Abstract
The standard argument for the existence of distinctively mathematical objects like numbers has two main premises: (i) some mathematical claims are true, and (ii) the truth of those claims requires the existence of distinctively mathematical objects. Most nominalists deny (i). Those who deny (ii) typically reject Quine’s criterion of ontological commitment. I target a different assumption in a standard type of semantic argument for (ii). Benacerraf’s semantic argument, for example, relies on the claim that two sentences, one about numbers and the other about cities, have the same grammatical form. He makes this claim on the grounds that the two sentences are superficially similar. I argue that these grounds are not sufficient. Other sentences with the same superficial form appear to have different grammatical forms. I offer two plausible interpretations of Benacerraf’s number sentence that make use of plural quantification. These interpretations appear not to incur ontological commitments to distinctively mathematical objects, even assuming Quine’s criterion. Such interpretations open a new, plural strategy for the mathematical nominalist.
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Notes
See e.g. Frege (1884), Benacerraf (1973), Wright (1983), and Hale (1988). Wagner (1996, pp. 84–86) presents Benacerraf’s argument as a hybrid that addresses both steps of the argument I have described above. I can find no textual evidence in Benacerraf (1973) of an argument for mathematical truth. But that is consistent with Benacerraf’s having presupposed such an argument to motivate his assumption of mathematical truth.
Sometimes the term ‘nominalism’ is used for the view that there are no abstract objects, or no abstract objects in some particular domain. (See, e.g., Burgess and Rosen 1997) My use of the term ‘nominalism’ excludes all views on which distinctively mathematical objects exist, whether or not the relevant objects qualify as abstract.
Quite a few nominalists are explicit about this ‘literal’ restriction; they deny that mathematics is literally true, and they implicitly or explicitly accept the truth-object connection for mathematical statements read literally. Nominalists who use the term ‘literal’ when taking such a position include, e.g., Hellman (1989, p. 2), Field (1989, p. 2), Chihara (2003, p. 294), Yablo (2002, p. 230), and Leng (2010, p. 14). Balaguer (2009, p. 135) also uses ‘literal’ in this context, but he is not clearly a nominalist; he seems to be agnostic between nominalism and ontological realism. But not all nominalist accounts that deny mathematical truth and accept the truth-object connection involve making a literal/non-literal distinction; the account Maddy (2011) calls ‘arealism’ does not.
Hellman’s (1989) modal structuralism and Chihara’s (2003) constructability theory, for example, both include intricate non-literal semantics for mathematical statements. And though Field’s main project lies elsewhere, he also briefly suggests a non-literal account on which statements like ‘\(2+2=4\)’ are true when placed inside the intensional operator ‘according to standard mathematics’ (see 1989, p. 3).
Strictly speaking, Benacerref presents it as an argument for platonism. That’s because Benacerraf thinks his semantic argument is to be paired with a “platonistic view of the nature of numbers” (1973, p. 664). But he offers no real reasons for construing numbers in this way, other than that they often are so construed. The conclusion that his argument supports is that mathematical objects—specifically numbers—exist. For the purposes of this paper, it is irrelevant whether or not putative mathematical objects are abstract.
See, e.g., Creath (1980). Field (1989, p. 229) observes, “if the belief that mathematics is true in some ‘correspondence’ sense leads to difficulties, perhaps the difficulties arise not from the belief in mathematics but from the belief in the notion of correspondence truth.” And Tait (1986, p. 347) disputes the relevance of a Tarskian theory, claiming, “It is difficult to understand how Tarski’s ‘account’ of truth can have any significant bearing on any issue in the philosophy of mathematics.”
Fregean arguments for ontological realism have two steps. First, it is argued that numerical expressions are used as singular terms in true sentences. Second, it is argued that singular terms in true sentences invariably refer to objects. The conclusion is that numerical expressions refer to objects: numbers. See, e.g., Hale and Wright (2001).
