Search This Blog

Showing posts with label Ramanujan. Show all posts
Showing posts with label Ramanujan. Show all posts

Sunday 18 February 2018

Ramanujan and Salam — what inspired them?

Pervez Hoodbhoy in The Dawn

SRINIVISAN Ramanujan (1887-1920) and Muhammad Abdus Salam (1926-1996), two intellectual giants of the 20th century, were born in the same corner of the world. Of humble origin and educated in local schools, they nevertheless rose to dizzying heights in the arcane world of theoretical science. Few others on the subcontinent enjoy their iconic status.

What I shall address below is that both attributed their works to some divine agency. Some of their devotees see this in validating their own respective belief system. With the rise of Hindutva in India, and the violent persecution of Ahmadis in Pakistan, these claims assume considerable importance. Hence a careful, impartial examination is called for.

No mathematician has a story more romantic than Ramanujan’s. Many books, plays, and movies — such as The Man Who Knew Infinity (2015) — dwell upon this enigmatic figure. Drawing upon deep intuition, Ramanujan created new concepts in the theory of numbers, elliptic functions and infinite series. Even full-blown mathematicians take years to grasp his complex ideas.


Exceptional genes plus fortunate circumstances is why some become maths-science superstars.


Born in Madras to a low-level clerk, this young Brahmin boy was steeped in tradition, sang religious songs, attended temple pujas, and was a strict vegetarian. But by age 12, he was inventing sophisticated theorems and unwittingly duplicating some results of European mathematicians of the previous century. He flunked college twice for lack of interest in anything but mathematics — in which he excelled. His awestruck teachers could not decide whether he was a genius or fraud.

At 16, encouraged by one of his teachers, Ramanujan sent off a letter to the renowned pure mathematician G.W. Hardy at Cambridge University. It was accompanied by theorems densely packed into nine pages. Hardy was stunned and arranged for him to travel to England. Ramanujan duly obtained permission from the family goddess Namagiri, consulting appropriate astrological data before his voyage overseas.

At age 32, Ramanujan was dead. He had returned to Madras exhausted, half-famished and fed up with English winters. But even on his deathbed, his pen scrawled out profound results. A century later these still intrigue the brainiest of mathematicians and string theorists. He attributed his exceptional qualities to the psychic visitations of Namagiri who would whisper equations to him. Sometimes, he said, “she wrote on my tongue”. He told colleagues, “An equation for me has no meaning unless it represents a thought of God.”

This was how Ramanujan saw it. But how does one explain that Euler, Bernoulli, Gauss, Cantor, Hilbert and Gödel were non-Brahmin mathematicians who stood still taller? The edifice of modern mathematics owes largely to them, not to Ramanujan. Some were ardent Christians, others agnostic or atheistic. Nobody knows how to explain their feats.

Curiously, Abdus Salam, then a 19-year-old student at Government College Lahore, wrote his very first paper proposing a simpler solution to an intriguing mathematical problem posed about 20 years earlier by Ramanujan. He ended his paper by triumphantly declaring: “His [Ramanujan’s] solution is much more laborious”.

This was Salam’s debut into the world of high mathematics. Born into a conservative religious environment in Jhang — then a village-town — this child prodigy rapidly outpaced his teachers. Fortunately they bore him no grudge and helped him move on to Lahore. The next stop was Cambridge, where he excelled. By the early 1960s, he was one of the world’s top particle physicists, ultimately winning 20 international prizes and honours including the Nobel Prize in 1979.

In his later years, Salam gave numerous public lectures and interviews, recorded on camera and in print, locating his source of inspiration in his religious belief. In particular he said the concept of unity of God powered his quest for the unification of nature’s fundamental forces as well as his search for ever fewer numbers of elementary particles.

For me, to engage on a sensitive matter with one so senior and superior was not easy. But sometime in 1986 I picked up the courage to ask Salam the obvious question: both he, who thought himself a believer, and Steven Weinberg, an avowed atheist, had worked independently on unifying two of nature’s four fundamental forces and yet had arrived at precisely the same conclusions. How?

