The Ramanujan Problem

st1\:*{behavior:url(#ieooui) } in january 1913, G.H.Hardy, perhaps England’s best pure mathematician at the time, received a bulky handwritten letter from a poor clerk in Madras who had three times failed to get into an Indian University. The correspondent confessed that  “since leaving school I have been employing the spare time at my disposal to work on mathematics” and wondered what G.H. Hardy thought of his efforts. Then followed ten pages full of weird theorems along with the claim that the author,  Srinivasa Ramanujan by name, had in his hands “an expression for the number of prime numbers less than N which very nearly approximates to the real result, the error being negligible.”             Hardy was completely knocked back by this letter, “the most extraordinary I received in my life” and eventually arranged for Ramanujan to come to Cambridge (without having to pass any examinations, of course), sponsoring him a year or so later for election to the Royal Society. So far this reads like a fairy tale but there is a sad ending. Ramanujan didn’t take to the English climate, awful cooking and stiff upper lipness : he tried to commit suicide once in the London Underground and ended up contracting  tuberculosis which, after his return to India, killed him before he was thirty-five. Those interested in his fascinating life story would be recommended to read “The Man who Knew Infinity” * by Robert Kanigel, Scribners 1991. My only fault with this book is the insufficient mathematical coverage, too scanty even for my modest level.              Ramanujan, even in his ‘maturer’ years gave very few proofs and those he did give were usually inadequate, and very occasionally actually wrong. He was a man who had received no more than the equivalent of ‘A’ level formal training in mathematics. I am not competent to read, let alone comment upon, Ramanujan’s mathematical output. But his reputation seems to be standing up  pretty well since his death and even underwent something of a  renaissance when his last Notebook was discovered in the eighties since some saw in it anticipations of string theory.  (Not that Ramanujan was at all interested in physics or indeed any applied mathematics.)              So how did he do it?             Patience and keen observation (of numbers) accounts for some of Ramanujan’s results. In the days when the PC was not even a pipe dream Ramanujan spent a lot of time trawling through seas of numbers, exactly the sort of drudgery Western mathematicians at the time rather looked down on. However, one can’t see observation alone producing         113          +     213        +      313         + ……          =     1               e 2p   1            e4p 1           e6p 1                                   24 or                coth p          +    coth 2p        +      coth 3p         + ……   =     19p7                    17                          27                          37                                 56700 just two of the results contained in this now celebrated letter.  (Note : p signifies ‘pi’.)  

