G. H. Hardy

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Godfrey Harold Hardy (7 February 1877 - 1 December 1947)


Hardy, called "Harold" or "G. H." by some, was a prominent British mathematician, known for his achievements in number theory and mathematical analysis. His 1940 essay, "A Mathematician's Apology," provided laymen with insights into the mind of a working mathematician.

Hardy was associated with the Bloomsbury group and friends G. E. Moore, Bertrand Russell, and J. M. Keynes. Hardy was a disciple of Bertrand Russell, not only in his interest in mathematical philosophy but also in his political views. He agreed with Russell's anti-war attitude.

Hardy held that mathematics was a purely intellectual endeavor, in its highest form devoid of all practical utility. He scorned the idea that any of his mathematical work might one day find utilitarian application.

Sayings by Hardy

From A Mathematician's Apology (London, 1941):

  • A mathematician, like a painter or a poet, is a maker of patterns. If his patterns are more permanent that theirs, it is because the are made with ideas. . . .The mathematician's patterns, like the painter's or the poet's, must be beautiful; the ideas, like the colours of 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.
  • It is not worth an intelligent man's time to be in the majority. By definition, there are already enough people to do that.
  • 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.
  • I am interested in mathematics only as a creative art.
  • 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.
  • In great mathematics there is a very high degree of unexpectedness, combined with inevitability and economy.
  • 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.
  • 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.
  • 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.
  • 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 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.

Hardy and Russell

According to The Mathematical Society Newsletter #218 (July 1994, p. 12):

"My friend, G. H. Hardy, who was professor of pure mathematics," Bertrand Russell recalled, "told me once that if he could find a proof that I was going to die in five minutes he would of course be sorry to lose me, but this sorrow would be quite outweighted by pleasure in the proof. I entirely sympathised with him and was not at all offended."

Hardy's New Year's Resolutions

Hardy in the 1940s wrote a postcard to a friend containing the following New Year's resolutions (Paul Hoffman, The Man Who Loved Only Numbers (1998, p. 81).

  1. To prove the Riemann hypothesis;
  2. To make a brilliant play in a crucial cricket match;
  3. To prove the nonexistence of God;
  4. To be the first man atop Mount Everest;
  5. To be proclaimed the first president of the U.S.S.R., Great Britain, and Germany; and
  6. To murder Mussolini.

Titchmarsh on Hardy

In an obituary in The Journal of the London Mathematical Society by E. C. Titchmarsh, Hardy was described as

the only son of Isaac Hardy, Art Master, Bursar and House Master of the preparatory branch of Cranleigh School. His mother, Sophia Hardy, had been Senior Mistress to the Lincoln Training College. Both parents were extremely able people and mathematically minded, but want of funds had prevented them from having a university training.
The future professor's interest in numbers showed itself early. By the time he was two years old he had persuaded his parents to show him how to write down numbers up to millions. When he was taken to church he occupied the time in factorizing the numbers of the hymns, and all through his life he amused himself by playing about with the numbers of railway carriages, taxi-cabs, and the like.
He and his sister were brought up by enlightened parents in a typical Victorian nursery, and, as clever children do, he agonized his nurse with long arguments about the efficacy of prayer and the existence of Sant Clause: "Why, if he gives me things, does he put the price on? My box of tools is marked 3s. 6d."

According to Titchmarsh,

  • Hardy always referred to God as his personal enemy. This was of course a joke, but there was something real behind it. He took his disbelief in the doctrines of religion more seriously than most people seem to do. He would not enter a relgious building, even for such a purpose as the election of a Warden of New College. The clause in the New College by-laws, enabling a fellow with a concientious objection to being present in Chapel to send his vote to the scrutineers, was put in on his behalf.
  • He was unmarried. He owed much to his sister, who provided him throughout his life with the unobtrusive support which such a man needs. . . . He was the author, or part author, of more than 300 original papers, covering almost every kind of analysis, which by their originality and quantity marked him as one of the leading mathematicians of his time.

