# Categorical Approaches to Reduction and Limits (5-6 September 2019)

In this meeting, we will discuss how categorical methods might be used to illuminate the notions of reduction and limits. On the one hand, we will seek to understand how existing criteria for reducing or limiting relationships might be expressed in category-theoretic terms; on the other, we will look at whether category-theoretic methods enable us to articulate new criteria. Specific questions include the following:

- What criteria for reduction and limits can be articulated in categorical terms?
- What can such categorical presentations teach us about the relationships between those criteria?
- How do such criteria relate to the criteria for equivalence?
- Is basic category theory all that is required, or does articulating these criteria require higher category theory or topos theory?
- What is the relationship between reduction in physical theories and reducibility or interpretability in formal theories?

## Speakers

The network members speaking at this meeting will be Patricia Palacios and Joshua Rosaler. We also have three invited guests speaking:

**Benjamin H. Feintzeig**is Assistant Professor in the Department of Philosophy at the University of Washington. His research focuses on the conceptual and mathematical foundations of physics, and the broader impacts of philosophy of physics on general issues in philosophy of science including interpreting physical theories, scientific representation, and scientific explanation. His primary focus is on the foundations of our best quantum theories and the classical theories that we “quantize” to arrive at them. He also has interests in philosophy of probability, epistemology, metaphilosophy, and the history of science.**Samuel C. Fletcher**is Assistant Professor in the Department of Philosophy at the University of Minnesota, Twin Cities, a Resident Fellow of the Minnesota Center for Philosophy of Science, and an External Member of the Munich Center for Mathematical Philosophy, Ludwig-Maximilians-Universität. Much of his work has concerned the foundations of physics and of statistics, and how problems in these fields inform and are informed by broader issues in the philosophy of science. He also has interests in the conceptual and physical basis of computation, metaphilosophy, and the history of physics and philosophy of science.**Christian List**is Professor of Philosophy and Political Science in the Department of Philosophy, Logic and Scientific Method at the London School of Economics. He works at the intersection of philosophy, economics, and political science, with a particular focus on individual and collective decision-making and the nature of intentional agency. He has long-standing interests in social choice theory and the theory of democracy. In recent years, a growing part of his work has addressed metaphysical questions, e.g., about free will, causation, probability, and the relationship between “micro” and “macro” levels in the human and social sciences.

Please note that Samuel C. Fletcher and Christian List will give their talks via Skype.

## Program

### Day 1 (Thursay, 5 September 2019)

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

12:30 - 13:00 | Coffee and Registration |

13:00 - 14:15 | Patricia Palacios: Intertheoretic Reduction in Physics Beyond the Nagelian Model |

14:15 - 14:30 | Break |

14:30 - 16:00 | Sam Fletcher (via Skype): Reduction and Limits: Topological and Categorical |

16:00 - 16:30 | Coffee |

16:30 - 18:00 | Ben Feintzeig: Why Care About Functorial Quantization |

19:00 | Workshop dinner (Picnic, Barer Str. 48) |

### Day 2 (Friday, 6 September 2019)

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

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

09:30 - 10:45 | Joshua Rosaler: Compound Reduction and Overlapping State Space Domains |

10:45 - 11:00 | Break |

11:00 - 12:30 | Christian List (via Skype): Levels, Supervenience, and Reducibility |

## Abstracts

### Benjamin H. Feintzeig: Why Care About Functorial Quantization?

In the philosophical literature, tools of category theory have primarily been used to aid our understanding of theoretical equivalence by comparing different formulations of a single theory. I will argue that these same tools can be extended to comparisons between different theories across the divide of theory change. In particular, I argue that answers to extant technical questions about the functoriality of quantization (understood as capturing the \hbar\to 0 limit) can help us compare classical and quantum physics. I believe this approach to comparison across theory change is important because it has the potential to bear on general issues in philosophy of science, including realism, reduction, and heuristics.top

