What Can Category Theory Do For Philosophy?

9-11 July 2013 University of Kent, Canterbury, UK
(location 9-10 July, SR16 Keynes College, building N6; location 11 July, Darwin Staff Common Room, building E5 on this map)

Organiser: David Corfield

 

 

INTRODUCTION

The first paper on category theory was published nearly seventy years ago, and has since that time profoundly altered the organisation of mathematics, in particular the fields most useful for modern physics, such as differential topology. These developments have prompted some philosophers of mathematics to formulate structuralist theses along category theoretic lines. Meanwhile, category theorists and theoretical computer scientists have worked out profound relationships with type theory, in particular the intensional dependent type theory of philosopher-logician Per Martin-Löf. Very recently this form of type theory has been found to be modelled by constructions in the branch mathematics known as homotopy theory, which has triggered a yearlong program at the Institute for Advanced Study in Princeton aiming to revolutionise the foundations of mathematics. These category theoretic Univalent Foundations promise to offer us exciting new ways to characterise the basic elements of mathematics as spatial.

 Now, since Martin-Löf dependent type theories are good for modelling natural language (cf. Arne Ranta, Type-Theoretic Grammar, Clarendon, 1995), there should be interesting areas for philosophers of language to explore. But more generally, there should be many new tools for metaphysicians. For example, the concept of identity is enriched by the intensional aspect of these new foundations to allow more subtle notions of equivalence.

 Category theory has also shown promising signs of being significant for a range of fields of philosophical interest, from modal logic to the natural sciences. A recent discovery by Awodey and Kishida has shown that sheaves, which may be thought of as spaces sitting over one another, provide a complete semantics for first-order modal logic. This work is driven by category theoretic reasoning. Elsewhere, we find category theory being used to reconceptualise quantum mechanics and gauge theory, and to allow the formulation of both major attempts towards quantum gravity, namely, string theory and loop quantum gravity. There are also signs of its usefulness for systems biology. (For a linked list see here.)

 The idea for this meeting is to bring together a group of people who are each working on some area where category theory is being used in an area of philosophical interest. One key aim is to discover what its multiple uses tells us about category theory itself, and about the relationships between its applications. For instance, already we can see that sheaf semantics for first-order modal logic bears a strong resemblance to principal bundles in gauge field theory. Does this tell us something about modal conceptions in physics?

Speakers: Brice Halimi, Hans-Christoph Kotzsch, Ralf Krömer, Constanze Roitzheim, Andrei Rodin, Kohei Kishida; Yoshihiro Maruyama, Karin Verelst; Michael Ernst; Koji Nakatogawa, Staffan Angere; David Corfield


TALKS