Subject: categories: Sammy

From: Peter Freyd <[email-masked]>
Date: Wed, 10 Jun 1998 17:49:10 -0400 (EDT)

The Editor of the AMS Notices, Tony Knapp, has kindly given me
  permission to post my part of Sammy's memorial collection. I trust
  that it's clear that it shouldn't be forwarded to any other list.
  For historical reasons (I guess) I've restored a number of
  paragraphs that were deleted from the final version -- they are
  marked with a | running down the left margin. And I've added a few
  footnotes. Let me take the opportunity to mention that Tony's
  editorial services were greatly appreciated, in particular, he
  caught a number of places where my original ms would have been much
  misunderstood.

  Thirty years ago I found myself a neighbor of Arthur Upham Pope, the
master of ancient Persian art. He had retired in his 90s to an estate
in the center of the city of Shiraz in southern Iran where I lived --
briefly -- across the street. I found an excuse for what has to be
called an audience and I mentioned that I was a friend of Samuel
Eilenberg.

  ``I don't know him.'' he said, ``I know _of_ him, of course. How do
you know him?''

  ``We work in the same area of mathematics.''

  ``You're talking about a different Eilenberg. I meant the dealer in
Indian art.''

  ``Actually, it's the same person. He's both a mathematician and a
collector of Indian art.''

  ``Don't be silly, young man. The Eilenberg I mean is not a
_collector_ of Indian art, he's the _dealer_ in Indian art. I know him
well. He established the historicity of one of the Persian kings.[1]
He certainly is not a mathematician.''

  End of audience.

                               *  *  *

  In later years even Arthur Upham Pope would have known. Eilenberg
became universally known as ``Professor'' in the art world; indeed, if
one walked with him in London or Zurich or even Philadelphia and one
heard ``Professor!'' it was always Eilenberg who was being hailed and it
was always the art world hailing him.

  If you heard ``Sammy!'', you knew it was a mathematician. 

                               *  *  *

  It was complicated, explaining that name. If you were my age and
knew him first through his works, it was hard to conceive of him as
``Sammy''. And when you met him for the first time, it was even
harder. You already knew that he was in charge of entire fields of
mathematics -- indeed, he had created a number of them -- and when you
met him you knew that he was in charge of the room you were in and it
didn't matter whose room it was. Sammy? The name didn't fit.

  But he had to have a name like Sammy. I said it was hard to explain.
Here was one of the most aggressive people you would ever meet. He'd
challenge almost anything. If you mentioned something about the
weather he'd challenge you -- once in California I heard him insist
that it wasn't weather; it was climate. But somehow it was almost
always clear: you'd could challenge him right back. Aggressive and
challenging, but not at all pompous. You can't be pompous with a name
like Sammy.[2]

                               *  *  *

| Once a gang of us spent an evening in a bar in San Antonio, Texas,
| on the occasion of an annual AMS meeting. Most of us, except for
| Sammy, ranged in age from very young to thirtysomething. As usual,
| it was a series of pitched battles between Sammy and the rest of us,
| and before the evening was out, the bartender became part of the
| crowd. The next day I told him that Sammy was a world-famous
| mathematician and when he wouldn't believe it I asked him to go to
| anyone in his bar who looked like a mathematician and ask about
| Samuel Eilenberg. He still wouldn't believe it. I don't know if he
| would have believed the part about Indian art.

                               *  *  *

  Sammy kept his two worlds, mathematics and art, at something of a
distance. But both worlds seemed to agree on one thing, the very one
that Arthur Upham Pope insisted upon. Sammy was the dealer. But the
two worlds only seemed to agree on this.

  Without question, Sammy loved playing the role of dealer. In the
days when mathematicians were in demand and jobs were easy to come by,
Sammy loved to tell about the math market he was going to create. The
trade would be in mathematician futures: ``This one's only done two
lemmas and one proposition in the last year; the most recent theorem
was two years ago; better sell this one at a loss.'' With his big
cigar (expensive) and his big gold ring (in fact a valuable Indian
artifact) he could enter his dealer mode at a moment's notice and one
always wondered just how many young mathematicians' careers were in
his hands.

