Categorical Equivalence in Philosophy of Science (30-31 July 2018)
This will be the first of three workshops associated with the grant: the purpose of this meeting is to evaluate recent work done on categorical approaches to theoretical equivalence. This is an area that has already received attention from philosophers, with the result that there is a valuable existing stock of results and case studies; but the time is ripe to synthesise the results so far, and look towards future developments. This meeting will also be a valuable opportunity to reflect on progress made since the conference on "The Semantics and Structure of Theories", held at the MCMP in June 2016, since there is some overlap between the topics of discussion.
The key questions we will address are as follows:
- What kinds of theories have equivalent categories of models?
- Is such categorical equivalence a plausible necessary or sufficient condition for physical equivalence?
- What are the problems with a (purely) category-theoretic approach to understanding equivalence?
- What is the relationship between equivalence and duality in physical theories?
- How might duality and equivalence be understood category-theoretically?
The network members speaking at this meeting will be Thomas Barrett, Erik Curiel, and Sarita Rosenstock. We also have two invited guests speaking:
- Hans Halvorson is the Stuart Professor of Philosophy at Princeton University. Prof. Halvorson is one of the leading experts on the applications of category theory in mathematical logic, and on several fields in the philosophy of physics. Recently, his work on the formalization of scientific theories has re-invigorated the debate between the Semantic and Syntactic View.
- James Owen Weatherall is Professor of Logic and Philosophy of Science at UC Irvine. Prof. Weatherall has written widely on topics in the philosophy of science and philosophy of physics, especially on the conceptual foundations of classical and quantum field theories.
Day 1 (Monday, 30 July 2018)
|13:00 - 13:30||Registration|
|13:30 - 14:45||Erik Curiel: The Categories of Physical Systems and Theories (Slides, 394 Kb)|
|14:45 - 15:00||Coffee|
|15:00 - 16:30||Hans Halvorson: 2-Categories of Theories (Slides, 114 Kb)|
|16:30 - 17:00||Coffee|
|17:00 - 18:15||Sarita Rosenstock: Category Theoretic Explanations in Data Science|
Day 2 (Tuesday, 31 July 2018)
|09:00 - 09:30||Registration and coffee|
|09:30 - 10:45||Thomas Barrett: How to Count Structure (Slides, 3,7 Mb)|
|10:45 - 11:00||Coffee|
|11:00 - 12:30||James Owen Weatherall: Why Not Categorical Equivalence? (Slides, 1,5 Mb)|
One way that philosophers have recently approached questions of theoretical equivalence is by looking to the “amount of structure” that the theories in question posit. The basic idea is that if two theories posit different amounts of structure — like the Galilean and Newtonian theories of spacetime, for example — then the theories must be inequivalent. It has also been suggested by physicists and philosophers that theories that posit less structure should be preferred. Jill North has explicitly endorsed this kind of structural parsimony principle in recent work. And in General Relativity from A to B, Geroch writes that “[a]lthough the evidence on this is perhaps a bit scanty, it seems to be the case that physics, at least in its fundamental aspects, always moves in this one direction [towards a theory that posits less structure]. It may not be a bad rule of thumb to judge a new set of ideas in physics by the criterion of how many of the notions and relations that one feels to be necessary one is forced to give up.”
There are a number of proposals currently on the table for how to best compare the amounts of structure that different theories posit. Many of these proposals employ (implicitly or explicitly) the same basic tools from category theory that have recently been used to answer questions of theoretical equivalence. My aim in this talk is to demonstrate how these proposals are related to one another, and to evaluate them on their own terms. In order to do so, I’ll present a collection of small results from model theory that are closely related to (and in a sense, generalizations of) Beth’s and Svenonius’ definability theorems.top
I find myself dragged in contrary directions. On the one hand, I am chary of formal methods in philosophy of science, as they too often seem to be used as ends in themselves; and this seems to me true to an unhealthy degree in the study of the structure and semantics of scientific theories. On the other, I believe that, when used with caution and when supplemented by substantive contact with and constraint by the real empirical content of science, they can be useful, even fruitful. I will attempt in this paper to submit to the second impulse. I will propose a way of representing a physical theory as a category, which is nothing new. I will also propose, however, a way of representing as a category the physical systems appropriately and adequately represented by a theory, and correlatively a way of representing as a category the family of measuring and observational practices used to bring the systems and the theory into substantive contact with each other. Various constructions on these categories, such as functors between them, will then be used to capture the idea that the theory represents those systems with propriety and adequacy, when the values of their physical quantities are determined by members of the given family of measuring and observational practices. This will yield a natural necessary and sufficient condition for the equivalence of two theories in a physically substantive sense.top
I describe a general framework for explicating relations between scientific theories -- a framework that can applied both to syntactic and semantic views. I focus on the notion of a "translation" between theories, with special attention to those translations that implement equivalences between theories, or the reduction of one theory to another.top
Philosophical work on how category theory gives insight into structure and equivalence in science has largely focused on the foundations of physics, but the basic arguments are general enough to apply to a wider range of scientific modeling practices. As a case study, I examine topological data analysis, a recent and popular approach to identifying “structure" in large data sets. I show that the existence of a particular functor between two categories of data models serves a central explanatory role in connecting the computational results in this field to their intended real-world applications. I investigate the extent to which philosophical work on functorial relationships in physics carries over to this vastly different context.top
In recent years, categorical equivalence has emerged among philosophers of science as a promising approach to understanding when two (physical) theories are equivalent. One the one hand, philosophers have discussed several examples of theories, including standard and geometrized Newtonian gravitation; Hamiltonian and Lagrangian mechanics; general relativity and Einstein algebras; and others, whose relationships seems to be clarified using these categorical methods. On the other hand, philosophers and logicians have studied the relationship, particularly in the first order case, between categorical equivalence and other notions of equivalence of theories, including definitional equivalence and generalized definitional (aka Morita) equivalence. In this article, I will express some skepticism about this approach, both on technical grounds and conceptual ones. I will not argue that the approach should not be pursued—it should—but rather that we need more control over precisely what categorical equivalence establishes and the conditions under which it can be applied fruitfully.top
Attendance is free, but space is limited; if you would like to attend, please email email@example.com. Please say whether you plan to attend the workshop dinner or not.
Main University Building
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