Ramanujan Quotes

An equation for me has no meaning, unless it expresses a thought of God.

For self-educated scientists and thinkers such as Charles Darwin, Srinivasa Ramanujan, Leonardo-da-Vinci, Michael Faraday, myself and many others, education is a relentless voyage of discovery. To us education is an everlasting quest for knowledge and wisdom.

They must be true because, if they were not true, no one would have the imagination to invent them.

Sometimes in studying Ramanujan's work, [George Andrews] said at another time, "I have wondered how much Ramanujan could have done if he had had MACSYMA or SCRATCHPAD or some other symbolic algebra package.

But what Ramanujan wanted more, more than anything, was simply the freedom to do as he wished, to be left alone to think, to dream, to create, to lose himself in a world of his own making.

Srinivasa Ramanujan was the strangest man in all of mathematics, probably in the entire history of science. He has been compared to a bursting supernova, illuminating the darkest, most profound corners of mathematics, before being tragically struck down by tuberculosis at the age of 33... Working in total isolation from the main currents of his field, he was able to rederive 100 years’ worth of Western mathematics on his own. The tragedy of his life is that much of his work was wasted rediscovering known mathematics.

{Replying to G. H. Hardy's suggestion that the number of a taxi (1729) was 'dull', showing off his spontaneous mathematical genius} No, it is a very interesting number; it is the smallest number expressible as a sum of two cubes in two different ways, the two ways being 13 + 123 and 93 + 103.

No mathematician should ever allow him to forget that mathematics, more than any other art or science, is a young man's game. … Galois died at twenty-one, Abel at twenty-seven, Ramanujan at thirty-three, Riemann at forty. There have been men who have done great work later; … [but] I do not know of a single instance of a major mathematical advance initiated by a man past fifty. … A mathematician may still be competent enough at sixty, but it is useless to expect him to have original ideas.

I still say to myself when I am depressed, and find myself forced to listen to pompous and tiresome people, 'Well, I have done one thing you could never have done, and that is to have collaborated with both Littlewood and Ramanujan on something like equal terms.

Plenty of mathematicians, Hardy knew, could follow a step-by-step discursus unflaggingly—yet counted for nothing beside Ramanujan. Years later, he would contrive an informal scale of natural mathematical ability on which he assigned himself a 25 and Littlewood a 30. To David Hilbert, the most eminent mathematician of the day, he assigned an 80. To Ramanujan he gave 100.

Don’t be so easy on yourself, it said.

Teaching Ramanujan was like writing on a blackboard covered with excerpts from a more interesting lecture.

You want self-knowledge? You should come to America. Just as the Mahatma had to go to jail and sit behind bars to write his autobiography. Or Nehru had to go to England to discover India. Things are clear only when looked at from a distance.

In the West, there was an old debate as to whether mathematical reality was made by mathematicians or, existing independently, was merely discovered by them. Ramanujan was squarely in the latter camp; for him, numbers and their mathematical relationships fairly threw off clues to how the universe fit together. Each new theorem was one more piece of the Infinite unfathomed. So he wasn’t being silly, or sly, or cute when later he told a friend, “An equation for me has no meaning unless it expresses a thought of God.

His academic failure forced him to develop unconventionally, free of the social straightjacket that might have constrained his progress to well-worn paths.

A pure mathematician must leave to happier colleagues the great task of alleviating the sufferings of humanity.

The heart in misery has turned upside down. The blowing gentle breeze is on fire. O friend moonlight burns like the sun.

Words come to me like equations came to Ramanujan and music came to Mozart.

Ramanujan was not the first foreigner to retreat into his shell in a new country; indeed, his was the typical response, not the exceptional one. One later study of Asian and African students in Britain observed that a sense of exclusion “from the life of the community … constituted one of the most serious problems with which they were confronted … [and had] a serious psychological effect” upon them. Another study, this time of Indian students in particular, reported that while 83 percent of them saw friends more or less every day back in India, just 17 percent did while in England.

