Rota's Mathematical World

Indiscrete Thoughts
By Gian-Carlo Rota
Forewords by Reuben Hersh and Robert Sokolowski
Edited and with an epilogue by Fabrizio Palombi
296 pages
Birkhäuser, Boston

We few, we happy few, we band of brothers. --- King Henry V

Now here is a book you will want to read. Here is a book that will grip you, amuse you, excite you, enrage you, confuse you, perplex you. You will applaud the author; you will want to throw bricks at him. Here is a book whose sentences will be clipped and quoted and anthologized for many years to come.

Two souls, or possibly three, reside in Gian-Carlo Rota's breast. Rota is a professor at MIT, both of mathematics and of philosophy. In mathematics, his contributions to combinatorics and probability theory are well known. In philosophy, he is a phenomenologist --- that is, he is of the school of Edmund Husserl and Martin Heidegger. Phenomenology is a little known and still less understand 20th-century phenomenon. Rota's third soul is the one that gives rise to an underlay (often camouflaged) of skepticism and despair.

For a long time, I have thought that if I were asked to elect a Mr. Mathematics USA (in the sense in which Senator Robert Taft was Mr. Republican), I would vote for Rota. His knowledge of mathematics, its application, its history is profound and extensive. His acquaintances in the world mathematical establishment are vast. His interests go far beyond proving theorems or creating new structures. His opinions are sharp and often surprising. His prose is scintillating and often brutal. In a centrist age, these last considerations would cost him the election. What we have here is an edited collection of, for the most part, previously published pieces, split into three parts: Persons and Places, Philosophy: A Minority View, and Readings and Comments. For the purposes of this review, I will group the first and third sections as easy bedtime reading. The second part is difficult to plough through; on a reading and a rereading, its meaning does not emerge readily. Of course, as with any philosophy, it's skippable. You would then be left with the pleasures of parts one and three, but you would have misses the whole man.

Persons and Places contains both biography and autobiography. Rota was an undergraduate at Princeton and a graduate student at Yale, and his experiences there have yielded portraits of Alonzo Church, William Feller, Emil Artin, Solomon Lefschetz, and Jack Schwartz, along with descriptions of how the young Rota interacted with them. His later close association with Stanislaw Ulam (mainly at the Los Alamos National Laboratory) has given rise to an extensive picture of this rare and often intuitive mathematician.

Rota opens his introduction to the book with the words The truth offends. Indeed it does -- quite often -- and Rota, who would not be my nominee for the Diplomatic Corps, seems to be a compulsive truth teller. We learn from Palombi's epilogue that Rota has lost valued friendships thereby.

Nonetheless, it may be pointed out that this tendency is quite in the biographical spirit of our iconoclastic age, which bugs bedrooms routinely, hauls down the statues of Lenin and Stalin, and reduces the statures of Washington and Jefferson. We live in a period of wholesale and grandiose character degradation, done often in the name of absolute truth, political correctness, or psychological purification. Rota writes so well that one has to forgive him (as he does for W.V.O. Quine, page 255) for being Rota.

Persons and Places also includes historic sketches of the development of linear operators, and of combinatorial and invariant theory. Rota views these fields from the top of the mathematical Alps. In these pieces, as in the biographies, he gives us a severely elitist view of mathematics as it has played out on grandiose themes. He has played out on grandiose themes. He has a strong inclination to assign or imply the assignment of cosmic grades to individuals and to fields. Elsewhere he asserts the opposite: Supermarket mathematics is also mathematics.

The view of mathematicians that emerges from Rota's book is not that of the middle-aged, childlike, cloud-nine geniuses of cliché. It is rather that of the deeply competitive, jealous, math macho, sometimes close to the loony bin, organized into discrete and largely non-intercommunicating bands of possibly happy brothers who identify joy with manic highs, and who, as they batten on each others' work, combine inspiration with nitpicking and have come up with as remarkable a corpus of intellectual material as exists under the sun.

The Reading and Comments section contains, among other things, Ten Lessons I Wish I Had Been Taught, A Mathematician's Gossip, and a selection of book reviews. The gossip is not so much gossip in the usual sense (although there's plenty of that in the first part) as a collection of aphorisms. As examples:

Mainstream mathematics is a name given to mathematics that more fittingly belongs on Sunset Boulevard.

Graph theory, like lattice theory, is the whipping boy of mathematicians in need of concealing their feelings of insecurity.

When I know the mathematics and the people he's talking about, I often agree with him (as, for example, in the case of his long and approvingb review of the Schaum Outline Series). His theorem on geriatric mathematics (last paragraph, page 203) is funny, true, and hits the nail right on the thumb.

But then, I run up against things like Pattern recognition is big business today. Too bad that none of the self-styled specialists in the subject ---- let us charitably admit that it is a subject ---- know mathematics..." To which I say, paralleling Bogie in Casablanca, that Rota has been misinformed.

I come now to the hardest part of my job: to write about Philosophy: A Minority View. Again, where I understand an individual paragraph or example, I often agree with Rota (indeed, I myself have been called a closet phenomenologist).

I agree that the notion of mathematical evidence is prior to and is rather more important than that of proof. I agree that the revelation of the human face of mathematics is important both philosophically and socially. These are important elements of Rota's phenomenology.

Where Rota asserts that beauty in mathematics is to be identified with enlightenment, I diverge. I have often found the beauty in an unexplained mystery to be diminished with enlightenment or trivialized by an explanation. Then, again, I have found, within my own experience, that there is no final enlightenment on any topic.

A further element in phenomenology, according to Palombi (a philosopher and a former student of Rota's who rakes his teacher over the coals in his epilogue) is the desire for intense study and possibly the formalization of the three words as, already, and beyond. What does formalization mean? Something like the mathematization of the if, then that occurs in formal logic, or something vastly different?

According to Sokolowski (a brother phenomenologist), "Rota calls for a widened mathematics that will incorporate such phenomena as evidence and Fundierung, as well as anticipation, identification, concealment, surprise and other forms of presentation that operate in our experience and thinking..."

Fundierung? What is the significance of this Germanism that looms large in Rota's philosophy? An example given in the book (page 105) is that of the computer scientist who is led by thought experiments... to a fundamental realization: a computer program's identity is related to the hardware by a peculiar relation philosophers call Fundierung. (The term is Husserl's.)

Is Rota calling for a formalization of Fundierung, for the creation of a mathematical object about which theorems can be proved?

Perhaps, as Reuben Hersh suggests in his foreword, you'll make fasterheadway with Gian-Carlo than with other writers on the subject. Light did break through for me occasionally, illuminating individual details. But I do not grasp the larger, deeper picture. I have the sense that what Rota is after is limited by his neglect of mathematics as it operates in our wider, everyday society, as it affects every one of us through chipified technology, economics, law--things that the public doesn't usually identity with mathematics. I suspect that the human face of Rota's mathematical world is his own face and not that of the larger community.

Here, in sum, is the soul of post-World War II mathematics filtered through the soul of one of its most distinguished representatives. In presenting his truths to the public, boldly and brashly, the author may have fulfilled his own maxim (page 199) that mathematicians are more likely to be remembered by their expository work.

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


Volume 44/Number 7, April 1997