From Plato and Aristotle to Armstrong, it has been hoped that entities known as "forms" or "universals" could do the job of explaining genuine similarity: why are X and Y both red? Because they both instantiate the universal redness. Various accounts of the nature of universals and the relation of instantiation have been given, and the most hopeful of the lot has been immanent realism: the doctrine that the universals are literally parts of the ordinary things that we interact with daily. Thus two things that are genuinely similar will literally have "something in common": X and Y are both red because both have the universal redness as a part. X, which I will call a "thick particular", consists of x, a "thin particular", and the universal redness (and probably some others besides); the thick particular Y consists of the thin particular y and the universal redness (and maybe some others). It is the fact that X and Y have the universal redness as a common part that explains their similarity.
The theory also needs a relation of instantiation, that holds between universals and thin particulars, in order to keep the world from being overpopulated with thick particulars. Suppose that every electron is the sum of a thin particular and the universal unit negative charge, and each proton a sum of a thin particular and the universal unit negative charge. Suppose that there is exactly one electron, E, consiting of thin particular e and unit negative charge, and one proton P, consisting of thin p and unit postive charge. [This of course is an unrealistic example, but bear with me for the purpose of exposition.] But what about the sum of p and unit negative charge: why isn't that an electron, too? And why isn't the sum of e and unit positive charge a proton? The answer, it seems, must be that e instantiates unit negative charge, whereas p does not, and p instantiates unit positive charge, whereas e does not.
Now for my challenge to immanent realism. Surely the fundamental monadic properties of physics are going to have to correspond to universals, for those properties make for genuine similarity, and are, ex hypothesi, not reducible to other properties. (A property P corresponds to a universal U iff everything that has P consists of U and a thin particular that instantiates U.) But the fundamental properties of physics are additive, i.e. orderable as relations to real numbers, so that if and x and y are wholly distinct, and Rxm and Ryn, then R(the mereological sum of x and y)(the arithmetic sum of m and n). For example, if e1 and e2 both have charge -1, then the sum of e1 and e2 has charge -2.
Suppose that unit negative charge is a universal, and consider two electrons e1 and e2. The sum of e1 and e2, call it e3, has charge -2. But the theory so far cannot explain this: for e3 has unit negative charge as a part, and a thin particular (consisting of the sums of the thin parts of e1 and e2) as a part; since it has the unit negative charge as a part, one would think that it would have charge -1. And it can't be in virtue of a new universal of -2 charge, for that isn't a part of e3 at all, by stipulation: e3 is merely the sum of e1 and e2.
The only answer I see forthcoming is that e3 has charge -2 because it instantiates unit negative charge twice: once by the thin part of e1 and once by the thin part e2. Now the only sense I can make out of a thing x bearing R to something y twice, is if x has distinct parts w and z, each of which bear R to y. E.g. I may be said to touch the keyboard twice, because I have two parts, viz. my right and left hands, each of which touch the keyboard. But if this is the explanation given of having charge -2, it follows that nothing atomic, i.e. having no proper parts, could possibly have charge -2. And this would seem possible, at least a priori.
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