The resident nominalist comments:
Nominalists say that the conception of an actual infinity of natural numbers depends on there being a set of all such numbers. But Ockhamists do not believe in sets. They say that the term ‘a pair of shoes’ is a collective noun which deceives by the singular expression ‘a pair’. Deceives, because it means no more than ‘two shoes’, and if there is only a pair of shoes, then there are only two things. But if a ‘pair’ of two things is a single thing, there are three things, the two things and the pair. Ergo etc.
I agree that there cannot be an actual infinity of natural numbers unless there is a (mathematical as opposed to commonsense) set of all such numbers. But of course this holds for all numbers, rational, irrational, transcendental, etc. Indeed, it holds for any category of item that is actually infinite. If there is an actual infinity of propositions, for example, then there must be a set of all propositions. I would point out however that there is nothing nominalistic about our friend's opening remark.
Nominalism kicks in with the claim that there are no sets. What there are are plural referring devices such as 'a pair of shoes' which fools us into thinking that in reality, i.e., extralinguistically, there are three things, a left shoe, a right shoe, and the pair, when there are only two things, the two shoes. The same goes for the following seemingly singular but really plural phrases: a gaggle of geese, a pride of lions, a parliament of owls, a coven of witches, etc.
This all makes good sense up to a point. When I put on my shoes, I put on one, then the other. It would be a lame joke were you to say to me, "You put on the left shoe and then the right one; when are you going to put on the pair?" To eat a bunch of grapes is to eat each grape in the bunch; after that task is accomplished there is nothing left to do. The bunch is not something 'over and above' the individual grapes that I still need to eat.
Consider now the Hatfields and the McCoys. These are two famous feuding Appalachian families, and therefore two pluralities. They cannot be (mathematical) sets on the nominalist view. But there is also the two-membered plurality of these pluralities to which we refer with the phrase 'the Hatfields and the McCoys' in a sentence like 'The Hatfields and the McCoys are families feuding with each other.'
If, however, a plurality of pluralities has exactly two members, as in the case of the Hatfields and the McCoys — taking those two collections collectively — then the latter cannot themselves be mere pluralities, but must be single items, albeit single items that have members. They must be both one and many. That is to say: In the sentence, 'The Hatfields and the McCoys are two famous feuding Appalachian families,' 'the Hatfields' and 'the McCoys' must each be taken to be referring to a single item, a family, and not to a plurality of persons. For if each is taken to refer to a plurality of items, then the plurality of pluralities could not have exactly two members but would many more than two members, as many members as there are Hatfields and MCoys all together. Compare the following two sentences:
1. The Hatfields and the McCoys number 100 in toto.
2. The Hatfields and the McCoys are two famous feuding Appalachian families.
In (1),'the Hatfields and the McCoys' can be interpreted as referring to a plurality of persons as opposed to a mathematical set of persons. But in (2), 'the Hatfields and the McCoys' cannot be taken to be referring to a plurality of persons; it must be taken to be referring to a plurality of two single items.
Or consider the following said to someone who mistakenly thinks that the Hatfields and the McCoys are one and the same family under two names:
3. The Hatfields and the McCoys are two, not one.
Clearly, in (3) 'the Hatfields and the McCoys' refers to a two-membered plurality of single items, each of which has many members, and not to a plurality of pluralities. And so we must introduce mathematical sets into our ontology.
My conclusion, contra the resident nominalist, is that we cannot scrape by on pluralities alone. (Man does not live by manifold alone! He needs unity!) We need mathematical sets or something like them: entities that are both one and many. A set, after all, is a one-in-many. It is not a mere many, and it is not a one 'over and above' a many. The nominalist error is to recoil from the latter absurdity and end up embracing the former. The truth is in the middle.
What I have given is an argument from ordinary language to mathematical sets. But there are also mathematical arguments for sets. Here is a very simple one. The decimal expansion of the fraction 1/3 is nonterminating: .33333333 . . . . But if I trisect a line, i.e., divide it into three equal lengths, I divide it into three quite definite actual lengths. This can be the case only if the the decimal expansion is a completed totality, an actual infinity, not a merely potential one. An even better example is that of the irrational number, the square root of 2 — it is irrational because it cannot be expressed as a ratio of two numbers, the numerator and the denominator of a fraction as in the case of of the rational 1/3. If the hypotenuse of a right triangle is units of length, that is a quite definite and determinate length. How could it be if the decimal expansion however protracted did not point to a completed totality, an actual infinity?
units of length, that is a quite definite and determinate length. How could it be if the decimal expansion however protracted did not point to a completed totality, an actual infinity?
REFERENCES
Max Black, "The Elusiveness of Sets," Review of Metaphysics, vol. XXIV, no. 4 (June 1971), 614-636.
Stephen Pollard, Philosophical Introduction to Set Theory, University of Notre Dame Press, 1990.
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