### Samuel C. Fletcher: Reduction and Limits: Topological and Categorical

Nickles (1973) first introduced into the philosophical literature a distinction between two types of intertheoretic reduction, one of which involved the notion of "limit" familiar to the physics community. In the first part of this presentation, I will present a positive account of limiting-type reductions, different from Nickles', as relations between classes of models endowed with extra, topological (or topologically inspired) structure that encodes formally how those models are relevantly similar to one another. This reveals an unnoticed point of philosophical interest, that the models of a theory themselves do not determine how they are relevantly similar: that must be provided from outside the formal apparatus of the theory, according to the context of investigation. Thus justifying why a notion of similarity is appropriate to a given context is crucial, as it may perform much of the work in demonstrating a particular reduction's success or failure. In the second part of this presentation, I show how this limiting notion of reduction can be understood in categorical terms, and address the question of when such categorical formulations yield conceptual benefits. My tentative answer is that conceptual benefits accrue to categorical formulations only when the theories being related need more than basic category theory to be well represented.top

### Christian List: Levels, Supervenience, and Reducibility

In this talk, I will first explain a simple, category-theoretic framework for representing levels, which can be interpreted in descriptive, explanatory, or ontological terms, and I will then use this framework to explore the relationship between supervenience and explanatory reducibility. I will sketch two arguments against the claim that supervenience implies explanatory reducibility: a formal combinatorial argument and an informal conceptual argument. A background paper is available at: http://philsci-archive.pitt.edu/13311/1/LevelsRevised.pdf top

### Patricia Palacios: Intertheoretic Reduction in Physics Beyond the Nagelian Model

Intertheoretic reductions play an important role in modern physics. But under what conditions a theory reduces to another, and what is achieved by reduction? Nagel (1961) famously attempted to offer a general structure of scientific reduction, whereby reduction is understood in terms of the logical deduction of the reduced theory from the union of the reducing theory and bridge laws. Despite its limitations, the Nagelian model, and revised versions of it, continues nowadays being regarded as the standard philosophical model of reduction in physics. In contrast to this view, I will argue that the Nagelian model does not suffice to explain the most important examples of reduction in physics, including the alleged reduction of thermodynamics to statistical mechanics. Thus, I will contend that in order to have a better understanding of reduction one needs to consider alternative approaches to reduction that emphasize the role of limits and approximations as well as the structural connection between the theories to be compared.top

### Joshua Rosaler: Compound Reduction and Overlapping State Space Domains

The purpose of this talk is to illustrate a particular methodology for chaining together distinct reductions between different models of the same physical system, and a particular consistency requirement between distinct "reduction paths" - i.e., distinct sequences of intermediate models that each serve to establish a link between the same pair of reduced and reducing models. In doing so, I consider the methodology of composing reductions based on the Bronstein cube; I underscore several flaws in this methodology and the particular manner in which it employs limiting relations as a tool for effecting reduction. An alternative methodology, based on a certain simple geometrical relationship between distinct state space models of the same physical system, is then described and illustrated with examples. Within this approach, it is shown how and under what conditions inter-model reductions involving distinct model pairs can be composed or chained together to yield a direct reduction between theoretically remote descriptions of the same system. Building on this analysis, I consider cases in which a single reduction between two models may be effected via distinct composite reductions differing in their intermediate layer of description, and motivate a set of formal consistency requirements on the mappings between model state spaces and on the subsets of the model state spaces that characterize such reductions. These constraints are explicitly shown to hold in the reduction of a non-relativistic classical model to a model of relativistic quantum mechanics, which may be effected via distinct composite reductions in which the intermediate layer of description is either a model of non-relativistic quantum mechanics or of relativistic classical mechanics. Some speculations are offered as to whether and how this sort of consistency requirement between distinct composite reductions might serve to constrain the relationship that any unification of the Standard Model with general relativity must bear to these theories. Connections to the field of category theory are then considered.top

## Fees

There are no fees to attend this workshop. The workshop dinner is at your own expense.

## 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

Prof-Huber-Platz 2

80539 München**Room W401**