Renowned mathematician Gian-Carlo Rota's book *Indiscrete Thoughts* is not about mathematics. It is about mathematicians, the way they think, and the world in which they live. It is 260 pages of Rota calling it like he sees it, sandwiched in between two forwards, an introduction by the author himself, and an epilogue by the editor. Fabrizio Palombi begins his epiloque with the statement, "As the editor of this book, I feel it is my duty to remark on a number of glaring issues in the preceding text." Rota begins his introduction with the observation, "The truth offends." You are certainly given fair warning about what to expect from this book.

The book consists of some twenty essays plus an additional chapter which gathers together various book reviews that Rota has scribed. Almost everything has appeared elsewhere, and each chapter can be read apart from the others. The book is divided into three sections--*Persons and Places*, *Philosophy: A Minority View*, and *Readings and Comments*--which loosely tie the various essays together.

*Persons and Places* is about mathematicians Rota has known, both personally and through his reading of history. His stated goal in these essays is to debunk the myth that "[e]very scientist must also be a good guy." The first two chapters recount the mathematicians Rota met as an undergraduate at Princeton and as a graduate student at Yale. His assessments of the Princeton mathematicians are scathing. Of logician Alonzo Church he writes, "his behavior must be classified as strange, even by mathematicians' standards." Church's lectures--verbatim repetitions from the manuscript of his well-known text--began with a ten-minute ritual of cleaning the blackboard until it was spotless and another ten minutes spent (in silence) waiting for the blackboard to dry. Topologist Solomon Lefschetz "despised mathematicians who spent their time giving rigorous or elegant proofs for arguments which he considered obvious." His textbook on topology "hardly contains one completely correct proof." Rota's description of the Yale mathematics department in the early fifties is a bit brighter. Much of the story centers around his advisor Jack Schwartz (whom he regards very highly) and the writing of Dunford and Schwartz' three-volume, 2592-page tome *Linear Operators*, known to every graduate student in mathematics since that time. The anecdotes about this book (such as an explanation of why not a single semicolon appears in the work) make the second chapter well worth reading.

Rota does his best biographical work when he is describing Stanislaw Ulam, whom he knew for some twenty-one years. Three chapters of the book are devoted to his recollections of Ulam. In "The Barrier of Meaning" Rota recounts in detail a single conversation the two had on a wintery day in Santa Fe. "Stan Ulam" is a brief obituary. "The Lost Cafe" is a lengthy description of Ulam's life as a mathematician, from his days at the now-famous Scottish Cafe in Lwow, Poland (where "he saw in a flash the truth of the Borsuk-Ulam theorem") to his time at Los Alamos Laboratory. Rota's portrayals of Ulam's relationship with John von Neumann (first as von Neumann's assistant at the Institute for Advanced Studies and then as his assistant in the Manhattan Project) and his competition with Edward Teller to design the hydrogren bomb (Ulam won) are particularly revealing. Alas, though Rota's affection and respect for Ulam are clear, he still has not pulled his punches. Ulam was "proverbially lazy" and "lived beyond his means." In the end, Palombi informs us that Ulam's widow has seen to it that Rota has been excluded from every meeting held in Ulam's memory.

There are countless other anecdotes scattered throughout the pages of this book, most wholly unknown to many students of mathematics and most giving a no-holds-barred assessment of the people involved. It is the sort of celebrity gossip that people are drawn to, and no doubt many of these stories will eventually become part of the mathematical lore passed down from generation to generation of mathematicians.

While Rota is known best for his mathematics, he has also published in philosophy. (His brief description--in "Ten Lessons I Wish I Had Been Taught"--of his experiences publishing in philosophy will certainly confirm the stereotypes of his mathematical readers.) *Philosophy: A Minority View* is devoted to his interest in phenomenology, a philosophical viewpoint characterized by its "refusal to yield to materialist or idealist reduction." (This is, of course, a tremendous mistatement of the goals of phenomenology, but these "reductionist" oversimplifications are the bugbears that Rota is fighting throughout his philosophical work.) Novices in this field of philosophy would be well-advised to begin with "The Primacy of Identity", where Rota presents a simple example of his phenomenological approach. Wisened by this example, the mathematician-reader can move on to the most alluring titles in this section: "The Phenomenology of Mathematical Truth", "The Phenomenology of Mathematical Beauty", and "The Phenomenology of Mathematical Proof". It is here that Rota most successfully turns his phenomenological lenses on the undertakings of mathematicians.

