• MBOs are often paraphrasable away with no felt loss of subject matter, e.g. ‘That was her first encounter with the green-eyed monster’ goes to ‘That was her first time feeling envious.’

• POs are often paraphrasable away with no felt loss of subject matter, e.g. ‘There is a possible world at which S’ goes to ‘It is possible that S’.

Impatience

• One is unmoved, even impatient, when someone suggests we should get worried about the fact that an MBO may not really exist. ‘I wasn’t thinking there was really a green-eyed monster.’

• One is unmoved, even impatient, when someone suggests that a PO or type of PO may not really exist. ‘So what if the models peter out above such and such a cardinality?’

Translucency

• About the best one can do with ‘What if there is no green-eyed monster?’ is to hear it as the (bizarre) suggestion that no one is ever really envious.

(p.170) • About the best one can do with ‘What if there are no God-worlds?’ is to hear it as the suggestion that God is not really possible.

Insubstantiality

• Unlike ordinary objects, MBOs tend to have not much more to them than what flows from our conception of them. The green-eyed monster has no ‘hidden substantial nature’; neither do the real estate bug, the blue meanies, the chip on my shoulder, etc.

• Unlike ordinary objects, POs often have not much more to them than what flows from our conception of them. All the really important facts about the numbers follow from (second-order) Peano’s Axioms. Likewise, mutatis mutandis for sets, functions, etc.

Indeterminacy

• Unlike ordinary objects, MBOs can be ‘indeterminately identical’. There is no fact of the matter as to the identity relations between the fuse I blew last week and the one I blew today, or my keister and my wazoo (‘I’ve had it up to the keister/wazoo with this paperwork’). The relevant game(s) leave it undecided what is to count as identical to what.

• Unlike ordinary objects, POs can be ‘indeterminately identical’. There is no fact of the matter as to the identity relations between the natural numbers and the Zermelo numbers, or the possible worlds and the maximal consistent sets of propositions, or the events and the property instantiations. It is left (partly) undecided what is to count as identical to what.

Silliness

• MBOs invite ‘silly questions’ probing areas the make-believe does not address, e.g. we know how big the average star is, where is it located? You say I have risen in your esteem, by how many inches? Do you plan to drop-forge the conscience of your race in the smithy of your soul?

• POs invite questions that seem similarly silly. What are the intrinsic properties of the empty set? Is the event of the water’s (p.171) boiling itself hot? Are universals wholly present in each of their instances? Do relations lead a divided existence, parcelled out among their relata?

Expressiveness (Representationality)

• MBOs show a heartening tendency to boost the language’s power to express facts about other, more ordinary, entities. ‘The average taxpayer saves an increasing fraction of his income.’

• POs show a strong tendency to boost the language’s power to express facts about other, more ordinary, entities. ‘This line here is p times as long as that line there.’

Irrelevance

• MBOs are called on to ‘explain’ phenomena that would not on reflection suffer by their absence. Even if the average taxpayer had not existed, the savings rate would still have increased.

• POs are called on to ‘explain’ phenomena that would not on reflection suffer by their absence. If all the one-one functions were killed off today, there would still be as many left shoes in my closet as right.

Disconnectedness

• MBOs have a tendency not to do much other than expressive work. As a result, perhaps, of not really existing, they tend not to push things around.

• POs have a tendency not to do much other than expressive work; numbers et al. are famous for their causal inertness.

Availability (Apriority)

• MBOs’ lack of naturalistic connections might seem to threaten epistemic access—until we remember that ‘their properties’ are projected rather than detected.

(p.172) • POs’ lack of naturalistic connections are widely thought to threaten epistemic access—the threat evaporates if ‘their properties’ are projected, not detected.

