All of us are born with some gifts (skill set, whether inborn or developed), but some are born more equal than others and as if by magic to mere mortals, they live to attain such mastery over a subject as to inspire feeling of incredulousness in observers. And they are the prime movers of the human thoughts, ideas and deeds for through their work, many life times worth of effort is laid on the ever forward march of our species to in their quest to epistemological immortality. For example, mathematician Gauss alone is said to have sped mathematics development by atleast 50 years by his genius (Had all his discoveries been published the impact would have been much greater!!). Same goes for the works of legendary geniuses and polymaths such Von Neumann, Leonard Euler, Herbert Simon.
It is also often said that what a "Newton" or "Einstein" achieved was a product of the time, as the ideas were ripe to be discovered. "No mortal force can stop an idea whose time has come" - said Voltaire, French philosopher and defender of human freedom nonpareil. It can be claimed that any exceptionally astute thinker of the caliber among the greatest could have done what a Newton or Einstein had achieved in their life time. A case in point is to examine the lives of Christian Huygens and Henri Poincare, contemporaries of Newton and Einstein respectively. Then even in the highly dignified list of such greatest geniuses, some still stood on a different level on their own and did their work seemingly beyond the reach of even the most gifted in many generations. It can be rightly said that Ramanujan was one among such singular geniuses the world has ever seen. Some of his theorems could never have been known if he hadnt mysteriously derived and wrote it in his famed notebooks, as if taken out of thin air.
He was born and raised in India and was a product of the best of how Indian system has molded the thinking of the some of the ablest men of sciences.
Mathematics has a platonic existence in the sense that its reality, the interlinking of relationships between mathematical concepts, their logical order, is independent of the task of discovery. The concept of numbers were immediately apparent to the thinking man when he was surmounted with the task of calculation of seasons for grain harvesting or the number of animals to be hunted for a daily meal for his growing family. But the elaborate edifice of number of theory and its many connections with other disciplines of mathematics is no man made object as the connections became apparent as and when the subject was explored by the intrepid mathematical explorers in their mental voyages. G H Hardy has likened the process of mathematical discovery as the process where a climber arduously scale a difficult peak and then survey the vast landscape laying below him and distant peaks lying ahead of him to be scaled later, while the large landscape of Mathematics remain a mysterious reality shrouded by the clouds of our limits of reasoning. Roger Penrose has beautifully and forcefully put forward the platonic reality of Mathematics in his discussion about how the fractals and its unique properties were discovered, and its computer based explorations was not able to scratch the surface of the immense complexity of the evolving patterns and their underlying relationships. Godel's famous theorem has put an end to the dream of mechanical theorem proving, which was akin to reducing the whole of mathematics sophisticated yet mechanical symbol play.
Here, Terence Tao's analogy also comes to mind. Mathematics as a subject consists of different levels of abstractions, all of which are attempts to model or approximate the reality of the physical world. At the base level would be the concept of the primitive objects as numbers and simple geometric concepts and then move up to sets, spaces, operations relations, functions and operators.
If mathematics is a universal construct independent of human existence, then it can be said that discovery of mathematics is also a process independent of humans. We do not yet know of existence of mathematical prowess in other animals, but is conjectured that whales and some species of Dolphins exhibit some rudimentary number skills. Some humans are better endowed with an uncanny skill to master mathematics. Infact geniuses are most commonly observed in Mathematics and Music. Legend is that, Gauss's famous pupil, Gotthold Eisenstein had remarked that he always knew Calculus. Many mathematics prodigies are also fabulous mental calculators. Familiarity and exceptional skills in number manipulation seems to be the one sure sign of giftedness in Mathematics. Also it is assumed that we come to this world with some hard wired logic and innate sense of the 3D world. This is also the case that we always believe when two statements of seemingly contradictory facts or logic are heard, we at once naturally think that only one of them should be correct.
