[hosh]

For most of my years in primary education, math was easy. Until it was not. I ran headlong into the idea of "instantaneous rate of change", and I was confronted with the existential crises that there is nothing inherently concrete or "real" about the idea of instantaneous rate of change. My mind just got stuck on there, and I nearly failed the course in high school.

Everything after that point was not easy, and somewhere, there was a belief forming that maybe I was not so good at math.

Two things I encountered changed my perspective on that. The first was reading a math graduate student's experience where they talk about struggling with concepts, like groping about in a dark room until one day, you find the light switch and you see it all come together. The other is Kalid Azad's "Math, Better Explained" series. His approach is to introduce the intuition first, before going into the rigor. Between those two, I realized that I had never really learned how to learn math.

Once I started with the intuition, I also realized why people get stuck on things. There are people who don't get negative numbers. They keep looking for something concrete, as if there were something intrinsic to reality that makes negative numbers "real". And there isn't. And yet, in my day-to-day life, I work with negative numbers. I am far better able to reason and navigate the modern world because I grok the intuition of negative numbers.

Then there are the cultures where there isn't even a notion of natural numbers. Their counting system is, "one", "two", and "many". In their day to day life, to echo your sentiment, they can go on their whole life without the "insight" that things can be countable. Many of them probably won't even care that they don't have the intuition of countable things. And yet, in this modern culture, I find myself introducing the idea to my toddler.

Ultimately, it's up to each individual's decision how far they want to go with this. Each culture and civilization has a norm of a minimum set of intuitions in order for that person to navigate that civilization. Category Theory is outside that norm, even among the software engineer subculture. Perhaps one day, it will be the norm, but not today. Beyond that, no one can make you grok a mathematical intuition, as useful as they are once grokked.

[mettamage]

I am not great at math. But I learned about complex numbers for fun. It took a bit to make them “real” for me, 3B1B helped a lot as did asking myself how I find negative numbers real (negative integer: if something in the future will exist, it won’t and the negative integer will be incremented, aka a debt to the future existence of whatever the number represents).

Complex numbers: the number line is just a metaphor. What would happen if you make it 2D? Oh you can solve negative roots. Whenever a number needs to spin or have a negative root, it’s a useful tool. Numbers are tools. Cool, that’s “real” enough for me.

I know I will never ever use it. Or at least, not in my current state of how I live my life. I liked learning it, there’s something to be said for what is beyond your horizon. But I do think just like complex numbers, category theory needs a motivation that is compelling. In retrospect, learning complex numbers is most likely not useful for me.

Oh wait a second, complex numbers helped me to understand trig. I never understood it in high school but 3B1B made it intuitive with complex numbers

Nevermind, I’ll just let this all stand here. It’s fun to be open and convinced by myself to the other side while writing my comment :’)

I’m sure if I would have learned category theory, I would have written a similar comment as I think there are a lot of parallels there.

[hosh]

Azad has this description of imaginary numbers that really tickled me: “numbers can rotate”.

I remember my high school teacher teaching imaginary numbers in the trig class, and one of the other kids asked what they could be used for. This was an honors class and the kid skipped a math grade. Our math teacher couldn’t talk about it off the bat, and unconvincingly told us about applications in electrical and electronic engineering. We still went through it, but none of use knew why any of it was relevant.

I think if he had said “numbers can rotate”, maybe some of us (maybe not me!) might have been intrigued by the idea. It would have been a way to describe how EM rotate.

My own personal motivation for pursuing CT has to do with working with paradigms, and how are they related, and how they are not. Categories and morphisms seem to talk about the kind of things we can talk about with paradigms, yet much more precisely.