Hardy’s Apology

G.H. Hardy A Mathematician’s Apology (with Foreword by C.P. Snow (1967)) Canto Edition 1992 (First Edition 1940)


[This is] the best account of what it was like to be a creative artist.

[The Cambridge] Mathematical Tripos … did not give any opportunity for the candidate to show mathematical imagination or insight or any quality that a creative mathematician needs. [This system] had effectively ruined serious mathematics in England for a hundred years.

[Hardy’s] purpose was to bring rigour into English mathematical analysis.


Any genuine mathematician must feel that it is not on these crude [practical ]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.


A mathematician, like a painter or a poet, is a maker of patterns. If his patterns are more permannet than theeirs, it is because they are made with ideas.


The ‘seriousness’ of a mathematical theorem  lies, not in its its practical consequences … but in the significance of the mathematical ideas which it connects … roughly … if it can be connected, in a natural and illuminating way, with a large complex of other mathematical ideas.


For example, Euclid’s proof of the existence of an infinity of prime numbers and …


Pythagoras’ proof of the irrationality of √2.


‘The certainty of mathematics’, says Whitehead [Science and the Modern World, p33.], ‘depends on its complete abstract generality.’

This sense of the word [generality] is important, and the logicians are quite right to stress it, since it embodies a truism which a good many people who ought to know better are apt to forget. It is quite common, for example, for an astronomer or a physicist to claim that he has found a ‘mathematical proof’ that the physical universe must behave in a particular way. All such claims, if interpreted literally, are strictly nonsense.

I can quote Whitehead … ‘it is the large generalization, limited by a happy particularity, which is the fruitful conception.’


   It seems that mathematical ideas are arranged somehow in strata, the ideas in each stratum being linked by a complex of relations both among themselves and with those above and below. The lower the stratum the deeper (and in general the more difficult) the idea.


If the theory of numbers could be employed for any practical and obviously honourable purpose … [but] mathematicians may be justified in rejoicing that there is one science at any rate, and that their own, whose very remoteness from ordinary human activities should keep it gentle and clean.


[Neither] physicists nor philosophers have ever given any convincing account of what ‘physical reality’ is, or of how the physicist passes , from the confused mass of fact or sensation with which he starts, to the construction of the objects which he calls ‘real’. Thus we cannot be said top know what the subject-matter of physics is; but this need not prevent us from understanding roughly what a physicists is trying to do. It is plain that he is trying to correlate the incoherent body of crude fact confronting him with some definite and orderly scheme of abstract relations, the kind of scheme which he can borrow only from mathematics.
A mathematician, on the other hand, is working with his own mathematical reality.


The great modern achievements of applied mathematics have been in relativity and quantum mechanics, and these subjects are, at present at any rate, almost as ‘useless’ as the theory of numbers. 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. Time may change this.


[Our] general conclusion must be that such mathematics is useful as is wanted by a superior engineer or a moderate physicist … .

One rather curious conclusion emerge, that pure mathematics is on the whole more distinctly useful than applied. … For what is useful above all is technique, and mathematical technique is taught mainly through pure mathematics.


Hogben (@Mathematics for the Million’) was a populariser of the notion that mathematics is ‘useful’.

[He] admittedly not a mathematician; he means by ‘mathematics’ the mathematics which he can understand, and which I have called ‘school’ mathematics. This mathematics has many uses … which we can call ‘social’ if we please … But he has hardly any understanding of ‘real’ mathematics … and still less sympathy with it. … ‘Real’ mathematics is to him merely an object of contemptuous pity.


There are then two mathematics … the real mathematics of the real mathematicians and … what I will call the ‘trivial’ mathematics, for want of a better word.

[The] trivial mathematics is, on the whole, useful, and … the real mathematics, on the whole is not … but we still have to ask whether either sort of mathematics does harm.

[There] are branches of applied mathematics, such as ballistics and aerodynamics, which have been developed deliberately for war and demand quite a deliberate techniques: it is perhaps hard to call them ‘trivial’, but none of them has any claim to rank as ‘real’. They are indeed repulsively ugly and intolerably dull … .

Dave Marsay


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