Urs Schreiber in The Reasoner

Urs Schreiber in The Reasoner:

In December 2008, I interviewed Tom Leinster for Kent’s in-house journal – The Reasoner. The latest version, September 2012, sees me do the same to Urs.

I’m posting this here so that any regulars, or anyone who comes here via The Reasoner, can discuss whatever strikes them in what Urs says.

The abrupt transition in The Reasoner from interview to succeeding article had people last time imagining Tom had decided to talk about ‘The Paradox of Omniscience’. Likewise, you need to guard against thinking Urs goes on to talk of ‘Personal taste ascriptions and the Sententiality assumption’.

Something I’d like to develop is the idea that transformations in the past in the relationships between logic, mathematics and physics were brought about by, and led to, a rethinking of the concepts of language, observation, world, knowledge, God (once upon a time), etc. Think Newton and the Leibniz-Clarke correspondence, or the transformations of, say, 1880-1930.

Regarding the latter, to quote a small passage from an article I’m reading at the moment, Logic, Mathematical Science, and Twentieth Century Philosophy: Mark Wilson and the Analytic Tradition, Michael Friedman characterises the thinking of early analytic philosophers as follows:

Just as Cantor, Weierstrass, Dedekind, and Peano had finally uncovered the “true” logical forms of the concepts of infinity and continuity, Frege and his followers could now embark on an analogous project of uncovering the “true” logical forms of all the concepts found in the mathematical-physical sciences–including, especially, the radically new science of space, time, motion, and matter emerging in the context of Einstein’s general theory of relativity. The crucial point, as I have emphasized, is that what we call modern set theory was not conceived as simply the most general purely mathematical theory comprising all possible purely mathematical structures; it was rather conceived as part of the most general framework for logic itself, and it was this, above all, that made it an appropriate general platform, in turn, for all properly philosophical conceptual analysis.

Should there not be an ongoing comparable, explicitly philosophical conversation taking place right now? I hope this interview can spark off such a thing.

A couple of possible starting points:

1) Is there anything in Per Martin-Löf’s philosophical background, perhaps his reading of Husserl and Frege, that gives rise to his type theory being taken up homotopy theoretically, in particular his constructive approach to identity judgements?

2) We can follow a line of thought which notes that physicists have been interested in certain kinds of theory, so that we ought in turn to be interested by a framework that’s able to capture important aspects of those theories. Cohesive homotopy type theory is such a framework. But is there not a way to run things in the other direction? Any interesting universe involves a dynamics. Dynamics must be encoded via a notion of the cohesion of adjacent elements of space. Cohesion is best encoded by the concept of a cohesive ∞-topos. Most suitably structured examples of cohesive ∞-toposes relate to smooth forms of cohesion. We could then be led to think that some parts of the form of current physics can be explained. What, though, if radically different kinds of cohesion were found?

DIGITAL JUICE