\u201cThe Man Who Knew Infinity (Part One)\u201d<\/a>:<\/p>\n\u00a0<\/p>\n
I carry on now with notes taken from (and\/or inspired by) Robert Kanigel, The Man Who Knew Infinity: A Life of the Genius Ramanujan<\/em> (New York and London: Washington Square Press, 1991):<\/p>\n\u00a0<\/p>\n
In the 1930s, E. T. Bell would remark that Ramanujan had broken the rules by which mathematicians evaluate their own.\u00a0 \u201cWhen a truly great [algorist, or formalist] like the Hindu Ramanujan arrives unexpectedly out of nowhere, even expert analysts hail him as a gift from heaven,\u201d he wrote, crediting him with \u201call but supernatural insight\u201d into hidden connections between seemingly unrelated formulas.<\/strong><\/span><\/p>\nSupernatural insight.<\/strong><\/em><\/span><\/p>\nA gift from Heaven.<\/strong><\/em><\/span><\/p>\nIt is uncanny how often otherwise dogged rationalists have, over the years, turned to the language of the shaman and the priest to convey something of Ramanujan\u2019s gifts. . . .\u00a0 And repeatedly they have been reduced to inchoate expressions of wonder and awe in the face of his powers, have stumbled about, groping for words, in trying to convey the mystery of Ramanujan.<\/strong><\/span><\/p>\n\u201cWe have no idea how he did the marvelous things he did, what led him to them, or anything else,\u201d says mathematician Richard Askey, a Ramanujan scholar at the University of Wisconsin in Madison.\u00a0 Says Bruce Berndt, after years of working through Ramanujan\u2019s notebooks:\u00a0 \u201cI still don\u2019t understand it all.\u00a0 I may be able to prove it, but I don\u2019t know where it comes from and where it fits into the rest of mathematics.\u201d\u00a0 He adds at another point, \u201cThe enigma of Ramanujan\u2019s creative process is still covered by a curtain that has scarcely been drawn.\u201d\u00a0<\/strong> (280)<\/span><\/p>\n\u00a0<\/p>\n
J. E. Littlewood essentially surrenders in perplexity when trying to understand Ramanujan\u2019s thought processes:<\/p>\n
\u00a0<\/p>\n
There is, indeed, a touch of real mystery [here].\u00a0 If only we knew [the result in advance], we might be forced, by slow stages, to the correct form of [Psi<\/em> sub q<\/em>, symbols that I can\u2019t reproduce on this blog].\u00a0 But why was Ramanujan so certain there was<\/em> one?\u00a0\u00a0Theoretical<\/em> insight, to be the<\/em> explanation, had to be of an order hardly to be credited.\u00a0 Yet it is hard to see what numerical instances could have been available to suggest so strong a result.\u00a0 And unless the form of\u00a0[Psi<\/em> sub q<\/em>] was known already, no numerical evidence could suggest anything of the kind \u2014 there seems no escape, at least, from the conclusion that the discovery of the correct form was a single stroke of insight.\u00a0<\/strong> (Cited at 280-281)<\/span><\/p>\n\u00a0<\/p>\n
To borrow terminology suggested by the Polish emigr\u00e9 mathematician Mark Kac, Ramanujan was actually a \u201cmagician,\u201d rather than an \u201cordinary genius\u201d:<\/p>\n
\u00a0<\/p>\n
An ordinary genius is a fellow that you and I would be just as good as, if we were only many times better.\u00a0 There is no mystery as to how his mind works.\u00a0 Once we understand what he has done, we feel certain that we, too, could have done it.\u00a0 It is different with the magicians.\u00a0 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.\u00a0 Even after we understand what they have done, the process by which they have done it is completely dark.<\/strong>\u00a0 (Cited at 281)<\/span><\/p>\n\u00a0<\/p>\n
Kanigel reacts to such statements in a striking way:<\/p>\n
\u00a0<\/p>\n
Mystery, magic, and dark, hidden workings inaccessible to ordinary thought; it is these that Ramanujan\u2019s work invariably conjures up, a sense of reason butting hard up against its limits.<\/strong><\/span><\/p>\nBut at reason\u2019s limits does<\/em> something else take over?\u00a0 Do we here flirt with spiritual or supernatural forces outside our understanding?\u00a0<\/strong> (281)<\/span><\/p>\n\u00a0<\/p>\n
Ramanujan himself plainly thought so.\u00a0 Kanigel cites reminiscences from fellow-Indians who knew him:<\/p>\n
