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
Many mathematical subjects study notions of "equivalence" that are of central interest to the particular area. A lot of times, this equivalence can be viewed as isomorphisms in a certain category. In topology, the equivalence to consider is "homotopy equivalence". To encode this in a category, one has to formally invert morphisms, which requires bypassing various set-theoretical issues. We give an overview of some general ideas in topology and of some relevant constructions.
This talk is intended to address the general question of the workshop not by presenting another example of a use of category theory in philosophy (or other fields), but by investigating the epistemological commitments of such uses from the perspective of the philosophy of category theory I developed in 'Tool and Object'. On this background, I will present a short epistemological analysis of Lawvere's proposal to use category theory for understanding Hegel (worked out and prolonged to other philosophers by Mélès), and Mazzola's uses of category theory in musicology.
Category theory, among other things, provides the general means and the global frameworks needed to study local phenomena. Tarskian semantics gave a first account of local universes of discourse, but without clear locus for those universes themselves. The category of all structures for a given formal language is such a locus. On the basis of that category, we shall explore how fibered categories can be introduced to cast a new light on basic notions of model theory, and to put forward a categorical notion of contextuality within logic. Time permitting, another example, related to sketch theory, will be considered. The general working hypothesis is that category theory is an essential support for the interaction between logic and all other branches in mathematics.
This paper illustrates several ways in which categorical methods can help semantic modeling of, and philosophical reflection upon, both propositional and first-order modal logic. As preliminaries, we first review propositional modal logic and its Kripke and other (less standard) semantics: topological, neighborhood, and ``impossible possible world'' semantics --- in this review we provide algebraic and categorical expressions that are useful in the model theory of these semantics. Then some of these expressions can be extended to Kripke's semantics for quantified modal logic. This extension sheds new light on the philosophical stance of the semantics and, in particular, on the conceptual role of the so-called converse Barcan formula: Taking advantage of the categorical expressions, we lay out an interpretation of the formula that is alternative to Kripke's widely accepted one. Moreover, as it turns out, this new interpretation leads to a range of other semantics for first-order modal logic of a different ontological makeup, and most notably David Lewis's in terms of his counterpart theory. Using categorical expressions as a tool of comparison, we lay out several reasons why we should fine-tune the semantics of Lewis's into ones in terms of bundles and moreover of sheaves over a space.
We will present a framework with respect to which it is possible to interpret a certain version of higher-order modal logic in an arbitrary topos. The semantics is essentially algebraic in nature. We will show how the notion of topos captures precisely the category-theoretic concepts needed to define this algebraic semantics abstractly, much in analogy with the well-known semantics for higher-order type theory. However, there are also significant differences from the latter. We will also show how this semantics unifies several categorical frameworks that have been developed previously.
Discussions on causality abound, but rare are the attempts at precise definition of what is meant. The reason might be that the concept in itself is intrinsically pluriform, but even then theories encompassing some notion of causation should exhibit certain common structural characteristics, otherwise the use of the common term would be absolutely pointless. Causation requires more than the mere transition of one state of a given system into another. State transitions imply changes in spatio-temporal relationships, so at least implicitly conservation of identity is required. I show that a fairly straightforward categorical characterisation of causation is possible when one takes both the history of the concept and Meyerson’s careful analysis of the relation between causation and time into account. When causal relations are interpreted as order relations, then causation appears as the Galois adjoint to identity and causality will be equivalent to the presence of physical law. Through Noether’s Theorem, this approach might be generalisable to other physically suitable notions of causation as well.
The shift of emphasis from objects to processes has been central in categorical advances, such as recent Categorical Quantum Mechanics. Category Theory has even delivered dynamic changes in our fundamental conceptions of space: notably, the notion of point has lost its primary role, as in Topos Theory. Philosophers have contemplated on conceptions of space as well. For instance, Wittgenstein asks, "What makes it apparent that space is not a collection of points, but the realization of a law?". In Whitehead's process philosophy, a point is understood not as a primary entity, but as being derived as the limit of shrinking regions; this gets quite closer to the idea of prime ideals as points in Locale Theory and Algebraic Geometry.
As these suggest, there are a number of intriguing similarities between mathematicians' and philosophers' conceptions of space. I argue that those different ideas of space may be classified into two categories: namely, the epistemological and the ontological. Duality often exists between them, as exemplified by syntax-semantics duality in Logic, and state-observable duality in Quantum Physics. I compare this idea of duality between the epistemological and the ontological, with Lawvere's duality between the formal and the conceptual, Cassirer's dichotomy between functions and substances, and Wittgenstein's dichotomy between arithmetical and geometrical space.
I derive a paradox for naive category theory in the application of categorial methods to graph theory. Analysis of the derivation illuminates the necessary assumptions. There are different costs depending on which assumptions one is disposed to jettison. However, the paradox guarantees that not all of them can be accepted. I use this fact to draw important consequences for Solomon Feferman's ongoing project on the foundations of unlimited category theory.
I will start with mentioning Reyes’s attempt ( La Palme Reyes M. and G. E. Reyes, 1989) to apply category theory to Montague Grammar, and then consider some problems related to the designations of an uncountable noun (in English) and of a Chinese character (in ancient Chinese and modern Japanese). Taking a doctor of Traditional Chinese Medicine (TCM) as an agent of decision making, I will try to give a sketch of (pre-)sheaf semantics to describe a certain type of (pre-judgmental) ‘reasoning’ used in the diagnosis process of TCM (or Kampo).
Lawvere's axiomatization of topos theory and Voevodsky's axiomatization of heigher homotopy theory exemplify a new way of axiomatic theory building, which goes beyond the classical Hibert-style Axiomatic Method. The new Axiomatic Method of Lawvere and Voevodsky revives some features of the traditional Euclid’s Axiomatic Method, which have been largely abandoned by Hilbert and his follows in the beginning of the 20th century. In my talk I shall describe this new-old Axiomatic Method and develop an analogy between Riemann’s idea of intrinsic geometry (of a manifold) and the concept of internal logic of a given category.
I present the outlines of a category-theoretic explication of the philosophically much over-used notion of "structure". Tools from concrete and fibred category theory are employed in order to arrive at (i) a single specific category of structures, and (ii) an explication of what "x instantiates structure S" might mean. Possible applications (if I have time to go through them) include theory change, categorical logic, structuralism in the philosophy of mathematics, and structuralism in contemporary philosophy of physics. The introduced framework also gives a category-theoretic view of ontology and metaphysics, although one centering on structures rather than the more traditional choice of properties or universals.
Category theory has shown itself to be closely related to the different forms of type theory. Currently, there is a huge amount of interest in the type theoretic and category theoretic communities in homotopy type theory, a form of dependent type theory. Were such a type theory to become an accepted foundational language, how might philosophy use it to reconsider some of its principles? In particular, does the notion of the context of a judgement point in favour of Michael Friedman's view of science over Quine's?
SCHEDULE
ACKNOWLEDGEMENTS
We are very grateful to the British Society for the Philosophy of Science, Mind Association and the University of Kent Centre for Reasoning for providing financial support.