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Friday, December 23, 2022

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I was asked this very question by a set theorist nearly 20 years ago. (Set theorists are arch realists).

Now nominalists can handle individual numbers without any problem. We point to expressions like 'a pair of shoes', which does not denote a third entity in addition to the two shoes, but rather quantifies over any two individuals.

However, "the Xs are greater in number than the Ys" is a little more tricky.

I must look at my notes.

I think your 2 is a little easier. One way, we could just have a name for every X of Ys, just as we have the name 'pair' and 'dozen', and indeed '100'. Even better, we could have a rule that allows us to construct such collective nouns, which indeed is what the decimal system is. The name '100' doesn't signify any concept to me, as humans don't have any concept of multiplicity beyond about 7.'100' is just a name that is related to '99' by the concept of adding 1. I.e. the name '99+1' signifies the same as '100' simply by the rule of constructing number names.

I think Ed is on the right track - the examples reduce the Hatfields/McCoys to a mathematical quantification. Thus, the entities are as real as any mathematical concept - which from an existence standpoint, is not very real (unless you buy into the Kantian synthetic a priori). The question, I think, is more ontological: the names Hatfields and McCoys each refer to a very definite phenomenal entity, the two different peoples that fight each other. These two complex entities contain their own unique list of predicates and produce their own real world consequences in time and space. Therefore, they are not reducible to the individual persons within the entity. QED, Hatfields and McCoys are names of real entities; or at least, entities as real as the individual members of each group.

I would say that in general we cannot analyse statements like (2) without resorting to a notion of counting. But in this case 100 can be thought of as 10 x 10, and 10 as 5 + 5, using only multiplicities less than 7, and not naming all numbers up to 100.

(1) can be explicated without counting and without resort to 'something in addition to individual Hatfields', etc. Given three empty rooms A, B, and C, we put the Hatfields in room A and the McCoys in room B and follow the following procedure. While both room A and room B remain non-empty, take a Hatfield from room A and a McCoy from room B and put them both in room C. The procedure halts when

room A is non-empty and room B empty: Hatfields outnumber McCoys;
room A is empty and room B non-empty: McCoys outnumber Hatfields;
both room A and room B are empty: Hatfields and McCoys are equinumerous.

A Merry Christmas and Happy New Year to one and all.

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