\u00a0<\/p>\n
T. K. Rajagopolan, a former accountant general of Madras, would tell of Ramanujan\u2019s insistence that after seeing in dreams the drops of blood that, according to tradition, heralded the presence of the god Narasimha, the male consort of the goddess Namagiri, \u201cscrolls containing the most complicated mathematics used to unfold before his eyes.\u201d<\/strong><\/span><\/p>\nR. Radhakrishna Iyer, a classmate at Pachaiyappa\u2019s College, recalled one day asking Ramanujan about his research only to have him reply, in Radhakrishna\u2019s words, \u201cthat Lord Narasimha had appeared to him in a dream and told him that the time had not come for making public the fruits of his research.\u201d<\/strong><\/span><\/p>\nAnd it was on a day about a year before he left for England, in 1912 or early 1913, that Ramanujan, showing his work to mathematics professor R. Srinivasan, made the statement in which he pictured equations as products of the mind of God.<\/strong>\u00a0 (281-282)<\/span><\/p>\n\u00a0<\/p>\n
Some, though \u2014 starting with Ramanujan\u2019s zealously atheistic mathematical mentor, the great G. H. Hardy \u2014 have sought to downplay or even to dispute Ramanujan\u2019s deep spirituality:<\/p>\n
\u00a0<\/p>\n
[M]ost Western observers, and some Indians, have wholly detached Ramanujan\u2019s mystical streak from his mysterious ability to forge new mathematical linkages.\u00a0 Hailing the one, they\u2019ve dismissed the other.\u00a0<\/strong> (283)<\/span><\/p>\n\u00a0<\/p>\n
Kanigel thinks that they are wrong to do so but, in the end, he too \u2014 a Western rationalist \u2014 wants to make Ramanujan\u2019s creativity purely internal, psychological.\u00a0 He believes that the Indian\u2019s credulity opened his mind in a unique way to intuition, to his own unconscious.\u00a0 Even G. H. Hardy was open, just a little tiny bit and rather reluctantly, to a role for the subconscious in mathematical thinking.\u00a0 But Kanigel isn\u2019t quite sure of his own position:<\/p>\n
\u00a0<\/p>\n
But it was Hardy, the dedicated atheist, who represented the extreme position, and Ramanujan who was more in line with the large body of belief and conviction, within the Western tradition as well as that of the East, that perceived links between creativity and intuition on the one hand and spiritual forces on the other.<\/strong><\/span><\/p>\nThe Greeks, for example, invoked the muses \u2014 goddesses to whom poets looked for inspiration.\u00a0 Both the English and French languages speak of \u201cdivining\u201d the truth.<\/strong>\u00a0 (286-287)<\/span><\/p>\n\u00a0<\/p>\n
Just as India was not alone in attributing creative insights to divine influence, Ramanujan was not alone among mathematicians in holding strong religious beliefs.\u00a0 Newton was an unquestioning believer, felt humility in the face of the universe\u2019s wonders, studied theology on his own.\u00a0 [The great eighteenth-century Swiss mathematician Leonhard] Euler, in E. T. Bell\u2019s words, \u201cnever discarded a particle of his Calvinistic faith,\u201d and grew more religious as he grew older.\u00a0 [The famous early-nineteenth-century French mathematician Augustin-Louis] Cauchy was forever trying to convert other mathematicians to Roman Catholicism.\u00a0 [The nineteenth-century French mathematician Charles] Hermite had a strong mystical bent.\u00a0 Even Descartes, that father of Enlightenment rationality, answered the call of the spirit:\u00a0 \u201cHis religious beliefs were unaffectedly simple in spite of his rational skepticism,\u201d writes Bell.\u00a0 \u201cHe compared his religion, indeed, to the nurse from whom he had received it, and declared that he found it as comforting to lean upon one as on the other.\u201d<\/strong>\u00a0 (287)<\/span><\/p>\n\u00a0<\/p>\n
But, of course, the issue isn\u2019t merely whether mathematicians can \u201chold strong religious beliefs.\u201d\u00a0 Obviously they can do so, and sometimes they actually do<\/em>.\u00a0 Ramanujan, however, wasn\u2019t merely religious.\u00a0 He believed that his mathematical insights came to him by revelation.\u00a0 And he has company, in that regard:<\/p>\n\u00a0<\/p>\n
