"** new foundations must give new math**"

from

by Giovanni Sambin, Oxford University Press, forthcoming

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Under the patronage of

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**

Nijmegen

**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.