The Bunges on categories
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. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
participants (1)
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Peter Freyd