On the Nature of Numbers

The following definition of the number 2 has been brought to my attention: "[t]he set of all pairs." Against this definition, I raise the following objections:

I. It seems viciously circular: if 2 is the set of all pairs, what are pairs other than a set of two objects? The same objection seems to hold for defining 3 as the set of all triples, etc.

II. Even if this kind of definition can be used for the natural numbers, I am at a loss to understand how numbers such as fractions, irrational roots, pi, imaginary numbers, and negative numbers are to be defined in terms of sets of objects.

III. If “pairs” is understood to mean “pairs of physical objects,” then it would seem that in a possible world with only two physical objects, numbers larger than two could not exist, since e.g. “the set of all triples” would be empty.

IV. Or again, take two possible worlds which each contain only twenty physical objects, with the stipulation that each world contains objects that are entirely different than the objects in the other. In this case, it would seem that “2” in one world would not be identical to “2” in the other, since the set defined as “the set of all pairs” would have different members in each possible world.