These parallels serve as evidence that abstract object talk is a kind of figurative language.9

the truth value of: is held to turn on:

argument A is valid the existence of countermodels

it is possible that B the existence of worlds

there are as many Cs as Ds the existence of 1-1 functions

there are over five Es the number of E’s exceeding five

he did it F-ly the event of his doing it being F

there are Gs which there being a set of Gs which

she is H her relation to the property H-ness

Platonic objects, Yablo points out, are deducible by what he calls overly easy existence proofs. Such overly easy proofs of the existence of Platonic objects trivialize the task of metaphysics. Yablo gives the following illustrations of discovering unexpected objects in statements’ truth conditions:

The entities denoted by the italicized terms in the right-hand column are said to be Platonic because the sentences in the left-hand column are not really about them.10 The italicized expressions on the right are therefore best taken as existential metaphors. Yablo claims that additional evidence for the metaphorical character of such abstract talk is that so construing it is the best explanation of why such overly easy existence proofs fail.

Ultimately, Yablo thinks that the decision to embrace Platonism or figuralism depends upon the answers to the following comparative questions:

1. (1) What does Platonism/figuralism help us to explain?

2. (2) What explanatory puzzles does Platonism/figuralism generate?

Consider, first, the explanatory puzzles generated respectively by each view. Yablo thinks that anti-Platonists have relied too heavily on the explanatory puzzles generated by Platonism (such as how we (p.173) could have epistemic access to abstract objects) as justification for anti-Platonism. But, unsurprisingly, Yablo takes no cognizance of the theological puzzle that drives our inquiry, namely, how the supposed existence of abstract objects can be reconciled with divine aseity and creatio ex nihilo. Given our theological commitments, some sort of anti-Platonism must be true.

As for explanatory puzzles generated by figuralism, the only problem that Yablo is aware of is the objection to pretence theory (to be discussed in Chapter 10) that abstract object talk and particularly mathematical discourse are not plausibly a matter of make-believe. This objection, however, is really an objection to a pretence theoretical analysis of figurative language, not to the fundamental figuralist claim that abstract object talk is figurative.

Consideration of such a puzzle is therefore better reserved for our discussion of pretence theory in Chapter 10. So with respect to the explanatory puzzles generated by Platonism/figuralism, Platonism clearly comes out with the short end of the stick.

So what about the explanatory benefits of Platonism/figuralism respectively? We have already seen the explanatory advantages which Yablo finds in figuralism’s handling of Platonic objects and overly easy existence proofs. That leaves the explanatory benefits of Platonism to be considered. The principal merit claimed on behalf of Platonism is that it provides a basis for the objective truth of mathematics.11 The assumption here is that mathematical objects

must exist in order for mathematical statements to be true. But why think that? Figuralism also affirms the truth of mathematical sentences, but since such sentences are figurative speech, they do not imply ontological commitment to objects. Just as ‘It’s raining cats and dogs!’ may be true without there being animals falling from the sky, so the truth that ‘1 + 1 = 2’ does not require the reality of numbers.

Still, we may wonder what the objective basis of mathematical truths is, if not the reality of the objects referred to or quantified over in such statements. Yablo maintains that what he calls the real content of mathematical truths is logical truths, which require no ontological foundation: ‘Arithmetic is, at the level of real content, a body of logical

truths—specifically, logical truths about cardinality—while set theory consists at the level of real content, of logical truths of (p.174) a combinatorial nature.’12 Since there are no numbers, the literal content of arithmetical truths is false; nevertheless, their real content is true by definition. When it comes to non-numerical mathematical statements

such as are comprised by set theory, Yablo takes the figurative language of sets to express certain combinatorial logical truths, that is, truths about what one gets when combining objects in different ways. Yablo recognizes that certain parts of set theory, for example, the Continuum Hypothesis, can at most be said to be relatively true, that is, true in some nice-looking models while false in others, but he thinks that the axioms of Zermelo-Fraenkel set theory are true in virtue of their real contents. Alternative set theories which do not represent sets as forming a cumulative hierarchy may also have real contents which are logical truths and therefore necessary and a priori; but such theories are not really about sets as we understand them. In sum, the realist has no advantage over the anti-realist in accounting for the objectivity of mathematical truth, since the real content of metaphorical statements about such imaginary entities as numbers and sets is logical truths. So once again, figuralism shows itself to be the superior view vis-à-vis Platonism in terms of its explanatory benefits.

