Abstract
In this paper, I argue that all expressions for abstract objects are meaningless. My argument closely follows David Lewis’ argument against the intelligibility of certain theories of possible worlds, but modifies it in order to yield a general conclusion about language pertaining to abstract objects. If my Lewisian argument is sound, not only can we not know that abstract objects exist, we cannot even refer to or think about them. However, while the Lewisian argument strongly motivates nominalism, it also undermines certain nominalist theories.
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Notes
Technically Lewis defines an external relation as one that is not internal and depends on the intrinsic nature of the fusion of the relata (p. 177), but as Lewis himself points out (p. 182, number 3), nothing is really lost just by thinking of an external relation as a relation that isn’t internal.
This is a modest simplification. Van Inwagen goes on to consider in more depth the possibility that membership is “range-internal” or “range-external”, but these complications are unnecessary to go into here.
A curious fact is that Lewis was bothered by van Inwagen’s tu quoque argument despite at times regarding impure sets as being located where their members are and capable of causally interacting with things. Lewis even says that pure sets are the most dispensable and metaphysically suspicious of them all Inwagen (1986, pp. 83–84). The reason for Lewis’ concern is how van Inwagen identifies the heart of the problem. For van Inwagen, the key is that we can only distinguish between sets by means of the membership relation. Even if they are spatiotemporally located and causally involved in the world, we don’t seem to be able to access the properties of impure sets that are relevant to the membership relation (Lewis 1991, pp. 29–35). In fact, we only come to know of a given set’s location and causal entanglements by first knowing what its members are. “Acquaintance” is a necessary condition to be able to grasp the intrinsic natures of things, but it isn’t a sufficient one. In the end, Lewis is ambivalent even about the locatedness and causal involvement of impure sets (1991, pp. 31–33). Since most abstract objects are not causally active, acquaintance is what I will focus on in my Lewisian argument. Still, the point is well-taken: the story of how an internal relation is understood doesn’t automatically stop once causal relations are introduced. A view like Penelope Maddy’s (1990) is not subject to the difficulty van Inwagen raises for identifying sets since Maddy explicitly views sets as being perceptible and having a variety of sensible qualities in addition to being spatiotemporally located and causally involved. Different sets, therefore, are apt to be distinguished by all sorts of properties on Maddy’s view.
Probably, perception requires causal interaction, but in case that is not always true I add it as an additional option for gaining understanding.
This assumes a standard platonic view of abstracta, wherein abstracta are not spatiotemporally located and are causally isolated from concreta. Those who hold, for example, that properties are spatiotemporally-located, perceptible constituents of concrete things will deny this point (e.g. David Armstrong (1989)), as will those who agree with platonists that abstracta are not located in space but hold that they causally interact with concreta (e.g. Julian Dodd (2007) and David Friedell (2020)). My opinion is that such views are flawed for independent reasons, but it should be noted that such views might have a way out of the Lewisian argument. However, causal interaction with the abstracta to which e pertains is not enough by itself to guarantee that e is intelligible: see fn. 3’s discussion of Lewis’ concerns about van Inwagen’s tu quoque.
One might think hallucination casts doubt on the principle that relevant perceptions or causal interactions are necessary for grasping the meaning of an internal expression. Take ‘color’. Suppose someone blind from birth had their brain stimulated by a scientist in such a way that they seemed to see a red sphere, a blue cone, and a green cube. If the scientist were to tell them ‘color’ referred to how these apparent objects looked and that there were other colors besides, it’s plausible that the blind person would understand what ‘color’ means even without having seen anything colored or having been causally related to colored things in a way relevant to concept acquisition. But even in this case, the blind person seemed to see a colored thing, and no one can seem to perceive an abstract object in this sense: abstract objects can no more be imagined or hallucinated than they can be perceived. Modifying the general principle so that it reads “For any n-ary expression that applies to some things in virtue of how they are intrinsically, to grasp it one needs to seem to perceive or causally interact with some things to which e applies (jointly or individually, as appropriate to the expression), or e needs to be analyzable in some way in terms of already familiar notions” would be sufficient to justify premise (2).
Though some philosophers would only say this is true for pure sets: see fn. 3.
This gloss assumes that ‘instantiates’ and ‘member of’ necessarily both exist and have their actual meaning if their actual application conditions are met. This assumption is false, but no matter: the real philosophical problem is posed by the biconditionals, not the gloss.
This assumes e is a binary predicate, but the point and schemas could easily be modified for any n-ary predicate.
This is only true for qualitative properties, but the schemas mentioned involve qualitative predicates on the left-hand side of the embedded biconditionals, not non-qualitative ones.
This is a factual point, not a logical one. There is no contradiction in supposing that there are some internal expressions that individually are applicable to pluralities of concrete objects, but some logical combination of which is not applicable to any plurality of concrete objects and is applicable to some plurality containing an abstract object. But it is obvious enough that no such internal expressions could be logically combined to define ‘member of’.
Setting aside the difficulties that some pluralities do not constitute a set and that some predicates lack a corresponding property as a concession.