            The early twentieth century was the era of rigour : Hardy himself disliked loose mathematical thinking and wanted to reform English mathematics to bring it up to the continental standard. And a stone’s throw away from Hardy’s rooms, Bertrand Russell was busily reducing the whole of mathematics to logic. Russell’s very definition of mathematics — “The science of drawing necessary conclusions” — seemingly excludes Ramanujan’s entire output. Unless, of course, one wants to argue that the reasoning that went on was largely unconscious. But this sort of talk wouldn’t have gone down very well with the early Russell’s positivist friends who tended to ridicule the very idea of the unconscious .              Most eminent pure mathematicians in the twentieth century have either been open or closet Platonists — Hardy himself was an open one. Mathematical Platonists believe that the truths of mathematics are true in an absolute sense : they are not human inventions, and cannot be refuted by an appeal to observation and experiment.   But Hardy was also a militant agnostic, a sort of mathematical Dawkins, and thus hostile to anything smacking of ‘mysticism’. The vision of the higher mathematical Sacred Grail required years of hard work, university training and self-discipline — there was no royal road to analysis. And here was a fellow who claimed to receive formulae for hypergeometric series and elliptic integrals in dreams and who attributed his mathematical achievements to his family’s tutelary goddess, Namagiri. This was Rider Haggard or worse.           If one takes a Formalist point of view, mathematics is invention, and belongs to the arts rather than to the sciences  — at least in principle. In practice, however, students of mathematics are never invited to devise their own symbolic systems in the way in which, for example, artists are invited (or more often obliged) to choose their own subjects for their paintings.  It is just about permissible, at least in popular books on mathematics, to speak of  ‘mathematical instinct’ and ‘inherent mathematical judgment’, but few writers on mathematics  even attempt to define such terms which belong to aesthetics or psychology. Obviously Ramanujan did have these elusive qualities but the trouble with the ‘creativity’ angle is that it leads us into the murky underwater channels of the unconscious, and the least that can be said of modern mathematicians is that they don’t fancy getting their feet wet. Also, it  does not explain why Ramanujan, working almost entirely on his own, homed in on so many of the great themes of nineteenth and twentieth century mathematics albeit from a rather different angle. One might have expected him to go off completely at a tangent, but he obviously didn’t or he wouldn’t have been elected to the Royal Society. So how did he do it?              Must we, after all, believe that Ramanujan was a sort of mathematical Joan of Arc? This is an explanation of sorts and has the merit of being the one Ramanujan himself preferred. In the 16th and 17th centuries Ramanujan would not have been such a misfit: even Descartes, the father of modern rationalism,  claimed to have been visited by the Angel of Truth.  There are nonetheless difficulties with this explanation, even for such an anti-rationalist as myself, principally the fact that Ramanujan was not invariably right. His claim to have in his hands a formula giving the distribution of the primes unfortunately turned out to be mistaken. (It has apparently since then been shown that no such formula can exist.) Of course, there is no reason why a goddess should not err on certain technical points but it is suspicious that the slips made by Ramanujan (or his source) were precisely the ones to be expected from someone not fully au courant  with  the very latest research into the divergence of  infinite series ¾ research that Ramanujan, in his Madras backwater, was unaware of.             I personally don’t have the sort of trouble Hardy and Russell (or today militant rationalists like Martin Gardner)  have with the idea that some people can tune in ‘directly’ to sources of knowledge most of us can’t, though I interpret the phenomenon more in terms of Jungian ‘Group Minds’ or ‘Collective Memories’ than in terms of goddesses and spirits. It may be that Ramanujan from the mysterious East connected up with planes of being invisible to us educated Westerners, had readier access to the Akashic Records of Mme Blavatsky, if you like. I certainly have less of a problem with this approach than that of   mathematical Platonism. The latter made good sense in the days when people viewed God as the Supreme Mathematician (as Kepler and Newton did) but cuts little ice today with anyone except professional mathematicians. For what it is worth, the consensus amongst physicists today is that the world we live in is not the result of intelligence and planning — it just happened. And the fact that mathematics has proved to be a useful tool in investigating the cosmos doesn’t in the least mean the cosmos is inherently mathematical. Is a cat mathematical? To actually model a predator pursuing its moving prey on the savannah, quite complicated mathematics involving differential equations is required, but no one in his right senses is going to suggest that a cheetah or a cougar knows what he or she is doing mathematically speaking, or needs to : trial and error and natural selection suffice. And, as far as we know, there’s nothing special about the values of the most important mathematical constants, G, c or  the fine structure constant: so far all attempts to derive such values by a priori reasoning ¾ as Eddington tried to do ¾  have been miserable failures. We just happen to be in a universe where these constants have the values they do and that’s ultimately all there is to it. And if there is something beyond and behind all possible and actual universes, the Matrix to end all matrixes, my feeling is that neither words  nor symbols nor numbers are going to be of any help here ¾ “The Tao that can be named is not the original Tao” (first line of the Tao Te Ching). As far as I am concerned mathematics deals strictly with what is measurable and, whatever ultimate reality is, it’s certainly not measurable or it wouldn’t be ultimate.             So how do I explain Ramanujan? As someone who believes that the origins of mathematics lie in our perceptions of the physical world in which we live, I must  admit Ramanujan worries me a bit. Because of the terseness of his results and his  air of absolute conviction he does, at first glance, look like  someone who has a window on a higher  reality, a strictly mathema
tical one, and that all he has to do is to transcribe  what he sees.  But then again part of the reason for this lies in his idiosyncratic working habits. In India at any rate — where he did most of his creative work — he did his mathematics with chalk and slate because he found paper too expensive. He rubbed out with his elbow as he progressed and only noted down the final result. So he  probably couldn’t remember the intermediate steps by the time he’d finished and had no means of checking. Maybe he even covered up his tracks on purpose : we don’t really want a magician  to reveal his secret as Cutter, the magician’s ingénieur, says in the film The Prestige
¾  it spoils our pleasure. Indian mathematics never was too much concerned with proofs anyway — there is the famous example of the ‘proof’ of Pythagoras’ Theorem by way of a diagram with the caption “Behold!”

