Professor W.W. Sawyer  




I first read this book about 50 years ago. I was forcibly struck by a passage in the opening paragraph;-

There is no scorn more profound, or on the whole more justifiable, than that of the men who make for the men who explain. Exposition, criticism, appreciation, is work for second-rate minds. [1.]

At the time I thought "He seems to have lost concentration here; let's hope it will be better later on". Since then I have met some mathematicians who agree with this view of Hardy's, and I am now convinced that this statement was not a temporary feeling, caused by Hardy's distress that illness had deprived him of the power to do the research in which he delighted, but - as is clear from statements that come a little later - was an opinion he had held firmly throughout.

That Hardy was a very great mathematician is beyond question, and I do not intend to question it in any way. However, when any person eminent in some field makes statements outside that field , it is legitimate to consider the validity of these statements, as any teacher must certainly wish to do in relation to the passage quoted above.


Of his own preferences Hardy writes

I hate 'teaching'....I love lecturing, and have lectured a
great deal to extremely able classes.

Here lecturing means imparting mathematical knowledge to those able to understand it with little or no difficulty; teaching means giving time and effort to make it accessible to those who require asistance. There is nothing wrong with Hardy's preference in this matter. I knew two teachers at a school. One was a mild clergyman who taught Greek successfully to the Classical Sixth, the other a good footballer who kept order and conveyed some learning to the boisterous spirits of the First Form. Both were providing a service; neither could have done the other's job. Good administration consists in appreciating the merits of a wide variety of individuals and combining them into an effective team. Now it is precisely this appreciation that Hardy lacks. He makes the extraordinary statement

Most people can do nothing at all well. [3.]

One gets the impression that he regards you as doing well only if you are one of the ten best in the world at this particular activity. The theorem that very few people do anything well is an immediate logical consequence.

However in life we continually depend on the co-operation of men and women far below this exacting standard. Even the publication of mathematical research depends on publishers, editors, printers, postmen and others, who need to be conscientious and competent, but rarely need to have truly exceptional qualities.

Indeed this is true even of the social process that links the great mathematicians of one generation to those of the next. There may of course be direct contact, as when Riemann was a student at Gottingen University under Gauss. But the fact that Gauss was able to reach university at all was due to two teachers, Buttner, who recognized Gauss' extraordinary ability when Gauss was only 10 years old, and Bartels, who not only worked at mathematics with Gauss but also told influential citizens about him, as a result of which support was provided for Gauss' further education.[4.]

In science the importance of the expositor is perhaps as great as that of the discoverer. Mendel's work in genetics remained unknown for many years because there was no one to publicize it and fight for it as Huxley did for Darwin.

In the arts, critics play a similar role. Once, when I was a student, Leavis spoke to a college society on recent poetry. Prior to this meeting I had regarded modern poetry as pretentious nonsense, much as I still do for - not all, but most - modern art. He explained the allusions in some obscure poems and showed that these conveyed, very powerfully, intense emotion. This is what good criticism does - it makes available to a wider audience artistic work that would be meaningless without such intervention.

It is interesting that Hardy had such a poor opinion of teachers when he himself possessed an important quality of a good teacher - the ability to recognize a correct idea behind a very imperfect statement of it. Some writers deal very scornfully with the work of analysts before the age of rigour. Hardy however finds essentially correct, even very modern, ideas in the work of Euler, even though these are sometimes expressed in a way that would not find favour to-day. [5.]


In Hardy's scheme of thought, ambition plays an important role. Now ambition has two sides. There is the intrinsic aspect, the desire to solve some significant problem, and the personal reflection of this - the hope that by so doing you will become famous and be remembered, perhaps for centuries. [6.]

The second aspect is certainly important for Hardy. He makes this curiously objective division of mankind into minds that are first-class, second class and so on; the ambitious person is concerned about where he fits into this scheme. There is no part of this that should be accepted as sound advice. If there is something you think worth doing, that you are able to do, that you have the opportunity to do, and that you enjoy doing, wisdom lies in getting on with it, and not giving a second's thought to what ordinal number attaches to you in some system of intellectual snobbery. As for concern with the self, you are both happiest and most effective when you are so absorbed in what you are doing that for a while you forget the limited being that is actually performing it.


Hardy is very anxious to show that the value of mathematics lies in its beauty, not in its practical consequences. Real mathematics is that "which has permanent aesthetic value". [7.] On the other hand, "It is what is commonplace and dull that counts for practical life." [8.] In connection with 'real' mathematics he writes:-

I was not thinking only of pure mathematics. I count Maxwell
and Einstein, Eddington and Dirac among 'real' mathematicians.

It is not surprising to find Einstein, Eddington and Dirac classified as remote from practical consequences; when this was written, atomic theory had produced neither bombs nor medical applications. The surprising inclusion is that of Maxwell. When, after a delay of 25 years, his at first derided prediction of radio was confirmed by the physicists, this had immediate and important practical consequences - for example, aiding rescues at sea - and later profound social consequences, brought by the coming of wireless. The thesis that only dull mathematics has practical consequences can surely not be maintained.

Mathematical Beauty.

Of great interest are Hardy's discussion of mathematical beauty, and the characteristics of a good mathematical theorem. Essentially he says that a good theorem is simple, surprising and fruitful. He contrasts very effectively mathematical theorems "that connect many different mathematical ideas" with the special puzzles of mathematical recreations. [10.] He believes that nearly everyone is capable of appreciating mathematical beauty, and cites the popularity of mathematical puzzles in newspapers;

What the public wants is a little intellectual 'kick', and nothing else has quite the kick of mathematics. [11.] It is interesting that the term, 'kick', used here is often heard in relation to drugs, and indeed there can be a danger of addiction to mathematics, the temptation to indulge in mathematics and neglect important, but much less exciting, mundane activities.

Mathematical and Physical Reality.

What Hardy says on this question is extremely interesting
and significant;-

I believe that mathematical reality lies outside us, that
our function is to discover or observe it, and that the
theorems which we prove, and which we describe
grandiloquently as our 'creations' are simply our notes of
our observations.

The other reality is

The other reality is

…. the physical world itself, which also has its structure or
pattern....The geometer offers to the physicist a whole set
of maps from which to choose. One map, perhaps, will fit the facts better than others, and then the geometry that
provides that particular map will be the geometry most
important for applied mathematics. 

Anyone who has ever worked at mathematics must know this feeling, that we are exploring something that exists outside ourselves, something we cannot change at will. We speak of the discovery of the calculus rather than its invention. To do this we do not need to believe in any Platonist nonsense. I believe that Hardy's recognition of the two structures, one in the universe and one in the mind, is the key to explaining our impression of mathematical reality. It is the existence of structure in the universe, which our minds somehow absorb and reflect, that gives us the power to use the structured subject, mathematics, to produce 'maps' that sometimes fit physical realities with such extraordinary success.


"AMA" denotes G.H.Hardy, A Mathematician's Apology. C.U.P. Cambridge Paperbacks, 1993.

[1.] AMA p. 61.
[2.] AMA p. 149.
[3.] AMA p. 67.
[4.] E.T.Bell, Men of Mathematics. Chapter 14. Simon and Shuster. New York.
[5.] For instance G.H.Hardy, Divergent Series, p.15. O.U.P. 1949.
[6.] AMA pp.73,76,80,81,83.
[7.] AMA p.131.
[8.] AMA p.132
[9.] AMA p.131.
[10.] AMA pp.113,103-105.
[11.] AMA pp.87-88
[12.] AMA pp. 123-124.
[13.] AMA p. 127.

Version: 26th November 2020