by Desmond Sander
Last night, watching an excellent SBS film about the mathematics genius Ramanujan, a thought occurred to me which might interest others who wonder about the effectiveness of Mathematics and Physics as ways of understanding what happens.
As the movie made plain, Ramanujan was most reluctant to supply the proofs of his theorems that Hardy and others demanded. I think it likely that, thanks to some fluke in the make-up of his unique brain/mind, Ramanujan was able to see his congruences directly, so to speak, and viewed them as not needing the proofs that are viewed as essential by Logicist Mathematicians, including the very distinguished Mathematician GH Hardy. As well as by Bertrand Russell, the inventor of modern logicism, who also featured in the film.
Ramanujan’s extraordinary insights are not so different in kind, it seems to me, from the insight of Pythagoras that led him to believe the Pythagoras Theorem long before it was proved. What those insights, separated by a couple of millennia, have in common is that they have to do with geometry, in particular the geometry of the circle.
In the Pythagoras case, all that was required to believe what might at that time have been called the Pythagoras conjecture was careful drawing (or imagining drawing), on a flat sheet, with straight-edge and compass and the kind of direct empirical insight that is nowadays disparaged as visual proof. Around a century later, Plato and Euclid between them had the idea that became logic, with its axioms and inference rules. So, I find myself wondering whether it might be that Ramanujan chose not to prove his results because he viewed those Logicist proofs as an easy waste of time.
I think that the easy proofs which currently pre-occupy the Logicist/Analytic Philosophers in leading Australian Universities nicely confirm the view that Logicist thinking is easy.
Mathematics/Physics, I would say, is much deeper than that.