All of my love,

Ryan E. Long

Having said all this about the tensions between us, I want also to emphasize how much I respected and
admired Ian always for his tough-minded adherence to his ideals in his life as a citizen, as a scholar, and as a member of our department. I should also mention that, however much he may have disapproved of my somewhat a priori style of doing history, I had only to hand him a draft of a paper on Plato or on Greek
mathematics to be assured of very substantial and valuable comments.

Some of the following I have from personal conversations with Ian, but a lot comes from a talk he gave to CFS (= Committee on the Conceptual Foundations of Science) in the 1970′s or early 1980′s|I can’t remember. Through that period, the committee was quite lively, with active participation from people from all of the divisions and semi-weekly colloquium meetings that were almost always worth attending. (That
committee died and another, CHSS, with quite a different agenda, is in its place.) Ian’s talk was one of a series by faculty members on the general topic \How come I’m doing what I do.” Ian began his career in my field, more or less. For his dissertation, he wrote on the continuum problem and, in particular, on Godel’s monograph The Consistency of the Continuum Hypothesis and the Axiom of
Choice in which Godel, in dreadful detail, describes his inner model of constructible sets. In 1964, (more or less) as Ian was finishing his dissertation, Paul Cohen made known his method of forcing, which immediately yielded, among other things, a proof of the consistency of denying the continuum hypothesis, settling the
question of independence. Ian said that he never really understood Cohen’s construction and, in thinking about it, he was finally led to doubt whether he understood the foundations of mathematics, its most
basic concepts, at all. I didn’t and still don’t understand this completely, but maybe it is something like an extended case of mental overload, where meaning simply breaks down for one. Anyway, he was led back to the study of the historical beginnings of the foundations of mathematics in classical Greece|in particular, to Euclid’s Elements. So, being Ian, he learned classical Greek|this while
teaching at the University of Illinois in Champaign-Urbana. (Years later, when I was in the midst of my own pathetic attempt to do the same, he mentioned to me that he learned using Fobes’ Philosophical Greek.)

It is interesting that Ian went back to Greek mathematics to seek foundation. When Hilbert and Bernays wanted a foundation on which to prove the consistency of axiomatically founded mathematics, they went back to an older conception of mathematics—mathematics as computation and construction—the
conception that Kant attempted to found in the Critique of Pure Reason, and Euclid’s geometry was their prime example of this ‘finitist’ conception. In his book on Euclid, Ian discusses the difference between the conception of geometry that appears to lie behind the Elements and Hilbert’s axiomatic foundation in his
Grundlagen der Geometrie. As I recall, although I am not entirely sure of this, Ian spent some time with Bernays in the late 1960′s or early 1970′s. What I am certain of is that he translated a number of works
of Bernays on the subject of `finitism’ in which the urexample of Euclid’s geometry is extensively discussed, notably \Die Philosophie der Mathematik und die Hilbertsche Beweisetheorie” and the first two sections of volume 1 of Hilbert and Bernays’s Grundlagen der Mathematik (written by Bernays). These are what Ian described as `rough translations’, which he executed entirely for the use of his students in courses on philosophy of mathematics; but they (chosen over other, published, translations) have served as the basis of the translations of these works soon to appear (God willing!) in a bilingual edition of Bernays’s philosophical works.

Incidentally, these translations are typical of the effort that Ian put into his courses: I have a file full of translations, lecture notes, transcriptions and glosses of texts in Greek philosophy, all composed for courses he was giving, including an eighty-two page set of notes for a course on the presocratics.

Of course, his scholarly contributions to the history of Greek mathematics and philosophy are central to these fields. Besides his book, three of my favorites are his paper on the completeness of Stoic logic, Aristotle’s conception of geometric objects and a paper on a fragment of the fifth century geometer Bryson. In the second, he argues that, for Aristotle, geometric statements are statements about sensible substances, but restricted to the language of extension. Ian later told me that he came to have doubts about this; but the textual evidence seems to me conclusive and I take it as gospel. (For what its worth, Godel also read Aristotle in that way.) Bryson was one of the fifth century types interested in squaring the circle. The third paper is a short discussion of one reading of a fragment from Bryson, in which he seems to be stating that the circle can be squared on the basis of a geometric version of the intermediate value theorem, and Aristotle’s criticism
of it. (IVT states: if f is a continuous function and f(a) ≠ f(b), then for every number e between f(a) and
f(b), there is an d between a and b such that f(d) = e.) Take f(x) = xx (the square on x), take a to be
the side of the square inscribed in the given circle C, and take b to be the side of the square circumscribed
about C. Then aa < (area of) C bb and so C = dd for some d between a and b. This would seem to be the first statement we have of the IVT. Aristotle criticizes Bryson’s argument, but there are several possible interpretations of what he meant. Ian argues that his criticism was that Bryson’s method of proof did not meet the demands of the problem, which was, in our terms, to give something like a Euclidean construction of the square = C. Interestingly, Bolzano (in 1817) in giving the first analytic proof of IVT, also quotes the same passage from Aristotle as authority for the view that a geometric proof of IVT is inappropriate, for it is a theorem of function theory and the geometric version is simply an application of it. Ian notes the
`non-constructive character of Bryson’s proof that C can be squared and refers to the 20th century version of constructivity in mathematics. This is of course is far more inclusive than Euclidean construction; but Ian's reference is nevertheless apt because, even in the wider sense, IVT is not constructively provable—at
least in any sense of constructivity that is consistent with the constructive numerical functions all being Turing computable.

I last saw Ian with Janel at the memorial party for John Haugeland at Joan's house. They both seemed pleased with the world (although I know that could never have been fully true of Ian!)|with plans for the
future and looking forward to the prospect of days of research in Regenstein.

RIP
Bill
2

I’m also heartbroken to see that Janel has lost such a great husband. I came to see the power of their relationship when Anne Eaton had organized a discussion on being an academic couple. The discussion was hosted by Ian and Janel in the old Anscombe lounge. What I remember most is how much Ian beamed in Janel’s presence and how evident it was that he and Janel were still madly in love. It was a wonderful demonstration of what a genuine partnerhood looked like and I thought we should all be so lucky to have what they have.