The Man Who Knew Infinity
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“a really searching analysis.” Hardy, “A Chapter from Ramanujan’s Notebook,” Proceedings of the Cambridge Philosophical Society 21 (1923): 503.
Polya. Berndt, Ramanujan’s Notebooks, part I, 14.
“not a light one.” G. N. Watson, “Ramanujan’s Notebooks,” Journal of the London Mathematical Society 6 (1931): 140.
a month to prove. Ibid., 150.
“at least a Jacobi.” Letter, Littlewood to Hardy, probably early March, 1913. Trinity College.
“Euler or Jacobi.” Hardy, Collected Papers, 720.
“greatest formalist of his time.” Hardy, Ramanujan, 14.
“delighted surprise.” Littlewood, Miscellany, 87.
“never heard of most of it.” Snow, Apology foreword, 36.
“without a rival.” Hardy, Collected Papers, 720.
“what a treasure I had found.” Hardy, Ramanujan, 1.
“I edited them very carefully.” Hardy, Collected Papers, 717.
the 2:15 P.M. train out of Cambridge. Hardy, Collected Papers, 751.
London Mathematical Society meeting. Minutes.
within the radius of a hydrogen atom. Jonathan M. Borwein and Peter B. Borwein, Scientific American, February 1988, 112.
“having no other business at the time.” Ibid., 114.
“new results.” Hardy, Collected Papers, 494.
“an exciting foxhunt.” Cambridge Daily News, 18 April 1914.
Caesar died. Cambridge Daily News, 19 April 1914.
“But he was a happy man.” Neville, “Ramanujan” (Nature 149), 294.
“no war in this country.” P. K. Srinivasan, 168.
muffled the echoes. First Eastern General Hospital Gazette, 6 July 1915, 135.
strung from the ceiling. Ibid., 22 June 1915, 100.
ambulance with a great red cross. Ibid., 97.
would see them on his way to Hall. Parry, 200.
“We have a new Cambridge.” Ibid., 95.
inches-deep mud on unasphalted roads. Keynes, 182.
On September 20. Butler, 200.
“The depravation of Germany.” Parry, 96.
“driven in front of the enemy.” Butler, 200.
“Germans set fire.” P. K. Srinivasan, 168.
“Now here’s a problem for you.” S. R. Ranganathan, 81. Mahalanobis didn’t cite the December 1914 issue, but the content of the problem, and the issue in which it appears, square perfectly with his recollection.
“The limitations of his knowledge.” Hardy, Collected Papers, 714.
“found a function which exactly represents.” Ramanujan, Collected Papers, 349.
“The stuff about primes is wrong.” Letter, Littlewood to Hardy, probably early March 1913, in Ramanujan, The Lost Notebook, 380.
“Euclid’s theorem assures us.” Hardy, Apology, 99.
De la Vallée-Poussin. A professor at the University of Louvain; many of his works were lost when the library was set to the torch in the first month of the war. Journal of the Indian Mathematical Society 10: 27.
“Proofs will be supplied later.” Journal of the Indian Mathematical Society 5 (1913): 50–51.
came to see where he had stumbled. See Hardy, Ramanujan, chapters 1 and 2.
“no complex zeroes.” Hardy, Collected Papers, 706.
Riemann hypothesis. For a nice introduction, accessible to the layperson, see Campbell and Higgins, vol. 2, 149–153.
Has the Riemann hypothesis been proved? This is one of many bits of Riemann hypothesis lore.
“the Analytic Theory of Numbers is not one of them.” Hardy, Collected Papers, 706.
“his achievement … is most extraordinary.” Littlewood, Miscellany, 87.
“more wonderful than any of his triumphs.” Hardy, Collected Papers, 706.
“His instincts misled him.” Hardy, Ramanujan, 38.
“as the climax of a conventional pattern of propositions.” Ibid., 16.
Take the sequence of integers. These illustrative examples are drawn from Ivars Peterson, “A Shortage of Small Numbers,” Science News, 9 January 1988.
One Hardy liked to cite. Hardy, Ramanujan, 16.
“the largest number which has ever served any definite purpose in mathematics.” Hardy, Ramanujan, 17.
The 2 = 1 “proof.” A high school classic.
slide into trouble in numerous ways. Examples due to mathematicians Clare Friedman, Richard Askey, and Bruce Berndt.
“He disregarded entirely all the difficulties.” Hardy, Collected Papers, 706.
“of what is meant by a proof.” Littlewood, Miscellany, 88.
“very much mitigated.” R. Clark, 176.
