Four Thousand---Or Possibly Thirty-seven Thousand---Years of Mathematics

November 21, 2000

Book Review
Philip J. Davis

The Norton History of the Mathematical Sciences: The Rainbow of Mathematics. By Ivor Grattan-Guinness, W.W. Norton, New York, 1998, 832 pages (hardcover), $35.95. (Paperback version: The Rainbow of Mathematics: A History of the Mathematical Sciences, By Ivor Grattan-Guinness, W.W. Norton, New York, 2000, 832 pages (softcover) , $19.95.)

In the process of reviewing a book, I generally look for a "platform" from which I can take off and present my own views, which don't necessarily overlap with what the author has had to say. Confronted now with a history of mathematics---designated in this review as "The Rainbow"---I find my mind teeming with so many possible platforms that I hardly know where to begin. Shall I compare The Rainbow with other histories? Shall I ask how one person is able to write about material that was developed over four millennia? Shall I say what I would have put in had I written the book? Shall I ask who the potential readers of general histories of mathematics are? Well, let's see what comes out of my word processor.

Ivor Grattan-Guinness is a distinguished historian of mathematics and science whose professional stamina is commensurate with his vast knowledge. In The Rainbow he has covered mathematics from its recorded beginnings to the end of World War I, with a few pages covering the post-World War I period. The "thirty-seven thousand years" in the title of this review is a reference to some Upper Paleolithic notches on bones that have been interpreted as recording the phases of the moon.

Allow me to conjecture why it is appropriate to make a break at World War I. In the early '60s, Alexander Ostrowski, a world-class mathematician who as a young man had done some editorial work on Felix Klein's voluminous Enzyklopaedie der Mathematischen Wissenschaften (1898-1935), asked me whether I would work with him toward the production of an updated version. I was flattered, of course, but after thinking it over said Thanks, but no thanks. I simply wilted when I confronted the incredible amount of significant mathematics that had been created after 1920. I am glad, however, that hardier souls than myself took up the challenge, for I am an enthusiastic reader of history. (As close in coverage and in depth to the spirit of the Enzyklopaedie as one could hope for is the ongoing Encyclopedia of Mathematics and Its Applications. Published initially by Addison Wesley and since 1976 by Cambridge University Press, it was edited until recently by the late Gian-Carlo Rota; more than 75 volumes have appeared to date.)

The end of World War I is probably the last time that any one individual could know in a substantial way essentially all the mathematics then around. Ostrowski told me so at the time of his proposal. More recently, I came across a confirmation in the Bulletin of the American Mathematical Society (Vol. 27, 1920); the Swiss-American Florian Cajori, commenting on Vorlesungen über der Geschichte de Mathematik (1880-1908), the earlier and still famous and still readable history of Moritz Cantor (no relation to Georg Cantor of set theory), wrote:

"A history of the desired size and accuracy can be secured only by the cooperative effort of many specialists."

Thus, for one person to write a comprehensive history of mathematics from its emergence to the present---to achieve the unified spirit, point of view, sense of importance that come from thorough understanding; to achieve the continuity of expression that distinguishes one author from another---is now an impossibility. Still, one needs shorter, more compact, occasionally impressionistic treatments. I take my hat off to Grattan-Guinness for his ability to write with zest and knowledge about what went on during 4000 years. I would guess that he was helped considerably by having at his fingertips the Companion Encyclopedia of the History and Philosophy of Mathematics (Routledge, 1994; two volumes, 1806 pages, 139 individual authors) that he himself edited. The Companion is an impressive and valuable work, and many sections of The Rainbow refer to it.

Did Grattan-Guinness have an agenda in writing The Rainbow? (Feminist, Marxist, sociology-oriented, anti-Eurocentric? Mathematics as the Science of the Infinite or as the Science of Structures? I use the word "agenda" in the following sense: I see things clearly, while you have an agenda.)

I think not. He occasionally wonders, as I often have, why mathematicians' interest in history is not

"deep, for their subject has shown long continuous concerns; after all, all the ancient roots are still with us. However, mathematicians often treat history with contempt (unsullied by any practice or even knowledge of it, of course)."

He suggests an explanation:

"One main reason lies in this continuity itself; since our predecessors had similar concerns to ours, let us rewrite their work as if we had done it. Such a re-reading can lead to perfectly respectable mathematics, but of course history itself is quite mutilated; time-warp takes over with cause and effect changing roles."

This leads him to a rainbow metaphor:

"Like the rainbow, mathematics may be admired, but---especially among intellectuals---it must be kept at a distance, away from real life and polite conversation."

Grattan-Guinness has set himself the goal of displaying the full range of mathematical concerns and accomplishments (another meaning of "rainbow"), during which display he intends to---I paraphrase his explicit statement---stress the 19th century; stress physical applications; distinguish different kinds of applications; stress probability and statistics; downplay a bit the Great Man approach; describe the mathematics as it was at the time, and not as the forerunner of what it came to be; describe changes in historical interpretation; stress national differences.

