> I agree about the natural numbers including 0. I'm trying to think of how to justify the strength of my opinion on this though. Usually I think notation should be adjusted to the situation to facilitate explanation. Why is the definition of the naturals more important though?
This is an interesting question. The natural numbers are the free monoid on one generator. The "bad" natural numbers, without 0, are the free semigroup on one generator. So, this question is related to another: "why are monoids better than semigroups?"
One answer is that, contrary to what you might think, the category of semigroups and semigroup homomorphisms embeds as a subcategory of the category of monoids and monoid homomorphisms in a very nice way. That is, there's a very real sense in which monoids are _more general!_
How does this work?
I'll let you ponder this puzzle. It's a special case of another puzzle: why do we demand categories have identity morphisms? Shouldn't we focus on more general "semicategories", where we don't demand the existence of identities?
When you put the puzzle this way, another answer rises into view: in a semicategory, you can't define what it means for objects to be isomorphic! (Well, okay, you can, since _some_ objects will have identity morphisms, and you can use those when they're available. But then you'll get objects that aren't isomorphic to themselves, which is annoying.)