In praise of mathematics
As part of my fundraising efforts for the Great Shave, I promised to write 500 words in praise of mathematics in return for some generations donations. I thought this would be the easiest of my promises to fulfil, but itâ€™s actually pretty hard to write in praise of something that is (to me) so obviously wonderful. Anyway, here goes.
The most striking single thing about mathematics is that a collective endeavour, pursued for thousands of years primarily because of its beauty and pure intellectual interest should turn out, in our time, to be so amazingly useful. To take perhaps the most striking example, the amazing fact that
or better still, with five fundamental constants
is, or ought to be, adequate reward for all the effort that went into the discovery of calculus, trigonometry and complex number theory, and the effort each new generation puts into learning these things. But, it gives us, free of charge, the amazingly useful Fourier transform, the basis of all kinds of modern communications, and much, much more.
(BTW, I made a big effort to instal a plugin that would make pretty math, trying and failing with both LatexRender and itextoMathML. Both require lots of fiddly modifications, and any wrong step crashes the blog completely. If anyone has an easy way to implement this, please advise me).
Although they didnâ€™t make much practical use of it, the ancient Greeks had already perceived the connection between mathematical reasoning and the workings of the universe, manifested in such observations as the golden ratio, and the relationship between mathematics and music. Mathematicians, then and now, are instinctive Platonists, thinking of themselves as discoverers of hidden truths rather than as inventors of new ideas.
As an aside, this kind of thought, extending into number mysticism, always used to be associated with Pythagoras. I was taught that, although Pythagoras himself was obscure, his school not only discovered the theorem that bears his name but followed it through to the scandalous discovery that irrational numbers have natural geometric representations (consider a right-angled triangle with two sides of length one. They square of the hypotenuse must be two, but itâ€™s easy to show that the square root of two cannot be rational). The person who leaked this terrible secret was supposedly put to death by the vengeful Pythagoreans. It turns out, as I read in a recent London Review of Books, that this is all legend and that there is no good evidence that the Pythagoreans discovered anything, although they were mystics. As is the way nowadays, the Wikipedia entry on Pythagoras had been updated before my copy of LRB had arrived in the post, and you can read all about it there.
Coming down to our own time, as late as the early 20th century, pure mathematicians could propose the toast â€œTo pure mathematics and may it never be of use to anyoneâ€?. GH Hardy is the archetype of this attitude, though I donâ€™t know if he ever actually made this statement. But economists and others use Hardyâ€™s work in inequalities all the time, and even his contributions to number theory are now hot topics in cryptography.
What Eugene Wigner has called â€˜the unreasonable effectiveness of mathematicsâ€™ remains a mystery. Maybe the Universe was designed by a mathematician. Or maybe we fit the mathematics we have to the universe, because no other mode of reasoning is remotely as powerful. Either way, mathematics is one of the great achievements of our species.