Assumptions of Physics - From Classical Mechanics to Field Theory

13:13 How do you define volume in an infinite dimensional phase space on a finite whiteboard? 

1:05:39 I enjoyed listening to this discussion. Thanks to everyone who took part. 

I was thinking about causal sets as modeling those second countable collections of empirically verifiable statements. That may not be how Sorkin sees it though! See Geometry from order: causal sets. 

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See also this discussion with Urs Schreiber on Gauge fields and configuration spaces at 44:33.

The earlier discussion he referred to on differential geometry and compact spaces is at 17:05.

He's saying that you do not want to restrict physics to describing only compact spaces. But you do want to restrict observations to compact spaces. At 26:03 you can get a coherent notion of field configuration spaces in a topos.

35:10 Lawvere started out in physics, but then he asked "What is a vector space, really?" and within a decade wrote a paper The Category of Categories as a Foundation for Mathematics. See Vladimir Voevodsky - What if Current Foundations of Mathematics are Inconsistent. 

Another relevant bit at 1:12:35 on the "Dialectica" interpretation of Kurt Gödel. See Gödel’s functional interpretation and its use in current mathematics by Ulrich Kohlenbach.  I think this is another example of the kind of stability you would want in a physical theory. So the idea of Lawvere is to set up such conditions in a "cohesion structure" and then effectively calculate the category of objects that satisfy it. Lawvere gave the example of Lie groups as on the one hand just abstract groups, and on the other as being something that would satisfy the Jacobi identity in the associated algebra.  

See internal logic in nlab and Nonstandard proof methods in toposes by José Siqueira and also Sheaf and duality methods for analyzing multi-model systems by Michael Robinson. Also Stefan Milius - Demystifying Codensity Monads through Duality. 

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