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Why I am learning category theory

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Re: Why I am learning category theory

#121

Post by Tainnor »

larve wrote: post #9
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…

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…

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.

Re: Why I am learning category theory

#122

Post by bmitc »

School-Cotton wrote: post #14
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…

> 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 Because people think “category theory” means “abstract math” in general, due to cargo-culting in and around the Haskell community.

I also have a math background but am now a software engineer, and I agree. One can learn category theory all by itself, but it's relatively unimportant in software and computer science as far as I can tell. It's really only powerful as a unifying, abstraction, and conceptual relating tool in upper mathematics where the divisions of mathematics begin to blur.

Re: Why I am learning category theory

#123

Post by larve »

bmitc wrote: post #120
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 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. :)

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).

Re: Why I am learning category theory

#125

Post by bmitc »

larve wrote: post #123
bmitc wrote: post #120

Earlier quoted context omitted.

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. :)

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…

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.

Re: Why I am learning category theory

#126

Post by Tainnor »

School-Cotton wrote: post #14
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…

> 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 Because people think “category theory” means “abstract math” in general, due to cargo-culting in and around the Haskell community.

I mean absolutely no disrespect to anyone doing serious work in CT or in Haskell - and I'm actually very interested in Haskell as a language per se and have been exploring it more in recent months - but having spent some time in Haskell communities, I have to agree.

There's a lot of empty posturing by people who don't really seem to understand a lot of mathematics but still seem to have very strong convictions about certain aspects of it (constructivism is another such thing).

Re: Why I am learning category theory

#127

Post by Tainnor »

zozbot234 wrote: post #19
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…

> or even just fundamental mathematical reasoning (definition, theorem, proof). Logic and proof theory is actually one of the areas in math where category theory is most clearly useful. Intuitively, this is because reflexivity ("A leads to A") and transitivity ("A leads to B and B leads to C, hence A leads to C") are well-known rules of any kind of proof writing or indeed of argumentation in general, and they exactly…

I know a fair bit about logic and none of it involves any CT. That's not to say you can't find it in there somewhere as a unifying concept, but the fundamental theorems such as completeness and compactness of FOL, basic model theory, Gödel's theorems, etc. do not require any CT.

Also, reflexivity, transitivity and symmetry just define an equivalence relation, no need for CT there.

Re: Why I am learning category theory

#128

Post by larve »

bmitc wrote: post #125
larve wrote: post #123

Earlier quoted context omitted.

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…

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.

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.

Re: Why I am learning category theory

#129

Post by charcircuit »

Mimmy wrote: post #12
charcircuit wrote: post #8

Honestly, this is a waste of time and is not going to help as much as the author thinks it is. There is no secret from category theory that will make one's design 10x better. Not everything is even applicable. >I hope that category theory will allow me to formalize my intuitive understanding of abstraction, put words to it, and use those words (or, better, words others have written) to explain my thinking. Note how t…

This is a bit unfair. The author never claimed it would make them a 10x engineer. Learning almost anything is useful in some capacity. Is everything you learn equally important? Maybe, maybe not, but you never know in advance how things might end up useful. Of all the things one can study, math seems to be a pretty safe bet in terms of ROI.

From reading the article he justified his motivation of learning category theory to help him learn how to express himself better. I feel he is looking for some knowledge he is missing from category theory.

I am not against learning category theory because you find it interesting, but one should be honest that it isn't going to benefit ones software engineering ability at least compared to studying good design directly.

>math seems to be a pretty safe bet in terms of ROI.

Ask most people how much math they have used of which they learned it university or high school. There are niches of math that are useful to niches. Study the wrong niche of math when you are in the wrong niche that doesn't use it, it will be a poor investment.

Re: Why I am learning category theory

#130

Post by charcircuit »

larve wrote: post #10
charcircuit wrote: post #8

Honestly, this is a waste of time and is not going to help as much as the author thinks it is. There is no secret from category theory that will make one's design 10x better. Not everything is even applicable. >I hope that category theory will allow me to formalize my intuitive understanding of abstraction, put words to it, and use those words (or, better, words others have written) to explain my thinking. Note how t…

author here, the way I see it is that nothing about my software per se will change. I do indeed like to write things in a "plain" fashion instead of littering my code with high-level constructs that only a few people understand. However, it is "generative" knowledge. Having a good intuition for monads is what allows me to break down a fair amount of the mundane tasks I have to do (say, breaking down and migrating inf…

>It also allows me to mine the literature or reuse things I've worked out in the past, for example exploiting monoidal structure for optimizing computation by leveraging associativity, for example.

This is an algebraic concept and it is possible to learn this idea directly in a software engineering context than a mathematical one.

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