Generating fewer explanatory puzzles while enjoying greater explanatory benefits than Platonism, figuralism is thus the preferable view.

Objections to Figuralism

How have Platonists responded to Yablo’s figuralism? The predominant complaint in the literature is that such a view either grossly misrepresents mathematical discourse or advocates without sufficient justification a radical revision of mathematical discourse. This complaint is rooted in a familiar distinction drawn by John Burgess and Gideon Rosen between what they call hermeneutic nominalism and (p.175) revolutionary

nominalism.13 Hermeneutic nominalism aims to offer an exegesis of mathematical discourse in order to determine

what mathematicians and scientists themselves mean by their mathematical assertions. The hermeneutic nominalist claims that the experts do not really take their assertions to be literally true and so ontologically committing to abstract objects. The revolutionary nominalist, by contrast, grants that mathematicians and scientists do take mathematical discourse to be literally true but offers in place of the standard interpretation a novel reinterpretation of such discourse which is not Platonistic and which the experts ought to adopt instead. Burgess and Rosen’s objection

is that neither hermeneutic nor revolutionary nominalism is plausible, and therefore both should be rejected in favour of the literal interpretation, which is taken to be ontologically committing to abstract objects.

Anti-realists have typically repudiated Burgess and Rosen’s distinction between hermeneutic nominalism and revolutionary nominalism as concocted in order to set up straw men as targets for criticism. Most anti-realists do not claim to know, in the absence of a sociological study, whether the majority of mathematicians and scientists take their statements to be Platonistic assertions about the existence of mathematical objects, nor would anti-realists be so audacious as to recommend a reform of mathematical practice predicated upon a new interpretation of mathematical

discourse.14 Their claim, rather, is that mathematical discourse can be understood in a non-literal way which is

ontologically non-committing and that such an interpretation is unobjectionable.

Still, Yablo does affirm that his project is hermeneutical in nature and that taking abstract object talk figuratively may well give the most plausible account of our practice.15 Two points may be made about his contention.

(p.176) First, Yablo has presented extensive and, I think, very persuasive grounds for thinking that abstract object talk in general is figurative rather than literal. Burgess and Rosen do not respond to the many intriguing parallels that Yablo draws between figurative discourse and talk of Platonic objects. Instead, they respond to three subsidiary arguments in Yablo’s work, not all of which are really arguments for figuralism.

First, they take Yablo’s claim that figuralism best explains the failure of overly easy existence proofs to be a version of an argument for anti-realism from the lightheartedness with which mathematicians introduce novel mathematical entities. The idea is that mathematicians, unlike natural scientists, are carefree about postulating previously unknown entities because they do not consider themselves to be making a genuine addition to the world’s ontology. Burgess and Rosen claim that the phenomenon of lightheartedness admits of other explanations. For example, mathematicians’ willingness to switch back and forth between affirming ‘There are n Fs’ and ‘The number of Fs

is n’ is because, for whatever reason, ‘mathematicians don’t regard the existence of numbers as problematic’.16 But

this cannot be right. For suppose the reason mathematicians take existential affirmations about numbers to be unproblematic is because they are not realists. Then the question will be what best explains their insouciance about Platonistic discourse, and in light of the parallels Yablo adduces, the answer could well be figuralism. Burgess and Rosen need to show that, for whatever reason, mathematicians do not find the Platonistic existence of numbers to be problematic and are therefore lighthearted about their existence assertions. Ironically, Burgess and Rosen themselves observe that number claims are not generally interpreted by mathematicians as ontologically committing to numbers, but are interpreted structurally, so that numbers are replaced by characterless positions in a mathematical structure. Moreover, Burgess and Rosen fail to take cognizance of the fact that Yablo’s claim about overly easy existence proofs is not limited to mathematics but, as we have seen, concerns a wide and varied range of so-called Platonic objects, some of which are concrete. Even if mathematicians are insouciant about overly easy existence proofs because they are, in fact, closet Platonists, such an explanation cannot (p.177) account for the full range of examples

