To a first approximation, physicalism is the view that everything is physical. Reaching a second approximation requires attending to two tasks. (1) specifying further what it is for something to be physical. (2) Specifying the range of the quantifier “everything”.
I don’t want to get involved here in detailed discussion of (1). I take it to be plausible that abstract objects—for instance, numbers—will not trivially be included in the domain of the physical. That’s not to say that they don’t play an essential role in theories in physics. But I take that to present a difficulty for simplistic accounts of the physical—for example, as whatever figures essentially in theories in physics—rather than as a reason for counting abstract objects as physical. It’s anyway natural to think that abstract objects will not be included except via an account of how talk about abstract objects is not to be taken at face value as talk about abstract objects, but is rather to be given some other treatment. Perhaps, for example, statements putatively about abstract objects are true, but are not really about abstract objects; rather, they are indirect characterisations of properly (that is, trivially) physical elements. Alternatively, perhaps statements about abstract objects are apt to be true or false of abstract objects but, despite appearances, are uniformly false, and their function in successful explanations is to be explained without appeal to their truth. As a third alternative, it might be argued that statements apparently about abstract objects are not statements at all; they are more like rules or regulations governing the practice of statement-making proper. I take it that the natural, default position is that no such reconstruction is required and that statements apparently about abstract objects and apparently true are about abstract objects and are true. For present purposes, I will assume that the natural, default position is correct.
If “everything” in the characterisation of physicalism meant everything, then the existence of abstract objects would provide a counter example to physicalism. The defender of that strength of physicalism would therefore be required to pursue a project of reconstruction: either one of those sketched above, or some other means of avoiding our apparent commitment to the existence of abstract objects. In fact, however, it’s common to avoid the need to pursue such a project by allowing that the existence of abstract objects is not an immediate threat to physicalism. Rather than defend the strong view on which absolutely everything is physical, many physicalists opt instead for a weaker view on which everything that is located in space-time, and/or that interacts causally, is physical.
Furthermore, standard arguments for physicalism support only the weaker view. Here is a version of a standard form of argument.
P1. Let e be an element such that (a) e impinges causally on some elements, p, that are trivially physical and (b) the status of e as physical or not is up for grabs.
P2. Wherever there is causal impingement on p, there are physical effects on, or involving, p. (Supposed to be underwritten by conservation laws in physics.)
P3. Wherever there are physical effects on, or involving, p, those physical effects are determined to obtain by other physical elements, q, together with physical laws governing the behaviours of p and q. (From P2.)
P4. Suppose that e is not physical.
P5. In that case, something non-physical impinges causally on p. (From P1.)
P6. But any effects that e has on p are determined by physical q, together with physical laws. (From P3.)
P7. It’s implausible/wrong to suppose that effects on, or involving, p are determined by q and also by independent e. (Supposed general metaphysical principle.)
P8. It’s implausible/wrong to suppose that e is independent from physical q (From P4, P6, P7.)
C9. e is, or is dependent upon, physical q.
Whether or not we think that this, or some successor, provides a good argument in favour of physicalism, it’s clear that it only applies to es that impinge causally on physical elements. So, unless we think that all existent abstract objects do that, the argument will not, on its own, support a form of physicalism that includes in its range all abstract objects. In order to make the argument bear on the status of all abstract objects, one would need a supplementary argument to the effect that all abstract objects impinge causally on the physical realm. Such a supplementary argument might appeal, for example, to a combination of the apparent fact that we know things about abstract objects together with a general principle—in need of detailed defence—to the effect that we can know about things only insofar as they impinge causally on the physical, e.g. on the physical elements that constitute our brains. However, I think that most physicalists who also accept the existence of abstract objects would not want to take that route. Since such a physicalist would have independent reason to hold that abstract objects are not physical, they would have to treat the conclusion either as a ground for viewing the argument with suspicion, or as a ground for giving up their commitment to the existence of abstract objects. Instead, I think most such physicalists would prefer to adopt a form of dualism, according to which there are two fundamental realms of being: the physical realm and the non-physical realm of abstracta.
Even the weaker form of dualist physicalism would be undermined if there were elements that were at least partially located in space-time, and that interacted causally with physical elements, but whose natures depended essentially on abstract objects. In the sequel, I’ll sketch, without attempting to defend, an argument to that conclusion.