I had better say something here about this question of age, since it is particularly important for mathematicians. No mathematician should ever allow himself to forget that mathematics, more than any other art or science, is a young man's game. To take a simple illustration at a comparatively humble level, the average age of election to the Royal Society is lowest in mathematics. We can naturally find much more striking illustrations. We may consider, for example, the career of a man who was certainly one of the world's three greatest mathematicians. Newton gave up mathematics at fifty, and had lost his enthusiasm long before; he had recognized no doubt by the time he was forty that his greatest creative days were over. His greatest idea of all, fluxions and the law of gravitation, came to him about 1666 , when he was twentyfour—'in those days I was in the prime of my age for invention, and minded mathematics and philosophy more than at any time since'. He made big discoveries until he was nearly forty (the 'elliptic orbit' at thirty-seven), but after that he did little but polish and perfect. Galois died at twenty-one, Abel at twenty-seven, Ramanujan at thirty-three, Riemann at forty. There have been men who have done great work a good deal later; Gauss's great memoir on differential geometry was published when he was fifty (though he had had the fundamental ideas ten years before). I do not know an instance of a major mathematical advance initiated by a man past fifty. If a man of mature age loses interest in and abandons mathematics, the loss is not likely to be very serious either for mathematics or for himself.

Ramanujan’s refrain was always the same—that his parents had made him marry, that now he needed a job, that he had no degree but that he’d been conducting mathematical researches on his own. And here … well, why didn’t the good sir just examine his notebooks. His notebooks were his sole credential in a society where, even more than in the West, credentials mattered; where academic degrees usually appeared on letterheads and were mentioned as part of any introduction; where, when they were not, you’d take care to slip them into the conversation. “Like regiments we have to carry our drums, and tambourinage is as essential a thing to the march of our careers as it is to the march of soldiers in the West,” Indian novelist and critic Nirad C. Chaudhuri has written of his countrymen’s bent for self-promotion. “In our society, a man is always what his designation makes him.” Ramanujan’s only designations were unemployed, and flunk-out. Without his B.A., one prominent mathematics professor told him straight out, he would simply never amount to anything.

When the Ramanujan function is generalized, the number 24 is replaced by the number 8. Thus the critical number for the superstring is 8 + 2, or 10. This is the origin of the tenth dimension. The string vibrates in ten dimensions because it requires these generalized Ramanujan functions in order to remain self-consistent. In other words, physicists have not the slightest understanding of why ten and 26 dimensions are singled out as the dimension of the string. It's as though there is some kind of deep numerology being manifested in these functions that no one understands. It is precisely these magic numbers appearing in the elliptic modular function that determines the dimension of space-time to be ten.

impetuous

In the work of Ramanujan, the number 24 appears repeatedly. This is an example of what mathematicians call magic numbers, which continually appear, where we least expect them, for reasons that no one understands. Miraculously, Ramanujan's function also appears in string theory. The number 24 appearing in Ramanujan's function is also the origin of the miraculous cancellations occurring in string theory. In string theory, each of the 24 modes in the Ramanujan function corresponds to a physical vibration of the string. Whenever the string executes its complex motions in space-time by splitting and recombining, a large number of highly sophisticated mathematical identities must be satisfied. These are precisely the mathematical identities discovered by Ramanujan. (Since physicists add two more dimensions when they count the total number of vibrations appearing in a relativistic theory, this means that space-time must have 24 + 2 = 26 space-time dimensions.)

Nash’s genius was of that mysterious variety more often associated with music and art than with the oldest of all sciences. It wasn’t merely that his mind worked faster, that his memory was more retentive, or that his power of concentration was greater. The flashes of intuition were non-rational. Like other great mathematical intuitionists — Georg Friedrich Bernhard Riemann, Jules Henri Poincaré, Srinivasa Ramanujan — Nash saw the vision first, constructing the laborious proofs long afterward.

The Riemann zeta function was a simple enough looking infinite series expressed in terms of a complex variable. Here, “complex” means not difficult or complicated, but refers to a variable of two distinct components, “real” and “imaginary,” which together could be thought to range over a two-dimensional plane. In 1860, Georg Friedrich Bernhard Riemann made six conjectures concerning the zeta function. By Ramanujan’s time, five had been proven. One, enshrined today as the Riemann hypothesis, had not

When I started training myself in Neurobiology, Psychology and Theology, mostly on the streets of Calcutta, at the book kiosks on the sidewalk, for I had no money to buy the books, I had no academic background - no college degree - no potential for earning a decent living - I was a direction-less canoe in the open sea. I did not come from a rich or learned family, nor did I have rich friends, so, as far as everybody else was concerned, my life was doomed. I come from the humblest of origins - like did Ramanujan, like did Tesla, like did many more legendary thinkers of human history. I didn't know the rules of academia - I didn't know the laws and the norms of the scientific community - all I knew was that I had to understand the humans if I were to unite them. Other than that, I had no clue to my future. I learnt by failing - I learnt by making errors - I learnt by moving slowly but surely, and by never losing my sense of awe. And that's really what science is about - it's about naivety, curiosity and awe.

