Online research in philosophy

A precise notion of ‘mathematical structure’ other than that given by model theory may prove fruitful in the philosophy of mathematics. It is shown how the language and methods of category theory provide such a notion, having developed out of a structural approach in modern mathematical practice. As an example, it is then shown how the categorical notion of a topos provides a characterization of ‘logical structure’, and an alternative to the Pregean approach to logic which is continuous with the modern structural approach in mathematics.

Category theory and topos theory have been seen as providing a structuralist framework for mathematics autonomous vis-a-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 many-topoi view and modal-structuralism. 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 uniform way, by relativization, sustaining insights of Zermelo.

Some mathematicians and philosophers contend that set theory plays a foundational role in mathematics. However, the development of category theory during the second half of the twentieth century has encouraged the view that this theory can provide a structuralist alternative to set-theoretical foundations. Against this tendency, criticisms have been made that category theory depends on set-theoretical notions and, because of this, category theory fails to show that set-theoretical foundations are dispensable. The goal of this paper is to show that these criticisms are misguided by arguing that category theory is entirely autonomous from set theory.

Formal Axiomatic method as exemplified in Hilbert’s Grundlagen der Geometrie is based on a structuralist vision of mathematics and science according to which theories and objects of these theories are to be construed “up to isomorphism”. This structuralist approach is tightly linked with the idea of making Set theory into foundations of mathematics. Category theory suggests a generalisation of Formal Axiomatic method, which amounts to construing objects and theories “up to general morphism” rather than up to isomorphism. It is shown that this category-theoretic method of theorybuilding better fits mathematical and scientific practice. Moreover so since the requirement of being determined up to isomorphism (i.e. categoricity in the usual model-theoretic sense) turns to be unrealistic in many important cases. The category-theoretic approach advocated in this paper suggests an essential revision of the structuralist philosophy of mathematics and science. It is argued that a category should be viewed as a far-reaching generalisation of the notion of structure rather than a particular kind of structure. Finally, I compare formalisation and categorification as two alternative epistemic strategies.

Two topics figure prominently in recent discussions of mathematical structuralism: challenges to the purported metaphysical insight provided by sui generis structuralism and the significance of category theory for understanding and articulating mathematical structuralism. This article presents an overview of central themes related to these topics.

The debate on structuralism in the philosophy of mathematics has brought into focus a question about the status of meta-mathematics. It has been raised by Shapiro ( 2005 ), where he compares the ongoing discussion on structuralism in category theory to the Frege-Hilbert controversy on axiomatic systems. Shapiro outlines an answer according to which meta-mathematics is understood in structural terms and one according to which it is not. He finds both options viable and does not seem to prefer one over the other. The present paper reconsiders the nature of the formulae and symbols meta-mathematics is about and finds that, contrary to Charles Parsons’ influential view, meta-mathematical objects are not “quasi-concrete”. It is argued that, consequently, structuralists should extend their account of mathematics to meta-mathematics.

This paper considers the nature and role of axioms from the point of view of the current debates about the status of category theory and, in particular, in relation to the “algebraic” approach to mathematical structuralism. My aim is to show that category theory has as much to say about an algebraic consideration of meta-mathematical analyses of logical structure as it does about mathematical analyses of mathematical structure, without either requiring an assertory mathematical or meta-mathematical background theory as a “foundation”, or turning meta-mathematical analyses of logical concepts into “philosophical” ones. Thus, we can use category theory to frame an interpretation of mathematics according to which we can be structuralists all the way down.

In this paper I argue that category theory ought to be seen as providing the language for mathematical discourse. Against foundational approaches, I argue that there is no need to reduce either the content or structure of mathematical concepts and theories to the constituents of either the universe of sets or the category of categories. I assign category theory the role of organizing what we say about the content and structure of both mathematical concepts and theories. Insofar, then, as the structuralist sees mathematics as talking about structures and their morphology, I contend that category theory furnishes a framework for mathematical structuralism.

This paper considers the nature and role of axioms from the point of view of the current debates about the status of category theory and, in particular, in relation to the “algebraic” approach to mathematical structuralism. My aim is to show that category theory has as much to say about an algebraic consideration of meta-mathematical analyses of logical structure as it does about mathematical analyses of mathematical structure, without either requiring an assertory mathematical or meta-mathematical background theory as a “foundation”, or turning meta-mathematical analyses of logical concepts into “philosophical” ones. Thus, we can use category theory to frame an interpretation of mathematics according to which we can be algebraic structuralists all the way down.

The popular view according to which category theory provides a support for mathematical structuralism is erroneous. Category-theoretic foundations of mathematics require a different philosophy of mathematics. While structural mathematics studies ‘invariant form’ (Awodey) categorical mathematics studies covariant and contravariant transformations which, generally, have no invariants. In this paper I develop a non-structuralist interpretation of categorical mathematics.