Yeah, I often wonder if people interested in "category theory" from a programming perspective are really just interested in "algebra".
Certainly most of the "category theory for programmers" tutorials I've seen have the flavor of an undergrad algebra course: starting out with a long introduction about why you should care about abstract structures, then giving a few definitions.
The only difference is that for some reason they dive right into esoteric definitions like "monoids" and "magmas" and "monads" instead of the bedrock useful things from algebra like "groups", "rings", "fields", etc.
By the way, "monoid" and "magma" are no more categorical concepts than "groups" or "vector spaces" or anything else, reinforcing my suspicion that categories are not the fundamental thing people care about here.
> Notice that there's more than one way to combine numbers. You can add them together, but you can also multiply them. Could there be a common abstraction for that? What about objects that can somehow be combined, even if they aren't 'number-like'? The generalisation of such operations is a branch of mathematics called category theory,
You would think the author was about to introduce the concept of a "group" here. Literally if you replace "category theory" with "group theory" here, the entire paragraph still makes sense. What in the world is the point of talking about categories here? Again, when people say "category theory" do they really just mean "algebra"?
It's weird. I really don't get how this caught on, and to be honest it feels a bit like cargo culting. As someone who studied math, it seems ludicrous to me that someone should care about category theory without having studied algebra first. It's like memorizing the C++ standard without ever having written a real program. The whole point of categories is to generalize algebraic structures; how can you appreciate the point of them without actually having seen any concrete algebraic structures?
> The only difference is that for some reason they dive right into esoteric definitions like "monoids" and "magmas" and "monads" instead of the bedrock useful things from algebra like "groups", "rings", "fields", etc.
A remark: from a programming perspective, monoids are not esoteric, and are indeed more important than any of the intro abstract algebra structures. This is because free monoids are ordered lists, the most fundamental data structure in all of programming.
You can have a long and productive career as a C programmer knowing nothing but how to create and iterate over arrays! Monoids are so important that every single modern language (from C++ and Java all the way to Haskell and Scala) devotes a HUGE amount of language and standard library surface area to render data structures as sequences.
They are so important that even a language like Haskell is willing to accept utterly nonsensical semantics (hello, Traversable) to better support sequences.
> You can have a long and productive career as a C programmer knowing nothing but how to create and iterate over arrays!
True.
> Monoids are so important that every single modern language (from C++ and Java all the way to Haskell and Scala) devotes a HUGE amount of language and standard library surface area to render data structures as sequences.
Yes and no. Yes, they devote a huge amount of language and standard library surface area to sequences. No, nobody in C cares about that being a monoid, or is thinking about monoids when they use sequences, just like no NBA players think about general relativity when they're shooting a basketball. With C++... maybe a few think about it that way.
> They are so important that even a language like Haskell is willing to accept utterly nonsensical semantics (hello, Traversable) to better support sequences.
I've mostly ignored Haskell for the past decade. What happened with Traversable?
That's a very helpful comment. I wonder if partly to blame is the fact the mathematics doesn't have great terminology here: it's taught in US Universities as "Abstract Algebra", but my impression is that actual mathematicians find that term silly. On the other hand, just calling it "algebra" is difficult for people with more humble mathematical backgrounds since for them "algebra" is solving simultaneous equations etc.
Indeed; in case it’s not clear from my comment, the above poster is right. I’m talking about abstract algebra, I.e. the study of algebraic structures, not about elementary algebra.
In French, the situation is even worse: number theory is also called “arithmetic”. According to legend, sometimes innumerate peasants wanting to learn how to do basic sums would pick up a book called “arithmetic” in a bookstore that turned out to be a dense theoretical monograph on number theory.
Certainly most of the "category theory for programmers" tutorials I've seen have the flavor of an undergrad algebra course: starting out with a long introduction about why you should care about abstract structures, then giving a few definitions.
The only difference is that for some reason they dive right into esoteric definitions like "monoids" and "magmas" and "monads" instead of the bedrock useful things from algebra like "groups", "rings", "fields", etc.
By the way, "monoid" and "magma" are no more categorical concepts than "groups" or "vector spaces" or anything else, reinforcing my suspicion that categories are not the fundamental thing people care about here.
> Notice that there's more than one way to combine numbers. You can add them together, but you can also multiply them. Could there be a common abstraction for that? What about objects that can somehow be combined, even if they aren't 'number-like'? The generalisation of such operations is a branch of mathematics called category theory,
You would think the author was about to introduce the concept of a "group" here. Literally if you replace "category theory" with "group theory" here, the entire paragraph still makes sense. What in the world is the point of talking about categories here? Again, when people say "category theory" do they really just mean "algebra"?
It's weird. I really don't get how this caught on, and to be honest it feels a bit like cargo culting. As someone who studied math, it seems ludicrous to me that someone should care about category theory without having studied algebra first. It's like memorizing the C++ standard without ever having written a real program. The whole point of categories is to generalize algebraic structures; how can you appreciate the point of them without actually having seen any concrete algebraic structures?