Why I am learning category theory

Re:Why I am learning category theory by larve on Wednesday November 30, 2022 @12:24PM (#33804072)

author here, for context, I do have a reasonable background in maths (algebra, analysis, statistics) at a CS master-ish level (self-taught and a long time ago, though), as well as spent quite some time with programming language / type theory when I was younger, and I do use monads quite a bit in my day to day programming. In fact, the fundamental algebra concepts (rings, groups, etc...) as well as fundamental CS theory (grammars, proof theory, semantics, ...) were the only maths I felt reasonably comfortable with, compared to say, calculus or statistics which really did my head in and ultimately caused me to drop out of university.

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Re:Why I am learning category theory by Karrot_Kream on Wednesday November 30, 2022 @02:47PM (#33806535)

How much do you think your predilection for algebra affected your interests? Most folks who do CT as a hobby that I know are much weaker in analytical concepts than any working mathematicians that I knew.

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Re:Why I am learning category theory by larve on Wednesday November 30, 2022 @02:50PM (#33806589)

Honestly I'm pretty weak in both, but they seem to reflect my intuitions, if that makes sense? I'm nowhere near a working mathematician, and proofs usually bore me. I prefer writing code.

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Re:Why I am learning category theory by Karrot_Kream on Wednesday November 30, 2022 @02:52PM (#33806634)

If you're looking for an alternate perspective, I would suggest taking an advanced undergraduate analysis book and working through it, proofs and all. It'll teach you the same kinds of stuff that you're getting out of CT but will give you "working" (lol) knowledge of how to apply it. A lot of mathematical work is about creating math objects and relations and proving properties of these objects through their relations. The exercises in a good algebra book will exercise this muscle and make your software organization better, at least IMO. Learning the definitions just isn't a substitute.

I can see why categories appeal if you're just looking at definitions, but CT often just abstracts/standardizes an approach that folks were already working with when working with rings, groups, topologies, etc. Going through the proofs will teach you when to apply what.

Aluffi (Algebra: Chapter 0) makes a great book that teaches the basics of CT and brings it up while teaching regular algebra. I highly recommend it.

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Re:Why I am learning category theory by larve on Wednesday November 30, 2022 @03:06PM (#33806856)

I studied CS in germany in 1999 and we had "analysis 1+2+3" which I think was a fairly rigorous analysis treatment, I definitely did more than my fair share of proofs. I think that foundation accompanied me through all those years of programming and helped me read and pick up concepts related to type theory, algebra and functional programming.

In parallel I developed this very "visual" way of thinking about structure that makes me really resonate with CT. In a way I'm less interested in learning new maths, more so than learning more about how people approach (complex) structures I already know with these very rudimentary concepts, and seeing how their formalizations match my "folk mathematics".

I'm having a lot of fun!

edit: thanks for the book recommendation! that looks like my kind of maths book

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Re:Why I am learning category theory by larve on Wednesday November 30, 2022 @03:20PM (#33807043)

i skimmed through aluffi, and it is indeed exactly the material I had at university in analysis and algebra, augmented with the category theory angle. this is great, thanks a lot. Now I have to find an affordable copy somewhere.

I also like the focus on quotient groups and galois fields, as well as using graphs as recurring examples, which have been the only times I really put the pedal to the maths metal in my programming (for some crypto and error correction stuff).

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Re:Why I am learning category theory by bmitc on Wednesday November 30, 2022 @05:18PM (#33808687)

How do you know analysis at a "CS master-ish level" (not sure what that means given it's a mathematical subject) but calculus did your head in? I'm not being judgmental but am just genuinely confused given that analysis is the foundation for calculus. :)

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Re:Why I am learning category theory by athrowaway12 on Wednesday November 30, 2022 @05:41PM (#33809011)

Yeah, this is some B.S. here. I have a degree in CS and math. It sounds like parent comment doesn't really know analysis. The exercises in Rudin 1 are way harder than memorizing some rules to evaluate integrals.

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Re:Why I am learning category theory by larve on Wednesday November 30, 2022 @05:55PM (#33809198)

That's how I remember it. I know I had to take calculus twice, but passed analysis the first time. To be fair, this is 20+ years ago and my memory sucks at the best of times.

Looking at it now, I think maybe I just found it too boring to remember the tricks, and too tedious to not make mistakes, while I found coming up with proofs more "interesting" and requiring less rote memorization.

