Saturday, May 11, 2019

On Intuition in the Life and Work of Ramanujan

These past few weeks I have been engrossed in reading about Srinivasa Ramanujan (1887-1920), the brilliant mathematician from South India whose prolific work in number theory lit up the academic world from the early 20th century onwards.

In my studies in this subject I had heard the name, seen references, but didn't use his work.  Even though he's famous in the mathematical world and in India, I wasn't minded to give much attention until I happened to chat over lunch with Thomas Bewley, who described his experiences of playing Prof. H. F. Baker in the film, The Man Who Knew Infinity.  I promptly ordered a DVD and enjoyed watching the film, so I then ordered the substantial biography of the same name by Robert Kanigel.

I'm writing this having just finished Kanigel's book.  It's extensively researched and very detailed, covering Ramanujan's life and work; whilst aimed at the general reader, quite a lot of mathematical material has been presented, in a generally convincing way – Kanigel is numerate (he has a degree in mechanical engineering) and he has evidently spent considerable time grappling with the material in conversation with scholars.

I particularly like the way he shows how the relevancy of Ramanujan's work and various applications has ebbed and flowed; in his early years he struggled to make known his findings, but gradually some friends and associates tried to help promote his cause until they were able to tap into the British colonial networks.  Eventually there arose opportunity to write to the scholars at Trinity College, Cambridge, Britains' foremost centre for mathematical research.  Even then, Ramanujan had to keep persevering until he succeeded at the third attempt when his genius was recognised by Hardy, who nurtured Ramanujan's talent by instilling the rigours of proof and dissemination of the various results.  After Ramanujan's passing, Hardy continued to promote his cause through papers and continued reference.  Subsequent decades saw changing foci, but recently his work has become of great significance – his “mock theta functions” have been integral to the development of mock modular forms, which are now used in astrophysics, even to model singularities (black holes, etc.).

Personally, I might like to explore his work in partition functions (having a natural interest in combinatorics).  However, I am mainly interested at the moment in Ramanujan's spirituality, what we might learn about intuition.  Kanigel attempts to explore this area, knowing full well that members of the public like especially to know about how a human being can navigate the vicarious aspects of life and its innumerable obstacles, triumphing over adversity – the indomitable spirit.  Kanigel dutifully delves into this with cultural sensitivity – undertaking fieldwork in the foreign lands and cultures of the British Isles and India.  Through the information he gathers from interviewees, Kanigel recreates at some length daily scenes in which Ramanujan lived and breathed mathematics - in Kumbakonam, his home town, and in various other places such as Triplicane (now Tiruvallikkeni) (with its historical sites such as the Arulmigu Sri Parthasarathyswamy Temple), and other areas in the then Madras Presidency (now Tamil Nadu).

Even in a volume as extensively researched as this, the whys and wherefores as to Ramanujan's  mathematical discoveries can't be fully explained when it seeks to explicate an entire life story and indicate especially its mathematical import today.  Even so it's noticeable that whilst Kanigel appears comfortable explaining material facts, he finds it exceedingly difficult to fathom Ramanujan's spiritual inspiration.  Hence overall he writes sympathetically, but when it comes to religious aspects, he doesn't have much to say, and even occasionally strikes an incredulous tone.

For example, referring to a gathering that developed into philosophical discussion he writes (pp. 31-2):
Another time, when he was twenty-one, he showed up at the house of a teacher, got drawn into conversation, and soon was expatiating on the ties he saw between God, zero, and infinity - keeping everyone spellbound till two in the morning. It was that way often for Ramanujan.  Losing himself in philosophical and mystical monologues, he'd make bizarre, fanciful leaps of the imagination that his friends did not understand but found fascinating anyway. So absorbed would they become that later all they could recall was the penetrating set of his eyes.

I don't suppose Ramanujan felt lost; if anything, he was finding deeper relationships in what the author describes as "bizarre" and "fanciful".  Whilst it might have been tantalising to his audience, at the same time the culture readily accepted this kind of expression.

In another chapter Kanigel writes (p.66):
Later, in England, Ramanujan would build a theory of reality around Zero and Infinity, though his friends never quite figured out what he was getting at. Zero, it seemed, represented Absolute Reality. Infinity, or ∞, was the myriad manifestations of that Reality. Their mathematical product, ∞ x 0, was not one number, but all numbers, each of which corresponded to individual acts of creation. 

Kanigel's tone conveys shades of incredulity, but these kinds of views are taken seriously in many parts of the world.  At least in recent centuries, they seem to be more naturally appealing to Asians – from all over, whether the South, South-East, Far East, and the North.  So I'm interested to read accounts from their perspective, particularly Indian perspectives and interpretations – and how do they interpret Ramanujan today?

As Ramanujan is a national hero, there's no shortage of material, a fair amount being helpfully referenced in Kanigel's book.  Arriving as a newcomer, I try to find where possible sources with first-hand accounts, ideally published by authorities who have some historical connection.  My starting point has been a broad selection made available by The Institute of Mathematical Sciences in Tamil Nadu, a national research centre.

They list five books in a section on books, including Kanigel's volume.  Among the others, I've been looking at  'Ramanujan - The Man And The Mathematician' by S. R. Ranganathan, part of Great Thinkers of India Series, published by Asia Publishing House in 1967.  The publisher is still registered, based in Mumbai, but I can find no website for it.  Dr Ranganathan, who was a mathematician and library information professional in India; an endowment in his name is associated with another publishing company, Ess Ess Publications limited, and copies of the book are readily available from them.  (But it's also not hard to find a free version online.)

