History Squared

January 13, 2004

Book Review
Philip J. Davis

Writing the History of Mathematics: Its Historical Development. Edited by Joseph W. Dauben and Christoph J. Scriba, Birkhäuser, Basel, 2002, 689 + xxxvii pages, $89.95 (paperback).

I first learned that mathematics had a history when I was in high school. Our geometry book had a historical appendix, where I read about Aryabhata of Kusumapura (c. 490) and other geometric worthies. I didn't learn that algebra had a history until a few years later. Our algebra book had no such appendix.

Over the decades, I've read and enjoyed many books on the history of our subject. Years ago, I even wrote a historical piece, "A Historical Profile of the Gamma Function" (American Mathematical Monthly, Vol. 66, 1959), for which I plumbed the stacks of the Library of Congress in search of first-hand information. I've sounded off in print (in the introduction to Mathematical People, edited by Albers and Alexanderson, 1985) as to why the history of mathematics is important and how it ought to be written. It may have been these accomplishments that, about twenty-five years ago, got me roped into the advisory board for the Springer-Verlag series History of Mathematics and Physical Science. Reading a variety of histories to be reasonably informed for that job, I noticed that the "take" or the slant differs from one history to another. I once asked a math historian whether there was a substantial history of the histories of mathematics. The answer was a firm "No."

There is now, and it is a landmark work. The book grew out of preliminary meetings at Oberwolfach in the early '90s, after which

"Joseph Dauben and Christoph Scriba agreed to accept editorial responsibility for a full-scale study of the history of the history of mathematics, a project undertaken with the support of the International Commission on the History of Mathematics."

The project also had the support of several other international organizations.

Such an enterprise, more than ten years in the making, required the interest
and cooperation of many people.

Among them were a working group of more than a dozen scholars from as many countries, an additional 32 experts who provided individual contributions, and editorial and publication assistants. The complexity of this job was truly mind-boggling.

First, a word about the relevance of histories of mathematics. Eric Temple Bell (1883-1960), a professor of mathematics at the California Institute of Technology, brilliant mathematician, historian of mathematics (if slightly maverick), and fiction writer, is quoted in the book under review (from the introduction to his very substantial Development of Mathematics):

"Without the devoted labors of these scholars [i.e., historians of mathematics], mathematicians would know next to nothing, and perhaps care less, about the first faltering steps of their science. Indeed, an eminent French analyst of the twentieth century declared that neither he nor any but one or two of his fellow professionals had the slightest interest in the history of mathematics as conceived by historians. He amplified his statement by observing that the only history of mathematics that means anything to a mathematician is the thousands of technical papers cramming the journals."

In my experience, the situation today is exactly as the unnamed analyst wrote perhaps three-quarters of a century ago. The editors of Writing are rather more sanguine about the current impact of the history of mathematics. They cite courses taught, and an abundance of historical publications of all sorts. I hope they are right.

The earliest history of mathematics appears to be by Proclus (c. 410-485), in his Commentary on Euclid. Among major early histories one must surely cite Histoire des Mathématiques (1758) of Jean-Etienne Montucla (1725-1799), Vorlesun-gen über Geschichte der Mathematik (1880-1908) of Moritz Cantor (who lived from 1829 to 1920, and was not related to Georg Cantor of set theory fame), Mémoires Scientifiques (1912-1950) of Paul Tannery.

Among the 20th-century general histories written in English, I keep handy as reference works those of Carl Boyer, Morris Kline, and Ivor Grattan-Guinness. Numerous fairly recent histories have been devoted to special mathematical topics. Examples include Detlef Laugwitz's Bernhard Riemann: Turning Points in the Conception of Mathematics (1999), Jesper Lützen's The Prehistory of the Theory of Distributions (1982), Andrew I. Dale's A History of Inverse Probability (1991), Scriba and Schreiber's 5000 Jahre Geometrie.

