A Mathematicians Apology

A Mathematicians Apology is the title of a book written by the great mathematician G H Hardy. Yes the same person who discovered our Ramanujan. Here he used the word apology in the meaning of defense, justification or explanation, not so much as regret or confession.  Here are some passages from the book for the benefit of my friends.


– – – “The beauty of a mathematical theorem depends a great deal on its seriousness, as even in poetry the beauty of a line may depend to some extent on the significance of the ideas which it contains. I quoted two lines of Shakespeare as an example of the sheer beauty of a verbal pattern, but


After life’s fitful fever he sleeps well


seems still more beautiful. The pattern is just as fine, and in this case the ideas have significance and the thesis is sound, so that our emotions are stirred much more deeply. The ideas do matter to the pattern, even in poetry, and much more, naturally, in mathematics; but I must not try to argue the question seriously.” – – –


– – – “It will be clear by now that, if we are to have any chance of making progress, I must produce example of ‘real’ mathematical theorems, theorems which every mathematician will admit to be first-rate. And here I am very handicapped by the restrictions under which I am writing. On the one hand my examples must be very simple, and intelligible to a reader who has no specialized mathematical knowledge; no elaborate preliminary explanations must be needs; and a reader must be able to follow the proofs as well as the enunciations. These conditions exclude, for instance, many of the most beautiful theorems of the theory of numbers, such as Fermat’s ‘two square’ theorem on the law of quadratic reciprocity. And on the other hand my examples should be drawn from the ‘pukka’ mathematics, the mathematics of the working professional mathematician; and this condition excludes a good deal which it would be comparatively easy to make intelligible but which trespasses on logic and mathematical philosophy.” – – –


– – – “Another famous and beautiful theorem is Fermat’s ‘two square’ theorem. The primes may (if we ignore the special prime 2) be arranged in two classes; the primes

5, 13, 17, 29, 37, 41,

which leave remainder 1 when divided by 4, and the primes

3, 7, 11, 19, 23, 31,

which leave remainder 3. All the primes of the first class, and none of the second, can be expressed as the sum of two integral squares: thus


5 = 12 + 22 , 13 = 22 +  32 ,

17 = 12 + 42 , 29 = 22 + 52 ;


but 3, 7, 11, and 19 are not expressible in this way (as the reader may check by trial). This is Fermat’s theorem, which is ranked, very justly, as one of the finest of arithmetic. Unfortunately, there is no proof within the comprehension of anybody but a fairly expert mathematician.” – – –


– – – “I wrote a great deal during the next ten years, but very little of any importance; there are not more than four or five papers which I can still remember with some satisfaction. The real crisis of my career came ten or twelve years later, in 1911, when I began my long collaboration with Littlewood, and in 1913, when I discovered Ramanujan. All my best work since then has been bound up with theirs, and it is obvious that my association with them was the decisive event of my life. 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 the thing you could never have done, and that is to have collaborated with both Littlewood and Ramanujan on something like equal terms.’ It is to them that I owe an unusually late maturity: I was at my best a little past forty, when I was a professor at Oxford. Since then I have suffered from that steady deterioration which is the common fate of elderly men and particularly of elderly mathematicians. A mathematician may still be competent enough at sixty, but it is useless to expect him to have original ideas.” – – –


Yes, Mr. Hardy discovered Ramanujan. But did he do enough to save him from his ailments? I am not so sure, even after reading two biographies on Ramanujan. Apart from obtaining an FRS, Ramanujan did not get anything substantial from the UK government. The fact remains, that if not for Hardy, Ramanujan might well have died as a clerk in Madras Port Trust.


(By the way, Fermat’s Two Square Theorem looks nice and simple. But why should its proof also be nice and simple? A subject for a future blog.)




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