By Kumar David –
Where do truths lurk before they are discovered by humans?
Suppose a mathematician discovers a theorem in 1916, does it mean that this truth was invalid before then? Of course not; it existed in 1915, in 1815, in BC 1915 and before. Where then did it hide? This discourse has overtones of Platonism but lies at the heart of the long arguments between Ramanujan and G.H. Hardy. At issue was the former’s religious faith and Hardy’s atheism strengthened by his friend Bertrand Russel’s rationalism. Russel, a Wrangler (so was Hardy), was a lecturer at Cambridge before he was kicked out for pacifism, rationalism and atheism in 1916.
What got me going on today’s ruminations was that on the Colombo-Bangkok leg of my journey to Hong Kong two weeks ago I watched The Man Who Knew Infinity, an enjoyable film about the Cambridge days of Sirinivasa Ramanujan, a poor Madras Tamil Brahmin now recognised as a mathematical genius. Ramanujan had no formal training in mathematics before joining Cambridge in 1914 at the age of 27 thanks to Hardy’s efforts; but his previous untutored solo work was stunning. This adds wonder to the question: “Where do ideas come from?”
There are many sides to the film that I cannot deal with; my focus is on the dispute about god and the sources of knowledge between Ramanujan (Dev Patel) and his mentor, another brilliant mathematician Godfrey Harlod Hardy (Jeremy Irons) who was only 10 years older. A second mentor who played a role in Ramanajun’s Cambridge days was John Edensor Littlewood, a Senior Wrangler (played by Toby Jones in the film) – Jeremy Northm plays Russel.
Hardy and Ramanujan
Hardy is said to have rated Ramanujan side by side with Euler, Jacobi and even Gauss and if he meant potential, not achievement – Ramanujan died of tuberculosis at the age of 32 – it is reasonable about the first two, both algorists like Ramanujan. But to include Carl Friedrich Gauss is a considerable stretch of Ramanujan’s potential had he lived to a ripe age. A scene in the film has Hardy reading out a note from Littlewood projecting Ramanujan’s latent potential as akin to Newton. This is baloney and probably the note was concocted by the film makers. Newton, for cognoscenti and laymen alike, is supreme among scientists; he was no mere mathematician. Nothing Ramanujan is likely to have achieved had he lived long could have stood beside the impact Newtonian physics (Classical Mechanics) has had on the practical world. Not just scientists but every engineer is reminded of this every working day.
Ramanujan was elected a Fellow of the Royal Society in 1917. Trinity which had previously rejected Hardy’s proposal to allow him to trample the grass was compelled in 1918 to let Ramanujan graze on its lawns. His recognition in India is extensive including a journal and societies bearing his name, and a special postage stamp with his likeness.
The god debate
Ramanujan was deeply religious; Hardy a hardened atheist but curious about how dazzling ideas entered his protégé’s head. To put things in context we must take account of Ramanujan’s magic method. Scientific researchers and mathematicians labour long over a problem, slice and scratch and go through much drudgery before reaching a conclusion or a theorem. Even flashes of insight only cut through difficulties that have long obstructed progress. Ramanujan was the polar opposite. Call it intuition or revelation or what you will, but his mind would in a flash see a result, an equation, or a theorem, and only afterwards would he sit down to brush it up, or bother about proving it. One thing that drove Hardy to paroxysms of frustration was Ramanujan’s reluctance, sometimes his refusal, to write out a complete proof without which publication was impossible. The impatient scholar would respond: “Of course it’s correct; I know it’s correct; why waste time writing out a proof”. Many theorems that Ramanujan left dangling have been proved by others later, sometimes much later. Only a few have been found to be erroneous.
So you see the kernel of Hardy’s curiosity. “How does it come into your mind? From where do these flashes of intuition arrive? What is the process; how does it happen?” Ramanujan was lost for an answer: “I don’t know; it just happens; it’s like a revelation; it just comes floating in”. And this is why Ramanujan came to say to Hardy “It must be God; it has to be the mind of God engaging with me, otherwise how to explain these revelations that I see in a flash?” Hardy, firm atheist that he was, did not buy the god story but he was shaken by Ramanujan’s all but miraculous gift as he admitted when he delivered a commemorative oration at Trinity when news of Ramanujan’s untimely death in 1920 came from India, a year after he had returned home.
Here is a sample. Mathematician Mahalanobis put a complex puzzle to Ramanujan. He reflected and responded with a ‘continued fraction’ but this was the solution to a whole class of problems. Mahalanobis, astounded, asked how he did it. “It is simple” Ramanujan replied, “the moment I heard the problem I knew the answer was a continued fraction. Which continued fraction, I asked myself and the answer came to my mind.”
Knowledge of the material world
I started by saying mathematical truths exist a priori, meaning that they existed before the mathematician discovered them; for this reason I was careful to use the word discover, never the word invent. America existed before man discovered it some 15,000 years ago, he did not invent it. The entities of mathematics, think of numbers, are created by man and therefore a mathematical theory or relationship is the product of the human mind, though logically its truth exists prior to its discovery by a mathematician. That much is common sense. But what about the real world which exists outside and separate from man and existed prior to man? (Such comments drive quantum scientists into a state of entanglement and encourage the likes of Gamini Kulatunga, in probable waves of particles suspended in superposition, to pile up texts for the edification of ignoramuses like this writer).
Mathematical entities are crisp and clear, they do not soil with time – think of prime numbers. Mathematical theorems are truths that do not degenerate with age and were true from before they were discovered – think of any fundamental theorem, for example the one that says any integer can be expressed as a product of primes (note, 2 is a prime) was/is forever true.
However, when we turn to the study of the real world it is collective human experiences, not mental abstractions that need to be explained and theorised. Definitions, quantities and relationships must pertain to experience and experiment, not to prima-face concepts alone. This imperative is central to science. The laws of nature and the conceptual building blocks for their expression change and evolve with the progress of observation and knowledge; their truth value is not eternal. Action at a distance in electromagnetism and gravitation is replaced by the concept of an all pervading field.
There is nothing like the pristine clarity of say a prime number or the unchanging eternity of a product-of-primes theorem in gravitation or electromagnetism. Thus the immaculate perfectness of the mind of god goes out of business. The answer to the question where does knowledge come from is different. Knowledge comes from the processing of real world material and observable information by the human mind which is endowed with rational and logical capabilities. How long before and after its discovery is knowledge true? It depends on the existential qualities of the phenomenon itself. The Ramanujan-Hardy debate, in so far as it relates to scientific knowledge, I avoid the word truth, is heavily weighted in Hardy’s favour.
[Reader Ringa Ranga Raja from London has written to complain that 14 March 1879 was Einstein’s birthday and 14 March 1883 Marx’s death. He had expected me to use the twin anniversary last week. Sorry, next year maybe].