A caveat should be made in that with the neo-Fregean’s examples, only a predicate is supposed to be grasped by the observation of truth-conditions, whereas in my extrapolations, a new predicate and new terms are proposed to be grasped by them. A neo-Fregean might say that only one expression can be imbued with meaning at a time given this device. If so, however, the neo-Fregean suggestion I’m making might already be a non-starter. It is hard to see how to use an abstraction principle to explain what ‘instantiates’ means without making use of terms for properties or how to explain what a term for a property means without invoking instantiation. The same goes for pretty much any pair e and s meeting the conditions mentioned in the exposition of the Lewisian argument.
This connects to the wider literature on incomplete stipulations. For example, Timothy Williamson (1997, pp. 222–227) and David Barnett (2008) offer interesting opinions about the relationship between meaning, incomplete stipulations, and those stipulations’ associated expressions. General theories of how they interrelate could bear on the ‘schmoogle’ case, and intuitions about the ‘schmoogle’ case might in turn bear on the plausibility of those theories. This merits deeper engagement that ought to be pursued in the future. For now I will just remark that I am sympathetic to views like Barnett’s and direct the reader toward my critique of some possible neo-Fregean rejoinders over the next several paragraphs.
Of course, one could stipulate a definition or precisification of ‘x schmoogles y and z’ if one already has expressions for abstracta in the lexicon: perhaps ‘y and z are members of x, y is spatiotemporal, and z is spatiotemporal’. The point is just that the principles given for schmoogling are not enough on their own.
Note that the question of whether schmoogling is boogling is nominalistic. To see this clearly, it can be put into the idiom of Cian Dorr (2016): is it the case that for any x and y, for x to schmoogle y is for x is to boogle y? Likewise, the question of whether schmoogling is doogling is nominalistic.
Saul Kripke (1980) provides the most well-known articulation and defense of the causal theory of reference.
The boogle and doogle case is reminiscent of the Julius Caesar problem for neo-Fregeanism. For an overview of the Julius Caesar problem, see Richard Kimberly Heck (1997). A fuller comparison of the boogle and doogle case and the Julius Caesar problem must await future work.
This is not to say that if the Lewisian argument is sound, no supervaluationist account of vague expressions can be correct. It is just that how vague predicates get a supervaluationist semantics can’t be via some explicit stipulation that involves quantifying over properties and relations.
Higher-order quantifiers can do the trick, but once higher-order quantifiers are permitted the Ramsey sentence solution already becomes available.
See Lewis (1984) for a defense of reference magnetism.
A plausible initial thought is that for every predicate there is a property, but such an approach leads to Russell’s paradox. Besides, it is at least an epistemic possibility that there are some properties for which no predicate has ever or will ever be introduced.
See Justin Clarke-Doane (2016) for discussion.
Michael D. Resnik (1997, Ch. 7) pursues this approach, among others.
At least—barring exceptional cases such as the one described in fn. 6 – this is true if pinkness is phenomenal pinkness rather than merely the disposition to reflect such-and-such wavelengths of light. But even if a reductionist account of color is correct, I must surely have been causally or perceptually related to light to know what ‘the largest pink object is self-identical’ means.
Note that Field takes himself to be understanding Benacerraf’s challenge in a new way. Clarke-Doane (2016) also discusses Field’s challenge.
A Field-style argument can also be used to motivate moral skepticism. See David Enoch (2010) for an exposition of this argument. Enoch proposes a solution that strikes me as unsatisfactory for reasons I cannot get into here. To my mind, that Field-style arguments can be generated in so many areas of inquiry is reason to think that something is wrong with the general demand for explanation that Field-style arguments reflect in the first place.
The Lewisian argument that I proposed above, of course, makes use of expressions in the sense of expression-types. It thus seems to presuppose some form of platonism about expressions. Happily, this is just convenient shorthand: it would be a simple matter to reframe the argument in terms of token expressions, albeit doing so would make the argument unnecessarily laborious to follow.
One might hope to get around this issue by interpreting the claims about abstract objects in the paraphrases through the lens of Lewis: that is, by taking the paraphrases as statements regarding what would obtain were (actual) concrete things how they actually are, modal realism true, Lewis right about properties and the like, and set theory reconstructed in the manner of the Appendix of Parts of Classes. Taken this way, no primitive abstract expressions would occur in the paraphrases when fully articulated. But this would raise several problems, not the least of which is that it is hard to evaluate a claim of the form “(actual concrete things are how they actually are ⋀ modal realism is true ⋀ [insert the ideas of Lewis regarding sets and other abstract objects]) □→ P”. What does one make of a claim like that? More simply, how does one evaluate a claim of the form “(actual concrete things are how they actually are ⋀ modal realism is true ⋀ Q) □→ P”, given that modal realism is false? Among other things, it seems at least epistemically possible that there is more than one actual spacetime, a possibility inconsistent with modal realism.
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Acknowledgments
I especially thank Jeff Speaks and my anonymous reviewers for their generous comments on this paper. I also thank Peter van Inwagen, Daniel Nolan, Geoffrey Hall, and Benjamin Middleton for related discussions that were instrumental to this paper’s improvement.
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Himelright, J. A Lewisian Argument Against Platonism, or Why Theses About Abstract Objects Are Unintelligible. Erkenn 88, 3037–3057 (2023). https://doi.org/10.1007/s10670-021-00489-4
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DOI: https://doi.org/10.1007/s10670-021-00489-4