Geoffrey Hellman

Geoffrey Hellman

What brought you to work or get interested in the Congress’ fields?

As a high school student in the 1950’s, my father gave me a ­collection of essays by Bertrand Russell (one of his ­intellectual heroes), which I read avidly. The most striking was his famous one, «What is Number?» in which he succinctly described ­Frege’s ­definition in the Grundlagen and went on to claim that Frege had really «solved the problem» of «What are the ­natural ­numbers?». This struck me as quite remarkable, yet puzzling. That wasn’t really like a mathematical problem with a definite right or wrong answer, was it? Do we really have to treat ­number-words as a­ctually ­referring to objects? That has always ­seemed to me an unnecessary, quasi-religious move, and I’ve never stopped ­wondering whether somehow it expressed a yearning for «something eternal, beyond this world» on the part of Russell, Goedel, and others inclined to make it.

At Harvard College, as an undergraduate, while I did get ­introduced to formal logic by Quine, I shied away from philosophy as too ­abstract, not closely enough connected with human affairs, and, in any case, dependent on a wealth of factual knowledge and experience which I had not yet acquired. Similarly with pure mathematics, which I didn’t pursue at the time beyond the calculus and some scientific applications. Instead I concentrated on music (I was already an accomplished classical pianist and performed solo and chamber recitals as well as the Mozart c minor concerto K 491 with the Bach Society Orchestra) and then on a pre-law curriculum. But just prior to entering Harvard Law School I took a course from Burton Dreben devoted to the Goedel incompleteness theorems, in which we read Goedel’s great 1931 paper, line by line. At that point, I realized my huge mistake in having enrolled in law school, but it was too late to start graduate school in Philosophy that year (1966), and, because of the Vietnam War, in which I refused to participate, I needed to remain enrolled in law school to avoid being drafted into the military. So I made the best of it, applying to graduate school in Philosophy, making the transition in the next year. (Yes, I passed my law school courses decently enough, just in case I should ever want to return.) At that point, the decision to «take the plunge» into philosophy seemed either courageous or foolhardy, but the path soon proved irreversible.

What has your participation to the Congress brought you?

This Nancy Congress provided a welcome opportunity to participate in a symposium devoted to a great topic, differing mathematical approaches to continuity, one to which I never thought I would be contributing anything new.  Thanks to Philip Ehrlich, who organized our symposium on this topic, I started pursuing in earnest some ideas I had earlier in the year about how to reconstruct classical continua without taking lower dimensional objects as primitive, i.e. the one-dimensional continuum without points, two-dimensional without points or lines, etc. Despite the fact that several precedents existed along such lines (Menger, Tarski, and others), I thought that the framework of atomless mereology and not been fully exploited. I had already discussed some of my ideas with Stewart Shapiro, who also was very interested in the topic and was planning to participate in the Nancy Congress, but had to withdraw due to a conflict with a prior commitment.

We had got far enough, however, so that I was able to ­present a talk on joint work in which we show how to derive the ­Archimedean principle from surprisingly elementary axioms, relying on no ­special induction principles beyond ordinary mathematical ­induction (applied to sequences of intervals). In a sense, we were showing that the classical real number system can be grounded in a theory that eschews points entirely, contrary to the common understanding of many who have pursued non-classical conceptions of continua such as smooth infinitesimal analysis. Our construction is «semi-Aristotelian»: lines are not «constituted of points»; however, the logic is fully classical, and we invoke «actual infinities», not merely «potential infinity», as a matter of course in our constructions (hence the «semi-»). The success of our symposium helped inspire us to pursue the program further, leading to a development of two-dimensional Euclidean geometry without points or lines, just now nearing completion. We anticipate that generalization from two to higher dimensions should be straightforward (although passing from one- to two-dimensions is not at all trivial, as the reader is welcome to experience for ­oneself by trying it). Stay tuned!

What is your opinion about your portrait?

The portrait is entirely different from any I’ve seen of myself. When Olivier Toussaint set me up with that pose, it felt quite unnatural. But in the portrait, I appear quite relaxed and the pose appears quite comfortable. I thank him for it.

 

Geoffrey Hellman - University of Minnesota