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Why I am learning category theory

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Re: Why I am learning category theory

#71

Post by larve »

Karrot_Kream wrote: post #67
larve wrote: post #9

Earlier quoted context omitted.

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 theo…

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.

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.

Re: Why I am learning category theory

#72

Post by khazhoux »

As a programmer and hobbyist math reader, I found category theory to be very unrewarding (and I gave up on it) because of the lack of interesting theorems and lemmas. My takeaway was that there's Yoneda lemma and really nothing interesting before you reach that. Like, CT describes a set of rules but very little emerges from those rules.

My complaint has nothing to do with whether CT is useful or practical. By contrast, abstract algebra can be taught (e.g., in Herstein's books) as pure abstraction (very dry and without presenting any real-world connections), but you reach Lagrange's theorem right away -- which is an simple-but-awesome result that will wake up your brain. You reach Cayley's theorem and others quickly, and each is more exciting that the last. And this is all while still in the realm of purely-abstract math.

Re: Why I am learning category theory

#73

Post by thr3ads »

I became very interested in CT from a biological perspective. The work originates from Robert Rosen who describes how biological/complex systems are not (completely) reducible to formal systems (aka any model of a natural system will always be incomplete+it is impossible to explicitly list out all the rules governing the system) because they "contain a model of themselves" to which they anticipate the future and act on towards their benifit.

Many relate his work as closely tied to Gödel's incompleteness theorem but for biology. The purpose of using category theory is that it is general enough to not talk about specific parts of an organism's construction, but rather how their general functional parts relate to each other.

Re: Why I am learning category theory

#74

Post by Karrot_Kream »

larve wrote: post #71
Karrot_Kream wrote: post #67

Earlier quoted context omitted.

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.

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.

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.

Re: Why I am learning category theory

#75

Post by solomatov »

khazhoux wrote: post #72

As a programmer and hobbyist math reader, I found category theory to be very unrewarding (and I gave up on it) because of the lack of interesting theorems and lemmas. My takeaway was that there's Yoneda lemma and really nothing interesting before you reach that. Like, CT describes a set of rules but very little emerges from those rules. My complaint has nothing to do with whether CT is useful or practical. By contras…

> As a programmer and hobbyist math reader, I found category theory to be very unrewarding (and I gave up on it) because of the lack of interesting theorems and lemmas. My takeaway was that there's Yoneda lemma and really nothing interesting before you reach that

One of the most interesting things in CT are adjoints. They happen literally everywhere. For example, static analysis via abstract interpretation is an example of adjoint (which in this case is called Galois connection). Free structures also give rise to adjoints.

Re: Why I am learning category theory

#76

Post by Koshkin »

BenoitP wrote: post #29
hosh wrote: post #25

Earlier quoted context omitted.

In Ruby and many other languages, you have this idea of a string concatenation: "foo" + "bar" -> "foobar" "foo" + "" -> "foo" That makes it a monoid. Instead of talking about OOP patterns, knowing that the "+" operator is a monoid for string objects lets us write code that is composable. Similarly, with arrays: [:foo,:bar] + [:baz] -> [:foo,:bar,:baz] [:foo,:bar] + [] -> [:foo,:bar] Some language platforms will imple…

I see no more than Groups in your comment. Which are very useful! Also having commutative operations makes for Abelian Groups, which enable operations to be reordered, and makes for implicit parallelism. Where is Category Theory?

Actually, it is weird to see groups mentioned in this discussion at all. Non-commutative (generally) groups are useful in dealing with symmetries, but what symmetries can we talk about here? Abelian groups (and modules in general), on the other hand, are completely different beasts (seemingly only useful in homological algebra, algebraic geometry, and topology).

Strings are a (free) monoid with respect to concatenation, sure, but it is easier to learn what a monoid is using strings as an example, rather to try and "learn" about strings by discussing monoids first. Why this is deemed useful by some is beyond me.

Re: Why I am learning category theory

#77

Post by adamwk »

hosh wrote: post #25
i_am_toaster wrote: post #21

I would be willing to drink the kool-aid if I saw it being used in a practical way. I always feel these posts are filled with category theory jargon without ever explaining why any of the jargon is relevant or useful. I’ve even watched some applied category theory courses online and have yet to feel I’ve gained anything substantive from them. However, as I started off with, I’m always willing to try something out or…

In Ruby and many other languages, you have this idea of a string concatenation: "foo" + "bar" -> "foobar" "foo" + "" -> "foo" That makes it a monoid. Instead of talking about OOP patterns, knowing that the "+" operator is a monoid for string objects lets us write code that is composable. Similarly, with arrays: [:foo,:bar] + [:baz] -> [:foo,:bar,:baz] [:foo,:bar] + [] -> [:foo,:bar] Some language platforms will imple…

What language only allows concatenating a non-empty array?

And it still doesn't explain why string concatenation being a monoid is useful in ruby. It's useful in Haskell because implementing one of their category-theory typeclasses means that type now inherits a stdlib-sized API that works for any monoid, not concrete types. But even Haskell hasn't proven that it's worth the jargon or abstraction after all these years; every other language remains productive without designing APIs at such a high level of abstraction.

Re: Why I am learning category theory

#78

Post by rssoconnor »

Tainnor wrote: post #7

As someone with a maths degree, yet who admittedly hasn't looked into category theory beyond some basic notions, I still don't quite understand why anyone would want to learn category theory before e.g. abstract algebra or even just fundamental mathematical reasoning (definition, theorem, proof). Maybe I'm missing something but it seems to me that all you can study monads in programming languages without having to al…

I agree with you. CT seems mostly useful as a way of devising entirely new abstractions, but once those abstractions are developed, you don't need CT to use them.

For example if it were 2008 and you want to be inventor Applicative functors for use in Haskell, then knowing lax monoidal functors from CT might be helpful. But if you want to just use Applicative functors, you don't need to learn lax monoidal functors first.

So programmers probably don't need to learn CT because they can just let the computer scientists devise their abstractions for them. But if you want to be a computer scientist and one day devise your own abstractions[0], then maybe CT would be helpful.

[0]https://news.ycombinator.com/item?id=33805923

Re: Why I am learning category theory

#80

Post by Koshkin »

solomatov wrote: post #75
khazhoux wrote: post #72

As a programmer and hobbyist math reader, I found category theory to be very unrewarding (and I gave up on it) because of the lack of interesting theorems and lemmas. My takeaway was that there's Yoneda lemma and really nothing interesting before you reach that. Like, CT describes a set of rules but very little emerges from those rules. My complaint has nothing to do with whether CT is useful or practical. By contras…

> As a programmer and hobbyist math reader, I found category theory to be very unrewarding (and I gave up on it) because of the lack of interesting theorems and lemmas. My takeaway was that there's Yoneda lemma and really nothing interesting before you reach that One of the most interesting things in CT are adjoints. They happen literally everywhere. For example, static analysis via abstract interpretation is an exam…

Sure, but what does the knowledge of something being an adjoint give you?
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