Peter Freyd on path Integral?

Question

In the issue Electronic Notes in Theoretical Computer Science Volume 29, 1999, Page 79 there is a very intriguing abstract by Peter Freyd.

Path Integrals, Bayesian Vision, and Is Gaussian Quadrature Really Good? Physicists know how to integrate over all possible paths, computer-vision experts want to assign probabilities to arbitrary scenes, and numerical analysts act as if some continuous functions are more typical than others. In these three disparate cases, a more flexible notion of integration is being invoked than is possible in the traditional foundations for mathematics. If allowed to enter a highly speculative mode, such as the intersection of category theory and computer science, we may bump into some solutions to the problem.

This was a special issue after the conference CTCS '99, Conference on Category Theory and Computer Science. Unfortunately, it seems to me that there is no additional material explaining what Freyd spoke about at the conference.

Q. Do we have any clue about what Freyd spoke about that day?

Answer 1

In his late paper Algebraic real analysis (published on Theory and Applications of Categories, Vol. 20, No. 10, 2008, pp. 215–306) he writes in the very last page what follows.

In September 1999 at an invited talk at the annual ctcs meeting (held that year in Edinburgh) I even characterized the mean value of real-valued continuous functions on the closed interval as an order-preserving linear operation that did the right thing to constants and had the property that the mean value on the entire interval equaled the midpoint of the mean-values on the two half intervals. I described it with a diagram that used (twice) the canonical equivalence between I and I ∨ I. But one equivalence, even used twice, doesn’t bring forth the general notion: it doesn’t prompt one to invent ordered wedges; without ordered wedges one doesn’t define zoom operators nor discover the theorem on the existence of standard models. One doesn’t learn how remarkably algebraic real analysis can become. What I needed was someone to kick me into coalgebra mode. Three months later two guys did just that and on December 22 I wrote to the category list: There’s a nice paper by Dusko Pavlovic and Vaughan Pratt. It’s entitled On Coalgebra of Real Numbers [115] and it has turned me on. A solution, alas, for the three motivating problems still awaits; but, at least, now I like analysis.

Very much "We must (not) know, we will (not) know" vibes, but at least mystery solved.