The BBC biography series Great Lives covered in its most recent episode Srinivasa Ramanujan. In the closing minutes of the programme, host Matthew Paris said this, which I found quite interesting (or at least, interestingly expressed):
I’m so far from understanding the mind of a mathematical genius that it’s simply inconceivable that you could tell a person an apparently random number and he could intuit or deduce the kind of fact that he deduced about that taxi license number. I mean, I can’t run a four-minute mile, but I once ran a five-minute mile, and I can extrapolate from my own experience, in a way understand how someone might just be a lot better than me at something that, in an inferior way, I can also do. But Ramanujan isn’t like that. It’s as though this man were a different species, not just a superior example of the same species. Can you learn to do this kind of thing? Could I, if I had applied myself? Or is it that goddess again, is it really just genius?
Answers on a postcard!
I’m reminded of the famous remark about Richard Feynman: that there are “ordinary” geniuses, clever people who one can imagine being like if only one was a bit cleverer and/or worked harder. And then there are the “magicians”, of whom Feynman was one, whose brains seem to be operating on an entirely different level to the rest of us.
Fields Medallists are generally in this second category. Ian Stewart said that a physicist friend once remarked that most serious physicists regard a Nobel Prize as something they can legitimately aspire to: if they work hard, get enough funding, are lucky with the projects they work on, then they’re in with a chance, at least in principle. Ian then observed that, in contrast, basically every mathematician knows that they’re never going to get a Fields Medal. Those who do seem to think at least an order of magnitude faster and better about mathematics than the rest of us, and also tend to be amazingly productive and talented people in general. For example: Manjul Bhargava also plays tablas to a high professional standard, and Martin Hairer writes award-winning audio editing software in his spare time.
I heard a recent interview of Sam Harris with David Deutsch. Deutsch recounts a conversation with Feynman where he drops a few crumbs of the work he’s been doing solidly for eighteen months. Feynman instantly grasps it and runs with it.
“I’m so far from understanding the mind of a mathematical genius that it’s simply inconceivable that you could tell a person an apparently random number and he could intuit or deduce the kind of fact that he deduced about that taxi license number.”
The taxi license number observation is numerical rather than mathematical, and it is a sign of being ‘far from understanding’ mathematics that this simple numerical observation is the canonical popular illustration of Ramanujan’s mathematical genius.
My engineering work colleagues are poor at mental arithmetic. I have over the years become familiar with some crude tools (some powers of small integers, Pythagorean triples, ‘the rule of 70’ for compound interest, some approximations to logarithms, etc.), which when applied may give results which seem to my uninformed observers to be ‘indistinguishable from magic’. So a simple and plausible explanation is that Ramanujan had a slightly more extensive toolbox, and knew his tools. There was no flash of intuition or deduction at the time of the taxi-cab observation.
The ‘different species’ examples for Ramanujan are probably the summation identities, which even Hardy said were ‘almost impossible to believe’. Conveying to a general non-technical audience an appreciation of how bizarre these identities are is a big task for a single episode of Great Lives, even assuming a listenership willing to concentrate.
Matthew Parris asked: “Can you learn to do this kind of thing? Could I, if I had applied myself?” For the taxi-cab observation, yes. For the summation identities, no chance…
Intuition is the quickest approach towards math problems.Ramanujam’s command over numbers is beyond imagination of a theorist.1729 as sum of cubes of two different numbers in different ways.. a flash reply we can expect from intuitive aspect only. other aspects like summation of series ,elliptic integrals also speak of his extraordinary ability and his own way (Different from a traditional style of solution finding) of getting solution.A gifted andb lessed with divine power aslo..