Yablo adduces. Yablo’s contention that figurative speech is a pervasive feature of natural language is surely correct and removes any presumption of literalness.17

A second argument for figuralism, to which Burgess and Rosen respond, has to do with what number statements are about. Yablo points out that someone who is concerned about the number of starving people in the world is concerned about people, not some mathematical object. Burgess and Rosen respond that intuitions of aboutness are unstable and context-dependent. For example, suppose someone says, ‘The number of cats in the yard is two.’ If we point out to him that his statement implies that the number of cats in the yard is a prime number, he would agree that this latter claim is about a number. Now if we return to his original statement and ask if it was not also about a number, Burgess and Rosen maintain, ‘it seems well within reason that he will now report that his original claim

was in part about the number 2, and not just about the cats’.18 I find it difficult to take such a reply seriously. Not

only is it highly implausible that an ordinary language user would admit that he was originally concerned about or talking about a number, but, more importantly, if intuitions of aboutness claims are shifting and context-dependent, that underscores the fact that one may be talking about people or cats in one context and then talking about a number in another context. Of course, we can talk about mathematical objects, but that goes no distance towards undermining the claim that in many contexts in which we employ number words, we are not talking about numbers but, plausibly, are using figurative speech to communicate a truth about something else.

The third argument for figuralism that Burgess and Rosen consider is the anti-realist claim that, since working mathematicians would be utterly unfazed if the Oracle of Philosophy (or God?) were to reveal (p.178) that mathematical objects do not, in fact, exist, they must not be committed to the literal truth of mathematical theorems postulating the existence of mathematical objects. This third argument is not really an argument for figuralism per

se, but rather is an argument for thinking that mathematicians do not generally take their existence theorems Platonistically. Burgess and Rosen respond to this argument by saying that, were such a revelation to be made, perhaps mathematicians would just refuse to believe it or perhaps they would adjust to the realization that they were wrong, so long as they were assured that mathematical theorizing could go on successfully under the pretence that such objects exist. Again, such a response seems frivolous. The point is that if mathematicians really took themselves to be describing an independently existing, Platonic realm, just as natural scientists are engaged in describing the independent, physical world, then if it were revealed that no such realm actually exists, they would be quite upset. Whether they could be so convinced or whether they could adjust to such a revelation is simply irrelevant. The point is that if mathematicians really thought themselves to be metaphysicians, then of course they would be initially shaken by the revelation that all their theorizing was empty. The fact that working mathematicians would not so react provides some reason to think that they do not take themselves to be doing metaphysics.

In addition to responding to these three arguments for figuralism, Burgess and Rosen also present an objection against figuralism, namely, the lack of direct evidence for it. They complain that we lack the best kind of direct evidence that the practice of mathematics and science involves something less than belief in the existence theorems of mathematics, namely, efforts to disavow such belief, warning students against believing them, and so on. But it seems to me that such insouciance on the part of mathematicians is consistent with an attitude of indifference towards such metaphysical conundrums—‘We’ll leave that question to the philosophers!’—or with the conviction that such theorems can be true without making ontological commitments. Since it just does not matter to mathematical or scientific practice whether mathematical objects exist, practitioners simply have no incentive to be bothered about them—unless, of course, they have taken a class in the philosophy of mathematics, in which case, well-informed practitioners may well be sceptical about either the literal truth of the existence theorems or else the ontological commitments of such truths. We would need a sociological study to tell.

(p.179) Burgess and Rosen protest:

Certainly in all clear cases of figurative language—and it is worth stressing that the boundary between figurative and literal is as fuzzy as can be—the non-literal character of the linguistic performance will be perfectly obvious as soon as the speaker is forced to turn attention to the question of whether the remark was meant literally.