I was familiar with Ramanujan's incredible story right from my school days. His story was an inspiration, of how genius can transcend the cultural and geographical barriers to its fullest expression. The old adage "the whole world conspiring to bring fruit to the man's effort" is apt in his case. Ramanujan was born in Kumbhakonam, Tamilnadu to a pious Hindu Brahmin family. His school days were filled with incidences of his irrepressible mathematics brilliance, stories that has become legends associated with genius in general. Other remarkable fact was the unfathomable level of concentration which Ramanujan was able to summon while doing mathematics, so much so that he forgot to study other subjects and failed them even though by all accounts he would have easily passed them with little extra effort if he cared. In this connection, I remember the statement by Erik Demaine of MIT, who is a prodigy and is the youngest professor ever appointed by MIT, said in an interview that - he considers himself remarkable only in his ability to devote long span of his undivided attention to a subject or solution of a problem. Discovery of Ramanujan's genius and his association with famous GH Hardy is well known to many, but less is known of the height of esteem with which Hardy regarded the mathematical talent of his protege. Hardy (as is said in the biography of Ramanujan) at once noticed the remarkable skill for algebraic manipulation of Ramanujan, and compared him only with universal geniuses Euler or Jacobi. Hardy had little doubt that, with proper application of Ramanujan's genius, Ramanujan could have been the best mathematician of the world, even surpassing the leadership of great David Hilbert or Henri Poincare, both last century's greatest mathematicians and acknowledged universal geniuses. In a note published in the Current Science Magazine (Vol 65. No.1, 94-95), famous Indian statistician P C Mahalanobis recounts his friendship with Ramanujan in Cambridge, when both were students in the mathematics department. We get the picture of Ramanujan as a very simple man of somewhat "shy and quiet disposition, dignified bearing and pleasant manners". On one occasion, Mahalanobis went to Ramanujan's room to have lunch with him. Mahalanobis had a copy of Strand Magazine which at the time used to publish a number of puzzles to be solved by its readers. Ramanujan was stirring something in a pan for lunch. Mahalanobis read out the puzzle about two British officers in Paris in a long street with houses on either side, the question was about the relationship between two of the house numbers, which were related in a special way. It took some trial and error effort to reach the answer, but the answer was not difficult to reason. Ramanujan promptly answered the solution as a continuous fraction, the first term of which was the solution that Mahalanobis had obtained. Each successive term were the successive solutions for same type of relation between two numbers as the number of houses in the street increases indefinitely. Mahalanobis was amazed and asked how Ramanujan got this solution, that too the most generalized one, in a flash. Ramanujan replied thus "Immediately when I heard the problem, it was clear that the solution should obviously be a continuous fraction, I then thought which continuous fraction and the answer came to my mind. It was as simple as that"
Though born and raised in a traditional Hindu Brahmin household, Ramanujan held progressive views about life and society. He was eager to work out a philosophical theory, which he termed the theory of reality based on the fundamental mathematical concepts of 'Zero' and 'Infinity', and the set of finite numbers. He spoke about Zero as the symbol of absolute, that is the part of reality to which no qualities can be attributed, which cannot be defined or described by words and is absolutely beyond reach of human mind. With zero this symbolizes the absolute negation of all attributes. According to him Infinity was the totality of possibilities which was capable of becoming manifest in reality and which was inexhaustible. According to Ramanujan, the product of zero and infinity would supply the whole set of finite numbers, akin to creation of all properties of the world associated with the finite numbers. We are not sure of how far Ramanujan made progress with this mathematical philosophy, but it was clear that Ramanujan held great prestige in working out a theory of reality than building conjectures or proving theorems about esoteric aspects of number theory, as former took real effort while the latter was nigh effortless and intuitive for him.
So looking back to the infinite and strange potential of his mind and the wonderful results he made in his short collaboration with Hardy, one could only wonder at the magnitude of loss suffered by mathematics community in losing Ramanujan at the young age of 32 to Tuberculosis. It might be that Ramanujan could have proved the famous Riemann conjecture (This was a life's quest for G H Hardy and many other famous mathematicians of the day. Hardy, himself had mistakenly thought that he managed to solve it after years of effort, only to be jolted back to reality when he uncovered a subtle flaw in one of the lemmas that invalidated the whole edifice of the proof. Together with Littlewood, he had proved many remarkable properties of the Zeros of Riemann function. Hardy would no doubt have prodded the genius in Ramanujan to solve the greatest unsolved problem in mathematics). Or his achievements could conceivably be much lesser than what we could imagine to be, as is human nature, we extol achievements and possibilities of geniuses to some unreasonable extent to be totally off reality.
