[tsimionescu]

> Category theory, as an example, has many more practical applications and leads into arithmetic.

Now this is a sentence I never expected to read.

I aheb thought before about the fact that derivatives are a much simpler and more useful concept than exponentials and logarithms, but I very much fail to see how category theory is useful for anything other than researching the foundations of math.

[myWindoonn]

Arithmetic is just decategorified set theory. The main advantage of teaching arithmetic via set theory is that it allows for building intuitions about counting.

It shouldn't be surprising that two sheep plus two sheep is four sheep. We can go further. Suppose that we have lots of longhair sheep and shorthair sheep. If we choose some sheep, how many ways can we have some shorthair and some longhair? This gives exponentiation intuitively, so that we could ask how many ways we could choose zero sheep from a collection of zero sheep, or in other words, why zero to the zeroth power is one.

The parts of category theory that you're imagining, with the morphisms and natural transformations, doesn't have to be taught before arithmetic. It can be taught when lambdas are first introduced, when we write "f(x)" on the board for the first time.

[tsimionescu]

But none of this modern theoretical understanding helps me see what is 131 + 121, or 2^10. Expressing arithmetic operations as set operations and counting elements is only useful for really small numbers - if you actually want to know the answer, you ignore the set-based or category based foundations and use an algorithm.

This is the major problem I have always had with examples of category theory use: they always sound nice and give very illuminating intuitions for certain mathematical structures, which is very useful for doing mathematics and furthering your understanding. But they never directly answer any practical questions - for those, you always abandon the abstractions and start getting into the nitty gritty of the specific domain. At best, they help you take a specific algorithm from domain 1 and apply it in domain 2.

Am I wrong? Is there some way to actually get the answer to how many ways you can combine longhair and shorthair sheep in sets of 10 sheep, other than simply counting all combinations, or using traditional arithmetic/geometry etc. (e.g. repeated multiplication, angle measurements)?

[myWindoonn]

I think that you're overreacting a little to an imagined strawman. We already teach kids a decategorified set theory in order to give them an understanding of counting. We don't take the next step into category theory, which is to explain mappings between sets: If I can trade one sheep for two goats, and I have five sheep which I all trade for goats, how many goats do I get?

Remember: Sets are 0-categories, so set theory is 0-category theory.

[bottled_poe]

I specifically mentioned arithmetic, which is often confused for mathematics.

[tsimionescu]

Yes, which is most bizarre, since school-level arithmetic is the first kind of mathematics ever invented, far before writing, specifically for its practical uses.

Are you really claiming that it's of more practical use to understand the properties of monoids and rings than it is to understand that 2 + 2 = 4? Or are you actually talking about arithmetic beyond what is normally taught in school?