PACT, Philosophically-Applied Category Theory:
or PhACT
The language of Category Theory has been under development since the 1940s and continues to evolve to this day. It was originally created as a formal language to capture common mathematical structures and inference methods across various branches of mathematics, and later found application outside of mathematics. By introducing arrows to mediate between objects, the language is designed to represent anything that can be perceived as a process - including processes of inference and physical processes. In other words, it captures not just the nouns but also the verbs of any situation.
The first applications of Category Theory outside of mathematics and logic were to physics and to computer science. There was also an early example of an application in biology too, Rosen.
But over the past decade we have seen researchers under the banner of Applied Category Theory take on a variety of novel subjects, addressing topics which include:
causality, probabilistic reasoning, statistics, learning theory, deep neural networks, dynamical systems, information theory, database theory, natural language processing, cognition, consciousness, explainable AI, systems biology, genomics, chemical reaction networks, neuroscience, complex networks, game theory, robotics, and quantum computing.
The central slogan is compositionality, building up complex wholes from their parts
There is now a substantial ACT community, which hosts annual conferences.
The work concerns both inferential and physical processes, and addresses topics of direct or indirect interest to philosophy. The premise of this proposal is that it is time for philosophers to interact with ACT, in what we might call Philosophically-Applied Category Theory, or PhACT.
…we should treat the use of categorical concepts as a natural part of transferring and integrating knowledge across disciplines. The restructuring employed in applied category theory cuts through jargon, helping to elucidate common themes across disciplines. (ACT 2018)
Where plenty of ACT practitioners reveal their encounters with philosophical, it is time to put this relationship on a more systematic basis.
The remit of PhACT will include both the direct application of category theory to philosophical topics and the use of philosophy to guide the application of category theory to any topic. PhACT offers an excellent opportunity for both sides to grow from the interaction: New tools and insights gained by the application of CT tools for philosophers, philosophical nuance for ACT practitioners working with concepts studied for decades or centuries by philosophers.
Regularity Theory: This approach states that an event is the cause of another event if and only if there is a constant and regular relationship between the two events. Regularity theory seeks to identify the necessary and sufficient conditions for causality, and it focuses on the relationship between events rather than on the nature of causation itself.
Probabilistic Theory: This approach claims that an event is the cause of another event if and only if the first event makes the second event more likely to occur. Probabilistic theory recognizes that causality involves uncertainty and probability, and it emphasizes the probabilistic relationship between events rather than a deterministic one.
Counterfactual Theory: According to this approach, an event is the cause of another event if and only if, in the absence of the first event, the second event would not have occurred. Counterfactual theory emphasizes the idea of a counterfactual dependence between events, and it focuses on the idea of necessity rather than sufficiency.
Interventionist Theory: This approach suggests that an event is the cause of another event if and only if the first event can be manipulated to produce the second event. Interventionist theory emphasizes the idea of intervention and control, and it is often used in the context of experimental design.
Process Theory: This approach claims that an event is the cause of another event if and only if there is a process or mechanism that connects the two events. Process theory emphasizes the idea of causal mechanisms, and it focuses on the underlying processes and structures that give rise to causation.
Pluralistic Theories:
Inferentialist Theory: There is no essence to causality. We have a range of expressions in science marking the influence of one entity on another, such as governs, disrupts, transduces, activates, eliminates, stimulates. What is in common is a certain family resemblance in the inferential practices that appeal to such terms to allow prediction, explanation and control.
Potential outcome reasoning
The tale wagged by the DAG: broadening the scope of causal inference and explanation for epidemiology Nancy Krieger, George Davey Smith International Journal of Epidemiology, Volume 45, Issue 6, December 2016, Pages 1787–1808, https://doi.org/10.1093/ije/dyw114
From the standpoint of IBE, ‘causal inference’ cannot be reduced to what the philosopher Stathis Psillos has termed ‘topic-neutral and context-insensitive’ algorithms (p. 441)79, whether involving deductive logic or Bayesian statistics. Core to IBE is the understanding that there are no clear-cut rules or short cuts that minimize the need to amass substantive expertise and to generate and think critically about contrastive hypotheses—but nor is it the case that ‘anything goes’.