What could Galois, Abel had done had they lived to a ripe age, are some similar musings of mathematics lovers. Basing on Probabilistic reasoning, it can be argued that greatest geniuses possible in any scientific discipline could have had obtained sufficient environment to fruitfully produce what they do the best and in that respect our world has seen its share of remarkable men and women, even with the untimely loss of a few, who could have been one among the greats but not too great as to have had such monumental impact as to make others contributions irrelevant. Yes, there are remarkable people who continue to toil in unremarkable positions in unfavorable environments, but such is the incandescent nature of true genius is that the whole world will conspire to allow the genius to flower in some ways.
Now coming back to speculation with respect to the independent reality of mathematics, with the case of Ramanujan it becomes plausible that he possessed an uncanny ability to "see" the equations as if dictated by his deity, which points to a notion not very much different from discovery of truths by intuition - which is possible if the truth is lying out there to be seen, only to be later connected with the mainstream mathematics by the laborious but essential process of proof. We live in such a complicated world that many of the seemingly easy questions elude answer, where the case may point to a sort of idea world with different structure than simply constituted of logic and that an otherworldly ability to see through ideas to truth would be essential to fathom the landscape to its fullest. In Ramanujan, we could have seen a genius who was able to do just that in a way, and thus showed to the world that with genius and patient application, this is possible. I also would like to add Terence Tao's assessment of unusual abilities of Ramanujan. According to Prof Terence Tao, Ramanujan's secret for coming out with strange theorems was his unusual mental felicity to do significant mental computations and his ability to drew intuition from patterns thus observed.
Hindu scriptures are full of accounts of teachings of ancient Rishis, who often with beautiful verses utter the most profound truths of the world. "Tatvamasi - I am what is the world" is a very famous utterance by a Aithreya Muni to Yaknjavalkyan and he explains the concept with seven different examples.
I would venture to say that such a profound thinking was unthinkable in a pre AD civilization (BC 500-350) with little training in logic and scientific thinking. It could only have been possible with this mystical superhighway of transcendental thinking to the heart of truth. We do not know whether this "Gods pathway" exists, but I would love to think that our capacity to learn is limitless and such a feature of our mind, connecting to the cosmic consciousness in a flash is conceivable.
It is also often said that what a "Newton" or "Einstein" achieved was a product of the time, as the ideas were ripe to be discovered. "No mortal force can stop an idea whose time has come" - said Voltaire, French philosopher and defender of human freedom nonpareil. It can be claimed that any exceptionally astute thinker of the caliber among the greatest could have done what a Newton or Einstein had achieved in their life time. A case in point is to examine the lives of Christian Huygens and Henri Poincare, contemporaries of Newton and Einstein respectively. Then even in the highly dignified list of such greatest geniuses, some still stood on a different level on their own and did their work seemingly beyond the reach of even the most gifted in many generations. It can be rightly said that Ramanujan was one among such singular geniuses the world has ever seen. Some of his theorems could never have been known if he hadnt mysteriously derived and wrote it in his famed notebooks, as if taken out of thin air.
He was born and raised in India and was a product of the best of how Indian system has molded the thinking of the some of the ablest men of sciences.
Mathematics has a platonic existence in the sense that its reality, the interlinking of relationships between mathematical concepts, their logical order, is independent of the task of discovery. The concept of numbers were immediately apparent to the thinking man when he was surmounted with the task of calculation of seasons for grain harvesting or the number of animals to be hunted for a daily meal for his growing family. But the elaborate edifice of number of theory and its many connections with other disciplines of mathematics is no man made object as the connections became apparent as and when the subject was explored by the intrepid mathematical explorers in their mental voyages. G H Hardy has likened the process of mathematical discovery as the process where a climber arduously scale a difficult peak and then survey the vast landscape laying below him and distant peaks lying ahead of him to be scaled later, while the large landscape of Mathematics remain a mysterious reality shrouded by the clouds of our limits of reasoning. Roger Penrose has beautifully and forcefully put forward the platonic reality of Mathematics in his discussion about how the fractals and its unique properties were discovered, and its computer based explorations was not able to scratch the surface of the immense complexity of the evolving patterns and their underlying relationships. Godel's famous theorem has put an end to the dream of mechanical theorem proving, which was akin to reducing the whole of mathematics sophisticated yet mechanical symbol play.