Despite Bell's (and my own) gloomy assessment of the regard in which professional mathematicians hold the history of their subject and of the slight financial support for such work, books of this sort have appeared steadily over the centuries. In recent years, we have had numerous biographies of the great and numerous special-topic books, targeted both for professionals and for the educated laity.

And let's not forget the plays with mathematical/scientific content that have hit Broadway and have won kudos. I admit that these are signs of increasing interest.

Now to Writing the History itself. I should stress again that this is not a history of mathematics. It is a history of such books and of those men and women who have written such books. (One contributor and correspondent of mine jocularly called the genre "history squared.") Close to two dozen scholars have written up the history of the histories that were published in their respective countries. The citations run from earliest times to essentially the present. To this have been prefaced and postscripted two articles describing the intent of the book and of the current position occupied by the history of mathematics in the world. Together, this material occupies about half the text.

There follow biographies of three hundred historians of mathematics and relevant bibliographic material. Some of these biographies are short: a paragraph or so. Others (e.g., those of Baldassarre Boncompagni, Bourbaki, Gino Loria, Paul Tannery, Bartel van der Waerden, Hieronymus Zeuthen) go on for pages. Many of these bios contain fascinating information that one does not easily come by; for me, this lifted the whole production out of the "telephone book" category. The bios occupy about a third of the text. The remainder of the book is filled with indices, additional bibliography, and photographs of twenty-five historians, of which I had previously seen only three.

***

A good dictionary will give three different meanings for the word "historiography." The first is simply the writing of history; the second is the corpus of historical writings. The third meaning---the principles or the methods of historical writing, or the slant of particular works---is the one emphasized by the editors of the book under review:

"History is what happened in the past; historiography is the analysis of 'history' as a discipline, an account of its assumptions, and the different approaches to which it has been subjected in the hands of different historians writing in different places in different times under varying constraints including (but not limited to) economics, politics, philosophy, religion, and even health and psychological states of mind."

I assume that in their charge to the contributors, the editors asked them particularly to stress the historiographic (in the above sense) aspect of the individual works they considered. But asking is one thing and getting is another; contributors usually write what they want to write, and editors can go hang. I know this from personal experience with books I've edited and others to which I've contributed. I grant that it is not always easy to extract the "take" of a history book that was written fifty or a hundred years ago. If the present compilation has a weakness, it lies in the fact that the historiographic mission is only occasionally and weakly fulfilled. Yet things can be gleaned from what there is.

A few samples (written by the respective biographers):

"The more substantial [historical notes of Bourbaki (20th century)] trace ideas and questions back to ancient times and follow their developments sometimes up to the twentieth century. All accounts are teleological in that the mathematics of recent years mould them; they are retrospective."

"It was [Joseph] Needham's (1900-1995) belief that one of the major reasons why modern science failed to develop in [ancient] China was due to the fact that there was no 'vivifying demand from the side of the natural sciences' for mathematics, and that it was a mercantile rather than an agrarian and bureaucratic civilization that was the necessary prerequisite for mathematized, experimental science."

Quido Vetter (1881-1960) "always emphasized the importance of understanding mathematics as a general component of cultural history."

"In the 1930's, together with S.A. Yanovskaya, M.Y. Vygodskii (1898-1965) undertook the creation of a specifically Marxist history of mathematics." This included "the [social] class nature of mathematics."

Hieronymous Zeuthen (1839-1920) "repeatedly underlined that one cannot evaluate or understand mathematics of an earlier period on the basis of the mathematics of today. On the contrary, he thought it indispensable to be acquainted with the techniques and symbolism of former times in order to contrast those tools and what they could be used for with what they had actually been used for."

What a job it must have been for the editors to get so many contributors to write their pieces and to pull them all together. I doff my hat. I hope that the International Commission on the History of Mathematics will find a way to keep this huge labor of love updated. My jocular correspondent would then be in a position to talk about a "History Cubed."

Philip J. Davis, professor emeritus of applied mathematics at Brown University, is an independent writer, scholar, and lecturer. He lives in Providence, Rhode
Island, and can be reached at philip_davis@brown.edu.


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