alt Hacker NewsThis is not a critque of the work, but I have now been often encountering category theory in many of the research topics I'm working on, and even as somebody who majored in math, I sort of feel like it doesn't add much. I know ML frameworks intimately, and you really don't need category theory to describe them. But this is maybe (probably) a failure of mine, because I have not yet groked what category theory is really bringing to the table.
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Category theory isn't usually intended to add things. Often, what category theory brings to the table is connecting different branches of math. (It's a bit similar to abstract algebra in that way, but at a different level.)
For example, the lambda calculus is the base for many useful programming languages. But the lambda calculus maps to a "closed Cartesian category". And many, many other interesting things in math can be mapped to a closed Cartesian category.
So now you can ask, "What if affine or linear logic were a programming language?" And the answer is, "You'd get a language with safe resource management, like Rust." Or you might ask, "What if probability were a programming language?"
Or on a smaller scale, a parameterized collection type with "map" is a functor. Add a single-element constructor and a "flatten" operation, and you have a monad. Functions with parameterized types are often natural transformations. And so on. This can then be directly analogized to constructs in different areas of math. Which might sometimes produce an interesting idea or two.
So category theory isn't always used to add something new and profound. Sometimes it's just a handy way to see something already there.
Category theory is all about relationships and structural patterns. so its useful when you want to interoperability and composition between systems i.e. invariants under transformations, etc.
Without reading too much into what this framework does, I'd say category theory could be useful for some ML problems (i.e. layer composition, gradient propagation, etc.) - but I'd think it would be more useful as an analytical tool than as actual lib/code structures.
You don't need category theory to describe the Result type. But the people who first introduced it to programming languages, were thing about category theory.
Category theory is rarely useful by itself, but it can be a mental scaffold when designing things like query languages. Microsoft's LINQ dsl within C# used category theory ideas to ensure consistency. That said, the applicability surface area in practice is typically quite limited in my experience. It's like formal methods -- elegant in practice, but a good problem fit is often rare. It's like writing a LEAN proof for your web app -- rarely needed, but if your web app needs a high degree of correctness, then indispensable.
This is John D Cook's take:
Category theory can be very useful, but you don’t use it the same way you use other kinds of math. You can apply optimization theory, for example, by noticing that a problem has a certain form, and therefore a certain algorithm will converge to a solution. Applications of category theory are usually more subtle. You’re not likely to quote some theorem from category theory that finishes off a problem the way the selecting an optimization algorithm does.
I had been skeptical of applications of category theory, and to some extent I still am. Many reported applications of category theory aren’t that applied, and they’re not so much applications as post hoc glosses. At the same time, I’ve seen real applications of categories, such as the design of LINQ mentioned above. I’ve been a part of projects where we used category theory to guide mathematical modeling and software development. Category theory can spot inconsistencies and errors similar to the way dimensional analysis does in engineering, or type checking in software development. It can help you ask the right questions. It can guide you to including the right things, and leaving the right things out. [1]
[1] https://www.johndcook.com/blog/applied-category-theory/