On the whole, his judgment of what constitutes the main tendencies of mathematics over the long ages is fairly standard, but new interpretations are presented when available. There are many references to specialized studies undertaken in the last generation.

What we have in The Rainbow is essentially "internal" history---the history of the theories considered grand and deep by practitioners of the discipline. The text follows fairly strictly a cross-the-board chronological order, rather than taking an individual idea, say the real number system, and describing its development from prehistory to the present, more or less as an isolated creation. Euclid, Archimedes, Newton, Galileo; Euler, Gauss, Fourier, Riemann, Poincaré, Hilbert, to name just a few of the Greats, are acknowledged as such and given more space in "production numbers" or in "set pieces." On the other hand, the Greats are not overemphasized and their treatment is balanced and modified by shorter discussions of the work of hundreds of others, occasionally underrated, often ignored, whose work serves to complete, bind, simplify, elaborate, anticipate, and not infrequently, to suggest new directions. Today's historian, writing about the past, cannot free him/herself entirely of the knowledge of what came later. The temptation to peek at the final pages of a great detective story or a theory is overwhelming. Thus,

"In wondering about the second focus of the parabola, he [Kepler] placed it infinitely far away along the axis, and approachable from either direction---in other (later) words, he introduced a double point of infinity."

There are many, frequently illuminating instances of this sort of looking ahead. By pointing to the deep roots of the contemporary, history makes us modest. The text contains a sufficiency of chronological tables, which are dreary in themselves but useful for reference or for giving bird's-eye views. Routine but necessary writeups are interlarded and spiced up with numerous personal selections not found in other histories. The reader will find Charles S. Peirce's Quincuncial Map of the World and Florence Nightingale's pie chart of morbidity data. To my great pleasure, mention is made of the Swedish astronomer Carl Sundman, about whose work on the three-body problem I wrote a minor thesis, and of Lippman Lipkin, who disproved his teacher Pafnuty Tschebyscheff's conjecture about linkages and whom I featured in my jeu d'esprit The Thread.

Spice---an integral part of the author's literary style---also occurs within gutsy opinions:

"A snobbish preference for pure mathematics over applications became more marked [after 1870]. . . . Purism became an explicit creed for the professional."

"The peak of Edgeworth's career was his appointment in 1891 to a chair at Oxford University, where economic theory surpassed even mathematics in dullness."

"Temperature and tempers over the axiom [of choice] gradually cooled, with a variety of attitudes being adopted; no general compromise was reached."

"Neglect [of Frege] during his lifetime is normally explained as due to his unusual notation . . . but I learnt it in 20 minutes and I cannot see why his contemporaries would have needed longer."

Much more than most general histories, The Rainbow is interested in mathematical historiography, i.e., the history of mathematical histories, journals, encyclopedias, publications, societies. National differences among the British, French, and German schools come across well. Philosophy, on the other hand, is given short shrift. In these different ways, then, The Rainbow Man has displayed his colours.

So who is The Rainbow for? As a steady reader of mathematical histories, I enjoyed it and learned much that I hadn't known. For example, I hadn't realized that Euclid treated incommensurable ratios via a kind of continued fraction expansion known as anthy-phairesis.

A reader needs---especially for the post-1650 material---a firm knowledge of the relevant mathematics in its current form. One cannot learn mathematics from histories, general or otherwise. One gets only a whiff of what's going on. The book is for the mathematically adept-knowledge at the level of an undergraduate major or better is required.

So as not to end on the limitations imposed by narrow specialization, I point out something suggested to me by a diagram in the book called "Philomaths." It derives from the 18th century and delineates the wide variety of applications of mathematics, including cartography, accountancy, assurance, navigation, music, optical instrument making, then in practice. Histories of mathematics traditionally go for the deepest theories and applications, as judged by the standards of the research community. Yet, in our highly mathematized civilization, the average person is subjected to and is affected by mathematics that may be utterly trivial by those standards. Political, social, and economic applications of mathematics are now multiplying like Fibonacci's rabbits. Many of the resulting systems are installed simply by fiat.

I have in mind such things as the U.S. Constitution, which is chock full of numbers, e.g., the 2/3 of the Senate required to overthrow a presidential veto. (Why 2/3? Why not 3/4?) Population counts. University admissions by percentages. Gambling systems (e.g., betting on horses via bookmakers as opposed to pari-mutuel.) Determination of national and state boundaries. The IQ. Standard of living indices. Money. The income tax. Product identification via zebra stripes.

The list can be extended indefinitely. Many of the items date back to pre-World War I days, and even the most trivial of the mathematical operations, say counting or symbol identification, can end up invoking---sometimes creating---deep theories. And now we have seen that final judgment on what mathematics has produced is possibly left to courts of law, e.g., the U.S. Census and DNA identifications.

I should like to see a history of mathematics that admits into the canon of applied mathematics not just the latest string theory, but all those everyday mathematical trivialities that have made, consciously or unconsciously, philo-or miso-maths of most people.

Philip J. Davis, professor emeritus of applied mathematics at Brown University, is an independent writer, scholar, and lecturer. He lives in Providence, Rhode Island.

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