[Eliezer]

2 is the only even prime?

Objection! This amounts to "2 is the only prime not divisible by 2"! How about 2 is the highest solution for N in any equation X^N + Y^N = Z^N?

3 is the number of spatial dimensions we live in?

Objection! This is a mere contingent empirical fact! This is a fact about our universe, not the number 3! You might as well say that your car has three seats in the back. How about "3 is the first number whose factorial does not equal itself"?

[ars]

3 dimensions is the smallest number of dimensions in which you can make a practical universe. For example photons could not exist in a 2 dimensional universe.

I do get your point that it's not a mathematical property of the number. (But I think you can find some geometrical something that only works in 3 dimensions.)