My professor told me that learning category theory would make me a better person but not a better programmer
Why I am learning category theory
191–200 of 224 posts
Re: Why I am learning category theory
#192Earlier quoted context omitted.
I’m a math major. I learned category theory in school. I think of category theory as an easy way to remember things. If some concept can be expressed as category theory, it can often be expressed in a very simple way that’s easy to remember. However, if you try to teach math using category theory from the beginning, it feels a little like trying to teach literature to someone who can’t read yet. Anything directly use…
Pardon me: what IS category theory?
Almost every field you can study in mathematics can be thought of as a “category”, and when you take a math class in undergrad, it usually focuses on one or two specific categories. Category theory is the study of categories in general. As in, “What if, instead of studying a specific field of math, you studied what these different fields had in common, and the relationships between them.”
The part where it gets wacky is when you realize that “categories” is, itself, a category.
Re: Why I am learning category theory
#193Earlier quoted context omitted.
I'm curious why you call out constructivism. I've only really seen constructivism talked about by people who actually have a strong math background. Because it is hard to speak out against, say, the classical notions of existence that say that there are more real numbers than rational ones unless you actually understand why the classical proofs don't work constructively. And not just as, "We don't allow that proof."…
> I've only really seen constructivism talked about by people who actually have a strong math background. Which I don't mind. If you know your maths, your logic and ideally even your philosophy it's perfectly fine to work in constructivism or even prefer it. But I've definitely seen some dumb, uninformed hot takes about how maths is just this big conspiracy that doesn't allow dissenting opinions because they've seen…
You can account for and "do" classical math perfectly fine as a constructivist+finitist. It just means that any statements about non-constructive existence or non-decidable choice must be phrased as negative statements in the logic; positive disjunction or existentials are reserved for anything that's decidable/constructible.
And uses of the axiom of choice are viewed with suspicion even by many who are doing perfectly classical math, so there's nothing wrong with pointing them out as assumptions ("if we admit of choice being applicable to X, we have Y").
Re: Why I am learning category theory
#194I worked through a popular Introductory Category theory text. Did all the problems and made sure I understood everything in there really. It became clear by the end why some mathematicians like category theory. It lifts some results in certain areas to a higher level of generality.
As for me it helped me with understanding some of what was being discussed on r/haskell. It didn't help one iota getting me a Haskell job though, failed every Haskell code pairing I've done.
Re: Why I am learning category theory
#195As 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. Like, CT describes a set of rules but very little emerges from those rules.
The fact it's so rare to find a counterintuitive fact in CT, so that you rarely find yourself proving something and you mostly spend your time constructing effective tools, it's a feature not a bug! McBride's "Don't touch the green slime!" is a great paper/saying about this principle. Work with the compiler, not against it. The compiler is very dumb so it understands only trivial things.
There's a legacy pseudomachist culture of 'hard is good' in mathematics (but really, it's transversal to science) which rewards working hard and not working smart.
Re: Why I am learning category theory
#196As 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. Like, CT describes a set of rules but very little emerges from those rules. The fact it's so rare to find a counterintuitive fact in CT, so that you rarely…
> 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.
This is correct. The only somewhat interesting result in Category theory is the Yoneda Lemma, everything else is machinery for diagram chasing. It's ubiquitous in the same way that short exact sequences are ubiquitous -- and about as interesting as short exact sequences.
I think most hobbyists or those who are intellectually curious would be better off studying physics or engineering first, and then picking up the math with applications along the way. For example, you can study multivariable calculus and then complex analysis, which naturally leads to questions of topology as you find try to solve integral equations, and then obstruction theory can come up. Lots of cool stuff to study that is actually interesting. I would never foist category theory on someone who finds math to be interesting and enjoyable -- that would be like throwing a match that just caught flame into the rain.
Re: Why I am learning category theory
#197Earlier quoted context omitted.
The way to run conjunctive SQL queries forward and backward is described in this paper, https://www.cambridge.org/core/journals/journal-of-functiona... , (also available on the arxiv), where they are referred to as query 'evaluation' and 'co-evaluation', respectively. We never would have been able to discover co-evaluation if not for category theory! The previous link includes this paper, and many others.
