G. H. Hardy's "Apology"

I am certainly among the perhaps one million readers of this classic who could never master the mathematics that Hardy taught. But the continuous popularity of this book speaks to the fact that our common culture values pure mathematics. It must seem like hubris for me to say that Hardy was wrong about pure mathematics. I enjoyed the book nevertheless and took my time through it, and read parts of it again. I also skipped the introduction by C. P. Snow at first. I wanted to know Hardy, not Snow. But I went back purposefully and accepted Snow on his own terms. 

Hardy asks, "Is this important and am I the person to do it?" On the second point, Hardy is naturally demur.  Even though he makes the case for ego, he does not spend much time advancing his own to the reader. He does say that when he was very young, excelling at maths was a way to beat others, to best them at something difficult. ([29] p. 144) Only later did he discover a different pride, a different merit collaborating with Littlewood and Ramanujan.

A Mathematician’s Apology by G. H. Hardy
(with a Foreword by C. P. Snow).
Cambridge University Press,
1940, 1967; 23^rdPrinting 2016. 

“Good work is not done by ‘humble’ men.” – G. H. Hardy. (Math. Ap. [2] p. 66)

“I am not suggesting that this is a defence which can be made by most people, since most people can do nothing at all well.—G. H. Hardy. (Math. Ap. [3] p. 67)

…perhaps five or even ten per cent of men can do something rather well.—G. H. Hardy. (Math. Ap. [3] p. 68; also, [5] p. 73)

… Poetry is more valuable than cricket, but Bradman would be a fool if he sacrificed his cricket in order to write second-rate minor poetry.”–G. H. Hardy. (Math. Ap. [3] p. 69)

“A mathematician, like a painter or a poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas.”–G. H. Hardy. (Math. Ap. [10] p. 84)

Hardy refers to cricket throughout the book. In the Foreword, C. P. Snow cites John Maynard Keynes who said that if Hardy had spent as much time with the stock market reports as he did with cricket scores, he would have retired a millionaire. The quote above was only one of very many allusions to the sport. It sent me (appropriately enough) googling. 

Sir Donald George Bradman, AC (27 August 1908 – 25 February 2001), often referred to as “The Don”, was an Australian international cricketer, widely acknowledged as the greatest batsman of all time. Bradman's career Test batting average of 99.94 has been cited as the greatest achievement by any sportsman in any major sport. -- Wikipedia.

“There are many highly respectable motives which may lead men to prosecute research, but three which are much more important than the rest. The first (without which the rest must come to nothing) is intellectual curiosity, desire to know the truth. Then, professional pride, anxiety to be satisfied with one’s performance, and shame that overcomes any self-respecting craftsman when his work is unworthy of his talent. Finally, ambition, desire for reputation, and the position, even the power or the money, which it brings. It may be fine to feel, when you have done your work, that you have added to the happiness or alleviated the sufferings of others, but that will not be why you did it. If a mathematician, or a chemist, or even a physiologist were to tell me that the driving force in his work had been the desire to benefit humanity, then I should not believe him (nor should I think better of him if I did).” ([7] p. 79)

Godfrey Harold Hardy 1887-1947. 
https://www.maths.ox.ac.uk/node/13895

Hardy is eloquent, making a strong case for his field of theoretical mathematics as having greater value than applied mathematics. He begins by laying out four theorems. The first is Euclid’s proof that no end exists for prime numbers, that an infinity of prime numbers exists. The second is Pythagoras’s proof that the square root of two is irrational. Those he proves for the reader. 

Two others are left unfinished. The Fundamental Theorem of Arithmetic says that any integer can be resolved in only one way into a product of primes. Hardy shows some examples, but says that the proof might be tedious for the general reader. Similarly, he tells us of Fermat’s “Two Square” Theorem which says that all prime numbers can be divided by 4 with a remainder of either 1 or 3; and that all of the first class and none of the second can be written as the sum of two squares. Again, he lists some examples, but leaves the proof for the reader. 

From section [8] of [29] throughout, Hardy argues on behalf of the beauty of pure mathematics. He says that the practical work that benefits civil engineering and other advances in civilization is not beautiful and therefore not permanent. It does not matter that pure math is not practical. In fact, he says, it is better that it not be. I disagree.

I believe that all mathematics is potentially practical, only that we have not found a specific application. I do not know how to prove that. Hardy says (and I agree) that a proof by enumeration of cases is the least attractive. But I point to irrational, negative, and imaginary numbers, all of which were denied as real and all of which have practical applications today. 

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