Salam gave his answer in the preface he wrote for my book on Islam and science (1990), where he stated: “I can confirm that he [Hoodbhoy] is right…”, and then went on to explicitly clarify that any bias towards the unification paradigm in his thinking was only unconsciously motivated by his religious background.

There is not the slightest doubt that Salam used exactly the same tools as Weinberg did — principally quantum mechanics and relativity theory — and did physics exactly as other physicists do (but better than most). His political and religious views were irrelevant to his work. Let’s note that although they are giants of physics, Salam and Weinberg stood on the shoulders of still greater giants — Einstein, Pauli, Dirac, Wheeler, and Feynman — whose personal philosophies of life vastly differed from each other.

Salam sourced his inspiration to his religious beliefs, while Ramanujan claimed direct transmission from his gods. These claims cannot ever be proved or disproved. It is also irrelevant here that Salam thought of himself as a Muslim whereas, by Pakistani law, he is not.

How can prodigious talent blossom in the absence of rigorous scientific training? Two factors explain Ramanujan’s and Salam’s successes. First, nature sometimes gifts an individual with exceptional innate mathematical ability. This is associated with brain circuits in the parietal lobe and acquired through genetic transmission. Second, by good fortune, Ramanujan and Salam managed to escape into a scholarly environment — Cambridge Uni­ver­sity — where their genius could flower. Had either stayed back home he would be unheard of today.
It is usual to take pride in the geniuses belonging to one’s own tribe. The ancient civilisations of China, India, Greece, Arabia, and modern European civilisation all claim superiority over others because of the creations of their most brilliant minds. But in fact an individual’s exceptional genes and fortunate circumstances — not some supreme transcendence — are the real reasons. While sources of inspiration do differ, empirically and logically deduced results don’t. Science and its heroes belong to all humankind, not to any one tribe.

Wednesday 4 July 2012

Higgs Boson - The Indian connection


The gods of the particles

The Higgs bit we know. But the boson? Western science is overlooking India's contribution to the discovery
Higgs bosun Bose
Satyendra Nath Bose (‘bosun') realised in 1924 the method used to analyse work on the thermal behaviour of gases was inadequate'. Photo: National Geographic/Alamy
With tomorrow's announcement of the latest findings in the search for the Higgs boson, the elusive particle is on everyone's mind. This kind of fame is relatively rare, even for important scientific discoveries; but the Higgs boson has been called, or miscalled, the God particle, enabling it to pass into the realm of popular scientific lore, like the discovery of the smallpox vaccine, the structure of DNA, or the theory of relativity.

It would be difficult for most people to understand its significance, just as it would be to comprehend the notion of relativity, but such problems are overcome by locating science in personalities as well as cultural and national traditions. The first thing that you and I know about the Higgs boson is that it's named after Peter Higgs, a physicist at Edinburgh University who made the discovery – although the original insight, in one of those recurrent back stories of science, was Philip Anderson's.

Still, we have Higgs, and Edinburgh, and western civilisation to fall back on. The rest – "the Higgs boson is a hypothetical elementary particle predicted by the Standard Model of particle physics. It belongs to a class of particles known as bosons ..." – we needn't worry too much about. But maybe we should worry just enough to ask, "What is a boson?", since the word tends to come up as soon as Higgs does. Is it, an ignoramus such myself would ask, akin to an atom or a molecule? It is, in fact, along with the fermion (named after Enrico Fermi), one of the two fundamental classes of subatomic particles.

The word must surely have some European, perhaps German, genealogy? In fact, "boson" is derived from Satyendra Nath Bose, an Indian physicist from Kolkata who, in 1924, realised that the statistical method used to analyse most 19th-century work on the thermal behaviour of gases was inadequate. He first sent off a paper on the quantum statistics that he perfected in Dhaka to a British journal, which turned it down. He then sent it to Albert Einstein, who immediately grasped its immense importance, translated the paper, and published it in a German journal. (And so our invented German provenance turns out to be not wholly inappropriate.) Bose's innovation came to be known as the Bose-Einstein statistics, and became a basis of quantum mechanics. Einstein saw that it had profound implications for physics; that it had opened the way for this subatomic particle, which he named, after his Indian collaborator, "boson". Few physicists would disagree with the suggestion that the Bose-Einstein statistics have had much wider consequences for physics than the Higgs boson has had.

