Gian-Carlo Rota. Indiscrete Thoughts. Edited by Fabrizio Palombi. xxiv + 280 pp., index. Boston/Basel: Birkhäuser, 1997. $36.50.

In his collection of essays Indiscrete Thoughts Gian-Carlo Rota first offers us an engaging, sentimental, hyperbolic, idiosyncratic history of mathematicians in the 1950s at Princeton, Yale, and, to a lesser extent, MIT and Los Alamos. Second, he defends mathematics against philosophical reductionism, whether the source be the friends or foes of mathematics. Third, on the basis of both his history and his philosophy, he provides avuncular advice for young mathematicians. In combining these apparently disparate elements, Rota's book has evoked some criticism: Steven Kratz (Notices of the American Mathematical Society, 1997, 44:824-827) finds disunity here, Jeremy Gray (Mathematical Reviews, 97i:01026), inconsistency. I, however, find both unity and consistency. Rota's advice follows from his philosophy of phenomenology, which in turn determines the mathematicians he praises, namely, those who were more than mere technicians.

Rota preaches Nietzsche's sermon to mathematicians: only that which has a history has a meaning. Mathematics, he argues, is historical in an essential way (p. 100). Rota's own history is unapologetically idiosyncratic--indeed, he calls it "gossip" (p. 38). It would read like an oral history except that Rota is overly modest. He writes about his friends, fellow addicts of mathematics. But they are in some ways mirrors of himself. Like Stanislaw Ulam, of the "sparkling manners" (p. 69), he himself is playful, with a Cheshire-cat grin. Like Alan Tucker (p. 23), he will write a good recommendation for you for a second-rate school if that's where he thinks you belong. Others whom Rota remembers in more than a passing way are Alonzo Church, William Feller, Emil Artin, Solomon Lefschetz, Jack Schwartz, Josiah W. Gibbs, and Alfred Young.

Rota's history fits the proverb "Faithful are the wounds of a friend." Eulogizing Ulam, for example, Rota cites his broad mathematical contributions, ranging from logic to physics, but he earns the wrath of Ulam's wife by pointing out Ulam's personal foibles. Similarly, he lavishes just praise on Paul Halmos for his expository genius while criticizing "plunderers" (p. 74) of a paper by C. J. Everett and Ulam on algebraic logic--plunderers who may possibly include Halmos.

According to Rota, middle-European Victorian culture confused meaning with precision and as a result promoted an embarrassing idea of progress, no less in mathematics than in the culture at large. Yet he pines for his roots in just such a culture (p. 68), even with its snobbery (p. 238). Rota has caustic words for historians of mathematics who know no mathematics (p. 243) and for artificial intelligence researchers who know nothing about phenomenology (p. 233). Yet he overstates his case: today, a defense of phenomenology (Herbert Dreyfus, Why Computers Can't Think [New York: Harper & Row, 1972]) is required reading in undergraduate classes on artificial intelligence. Similarly, although Rota claims no "mathematical beauty appreciation" courses exist (p. 121), there are in fact many of them, including a renowned geometry-based course at his beloved Princeton. Only because he has such high standards and exaggerates deficiencies does Rota fail to recognize honest attempts of mathematical mentors to produce more than technicians. But insofar as the current generation of mathematicians do go beyond mere technique, Rota deserves credit for helping them to do so.

As for technicalities, the editor, Fabrizio Palombi, provides valuable references to obscure journals. Unfortunately, he does not indicate where the essays themselves were previously published. Further, the versions printed here are incomplete: for example, Chapter 2 has none of the pictures that originally appeared in the Mathematical Intelligencer, 1996, 18:44-51. All footnotes are missing from Chapter 12. This collection has a modest amount of redundancy: Rota's essay on Ulam appeared in a previous anthology, and in this volume two of the pedagogical essays overlap considerably.

GENE B. CHASE

Gene B. Chase is Professor of Mathematics and Computer Science at Messiah College, the author of Bibliography of Christianity and Mathematics: 1910-1983 (Dordt College Press, 1983), the editor of A Seventh Conference on Mathematics from a Christian Perspective (Wheaton College Press, 1990), and a former student of Gian-Carlo Rota.

This review is from Isis, 89, 2, June 1998: 358-359.