We further submit that mathematical discourse fails this test for non-literalness.19

They offer no evidence in favour of their claim that mathematical discourse fails the test, but never mind. The more important point is that this objection, if sound, at best proves that mathematical discourse is not a clearcase of figurative language, a hardly surprising result. What does not follow is that mathematical discourse does not lie somewhere in that fuzzy area between clearly figurative and clearly literal expressions, just as Yablo claims.20

All this concerns my first point, that Yablo has, in fact, presented extensive and pretty persuasive reasons for his hermeneutic claim that abstract object talk in general is figurative rather than literal. But my second and more important point about figuralism is that, while Yablo espouses it as a hermeneutic thesis, there is no reason the anti- realist has to present it as such. Like most anti-realists, the figuralist can, in the absence of linguistic and sociological studies of the mathematical and scientific community, remain agnostic about hermeneutical questions and present the figurative interpretation as one reasonable way of understanding abstract object talk without commitment to abstract objects. If such a figurative interpretation is, indeed, reasonable, as it seems to be, then the Indispensability Argument has been defeated.

Concluding Remarks

A key plank in the Indispensability Argument for the existence of abstract objects is the claim that sentences quantifying over or (p.180) containing singular terms referring to abstract objects are not merely true, but are literally true. Unlike neo-Meinongians, neutralists, and free logicians, the figuralist is willing to concede that the literal truth of mathematical statements would be ontologically committing to abstract objects. We therein see revealed the figuralist’s tacit agreement with the neo-Quinean criterion of ontological commitment embodied in the first premiss of the Indispensability Argument. But in contradiction to the argument’s second premiss, the figuralist sees no reason to take abstract discourse to be literally true. That does not imply that such statements are false, as the fictionalist claims. Rather, such statements are figuratively true.

Figuralism strikes me as a perfectly reasonable interpretation of abstract discourse. The parallels between figurative speech and Platonistic discourse are striking and many. Figuralism also does a good job of explaining why overly

easy existence proofs fail. So long as one eschews a hermeneutic claim about how most mathematicians understand their discourse and offers a figurative interpretation as one reasonable way of construing such discourse, it seems to me that anti-realist figuralism cannot be gainsaid.

Figuralism therefore presents itself as a different, viable alternative for the classical theist concerned to meet the challenge of Platonism to divine aseity. One may accept the criterion of ontological commitment assumed by realists but deny that the truth of sentences quantifying over or containing singular terms referring to abstract objects commits us to the reality of said objects because such sentences may be understood to be, not literally, but figuratively true.

Notes:

(1) Stephen Yablo, ‘Go Figure: A Path through Fictionalism’, in Peter A. French and Howard K. Wettstein (eds), Figurative Language (Oxford: Blackwell, 2001), 72–102.

(2) Ibid., 85.

1. Stephen Yablo, ‘Does Ontology Rest on a Mistake?’, Proceedings of the Aristotelian Society (Supplement), 72 (1998): 255, 259.

   
   
   

2. Ibid., 229. Cf. Stephen Yablo, ‘A Paradox of Existence’, in Anthony Everett and Thomas Hofweber (eds), Empty Names, Fiction, and the Puzzles of Non-Existence (Stanford, CA: Center for the Study of Language and Information, 2000), 304–5.

   
   
   

3. Yablo, ‘Paradox of Existence’, 293.

   
   
   

4. Stephen Yablo, ‘The Myth of the Seven’, in Mark Eli Kalderon (ed.), Fictionalism in Metaphysics (Oxford: Clarendon Press, 2005), 98.

