Ramanujan, the self-taught Indian mathematician, is being celebrated this year. Jeremy Irons and Dev Patel (Slumdog millionaire) star in a film being made in Cambridge as I write, summer 2014. Another film about his life has just been completed.
Ramanujan wrote down, worked on, and arrived at thousands of results, the vast majority of which were correct. He came to Cambridge for five years towards the end of his short life and became a Fellow of the Royal Society. Born in Madras in 1887, he died there of TB in 1920, aged 32.
This little anecdote below, however, from the 1940 book “Ramanujan” by Professor G Hardy (Chapters I and II), who helped bring Ramanujan to Cambridge, shows how hard it is to get it right in mathematics.
In the theory of prime numbers, there was a ‘prime number theorem’ for the number of primes up to a given number, call it x, which, brilliantly, Ramanujan independently rediscovered. Gauss, Legendre and Dirichlet had “endorsed” it prior. The theorem however “errs by defect” – which means if you stress test it enough, i.e. go to high enough examples, you’ll see it deviate from the truth, and increasingly so.
Up to x = 1 billion, the expression is good!
But another mathematician, Littlewood, went on in 1912 to show that there are an infinite number of (very high) values of x where the formula is shown to be incorrect. He did so by looking at its difference from a better, proven formula for the number of primes up to a given value, and an example found by a man called Skewes was 10 to the power of ten to the power of ten to the power 34. A large number! Hardy thought this was the “largest number ever to have served a definite purpose in mathematics”.
When Einstein wrote down Special and General Relativity and these were later tested, this showed that under certain conditions (strong gravitational fields or velocities close to the speed of light) the Newtonian theory was very wrong. Quantum mechanics does the same thing at the small scale to Newton’s theories.
A conjecture, a model, a hypothesis remains just that until you prove it, as in mathematics, e.g. prime numbers, or do experiments to check it, as in gravitation, or in small-scale physics.
Proof is extremely difficult.