The aim of this one-day meeting^{1} was to discuss applications of category theory in philosophy and physics.
9:00
Marek Kuś: Logical and categorical approaches to no-signaling theories
10:00
Krzysztof Wójtowicz: The possible explanatory role of category theory
11:00
Bartłomiej Skowron: A defense of the theory of ideas
12:00
Wiesław Kubiś: Categories with norms
Marek Kuś: Logical and categorical approaches to no-signaling theories
We analyze the structure of the so-called no-signaling theories respecting relativistic causality, but allowing correlations violating bounds imposed by quantum mechanics. We use a quantum logic approach to discuss the relations among such theories, quantum mechanics, and classical physics and reconstruct a probability theory adequate for the simplest instance of a no-signaling theory, in particular we give a rigorous justification of using term probability in this context. In frames of the category theory we adapt to no-signaling models a categorical “Bohrification” formalism that, originally, was used to find an alternative logical foundation for the orthodox quantum theory, replacing the usual non-distributive orthomodular lattices with the distributive logic of a point-free space. We show that not only no-signaling theories are to the process of Bohrification, but furthermore they acquire a natural logical structure, compatible with their probabilistic content. In the light of the obtained results, we show that the discussed models cannot be considered as a generalizations of quantum mechanical ones and seeking for additional principles that could allow to “recover quantum correlations” in such models is, at least from the fundamental point of view, pointless.
Krzysztof Wójtowicz: The possible explanatory role of category theory
The talk is not technical, but has a philosophical character. It addresses the problem of explanation in mathematics, which is a much discussed subject recently. I propose to consider it within the context of category theory. The problem of explanation in mathematics arises in many different contexts and on different levels of generality: it concerns some “local” phenomena (e.g. particular theorems or definitions), or some “global” issues (the most general being probably the issue of the coherence of mathematics as a whole). One of important issues is the explanatory value of mathematical proofs. After presenting the problem, I set forth some questions concerning the potential explanatory value of category theory. I also explore some possible objections to the thesis, that category theory has a genuine “explanatory input” for (loosely speaking) the mathematics outside category theory itself.
Bartłomiej Skowron: A defense of the theory of ideas
I propose the new incarnation of the theory of Ideas and I try to defend the theory against traditional counterarguments. The starting point are the theories of Ideas of Plato and Ingarden and an ontology of Ideas proposed by Kaczmarek; these theories are paraphrased—using a modified method of semantic paraphrases of Ajdukiewicz—and presented in terms of the basic concepts of category theory. To paraphrase Ideas as categories I propose recognized category theory as a pattern for the theory of Ideas. This recognition is based on an analogy between mathematical structures and philosophical structures. It could also be called a mathematical philosophy or mathematical modeling in metaphysics. I invoke an arrows-like, i.e. no-object-oriented, formulation of a category and I base the proposed theory of Ideas on that formulation. The components of an Idea are arrows and their compositions (equivalents of changes and transformations); objects in this approach are special arrows namely the identity arrows. Using the category of higher dimensions I introduce the concept of the dimension of an Idea (and other concepts) which allows me to refute the argument of the ”third man”.
Wiesław Kubiś: Categories with norms
We will discuss a metric-like structure on a category, called a norm. Namely, to each arrow a nonnegative real value is assigned, and some natural axioms are imposed. It turns out that several well-known categories have natural norms. A norm on a category allows to extend the concept of a Cauchy sequence and Cauchy completeness. We will also present a category-theoretic version of the Banach Fixed Point Principle, showing that in our setting it becomes merely trivial.
^{1} Partially supported by GAČR project Generic objects (GA17-27844S).
Organizers: | Wiesław Kubiś (IM CAS), | Bartłomiej Skowron (ICFO) |
Last updated: 28.6.2017