Here's just one of Ramanujan's many provocative formulas: (1+1/2^4) * (1+1/3^4)*(1+1/5^4)*(1+1/7^4)*(1+1/next prime number^4)x...= 105/pi^4. The infinite product on the left side of this equation is based on successive prime numbers raised to the 4th power. Primes are integers greater than 1 that are evenly divisible only by themselves and 1. Thus, 3 is a prime, but 4 isn't because it's evenly divisible by 2. The first nine primes are 2,3,5,7,11,13,17,19, and 23. The primes go on forever, which accounts for the ellipsis at the end of the product in Ramanujan's formula. This formula shows a deep connection between pi and the prime numbers.

I have no control over my words, including the titles. Every time a title appears in my mind it sends a chill down my spine. The entire body shakes up in momentary tremor followed by the rush of an immense tranquility. And that's it! Right or not, that is the title. The same happens whenever I come up with a radical statement. Just like Ramanujan used to have visions of numbers, I have visions of words, that too, in the most socially relevant manner possible.

Every positive integer is one of Ramanujan's personal friends.

Every positive integer is one of Ramanujan’s personal friends. John E. Littlewood

Part of Ramanujan's genius was to find a way to get values out of the zeta function for negative values. As we saw before, these sums diverge and go racing off to infinity, but Ramanujan was able to extract the bit of the answer which explodes and leave the important bit behind. Using Bernoulli numbers, he could produce values for the negative half of the zeta function-which, eventually, gives us a complete plot. This is the zeta function in graphical form.

If you recall, in Ramanujan's letters to English mathematicians, he claimed that 1 + 2 + 3 +...= -1/12. He was so surprised when Hardy took him seriously that he replied on 27 February 1913 in the following words: 'I was expecting a reply from you similar to the one which a Mathematics Professor at London wrote asking me to study Infinite Series and not fall into the pitfalls of divergent series. If I had given you my methods of proof I am sure you will follow the London Professor. I told him that the sum of an infinite number of terms of the series: 1 + 2 + 3 + 4 +...= -1/12 under mu theory. If I tell you this, you will at once point out to me the lunatic asylum as my goal.' It turns out that not only had Ramanujan independently rediscovered the Bernoulli numbers, but he may have found more than one way to prove that 1 +2 +3 + 4 ...= -1/12. This is now called Ramanujan summation and gives us an insight into the ways in which the sum of a sequence can be divergent. Of course, the sum of all the positive whole numbers is infinite, but if you can somehow peel that infinity back out of the way and look at what else is going on, there's a -1/12 in there.

Just like Ramanujan used to have visions of numbers, I have visions of words, that too, in the most socially relevant manner possible.

Various factors, it seems, contribute to what we call genius and the forms it may take: speed of thought (at which von Neumann, by all accounts, was exceptional), depth of understanding (at which, according to Wigner, Einstein excelled), originality, creativity, and so forth. Sometimes, too, genius may be narrow in its focus – as in the case of Einstein or Ramanujan – while at other times, as illustrated by von Neumann, and to an even greater extent by some Renaissance figures such as Leonardo da Vinci, it can range over many subjects.

