- 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.
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.
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?
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. :)
