The principle of general tovariance

Abstract

We tentatively propose two guiding principles for the construction of theories of physics, which should be satisfied by a possible future theory of quantum gravity. These principles are inspired by those that led Einstein to his theory of general relativity, viz. his principle of general covariance and his equivalence principle, as well as by the two mysterious dogmas of Bohr's interpretation of quantum mechanics, i.e. his doctrine of classical concepts and his principle of complementarity. An appropriate mathematical language for combining these ideas is topos theory, a framework earlier proposed for physics by Isham and collaborators. Our "Principle of general tovariance" states that any mathematical structure appearing in the laws of physics must be definable in an arbitrary topos (with natural numbers object) and must be preserved under so-called geometric morphisms. This principle identifies geometric logic as the mathematical language of physics and restricts the constructions and theorems to those valid in intuitionism: neither Aristotle's principle of the excluded third nor Zermelo's Axiom of Choice may be invoked. Subsequently, our "Equivalence principle" states that any algebra of observables (initially defined in the topos Sets) is empirically equivalent to a commutative one in some other topos.

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Klaas Landsman
Radboud University Nijmegen

Citations of this work

Continuity and geometric logic.Steven Vickers - 2014 - Journal of Applied Logic 12 (1):14-27.

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References found in this work

A Primer of Infinitesimal Analysis.John L. Bell - 1998 - Cambridge University Press.
Between classical and quantum.Nicolaas P. Landsman - 2007 - Handbook of the Philosophy of Science 2:417--553.
The uses and abuses of the history of topos theory.Colin Mclarty - 1990 - British Journal for the Philosophy of Science 41 (3):351-375.

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