The Bunges on categories

The Bunges on categories

[Peter Freyd]

Mario Bunge has a great new book, "Between Two Worlds," with an
appendix by Marta and with a number of category comments. Hare are
some of those comments (indeed, all those I could find with the
help of the indexes).


p103-104

    [P]hilosophers of mathematics may know all about the paradoxes in
    set theory that caused so much worry a century ago, but they are
    unlikely to know that category theory replaced set theory as the
    foundation of mathematics half a century ago.

p160  Freyd

    [T]he rising mathematical star Peter Freyd, who would have a
    decisive influence on Marta's career -- and who perched on the best
    chair in the room [c1965], pontificated cleverly on all topics.

p175  Adelman, Bunge, Lambek (none of whom appears in the index)

    During that year [1966] we hosted...Marta's former classmate Murray
    [A]delman -- who had taken her place in the graduation ceremony at
    Penn -- and the German-Canadian mathematician Jim Lambek who offered
    Marta a postdoctoral position.

    Jim, whom we had met 2 years earlier at the Jerusalem congress, had
    been forced to flee from Germany to England at 17. Despite being
    Jewish, he was regarded as an enemy alien, and was locked in a
    Canadian prisoners' camp along with some hardened Nazis. While in
    the camp, he borrowed books from McGill University, where he went to
    study at the end of the war. He got his PhD in mathematics, and had
    a brilliant academic career. (To earn a few dollars he wrote for
    classmates term papers in philosophy, that got better grades than
    his own.) When Jim visited us in Freiburg he was taking his first
    steps in category theory, and had decided to set up at a Category
    Center at McGill, an aim that he fulfilled. Over a couple of decades
    Montreal was a world center in category theory.

p358-359 Eilenberg, Mac Lane (both names are in the index, but not for
this)

     ...Budapest, where Marta and I attended the colloquium of the
     Acad\'emie Internationale de Philosophie des Sciences on the
     foundations and philosophy of mathematics [1993]....The eminent
     mathematician Saunders Mac Lane argued that mathematics is the
     study of structures, and chided the logicians who did not keep up
     to date with the foundations of mathematics. The logicians in
     attendance, particularly Charles Parsons, felt offended when
     Saunders told them that they had remained in the time, more than
     half a century earlier, when set theory had become settled and
     G\"odel's proof was still a novelty. He informed them that the
     theories of categories and topos had replaced set theory as the
     foundation of mathematics. Saunders knew because he and Sammy
     Eilenberg had founded category theory half a century earlier and I
     knew because Marta had studied under two of Saunders's top
     students.

p399  Johnstone

    Professor Peter Johnstone, Marta's famous colleague, took us for
    supper at at his College, as well as for a stroll around its
    beautiful gardens. [2004] At the sight of a flock of loud Canadian
    geese busy fertilizing the centuries-old lawn, Peter exclaimed with
    patriotic zeal: "They have no right to be here!" The nerve of those
    illegal immigrants! [Since then it has become legal in England to
    kill Canada geese to protect lawns.]


And from Marta's appendix, a section labeled "CATEGORY THEORY"

p414-415 Eilenberg, Freyd, Lawvere (whose name does not appear in the
           index!), Mac Lane

    Until the year 1964, my intention to return to philosophy after what
    was meant to be a more mathematical incursion was still standing,
    that very year something made me change my mind. During the
    international Congress on logic, History and the Philosophy of
    Science in Jerusalem I would meet a person that would influence my
    almost as much as Mario Bunge in my future career. That person was
    F. William Lawvere, the most brilliant student at Columbia
    University of the famous mathematician Samuel Eilenberg ("Sammy" for
    the mathematicians). Lawvere, who had obtained his doctorate in 1963
    was one of the few mathematicians invited to give one-hour lectures
    at this congress. My interest in the theory of categories, founded
    by Sammy Eilenberg and Saunders MacLane in 1945 in order to better
    present and understand algebraic topology, and with which I was
    already acquainted through courses given by Peter Freyd at Penn,
    grew even more out of conversations with Bill Lawvere in Jerusalem.

    From both Freyd and Lawvere I had learned enough category theory to
    realize that, by making it my area of concentration, I would not
    have to abandon, if not philosophy, at least the foundations of
    mathematics. I had already become a doctoral student of Peter Freyd
    at Penn, but the opportunity to work under the direction of Bill
    Lawvere would luckily soon present itself. On the one hand Lawvere
    intended to spend a couple of years at the E.T.H. in Zurich,
    Switzerland, while on the other, Mario had been awarded a generous
    fellowship of the Humboldt Foundation to spend a year anywhere in
    Germany working on the foundations of physics. At my request, Mario
    chose Ferber's im Breast, as being the nearest to Zurich. These
    events led me to travel weekly by train to Switzerland in order to
    participate in the Benno Eckmann seminar at the Forshungsinstiitut
    f\"ur Mathematik of the E.T.H. and to have long discussions, with
    Lawvere on the subject of my thesis, which he had suggested. This
    alone shows the generosity and support that Mario had given me in my
    career.

    In Freiburg, Mario had interesting interactions with physicists and,
    equally important perhaps, he did not have to cross paths with
    Martin Heidegger, whom he despised for both his empty and enigmatic
    philosophy and for his Nazi affiliations. When Mario could not have
    imagined, however, was that choosing Freiburg, he would be making,
    in Lawvere, a formidable intellectual rival. Lawvere and (and still
    is) a deeply convinced Marxist, but at the same time a notably
    original mathematician without whose contributions the theory of
    categories would possibly have taken quite a different path that the
    one it actually did as an area independent from the algebraic
    topology that had inspired it, changing radically as well the way to
    view logic and algebra as well as functional analysis and
    differential geometry.

    In his mathematics, Lawvere employed a terminology taken from the
    dialectics that inspired him, but the mathematical concepts that he
    introduced stood on their own, and could be understood and accepted
    (or rejected) by anybody with any knowledge of or allegiance to
    Marxism. This at least is how it appeared to me. My fascination with
    his ideas and projects overtook all my previous interests. Mario,
    however, did not see it that way, and argued with me, and with
    Lawvere, owing principally to the Hegelian impression which his
    mathematics gave him. What Mario did not realize was the this aspect
    was negligible considering the amazingly clear and concise concepts
    that allowed Lawvere to advance mathematics and to become the
    unquestionable leader of tn entire generation of mathematicians, to
    which I was lucky to belong.

    ...

    My work in mathematics since my doctoral thesis consisted in
    developing aspects of the theory of categories as utilizing
    categories as a foundation for areas as varied as set theory, model
    theory. differential geometry and topology, theoretical computer
    science, algebraic topology, and functional analysis. I will not
    mention here my published work or the students whom I have formed
    because these are not relevant to a tribute to Mario but what I will
    say is that he helped me in various ways throwout my career as a
    mathematician. For his constant faith in me, I am deeply grateful.


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