In general, the whole "oh yeah CS people should know some category theory and abstract algebra" is pretty hilarious.
First, it's a bit like saying "oh yeah CS people need to know the undegraduate basics and also the generalization that most mathematicians don't encounter until a couple years into grad school."
Second, most people who say this really mean "a conceptual grasp on different types of morphisms is useful". But that's like saying you need calculus in order to drive a car; or, in the case of categories, it's like saying you need two semesters of real analysis in order to drive a car.
Why not just say "knowing about different sorts of mappings is pretty useful in functional programming"? Knowing how this generalizes to more abstract mathematical objects is totally unnecessary.
By analogy, you don't need to know abstract algebra and category theory to write Haskell. But as in the other cases, knowing helps.
"Different sorts of mappings" is NOT synonymous with "category theory". Not even close. Heck, Euclid knew about "different sorts of mappings".
Most everything in Gamma et al is arguably useful for everyday programming in Java. Maybe 5-10 pages of MacLane is useful for everyday programming in functional languages.
Unless by "Category Theory" you mean "5-10 pages of MacLane", Category Theory -- on the whole -- is a horrendously inefficient way of teaching about "different sorts of mappings useful in functional programming."
Unless you want to use functional programming as an environment for doing pure mathematics, there's no reason to actually study actual Category Theory.
I've really never needed an abstraction for semigroups, monoids, meet-semi-lattices, monads, comonads, arrows or catamorpisms in Clojure, Common Lisp, Scheme, or Hy.
These concepts become more relevant when I program Haskell, Agda, Isabelle/HOL, or Coq.
I'd say a stronger analogy can be made between reading MacLane's Catagories for the Working Mathematician and reading Hoyte's Let Over Lambda; you really only need to read a little bit of these books to get the core concepts. That being said, depending on what sort of functional programming you're doing, a strong background in category theory or meta-programming can enabling (or not).
That's fair. Although Haskell is a bit of an odd man out in that list, both in terms of its nature and in terms of its typical use case.
> That being said, depending on what sort of functional programming you're doing, a strong background in category theory or meta-programming can enabling (or not).
This is where the analogy between the two books breaks down. When you're using a functional programming language as a proof assistant, category theory can be helpful. But this is far less common than meta-programming.
edit: 2nd paragraph.
You can teach about "different sorts of mappings" in just about any setting. In fact, that's kind of the whole reason Category Theory exists. So why teach the general result when all you care about is its application to functional programming?
Given the investment necessary to make this jump, I believe that the benefit outweighs the cost for most.
I have argued that almost all numerical mathematics, in some form, can be modeled as a linear algebra problem. Google's original page rank algorithm is linear algebra. Remember that Netflix challenge? All linear algebra. Optimisation? Linear algebra. Want to do any engineering of a system? Behind the scenes it is all linear algebra as every single numerical technique for solving PDEs, that I am aware of, can be thought of in a linear algebra sense. Fluid modeling is all linear algebra. A lot of the machine learning I've seen is just linear algebra. I would also that almost every simulation running on the world's supercomputers involves linear algebra.
Do you absolutely need linear algebra to do this things? No. Just like I don't need to understand how my car works to drive it. But having an understanding how these systems operate can really help you use them in a more logical way.
I'm asking not to rebut but because I hope to prompt the sort of "sell us on linear algebra" statement that will make me study harder. :)
Still, many folks use the APIs without really grokking that, so in practice it can be a bit cookie-cutter. I think of it similarly to how people can use crypto APIs without, say, really understanding what's going on under the hood.
BTW, projective space is also intimately related to elliptic curves (as you may or may not know — not implying anything!). So that darn linear algebra is lurking all over the place.
Likewise, any time you're talking about fields (even finite fields), vector spaces and linear algebra are right around the corner.
http://www.hyperelliptic.org/EFD/g1p/index.html
Curves are what made algebra 'click' for me, getting me from my high school understanding of "algebra is math about unknown variables" to "algebra is about sets of related objects with operators that have identities and inverses".
However, I do think you can find plenty of examples in academia. For example, applying PCA
http://cmp.felk.cvut.cz/~hlavac/TeachPresEn/11ImageProc/15PC...
(Perhaps this is just the result of crossover between CG and ML/Statistics here).
Not a good sell, but perhaps someone with more experience can chime in.
- You write a recursive program the same way you write an inductive proof
- Abstract algebra and category theory are likely relevant, especially for metaprogramming. My math education hasn't included this, so I can't say much more.
- Linear algebra is just ridiculously important
- Statistics for machine learning. Also for figuring out how to combine data in a meaningful way. There are also a lot of people asking statistical questions directly, and writing programs is how you get those kinds of answers in a reasonable timeframe.