  But his two worlds, mathematics and art, perceived this role quite
differently. In mathematics we understood that it was a role he loved
playing, but that he was only playing. His being as a mathematician
was what counted and he would have been the same mathematician whether
or not he played the dealer, indeed, whether or not he played -- and
he did -- a high-stakes poker. This was not so clear in his other
world.

|   In 1981 Sammy was briefly my guest at St John's College, Cambridge,
| where I was, for the year, an Overseas Fellow. In due course, he was
| given a tour of the Asian collection at the Fitzwilliam Museum by 
| the appropriate functionary, and the two of them landed at high
| table for dinner. There seems to be a stock of stories about
| mathematicians not being appreciated at high tables, the best known
| being Heinz Hopf's (unwitting) success in convincing his fellow
| diners that he taught at a technical high school in Zurich. The
| Fitzwilliam functionary was treating Sammy as some sort of rich
| dealer who apparently taught mathematics on the side. Sammy went
| along. But then the doyen of the Cambridge philosophy community let
| it be known that he viewed Sammy as one of the two leading
| mathematicians in the world. Next came an American from Missouri
| who realized that Sammy was the same Eilenberg for whom a part of
| the local museum was named. It went on from there.[3]

                               *  *  *

  It was usually frustrating trying to explain to others how Sammy was
perceived by his fellow mathematicians. Sammy had an unprintable way
of saying that mathematics required both intelligence and
aggression.[4] But imagine not knowing how his mathematics -- when he
had finished -- would totally belie that aggression. Imagine not
knowing how remarkably well-behaved his mathematics always was.
Imagine not knowing how his mathematics -- when he had finished --
always seemed pre-ordained and how it seemed no more aggressive than,
say, the sun rising at its appointed sunrise time.

  Forty years ago Sammy hoped to do the same for Indian bronzes. He
had already acquired the reputation of being the best detector of fakes
in the business and he believed he could axiomatize the process. He
even had a provisional list of axioms and it was truly an elegant
list.

  A few years later we found ourselves at a small French-style bistro
in La Jolla, California. We had been out of touch: there had been an
argument about mathematical ethics and somehow we had resolved it; the
dinner was something of a celebration of the resolution. I asked him
about his book on bronzes.

  ``The axioms failed.''

  ``What does that mean?''

  ``It means that I've been taken. I bought a fake.''

  He had suspected it only after it had been in his bedroom for a few
weeks. He had the pleasure, at least, of investigating until he found
out the master faker and he had the pleasure of going to the master
faker's studio, not to berate hem but to congratulate him.

                               *  *  *

  After that, Sammy made a point of not building bridges between his
two worlds. I recall just one exception. He moved from a conversation
about sculpture to one about mathematics. Sculptors, he had said,
learn early to create from the inside out: what finally is to be seen
on the surface is the result of a lot of work in conceptualizing the
interior. But there are a few others: there are those for whom the
interior is the result of a lot of work on getting the surface
right. ``And,'' Sammy asked, ``isn't that the case for my
mathematics?''

  Style was only one part of his mathematics -- as, of course, he knew
-- but there are, indeed, wonderful stories about Sammy, by attending
only to what seemed the most superficial of stylistic choices,
succeeding in restructuring an entire subject on the spot.

  Many have witnessed this triumph of style over substance,
particularly with students. But the the most dramatic example had a
stellar cast.  D.C.Spencer gave a colloquium at Columbia in the Spring
of '62, and Sammy decided it was time to demonstrate his
get-rid-of-subscripts rule: ``If you define it right, you won't need a
subscript''. Spencer, with the greatest of charm -- it was for good
reason that he was already affectionately known as ``Uncle Don'' --
followed Sammy's orders and proceeded to restructure his subject while
standing there at the board. One by one, the subscripts disappeared,
each disappearance preceded by a Sammy-dictated redefinition. He had
virtually no idea of the intended meanings of any of the symbols. he
was operating entirely on the surface, looking only at the shape of
the syntax.

  The process went on for several minutes, until Sammy took on the one
proposition on the board. ``So now what does that say?''

  ``Sammy, I don't know. You're the one making all the definitions.''

  So Sammy applied his definitions and one by one the subscripts
continued to disappear, until finally the proposition itself
disappeared: it became the assertion that a thing was equal -- behold
-- to itself.

                               *  *  *

|   When he received an honorary doctorate in 1985, the University of
| Pennsylvania cited him as ``our greatest mathematical stylist''.[5]
| 
|                              *  *  *
| 
| On that occasion, Sammy was just a little put out that one of his
| fellow honorees seemed to be there for entirely political reasons.
| At the commencement eve banquet he got into one of his better moods
| and told the university president, ``When you choose your honorary
| degrees correctly it is the university that is honored; when you
| choose incorrectly it is they who are honored."  The next day the
| president -- not catching Sammy's subtext (fortunately, I guess) --
| incorporated these "inspiring words of Professor Eilenberg's'' into
| his commencement address. It wasn't easy to keep a straight face.

