The ‘invincible originality’ of Srinivasa Ramanujan
GS Paper 1: Contributions of important personalities
Important for
Prelims exam: About Srinivasa Ramanujan
Mains exam: Contribution of Srinivasa Ramanujan
Why in News?
National Mathematics Day is observed on December 22 every year. This date marks the birth anniversary of legendary mathematician Srinivasa Ramanujan.
About Srinivasa Ramanujan
- Srinivasa Ramanujan was born on December 22, 1887, in Tamil Nadu’s Erode to a Brahmin Iyengar family. He had developed a liking for mathematics at a very young age, mastering trigonometry at 12 and was eligible for a scholarship at the Government Arts College in Kumbakonam.
- He studied at the Government College in Kumbakonam in 1903. Due to his dislike for non-mathematical subjects, he failed exams there. He had enrolled in Madras’ Pachaiyappa College at the age of 14.
- In 1912, Ramanujan started working as a clerk in the Madras Port Trust. There, his mathematical genius was recognised by some of his colleagues and one of them referred him to Professor GH Hardy of Trinity College, Cambridge University. He met Hardy in 1913, after which he went to Trinity College.
- In 1916, Ramanujan received his Bachelor of Science (BSc) degree. He went on to publish several papers on his subject with Hardy’s help. The two even collaborated on several joint projects.
- Ramanujan was elected to the London Mathematical Society in 1917. Next year, he was elected to the prestigious Royal Society for his research on Elliptic Functions and theory of numbers. He was also the first Indian to be elected a Fellow of the Trinity College.
- Despite not receiving any formal training in pure maths, Ramanujan made impactful contribution to the discipline in his short life. His areas of work include infinite series, continued fractions, number theory and mathematical analysis.
- He also made notable contributions like the hypergeometric series, the Riemann series, the elliptic integrals, the theory of divergent series, and the functional equations of the zeta function. He is said to have discovered his own theorems and independently compiled 3,900 results.
- In 1919, Ramanujan returned to India. A year later, on April 26, he breathed his last owing to deteriorating health. He was just 32 years old. His biography ‘The Man Who Knew Infinity’ by Robert Kanigel depicts his life and journey to fame.
Key Points
- The English computer scientist Alan Turing’s “imitation game” or “Turing test” takes the stance that a machine successfully represents a human if it responds to a question like a human mind does.
- From Plato’s mimesis (representation) as the principle of art, to the Turing test, and to recent progresses in artificial intelligence, “the key to artificial intelligence has always been the representation” (Jeff Hawkins).
Taking representation further, can a machine “create” new things?
- Latest advancements in technology have attempted to “create”.
- The recent buzz around ChatGPT (Chat Generative Pre-trained Transformer), a software tool that can answer questions on almost any topic, carry on conversations with humans, write poems, computer programs and perform many more complex tasks that require intelligence, is testimony that artificial intelligence can “create”.
- Google’s product LaMDA (Language Model for Dialogue Applications) that is similar to ChatGPT and other sophisticated products (Dall E) that can create image from verbal descriptions.
- Broadly speaking, the products mentioned above learn to be creative from already existing information: human conversations and documents and pictures, to synthesise and create.
Can a machine do research?
- In early 2021, a team of Israeli scientists announced a software tool called The Ramanujan Machine that creates mathematical conjectures which are equations without proof.
- Mathematicians then prove or disprove these conjectures, thereby establishing theorems.
- Conjectures in mathematics shed light on newer frontiers that otherwise lurk in tenebrous corners.
- Srinivasa Ramanujan was famous for such conjectures. From 1904 till his passing in 1920, Ramanujan, recorded more than 3,000 equations that were mostly conjectures because he did not supply proof.
- The American mathematician Bruce C. Berndt, an expert on Ramanujan’s works, says that most of Ramanujan’s conjectures are correct as established by the proofs provided pari passu with their generalisation in the last 100 years.
How does the Ramanujan Machine imitate Ramanujan?
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- The Ramanujan Machine adopts a different approach to compute formulae for mathematical constants such as ℼ, e, etc.
- It does so by taking a mathematical structure such as a continuous fraction on one side of an equation and the mathematical constant such as on the other side, with the two sides yet to be matched.
- The software then employs high computing power and algorithms to iteratively match both sides and a conjecture is discovered. In the last couple of years, dozens of conjectures have been discovered this way.
Is this advancement a pronouncement that Ramanujan’s creativity was held high solely by the stanchions of his calculating speed and memory?
- Mathematicians aver that Ramanujan’s intuition, as seen in his works, was non pareil; Ramanujan could connect different mathematical domains deeply with “invincible originality” (G.H. Hardy).
- The ramifications of such connections between mathematical abstractions that Ramanujan was prescient of have continued to unravel for over a century now, leading to remarkable technological applications.
- For example, elliptic curves are widely studied in number theory because of their properties that have applications in cryptography which make computer network communications secure.
- Ramanujan’s contributions here are significant: he developed certain equations called class invariants which generate elliptic curves suitable for encryption and this led to Elliptic Curve Cryptography, an efficient cryptographic technique, in 1985, many decades after his demise.
A leap of genius
- Another instance of his originality was his discovery of mock-theta functions, his swan song; working on it might have been a palliative to him in his painful last days.
- George Andrews, an American mathematician and an expert on mock theta function, elegantly surmised that Ramanujan might have discovered it by starting from hypergeometric series whose consecutive terms form ratios that follow a pattern.
- George Andrews adds that it is a plausible method to arrive at mock-theta functions “provided you had the genius to recognize”.