Eugenia Cheng, in a March 2023 interview-poscast by Steven Strogatz titled "Is There Math Beyond the Equal Sign?", describes the purpose of Category Theory:

... it’s really about spotting similarities between different situations, and then finding a unified way to think about them — essentially so that we can use our brains better. Because our poor, finite brains are really very small compared with the complexity of the world that we are trying to understand. And one way to deal with that complexity is just to willfully ignore parts of it. And unfortunately, that is quite a common way of doing it.

But I believe that a better way to deal with the complexity of the world is to take a broader view of things and to find similarities between different situations, so that you can study many things at the same time at a certain level. And I think that’s what you mean by unification. We’re not trying to declare things are the same when they’re not. We’re trying to find some deep essence about them that they have in common, so that we can at least study that part as the same thing before then zooming back in on the individual situations to look at the details. ...

... and the joy of abstraction:

... It’s not the usefulness of abstraction — it’s the joy of abstraction. Because to me, it really is a joyful process. It’s like shining light on things. You know, I just went outside today, and the sun was shining so brightly, and that just gave me joy, not because I’m looking for something, and I can see it better. You know, if I dropped something on the street and I was looking for it, then it, yes, it would be helpful that the sun was shining. But it’s just nice, isn’t it, to be able to see things clearly. And that’s what I love about abstraction. What it feels like to me is that there’s kind of fog everywhere. And then when you perform an abstraction, you’re clearing away irrelevant details, as you say, and then you can see things more clearly. And that, to me is joyful. ...

... and on how different people have different styles of learning, some preferring to generalize from the specific, and others to go vice versa:

... I felt personally that it would have helped me understand all the other things if I had done category theory first, rather than using the other things as a jumping point to understand category theory. And the thing is that — radical thought — everyone’s different. And so some people understand things via the other parts of mathematics. And some people understand other parts of mathematics via category theory.

And I think one of the big problems with the way that mathematics education is at the moment is that there’s a belief that there’s a certain order that you have to do mathematics. And so I’ve described it in my book at the beginning as a series of hurdles. If you think that math is a series of hurdles, and that they get higher and higher as you go along, then yes, indeed, there’s not much point trying to get over a higher hurdle if you can’t get over the lower hurdle.

But the thing is, math is not actually a series of hurdles. It’s an interconnected network of ideas. And so there are many different paths around that. Because everything is connected. You can go in all sorts of different routes around that network. And here’s the radical thought, again: Different routes will suit different people in different ways. And it’s like just how you present mathematics in the first place. Some people like seeing specific examples first, and then looking at the general theory, based off their understanding of their specific examples.

But I prefer seeing general theory, and then using the specific examples afterwards, to help me while using the general theory to help me understand the specific examples. And when I go to research seminars, when the seminar starts with the examples and does the theory afterwards, I always feel like I want to watch the seminar backwards. So then I have to, I have to hold the examples in my brain and ignore them, listen to the general theory, and then rapidly try and rewind and go over the examples afterwards. ...

^z - 2023-03-26