¹ van Inwagen, Peter, ‘God and Other Uncreated Things,’ in Timpe, Kevin (ed.) Metaphysics and God (London: Routledge, 2009)Google Scholar, 19.

² van Inwagen, Peter, ‘A Theory of Properties,’ in Zimmerman, Dean (ed.) Oxford Studies in Metaphysics, vol. 1 (Oxford: Clarendon Press, 2004)Google Scholar, 107.

³ Ibid., 110–11.

⁴ Ibid., 107.

⁵ Van Inwagen, ‘God and Other Uncreated Things’, 19.

⁶ Øystein Linnebo identifies the insouciant platonism of which I speak as a species of what he calls ‘lightweight platonism’ (Stanford Encyclopedia of Philosophy, s.v. ‘Platonism in the Philosophy of Mathematics [§§5.2–3],’ by Øystein Linnebo, http://plato.stanford.edu/entries/platonism-mathematics/ July 18, 2009; cf. David Chalmers' characterization of such views as ‘lightweight realism,’ and David Manley's ascription to neo-Fregeans of the use of ‘lightweight quantification’ [Chalmers, David J., ‘Ontological Anti-Realism,’ in Metametaphysics: New Essays on the Foundations of Ontology (Oxford: Clarendon Press, 2009), 78, 95–101Google Scholar; David Manley, ‘Introduction: A Guided Tour of Metametaphysics,’ in Metametaphysics, 19, 25]). Linnebo thinks such lightweight semantic platonism falls short of platonism by denying the mind-independence of abstract objects, while I suspect these would-be platonists of failing to affirm the existence of abstract objects. Of them it could be said with Russell, ‘In such theories, it seems to me, there is a failure of that feeling for reality which ought to be preserved even in the most abstract studies’ (Russell, Bertrand, ‘Descriptions,’ in Moore, A.W. (ed.) Meaning and Reference (Oxford: Oxford University Press, 1993)Google Scholar, 47). N.B. that Linnebo admits that lightweight semantic platonism meets the standard criterion for mind-independence with respect of mathematical objects, viz., there would still be such objects if no persons existed, but he thinks that we need a different criterion because lightweight platonists reject the analogy between physical objects and mathematical objects. As we shall see, that is not true: they think that such objects are as real as hurricanes, holes, and the Eiffel Tower. Whence, then, their insouciance? It is, I suspect, because they do not think these things truly exist.

⁷ Sider, Theodore, ‘Neo-Fregeanism and Quantifier Variance’, Aristotelian Society Supplementary Volume 81 (2007): 201–32CrossRefGoogle Scholar; Eklund, Matti, ‘Neo-Fregean Ontology’, Philosophical Perspectives 20 (2006): 95–121CrossRefGoogle Scholar.

⁸ Bob Hale and Crispin Wright, ‘The Metaontology of Abstraction’, in Metametaphysics, 181–6.

⁹ Ibid., 192.

¹⁰ Ibid., 193.

¹¹ Ibid., 194.

¹² Ibid.

¹³ Ibid., 209.

¹⁴ Hale, Bob, Abstract Objects, Philosophical Theory (Oxford: Basil Blackwell, 1987)Google Scholar, 3.

¹⁵ Ibid., 3–4.

¹⁶ Ibid., 4; cf. Hale and Wright, ‘Metaontology of Abstraction’, 207.

¹⁷ Hale, Abstract Objects, p. 4; cf. the last paragraph on 26.

¹⁸ Chihara notes that ‘The classical Logicists, Frege and Russell, thought that there was some ontologically (or logically) basic totality-‘objects’ for Frege and ‘individuals’ for Russell-that the lowest level variables were supposed to range over,’ a view which he finds widely doubted in contemporary Anglo-American philosophy (Chihara, Charles S., Constructibility and Mathematical Existence (Oxford: Clarendon Press, 1990)Google Scholar, 69). For we should not think of an object or individual as ‘a particular kind of thing; it is a particular role that things of any kind may occupy: the role of subject of predication. To accept the semantics for quantification theory is not to accept any particular metaphysics of individuals’ (Ibid., 70). Chihara thus questions Quine's criterion of ontological commitment because Chihara is not sure what an entity is on Quine's view. Similarly, Hale seems to have stripped objects of any ontological significance.

