Read This! The MAA Online book review column The Search for Mathematical Roots, 1870-1940 Logics, Set Theories and the Foundations of Mathematics from Cantor through Russell to Gödel by I. Grattan-Guinness Reviewed by Marvin Schaefer

As in any good story, it is necessary for the reader to be aware of the setting. In this case, Grattan-Guinness must start from the 1790s in the France of Napoléon, Condillac, Condorcet, Lagrange and Cauchy, whence he moves forward rapidly to recap the contributions throughout Europe of Babbage, Herschel, Boole, De Morgan, Bolzano, Dodgson (Lewis Carroll), Kant and others. What a prelude that is! Syllogistic logic was not yet rigorously developed, there was still an imprecision in understanding the distinction between necessary and sufficient conditions, confusion over the logical basis for the rigorous application of instantiation and generalization, and the hypotheses of theorems were not yet being reduced to their bare bones. I was intrigued to read the many starts and false-starts that were occurring over this period.

Once the stage has been set, the book follows its subtitle by moving through the shaping Mengenlehre of Cantor, the paradoxes of the infinite and infinitessimal, the paradoxes and Principia of Russell and Whitehead, and on to Gödel's incompleteness theory. Grattan-Guinness does so by examining the details and critiques of an incredible cast of participants: here we find the contributions of Venn, Heine, Borel, Dedekind, Peano (and the Péanists!), the Peirces, Schröder, Hilbert, Weistrauss, Poincaré, Wittgenstein, Ramsey, Brouwer, Carnap, Quine, Zermelo, Burkamp, Schlipp, Skolem, Veblen, and others.

This book is chock-full of little facts and factoids: One learns much more than one would have suspected about the rôles of infinitessimals, measure theory and orders of infinity in the development of modern understanding of the real number system. In 1903, in his response to a letter from Frege, Hilbert revealed that Zermelo had anticipated the Russell Paradox by at least three years! In 1890, we learn that Bettazzi and Peano anticipated the axioms of choice. Grattan-Guinness treats us to a heartrending biography of Charles Sanders Santiago Peirce (1839-1914), a Joe Bfstplk-like Harvard philosopher and son of Benjamin Peirce. Peirce's valuable work, which sought to unify and expand the logics and algebras of Boole and De Morgan, was nearly lost until its "rediscovery" in 1960s by Arthur Burks' UCLA dissertation research. Peirce's obscurity was largely due to active maltreatment, often by relatives and by opposition to its publication by none other than his department at Harvard University. Grattan-Guinness tells us that Peirce "died Hollywood style without the music, on a cold April day without a stick of firewood in the box or scrap of food in the larder."

The Search for Mathematical Roots comes with a remarkable 75-page bibliography which cites published works, articles, letters, and notes. The book ends with a very helpful summary chart that shows the interconnections and interdependencies between mathematical and algebraic logics, set theory and formalisms. This bibliography is very heavily referenced in this book, there being several citations in nearly every paragraph in the book's eleven chapters. This is both a strength and a weakness of this book. On the one hand, it makes the text a tremendous asset to the scholar who wants to study the evolution of a particular concept or trend in mathematical logic. On the other hand, reading this book can become a tedious ordeal because of the author's disciplined use of citations, footnotes, and contemporaneous notational conventions.

For example, a short table of notation is presented in the introduction and is frequently followed. But Grattan-Guinness shows the development of theory historically and often needs to show exemplary formulations in their original notation. Necessarily, this results in a significant overloading of symbols — indeed, the treatment of Frege's and Peano's works shows these men using symbols that were overused with as many as five different meanings in a given work, depending on context to make their use "clear". Often, the symbols were used with yet different meanings in later works by the same author. Frege is quoted as saying "The comfort of typesetters is not yet the highest of possessions."

This book's major blemish is its scholarship. It is not an easy book to read. The author reserves the use of page numbers for the referenced works [e.g., De Morgan (1862a, 307) refers to page 307 of the ath 1862 publication of De Morgan] using an equation and expressing numbering scheme for cross-referencing within the book itself [e.g., (255.3) is used to indicate the third equation in §2.5.5]. As an example of Grattan-Guinness' style, here is part of a randomly-chosen paragraph from page 67 of the chapter on Cauchy's contributions:

In Britain De Morgan produced a large textbook on The differential and integral calculus. In a Cauchyan spirit he began with an outline of the theory of limits and gave versions of (272.1-2) as basic definitions; but he made no mention of Cauchy in these places (De Morgan 1842a, 1-34, 47-58 (where he even used Euler's name 'differential coefficient' for the derivative!) and 99-105). He even devoted some later sections to topics consistent with his philosophy of algebra (§2.4.0) but which Cauchy did not tolerate, such as pp. 328-340 on Arbogast's calculus of 'derivations' (an extension of Lagrange's approach to the calculus which influenced Servois in §2.2.5), and ch. 19 or 'divergent developments' of infinite series....

So here we have a marvelous resource that covers a particularly fecund period in mathematical history. Large and detailed as it is, the text can only touch on the major themes and outline the details. The interested reader will need to have access to a good university library in order to follow up on the mathematical details. This book can serve as a supplementary reference for a graduate foundations course and as a needed resource for serious research pursuits.

Publication Data: The Search for Mathematical Roots 1870-1940, by I. Grattan-Guinness. Princeton University Press, 2001. Softcover, 624pp, $45.00. ISBN 0-691-05858-X.

Marvin Schaefer (bwapast@erols.com) is a computer security expert and was chief scientist at the National Computer Security Center at the NSA, and at Arca Systems. He has been a member of the MAA for 39 years and now operates an antiquarian book store called Books With a Past.

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