Azzouni’s characterization of ontological independence actually is not metaphysical; it is epistemic. It includes, e.g., that we must use non-trivial methods for determining truths about ontologically independent posits. For details of his characterization of this notion, see his (2004).
Bueno and Zalta (2005) follow Azzouni in denying Quine’s criterion of ontological commitment. They offer a deflationary account of ‘object’ as a possible way of fleshing out Azzouni’s theoretical posits. On their view, arithmetical statements like (2) are ambiguous between two literal readings, each with the general form of (3). One reading incurs ontological commitments; (2) is false on this first reading. The other reading merely incurs commitments to the deflationary objects, and (2) is true on this second reading.
My numbering is the same as Benacerraf’s.
I would guess that Hofweber (2005) would also be concerned about this assumption. But he does not address sentences like (2), and it is not clear how he would handle them. Hofweber does not explain how his account handles predication in arithmetic; he never explains how to interpret claims like ‘28 is a perfect number’. He says this explicitly in the case of the predicative sentence ‘Two is a prime number’: “I will not be able to discuss these uses here, but leave them for another occasion” (2005, p. 210). As far as I can tell, he has not addressed the issue of arithmetical predication elsewhere.
Thanks to an anonymous referee for suggesting this line of thought.
The same holds if (4) is used as a generic. (Perhaps generics are not about any and all instances of the kind, but only most or typical ones.) But for the rest of this paper, I will set generics aside. I am using (4) to open up possible interpretations of (2), and it strikes me as implausible to interpret (2) as a generic. Thanks to another anonymous referee here.
Helen Cartwright (1970) points out that the volumes, weights, etc. of what she calls quantities of stuff can vary; for example, the same quantity of gold will weight less at the top of a mountain than it does at a lower altitude, and the same quantity of coffee has a greater volume when hot than it has when cool. Because of these kinds of variations, she thinks that it is possible to compare two different quantities of the same kind of stuff in a way that it is not possible to compare two quantities of different kinds of stuff. This observation seems to pose further problems for taking ‘heavier than’ to be a relation between masses of e.g. helium and krypton. What unit of measurement do you use? The obvious candidates, like volume, are not related to weight in a consistent way (since e.g. increasing pressure will decrease the volume of a given quantity of a gas like helium or krypton).
Burgess (1990, p. 7) argues in a similar way. On the basis of the fact that linguists have not offered a nominalistic reconstruction of (2), he assumes that it must have a platonistic interpretation.
Some might follow Boolos (1984) in thinking that second-order quantifiers do not involve the same kind of ontological commitment as first-order quantifiers. If they are correct, so much the better for nominalistic interpretations.
Yi (2006) points out that Tarski’s semantics cannot handle plurals, so what follows will depend on a non-Tarskian semantics. But I am still assuming that first-order singular quantifiers, second-order singular quantifiers, and singular terms are ontologically committing.
In this paper, I assume that plural quantification (e.g.) over the Fs incurs ontological commitment to each of the Fs, but nothing beyond that; plural quantification is ontologically innocent in that it incurs no ontological commitment to a set, sum, plurality, etc. of the Fs. This is the standard view. Arguments for this view can be found in Boolos (1984), Lewis (1991), McKay (2006), Rayo (2002), and Yi (2002, 2005). But before the publication of many of the arguments referenced above, there were, as Linnebo (2003, p. 71) puts it, “some isolated voices of dissent” about the ontological innocence of plural quantification. These include Resnik (1988), Parsons (1990), and Hazen (1993).
More recently, additional concerns have been raised about the ontological innocence of plural quantification. Linnebo (2003) and Florio and Linnebo (2015) have floated the idea that the debate over the ontological commitment of plural quantification is just a verbal dispute. Florio and Linnebo also claim that there are both narrow and broad notions of Quinean ontological commitment, and that plural quantification incurs ontological commitments to pluralities on the broad notion. But they do not argue for the broad notion of ontological commitment over the narrow one, and hence do not argue that plural quantification does incur ontological commitments to pluralities.