This book records some accounts by those who knew and met with Ramanujan.  One of the respondents is Dr. Mahalanobis, who was one of those who was there in the late night discussion that Kanigel refers to.  He recalls:
He sometimes spoke of “zero” as the symbol of the Absolute (Nirguna-Brahmam) of the extreme monistic school of Hindu philosophy, that is, the reality to which no qualities can be attributed, which cannot be defined or described by words, and which is completely beyond the reach of the human mind. According to Ramanujan, the appropriate symbol was the number “zero”, which is the absolute negation of all attributes.  He looked on the number “infinity” as the totality of all possibilities, which was capable of becoming manifest in reality and which was inexhaustible. 
(MN Reminiscences of Dr P C  Mahalanobis FRS,
Member of the Planning Commission of India: MP1, p.82)

Ramanujan starts with that philosophical position and then gives it mathematical expression, based on numerical properties that can exhibit the transcendent qualities.  These are not just made up fancifully, but rather there are references to a philosophical school.   Nirguna-Brahmam (or Para Brahman), is described in Hindu texts as the highest spiritual state, the formless Brahman, specifically in the sense of being absent of Maya, illusion.  It's a core belief in the Advaita Vedanta tradition.

Mahalanobis continued:
According to Ramanujan, the product of infinity and zero would supply the whole set of finite numbers.  Each act of creation, as far as I could understand, could be symbolised as a particular product of infinity and zero, and from each such product would emerge a particular individual of which the appropriate symbol was a particular finite number. I have put down what I remember of his views. I do not know the exact implication. 

Whilst Mahalanobis lacked understanding of the finer points, he could gain a general sense of what lay behind Ramanujan's words – there was valid and useful communication.  If they had been completely incomprehensible, then Ramanujan probably would not have sustained interest for so long.  Perhaps more significant still as an indication of the importance of this spiritual view, was the following reflection:
He seemed to have been perhaps emotionally more interested in his philosophical ideas than in his mathematical work. He spoke with such enthusiasm about the philosophical questions that sometimes I felt he would have been better pleased to have succeeded in establishing his philosophical theories than in supplying rigorous proofs of his mathematical conjectures.

This is a significant passage as it points to how important to him was his underlying spirituality of which mathematics was an expressions.  I think we see the deleterious effects of denying him support for this spirituality when Kanigel describes how to many Ramanujan appeared a much-changed man on his return to India in 1919.  Mentally and emotionally he was a different person: whereas previously he was full of fun and sociable in small groups, on his return he appeared withdrawn and angry.  It seems England was able to support his mathematics, but it came at the price of his Brahmin caste (at least for those who did not allow any exemptions to Samudrolanghana, the offence of crossing the sea) and his wellbeing.  There are areas that the book perhaps understates this sacrifice – which was more than the decline in his physical health.  Yet Ramanujan foresaw his own death (“I won't reach 35 years of age”), so the speculation around what might have been regarding alternative life paths and treatments of his tuberculosis should be set against that.

In modern times, we can still find views from India, especially religious teachers, who can give some indications of Ramanujan's spirituality.  Even though they might not have any formal background in mathematics and may lack rigorous language, they can express the 'inner voice', as it were.  For example, in his talk at SRCC College, available in a YouTube video, entitled The Secret of Ramanujan's Genius,  Sadhguru likens deities or, more specifically, murtis (forms) to energetic machines that are able to enhance particular faculties; unlike mechanical devices, such machines don't have moving parts, are easy to maintain, and are available all day and every day.  Ramanujan knew how to use the murti known as the goddess or deity Namagiri to receive mathematical insights and he seemed to be working continuously.  In the short excerpt, it's not explained how one cultivates practice of utilising these murtis, but in India it is typically through yogic or meditative training, and, as for most yogis, Sadhguru gives instruction in these, such as Isha Kriya.

Another perspective is shared in a presentation on teachings by Sri Aurobindo & The Mother: the quality of beauty is highlighted in a post where Sandeep, the author, asks: Where does Mathematics come from?   According to teachings in this tradition, having some correlation to the energetic machines, it as though humans have inner beacons of light that can be directed towards specific arts; an agile mind can shine the light in different directions.  But here, this longer article also emphasizes  development (I'd choose the word bhavana) of the capacity of attention and concentration.  Other posts on that informative site, including one that considers some views of Roger Penrose, describe how a prerequisite is knowing how to bring the mind into stillness (once the mind is at a standstill one can move easily in any direction); bringing the mind to a standstill is  key to allowing novel ideas to arise.

From my own Buddhist perspective, I would highlight that Ramanujan's superlative ability can only come through sustained kusala karma (skilful intentional actions), usually over many lifetimes.  In this way he would have generated puñña (merit), a kind of energetic fuel that with continued cultivation crystallizes as paramis (perfections) – puñña gives you the capacity to achieve, paramis enable that capacity to be readily and instantly available.  Perhaps Ramanujan refrained strictly from intoxicants leading to great clarity and receptivity of mind – certainly even in such a foreign environment he practised strictly as a Brahmin, so he retained that quality of mind seeking perfection.

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