Good tale, weak maths

Number
March 10, 2006

" This is beyond doubt the most interesting book on the evolution of mathematics which has ever fallen into my hands " - Albert Einstein

Seeing this quote on the cover of Number made me wonder how many did fall into his hands: there were not many such books around in 1930, when Number was first published. This reprint of the fourth, 1953, edition, has many irritating features: the orotund style must already have seemed old-fashioned by 1953; more serious problems with the content I will detail below. Out of this 400-page tome one could probably extract a rather good book of maybe half the size.

The first chapter is not promising. Beginning with interesting observations about "number sense" in animals, it proceeds to speculate on the development of counting by humans. The evidence cited, on the number languages of "primitive tribes", looks, to me at least, both dubious and patronising. Still, if you read it quickly and do not take it too literally, you get a good idea of Dantzig's first theme: the tremendous intellectual leaps that must have taken place before numbers became a standard part of our mental repertoire.

The interdependence at a basic level between the distinct concepts of cardinal and ordinal numbers, and the related role of the one-to-one correspondence, is well conveyed, preparing the reader for later developments.

In subsequent chapters the development of ideas gets more focused. Zero appears, virtually at first, in the form of an empty column on Hindu counting-boards. With the development of algebra emerges the idea of using letters as symbols for arbitrary numbers; it opened the way to the extension of the number concept. This is Dantzig's main theme, and its ramifications are traced through the rest of the book. A key idea is what he calls "the principle of permanence": to the familiar natural numbers (1, 2, 3 and so on) we may adjoin "ideal" (or "imaginary") numbers, as long as they obey all the same rules (such as commutative addition). Each such extension took centuries to achieve, and arose out of a pressing need: negative numbers for solving equations a+x=b , rational numbers for equations a.x=b , and - later - complex numbers for solving polynomial equations. An arithmetical definition of the real numbers, the lengths of Euclidean line segments, was not achieved until the 19th century with the work of Cantor and Dedekind.

This is a long and intricate tale, told with gusto. Interwoven is another fascinating tale, the gradual taming of infinity. Along the way we are made familiar with a lot of important mathematical ideas, and learn about their often severe birth-pangs: mathematical induction, continuity, the arithmetisation of geometry, the development of calculus and a good selection of classic problems of number theory.

The appendices pick up on a few of the topics discussed informally in the text and show some hands-on mathematics. Some of the topics - the pigeonhole principle, Cantor's diagonal argument - are well chosen; some seem to have been picked rather randomly, while others verge on metaphysical ramblings.

The exposition of mathematical detail is sometimes sloppy. Dantzig's "principle of permanence", explicitly articulated to illustrate the use of precise definitions, is not used in a precise manner (for example, he applies it to impose an - impossible - ordering on the complex numbers).

The "principle" itself seems, to a modern mathematician, rather oddly chosen. Such details are easily corrected by an expert; but, as any teacher of maths knows, they have a discouraging effect on newcomers who are made to feel stupid because they cannot follow the (flawed) argument.

Another unfortunate feature is the use of certain words (compact, continuous) in a technical sense that differs from the usual one.

Similarly, many of the diagrams, while not wrong, are dauntingly intricate and are likely to intimidate not illuminate. A puzzled reader may find it worthwhile to consult the notes added at the end by the editor. These are helpful, but there should be more. However, my advice would be, don't worry about the detail, just enjoy the broad sweep.

There is an enthusiastic preface by Barry Mazur, one of the world's leading mathematicians. He writes: "Dantzig's book captures both soul and intellect; it is one of the few great popular expository classics of mathematics truly accessible to everyone."

I would call it an entertaining white elephant; idiosyncratic, rambling, sprinkled with laughably portentous asides ("Man is the measure of all things"). Behind the verbiage, though, there is an inspiring story that everyone interested in the development of thought should be exposed to. The editor has added a good list of further reading, and some of the books mentioned may be more palatable than Number to an impatient modern reader.

Dan Segal is senior research fellow, All Souls College, Oxford.

Number: The Language of Science

Author - Tobias Dantzig
Editor - Joseph Mazur
Publisher - Pi Press
Pages - 396
Price - £16.99
ISBN - 0 13 1856 8

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