[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.