            One thing that’s certain is that  Ramanujan was born in the right place and time and that maybe accounts for a lot about his mathematics. India was, at the end of the nineteenth century, a country looking in two directions. It was still immersed in mysticism, the occult, philosophic and religious speculation. But at the same time it had an advanced educational system modelled on the British, and was encouraged,  to send particularly  bright pupils to Oxford and Cambridge. The rational plus the irrational (or supra-rational) is  a heady and treacherous mixture but it suits certain types of minds perfectly. Kepler, astrologer and astronomer, mystic and painstaking observer, was a child of a similar place and time, Renaissance Germany. The dangers of irrationalism have been trumpeted in our ears for two centuries already , but there are equally grave dangers attendant upon the exclusive use of ‘reason’. There has always been something threatening and, above all, puritanical in rationalism. Hardy wrote of one of his contemporaries, “Bromwich would have had a happier life, and been a greater mathematician if his mind had worked with less precision”. The Houhnhms, the strictly rational beings of Swift’s “Gulliver’s Travels”, are not only rather dull but not even very congenial since they entirely lack spontaneity, tenderness and enthusiasm.             Much has been said about Ramanujan’s lack of adequate mathematical training. But it was, on the contrary, very suitable — for him. He was given about as much as he needed to get going, namely groundings in most areas including calculus (still little taught in schools at the end of the nineteenth century). He didn’t make it to university but he did get to know several eminent  Indian savants and his immediate superior in his office was an excellent amateur mathematician. So Ramanujan had people he could talk to about mathematics, and it was in many respects an advantage that such persons did not know more than they did, or more than he did  — for they would have put him off following down certain pathways. It is an open question whether even Hardy, who discovered him, had, in the last analysis, a good or a bad influence on him.            Mathematics has, in the last two centuries, become a matter of solving the great problems, and rigorously proving the great theorems, bequeathed to us by the previous generation. It has become grimly serious and has long since ceased to be the carefree exploration  of virgin territory that it was in the time of Fermat and Euler. Ramanujan was not a prover nor  even especially  a problem solver :  he was an explorer. In his youth, after giving up the idea of getting into college, he spent five happy years supported by his poor parents doing nothing except sitting on a wooden bench in the sun in front of the family house working at mathematics, his choice of mathematics. After his excursion into Europe he returned to this mode of life in his last years, exploring peculiar things he called “mock-theta functions”. The best thing to do with such a person is to let him get on with it and have someone check up on his results later. But that wouldn’t do in the contemporary era, it sounds far too lax, ‘libertarian’ ¾ people might actually get to enjoy mathematics if they were allowed to follow down pathways that caught their fancy.              In this era of “education, education and education” it is worth pointing out that, though lack of knowledge renders people impotent, too much knowledge available at the drop of a hat makes one lazy, blasé and unimaginative. It is indeed often salutary to be deprived of knowledge.  If Pascal’s father had not forbidden him to study geometry, he would not have got off to such a good start by re-discovering whole chunks of Euclid unaided.  Ramanujan kicked off with an out of date pot-boiler, Carr’s Synopsis,  which is apparently all formulae and no proof. The author was an enthusiast for his subject, however, and managed to communicate this to his readers. According to Kanigel, the book has a certain flow and movement ¾ indeed I’d like to read it myself and I’m sure I’d get a lot out of it.            Now you can’t teach ‘exploration’ but it can be encouraged. In contemporary schools and colleges it practically never is. What we get is the  message to the world delivered by Head of the American Patent Office in 1890 : “Everything worth discovering has already been discovered” (he actually said that)    with the exception perhaps of a few abstruse issues that require ten or fifteen years of preparatory training in a college of higher education. As it happens, one of the most exciting mathematical events in the last twenty-five years has been the discovery (or rather invention) of fractals. But they were turned up by an explorer of mathematics, Mandelbrot, who worked at the time for IBM, not Princeton University  — I gather that  even  today the snobbish pure mathematical fraternity in America does not accept Mandelbrot as being part of the club. And it all came out of looking into a simple function that goes back to Newton and is known to most sixth formers.             The great objection to exploration is that there’s no point in re-inventing the wheel. But there is. Invention or re-discovery gives you a thrill that  answering  routine questions set by someone else never does. Secondly, it gets you into the habit of inventing and “If you want to be a blacksmith, go and work in the forge”. Once you’ve started inventing, you may well end up with something that really is original since discovering something for yourself is much more likely to lead on to further discoveries. A retired civil engineer of my acquaintance, Henry Jones by name, with no mathematical training beyond ‘O’ level, produced a weird-sounding definition of an ellipse, namely the locus traced out by a point on the circumference of a revolving circle, the centre of which is revolving around a fixed centre at half the speed of the point in question. This is in effect the parametric equation of the ellipse which goes back to Copernicus though Jones did not know it. This hopelessly old-fashioned geometric definition suggested an immediate application — which the algebraic definition doesn’t— and Jones went on to design a compass which could draw ellipses, as well as circles and straight lines, since the circle and the straight line are, mathematically speaking, limiting cases of the ellipse. (Although he took out a p
atent I believe the Jones elliptical compass was never manufactured, though it deserved to be.)   
            On the basis of Kanigel’s book, I don’t think I am able to subscribe to the conventional wisdom that “if only Ramanujan had had the proper training what a great mathematician he would have been!” More likely strict training would have turned him, or killed him, off. Einstein himself, a mediocre physics student who found himself obliged to borrow the notes of his friend Besso when preparing for his final examinations, only just survived the academic obstacle course3.    Ramanujan had the sort of education suitable for a bold and imaginative person, more would have weakened his self-confidence and destroyed his enthusiasm for the subject. His very failures were glorious. Although Ramanujan’s claim that he had a function giving the distribution of the primes fails for very large numbers, it is for all that a tremendous achievement. “Of the first nine million numbers, 602, 489 are prime. Ramanujan’s formula gave a figure off by just 53 — closer than the canonical version of the prime number theorem.” (Kanigel, op. cit.) This really is David against Goliath, on the one hand a hundred or more years of research from the cream of the West’s pure mathematicians with all the data available and on the other a man with a slate and a piece of chalk who had never even heard of the Cauchy Integral Theorem. If he’d done nothing else the man deserves a name in the history books — and this was one of his errors!  


 * The title of Kanigel’s book, The Man Who Knew Infinity,  is a misnomer and would be more applicable to a biography of Cantor. To my knowledge  Ramanujan never showed any interest in Transfinite Ordinals and, when he came to England, does not seem to have even heard of Set Theory. The Man who Knew and Loved Ordinary Numbers  would have been a more suitable, but less eye-catching,  title for a biography of Ramanujan. .   

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