Review of The Man Who Knew Infinity
By Robert Kanigel
Charles Scribner's Sons, New York 1991, 438 pages, $27.95
The famous playwright Moss Hart, in the flush of his first grand success on Broadway, bought a yacht and invited his family and friends to a party on board. His mother saw him strutting on the deck wearing a captain's hat. This upset her. And she said to her son, "Moss, you think you're a captain. And me? Well, for me you're a captain. But when it comes to the captains, do they think you're a captain?"
The story of Ramanujan is fairly well known, but a brief outline may be in order. Late in 1912, Cambridge mathematicians H.F. Baker (algebraic geometry) and E.W. Hobson (theory of functions of a real variable, spherical harmonics) received letters from India that included samples of newly worked out mathematics. They ignored the letters.
In January 1912, Geoffrey Harold Hardy, who at that time was considered to be among the most brilliant of English mathematicians, received a similar letter. Hardy, who was closer to the material of the samples than either of the first two men, did not ignore it. He realized that the samples were not only inspired but devilishly difficult. Through Hardy, the author of the letter, a young Indian by the name of Srinivasa Ramanujan, received a fellowship to work in Cambridge. He remained there off and on, working with Hardy, suffering from tuberculosis. Finally, he returned to Madras, where he died in 1920 at the age of 32, a fellow of the Royal Society.
The legend of Ramanujan then began in earnest. In certain quarters, studies in Ramanujanian mathematics are popular today, perhaps even more so than they were a few years ago.
What are some of the raw materials that contribute to this legend? Ramanujan was born a high-caste Brahmin of modest economic status. In his early years, he lived the traditional life of a Brahmin. He wore the topknot; his forehead was shaved. He was a strict vegetarian. He was spiritually and ceremonially religious. Every year he went to a local temple at the full moon of Sravanam to renew his investiture with the Sacred Thread of the Brahmin caste. When he went to England, it was only after much soul searching and in defiance of Brahmin tradition. Throughout his life he recited the Vedas; he believed is palmistry and in dreams, and he interpreted what he saw. He talked till the wee hours about the intimate relationship among God, zero, and infinity (as do some of our contemporary semioticians of mathematics!) As to his purely mathematical life, however, Ramanujan's formal education in mathematics seems hardly to have gone beyond Sidney Loney's Trigonometry (CU Press, 1893), which he mastered by the age of 13. He seems to have had little conception of the theory of complex analytic functions, in which some of his identities would later be embedded. He seems to have had little conception of what a "rigorous" proof was, how it functioned, what its purposes where. And yet, this mystic mathematical autodidact, if we disregard a few errors and some trash, came up with some of the most remarkable of mathematical theorems and identities. The problem for human cognition: How did he do it? The problem for Hardy: How to place this wonderful stuff firmly within the canon; how also to move beyond the material that could be grasped with Ramanujan as a guide to where there would be yet more wonderful things.
As part of his effort to educate him, Hardy tried to teach Ramanujan what a proof was, what the theory of functions of a complex variable was --- in short, to teach him how to behave and think and conceptualize in the manner of a proper Western mathematician of the period, to recreate him in his own image.
In the social sphere, Hardy tried to make Ramanujan "clubbable", to bring him into the Cambridge milieu somewhat in the manner in which Professor Higgins worked with Eliza Doolittle. He wanted to bring this eater of rice, yogurt, and sambar to the roast beef and mutton of the Trinity High Table, and after dinner, to the port, apples, and little Conversaziones in the Combination room. Where the fictitious Higgins succeeded, Hardy largely failed. Despite this failure, the younger man was not only to inspire the older man mathematically, but to provide him with a decided dash of romantic color in a gray bachelor life whose only elements seemed to be mathematics and cricket.
This, in brief, is the story, and it is also the background for a remark made to me recently by a Cambridge don: "Of course we get a lot of fat, unsolicited envelopes. Most of the contents are rubbish. But when the envelope bears a postmark from India, we tend to look at the rubbish rather more carefully."
Robert Kanigel, a science writer and a teacher of literary journalism at Johns Hopkins University, has written a splendid biography of Ramanujan. Carefully researched and documented, written lightly with hardly a tedious page in it, the book is very much in the spirit of today's biographical craft: detailed, and with little respect for the "privacy" of the biographer. Although today's biographies have a marked tendency toward simultaneous hero-worship and hero-bashing, Kanigel wisely maintains respect for his subject.