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Re:Why I am learning category theory by athrowaway12 on Wednesday November 30, 2022 @05:59PM (#33809240)

Ah, for sure pure math is much more enjoyable to learn. I'm glad you find math interesting. Especially as a writer; that's really cool. Sorry for my previous comment.

I wish the university calculus changed the curriculum. I didn't find it particularly useful to learn how to integrate e.g. inverse trig given that WolframAlpha exists. There are lots of other examples of teaching memorization of silly tricks like that. I guess they probably do it because it's easy to test.

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Re:Why I am learning category theory by larve on Wednesday November 30, 2022 @06:04PM (#33809285)

No offense taken, I totally understand that it can come across as odd. I hope my clarification makes more sense. Once I learned more about say, why leibniz and newton came up with their formulations, and what say, the inverse of the square root of the determinant of the jacobian actually allows one to formulate (how a space "dilate" when a continuous distortion is applied to it, for example), calculus (applied calculus) became a much more interesting topic for me.

While I only got so far in proving stuff about a cyber-physical I was working on, but this was really fun to learn: https://github.com/LS-Lab/KeYmaeraX-release

See also: http://lfcps.org/lfcps/

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Re:Why I am learning category theory by larve on Wednesday November 30, 2022 @05:23PM (#33808753)

Calculus (at least our exams and lecture) were so much about "calculating" and applying rules and clever tricks on complex functions and multiple dimensions which just... didn't work for me. I had a lot of other things going on, so this recollection is probably off. Calculus and diff eqs started making sense to me when I started working in mathematica and systemmodeler, and realizing that it was a brilliant tool. It probably also came from the way it was taught, which was completely devoid of any historical or practical context. It was just a lot of "compute the laplacian of e^i_sin(phi) or whatever.

I found groups, limits, continuity, fields to be much more amenable to my thinking. It felt way more similar to the things I really enjoyed (building software in scheme and C and writing exploits in assembly).

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Re:Why I am learning category theory by bmitc on Wednesday November 30, 2022 @05:27PM (#33808817)

Oh, I see. So it seems it was more about the presentation and interest rather than the concepts. You may like Calculus by Michael Spivak if you're interested in it again. It is much more analytical than computational.

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Re:Why I am learning category theory by larve on Wednesday November 30, 2022 @05:35PM (#33808927)

That's one of the books I used when I got back into it. These days, there is so much great material on youtube and edx and coursera (plus maths stackexchange and discords) that it's absolutely fantastic to learn maths, as fantastic instructors are immediately available.

For me, I realized thst I need to write the code to accompany the maths, because it helps me disambiguate mathematical notation and put "types" on things.

It helped me a lot to work alongside real maths PhDs and realize that they have a very "intuitive" approach to maths, and don't necessarily yeet lemmas and proofs around all day, instead spending a lot of time brainstorming on the whiteboard and discussing things out, very similarly to software design. It was one of them telling me that my thinking was very mathematical that helped me "deconstruct" the fear of maths that I had built up.

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Re:Why I am learning category theory by bmitc on Wednesday November 30, 2022 @06:08PM (#33809326)

I agree that using programming to understand a domain is very useful, as it’s something I’m trying to do a lot of these days. Although sometimes, there needs to be some simplification to do that that maybe skirts around the meat of the matter. But as always, a diverse approach is more powerful than a singular one.

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Re:Why I am learning category theory by Tainnor on Wednesday November 30, 2022 @05:19PM (#33808700)

It's a common observation that many mathematicians and maths students tend to fall into either the algebraic or the analytic camp, although of course, there are more nuances and some people are genuinely good at both.

I myself find beauty in some parts of analysis, but algebra definitely speaks more to me (e.g. I find a proof of Lagrange's Theorem more beautiful than one of the Intermediate Value Theorem). So I can relate.

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Re:Why I am learning category theory by SleekEagle on Wednesday November 30, 2022 @07:05PM (#33810050)

> It's a common observation that many mathematicians and maths students tend to fall into either the algebraic or the analytic camp

(we don't talk about the topologists)

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Re:Why I am learning category theory by Koshkin on Wednesday November 30, 2022 @08:47PM (#33811178)

Aren’t all topologists algebraic topologists today?

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Re:Why I am learning category theory by Karrot_Kream on Thursday December 01, 2022 @06:31PM (#33824601)

No way? It's a pretty even split between analysts and algebraists from what I can tell. Really depends on what you're researching. Only the internet and online CS-adjacent communities have these crazy algebra-dominated communities.

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