Short on definitions in mathematics

I don’t like when one says “natural numbers are Peano numbers”. The axioms are kinda okay but do we really need to derive commutativity and associativity of addition and multiplication if we aren’t interested in foundations stuff?

I don’t like when one says “natural numbers are finite von Neumann ordinals”. Really, offloading semantics to set theory? And still using the successor operation as Peano axioms do? Again, that’s okay for foundations, but does the majority of mathematically-inclined people need to go through this?

What I ended up liking for now is that ℕ is the initial semiring. That means we postulate semiring axioms, which are in my opinion richer than Peano-like axioms, and we state that ℕ is kind of a term algebra over (0, 1, +, ⋅); expressions over this semiring language denote naturan numbers, and two names denote the same if the semiring axioms allow to prove their equality—say, (1 + (0 + 1)) ⋅ 1 = 1 + 1.

This means initiality: in any semiring, these expression would still mean its elements, and identities like 1 + 2 = 3 will still hold, if we treat 2 as 1 + 1 and 3 as 1 + 1 + 1. Even when some of these expression would be conflated, we can’t make equal things unequal. So there’s a morphism from ℕ to every semiring, just like there is a morphism from ℤ to every (unital) ring or from ℚ to every field of characteristic zero.

And we can use these latter facts as universal definitions for ℤ and ℚ if we want (we might well want it!).

Initiality of ℕ is also really important to me because it says: look, you can name so many things 0, 1, 2, 3, … and you won’t get inconsistent because of that. I’m surprised people still call their identity operators or matrices E or I even when they’re agreeing to call the zero matrix 0 (and not, say, O). It really shouldn’t add much type safety or distinguishability: there are scalar operators that multiply a vector by a scalar λ—then why not call them λ? We even have notation that makes it clear that there is no difference between λ v (multiplication) and λ v (application of a scalar operator which does the same). It really is a boon and not a pitfall.

And when suddenly 2 = 0 in fields of characteristic 2, it’s also okay. It’s as if we’ve forgotten modulo arithmetic for some reason, if we shy from being able to call the same thing both 0 and 2 (and 32, and −80). Yes, in many cases we want canonical representatives of things. But that’s not universally desired nor always possible.

Math is about freedom from what we can’t and shouldn’t control, from what we don’t need to discern. When we want to discern a new property, we can always add it into the mix; it is forgetting properties that’s harder.