Still, science and the west are largely synonymous and coeval: they are two words that have the same far-reaching meaning. Just as Van Gogh and Toulouse-Lautrec's paintings consume and digest the Japanese prints they were responding to so that we don't need to be aware of Japanese prints when viewing the post-impressionists, western science is pristine, and bears no mark of what's outside itself.

The last Indian scientific discovery that is fairly universally acknowledged is the zero. As Carol Vorderman has pointed out, Indians are very strong at maths, and the only modern Indian who's remotely part of the popular western mythology of science isSrinivasa Ramanujan, equally well known for his Hindu idiosyncrasies and his agonised stay in Cambridge as he is for his mathematical genius.

Indians can be excellent geeks, as demonstrated by the tongue-tied astrophysicist Raj Koothrappalli in the US sitcom Big Bang Theory; but the Nobel prize can only be aspired to by Sheldon Cooper, the super-geek and genius in the series, for whom Raj's country of origin is a diverting enigma, and miles away from the popular myth of science on which – along with solid scientific background research – Big Bang Theory is dependent.

Bose didn't get the Nobel prize; nor did his contemporary and namesake, J C Bose, whose contribution to radio waves and the fashioning of the wireless predates Marconi's. The only Indian scientist to get a Nobel prize is the physicist C V Raman, for his work on light at Kolkata University, called the Raman effect. Other Indians have had to become Americans to get the award.

Conditions have always been inimical to science in India, from colonial times to the present day; and despite that, its contributions have occasionally been huge. Yet non-western science (an ugly label engendered by the exclusive nature of western popular imagination) is yet to find its Rosalind Franklin, its symbol of paradoxical success. Unlike Franklin, however, these scientists were never in a race that they lost; they simply came from another planet.

Wednesday 18 January 2012

Ian Stewart's top 10 popular mathematics books

Ian Stewart is an Emeritus Professor of Mathematics at Warwick University and a Fellow of the Royal Society. He has written over 80 books, mainly popular mathematics, and has won three gold medals for his work on the public understanding of science. In collaboration with Terry Pratchett and Jack Cohen he wrote the Science of Discworld series. His new book, 17 Equations That Changed the World, is published by Profile.
  1. Seventeen Equations that Changed the World
  2. by Ian Stewart
  3. Buy it from the Guardian bookshop
  1. Tell us what you think: Star-rate and review this book
Buy 17 Equations That Changed the World from the Guardian bookshop
"'Popular mathematics' may sound like a contradiction in terms. That's what makes the genre so important: we have to change that perception. Mathematics is the Cinderella science: undervalued, underestimated, and misunderstood. Yet it has been one of the main driving forces behind human society for at least three millennia, it powers all of today's technology, and it underpins almost every aspect of our daily lives.
"It's not really surprising that few outside the subject appreciate it, though. School mathematics is so focused on getting the right answer and passing the exam that there is seldom an opportunity to find out what it's all for. The hard core of real mathematics is extremely difficult, and it takes six or seven years to train a research mathematician after they leave school. Popular mathematics provides an entry route for non-specialists. It allows them to appreciate where mathematics came from, who created it, what it's good for, and where it's going, without getting tangled up in the technicalities. It's like listening to music instead of composing it.
"There are many ways to make real mathematics accessible. Its history reveals the subject as a human activity and gives a feel for the broad flow of ideas over the centuries. Biographies of great mathematicians tell us what it's like to work at the frontiers of human knowledge. The great problems, the ones that hit the news media when they are finally solved after centuries of effort, are always fascinating. So are the unsolved ones and the latest hot research areas. The myriad applications of mathematics, from medicine to the iPad, are an almost inexhaustible source of inspiration."

