In the issue **[Electronic Notes in Theoretical Computer Science](https://www.sciencedirect.com/science/article/pii/S1571066105803081?via%3Dihub)** *Volume 29, 1999, Page 79* there is a very intriguing abstract by *Peter Freyd*.



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> **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.
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This was a special issue after the conference *[CTCS '99, Conference on Category Theory and Computer Science](https://www.sciencedirect.com/journal/electronic-notes-in-theoretical-computer-science/vol/29/suppl/C)*. Unfortunately, it seems to me that there is no additional material explaining what Freyd spoke about at the conference.



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> **Q.** Do we have *any clue* about what Freyd spoke about that day?
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Peter Freyd on path Integral?
In his late paper [**Algebraic real analysis**](http://www.tac.mta.ca/tac/volumes/20/10/20-10.pdf) (published on *Theory and Applications of Categories, Vol. 20, No. 10, 2008, pp. 215–306*) he writes in the very last page what follows.



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


