The debates

Rigour versus intuition

He stated many theorems and proved few. Frank Aiello writes that he had only the vaguest notion of proof. His theory of primes was wrong because he did not know complex function theory. Hardy also said he leaned on divergent series and that some results were false. Critics see a gift that was never fully disciplined.

Hardy judged the 1913 theorems true because no one could have had the imagination to invent them. He called Ramanujan the greatest formalist of his day. Berndt, Andrews and others have since proved most of the claims in the notebooks. His intuition anticipated mock modular forms by decades.

Frank Aiello: Ramanujan’s Thoughts from God (arXiv 1707.03379) · Encyclopedia.com: Ramanujan, Srinivasa Aaiyangar · Springer: Ramanujan’s Notebooks Part V by Bruce C. Berndt (1998) · Wikipedia: Srinivasa Ramanujan · Jeffrey A. Harvey (University of Chicago): Ramanujan’s influence on string theory, black holes and moonshine (arXiv 1909.11477)

The myth of the untrained genius

He is often called self-taught. Hardy wrote that he had never seen a French or German book. Yet he had Carr’s Synopsis and Loney’s Trigonometry and access to the Kumbakonam school library. Ramaswamy Aiyer and the Indian Mathematical Society published him before Hardy did. Patrons in Madras such as S. Narayana Iyer backed him. The lone-genius story hides that network.

The training he lacked was real. He never completed a degree in India. He rediscovered results long known in Europe and missed whole fields such as complex analysis. Hardy said he had to teach him modern mathematics. Both points hold: he lacked formal training but not support.

Obituary of Srinivasa Ramanujan, Proceedings of the London Mathematical Society (1921) · Encyclopedia.com: Ramanujan, Srinivasa Aaiyangar · MacTutor History of Mathematics (St Andrews): Srinivasa Aiyangar Ramanujan · ThePrint: Srinivasa Ramanujan, self-taught mathematician whose genius survives more than a century on (22 December 2020) · Krishnaswami Alladi: A pilgrimage to Ramanujan’s hometown (2005) · IMSc (Chennai): Namagiri and Ramanujan’s visit to Namakkal

His diet and treatment in England

He was a strict vegetarian in wartime England. Food was short and he cooked for himself at odd hours. He sometimes worked for 30 hours and slept for 20. The cold was a shock. D. A. B. Young argues he was malnourished and that his illness was amoebic and curable. Novelist David Leavitt says Hardy wondered for the rest of his life whether bringing him was right.

Hardy found him the best medical opinion and sanatoria. He pressed for his honours because he feared for his health. He wrote that Ramanujan was hard to get to take care of himself. He kept the observances of his caste with unusual strictness by his own choice. Wartime made a return home impossible until 1919.

Today in Science History: The mystery of Ramanujan’s illness · IMSc: D. A. B. Young on Ramanujan’s illness (after Notes and Records of the Royal Society, 1994) · NPR: David Leavitt on The Indian Clerk (2007) · Krishnaswami Alladi (University of Florida): review of The Man Who Knew Infinity (Inference) · G. H. Hardy: Obituary of Srinivasa Ramanujan, Proceedings of the Royal Society A (1921) · MacTutor History of Mathematics (St Andrews): Srinivasa Aiyangar Ramanujan

Did a goddess give him his formulas?

Indian accounts say the goddess Namagiri gave him formulas in dreams. He credited her with his gifts and is reported to have said that an equation meant nothing unless it expressed a thought of God. Some writers treat this as the key to his mind.

Hardy wrote that his religion was a matter of observance and not of intellectual conviction. He called the dream story a fable. The DNB entry notes that other evidence suggests he was devout. The quote comes through his biographer Ranganathan and not from his own writing.

Krishnaswami Alladi: A pilgrimage to Ramanujan’s hometown (2005) · MacTutor: Ramanujan, Srinivasa (Dictionary of National Biography entry) · G. H. Hardy: Obituary of Srinivasa Ramanujan, Proceedings of the Royal Society A (1921) · Krishnaswami Alladi (University of Florida): review of The Man Who Knew Infinity (Inference) · Hinduism Today: A Baffling Mind (October 2003)

Legend and the record

Films and popular accounts add drama. The 2015 film shows a hostile Cambridge, an ultimatum from Hardy and an assault on Ramanujan. Even the cause of his death is still argued between tuberculosis and hepatic amoebiasis.

Krishnaswami Alladi says several of those scenes never happened, though he praises the film. Hardy’s obituaries, Rankin’s 1984 medical papers and Young’s 1994 study give a sourced record. Kanigel’s biography is judged accurate.

Krishnaswami Alladi (University of Florida): review of The Man Who Knew Infinity (Inference) · IMSc: D. A. B. Young on Ramanujan’s illness (after Notes and Records of the Royal Society, 1994) · Today in Science History: The mystery of Ramanujan’s illness

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