Even among mathematicians not religiously minded, one finds evidence of at least respectful allusion to the dark terrain between faith and reason.\u00a0 [The preeminent nineteenth-century German mathematician Carl Friedrich] Gauss, for example, once proved a theorem, as he wrote, \u201cnot by dint of painful effort but so to speak by the grace of God.\u201d\u00a0 James Hopwood Jeans, Hardy\u2019s Cambridge classmate and a famous applied mathematician, wrote:\u00a0 \u201cFrom the intrinsic evidence of his creation, the Great Architect of the Universe now begins to appear as a pure mathematician.\u201d\u00a0 Even Littlewood, commenting on an incident in which \u201cmy pencil wrote down\u201d the solution to a particularly bedeviling problem, could write, \u201cIf we may reject divine bounty, it happened exactly as if my subconsciousness knew the thing all the time.\u201d\u00a0 Littlewood\u2019s, of course, was the usual safe, ironic Cambridge skepticism, perhaps no more than a stylistic device \u2014 just as the other statements may be seen as no more than metaphor,\u00a0 Yet each contained the barest breath of ambivalence or humility in the face of the mysterious origins of human creativity.<\/strong>\u00a0 (287-288)<\/span><\/p>\n\u00a0<\/p>\n
[Another excellent example that might have been included here is that of Blaise Pascal.\u00a0 I\u2019ll write and post something about his case fairly soon.\u00a0 Perhaps also Henri Poincar\u00e9, if I\u2019m not misremembering.]<\/p>\n
\u00a0<\/p>\n
There is much to think about here:\u00a0 A Latter-day Saint perspective would certainly allow, and perhaps insist on, the likelihood of genuinely prompting divine revelation in such cases.<\/p>\n
\u00a0<\/p>\n
And there is matter to think about not just on that particular subject:\u00a0\u00a0Robert Kanigel explicitly raises one question in this next paragraph, about the great loss suffered by mathematics because Ramanujan went so long without adequate mathematical education and because he died so very, very young, and indirectly suggests another:\u00a0 Isn\u2019t it astonishing that purely theoretical mathematical ideas that have been worked out in isolation by a genius who knew nothing about chemistry, computers, or oncology can be found relevant to those fields (and others) decades after his death?\u00a0 This is precisely the issue raised by the famous 1960 essay \u201cThe Unreasonable Effectiveness of Mathematics in the Natural Sciences\u201d by the Hungarian-American theoretical physicist Eugene Wigner, who would go on to win the 1963 Nobel Prize in Physics.<\/p>\n
\u00a0<\/p>\n
Writes Kanigel:<\/p>\n
\u00a0<\/p>\n
\u201cSrinivasa Ramanujan,\u201d an Englishman would later say of him, \u201cwas a mathematician so great that his name transcends jealousies, the one superlatively great mathematician whom India has produced in the last thousand years.\u201d\u00a0 His leaps of intuition confound mathematicians even today, seven decades <\/strong>[now almost exactly 101 years] after his death.\u00a0 His papers are still plumbed for their secrets.\u00a0 His theorems are being applied in areas \u2014 polymer chemistry, computers, even (it has recently been suggested) cancer \u2014 scarcely imaginable during his lifetime.\u00a0 And always the nagging question: What might have been<\/em>, had he been discovered a few years earlier, or lived a few years longer?<\/strong>\u00a0 (3)<\/span><\/p>\n\u00a0<\/p>\n
Another question that Ramanujan\u2019s story raises is, How many geniuses \u2014 with enormous gifts not only for mathematics but for business and literature and law and physics and statecraft and art and genetics and medicine and economics and scores of other fields \u2014 go utterly unrecognized, are lost in poverty, die too early, leaving their enormous potential untapped and even unknown?\u00a0 And what cures for cancer were lost at Bergen-Belsen and Dachau?\u00a0 What great novels and plays were snuffed out in the Gulag or the Cambodian \u201ckilling fields\u201d before their authors ever put pen to paper or, perhaps, even conceived them?\u00a0 What contributions to science and art, literature and government, commerce and medicine have been lost in abortion clinics?<\/p>\n