¹⁹ I note that this is how van Inwagen also understands the word, for he says that if a table were to exist, ‘it would be real, a true object, actually a thing, a substance, a unified whole’ (van Inwagen, Peter, Material Beings (Ithaca, NY: Cornell University Press, 1990)Google Scholar, 100).

²⁰ A deflationary theory of reference developed along the lines limned by Horwich, Paul, Meaning (Oxford: Clarendon Press, 1998)CrossRefGoogle Scholar or, better, Båve, Arvid, ‘A Deflationary Theory of Reference’, Synthèse 169 (2009): 51–73CrossRefGoogle Scholar allows us to use singular terms non-vacuously even though there are no objects in the world correlated with those terms. Noneists like Richard Routley have vigorously protested what he calls the Ontological Assumption (to wit, the assumption that a statement has the value true and is about something only if the subject of the statement refers to an existent object) underlying most contemporary theories of reference (Routley, Richard [Sylvan], Exploring Meinong's Jungle and Beyond: An Investigation of Noneism and the Theory of Items (Canberra: Australian National University Research School of Social Sciences, 1979)Google Scholar, 44; cf. 17, 22). Unlike deflationists, Noneists still share the belief that there must be an object to which reference is made, if reference is to be successful-hence, their belief in non-existent objects. Some have accused neo-Meinongians of being closet platonists; but my suspicion is quite the reverse: that insouciant platonists may, in fact, be crypto-Meinongians of some sort. For they hold that some singular terms refer to objects whose existence they deny or whose existence is widely denied.

²¹ For example, van Inwagen considers the postulation of events to be ‘ontologically profligate.’ ‘There are, I would say, no events. That is to say, all statements that appear to involve quantification over events can be paraphrased as statements that involve objects, properties, and times-and the paraphrase leaves nothing out’ (Van Inwagen, ‘God and Other Uncreated Things’, 14). Theodore Sider compares talk of properties in a nominalistic understanding to talk of holes:

N.B. that abstract objects would similarly be entities made up of nothingness and, unlike holes, lacking even liners.

²² Hale, Abstract Objects, 4.

²³ On which see Dummett, Michael, Frege: Philosophy of Mathematics (Cambridge, Mass.: Harvard University Press, 1991)Google Scholar, chap. 10. Dummett identifies §62 of Frege's Die Grundlagen der Arithmetik (1884) as the first example of the linguistic turn in philosophy. In what Dummett deems ‘the most pregnant philosophical paragraph even written’, Frege construes the question of how mathematical objects are given to us as a question concerning how the meaning of sentences containing singular terms for mathematical objects is to be fixed. Similarly, Hale is preoccupied with whether singular terms take abstracta as their objects, an approach which seems to me to obfuscate rather than elucidate ontology, since the notion of object as a semantic category is said to be so different than that of an object as a category of ontology. E. J. Lowe distinguishes a ‘linguistic’ and a ‘metaphysical’ answer to the question, ‘What is an object?’ The linguistic answer is anything that can be referred to at all, the reference of a singular term or the value of a variable of quantification. The metaphysical answer is any item that enjoys determinate identity conditions and so falls under a sortal concept (Lowe, E. J., ‘Objects and Criteria of Identity’, in Hale, Bob and Wright, Crispin (eds.) A Companion to the Philosophy of Language, Blackwell Companions to Philosophy 10 (Oxford: Blackwell, 1997)Google Scholar, 616). Lowe thinks that properties, facts, and propositions are objects merely in the linguistic sense and that mathematical objects count as objects in the metaphysical sense. What does Hale think? Since he denies that abstract objects are objects in the common sense of that word and focuses on their role as referents of singular terms, it is hard to tell.