So what is coevaluation and why is it useful? Please don't just point at the paper again.
Bi-directional data exchange has many uses. For example, given a set of conjunctive queries Q, because coeval_Q is left adjoint to eval_Q, the composition coeval_Q o eval_Q forms a monad, whose unit can be used to quantify the extent to which the original query Q is "information preserving" on a particular source (so query/data quality). As another example, we use the technique to load data into OWL ontologies from SQL sources, by specifying an OWL to SQL projection query (tends to be easy) and then running it in reverse (tends to be hard). But no doubt more applications await!
Re: Why I am learning category theory
#198Reading at the comments, people confuse concepts like monoids and functors with category theory.
> functor Google search first result: Functor - Wikipedia https://en.wikipedia.org › wiki › Functor In mathematics, specifically category theory, a functor is a > monoid https://en.wikipedia.org/wiki/Monoid_(category_theory)
Yes, these words do have a meaning in category theory. But for a haskell programmer, a functor is a type with a map operation, for a ocaml program it's a module parameterised by a module signature. A monoid a type with a function T x T -> T. That's it.
You can know these things and still know nothing about category theory.
Re: Why I am learning category theory
#199Earlier quoted context omitted.
Sure, but what does the knowledge of something being an adjoint give you?
There are a lot of interesting properties: - Adjoints preserve limits/colimits. - Adjoint functors give rise to a monad - They are connected to universal morphisms
Where do adjoint functors occur in CS? They occur in advanced algebra, and they occur in topology, but where else? And indeed, the fact that they preserve limits/colimits may help speed up communication and thinking. But I'm not seeing CS connections here.
Re: Why I am learning category theory
#200Earlier quoted context omitted.
There are a lot of interesting properties: - Adjoints preserve limits/colimits. - Adjoint functors give rise to a monad - They are connected to universal morphisms
My problem with category theory (my limited study of it, several years ago) was that it describes and defines a list of properties , but those properties don't combine to reveal any unexpected, exciting results. Again, with my abstract algebra example from above: after just a couple of basic abstract algebra definitions, you learn about subgroups. Simple enough, and not particularly exciting so far. But then you quic…
I think it's just compression:
- An operation may be "functorial", meaning that it preserves more structure than perhaps originally thought. For instance, the "fundamental group" operation is indeed functorial, which means that it acts on continuous functions in a nice way as well as topological spaces. Other examples are tensor-product, vector-space-duality, forming function-spaces in certain categories, etc.
- Two categories may be isomorphic, but this is trivial.
- Two categories may be equivalent, which while a weaker notion than isomorphism, is sufficient for most things which can be expressed in categorical language to be true for both categories. This is helpful when one category is well-understood and the other one is an object of present interest. (One application is showing that the category of representations of a fixed quiver Q is Krull-Schmidt, by showing that it's equivalent to another category with the Krull-Schmidt property).
- A functor between two categories may admit a left-adjoint. It then immediately preserves all limits (it's "continuous") which immediately means that a great deal of structure gets preserved by it.
- A functor between two categories may preserve all limits. It may (under some circumstances, expressed by the "adjoint functor theorem") therefore admit a left adjoint. This may be a non-trivial fact of interest in its own right. It's related to dualities in optimisation and game theory.
- There's isolated results like the Seifert Van-Kampen Theorem (which states that the Fundamental-Group functor preserves pushouts) which would be difficult to express without categorical language.
Ultimately, Category Theory appears to be a language for compressing complicated facts about structure-preservation of operations.
Category theory is helpful in advanced algebra, and helpful too in advanced topology, and is in its absolute element in any area which combines the two, like algebraic topology and algebraic geometry. In the latter two areas, you've got lots of functors between algebraic categories, lots of functors between topological categories, and even functors going between algebraic categories and topological categories.
There's also categorical logic, which is where the CS-adjacent stuff seems to be found. But this is of little interest to everyday programming, and is very forbidding for people who lack the requisite mathematical maturity. Only the most dedicated should enter its harsh plains, and should expect to gain nothing without tremendous efforts and sacrifice.