Hardy and Ramanujan

From 1914 onwards, Hardy was mentor of the Indian mathematician Srinivasa Ramanujan, a Tamil Brahmin whom he found had a mind of untutored brilliance. Although his first Indian biographers described Ramunujan as rigorously orthodox, Hardy found him essentially agnostic as far as metaphysical matters were concerned. Suffering from a severe illness that made his work in mathematics difficult, he in frustrated agony while in his death throes is said to have denied any belief in God. Hardy reported, however, that Ramanujan believed all religions are equally correct. One biographer, Kanigel, had a negative view of Hardy, writing that Ramanujan would not have shown Hardy his religious side, that Ramanujan had often said, "An equation for me has no meaning, unless it represents a thought of God."

Once, when Hardy remarked that he had taken taxi number 1729, a singularly unexceptional number, Ramanujan immediately responded that this number was actually quite remarkable:

it is the smallest integer that can be represented in two ways by the sum of two cubes: 1729 = 1 to the 3rd power+12 to the 3rd power = 9 to the 3rd power +10 to the 3rd power.

Of him, Hardy wrote,

Srinivasa Ramanujan was a mathematician so great his name transcends jealousies, the one superlatively great mathematician whom India has produced in the last thousand years.
His leaps of intuition confound mathematicians even today, seven decades after his death. ..the brilliant, self-taught Indian mathematician whose work contains some of the most beautiful ideas in the history of science. His legacy has endured. His twenty-one major mathematical papers are still being plumbed for their secrets, and many of his ideas are used today in cosmology and computer science. His theorems are being applied in areas - polymer chemistry, computers, cancer research - scarcely imaginable during his lifetime.

Snow on Hardy

[Hardy commented to me (The Two Cultures)]: Have you noticed how the word `intellectual' is used nowadays? There seems to be a new definition which certainly doesn't include Rutherford or Eddington or Dirac or Adrian or me. It does seem rather odd, don't y'know.

Dyson on Hardy

Freeman Dyson called Hardy not just an ordinary atheist (who does not believe in God) but a passionate atheist (one who considers God to be their personal enemy). Illustrating this, he described how, at Trinity College when Hardy was his mentor, a Professor Simpson died and, a religious believer, he left instructions that his cremated ashes should be scattered on the bowling green in the fellows' garden where he loved to walk and meditate. "A few days after he died," Dyson wrote,

a solemn funeral service was held for him in the college chapel. His many years of faithful service to the collgege and his exemplary role as a Christian scholar and teacher were duly celebrated. In the evening of the same day I took my place at the high table. One of the neighboring places at the table was empty. Professor Hardy, contrary to his usual habit, was late for dinner. After we had all sat down and the Latin grace had been said, Hardy strolled into the dning hall, ostentatiously scraping his shoes on the wooden floor and complaning in a loud voice for everyone to hear, 'What is this awful stuff they have put on the grass in the fellows' garden? I can't get it off my shoes.' Hardy, of course, knew very well what the stuff was. He had always disliked religion in general and Simpson's piety in particular, and he was taking his opportunity for a little revenge.

Comments About

Timothy Haugh (reviewing David Auburn's Proof, as performed in New York City at the Manhattan Theater Club:

This is a cleverly written piece. Unlike "Copenhagen," this play really isn't about mathematicians and scientists. It is just framed around them. No math skills are necessary to enjoy this play. Instead, it is an examination of love, trust, madness and genius presented through the lives of mathematicians. In fact, the only weakness in this play is when real mathematics comes up. I cringed when I heard the famous exchange between mathematicians G.H. Hardy and Srinivasa Ramanujan put in the mouth of Robert and Catherine, the father/daughter mathematicians at the heart of this play. It just rubbed me the wrong way.

David Auburn [I was reading Hardy's A Mathematician's Apology by] a Cambridge mathematician who wrote very eloquently about the pleasure, passion, and joy of doing the work. This emotional involvement of the mathematician with his work fed into the characters of Catherine and her father in Proof, giving the audience a sense of the fine line between madness and genius, showing that obsessive math is not done by geeks, but by passionate people who feel as much as they think.

Hardy's Orientation

Hardy collaborated with J. E. Littlewood and others, covering all areas of mathematics. Littlewood called Hardy "a non-practising homosexual." His sexual activities, however, have not been documented, either by Bloomsbury group members or others.

Hardy's Influence

Hardy is credited with reforming British mathematics, bringing it from the tradition of applied mathematics to the cours d'analyse methods dominant in France. He formulated the Hardy-Weinberg principle, which is a basic principle of population genetics.

Friends talk of Hardy's avid interest in cricket and in teaching pupils rather than just professing about the subject of mathematics.