Take "The Phenomenology of Mathematical Proof". Every mathematician knows (or thinks he knows) what constitutes a mathematical proof: a linear sequence of deductions from a given axiomatic system using universally-acknowledged rules of inference. A quick glance through any serious mathematical research journal will confirm that this is the universal discourse of mathematicians. But Rota sets out to analyze the hidden sense of the notion of mathematical proof--to find out what mathematicians really mean when they think about a mathematical proof. The first stop is the role of verification in proof. No doubt a lengthy verification of all possible cases (such as Haken and Appel's proof of the four-color conjecture) is a valid proof, but it is unsatisfying because it does not automatically yield *the reason* for the truthfulness of the theorem. What then makes one proof more valuable than another? Rota settles on the notion of possibility. For instance, Wiles' proof of Fermat's last theorem is valuable not for what it proves, "but in what it has *opened up*, in what it will make possible." This reveals the mathematician's task:

A realistic look at the development of mathematics shows that the *reasons* for a theorem are found only after digging deep and focusing on the possibilities of the theorem. The discovery of hidden reasons is the work of the mathematician. Once such reasons are found, the choice of particular formal statements to express them is secondary.

As Rota notes, the implications of formally enriching our notion of mathematical proof are troubling. Mathematics prides itself on being able to understand even the action of proving theorems from a mathematical perspective. This is the upshot of modern meta-mathematics. However, the price paid for this rigorous understanding of mathematics is that everything except the axiomatic method of deduction is dismissed as secondary and unimportant. Formalizing the notion of possibility (and other fundamental notions) in the the definition of proof would be an undertaking that simply cannot be accomplished with our present logic.

In *Readings and Comments*, the final section of the book, Rota assumes the role of elder statesman of mathematics and imparts his collected wisdom with all of the authority due to an accomplished research mathematician. In "Ten Lessons I Wish I had Been Taught", Rota pokes fun at his stature in the mathematics community, advising his listeners on everything from blackboard technique ("It is particularly important to erase those distracting whirls that are left when we run the eraser over the blackboard in a non uniform fashion"), to how to cope with old age ("You must realize that, after reaching a certain age, you are no longer viewed as a person. You become an instution, and you are treated the way institutions are treated.") Even the book reviews reprinted in this volume have something to offer: Here he reveals the mystery of why *Schaum's Outlines in Mathematics and Statistics*--widely regarded as "inferior products" by mathematicians--have such universal appeal.

The highlight of this section is "A Mathematician's Gossip", where Rota brings together a collection of his proverbs and prophecies about the mathematical community. These are undoubtedly jottings that Rota has made over the years, but never had the opportunity to flesh out. Some favorites:

Mathematicians have to attend (secretly) physics meetings in order to find out what is going on in their fields. Physicists have the P.R, the savoir-faire, and the chutzpah to write readable, or at least legible accounts of subjects that are not yet obsolete, something few mathematicians would dare to do, fearing expulsion from the A.M.S.

It is bad luck to title a book "Volume One".

Textbooks in algebraic geometry should be written by Italians and corrected by Germans.

Hypergeometric functions are one of the paradises of nineteenth-century mathematis that remain unknown to mathematicians of our day. Hypergeometric functions of several variables are an even better paradise. They will soon crop up in just about everything.

Readers are bound to find his observations amusing, if not insightful.

Gian-Carlo Rota has written the sort of book that few mathematicians could write. What will appeal immediately to anyone with an interest in research mathematics are the stories he tells about the practice of modern mathematics. But eminent research mathematicians who can command the attention of the mathematical community are not such rare birds, and every one of these mathematicians is capable of writing a book of anecdotes about other well-known mathematicians. What sets this book apart is Rota's philosophy and the realistic view of mathematics that he derives from this philosophy. Against the convenient image of mathematics continually moving upward based on the power of the axiomatic method, Rota again and again reveals a world where mathematicians are enmeshed in subtle social and psychological forces that they are only beginning to understand.

Why should mathematicians care? The mathematical community does not exist in isolation. In fact, its existence must be continually justified to those unfamiliar with its hidden ways. If we have such a limited understanding of what mathematics is, how can we expect others to see its worth? Rota hints at this problem himself:

The mistaken identification of mathematics with the axiomatic method has led to a widespread prejudice among scientists that mathematics is nothing but a pendantic grammar, suitable only for belaboring the obvious and for producing marginal counterexamples to useful facts that are by and large true.

Rota's collection of essays covers many distinct topics, but perhaps there is one important overall lesson that we can learn from his book: As mathematicians we must begin to understand more deeply what we do and learn to communicate that understanding to the world at large.

Andrew Leahy is Assistant Professor of Mathematics at Knox College.

## Comments

## Fernando Q. Gouvêa

This is indeed an excellent book that mathematicians should read. Rota is always insightful and provocative, worth listening to even when one is fairly sure he is wrong. I'm not quite sure what to make of the "epilogue" by the editor; it is weird, and probably best ignored.

The collection of vatic sayings in the section called "A Mathematician's Gossip" is fascinating. Someone should write a commentary on these, trying to figure out what Rota was getting at, explaining the mathematics he is assuming we know, and assessing whether he was right!

Ten years or so on, this is still a book that can be strongly recommended.

Tolle lege!