1. The Man Who Knew Infinity by Robert Kanigel


The self-taught Indian genius Srinivasa Ramanujan had a flair for strange and beautiful formulas, so unusual that mathematicians are still coming to grips with their true meaning. He was born into a poor Brahmin family in 1887 and was pursuing original research in his teens. In 1912, he was brought to work at Cambridge. He died of malnutrition and other unknown causes in 1920, leaving a rich legacy that is still not fully understood. There has never been another mathematical life story like it: absolutely riveting.

2. Gödel, Escher, Bach by Douglas Hofstadter


One of the great cult books, a very original take on the logical paradoxes associated with self-reference, such as "this statement is false". Hofstadter combines the mathematical logic of Kurt Gödel, who proved that some questions in arithmetic can never be answered, with the etchings of Maurits Escher and the music of Bach. Frequent dramatic dialogues between Lewis Carroll's characters Achilles and the Tortoise motivate key topics in a highly original manner, along with their friend Crab who invents the tortoise-chomping record player. DNA and computers get extensive treatment too.

3. The Colossal Book of Mathematics by Martin Gardner


In his long-running Mathematical Games column in Scientific American, Gardner – a journalist with no mathematical training – created the field of recreational mathematics. On the surface his columns were about puzzles and games, but they all concealed mathematical principles, some simple, some surprisingly deep. He combined a playful and clear approach to his subject with a well-developed taste for what was mathematically significant. The book consists of numerous selections from his columns, classified according to the mathematical area involved. Learn how to make a hexaflexagon and why playing Brussels sprouts is a waste of time.

4. Euclid in the Rainforest by Joseph Mazur


A thoroughly readable account of the meaning of truth in mathematics, presented through a series of quirky adventures in the Greek Islands, the jungles around the Orinoco River, and elsewhere. Examines tricky concepts like infinity, topology, and probability through tall tales and anecdotes. Three different kinds of truth are examined: formal classical logic, the role of the infinite, and inference by plausible reasoning. The story of the student who believed nothing except his calculator is an object lesson for everyone who thinks mathematics is just 'sums'.

5. Four Colours Suffice by Robin Wilson


In 1852 Francis Guthrie, a young South African mathematician, was attempting to colour the counties in a map of England. Guthrie discovered that he needed only four different colours to ensure that any two adjacent counties had different colours. After some experimentation he convinced himself that the same goes for any map whatsoever. This is the remarkable story of how mathematicians eventually proved he was right, but only with the aid of computers, bringing into question the meaning of "proof". It contains enough detail to be satisfying, but remains accessible and informative throughout.

6. What is Mathematics Really? by Reuben Hersh


The classic text What is Mathematics? by Richard Courant and Herbert Robbins focused on the subject's nuts and bolts. It answered its title question by example. Hersh takes a more philosophical view, based on his experience as a professional mathematician. The common working philosophy of most mathematicians is a kind of vague Platonism: mathematical concepts have some sort of independent existence in some ideal world. Although this is what it feels like to insiders, Hersh argues that mathematics is a collective human construct – like money or the Supreme Court. However, it is a construct constrained by its own internal logic; it's not arbitrary. You choose the concepts that interest you, but you don't get to choose how they behave.

7. Magical Mathematics by Persi Diaconis and Ron Graham


Both authors are top-rank mathematicians with years of stage performances behind them, and their speciality is mathematical magic. They show how mathematics relates to juggling and reveal the secrets behind some amazing card tricks. Here's one. The magician mails a pack of cards to anyone, asking them to shuffle it and choose a card. Then he shuffles the cards again, and mails half of them to the magician—not saying whether the chosen card is included. By return mail, the magician names the selected card. No trickery: it all depends on the mathematics of shuffles.

8. Games of Life by Karl Sigmund


Biologists' understanding of many vital features of the living world, such as sex and survival, depends on the theory of evolution. One of the basic theoretical tools here is the mathematics of game theory, in which several players compete by choosing from a list of possible strategies. The children's game of rock-paper-scissors is a good example. The book illuminates such questions as how genes spread through a population and the evolution of cooperation, by finding the best strategies for games such as cat and mouse, the battle of the sexes, and the prisoner's dilemma. On the borderline between popular science and an academic text, but eminently readable without specialist knowledge.

