"new foundations must give new math"

from "The Basic Picture. Structures for constructive topology"

by Giovanni Sambin, Oxford University Press, forthcoming


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Dipartimento di Matematica Pura ed Applicata, Università degli Studi di Padova, Italy

University of Padova, Italy

AILA (Italian Association of Logic and Applications), whose first president was Giovanni Sambin (from 1987 to 1994).

Abstracts » Spitters

Bas Spitters

A topos for algebraic quantum theory
(jww Chris Heunen and Klaas Landsman)

We show how a C*-algebra naturally induces a topos in which the family of its commutative subalgebras becomes a commutative C*-algebra.
Its internal spectrum is a compact regular locale, and the Kochen-Specker theorem is equivalent to this spectrum having no points. (Quasi-)states become integrals, and self-adjoint elements become functions to the pertinent generalised real numbers (the interval domain). This provides a probabilistic interpretation of propositions in quantum theory.

The topos-theoretic truth value of such a proposition is the collection of pure states of commutative subalgebras that make it true; in a physical interpretation these are the pure states for a classical observer making the proposition true.

These results were motivated by a topos-theoretic approach of the Kochen-Specker theorem by Isham and co-workers. Our main tool is the use of the internal mathematics of a topos, such as the constructive Gelfand duality of Banaschewski and Mulvey, which simplifies the computations and provides very natural connections between internal and external reasoning.

Connections between formal topology and geometric reasoning will be emphasized.