Here, Terence Tao's analogy also comes to mind. Mathematics as a subject consists of different levels of abstractions, all of which are attempts to model or approximate the reality of the physical world. At the base level would be the concept of the primitive objects as numbers and simple geometric concepts and then move up to sets, spaces, operations relations, functions and operators.
If mathematics is a universal construct independent of human existence, then it can be said that discovery of mathematics is also a process independent of humans. We do not yet know of existence of mathematical prowess in other animals, but is conjectured that whales and some species of Dolphins exhibit some rudimentary number skills. Some humans are better endowed with an uncanny skill to master mathematics. Infact geniuses are most commonly observed in Mathematics and Music. Legend is that, Gauss's famous pupil, Gotthold Eisenstein had remarked that he always knew Calculus. Many mathematics prodigies are also fabulous mental calculators. Familiarity and exceptional skills in number manipulation seems to be the one sure sign of giftedness in Mathematics. Also it is assumed that we come to this world with some hard wired logic and innate sense of the 3D world. This is also the case that we always believe when two statements of seemingly contradictory facts or logic are heard, we at once naturally think that only one of them should be correct.
I was familiar with Ramanujan's incredible story right from my school days. His story was an inspiration, of how genius can transcend the cultural and geographical barriers to its fullest expression. The old adage "the whole world conspiring to bring fruit to the man's effort" is apt in his case. Ramanujan was born in Kumbhakonam, Tamilnadu to a pious Hindu Brahmin family. His school days were filled with incidences of his irrepressible mathematics brilliance, stories that has become legends associated with genius in general. Other remarkable fact was the unfathomable level of concentration which Ramanujan was able to summon while doing mathematics, so much so that he forgot to study other subjects and failed them even though by all accounts he would have easily passed them with little extra effort if he cared. In this connection, I remember the statement by Erik Demaine of MIT, who is a prodigy and is the youngest professor ever appointed by MIT, said in an interview that - he considers himself remarkable only in his ability to devote long span of his undivided attention to a subject or solution of a problem. Discovery of Ramanujan's genius and his association with famous GH Hardy is well known to many, but less is known of the height of esteem with which Hardy regarded the mathematical talent of his protege. Hardy (as is said in the biography of Ramanujan) at once noticed the remarkable skill for algebraic manipulation of Ramanujan, and compared him only with universal geniuses Euler or Jacobi. Hardy had little doubt that, with proper application of Ramanujan's genius, Ramanujan could have been the best mathematician of the world, even surpassing the leadership of great David Hilbert or Henri Poincare, both last century's greatest mathematicians and acknowledged universal geniuses. In a note published in the Current Science Magazine (Vol 65. No.1, 94-95), famous Indian statistician P C Mahalanobis recounts his friendship with Ramanujan in Cambridge, when both were students in the mathematics department. We get the picture of Ramanujan as a very simple man of somewhat "shy and quiet disposition, dignified bearing and pleasant manners". On one occasion, Mahalanobis went to Ramanujan's room to have lunch with him. Mahalanobis had a copy of Strand Magazine which at the time used to publish a number of puzzles to be solved by its readers. Ramanujan was stirring something in a pan for lunch. Mahalanobis read out the puzzle about two British officers in Paris in a long street with houses on either side, the question was about the relationship between two of the house numbers, which were related in a special way. It took some trial and error effort to reach the answer, but the answer was not difficult to reason. Ramanujan promptly answered the solution as a continuous fraction, the first term of which was the solution that Mahalanobis had obtained. Each successive term were the successive solutions for same type of relation between two numbers as the number of houses in the street increases indefinitely. Mahalanobis was amazed and asked how Ramanujan got this solution, that too the most generalized one, in a flash. Ramanujan replied thus "Immediately when I heard the problem, it was clear that the solution should obviously be a continuous fraction, I then thought which continuous fraction and the answer came to my mind. It was as simple as that"
Though born and raised in a traditional Hindu Brahmin household, Ramanujan held progressive views about life and society. He was eager to work out a philosophical theory, which he termed the theory of reality based on the fundamental mathematical concepts of 'Zero' and 'Infinity', and the set of finite numbers. He spoke about Zero as the symbol of absolute, that is the part of reality to which no qualities can be attributed, which cannot be defined or described by words and is absolutely beyond reach of human mind. With zero this symbolizes the absolute negation of all attributes. According to him Infinity was the totality of possibilities which was capable of becoming manifest in reality and which was inexhaustible. According to Ramanujan, the product of zero and infinity would supply the whole set of finite numbers, akin to creation of all properties of the world associated with the finite numbers. We are not sure of how far Ramanujan made progress with this mathematical philosophy, but it was clear that Ramanujan held great prestige in working out a theory of reality than building conjectures or proving theorems about esoteric aspects of number theory, as former took real effort while the latter was nigh effortless and intuitive for him.