To round out the picture, Kanigel has done much more. As minor themes, he has given us a parallel biography of G.H. Hardy and travelogues into the minds and customs of Madras, India and of Cambridge, England. And there is even more: He has tried to tell the general reader (I'm not sure how successfully) what Ramanujanian mathematics is all about.
The literary world has been deluged by material on the Cambridge scene. There are now dozens (it seems) of books on the Cambridge spies, moles, and double agents. Even this reviewer has contributed to the Cambridge industry with a satire on the Cambridge Common Room (Thomas Gray, Philosopher Cat). Kanigel's work fills some of the remaining gaps, and the world of Cambridge mathematics now stands quite adequately fleshed out. Madras, on the other hand, is unfamiliar, exotic, and more than a bit incomprehensible.
The basic as yet unanswered question is: How did Ramanujan do it? Sheer formal computational strength, as some have conjectured? Or does there exist an intuitive grasp of mathematical matters, a grasp of such super penetrability that it is possessed by only one or two geniuses in a millenium?
The late Mark Kac gave as good an answer as can be found. Kanigel quotes it:
An ordinary genius is a fellow that you or I would be just as good as, if we were only many times better. There is no mystery as to how his mind works. Once we understand what he has done, we feel certain that we, too, could have done it. It is different with the magicians. They are, to use mathematical jargon, in the orthogonal complement of where we are, and the working of their minds is, for all intents and purposes, incomprehensible. Even after we understand what they have done, the process by which they have done it is completely dark.
Swimming along with today's psychologizing, social constructivist mode of biography, we may ask what, if any of the pieces of local color that Kanigel provide, were critical. Did it matter that Ramanujan ate pickles and chilies while he was tubucular? Did his communion with the family goddess Namagiri of Namakkal help his mathematics? Were his firm beliefs in palmistry and astrology part of a unified picture that included mathematics? In the Hardy-Ramanujan collaboration, what, if anything, was the role of the suppressed homosexuality sensed in Hardy by his close collaborator J.E. Littlewood.
The Ramanujan story raises many kinds of questions. What is mathematical intuition, insight? Can they be cultivated? What is genius? Can education foster it? Or is genius, bu definition, that which is beyond fostering? What is proof? Why should one care? Philosophically, does the reality of mathematical results depend on their being accommodated to an establishment view? Is there such a thing as mathematical luck? What is interesting, dull, significant in mathematics. (Most mathematicians couldn't care less about Ramanujan's discoveries.)
The succès fou of this particular unorthodoxy raises another question. How would you answer if I asked whether Ramanujan was a professional? I ask because several years ago, a prominent number theorist in the U.K. told mem that in his opinion Louis Mordell was not a professional. Mordell? Of the Mordell-Weill theorem? Sadlerian Professor of Mathematics at Cambridge and an FRS? Come off it!
I am not sure what was implied by the denial, but it got me thinking about the word "professional". I would have said, following the practice in athletics, that professionals are simply people who make money out of whatever it is that they do. But there is a counter definition: "A hired man is not a professional man. The essence of a professional man is that he is answerable for his professional conduct only to his professional peers." Thus saith H.L. Mencken (Prejudices: Sixth Series, 1927), and I believe that Lily Hart of my lead-in anecdote would have agreed.
With this in mind, I would hope that today's mathematical establishment could exhibit some of the flexibility as regard mathematical conduct that Hardy did --- and that wasn't too much. Flexibility seems often today to be a rare commodity. After all, in Hardy's time the name of the game was Proof; but that, really, is only part of the game. I know a number of brilliant people in several countries whose work falls "between the cracks" --- and who have suffered greatly from cross-disciplinary intolerance. Their interests are viewed as peculiar, their methodologies unorthodox, their results strange, their personalities cranky. They have been filtered out of university mathematics teaching as misfits, and they make their livings as they can. Think of Charles Sanders Peirce and you will have an example of the sort of person I am talking about.
The Established Church always has great trouble with its Saints --- until, of course, they are canonized. Saints are a difficult, nonstandard lot, and anyone who claims to have a private hot line to God is prima facie suspect. Such people are very lucky indeed when they have a Hardy to put forward their names in nomination.
Philip J. Davis is a professor in the Division of Applied Mathematics at Brown University.