\u00a0<\/p>\n
Thus Kanigel:<\/p>\n
\u00a0<\/p>\n
It is a story of one man and his stubborn faith in his own abilities.\u00a0 But it is not a story that concludes, Genius will out<\/em> \u2014 though Ramanujan\u2019s, in the main, did.\u00a0 Because so nearly did events turn out otherwise that we need no imagination to see how the least bit less persistence, or the least bit less luck, might have consigned him to obscurity.\u00a0 In a way, then, this is also a story about social and educational systems, and about how they matter, and how they can sometimes nurture talent and sometimes crush it.\u00a0 How many Ramanujans, his life begs us to ask, dwell in India today, unknown and unrecognized?\u00a0 And how many in America and Britain, locked away in racial or economic ghettos, scarcely aware of worlds outside their own?<\/strong>\u00a0 (3-4)<\/span><\/p>\n\u00a0<\/p>\n
And, finally, on the question of whether mathematical facts or objects really objectively exist, of whether mathematicians \u201cinvent\u201d them or \u201cdiscover\u201d them \u2014 Ramanujan plainly sided with those who believe them to be objectively real \u2014 G. H. Hardy offered a strikingly strong statement of what seems to be a realist position.\u00a0 In a 1928 Cambridge lecture, he said the following:<\/p>\n
\u00a0<\/p>\n
I have always thought of a mathematician as in the first instance an observer<\/em>, a man who gazes at a distant range of mountains and notes down his observations.\u00a0 His object is simply to distinguish clearly and notify to others as many different peaks as he can.\u00a0 There are some peaks which he can distinguish easily, while others are less clear.\u00a0 He sees A sharply, while of B he can obtain only transitory glimpses.\u00a0 At last he makes out a ridge which leads from A, and following it to its end he discovers that it culminates in B.\u00a0 B is now fixed in his vision, and from this point he can proceed to further discoveries.\u00a0 In other cases perhaps he can distinguish a ridge which vanishes in the distances, and conjectures that it leads to a peak in the clouds or below the horizon.<\/strong>\u00a0 (cited at 285)<\/span><\/p>\n\u00a0<\/p>\n
It seems, as was suggested in Part One of this entry, almost a religious vision.\u00a0 I\u2019m happily inclined to believe that G. H. Hardy was religious, or spiritual, in his own way, as Srinivasa Ramanujan certainly was.\u00a0 May God bless them both.<\/p>\n
\u00a0<\/p>\n
Posted from Park City, Utah<\/p>\n
\u00a0<\/p>\n
\u00a0<\/p>\n<\/body><\/html>\n","protected":false},"excerpt":{"rendered":"
\u00a0 \u00a0 *** \u00a0 First of all, I\u2019ve just become aware that Edward Idris Cardinal Cassidy passed away a week ago today, 10 April 2021, at the age of 96.\u00a0 I had the privilege of meeting with Cardinal Cassidy on two separate occasions, once in Rome (or, more precisely, in Vatican City) and once in […]<\/p>\n","protected":false},"author":1019,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[1014,3799,1842,7102,22505,22529,22499,3994,3451,11830,22370,22511,22508,22502,5445,4051,22361,22532,22358,11846,3304,861,8576,22373,22493,22367,16945,69,22352,22496,22514,22517,22355,1839,22364,22349,1848,2454,3448,17363],"class_list":["post-90913","post","type-post","status-publish","format-standard","hentry","category-uncategorized","tag-atheism","tag-atheist","tag-australia","tag-cambridge","tag-cardinal","tag-carl-friedrich-gauss","tag-cassidy","tag-catholic","tag-catholicism","tag-college","tag-dev-patel","tag-edward-cardinal-cassidy","tag-edward-idris-cardinal-cassidy","tag-edward-idris-cassidy","tag-england","tag-english","tag-g-h-hardy","tag-gauss","tag-hardy","tag-hindu","tag-hinduism","tag-india","tag-indian","tag-jeremy-irons","tag-kanigel","tag-madras","tag-mathematician","tag-mathematics","tag-ramanujan","tag-robert-kanigel","tag-roman-catholic","tag-roman-catholicism","tag-srinivasa-ramanujan","tag-sydney","tag-tamil-nadu","tag-the-man-who-knew-infinity","tag-trinity","tag-university","tag-vatican","tag-vatican-city"],"yoast_head":"\n
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