Mathematics as an art

The following is the draft of a lecture that Arthur Hinton Read delivered.

Mathematics as an art

I must begin with an apology for letting the terrifying word Mathematics loose among this company. But it is my hope that those of you who jib at discussing mathematics may be provoked into having their say on the theory of aesthetics.

In my talk I shall try to illuminate the claim that Mathematics has many of the essential features of an art. This claim has been repeatedly made, implicitly or explicitly, by the greatest of creative mathematicians. Newton, as we know, pictured himself as a boy playing on the seashore and diverting himself in now and then finding a smoother pebble or a prettier shell than ordinary, while the whole ocean of truth lay all undiscovered before him. Others have spoken more directly. Weierstrass, who is sometimes hailed as the father of modern mathematical analysis, once said:

A mathematician who is not at the same time something of a poet will never be a full mathematician.

Henri Poincaré, who in this age of specialisation was perhaps the last mathematician who could produce work of the first rank in practically every field of mathematics, wrote:

It may seem strange to see emotional sensibility invoked a propos of mathematical demonstration which it would seem can interest only the intellect. This would be to forget the feeling of mathematical beauty, of the harmony of numbers and forms, of geometrical elegance. This is a true aesthetic feeling that all true mathematicians know ...

Finally the great Cambridge mathematician G H Hardy has written:

The mathematician's patterns like the painter's or the poet's must be beautiful: the ideas like the colours or the words must fit together in a harmonious way. Beauty is the first test: there is no place in the world for ugly mathematics.

In the light of such remarks by such men one cannot deny the existence of aesthetic satisfactions on the part of the mathematician. But one may well ask: are these satisfactions on the part of the mathematician less those of a creative artist than of the beholder of a beautiful sunset. The mathematician, one might suggest, is able to appreciate beauty in patterns of abstract ideas - but is he creative in the sense that a painter or composer is creative? Indeed I can see that some may be inclined to expostulate: "How can the mathematician be creative when he is contained in a strait jacket of logical necessity?"

Clearly an artist, if he is to be an artist, must be free - free to choose between alternate forms of creation. And the first thing I want to insist on is that the mathematician, in spite of what you might thing, has an enormous degree of freedom. He is free in brief to choose between infinitely many logically correct trains of mathematical reasoning, and I believe the musical composer has no greater freedom than this. Actually the extent to which the mathematician is free was only slowly realised by mathematicians themselves: this makes an interesting chapter of mathematical history.

I am sure you have all had a solid grounding in Euclidean geometry at school. You will remember that at the outset Euclid sets out a system of postulates. These he regarded as self-evident propositions which he made no attempt to rove: but the subsequent theorems were supposed to be logically derived from these postulates without using any other unproved geometrical intuitions. A typical postulate is the one which says that through two given points there exists exactly one straight line. Now one of Euclid's postulates was always regarded as having a rather smaller degree of self-evidence than the others. This was the celebrated parallel postulate: it states that given a straight line and a point outside it, there exists one and only one straight line through the point and not meeting the given line. Mathematicians made repeated attempts to deduce this postulate from its fellows. But these attempts were uniformly unsuccessful. The most noteworthy attempt was that made by the Italian geometer Sacheri (already 2000 years after Euclid). His idea was to employ the celebrated reductio ad absurdum method. He started by assuming that the parallel postulate was false. In this way he expected to fall sooner or later upon a contradiction. But although he developed quite a long train of theorems based on his denial of the postulate, he was unable to detect any inconsistency in his work.

It was probably Gauss who first really envisaged the possibility that a perfectly self-consistent mathematics could be built upon the denial of the parallel postulate. But so engrained was the belief in the absolute necessity of the parallel postulate that even Gauss, the foremost mathematician of his time, was afraid to publish his ideas on the subject. It was left to the Russian Lobachevsky to construct the first Non-Euclidean geometry. He built a system of geometry in which he assumed that more than one parallel could be drawn through the given point. Indeed he did little more than Sacheri had done before him: the vital difference, which was to make so much difference to the development of mathematics, was while Sacheri still believed the parallel postulate to be an essential part of mathematics, Lobachevsky realised that the mathematician was perfectly free to change the postulate if a self-consistent mathematics resulted. We now know that if Euclidean geometry is self-consistent then so in Lobachevskyan geometry. The idea that axioms are at the disposition of the mathematician, and do not depend on anything in the physical world, has now spread throughout mathematics.

I suppose I have now reached the point at which I shall become controversial. I am going to suggest that the fundamental activity of every artist is choice. Out of an infinite number of alternatives the artist chooses what his aesthetic judgment leads him to regard as beautiful. In some cases the situation is complicated by the presence of certain types of physical dexterity, but even the painter is choosing where to put his paint and choosing how to mix his colours. My contention will perhaps be clearer in the case of the composer or writer where no form of muscular skill is involved, and I can lean upon the writing of the poet Paul Valery who said: It takes two to invent anything. The one makes up the combinations; the other chooses, recognises what he wishes and what is important to him in the mass of things which the former has imparted to him. What we call genius is much less the work of the first one than the readiness of the second one to grasp the value of what has been laid before him and to choose it.

I have emphasised already that the mathematician enjoys a very great freedom of choice, and I have suggested by quotation from some first rank mathematicians that there are aesthetic criteria by which he makes his choice. These quotations are certainly good evidence for my case, but I'm sure you would be much more convinced if I could present you with some mathematics whose aesthetic content you could appreciate. here is no little task. For you will not appreciate the elegance of a pattern of logical thought unless you can follow the logic, and you are not professional mathematicians. Fortunately this problem has been tackled by a mathematician more accomplished than I. G H Hardy in his book A Mathematician's Apology selected two examples of simple mathematics which he considered to have mathematical elegance.

Last Updated September 2023