So looking back to the infinite and strange potential of his mind and the wonderful results he made in his short collaboration with Hardy, one could only wonder at the magnitude of loss suffered by mathematics community in losing Ramanujan at the young age of 32 to Tuberculosis. It might be that Ramanujan could have proved the famous Riemann conjecture (This was a life's quest for G H Hardy and many other famous mathematicians of the day. Hardy, himself had mistakenly thought that he managed to solve it after years of effort, only to be jolted back to reality when he uncovered a subtle flaw in one of the lemmas that invalidated the whole edifice of the proof. Together with Littlewood, he had proved many remarkable properties of the Zeros of Riemann function. Hardy would no doubt have prodded the genius in Ramanujan to solve the greatest unsolved problem in mathematics). Or his achievements could conceivably be much lesser than what we could imagine to be, as is human nature, we extol achievements and possibilities of geniuses to some unreasonable extent to be totally off reality.
What could Galois, Abel had done had they lived to a ripe age, are some similar musings of mathematics lovers. Basing on Probabilistic reasoning, it can be argued that greatest geniuses possible in any scientific discipline could have had obtained sufficient environment to fruitfully produce what they do the best and in that respect our world has seen its share of remarkable men and women, even with the untimely loss of a few, who could have been one among the greats but not too great as to have had such monumental impact as to make others contributions irrelevant. Yes, there are remarkable people who continue to toil in unremarkable positions in unfavorable environments, but such is the incandescent nature of true genius is that the whole world will conspire to allow the genius to flower in some ways.
Now coming back to speculation with respect to the independent reality of mathematics, with the case of Ramanujan it becomes plausible that he possessed an uncanny ability to "see" the equations as if dictated by his deity, which points to a notion not very much different from discovery of truths by intuition - which is possible if the truth is lying out there to be seen, only to be later connected with the mainstream mathematics by the laborious but essential process of proof. We live in such a complicated world that many of the seemingly easy questions elude answer, where the case may point to a sort of idea world with different structure than simply constituted of logic and that an otherworldly ability to see through ideas to truth would be essential to fathom the landscape to its fullest. In Ramanujan, we could have seen a genius who was able to do just that in a way, and thus showed to the world that with genius and patient application, this is possible. I also would like to add Terence Tao's assessment of unusual abilities of Ramanujan. According to Prof Terence Tao, Ramanujan's secret for coming out with strange theorems was his unusual mental felicity to do significant mental computations and his ability to drew intuition from patterns thus observed.
Hindu scriptures are full of accounts of teachings of ancient Rishis, who often with beautiful verses utter the most profound truths of the world. "Tatvamasi - I am what is the world" is a very famous utterance by a Aithreya Muni to Yaknjavalkyan and he explains the concept with seven different examples.
I would venture to say that such a profound thinking was unthinkable in a pre AD civilization (BC 500-350) with little training in logic and scientific thinking. It could only have been possible with this mystical superhighway of transcendental thinking to the heart of truth. We do not know whether this "Gods pathway" exists, but I would love to think that our capacity to learn is limitless and such a feature of our mind, connecting to the cosmic consciousness in a flash is conceivable.
