Monthly Archives: May 2015

Beautiful Minds

Having studied engineering once, I’ve had minimal contact with mathematics- real mathematics, that is. I vaguely remember learning the basics of abstract algebra – just about enough to pass an exam – and the minimum that they teach you in a CS course: graph theory, sets, logic. So when I stared at the first page of G.H. Hardy’s essay A Mathematician’s Apology, I was tentative. A curious layman who wants to glance into the mind of a mathematician without the bother of understanding his works – was I anywhere near the kind of audience its author had in mind?

Godfrey Harold Hardy (1877-1947) is not a well-known figure. If at all the general public has ever heard of him, it would be likely due to his far more famous co-worker, Srinivasa Ramanujan. One of the first people to realize Ramanujan’s genius, Hardy was no minor mathematician himself. But apart from his work, he was not world-famous in the way an Einstein or a Bertrand Russell was. He wrote no philosophy and he did not particularly seem to care about philanthropic activity. He did not suffer from a dramatic mental illness or physical ailment, or hail from an impoverished family. To the wider world, he must have seemed a typical, eccentric, slightly boring elderly professor, with an inexplicable talent for cricket.

And yet he had a magical way with words. And at sixty-three, with most of his mathematical work behind him, he wrote a charming essay called The Mathematician’s Apology (apology here means justification rather than an expression of regret). In it, he set out to establish that his life’s work – the study of pure mathematics – was not futile.

The context of the essay lies in the two major views of technology that were current then, which remain (with minor changes) those held by most people today. The first is the commonsense view that technology (and science and maths) are useful because they promote the wellbeing of mankind. The second accepts the view of maths and science as mere handmaidens of technology, but sounds a note of warning: science is not always a force for good. (Think anti-GMO campaigners, ecological groups, anti-nuclear power groups and so on). Hardy rejects both these views and sets out a third alternative.

What makes some brilliant people become mathematicians over, say, Wall Street investment bankers ? Hardy claims they are driven by two major forces: first, intellectual curiosity, and second, ambition. Ambition might seem puzzling, until he explains what he means in more detail:

What we do may be small, but it has a certain character of permanence; and to have produced anything of the slightest permanent interest, whether it be a copy of verses  or a geometrical theorem, is to have done something utterly beyond the powers of the vast majority of men.

In other words, mathematicians such as himself are motivated by the need for fame of a different kind, the need for their names to be remembered down the ages. It sounds too idealistic to be true, yet while you are reading the essay you find yourself empathizing with him.

He then develops the major point of the essay: that mathematics is an intrinsically worthwhile human activity, regardless of any benefits it brings to humanity in general. Hardy’s position here is strictly “art for art’s sake”. He claims not to care about the practical applications of differential calculus or the theory of primes. In fact, he remarks that from the mathematical point of view, “it is the dull and elementary parts of applied mathematics, as it is the dull and elementary parts of pure mathematics, that work for good or ill”.

It is not the application, but the cold beauty of what he calls “mathematical reality” that he finds most attractive. Mathematical reality, he says provocatively, cannot be distorted by the observer unlike physical reality:

317 is a prime, not because we think so, or because our minds are shaped in one way rather than another, but because it is, because mathematical reality is built that way.

So Hardy leaves the question of usefulness open, preferring to concentrate on the inherent beauty of the subject. Perhaps he felt himself unequal to the philosophical task of defending “aesthetics” in general against “utility”. But more likely, perhaps his writing the essay was a justification for himself, a defense of the worth of his own life.

Four years after writing the essay, Hardy attempted suicide. Was he accused one time too many of the crime of being an ivory-tower intellectual? We do not know.

Famously, Hardy completely failed to imagine a world where the theory of prime numbers and modular arithmetic might become entangled with everyday life. We interact with these terrifying-sounding concepts every time we purchase a book online, complete a netbanking transaction or even log in to our Gmail accounts. In an irony that Hardy himself would have relished, his own work in number theory has helped in driving its numerous applications today.

GH Hardy

The (Cambridge) don.

“I still say to myself when I am depressed, and find myself forced to listen to pompous and tiresome people,”Well, I have done one thing you could never have done, and that is to have collaborated with both Littlewood and Ramanujan on something like equal terms.”- GH Hardy


  1. “A Mathematician’s Apology” by G.H Hardy (1940) : Available at : (PDF)
  2. The Man Who Knew Infinity: A Life of the Genius Ramanujan by Robert Kanigel (1991)

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Survival of the Fittest

The koels were the first to be silenced. Spring dissolved into summer and summer soon grew intolerable; but walk into the densest of jungles or stroll in the tamest of colony parks, you could not hear a single note from that sulky singer.

Then it was the babbler’s turn. Usually, even the smallest park is fair game to them. They gather in groups of seven and set up their strange chorus, a lone individual starting, the others successively adding their inquiring notes. Babblers believe in joint families. Strange though it would have been to see a solitary member of the species, it was stranger still to see none.

Meanwhile we continued to cut down trees for better roads and better trains and larger buildings. We needed to go places faster and in greater comfort. We needed more things. We needed more services.

The doves were the next to go. But no one noticed their silence- they’re so understated anyway.

Something happened to the barbets next. Their slightly unbalanced-sounding calls are usually an annoying disturbance on May afternoons, but now you almost missed their insistent warbling…It’s summer, it’s summer, it’s summer….The birds seemed to have gone on strike entirely. It was as unexpected as Indian politicians saying nothing laughable for an entire day. But we had better things to do, so we thought little of it at the time.

And all the while, we were getting better roads, and better skyscrapers, and faster internet connectivity – the last of which we used, among other things, to watch videos of birds calling in their natural habitats.

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