“hardened to some extent.” Hardy, Ramanujan, 10.
“I have changed my plan.” P. K. Srinivasan, 15.
“My notebook is sleeping in a corner.” Ibid., 21.
“no symptom of abatement.” Hardy, Collected Papers, 715.
“a very curious function.” P. K. Srinivasan, 23.
“he sets to work to manufacture a proof.” Hardy, Ramanujan, 16.
“first-rate importance.” Littlewood, Miscellany, 88.
“Mathematics has been advanced.” Klein, in Mordell, “Ramanujan,” 647.
an informal scale. Berndt, Ramanujan’s Notebooks I, 14.
“break the spell of his inspiration.” Hardy, Collected Papers, 715.
“excerpts from a more interesting lecture.” Young, 282.
“At Cambridge we are in darkness.” Parry, 98.
“In France and Flanders we make no progress.” Ibid.
Cambridge schoolgirl. Keynes, 62.
“the front was like a first-rate club.” Parry, 96.
“cheerful indifference.” Williams, 15.
“Even Littlewood.” Hardy, Apology, 140.
“did not fill all of Littlewood’s working hours.” Burkill, 63.
Hardy deemed “unfit” to serve. On what basis I have not learned.
at least one obituary. Norbert Wiener, “Godfrey Harold Hardy,” Bulletin of the American Mathematical Society 55: 72–77. The record was set straight in a subsequent letter to the Bulletin from Littlewood and others.
“I don’t like conscientious objectors as a class.” Letter, Hardy to Jenkinson, 15 June 1918. Cambridge University.
Hardy “wrote passionately.” Littlewood, “Reminiscences,” 13.
Its first public meeting. Hardy, Russell, 12.
the authorities moved to block it. Ibid., 19.
risen 32 percent. Marwick, 125.
three or four times a month. Family Record, 55.
a parcel full of books. P. K. Srinivasan, 162.
preoccupied … with seeing his work in print. The letters are all in P. K. Srinivasan, 7, 11, 13, and 32, respectively.
he went straight to Hardy. S. R. Ranganathan, 77.
the following year. P. K. Srinivasan, 28.
“no help nor references in Madras.” Ibid., 29.
students sometimes taunted him. S. R. Ranganathan, 71.
“a thrill to me to discover.” Ibid., 78.
Ramanujan at tea parties. Ibid., 76.
reserved in large groups. Ibid., 83.
Ramanujan during vacations. Ibid., 76.
Charley’s Aunt. Suresh Ram, 43.
By mid-October. Trinity College records.
Highly composite numbers paper. See Ramanujan, Collected Papers, 78–128.
“I gave up the struggle earlier.” W. N. Bailey, quoted in George E. Andrews, Q-Series.
“of an elementary but highly ingenious character.” Hardy, Collected Papers, 497.
“extraordinary insight and ingenuity.” Ibid., 499.
perhaps the most brilliant. Barnes, 33.
“entirely justifying the hopes.” P. K. Srinivasan, 66.
“the most remarkable mathematician I have ever known.” Seshu Iyer and Ramachandra Rao, xvii.
Spring wrote Dewsbury. This and other correspondence concerning Ramanujan’s possible return to India is in Madras Port Trust.
/> about 75 pounds per year. Marwick, 23.
threshold for paying income taxes. Ibid., 21.
“almost ludicrously simple tastes.” Hardy, Collected Papers, 714.
Ramanujan, B.A. Rankin, “Ramanujan’s Manuscripts and Notebooks, II,” 364.
Ramanujan didn’t return to India. But there is a third reason, beyond the two given. See chapter 7.
“most unfortunate.” Hardy, Collected Papers, 491.
four thousand casualties per day. Marwick, 133.
CHAPTER SEVEN
Chatterji dinner. Suresh Ram, 40–42. Details about Chatterji and the other guests come from Who’s Who in India, 1937–38. Chatterji was apparently still in India, at Punjab University, until late 1915, and he was married in 1916, leading to my guess on the date of the dinner.
Kasturirangar’s appraisal. S. R. Ranganathan, 90.
“instinctive perfection of manners.” Neville, “Ramanujan” (Nature 149), 294.
the tablecloths of peacetime were gone now. Told to me at High Table.
“one does not want to work at conversation.” Littlewood, “Reminiscences.”
little book kept over the years. Trinity College Kitchens: Suggestions and Complaints, 1909–1955. Trinity College.
unusually strict in his orthodoxy. S. Chandrasekhar has estimated that even back in India, Ramanujan’s family would have been in the top quarter of some imaginary Orthodoxy Scale. Those traveling to England were typically far less orthodox.