“usatives represent a myriad of metaphysical kinds’‘ Reiss
governs, disrupts, transduces, activates, eliminates, stimulates,
Reiss’ inferentialist account
Structural mechanisms - context (Strevens 2012); Process-interaction causal activities; Mackie INUS - support factors.
Taco Cohen, Categorical causality, (talk)
Reiss, Cause, causatives and theories of causation’.
Aleks Kissinger, Sander Uijlen , A categorical semantics for causal structure, Logical Methods in Computer Science, 2019
Robin Lorenz, Sean Tull, Causal models in string diagrams (arXiv:2304.07638)
Pim de Haan, Cyclic Causal Networks via Partial Markov Categories
Robin Piedeleu, Basic causal inference via string diagrams
Bart Jacobs, Aleks Kissinger, and Fabio Zanasi. Causal inference via string diagram surgery. Math. Structures Comput. Sci., 31(5):553–574, 2021. arXiv:1811.08338.
Tobias Fritz and Andreas Klingler. The d-separation criterion in categorical probability. J. Mach. Learn. Res., 24(46):1–49, 2023. arXiv:2207.05740.
Julian Reiss, ‘http://www.jreiss.org/Presentations/HK_Cause%2C%20Causatives%2C%20Causa-tion.pdf
Cartwright, middle-range law mechanisms
We have Judea Pearl develop the do-calculus for Bayesian networks. ACT has
Free gs-monoidal categories and free Markov categories Tobias Fritz, Wendong Liang
ACT has developed considerable tools to represent the internal functioning of a dynamical system and its interface with its environment, systems can interact through well-defined interfaces.
I have been working on a philosophical understanding of Category Theory since my Masters degree over 30 years ago. A chapter in my 2003 book treated higher category theory. In 2006 I co-founded a blog, The n-Category Cafe, with two mathematical physicists to explore the implications of category thoery and higher category theory for mathematics, physics and philosophy. From the early days we considered topics in probability/
Symposium on Compositional Structures, Invited to SYCO 1, invitation to Topos Institute.
Split time between Topos Institute and Oxford
One way in which category theory can be applied to the study of causality is by providing a mathematical framework for understanding causal relationships between events. This involves identifying the causal structure of a system, which can be modeled as a category, with objects representing events and morphisms representing causal relationships between them. In this way, category theory can provide a way to represent and reason about causal relationships in a rigorous and formal manner.
Another application of category theory to causality is in the study of complex systems, where causal relationships between events are often difficult to identify and analyze. Category theory provides a way to model complex systems as categories and to analyze their structure using mathematical tools such as functors and natural transformations. This can help to identify and understand the causal relationships between events in a complex system, as well as to identify emergent properties that arise from the interactions between events.
Category theory can also be applied to the study of causality in the context of quantum mechanics, where traditional notions of causality are challenged by the non-local and probabilistic nature of quantum systems. In this context, category theory can provide a way to model and analyze the causal relationships between events in a quantum system, as well as to develop new mathematical tools for understanding and predicting quantum phenomena.
• Sophie Libkind, Andrew Baas, Micah Halter, Evan Patterson and James Fairbanks, An algebraic framework for structured epidemic modeling, Philosophical Transactions of the Royal Society A 380 (2022), 20210309.
John Baez, Xiaoyan Li, Sophie Libkind, Nathaniel D. Osgood and Eric Redekopp, A categorical framework for modeling with stock and flow diagrams. https://arxiv.org/abs/2211.01290
Andrew Baas, James Fairbanks, Micah Halter, Sophie Libkind and Evan Patterson, An algebraic framework for structured epidemic modeling. https://arxiv.org/abs/2203.16345
Last revised on June 28, 2023 at 06:33:24. See the history of this page for a list of all contributions to it.