Category Theory

Welcome to the Active Inference Institute. This is the participatory online laboratory. Today is Tuesday, September 13th, and we are here with Math Stream 4.1 with David Spivak. And we are welcome to take any questions in the YouTube chat. So if you have any questions, please write them there and we will have time to ask our presenter during the course of this discussion. David is here to talk to us today about dynamic interfaces and arrangements and algebraic framework for interacting systems. Please take it away. David Spivak Okay, yeah, thanks a lot for having me. So I, I'm going to talk for around 40 minutes, but maybe almost an hour, I don't know. But feel free to interrupt with whatever. And I wanted to say at the outset that I'm not really talking much about active inference itself here. I'm mainly saying like another perspective on interacting systems, or systems that might learn or something like that. And I'm more like waiting for the discussion time to kind of find these links. So yeah. David Spivak Okay, so I want to talk about dynamic interfaces and arrangements and what that means and this applied category theory point of view on things. So, so why am I here? David Spivak I'm here. I'm here to seek a valuable exchange of ideas, because active inference offers first time I heard about it. It's compelling, and most people apparently find it unusual, but for some reason, it instantly meshed with my own intuition about things that I've gotten from applied category theory. And so I think there's a lot we can learn from each other. David Spivak So applied category theory emphasizes structure rather than quantitative analysis. So most of math and science today reduce all of our experience to plays of quantity. So you have graphs, and you have numbers, and you measure things and numbers, and you have probabilities, and those are numbers. And so everything is just a number, but experience what's happening now. I mean, what number would you put on, you know, the sound of my voice or the, or what numbers you could put numbers on this slide, but would they David Spivak Tell you what the slide was about. And so what category theory emphasizes is still math, but it emphasizes structure. It emphasizes relations between things like the relationship between what's on this slide, and other things on the slide, and what's on the rest of the talk, the relationships between that and me, the relationship between that and you, and the kind of coherence between all that, like why are we really here? What is it? How does it cohere? How does it come together? So applied category theory aims to bring that math to relational disciplines outside math. So I want to tell you about applied category theory, specifically in reference to interfaces, because I'm going to say that we interact through our interfaces. When you hear the word interface, you might think Markov blanket, I don't know enough to be sure that that's exactly what I mean. So that's kind of where this is fitting in maybe into an active inference framing, but I'd rather you just think about interfaces. So I'm asking in this talk, what sort of algebra could support us in thinking about interaction and stuff like that. I think of mathematical fields as accounting systems. So I like that the word count is in there. And I like how kind of humble it is. It's like an accounting system. So if you want to, if you have arithmetic, that's a mathematical field. And it accounts for the flow of quantities like in finance. So if you want to, if you want to, if you want to, if you want to, if you want to write arithmetic account, you'd write an arithmetic account of it. You'd say, and they could, someone could point that and say, what's this $12.83. And you could explain. So there's numbers and you can account for stuff. What if you wanted to account for the teleportation protocol of quantum mechanics or something like that. So Hilbert spaces is a, is a branch of math and it helps you make…