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Hi,

I am a mathematician (less so) and a cognitive scientist (more so) and I have been participating in and arguing with David and Brendan in their applied category theory seminar at MIT for the last couple of month so it will be a little strange to think through their ideas in their absence.

I have two ideas that I have been playing with that I am hoping is of interest to this community as well:

- Categorical foundations of cognition. In other words, is category theory the right framework for modeling (some, if not all) cognitive phenomena? I have myself used category theory to understand how language and perception might be related to each other. Is it uniquely suited to do so? How is it better than other tools and frameworks?
- Philosophical category theory. I don't mean category theory for philosophers or the philosophy of category theory, but something more like philosophical logic, with "logic" replaced by "category theory." Just as philosophical logic is the philosophical exploration of logical ideas and principles (and in that guise much older than mathematical logic), we might want to ask if there are philosophical ideas and principles that are closely tied to category theory - identity is an obvious example.

Looking forward to learning more from John and everyone else.

Rajesh

## Comments

Hi Rajesh, thanks for sharing these interesting ideas.

I don't know much about cognitive science, but I do find it intriguing, so I'll be curious to see how this pans out.

Although I'm not sure what you mean by "philosophical category theory," a lot has been written about the philosophical implications of category theory (e.g., see the references in the SEP entry).

`Hi Rajesh, thanks for sharing these interesting ideas. 1. I don't know much about cognitive science, but I do find it intriguing, so I'll be curious to see how this pans out. 2. Although I'm not sure what you mean by "philosophical category theory," a lot has been written about the philosophical implications of category theory (e.g., see the references in the [SEP entry](https://plato.stanford.edu/entries/category-theory/#3)).`

Hi, Rajesh! What sort of talks are there at Brendan and David's seminar? Brendan hasn't told me about it..

Here are some shoot-from-the-hip comments about your ideas:

I think there's great potential for using category theory in

allbranches of science, because it's sort of "Set theory 2.0" - that is, a comprehensive framework for formalizing things, which subsumes and goes far beyond set theory. But it takes a lot of work to exploit this potential: a good solid knowledge of category theory, a good solid knowledge of the intended application, and a lot of time spent figuring out how to connect them. It could work better with two people - a category theorist who knows a bit about the intended application and an expert in the intended application who knows some category theory - but they would need to be committed to it. I've been focusing on applications where I can play both parts.The categorical approach to logic is deeply philosophical at its core: it's based on new insights that go beyond the early 20th century work by Russell, Zermelo, Frenkel, von Neumann, Gödel

et al. Some of the ideas are introduced in Elaine Landry's bookCategories for the Working Philosopher. Have you read that?`Hi, Rajesh! What sort of talks are there at Brendan and David's seminar? Brendan hasn't told me about it.. Here are some shoot-from-the-hip comments about your ideas: 1. I think there's great potential for using category theory in _all_ branches of science, because it's sort of "Set theory 2.0" - that is, a comprehensive framework for formalizing things, which subsumes and goes far beyond set theory. But it takes a lot of work to exploit this potential: a good solid knowledge of category theory, a good solid knowledge of the intended application, and a lot of time spent figuring out how to connect them. It could work better with two people - a category theorist who knows a bit about the intended application and an expert in the intended application who knows some category theory - but they would need to be committed to it. I've been focusing on applications where I can play both parts. 2. The categorical approach to logic is deeply philosophical at its core: it's based on new insights that go beyond the early 20th century work by Russell, Zermelo, Frenkel, von Neumann, Gödel _et al_. Some of the ideas are introduced in Elaine Landry's book _[Categories for the Working Philosopher](https://global.oup.com/academic/product/categories-for-the-working-philosopher-9780198748991?cc=us&lang=en&)_. Have you read that?`

Thanks for the comments, Evan and John. Brendan and David's seminar is an informal group that meets every week. It's mostly been one of them presenting other people's work or (less often) their own work - think of it as an expanded journal club. You can see my response to one of the seminars here.

Some quick further thoughts:

Category Theory in Cognitive Science has a longish but mostly neglected history - Michael Arbib wrote a book on Category Theory which is perhaps the first for a neuroscientist. Then there's the book edited by McNamara and Reyes on the Logical Foundations of Cognition, but with quite a bit of category theory in it. There aren't that many examples of what John calls "a good solid knowledge of category theory, a good solid knowledge of the intended application, and a lot of time spent figuring out how to connect them"

I think what I am trying to get to - purely speculatively for now - isn't a categorical take on logic but a C-theory (think M-theory but with a C) that uses Category Theory to model C-phenomena that may be quite different from logical phenomena. That said, I do have the Elaine Landry edited volume with me and since I have barely started the introduction, I should read the book first.

`Thanks for the comments, Evan and John. Brendan and David's seminar is an informal group that meets every week. It's mostly been one of them presenting other people's work or (less often) their own work - think of it as an expanded journal club. [You can see my response to one of the seminars here](https://categoricalmodeling.wordpress.com/2018/02/22/categorical-modeling/). Some quick further thoughts: 1. Category Theory in Cognitive Science has a longish but mostly neglected history - [Michael Arbib](https://en.wikipedia.org/wiki/Michael_A._Arbib) wrote a book on [Category Theory](https://en.wikipedia.org/wiki/Michael_A._Arbib) which is perhaps the first for a neuroscientist. Then there's the book edited by McNamara and Reyes on the [Logical Foundations of Cognition](https://books.google.com/books?id=Outqx9VKqIMC&printsec=frontcover#v=onepage&q&f=false), but with quite a bit of category theory in it. There aren't that many examples of what John calls "a good solid knowledge of category theory, a good solid knowledge of the intended application, and a lot of time spent figuring out how to connect them" 2. I think what I am trying to get to - purely speculatively for now - isn't a categorical take on logic but a C-theory (think M-theory but with a C) that uses Category Theory to model C-phenomena that may be quite different from logical phenomena. That said, I do have the Elaine Landry edited volume with me and since I have barely started the introduction, I should read the book first.`