9. Mathenauts: Tales of Mathematical Wonder edited by Rudy Rucker


A collection of 23 science fiction short stories, each of which centres on mathematics. Two are by Martin Gardner, and many of the great writers of SF are represented: Isaac Asimov, Gregory Benford, Larry Niven, Frederik Pohl. The high point is Norman Kagan's utterly hilarious "The Mathenauts", in which only mathematicians can travel through space, because space is mathematical – and, conversely, anything mathematical can be reality. An isomorphomechanism is essential equipment. Between them, these tales cover most of the undergraduate mathematics syllabus, though not in examinable form.

10. The Mathematical Principles of Natural Philosophy by Isaac Newton


There ought to be a great classic in this top 10, and there is none greater. I've put it last because it's not popularisation in the strict sense. However, it slips in because it communicated to the world one of the very greatest ideas of all time: Nature has laws, and they can be expressed in the language of mathematics. Using nothing more complicated than Euclid's geometry, Newton developed his laws of motion and gravity, applying them to the motion of the planets and strange wobbles in the position of the Moon. He famously said that he "stood on the shoulders of giants", and so he did, but this book set the scientific world alight. As John Maynard Keyes wrote, Newton was a transitional figure of immense stature: "the last of the magicians … the last wonderchild to whom the Magi could do sincere and appropriate homage." No mathematical book has had more impact.

Saturday 5 November 2011

Lord Ram’s Story: Many Tellings


By Ram Puniyani
04 November, 2011
Countercurrents.org

Recently Delhi University Academic Council (Oct 2011) decided to drop the scholarly essay “Three Hundred Ramayanas” of A.K.Ramanujan, on different telling of Ram’s story from the syllabus of ‘Culture in India’ for BA Honors students. Of the four experts on the committee, one of them, whose opinion was finally accepted, said that undergraduate students will not be able to tolerate the portrayal of divine characters in the different versions given in the essay. In response to the ban while Akhil Bhartiya Vidyarthi Parishad, which is an affiliate of RSS, and company celebrated, the staff and many students protested against this ban. Just to recall earlier in 2008 ABVP activists had protested against the introduction of this essay, and indulged in vandalism on the issue.

This essay by the much acclaimed scholar, A.K.Ramanujan is part of his "The Collected Essays of A.K.Ramanujan (Oxford 1999). Earlier in the aftermath of Babri demolition, a Sahmat exhibition on different versions of Ramayana was attacked by RSS combine's goons in Pune in 1993. This was done on the pretext that one of the panels based on Jataka (Buddhist version) showed Ram and Sita as brother and sister, and it is an insult to their faith. Ramanujan's essay talks of different versions and presents five of them as an example.

It is known that there are hundreds of versions of Ramayana, Buddhist, Jain, Valmiki etc. Paula Richman in her book Many Ramayana's (Oxford) describes several of these. And again there are different interpretations of the prevalent Valmiki Ramayana, many of which are not to the liking of those who are indulging in politics in the name of their faith. Surprisingly all this intolerance is shown by those who assert that Hinduism is tolerant and other religions are intolerant.

It is a fascinating exercise to go through various tellings and interpretations of Ramayana. Even the other renderings acceptable to this intolerant but currently dominant political force are not uniform. Valmiki, Tulsidas and later the one adopted by Ramanand Sagar for his serial Ramayana have their own subtle nuances, which are very different from each other.

Ramayana has been rendered in many languages of Asia in particular. Ramanujan points out that the tellings of Ram story has been part of Balinese, Bengali, Kashmiri, Thai, Sinhala, Santhali Tamil, Tibetan and Pali amongst others. There are innumerable versions in Western languages also. The narrative in these is not matching. Those opposing this essay take Valmiki as the standard and others as diversions which are not acceptable to them for political reasons. The version of Ramayana, the communalists want to impose has the caste and gender equations of pre-modern times so it is hung up upon only that version as the only one acceptable to it.