                               *  *  *

  ``My mother's father had the town brewery and he had one child, a
daughter. He went to the head of the town yeshiva and asked for the
best student.'' Sammy told me one day. ``So my future father became a
brewer instead of a rabbi.''

   Sammy regarded pre-war Poland with some affection. He felt that he
had been well nurtured by the Polish community of mathematicians and
he told me of his pleasure on being received by Stefan Banach,
himself, a process of being welcomed to the holy of holies, the cafe
in which Banach spent his time during the annual Polish mathematical
conferences. By the time he came to the U.S. in his mid 20s Sammy was
a well-known topologist.

  When I questioned him on his attitude about pre-war Poland, he
answered that one must ``watch the derivative'': don't judge just by how
good things are, but by how fast they're becoming better.[6]

  Sammy's view of Poland since the war was more complicated. It was
particularly complicated by what he viewed as its treatment of category
theory as a fringe subject.

                               *  *  *

  In the late 1950s, Sammy began to concentrate his mathematical
activities -- both research and dealing -- on category theory. He and
Mac Lane had invented the subject, but to them, it was always an
applied subject, not an end in itself. Categories were defined in
order to define functors, which, in turn, were defined in order to
define natural transformations, which were defined, finally, in order
to prove theorems that could not be proved before. In this view,
category theory belonged in the mainstream of mathematics.

  There was another view, the categories-as-fringe view. It said that
categories were defined in order to _state_ theorems that could not be
stated before, that they were not tools but objects of nature worthy
of study in their own right. Sammy believed this counter-view was a
direct interference with his role as the chief dealer for category
theory. He had watched many of his inventions become standard
mathematics -- singular homology, obstruction theory, homological
algebra -- and he had no intention of leaving the future of category
theory to others.

  Today the language of category theory has permeated a good part of
mathematics and is treated with some respect. It was not ever
so. There were years before the words "category" and "functor" could
be pronounced unapologetically in mixed mathematical company. One of
my fonder memories comes from sitting next to Sammy in the early 60s
when Frank Adams gave his first lectures on how every functor on
finite dimensional vector spaces gives rise to a natural
transformation on the K-functor. Frank used that construction to
obtain what are now called the Adams operations and he used those to
count how many independent vector fields there could be on a sphere.
It was not until then that it become permissible to say "functor"
without a little snort.

  In those years, Sammy was a one-man employment agency for a fresh
generation of mathematicians who viewed categories not just as a
language but as potentially central mathematical subject. For the
next 35 years he went to just about every category theory conference,
and much more important, he used his masterly expository skills to
convey categorical ideas to other mathematicians. Sammy's efforts
succeeded for the language of category theory and he never gave up
with his efforts for the theory itself. He was confident that the
categorical view would eventually be the standard mathematical view,
with or without his salesmanship. Its inevitability would be based on
the theorems whose proofs required it. That was obvious to Sammy. He
wanted to make it obvious to everyone else.

[1] Sammy published a paper showing that some gold coins he had were
    produced during the king's reign.

[2] In the final version the use of the 2nd person in the last three
    paragraphs was replaced with something more standard, as were the
    contractions.

[3] The dinner was actually at Kings, the functionary was the relevant
    curator, the philosopher was R.B.Braithwaite. I wasn't there --
    Sammy told me about it the next day and with great relish.

[4] What was found to be unprintable was already a bowdlerization: 

      Sammy liked to say that when you did mathematics you were using
      not just your brains but your guts. (Well, all right, it wasn't
      guts, but another plural body part far from the brain; I like to
      believe that Sammy would, in time, have agreed to my
      translation.)

    Dick Kadison tells me that in the very first utterance was in
    reply to a lunch-time question put by the physicist, Polykarp
    Kusch, "Sammy, what is it you use to do mathematics?" And in that
    very first utterance (as opposed to the way Sammy enjoyed saying
    it thereafter) the body-part in question was not plural but quite
    singular.

[5] This sentence will appear in Hy Bass's opening narrative vita for
    the Notices' memorial collection.

[6] A paragraph I used in an earlier draft as a bridge into the
    category material:

      Thus, in 1961 Sammy was one of the first American scientists to
      journey to Russia. A.G.Kurosch reciprocated with a visit to the
      U.S. and because the two visits ended up being at the same time,
      the two did not meet. Pity. We wondered would have time happened
      if Sammy had heard Kurosch's opening words at his colloquium
      lecture: ``We in Russia believe that category theory is destined
      to be as important as lattice theory.'' As it was, no one
      relished the task of repeating those words to Sammy.  Actually,
      now that I think about it, everybody looked forward to seeing
      Sammy's reaction when the words were repeated.