Indiscrete Thoughts. By Gian-Carlo
Rota, forewords by Reuben
Robert Sokolowski, edited and with an epilogue by Fabrizio Palombi. Birkhäuser, Boston, 1997, 296 pages, $36.50.
"We few, we happy few, we band of
brothers."¡ª¡ªKing 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
breast. Rota is a
professor at MIT, both of mathematics and of philosophy. In
his contributions to combinatorics and probability theory are
well known. In
philosophy, he is a phenomenologist¡ª¡ªthat is, he is of the
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
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
application, its history is profound and extensive. His
acquaintances in the
world mathematical establishment are vast. His interests go
proving theorems or creating new structures. His opinions are
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
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
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,
experiences there have yielded portraits of Alonzo Church,
William Feller, Emil Artin, Solomon Lefschetz, and Jack Schwartz, along with
of how the young Rota interacted with them. His later close
Stanislaw Ulam (mainly at the Los Alamos National Laboratory)
rise to an extensive picture of this rare and often intuitive
Rota opens his introduction to the book with the words
"The truth offends." Indeed it does¡ªquite often¡ªand Rota, who would not be my
the Diplomatic Corps, seems to be a compulsive truth teller.
We learn from
Palombi's epilogue that Rota has lost valued friendships
Nonetheless, it may be pointed out that this tendency is quite
biographical spirit of our iconoclastic age, which bugs
hauls down the statues of Lenin and Stalin, and reduces the
Washington and Jefferson. We live in a period of wholesale and
character degradation, done often in the name of absolute
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
has a strong inclination to assign or imply the assignment of
to individuals and to fields. Elsewhere he asserts the
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 clich¨¦. It is
rather that of the
deeply competitive, jealous, math macho, sometimes close to
the loony bin, organized into discrete and largely non-intercommunicating
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
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
need of concealing their feelings of insecurity.
When I know the mathematics and the people he's talking about,
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
I come now to the hardest part of my job: to write about
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
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
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
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 AM188000@brownvm.brown.edu.