AKKA MAHADEVI Around nine hundred years ago in southern India, there lived a female mystic called Akka Mahadevi. Akka was a devotee of Shiva. Ever since her childhood, she had regarded Shiva as her beloved, her husband. It was not just a belief; for her it was a living reality. The king saw this beautiful young woman one day, and decided he wanted her as his wife. She refused. But the king was adamant and threatened her parents, so she yielded. She married the man, but she kept him at a physical distance. He tried to woo her, but her constant refrain was, “Shiva is my husband.” Time passed and the king’s patience wore thin. Infuriated, he tried to lay his hands upon her. She refused. “I have another husband. His name is Shiva. He visits me, and I am with him. I cannot be with you.” Because she claimed to have another husband, she was brought to court for prosecution. Akka is said to have announced to all present, “Being a queen doesn’t mean a thing to me. I will leave.” When the king saw the ease with which she was walking away from everything, he made a last futile effort to salvage his dignity. He said, “Everything on your person—your jewels, your garments—belongs to me. Leave it all here and go.” So, in the full assembly, Akka just dropped her jewelry, all her clothes, and walked away naked. From that day on, she refused to wear clothes even though many tried to convince her otherwise. It was unbelievable for a woman to be walking naked on the streets of India at the time—and this was a beautiful young woman. She lived out her life as a wandering mendicant and composed some exquisite poetry that lives on to this very day. In a poem (translated by A. K. Ramanujan), she says: People, male and female, blush when a cloth covering their shame comes loose. When the lord of lives lives drowned without a face in the world, how can you be modest? When all the world is the eye of the lord, onlooking everywhere, what can you cover and conceal? Devotees of this kind may be in this world but not of it. The power and passion with which they lived their lives make them inspirations for generations of humanity. Akka continues to be a living presence in the Indian collective consciousness, and her lyrical poems remain among the most prized works of South Indian literature to this very day. Embracing

Young people need looking after,” she said. “Think of that beautiful boy Galois. People felt there was something secret in his character. They were right. The secret was mathematics. His father a suicide. His own death a horrible farce. Dawn in the fields. Caped and whiskered seconds. Sinister marksman poised to fire.” I need all my courage to die at twenty. “Then there was Abel, not much older, desperately poor, Abel in delirium, hemorrhaging. So often mathematical experience consists of time segments too massive to be contained in the usual frame. Lives overstated. Themes pursued to extreme points. Adventure, romance and tragedy.” I will fight for my life. “Look at Pascal, who rid himself of physical pain by dwelling on mathematics. He was just a bit older than you when he constructed his mystic hexagram. The loveliest aspect of the mystic hexagram is that it is mystic. That’s what’s so lovely about it. It’s able to become its own shadow.” Keep believing it. “The tricky thing about mathematical genius,” she said, “is that its sources are so often buried. Galois for one. Ramanujan for another. No indication anywhere in their backgrounds that these boys would one day display such natural powers. Figures jumping out of sequence. Or completely misplaced.” (...) “Numbers have supernatural harmonies, according to Hermite. They exist beyond human thought. Divine order through number. Number as absolute reality. Someone said of Hermite: ‘The most abstract entities are for him like living creatures.’ That’s what someone said.” “People invented numbers,” he said. “You don’t have numbers without people.” “Good, let’s argue.” “I don’t want to argue.” “Secret lives,” she said. “Dedekind listed as dead twelve years before the fact. Poncelet scratching calculations on the walls of his cell. Lobachevski mopping the floors of an old museum. Sophie Germain using a man’s name. Do I have the order right? Sometimes I get it mixed up or completely backwards. (...) “Tell me about your mathematical dreams.” “Never had one.” “Cardano did, born half dead, his inner life a neon web of treachery and magic. Gambler, astrologer, heretic, court physician. Schemed his way through the algebra wars.” “Can I see the baby?” “Ramanujan had algebraic dreams. Wrote down the results after getting out of bed. Vast intuitive powers but poor education. Taken to Cambridge like a jungle boy. Sonja Kowalewski wasn’t allowed to attend university lectures. We both know why. When her husband died she spent days and days without food, coming out of her room only after she’d restored herself by working on her mathematics. Tell me, was it Kronecker who thought mathematics similar to poetry? I know Hamilton and many others tried their hands at verse. Our superduper Sonja preferred the novel.

Humanitarian Arithmetic (Sonnet 1354) If it takes $300bn to end world hunger, and 7 trillion to fund the next AI wonder, how many people have to starve to death, to feed the appetite of the cyberworld? If Britain's NHS costs about $200bn, and US military costs 800 billion dollars, how many have to suffer from sickness, for the tribal chiefs to feel secure? If it takes $20bn to end homelessness in the US, and trillions to colonize Mars, how many have to sleep in cardboard boxes, for heirs of billionaires to breed on Mars? You don't need to be a Ramanujan or Euler, to solve this simple arithmetic equation. But you do need a living human heart, to take responsibility for the solution.

