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11
Does category theory provide a framework for mathematical structuralism?
 PHILOSOPHIA MATHEMATICA
, 2003
"... Category theory and topos theory have been seen as providing a structuralist framework for mathematics autonomous vis à vis set theory. It is argued here that these theories require a background logic of relations and substantive assumptions addressing mathematical existence of categories themselves ..."
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Cited by 10 (3 self)
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Category theory and topos theory have been seen as providing a structuralist framework for mathematics autonomous vis à vis set theory. It is argued here that these theories require a background logic of relations and substantive assumptions addressing mathematical existence of categories themselves. We propose a synthesis of Bell’s “manytopoi” view and modalstructuralism. Surprisingly, a combination of mereology and plural quantification suffices to describe hypothetical large domains, recovering the Grothendieck method of universes. Both topos theory and set theory can be carried out relative to such domains; puzzles about “large categories ” and “proper classes ” are handled in a
Quantum Observables Algebras and Abstract Differential Geometry, preprint
, 2004
"... We construct a sheaf theoretical representation of Quantum Observables Algebras over a base Category equipped with a Grothendieck topology, consisting of epimorphic families of commutative Observables Algebras, playing the role of local arithmetics in measurement situations. This construction makes ..."
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Cited by 2 (1 self)
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We construct a sheaf theoretical representation of Quantum Observables Algebras over a base Category equipped with a Grothendieck topology, consisting of epimorphic families of commutative Observables Algebras, playing the role of local arithmetics in measurement situations. This construction makes possible the application of the methodology of Abstract Differential Geometry in a Category theoretical environment, and subsequently, the extension of the mechanism of differentials in the Quantum regime. 1
Categorical Foundations of Quantum Logics and Their Truth Values Structures
, 2004
"... We introduce a foundational sheaf theoretical scheme for the comprehension of quantum event structures, in terms of localization systems consisting of Boolean coordinatization coverings induced by measurement. The scheme is based on the existence of a categorical adjunction between presheaves of Boo ..."
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Cited by 1 (1 self)
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We introduce a foundational sheaf theoretical scheme for the comprehension of quantum event structures, in terms of localization systems consisting of Boolean coordinatization coverings induced by measurement. The scheme is based on the existence of a categorical adjunction between presheaves of Boolean event algebras and Quantum event algebras. On the basis of this adjoint correspondence we prove the existence of an object of truth values in the category of quantum logics, characterized as subobject classifier. This classifying object plays the equivalent role that the twovalued Boolean truth values object plays in classical event structures. We construct the object of quantum truth values explicitly and argue that it constitutes the appropriate choice for the valuation of propositions describing the behavior of quantum systems.
Russell’s Absolutism vs.(?)
"... Along with Frege, Russell maintained an absolutist stance regarding the subject matter of mathematics, revealed rather than imposed, or proposed, by logical analysis. The Fregean definition of cardinal number, for example, is viewed as (essentially) correct, not merely adequate for mathematics. And ..."
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Along with Frege, Russell maintained an absolutist stance regarding the subject matter of mathematics, revealed rather than imposed, or proposed, by logical analysis. The Fregean definition of cardinal number, for example, is viewed as (essentially) correct, not merely adequate for mathematics. And Dedekind’s “structuralist” views come in for criticism in the Principles. But, on reflection, Russell also flirted with views very close to a (different) version of structuralism. Main varieties of modern structuralism and their challenges are reviewed, taking account of Russell’s insights. Problems of absolutism plague some versions, and, interestingly, Russell’s critique of Dedekind can be extended to one of them, ante rem structuralism. This leaves modalstructuralism and a category theoretic approach as remaining nonabsolutist
Contents: §1. Locally Noetherian Schemes
, 2002
"... Let X log be a fine (cf. [Kato1], §2.3) saturated (cf. (the evident étale generalization ..."
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Let X log be a fine (cf. [Kato1], §2.3) saturated (cf. (the evident étale generalization
TOPOSTHEORETIC RELATIVIZATION OF PHYSICAL REPRESENTABILITY AND QUANTUM GRAVITY
, 2006
"... In the current debate referring to the construction of a tenable background independent theory of Quantum Gravity we introduce the notion of topostheoretic relativization of physical representability and demonstrate its relevance concerning the merging of General Relativity and Quantum Theory. For ..."
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In the current debate referring to the construction of a tenable background independent theory of Quantum Gravity we introduce the notion of topostheoretic relativization of physical representability and demonstrate its relevance concerning the merging of General Relativity and Quantum Theory. For this purpose we show explicitly that the dynamical mechanism of physical fields can be constructed by purely algebraic means, in terms of connection inducing functors and their associated curvatures, independently of any background substratum. The application of this mechanism in General Relativity is constrained by the absolute representability of the theory in the field of real numbers. The relativization of physical representability inside operationally selected topoi of sheaves forces an appropriate interpretation of the mechanism of connection functors in terms of a generalized differential geometric dynamics of the corresponding fields in the regime of these topoi. In particular, the relativization inside the topos of sheaves over commutative algebraic contexts makes possible the formulation of quantum gravitational dynamics by suitably adapting the functorial mechanism of connections inside that topos.
The Uses and Abuses of the History of Topos Theory
"... The view that toposes originated as generalized set theory is a figment of set theoretically educated common sense. This false history obstructs understanding of category theory and especial1 y ol categorical foundations for mathematics. Problems in geometry, hpology, and related algebra led to cate ..."
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The view that toposes originated as generalized set theory is a figment of set theoretically educated common sense. This false history obstructs understanding of category theory and especial1 y ol categorical foundations for mathematics. Problems in geometry, hpology, and related algebra led to categories and toposes. Elementary toposes arose when Lawvere's interest in the foundations of physics and Tierney's in the loundations of topology led both to study Grothendieck's foundations for algebraic geometry. I end with remarks on a categorical view of the history of set theory, including a false history plausible from that point of view that