Interestingly one can see the correlation between the class-caste aspirations of the narration-interpretation. In Buddhist Dasharath Jataka, Sita is projected both as sister and wife of Ram. As per this version Dashrath is King not of Ayodhya but of Varanasi. The marriage of sister and brother is part of the tradition of glorious Kshtriya clans who wanted to maintain their caste and clan purity. This Jataka tale shows Ram to be the follower of Buddha. Similarly Jain versions of Ramayana project Ram as the propagator of Jain values, especially as a follower of non-violence. What do both Buddhist and Jain versions have in common is that in these Ravan is not shown as a villain but a great spiritual soul dedicated to quest of knowledge, endowed with majestic commands over passions, a sage and a responsible ruler. Popular and prevalent "Women's Ramayana Songs" of Telugu Brahmin Women, put together by Rangnayakamma, keep the women's concern as the central theme. These songs present Sita as finally victorious over Ram and in these, Surpanakha succeeds in taking revenge over Ram.

In Thai Ramkirti, or Ramkin (Ram's story), there is a twist in the tale and Shurpanakh's daughter decides to take revenge attributing her mother’s mutilation primarily because of Sita. More interestingly here the focus is on Hanuman, who in this telling is neither devout nor celibate but quite a ladies’ man, looking into the bedrooms of Lanka. In Valmiki, Kampan and Tamil tellings Hanuman regards seeing another man’s sleeping wife as a sin, but not in this Thai version. Incidentally he is a very popular Thai hero even today. Also like Jain Ramayana this Thai telling focuses on genealogy and adventures of Ravana and not of Ram.

In recent times Jotiba Phule who stood more with the interests of Dalits and women, was amongst the first to interpret this mythological tale from the perspective of those subjugated by caste-varna-gender hierarchy. Phule points out that upper castes were descendents of conquering Indo-Europeans who overturned the original egalitarian society and forbade the conquered from studying texts. His mythology is woven around King Bali, who could invoke the image of peasant community. Needless to say his murder by Lord Ram from behind is condemned and is seen as an act of subjugation of lower castes by the upper castes. And Ram is seen as an avatar of Vishnu out to conquer the land from the Rakshasas (those protecting their crops) for establishing the hegemony of upper caste values of caste and gender hierarchy.

Dr. Ambedkar and Periyar's commentaries are more an alternative reading of the Valmiki's text rather than a separate version. There is a good deal of overlap in the interpretation of both. Dr. Ambedkar focuses his attention on the issues pertaining to Ram's killing of Shambuk for violating the prevalent norm where a low caste has no right to do penance, tapasya. Like Phule he also castigates Lord Ram for murdering the popular folk king Bali. He questions Ram's act of taking Sita's agnipariksha, trial by fire, and his patriarchal attitude towards her. After defeating Ravan he tells Sita that he had done all this battle not to get her released for her own sake but to restore his honor, and his banishing her in response to the rumors about her chastity when she was pregnant comes for severest criticism from Ambedkar.

Periyar is basically taking the same line but in his interpretation the North Indian upper caste onslaught-South Indian resistance becomes the central theme. Periyar the initiator of ‘Self Respect Movement’ was the pioneer of caste and gender equality in Tamilnadu. In one of the movements, which is very less known, on the lines of Dr. Ambedkar burning Manusmriti, he planned to burn the photo of Ram, as for him Ram symbolized the imposition of upper caste norms in South India. This was a part of his campaign against caste Hinduism. Periyar also upheld Tamil identity. According to him the Ramayana story was a thinly disguised historical account of how caste ridden, Sanskritic, Upper caste North Indians led by Ram subjugated South Indians. He identifies Ravan as the monarch of ancient Dravidians, who abducted Sita, primarily to take revenge against the mutilation and insult of his sister Surpanakha. In his interpretation Ravana is practitioner of Bhakti, and is a virtuous man.

It seems the dropping of the essay from syllabus is under indirect political pressure of communal forces. RSS and affiliates who have reaped rich benefit from the campaign around Lord Ram are also giving the political message of caste and gender hierarchy, through the version upheld by them, the one of Valmiki and presented in current times by Ramanand Sagar’s tele serial Ramayana. And the politics claiming to be tolerant is intolerant about scholarly renderings of ‘Many Rams: Many Ramayanas’ prevalent World over!