David Guest

Source: *The Labour Monthly*, Vol. 19, October 1937, No. 10, pp. 644-646, (1,263 words)

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*Men of Mathematics*

By Prof. E. T. Bell.

(Gollancz.)

12s. 6d.

This book sets out to be a story of the lives of mathematicians over the last 200 years, not a *history* of mathematics. But in spite of its author’s intentions it becomes in fact a history, and perhaps inevitably suggests the “great man” explanation of this history.

If we discount this bias there is much of tremendous interest in the story which unfolds. How few of the great mathematicians have lived up to the popular idea of the scientific recluse? Most of them were buffeted about in the social storms of their epoch, and some took an active part in these struggles. Even the absent-minded Newton, according to Bell,

Grew up with a fierce hatred of tyranny, subterfuge, and oppression, and when King James later sought to meddle repressively in University affairs, the mathematician and natural philosopher did not need to learn that resolution and a united front on the part of those whose liberties are endangered is the most effective defence against a coalition of unscrupulous politicians. (p.514.)

Of course not all mathematicians have been on the Left, although, according to Bell, there has been a tendency in this direction. Some have held the most reactionary views, both political and philosophical, which are typified in the galaxy of idealist-tinged *obiter dicta* which preface the book. As a contrast, how refreshing is the materialist spirit in the following declaration of Fourier, the great mathematician of the French Revolution.^{1}

The profound study of nature is the most fecund source of mathematical discoveries. Not only does this study, by offering a definite goal to research, have the advantage of excluding vague questions and futile calculations, but it is also a sure means of moulding analysis itself and discovering those elements in it which it is essential to know and which science ought always to conserve. These fundamental elements are those which recur in all natural phenomena. (p.596)

In the French Revolution we can clearly trace the liberating effect of a great political upheaval on human thought. This was the time, according to Hegel, when the world was stood upon its head, when “the reasoning intellect was applied to everything as the sole reason.” (Engels in *Anti-Dühring*.)

It was certainly a time of the most splendid efflorescence of French mathematics. Fourier and the Convention were responsible for an important reform in the teaching of French mathematics.

Remembering the deadly lectures of defunct professors, memorised and delivered in identical fury year after dreary year, the Convention called in *creators* of mathematics to do the teaching, and forbade them to lecture from any notes at all. The lectures were to be delivered standing (not sitting half asleep behind a desk), and were to be a free interchange of questions and explanations between the professor and his class. It was up to the lecturer to prevent a session from degenerating into a profitless debate. (p.223.)

If only some modern University departments would revive this method!

Perhaps it is not altogether fanciful to see in the England of the 17th century a like stimulating influence of revolutionary storms on creative mathematics. And something of the same sort may be found in the “Sturm-und-Drang” period of German history before 1848.

However this may be, such things have to be read between the lines in this book, for its author is not concerned with mathematics as “a mirror of civilisation” (to use Hogben’s phrase), but only as an activity of mathematicians. This is one of the fundamental limitations of the book, as already stated.

Another limitation comes from the very character of the advanced mathematics with which it mainly deals. While Hogben’s “Mathematics for the Million” is easily accessible to the general reader, and it is even possible to learn mathematics from it, the same cannot be said about this book. Critics have complained that Hogben’s book deals only with mathematics before 1750. That is probably one of its great merits.

For in spite of a disarming introduction in which Bell informs us that “the basic ideas of modern mathematics, from which the whole vast and intricate complexity has been woven by thousands of workers, are simple, of boundless scope, and well within the understanding of any human being with normal intelligence” (p.19), it must be stated that the general reader is not likely to see the *why* and *wherefore* of these ideas. There is a danger that the book will merely produce a state of “gaping wonderment,” during which the reader is too stunned by unfamiliar words to realise that he is not following the argument. If this “stunning” danger can be averted the book is likely to stimulate those who have time to take a real look at modern mathematics.

The book ends, appropriately enough, with a chapter on the “Crisis” in mathematics, and the struggle that is going on between different schools over the “foundations” of the subject to-day.

The row began in a fight over the theory of the Infinite which Cantor had pioneered, despite fierce attacks by the mathematicians of his day, to a recognised place in the late 19th century world of mathematics. Right from the start, cracks and crevices were observed in this structure. Fatal contradictions appeared which seemed to endanger the whole building. Mathematicians were divided into two camps. Those who regarded the theory of the Infinite as a “disease from which mathematics had to recover” (Henri Poincaré), and those who said with Hilbert: “No one shall expel us from the Paradise which Cantor has created for us.”

The latter view led to the attempt of the *formalist* school of Hilbert in Germany, and of the logistic school of A.N. Whitehead and Bertrand Russell in England, to build mathematics on a “solid foundation.” The most monumental achievement of this period was the tremendous *Principia Mathematica* of Whitehead and Russell, which appeared just before the World Crash of 1914.

In a sense, this work was the last great attempt to build classical mathematics on a completely firm and unassailable basis. For it must be realised that it was not only the newly-discovered Infinite Realms of Cantor that were at stake, the contradictions now revealed struck at the root of the whole of Classical Analysis, that is, of precisely that branch of mathematics which had been so essential in the development of Physical Science.

After the war attempts were made to “patch up” the *Principia Mathematica*, and to develop further the formalist basis of mathematics. But such “reformism” was doomed to failure. And soon we had the revolutionary challenge of the “Intuitionists,” Brouger and Weyl, demolishing whole stories of the mathematical building, and discarding venerable modes of reasoning. Finally they went to the root of the whole matter by assailing the *fundamental principle* of formal logic—the famous “law of the excluded middle”—perhaps best described as the rule of “either—or.”

But what is this but the spirit of dialectics breaking through the hard shell of formal logic, within which so much scientific thought has been imprisoned in the past! In the midst of much that is still under in detail, a Marxist cannot help feeling that here vast realms of thought await a dialectical understanding. But such things cannot be proved by dogmatic assertion. The only “proof” that will satisfy, the “proof” that is so badly needed by mathematics to-day, is the carrying through of the work of clearing up its foundations. To do this would be a tremendous achievement for the dialectical materialist approach to science.

DAVID GUEST

1. Not to be confused with Charles Fourier, the Utopian Socialist pioneer of the same period.