   
   
   

5. Yablo, ‘Paradox of Existence’, 295.

   
   
   

6. It should be noted that Yablo has an idiosyncratic understanding of what a Platonic object is. Normally, a Platonic object is taken to be an abstract object. But Yablo defines an object as Platonic relative to an area of discourse, just in case the discourse depends on how that object behaves, even though the discourse is not really about that object. For example, someone who expresses concern about the number of starving people in the world is concerned about people, not some abstract mathematical object. Notice that even though Yablo does not identify Platonic objects as abstract objects, nevertheless, most of the objects which he identifies as Platonic are plausibly construed to be abstract objects. In fact, Yablo himself says, ‘the existence of abstract objects is straightforwardly deducible from premises that few would think to deny’ (Yablo, ‘Paradox of Existence’, 276). Since our interest is in the existence of abstract objects, I shall take Yablo’s Platonic objects to be abstract.

   
   
   

7. Yablo, ‘Paradox of Existence’, 302–4; cf. Yablo, ‘Go Figure’, 89–90; Stephen Yablo, ‘Abstract Objects: A Case Study’, in Ernest Sosa and Enrique Villanueva (eds), Realism and Relativism (Boston: Blackwell, 2002), 227–30.

   
   
   

8. Yablo, ‘Paradox of Existence’, 277.

   
   
   

9. Yablo, ‘Go Figure’, 88; Yablo, ‘Paradox of Existence’, 286–90.

   
   
   

10. Yablo, ‘Myth of the Seven’, 99; Yablo, ‘Abstract Objects’, 230–2. For example, the real content of arithmetical truths like 2

    + 3 = 5 is the first-order logical truth (∃2xFx & ∃3yGy & ¬∃z (Fz & Gz)) → ∃5u (Fu v Gu). This states that, if there are two things that are F and three things that are G, and there is nothing that is both F and G, then there are five things that are either F or G.

    
    
    

11. See John P. Burgess, ‘Why I Am Not a Nominalist’, Notre Dame Journal of Formal Logic, 24/1 (1983): 93–105; John P. Burgess and Gideon A. Rosen, A Subject with No Object (Oxford: Clarendon, 1997), 6–7, 205–37; John P. Burgess, ‘Mathematics and Bleak House’, Philosophia Mathematica, 12/1 (2004): 23–30; Gideon Rosen and John P. Burgess, ‘Nominalism Reconsidered’, in Stewart Shapiro (ed.), The Oxford Handbook of Philosophy of Mathematics and Logic (Oxford: OUP, 2005), 515–28.

    
    
    

12. See, e.g. Charles S. Chihara, ‘Nominalism’, in Stewart Shapiro (ed.), The Oxford Handbook of Philosophy of Mathematics and Logic (Oxford: OUP, 2005), 506; Geoffrey Hellman, ‘On Nominalism’, Philosophy and Phenomenological Research, 62/3 (2001): 703.

    
    
    

13. Yablo, ‘Go Figure’, 85, 87.

14. Rosen and Burgess, ‘Nominalism Reconsidered’, 531.

    
    
    

15. The literary figure C. S. Lewis similarly claims that the greater part of our language is metaphorical rather than literal. Lewis was an anti-realist about mathematical discourse, taking it to be metaphorical and its objects unreal. Lewis thinks that mathematicians themselves realize that their discourse is not literal, but metaphorical. Lewis was apparently also an anti-realist about other abstract objects, such as universals. Indeed, it is likely that he took the whole Platonic host to be creatures of metaphor, for he writes, ‘open your Plato, and you will find yourself among the great creators of metaphor, and therefore among the masters of meaning’. C. S. Lewis, ‘Bluspels and Flalansferes: A Semantic Nightmare’, in Walter Hooper (ed.), Selected Literary Essays (Cambridge: CUP, 1979), 265; see all of pp. 260–5. Lewis is thus an example of a theistic figuralist.

    
    
    

16. Rosen and Burgess, ‘Nominalism Reconsidered’, 532.

(19) Ibid., 533.

(20) Burgess and Rosen themselves explain that on Yablo’s view, ‘an existence theorem is ambiguous between a literal and a figurative sense’ (ibid., 528). I am not sure how seriously they take this ambiguity.