Nanaki would be besotted for many hours. She would watch like a novice, like she was in a foreign country, with fresh eyes. Like starting all over again. Like wiping clean a film of experience from eyes and starting afresh, like a child. She would then do a very Chandigarh thing - buy herself a tub of buttered popcorn and continue observing. On days she would get so late that the blue of the sky would deepen into a flush of Prussian. Poor selling boys launched neon frisbees to attract little children taking a walk with parents. The sodium pole lamps would be lit and the water of the bird fountain would become a psychedelic pink. She would continue to observe- not in a way that would make people uncomfortable but in a detached, wholesome way, like she was part of the surroundings. This was also one of the early lessons by her favourite Prof Ramanujan at DCA, who always said that observation was the key. Nature or culture.

The cards are stacked, against any original mind, and perhaps properly so.

Ramanujan had lost all his scholarships. He had failed in school. Even as a tutor of the subject he loved most, he'd been found wanting. He had nothing. And yet, viewed a little differently, he had everything. For now there was nothing to distract him from his notebooks- notebooks, crammed with theorems, that each day, each week, bulged wider.

Viewed one way, then, for at least five years between 1904 and 1909, Ramanujan floundered- mostly out of school, without a degree, without contact with other mathematicians. And yet, was the cup half-empty or half-full?

He received no guidance, no stimulation, no money beyond the few rupees he made from tutoring. But for all the economic deadweight he represented, his family apparently discouraged him little- not enough, in any case, to stop him.

Like an Elephant Translated by A.K. Ramanujan Like an elephant lost from his herd suddenly captured, remembering his mountains, his Vindhyas, I remember. A parrot coming into a cage remembering his mate, I remember. O lord white as jasmine show me your ways. Call me: Child, come here, come this way.

was convinced that he had discovered a mathematical genius who needed proper exposure and guidance to flourish in mathematics. He therefore sent one of his mathematician colleagues, E. H. Neville, to India on the pretext of a lecture tour and assigned him the task of bringing Ramanujan to Cambridge. However, when Neville met Ramanujan after a lecture in Madras, he was pleasantly surprised to find the latter willing to come to England! This sea change in Ramanujan’s outlook towards the England visit came only

Years of practice in mathematics hardwires a person's brain for subconscious solving of mathematical problems - this is how Srinivasa Ramanujan came up with most of his discoveries in mathematics. Years of practice in the study of human behavior hardwires a person's brain for the subconscious solving of the puzzle of human nature - this is how I come up with many of my insights on human cognition and behavior. There is no magic or supernatural intervention involved here. It’s just that, when you foster a forte in a specific activity, your brain keeps working on it, beneath the surface of your conscious awareness, even if you are consciously engaged in other activities or even if you are asleep.

Para los científicos y pensadores autodidactas como Charles Darwin, Srinivasa Ramanujan, Leonardo-da-Vinci, Michael Faraday, yo y muchos otros, la educación es un viaje incesante de descubrimiento.

Then, too, it seems certain, in light of future events, simple racism was a factor; Ramanujan, after all, was a black man.

A tangent that departs from the real to the imaginary: pure consciousness does and does not transcend the body, and I believe this after hearing that my mother felt suicidal after she took her medicines for weight loss and her biggest regrets in life came crushing down on her for three days in a row. This is the best of what I have learnt in my years of fascination for science and knowledge, and to make you grasp this takes fullness of life: in hydrology, the wet and the dry, and the hot and the cold always co-exist, but they are also in flux and are also stable: all depending on the reference point of analysis. Consciousness beyond matter, and consciousness tied to matter co-exist in everyplace at different scales, and sometimes even in the same scale. Tao te ching (the way and its power) that fascinated Lao Tzu; the calculus of infinitesimals; the wonderful infinity of the number line and fractals that fascinated Ramanujan and Mandelbrot; the horn of the rhinoceros that fascinated Dali, thermodynamic and hydrodynamic equilibriums that fascinate all scientists, the surety of a fading perfume smell or the permanence of a shattered mirror that is easy to understand to anyone; the concepts of anti-fragility, entropy, volatility, randomness, disorder are all intimately tied to this. Consciousness is constantly attainted and broken all around us all the time, and we rarely stop to think about this because it infinitesimally evades us. Here is where I begin to stretch this and I can't understand it and it is very discouraging -- prudence, temperance and courage -- some of the highest virtues may also be related to this. When you are prepared, it is consciousness. When we are unprepared for it, and this hits you without hurting you, it is magic and strength. Else, perhaps death.