²⁴ Hale later agrees with Dummett that the debate over mathematical platonism must be about the question, ‘Are there true statements whose proper analysis discloses expressions purporting reference to numbers?’ Although it might seem tendentious to ignore the ontological dispute in favor of the truth-value dispute, Hale finds plausible Dummett's suggestion that a dispute over the existence of certain abstract entities might be represented as a truth-value dispute by taking the disputed class of statements to consist of statements purporting reference to those entities. Indeed, the dispute is best elucidated in terms of the objective truth of statements purporting reference to such entities (Bob Hale, ‘Realism and its Oppositions’, in Companion to the Philosophy of Language, 272–3, 284–5). The general endorsement of this approach to questions of ontology, he says, admits to acceptance of Frege's Context Principle which warns against asking after the reference of substantial expressions outside the context of complete sentences. For the implications of this approach see Dummett's comments in note 28 below.

²⁵ Whether or not semantically determined objects belong in one's ontological inventory will depend on one's theory of reference. Dummett muses that Frege had a ‘thin’ theory of reference analogous to the redundancy theory of truth which was insufficient to bear the weight of a realistic interpretation of those terms (Ibid., 195–8; cf. note 28 below). Which theory of reference one prefers is apt to depend on what one thinks exists. I am therefore inclined to agree with Achille Varzi that linguistic analysis is pretty useless as a tool for drawing up an ontological inventory (Varzi, Achille C., ‘Words and Objects’, in Bottani, Carrara, Giaretto (eds.) Individuals, Essence, and Identity, Topoi Library 4 [Dordrecht: Kluwer Academic Publishers, 2002], 49–75CrossRefGoogle Scholar).

²⁶ Dummett, Frege: Philosophy of Mathematics, 231.

²⁷ Ibid., 181.

²⁸ Cf. Dummett's earlier reflections on Frege's platonism:

²⁹ Dummett, Frege: Philosophy of Mathematics, 182.

³⁰ Ibid., 80.

³¹ Rosen, Gideon and Burgess, John P., ‘Nominalism Reconsidered’, Shapiro, Stewart (ed.) in The Oxford Handbook of Mathematics and Logic, Oxford Handbooks in Philosophy (Oxford: Oxford University Press, 2005)Google Scholar, 525.

³² Ibid.

³³ Ibid.

³⁴ Van Inwagen, Material Beings, 107.

³⁵ Burgess, John P., ‘Mathematics and Bleak House,’ Philosophia Mathematica 12 (2004): 30–1CrossRefGoogle Scholar.

³⁶ Ibid., 19. Some of Burgess' remarks suggest that he is a sort of Carnapian conventionalist or ontological pluralist with respect to abstract objects; others of his remarks suggest at most agnosticism about what really exists. But his theological perspective–‘the only way to make sense of questions of ontological metaphysics’–yields a clear, negative answer. Unlike van Inwagen, Burgess could recite the Nicene Creed without mental reservation–at least on grounds of platonism.

³⁷ Ibid.

³⁸ Quine, W. V. O., ‘A Logistical Approach to the Ontological Problem’, in The Ways of Paradox, rev. ed. (Cambridge, Mass.: Harvard University Press, 1978)Google Scholar, 202.

³⁹ Van Inwagen, ‘Theory of Properties’, 131.

⁴⁰ Armstrong, D. M., A Theory of Universals: Vol. 2: Universals and Scientific Realism (Cambridge: Cambridge University Press. 1978), 7–12Google Scholar. Cf. Hale and Wright, ‘Metaontology of Abstraction’, 207–9, where they compare favorably their minimalist conception of objects and reference with the ‘abundant’ as opposed to sparse view of properties, in contrast to ‘the anxious metaphysician’ who thinks of the issue analogously to the existence of sparse properties, worrying whether one is referring to ‘real properties’ in the metaphysical World.