-1985) of Zürich, Norbert Wiener (1894-1964) of MIT and Harald Cramér (1893-1985) of Stockholm-appropriately Wiener and Cramér first met when visiting Hardy in 1920.
- Aldrich, John (December 2009). "England and Continental Probability in the Inter-War Years".
*Electronic Journal for History of Probability and Statistics***5**(2).

- Aldrich, John (December 2009). "England and Continental Probability in the Inter-War Years".

- In 1933 Landau was dismissed from his [University of Göttingen] chair on the grounds of his race. An important colleague... Ludwig Bieberbach ...wrote the following lines in a treatise on
*Personality structure and mathematical creativity*:

"In this way... the ultimate reason behind the courageous rejection which the students at Göttingen University meted out to a great mathematician, Edmund Landau, was that his un-German style in research and teaching had become intolerable to German sensitivities. A people which has seen how alien desires for dominion are gnawing at its identity, how enemies of the people are working to impose their alien ways on it, must reject teachers of a type alien to it."

The English mathematician Godfrey H. Hardy... responded to Bieberbach...

"There are many of us, many English and many Germans, who said things during the (First) War which we scarcely meant and are sorry to remember now. Anxiety for one's own position, dread of falling behind the rising torrent of folly, determination at all costs not to be outdone, may be natural if not particularly heroic excuses. Prof. Bieberbach's reputation excludes such explanation for his utterances; and I find myself driven to the more uncharitable conclusion that he really believes them true."- Jörg Arndt & Christoph Haenel,
*Pi - Unleashed*(2012)

- Jörg Arndt & Christoph Haenel,

- To illustrate to what extent Hardy and Littlewood in the course of the years came to be considered as the leaders of recent English mathematical research, I may report what an excellent colleague once jokingly said: 'Nowadays, there are only three really great English mathematicians: Hardy, Littlewood, and Hardy-Littlewood.'

- Hardy in his thirties held the view that the late years of a mathematician's life were spent most profitably in writing books; I remember a particular conversation about this, and though we never spoke of the matter again it remained an understanding.
- John Littlewood, Preface to Hardy, G. H..
*Divergent Series*. Oxford University Press.

- John Littlewood, Preface to Hardy, G. H..