### A Mathematician's Apology, by GH Hardy

A Mathematician's Apology is an interesting insight into the mind of a mathematician, an investigation of what mathematics 'really' is, and why one would want to mess about with it. It's a relatively brief work, written in 1940 when Hardy was in his 60s and, he sadly concluded, was quite finished as a mathematician. Probably the most famous quote from the work is Hardy's dictum that mathematics is a "young man's game." The edition I have on the Kindle also includes an introduction by CP Snow that is nearly as long as the work itself, and provides a lot more biographical detail, including details of his student life:

One of Hardy's claims to fame is having discovered the self-taught & idiosyncratic Indian mathematician Ramanujan, who had sent some of his bizarre discoveries to him. "[Hardy] was accustomed to receiving manuscripts from strangers, proving the prophetic wisdom of the Great Pyramid, the revelations of the Elders of Zion, or the cryptograms that Bacon had inserted in the plays of the so-called Shakespeare.

So Hardy felt, more than anything, bored. He glanced at the letter, written in halting English, signed by an unknown Indian, asking him to give an opinion of these mathematical discoveries."

Hardy was soon intrigued, and took the time to puzzle some of out. An interesting detail of which I was unaware is that...

"But I mentioned that there were two persons who do not come out of the story with credit. Out of chivalry Hardy concealed this in all that he said or wrote about Ramanujan. The two people concerned have now been dead, however, for many years, and it is time to tell the truth. It is simple. Hardy was not the first eminent mathematician to be sent the Ramanujan manuscripts. There had been two before him, both English, both of the highest professional standard. They had each returned the manuscripts without comment. I don't think history relates what they said, if anything, when Ramanujan became famous."

Snow also talks of Hardy's abiding love of cricket, and how he (Snow) would have to study up on the latest scores before visiting Hardy, in order to help cheer Hardy up in his later years of illness.

But to finally get to the man himself in his own words, Hardy more or less rejected the idea that the pursuit of mathematics is justified by its technological fruits:

"The mass of mathematical truth is obvious and imposing; its practical applications, the bridges and steam-engines and dynamos, obtrude themselves on the dullest imagination. The public does not need to be convinced that there is something in mathematics.

All this is in its way very comforting to mathematicians, but it is hardly possible for a genuine mathematician to be content with it. Any genuine mathematician must feel that it is not on these crude achievements that the real case for mathematics rests, that the popular reputation of mathematics is based largely on ignorance and confusion, and that there is room for a more rational defence."

He also considered that, even if mathematics was unimportant, it might well be right for those with an aptitude to pursue it. "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."

There is virtually no mathematics in the work, but there are occasional allusions to things that were unfamiliar to me: "Farey is immortal because he failed to understand a theorem which Haros had proved perfectly fourteen years before; the names of five worthy Norwegians still stand in Abel's Life, just for one act of conscientious imbecility, dutifully performed at the expense of their country's greatest man." [I haven't the faintest idea what that refers to.]

Getting back to heart of the matter: "THERE are then two mathematics. There is the real mathematics of the real mathematicians, and there is what I will call the `trivial' mathematics, for want of a better word. The trivial mathematics may be justified by arguments which would appeal to Hogben, or other writers of his school, but there is no such defence for the real mathematics, which must be justified as art if it can be justified at all."

And here's one last awkwardly timed prediction: "There is one comforting conclusion which is easy for a real mathematician. Real mathematics has no effects on war. No one has yet discovered any warlike purpose to be served by the theory of numbers or relativity, and it seems very unlikely that anyone will do so for many years."

Hardy had decided-I think before he left Winchester-that he did not believe in God. With him, this was a black-and-white decision, as sharp and clear as all other concepts in his mind. Chapel at Trinity was compulsory. Hardy told the Dean, no doubt with his own kind of shy certainty, that he could not conscientiously attend. The Dean, who must have been a jack-in-office, insisted that Hardy should write to his parents and tell them so. They were orthodox religious people, and the Dean knew, and Hardy knew much more, that the news would give them pain-pain such as we, seventy years later, cannot easily imagine.

Hardy struggled with his conscience. He wasn't worldly enough to slip the issue. He wasn't even worldly enough-he told me one afternoon at Fenner's, for the wound still rankled-to take the advice of more sophisticated friends, such as George Trevelyan and Desmond MacCarthy, who would have known how to handle the matter. In the end he wrote the letter. Partly because of that incident, his religious disbelief remained open and active ever after. He refused to go into any college chapel even for formal business, like electing a master. He had clerical friends, but God was his personal enemy.

One of Hardy's claims to fame is having discovered the self-taught & idiosyncratic Indian mathematician Ramanujan, who had sent some of his bizarre discoveries to him. "[Hardy] was accustomed to receiving manuscripts from strangers, proving the prophetic wisdom of the Great Pyramid, the revelations of the Elders of Zion, or the cryptograms that Bacon had inserted in the plays of the so-called Shakespeare.

So Hardy felt, more than anything, bored. He glanced at the letter, written in halting English, signed by an unknown Indian, asking him to give an opinion of these mathematical discoveries."

Hardy was soon intrigued, and took the time to puzzle some of out. An interesting detail of which I was unaware is that...

"But I mentioned that there were two persons who do not come out of the story with credit. Out of chivalry Hardy concealed this in all that he said or wrote about Ramanujan. The two people concerned have now been dead, however, for many years, and it is time to tell the truth. It is simple. Hardy was not the first eminent mathematician to be sent the Ramanujan manuscripts. There had been two before him, both English, both of the highest professional standard. They had each returned the manuscripts without comment. I don't think history relates what they said, if anything, when Ramanujan became famous."

Snow also talks of Hardy's abiding love of cricket, and how he (Snow) would have to study up on the latest scores before visiting Hardy, in order to help cheer Hardy up in his later years of illness.

But to finally get to the man himself in his own words, Hardy more or less rejected the idea that the pursuit of mathematics is justified by its technological fruits:

"The mass of mathematical truth is obvious and imposing; its practical applications, the bridges and steam-engines and dynamos, obtrude themselves on the dullest imagination. The public does not need to be convinced that there is something in mathematics.

All this is in its way very comforting to mathematicians, but it is hardly possible for a genuine mathematician to be content with it. Any genuine mathematician must feel that it is not on these crude achievements that the real case for mathematics rests, that the popular reputation of mathematics is based largely on ignorance and confusion, and that there is room for a more rational defence."

He also considered that, even if mathematics was unimportant, it might well be right for those with an aptitude to pursue it. "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."

There is virtually no mathematics in the work, but there are occasional allusions to things that were unfamiliar to me: "Farey is immortal because he failed to understand a theorem which Haros had proved perfectly fourteen years before; the names of five worthy Norwegians still stand in Abel's Life, just for one act of conscientious imbecility, dutifully performed at the expense of their country's greatest man." [I haven't the faintest idea what that refers to.]

Getting back to heart of the matter: "THERE are then two mathematics. There is the real mathematics of the real mathematicians, and there is what I will call the `trivial' mathematics, for want of a better word. The trivial mathematics may be justified by arguments which would appeal to Hogben, or other writers of his school, but there is no such defence for the real mathematics, which must be justified as art if it can be justified at all."

And here's one last awkwardly timed prediction: "There is one comforting conclusion which is easy for a real mathematician. Real mathematics has no effects on war. No one has yet discovered any warlike purpose to be served by the theory of numbers or relativity, and it seems very unlikely that anyone will do so for many years."