Traveler’s story. Interview, T. M. Srinivasan.
Fried in lard. Suresh Ram, 42. See also S. R. Ranganathan, 78.
a little gas stove. Ibid., 76.
stirring vegetables over the fire. See also Snow, Apology foreword, 35: “Hardy used to find him ritually changed in his pyjamas, cooking vegetables rather miserably in a frying pan in his own room.”
“think of my home and country” Gandhi, 40.
about twenty Indians per year. These figures are derived from the “government studies” cited. Fuller, 186, gives a figure of about seventeen hundred for the whole United Kingdom.
“The initial difficulty.” Amar Kumar Singh, Indian Students in England.
A still later student. Interview, Rajiv Krishnan, Christ’s College, Cambridge. dream of dying. Ibid.
“the reticence of his English friends.” Neville, Reading Manuscript.
“detachment almost amounting to indifference.” Quoted in Barbara Tuchman, The Proud Tower (New York: Macmillan, 1966), 66.
“without the slightest suggestion of discourtesy.” Chaudhuri, 90.
“Oh, were we introduced?” Interview, S. Sankara Narayanan. A similar story is told by Sud in How to Become a Barrister and Take a Degree at Oxford or Cambridge.
“an air of indifference.” Satthianadhan, 28.
“that they were not greatly interested.” Barnes, 35.
“his face dropped.” Young, 271.
“no one will blink an eyelid.” Bollobás, Littlewood’s Miscellany, 149.
ritually pure dhoti. Interview, T. V. Rangaswami.
could not long forget his foreignness. See also S. R. Ranganathan, 47.
“you bathe only once a day?” David E. Fisher, Fire & Ice (New York: Harper & Row, 1990), 46. Told to Fisher, he reports, by a Polish mathematician, at Cornell, in the 1960s.
“very rarely seen.” B. M. Wilson. Unpublished notes. Trinity College. in the cool of the night. Suresh Ram, 40.
“a serious psychological effect.” Colonial Students in Britain, 94.
just 17 percent did while in England. Amar Kumar Singh, Indian Students in England.
Trinity Sunday Essay Society. Trinity College.
Ramanujan not in Majlis. Suresh Ram, 44.
“their nationalism and radicalism.” Nandy, 125.
“wrapped in a medieval gloom.” W. E. Heitland, After Many Years, quoted in Cambridge Commemorated, ed. Laurence and Helen Fowler (Cambridge University Press, 1984), 283–284.
“greatest power and skill to resolve.” Norbert Wiener, Bulletin of American Mathematical Society 55 (1949): 77.
“A moment’s consideration.” Hardy, Lectures by Godfrey H. Hardy on the Mathematical Work of Ramanujan [Notes by Marshall Hall] (Ann Arbor, Michigan, 1937), 25.
“unable to discover.” Ramanujan, Collected Papers, 277.
“as Ramanujan imagined.” Hardy, Ramanujan, 9.
“an extremely obvious one.” Hardy, Quatrième Congrès des Mathématiciens Scandinaves, 46.
“hardly the type to be chosen by Central Casting.” Gian-Carlo Rota, Introduction to Collected Papers of P. A. MacMahon, ed. George Andrews, xiii.
and regularly thrash him. Ibid., xiv.
“Then the problem is completely solved.” Bollobás, Littlewood’s Miscellany, 98.
“a very great step.” Littlewood, Miscellany, 89.
“or even bounded.” Ramanujan, Collected Papers, 283.
“we proceeded to test this hypothesis.” Ibid.
Selberg’s argument. Atle Selberg, “Ramanujan and I,” Science Age, February 1988, 39.
two men. Hindu, 22 December 1987.
“not the only mathematician.” Transcript, Nova 1508, 12.
“a singularly happy collaboration.” Littlewood, Miscellany, 90. See also A. Ostrowski, in Experentia 5 (1949): 131–132: “They were very different scientific personalities … On the one side, a man who had completely given himself up to calculation, who guessed intuitively and worked out the most complicated and hidden formulae, who overflowed with such discoveries, only to become entangled again and again in the net of analytical difficulties. On the other side, a man who was a supreme master of all the finesses and most subtle arguments of modern analysis, and who was continually enriching them by new inspirations, devices, and subtleties.”
“any man I have ever known.” Snow, Variety of Men, 194.
“His sense of excellence was absolute.” Lionel Charles Robbins, in Buxton and Williams, 117. See also Philip Snow, C. P. Snow’s brother: “Hardy was a stern critic of the irrelevant, the ignorant, the brash.” Philip Snow, 44.
