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The Frege/Russell definition of numbers is pretty cool from that sort of perspective. The general style of their definitions is to temporarily sidestep the question of what X type of number is, and first ask to what sort of thing it applies to, and what is the relation of having the "same number" among such things. Then that sort of "number" is just defined as the equivalence classes of whatever the identified "same number" relation is. There are two different definitions of the Naturals provided, that of finite Cardinal and Ordinal numbers.
The first applies to sets, grouping them by "same cardinality" or same size. Thus, the cardinal number n is identified with the set of all sets with cardinality n.
The second applies to well-ordered binary relations, and groups them by what we would call order-isomorphism. This might sound complicated, but the end result is that the ordinal number n becomes the set of all well-orders of length n (you can soft of think of this as the set of sequences of length n, at least for the finite case).
Amusingly, both of these notions are too general to correspond to just the natural numbers, since they don't discriminate between finite and infinite numbers. Thus, the 'natural number' portion of each is actually defined as the smallest initial subset for which mathematical induction is valid. Anyhow, the Principia Mathematica is pretty fun if you're into that sort of thing. It builds up a lot of neat and weird representation of all sorts of different numbers, up from the naturals, integers, ratios, and reals. It even provides its own weird definition of vectors, and gives a kind of analysis of "signed magnitudes" (like weight, height, temperature etc) and how their definition of real numbers as pure mathematical objects relate to them, providing a kind of abstract interpretation of what we mean when we measure something in the real world. |