Paul Lessard: observation

Transcript context

…But you're probably thinking, why do we need to use such abstract mathematics in the first place? Well, this kind of structured thinking can actually help us see what actually matters. To introduce the notion of structuralist mathematics, and then the best known example of structure structuralist mathematics category theory, I would like to begin with the distinction between analytic mathematics and synthetic mathematics. And the usual 2 examples that are given for this is the geometry of Descartes versus the geometry of Euclid. For Descartes, lines are solution sets to equations, whereas for Euclid, lines are precisely that, which is stretched out between 2 points. What's the essential distinction here? In analytic mathematics, stuff is made of stuff. Right? There's always some question of, okay, I have to have some common foundation from which everything is built, and all of my lemmas, theorems, etcetera, everything eventually boils down to a computation in that more basic substance. On the other hand, in synthetic mathematics, it's like I don't need to go I don't need to know what, like, the inside of a line is. That doesn't matter. The point is I only abstract what are the principles by which I can make inference on lines and their relationships to each other. Right? The point is you get rid of everything that is inaccessible to your logic. Right? So you get rid of all of this stuff that you might call detail, but it's not even detail, it's noise. It because it doesn't have any content for that which you can know. Right? And so therefore, it's completely irrelevant. So synthetic mathematics gets rid of all of that and focuses just on how you can produce more knowledge. Right? And so then to explain at least what I mean by a structuralist mathematics, I want a synthetic mathematics of structure. Right? What is structure? You know, in the context of machine learning, we've got lots of notion. Everyone says structure. Right? The best known and best studied example of that structure is group actions. Right? All of geometric deep learning is about group actions. But that's only 1 That's like 1 pinprick in the vault. Right? That's just like 1 thing. Right? There are lots of other things, say, that come from theoretical computer science. The notion of thing things being lists, or, you know, things being trees. All of these other various, like, algebraic structures. So you want, like, a single, you know, language in which all of these various kinds of structure can be described elegantly. This is exactly why we appeal to category theory. In in the simplest possible terms, what do you mean by 2 category?…

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