The following is partially in response to Jody Azzounis recent post, via Eric Schliesser, on New APPS. However, points 15 and 811 below stand on their own; I refer to Azzouni only in points 6 and 7.
In point 7, especially, I read a lot into Azzounis words. I may have misunderstood his motivations. Even if my response does not apply to him, however, I suspect it applies to plenty of others.
1. Realism about the object of some representation R is the doctrine that the entity object to R is per se a res: that it has its realitas qua object of R , or, in other words, that R represents it with respect to its essence. Nominalism, strictly speaking, is the denial of realism in this sense. So far this has nothing to do with abstraction.
2. If the object of R is not per se a res, then R must be a mediate representation. It represents an entity as viewable under a certain aspect, so to speak, but it does not determine which entity that is. So it is parasitic on some at least possible immediate representation R0 of the same entity. Prior to Kant it was assumed that R0 itself must be a representation whose object is per se a res. Under that assumption (which Kant calls transcendental realism), the representation R about which we are nominalists must always be the non-essential representation of an entity which is essentially represented by R0 .
It is here that abstracta come in. If the nominalist can maintain that R is a mediate representation of this kind, the realist must say in opposition that R represents a mediate entity. For example: where the nominalist about position maintains that Socrates sitting is just a mediate representation of the primary substance, Socrates, and hence dependent on the immediate representation Socrates, the realist about position will account for the same facts by claiming that Socrates sitting is an entity which can exist only by inhering in a primary substance that is, an accident. So the alleged entity whose existence is denied by the nominalist (as a multiplication of entities without necessity) is always an abstractum, in the strict sense that it is not self-subsistent.
3. But what does this have to do with what is currently called nominalism? I fear that the answer involves a confusion. The term abstractum has come to be used in a new sense, to mean, roughly speaking, an incorporeal primary substance. This is a bad way to use the term, since it is not only historically and etymologically baseless, but also mutually exclusive with the old, proper sense: an abstractum in the old sense is never self-subsistent and hence never a primary substance of any kind. As of now I cant say exactly how this new, bad sense arose, but I suspect the story revolves around Goodman and Quine, so that the terms abstract and nominalist were always on the table at the same time. In any case, the result seems to be that people know nominalism is about denying the existence of abstracta, but no longer have the appropriate understanding of what abstracta (allegedly) are.
4. Nominalism thus appears as the attempt to explain away what looks like successful representation of incorporeal substances under the assumption that all substances are corporeal. There are various different approaches to doing that (fictionalism, etc.), which neednt detain us. The main problem, common to all such approaches, is that substance and incorporeal are terms of Aristotelian metaphysics, whose application has now grown difficult, if not wholly illegitimate. The motivation for such a nominalism must, then, be a metaphysical thesis which is difficult, if not wholly impossible, to reconcile with modern empirical science: difficult, not because science seems to indicate that the thesis is false, but rather because science seems to indicate that the thesis is put in empirically illegitimate terms.
So there are two points where I worry about Azzounis post.
6. First, I cant agree that the nominalists scruples at the end of the day are empiricist ones. Nominalism, in either the old or the new sense, is in no obvious way connected to empiricism. As far as I can tell, the feeling to the contrary comes about because nominalists dont believe in spooky entities, and scientists dont believe in spooky entities, and science is empirical. But there is no consistent way to define spooky which makes both of the first two clauses true, and the sense of spooky required by the first clause is empirically unsound.
7. Second, although there is indeed a difference between Field et al., and Azzounis genuine nominalist, it is not clear to me that the difference is to the genuine nominalists advantage. Field and company are guilty of letting the original definition of spookiness drift away into something vaguer; but, since the original definition was the problem, that is not in itself a vice. The genuine nominalist, I fear, may be precisely one who doesnt allow such drift.
In Azzounis post the question comes to a head in the sentence: Scientific facts about such particles and their properties has been established not merely on the basis of the simplicity of the scientific theories about such, but on the basis of actual physical probings of nuclei. What is actual physical probing, and why should a nominalist/empiricist favor it over mere considerations of simplicity? Im afraid the answer is that both nominalism (in the new, improper sense) and something one might think of as empiricism are being derived from metaphysical corpuscularism. We should be nominalists because everything other than big or little balls of stuff (cf. Azzounis previous example of a handball) is spooky; and we should be empiricists because big balls of stuff can know about other balls of stuff only because there are little balls which interact with the latter and then return to bounce off the sense organs of the former. So empiricism entails rejecting any theory, whatever its virtues, unless it portrays the world and our relationship to its constituents in this way. In particular, Azzouni points out that, given the axiom of choice, regions of topologically continuous spaces cant be identified with little and big balls; so, he concludes, a nominalist/empiricist cant be happy with any theory which describes the world as containing such regions.
I wont say that this is a simplistic picture of physical theory, but I do think its false. In fact, to my mind the above conclusion is a reductio ad absurdum.
8. The business of the science of physics is not simply to describe the world, using terms prescribed by metaphysics. Rather, a fundamental physical theory is above all a thesis about the appropriate terms in which the world is to be described. If actual physical probing is to mean: the type of probing which actually serves the science of physics, it must be a probing in which the question: what is a physical description? is raised and potentially given a new answer. Physicists are empiricists because and to the extent to which they do not allow traditional concepts such as body and motion to constrain the possible answers to that question.
9. Since observation and measurement, if there are such things at all, must in some way be components of the world, a change in physical theory in principle brings about a change in the physical description of observation and measurement, or in other words in the empirical significance of the term empirical. Or it would produce such a change, rather, if there were ever such a thing as a physical description of observation and measurement. An observations qua observation or a measurement qua measurements is not of interest to physicists: i.e., these are not, as such, objects of the science of physics. (I even think this could serve as the definition of the science.) So in fact there never is a physical description of observation and measurement as such, and empirical never has a well-understood empirical significance.
10. A philosophical understanding of empiricism must, therefore, focus on the transcendental, rather than the empirical, significance of the term. And this brings us back to the topic of nominalism, properly speaking. I see no call to turn back from Kant. In the pre-Kantian version, to repeat, the claim that the object of some representation R is not per se real is connected to the claim that the same entity which is object to R is also represented qua res by some other representation, R0 . R0 represents the object immediately, whereas R represents it via the mediation of R0 . But Kant rejects this view precisely on the basis of a principle of empiricism, in a sense which abstracts from all empirical content of the term empirical: the principle, namely, that, for finite beings like ourselves, the immediate representation of an object is not intellectual. Since the representation of an object as possessing determinate realitas that is, as limited by possession of an essence is necessarily intellectual (for us: conceptual), this means that the object of cognition for us is always real only qua object of mediate representation. The object is not a res per se (Ding an sich), then, because intellectual representation is, in itself, the representation of something real, but is only mediately the representation of an object. Empirical realism is transcendental idealism.
11. This view faces severe difficulties in accounting for progress in physical science. Without going into all the details here, it probably should not surprise anyone if I say that they revolve around what Kant calls the pure form of intuition. Specifically, the challenge is to show how we can learn empirically that what appeared to be a feature of this pure form in general was actually specific to a degenerate case (as the absoluteness simultaneity, for example, follows from the degenerate metric of Galilean spacetime). But I think this problem is the right problem: that is, I think this is what we in fact dont understand about progress in physics.