“on a high plane.” Interview, Charles Burkill.
Polya’s zoo story. George Polya, “Some Mathematicians I Have Known,” American Mathematical Monthly 76 (1969): 746–753.
“a morning’s work gone west.” Letter, Hardy to Mordell, undated. St. John’s College.
a personality of expectations. The expectations applied fully to himself: “In Hardy’s letters [to Littlewood], ‘Please check,’ ‘Please check very carefully,’ seems like a theme song, and when he was wrong he swore, which confirms my impression that he really minded very much when he himself made bad mistakes.” Mary Cartwright, “Some Hardy-Littlewood Manuscripts,” Bulletin of the London Mathematical Society 13 (1981): 273–300.
“some splendid problems to work at.” Ramanujan, The Lost Notebook, 389.
Oliver Lodge. P. K. Srinivasan, 143.
“the bath rooms are nice and warm.” Bollobás, “Ramanujan—A Glimpse,” 79.
For thirty hours at a stretch. Unpublished notes, B. M. Wilson. Trinity College.
“Only cabbages have no nerves.” Bell, 329–330.
“Meals were ignored or forgotten.” Bell, 109.
“their mental development.” Satthianadhan, 38–39.
“study-worn, consumptive-looking.” Ibid.
toward the end of their second year. Lytton Committee, testimony of S. M. Burrows.
“inclined to develop tuberculosis.” Lytton Committee, testimony of S. S. Singara.
“a regime unsuited to … conditions.” Lytton Committee, testimony of Sir R. W. Philip.
“to cook one or two things myself.” P. K. Srinivasan, 5.
powdered rice in tin-lined boxes. Bharathi, 48.
Ramanujan’s eating habits. Interview, Janaki.
weird hours in the early morning. S. R. Ranganathan, 39.
“a little salt and lemon juice.” P. K. Srinivasan, 27.
“ ‘infants, India
ns and invalids.’ ” Basil Willey, Spots of Time: A Retrospect of the Years 1897–1920 (London: Chatto and Windus, 1965), 118.
on a cricket ground. Historical Register of the University of Cambridge, Supplement, 1911–1920 (Cambridge: Cambridge University Press, 1922), 203.
Marching boots. Cambridge Magazine, 5 May 1917.
“our pride and sad privilege.” Larmor address, 2 November 1916. Proceedings of the London Mathematical Society, 2d series, 16 (1917): 6.
“for the most part wasted.” Barnes, p. 35.
Robert Collins. Seymour & Warrington, caption, fig. 41. The death of Graham is recorded in Madras Times, 6 April 1919.
“memorial services.” Wootton, In a World I Never Made, quoted in Fowler and Fowler, 284.
“Things here are sad and sorrowful.” Parry, 101.
two-fifths of her meat. Marwick, 18.
Kipling poem. Cited in Marcus, 41.
one sure way to tell rich from poor. Marwick, 24: “Poor dieting gave many members of the lower orders yet another characteristic which distinguished them from the rest of society: their small stature.”
“rosy, well-covered men.” Russell, 51.
had climbed 65 percent. Marwick, 125.
“At Wrexham.” Quoted in Marwick, 192.
“portions strangely dwarfed.” Parry, 107.
“good milk and fruits.” P. K. Srinivasan, 5.
“overwork, overplay, overworry.” R. C. Wingfield, Modern Methods in the Diagnosis and Treatment of Pulmonary Tuberculosis, 1924. Quoted in Bryder, 166.
“We travelled in convoy.” Russell, 54–55.
Nursing home on Thompson’s Lane. Rankin, “Ramanujan’s Manuscripts and Notebooks,” 95.
by special dispatch. P. K. Srinivasan, 73.
Hardy … nursed him. K. R. Rajagopalan, 41.
“to take proper care of himself.” P. K. Srinivasan, 69.
“Ramanujan was not exemplary.” S. R. Ranganathan, 74.
“A difficult patient.” Seshu Iyer and Ramachandra Rao, xviii.
“Whenever he started with a doctor.” Béla Bollobás, in Nova 1508, 14. The transcript attributes this statement to S. Chandrasekhar, but is in error.
Few doctors could abide Ramanujan. Letter, A. S. Ramalingam to Hardy, in Rankin, “Ramanujan as a Patient,” 87.
Probably around October. Ibid., 81.
Chowry-Muthu. I am guided in these assertions by Rankin, Ibid. See also P. K. Srinivasan, 110.