Azzouni (2015) goes a bit further. He argues that plural quantification does incur ontological commitments to some kind of set, sum, plurality, or such beyond the individuals. He does so on the grounds that it is a notational variant of a logical system he calls ‘(2m)truncated logic’. For the sake of argument, let’s suppose that Azzouni is correct about the ontological commitments of plural quantification. He claims that the Quinean is (or ought to be) committed to some entity that is, e.g., the Cheerios taken together. But, he insists, “nothing requires these things be sets” (p. 15). Other possible Cheerios-taken-together entities he suggests are a bunch, an aggregate, a fusion, a lot, a group, and a heap. Let’s take one of his suggestions: the Cheerios taken together are a fusion. If so, then the Cheerios are akin to a mass of physical stuff. That is, suppose that plural quantification amounts to quantification over masses of stuff. Then accounts of number that rely on plural quantification, such as the accounts that I suggest later in this paper, involve ontological commitment to masses of stuff. But masses of stuff are not distinctively mathematical entities. So, even supposing that Azzouni’s arguments succeed, his conclusion is consistent with nominalistic plural interpretations of (2). Thanks to an anonymous referee for raising this issue.
Again, I am setting aside generics. See note 15.
Moltmann offers a different account of semantics of the very similar sentence ‘\(2+2=4\)’, which she takes to be an identity relation between the abstract objects that are the referents of ‘\(2+2\)’ and ‘4’, respectively. So her overall view is a kind of ontological realism.
Yablo (2002) also suggests an interpretation that reads a bit like a plural property view: “Consider now ‘\(7 < 11\)’. To most (!) people, most of the time, it means that seven somethings are fewer than eleven somethings” (p. 230). But Yablo’s intended interpretation involves nested singular existential quantifiers, not pluralities (see Yablo 2002, p. 231). And Yablo denies that this interpretation captures the literal content of ‘\(7<11\)’ anyways (see Yablo 2001, p. 78); he takes that to involve quantification over putative number objects. See note 3.
At least, they are the only ones I can find; they are the only sentences he discusses in print.
Linnebo and Nicolas (2008) provide different examples of the use of superplurals in natural language.
For an example of how a semantics for plurals, superplurals, and higher-order plural quantification might work, see Rayo (2006).
It might be ontologically misleading to use the term ‘plurality’ in this way. The term sounds like it involves ontological commitment to a set of things, rather than just to the things themselves. Because I merely use the term as shorthand, I hope to avoid the ontological connotations associated with less artificial uses of the term.
I read the phrasing ‘at least n’ as pushing in a quantificational direction. But a quantificational reading of a numerical claim might seem a bit incongruous with an account of numbers as plural properties. One might want, e.g., ‘there are five prime numbers less than 12’ to attribute the plural property being five to the prime numbers less than 12, taken together. But notice that 5 itself is one of the prime numbers less than 12. This suggests a potentially vicious circularity of reference.
On the superplural account, number-words refer superplurally. Such an account could allow being five to apply to the prime numbers less than 12 by giving a type theory of the higher-order plural reference of number-words. At the first level, ‘5’ would superplurally refer to any objects that, taken together, have the property of being five [objects]. At the second level, ‘5’ would refer to any things taken together that, taken together, have the property of being five [takings-together-of-objects]. And so on. Rayo’s (2006) semantics of higher-order plurals can support a type theory such as this, but cannot support an untyped theory of superplural reference for number-words.
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Acknowledgements
Thanks to Ben Caplan for discussion. Thanks also to Otávio Bueno and two anonymous referees for feedback on this paper. For feedback on a shorter version of this paper, or on parts of this paper that were once contained within a different paper, thanks to Patricia Blanchette, Salvatore Florio, Graham Leach-Krouse, Corey Maley, and Stewart Shapiro, and also to audiences at McGill University and the University of Kansas.
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Nutting, E.S. Ontological realism and sentential form. Synthese 195, 5021–5036 (2018). https://doi.org/10.1007/s11229-017-1446-4
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DOI: https://doi.org/10.1007/s11229-017-1446-4