The “uselessness” of mathematics

One of the most fascinating aspects of mathematics is the abstraction they made of infinity. For example, everybody intuitively “feels” that the number of prime numbers, that is, the ones which are divisible only by themselves and 1 (3, 5, 7,…) should be infinite, but from here to conclude it with absolute certainty mediates a great mathematical mind, like Euclid’s, for instance, who more than two thousand years ago proved it with a sublime simplicity.

Euclid told himself: if the quantity of prime numbers were finite, let’s multiply all of them by each other, add 1 to the resulting product, and name x the result. Obviously, x is divisible by none of the known primes, as we would have a residue of 1 for each one of them. But for a well-known “principle” by then, every integer number can be univocally expressed as a product of its prime factors (fundamental theorem of arithmetic), in particular, x, which we already have seen has none of the known primes so far as its factors, from where it follows that it has a prime necessarily different from these as one of those factors. Therefore, the number of primes is infinite, as we could repeat this process indefinitely.

If not for other reason, we should recognize in this beauty the usefulness of mathematics, like in any other of the fine arts.

All this comes to my mind for what once the eminent British mathematician G. H. Hardy wrote: “I have never done anything ‘useful’. No discovery of mine has made, or is likely to make, directly or indirectly, for good or ill, the least difference to the amenity of the world. I have helped to train other mathematicians, but mathematicians of the same kind as myself, and their work has been, so far at any rate as I have helped them to it, as useless as my own. Judged by all practical standards, the value of my mathematical life is nil; and outside mathematics is trivial anyhow. I have just one chance of escaping a verdict of complete triviality, that I may be judged to have created something worth creating. And that I have created something is undeniable: the question is about its value.”

This moving story is evidenced in the book by the same Hardy A mathematician’s apology (Cambridge University Press, Canto edition, 1992).

Hardy says he sometimes thinks pure mathematicians (as opposed to applied mathematicians) glory in the useless of its work and they feel proud it does not have any practical application, and suggests it was Gauss who said that if pure mathematics are the queen of sciences by its useless, then number theory is the queen of mathematics by its supreme useless. Needless to say Hardy devoted his entire life to this theory, while everybody knows Gauss tackled brilliantly many other subjects.

Hardy points he has no evidence of Gauss saying that, but he doubts that if any practical and honorable application could be found for the theory of numbers Gauss or any other mathematician would have been so foolish in not rejoicing by it.

One of the reasons of Hardy’s depression was the military use attached to applied mathematics, and the clear evidence of it were the two wars he survived (he was born in 1877 and died in 1947), and in which some of his colleagues collaborated. But the true causes were the decline of his mathematical skills (it is known a mathematician gives the best of himself long before his 50s), a coronial thrombosis in 1939, which obliged him to quit the practice of sports, and the tragic death of one of his closest friends. This, together with the somber vision of his profession as pure mathematician, took him to write this apology in 1940, at 63.

C. P. Snow, famous British chemist and novelist, who wrote the forward to Hardy’s book, says: “Three or four years before his interest in everything was so sparkling as sometimes to tire us all out. ‘No one should ever be bored’, had been one of his axioms. ‘One can be horrified, or disgusted, but one can’t be bored.’ Yet now he was often just that, plain bored.”

Hardy’s depression became so intense that he attempted to commit suicide by an overdose drug in 1947 early summer, attempt from which he didn’t recover completely, and he finally died the morning of December 1 that year.

Ironically, Hardy’s words about Gauss were a premonition on himself, since as John Derbyshire points (Prime obsession, Joseph Henry Press, 2003): “Beginning in the late 1970s, prime numbers began to attain great importance in the design of encryption methods for both military and civilian use… Theoretical results, including some of Hardy’s, were essential in these developments, which, among other things, allow you to use your credit card to order goods over the internet.”