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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
The formal method known as B and a sketch for its implementation
, 2002
"... This thesis provides a reconstruction of the Bmethod and sketches an implementation of its tool support.For background, this work investigates the field of formal methods in general and the relevance of formal methods to software engineering in particular. Formal (firstorder) logic is also conside ..."
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Cited by 2 (2 self)
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This thesis provides a reconstruction of the Bmethod and sketches an implementation of its tool support.For background, this work investigates the field of formal methods in general and the relevance of formal methods to software engineering in particular. Formal (firstorder) logic is also considered: both its development and important points relevant to formal methods. Automated reasoning, particularly its theoretical limits as well as unification and resolution, is discussed. The main part of this thesis is a systematic reconstruction of the Bmethod, starting from its version of untyped predicate calculus and typed set theory, continuing with the Generalized Substitution Language (GSL) and finishing with the Abstract Machine Notation (AMN). Specification, refinement and implementation of a simple example using the Bmethod is presented. Both validation and verification of specifications, refinements and implementations using the Bmethod is discussed. The thesis concludes with a report of the current state of the effort (by the author) to implement the tool support of the Bmethod, as the Ebba Toolset. The main design decisions are discussed. The use of Unicode as the primary input encoding of AMN and GSL in Ebba is described.
Structuralism
"... With the rise of multiple geometries in the nineteenth century, and in the last century the rise of abstract algebra, of the axiomatic method, the settheoretic foundations of mathematics, and the influential work of the Bourbaki, certain views called “structuralist ” have become commonplace. ..."
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With the rise of multiple geometries in the nineteenth century, and in the last century the rise of abstract algebra, of the axiomatic method, the settheoretic foundations of mathematics, and the influential work of the Bourbaki, certain views called “structuralist ” have become commonplace.