# Foundations of Categorical Philosophy of Science (26-27 April 2019)

The purpose of this meeting will be to determine what kinds of category-theoretic resources are most appropriate for application to philosophy of science. We aim to address the following questions:

- What is the role for higher category theory and topos theory in articulating the structure of scientific theories?
- To what extent can these more powerful category-theoretic resources capture the full structure of a theory?
- Does the topos-theoretic approach to quantum mechanics admit of any natural generalisation to other theories?
- What (if any) is the relationship between the use of topos theory for foundations of quantum mechanics and its use in general philosophy of science?

## Speakers

The network members speaking at this meeting will be John Dougherty, Benjamin Eva, and Laurenz Hudetz. We also have one invited guests speaking:

**Hajnal Andréka**is research professor emeritus at Renyi Institute of Mathematics, Hungarian Academy of Sciences. Her research interests lay mostly in the crossroads of logic, algebra and geometry. She has strong scientific ties with Tarski’s school in logic and methodology of science. She was head of the Algebraic Logic Division in the Renyi Institute, whose research group has been working on logical analysis of relativity theory since a while.

## Program

### Day 1 (Friday, 26 April 2019)

Time | Event |
---|---|

13:00 - 13:30 | Registration |

13:30 - 14:45 | Laurenz Hudetz: Defining Internal Structure in Terms of Morphisms |

14:45 - 15:15 | Coffee |

15:15 - 16:45 | Hajnal Andréka, Judit Madarász, István Németi, and Gergely Székely: Conceptual Structure of Spacetimes, and Category of Concept Algebras |

16:45 - 17:15 | Coffee |

17:15 - 18:30 | John Dougherty: Equivalence Within and Among Theories |

19:00 | Dinner |

### Day 2 (Saturday, 27 JApril 2019)

Time | Event |
---|---|

09:00 - 09:30 | Registration and coffee |

09:30 - 10:45 | Benjamin Eva: Indiscernibility, Categorically |

10:45 - 11:15 | Coffee |

11:15 - 12:45 | Closing Remarks and Roundtable |

## Abstracts

### Hajnal Andréka, Judit Madarász, István Németi, and Gergely Székely (Renyi Institute of Mathematics, Hungarian Academy of Sciences): Conceptual Structure of Spacetimes, and Category of concept Algebras

In the first part of the talk, we explore the first-order logic conceptual structure of special relativistic spacetime: We describe the algebra of explicitly definable relations of Minkowski-spacetime, and draw conclusions such as “the concept of lightlike-separability can be defined from that of timelike-separability by using 4 variables but not by using three variables”, or “no non-trivial equivalence relation can be defined in Minkowski-spacetime”, or “there are no interpretations between the classical (Newtonian) and the relativistic spacetimes, in either directions”.

In the second part of the talk, we generalize the notion of a concept algebra from first-order language/logic to any language. A duality between algebra and category theory emerges here quite nicely. Namely, category theoretic properties of the category of all concept algebras shed light on definability properties of the language. For example, “all implicitly definable concepts are explicitly definable (Beth definability property) if and only if epimorphisms are surjective in the category of concept algebras”, or “all existence-requiringly implicitly definable concepts are explicitly definable (weak Beth definability property) if and only if there is no proper epi-reflective subcategory of the category of concept algebras that contain the so-called full concept algebras”. Connections with category theoretic injectivity logic will be pointed out.top

### John Dougherty (MCMP/LMU Munich): Equivalence Within and Among Theories

I refine Barrett and Halvorson’s notion of Morita equivalence to a 2-categorical context, and apply this refinement to questions of theoretical equivalence. I argue that equivalence of categories has a valuable role to play in questions of theoretical equivalence, despite Barrett and Halvorson’s results. I distinguish two roles that equivalence could play: an external role and an internal role. On some kinds of generalizations the only obvious role for equivalence of categories is external: they can be used to compare the underlying categories of the ultracategories of models for two theories. In this case we should expect Barrett and Halvorson’s results to generalize as well, and to show that ultracategories of models for this more general logic involve more structure than their underlying categories of models do. But on another kind of generalization—specifically, when the generalization is to a logic whose models are structured categories, rather than structured sets—equivalence of categories can play an internal role, modeling equivalences within a theory. This use of equivalence of categories is distinct from the use to which Morita equivalence is meant to be put, and so Barrett and Halvorson’s results don’t generate any worries about it.top

### Benjamin Eva (University of Konstanz/MCMP): Indiscernibility, Categorically

Leibniz’s principle of the identity of indiscernibles (PII) prohibits the existence of distinct indiscernible entities. Numerous prospective counterexamples to PII have been forwarded in the mathematical domain. Typically, these counterexamples proceed by demonstrating that in the context of a fixed mathematical structure (a graph, the complex numbers etc), distinct elements of the structure can be indiscernible. In this talk, we argue that counterexamples of this form aren’t really problematic for advocates of PII. What the critic of PII needs to establish is that distinct elements can be indiscernible when the background structure is not an arbitrary mathematical structure, but rather *the whole mathematical universe*. We then consider whether counterexamples of this form exist by developing and utilising novel categorical formalisations of PII.top

### Laurenz Hudetz (LSE): Defining Internal Structure in Terms of Morphisms

Consider categories which have models in the model-theoretic sense as objects (e.g., categories of models of scientific theories). For some categories of models, it seems possible to reconstruct the internal structure of any model in terms of morphisms in the category. The category of vector spaces is a well-known example. Given a vector space V, one can view vectors in V as linear maps from the one-dimensional vector space IR to V. And one can also reconstruct the vector space operations in terms of morphisms. Although this idea is not new and has already received attention in formal philosophy of science, it has not yet received a proper explication. The aim of this talk is to address this issue. I put forward a definition of what exactly it means to reconstruct the internal structure of models in terms of morphisms. A rigorous definition is important because, without one, we cannot make principled judgments about whether or not the morphisms in a category of models encode the internal structure of the models. Being able to make such principled judgments is relevant for the category-theoretic analysis of scientific theories. If we represent a scientific theory by a category of models and the morphisms in the category encode the internal structure of the theory's models, we do not lose relevant information by adopting a purely category-theoretic point of view. On the other hand, if a theory's category of models does not have this property, there is risk that a purely category-theoretic analysis neglects some (representationally) relevant features of models.top

## Fees

## Registration

Attendance is free, but space is limited; if you would like to attend, please email neil.dewar@lrz.uni-muenchen.de. Please say whether you plan to attend the workshop dinner or not.

## Venue

Main University Building

Ludwigstraße 28 (rear building)

D-80539 München**Room 025**

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