Goodman and Quine's "Steps Toward a Constructive Nominalism" is an ingenious attempt to show how one might begin to give nominalist analyses of claims that prima facie involve abstracta like functions, numbers or types. For instance, the analysis of "There are more cats than dogs" would be that there is no one-to-one function pairing every cat with a different dog. Here is the clever analysis. Say that an object is a bit iff it is exactly as big as the smallest animal among the cats and dogs. Say that an object is a bit of z iff it is a bit and it is a part of z. Now, there are more cats than dogs iff every object that has a bit of every cat is bigger than some object that has a bit of every dog.
The first issue here is that this—like many of Goodman and Quine's definitions—only works given mereological universalism. We need to be able to form an object that has a bit of every dog. But while the theory requires mereological universalism, it is not clear that one can state the relevant mereological universalism without adverting to sets or properties. For instance, it seems that the account of "There are more cats than dogs" only works if from the fact that every dog has a bit we can infer that there is a minimal object among the objects that have a bit of every dog. The relevant axiom is something like this: given non-overlapping Fs and an object x no bigger than any F, there is a minimal object that has an x-sized portion of every F. But this axiom seems to quantify universally over kinds F. Without such quantification, we will simply have a separate axiom for each kind, and then we cannot state the fact that the counting method works "in general".
The second problem is technical. There might not actually be a smallest cat-or-dog if there is an infinite chain of smaller and smaller cat-or-dogs. Perhaps the definition only works if there are only finitely many cat-or-dogs. But it is not clear how one can say that there are only finitely many cat-or-dogs on the theory. A natural suggestion is that the fusion of all the cat-or-dogs has finite size, i.e., no proper part of it is the same size as the whole. But that won't do, because it could be that the fusion of the cat-or-dogs is of finite size, while there are infinitely many cat-or-dogs. In fact, this point suggests that in general the notion of finitude is going to be difficult for Goodman and Quine to express.
The third issue is that the crucial basic concept here is that of being "bigger than". One sense of "y is bigger than x" is that a rigid motion could bring x wholly within the space occupied by y. This sense won't do here, however. For there need not be a smallest cat-or-dog in this sense, as the shapes might not nest. The relevant sense of "y is bigger than x" is that y has greater mass or volume than x. But, now, how do we understand that in a nominalist way? Here is one problem: mass and volume are relative to a reference frame. Maybe, though, we fix some particle r and then understand mass or volume relative to r. This would require a ternary relation BiggerThan(y,x,r). I suppose this is doable. But what about massless and volumeless objects, like photons? Well, maybe then we need the notion of energy comparison instead. However, now we have the oddity that the concept of counting depends on the concept of energy. But surely there could be more As than Bs in a world whose laws were so different from our world's laws that there could be no concept of energy there.
It is hard to be a thorough-going nominalist.