8

8

Useful Fictions

Fictionalism accepts the customary criterion of ontological commitment that underlies the Indispensability Argument but denies that sentences involving quantification over or singular terms referring to abstract objects are true. Field and Balaguer’s versions of fictionalism are exposited. The three principal objections to fictionalism raised in the literature—namely, the obvious truth of elementary arithmetic, the indispensability/utility of mathematics, and the self-referential incoherence of fictionalism—are critically examined. Insofar as these objections cast doubt on the fictionalist’s claim that mathematical statements are untrue, they cast doubt, not on anti-realism, but on the criterion of ontological commitment assumed by the fictionalist, which would force on us Platonistic ontological commitments as a result of seemingly innocuous truths. Nevertheless, granted the neo-Quinean criterion, fictionalist claims can be formulated in such a way as to seem not only coherent but even plausible. Fictionalism is found to be a viable option for theists who accept the traditional criterion of ontological commitment.

In the previous two chapters, we saw that the criterion of ontological commitment that lies at the heart of the Indispensability Argument for Platonism has been plausibly challenged, in part or in whole, by neo-Meinongians, neutralists, and free logicians, so that it can hardly be said to be incumbent upon the classical theist. Premiss (I) of that argument may therefore be plausibly rejected. This fact alone suffices to defeat the Indispensability Argument for Platonism.

But what about premiss (II) of that argument?

If there are plausible ways of regarding such sentences as not literally true, then the Indispensability Argument will be doubly defeated, so to speak. In the next three chapters, we shall examine three anti-realisms which challenge premiss (II), namely, fictionalism, figuralism, and pretence theory (Figure 1). Fictionalism holds that abstract object talk is not true. Figuralism holds that abstract object talk is figurative and therefore not literally true. Pretence theory holds that abstract object talk is a matter of make-believe and therefore need not be taken to be true.

In this chapter, we want to explore the boldest of these three views, fictionalism. Fictionalists flatly deny that mathematical sentences and other abstract sentences are true, period. Statements involving quantification over or reference to abstract objects are false (or at least untrue). Abstract objects are merely useful fictions; that is to say, even though no such objects exist, it is useful to talk as though they did. Hence, the name fictionalism.

(p.145) Contemporary Fictionalism

Although contemporary fictionalism has historical precursors, particularly Jeremy Bentham’s ‘theory of fictions’ and Hans Vaihinger’s ‘philosophy of “as if”’,1 the fount of contemporary fictionalism is Hartry Field’s Science without Numbers (1980). Field questions why we should regard standard mathematics as a body of truths. The fact that the theorems of mathematics are logically derived from a consistent body of axioms does not guarantee their

truth, for we may still ask, ‘why regard the axioms as truths, rather than as fictions that for a variety of reasons mathematicians have become interested in?’2 Since the truth of mathematical theories, taken at face value, would commit us to the existence of a variety of abstract objects, ‘an anti-platonist’, Field advises, ‘should embrace fictionalism about mathematics—or at least fictionalism about mathematics-taken-at-face-value’.3

Field’s brand of fictionalism is thus compatible with views like figuralism, which treats mathematical statements as figurative discourse. The figuralist agrees that taken literally, or at face value, mathematical statements are not true, but he denies that they should be taken literally. A good many fictionalists, however, would say that mathematical

statements should be understood as literal assertions, not as figurative discourse. Mark Balaguer thinks that fictionalism enjoys an advantage over other anti-realist views precisely because it shares with Platonism a common and customary semantics for abstract discourse.4 Fictionalists agree with Platonists that sentences involving either quantification over or singular terms referring to (p.146) abstract objects, if true, are ontologically committing; but they differ from Platonists in thinking such sentences to be untrue.

What would be the consequences for physical science of taking mathematical sentences to be untrue? Field thinks that the falsity of mathematical sentences does not undermine natural science because mathematics is ultimately dispensable for science. Balaguer thinks that the falsity of mathematical sentences does not undermine natural science because what he calls the nominalistic content of scientific theories is independent of and, hence, unaffected by the falsity of their Platonistic, mathematical content. Balaguer is willing to concede Quinean claims about the indispensability of mathematics to physical science, but he maintains that while the nominalistic content of empirical science is (mostly) true, its Platonistic content is not. Because abstract objects are causally unconnected to the physical world, there must be a nominalistic content of scientific theories, which, even if inexpressible by us due to mathematics’ indispensability, is made true by the physical world wholly independently of whether abstract objects exist. Thus, says Balaguer, while Field’s claim that mathematics is scientifically dispensable is ‘highly controversial’, the claim that empirical science has a nominalistic content that captures its complete picture of the physical world ‘is no more controversial than the claim that abstract objects (if there are such things) are causally

inert’.5

In support of the plausibility of nominalistic scientific realism, Balaguer defends the thesis

(TA) Empirical theories use mathematical object talk only in order to construct descriptive frameworksin which to make assertions about the physical world.

Balaguer takes it to be obvious that mathematics does function in the way described by (TA) in physical theories. Consider, for example, the use of mathematics in quantum mechanics (QM):

it seems entirely obvious that (TA) applies to QM, that is, the reason we refer in QM to things like vectors and subspaces and real numbers is that this provides us with a convenient way of describing quantum phenomena. (Indeed, what else could we say here? We certainly (p.147) wouldn’t want to claim that we refer to these objects in QM because we simply want to state facts about them, or because we think they are partly responsible for the operation or state of the quantum level of the physical world. It just seems obvious that the reason we refer to these objects is that this provides us with an easy way of saying what we want to say about quantum phenomena.)6 Whether or not mathematics is dispensable to physical theory, (TA) accurately describes, in Balaguer’s view, the function of mathematics in scientific theories. But such a role is perfectly compatible with fictionalism because descriptive aids can be useful without being genuinely referential. We could use mathematical object talk to depict accurately the physical world even if no mathematical objects existed. In short, ‘the reason nominalistic scientific realism is a sensible philosophy of science is that the nominalistic content of empirical science is all empirical

science is really “trying to say” about the world. Its platonistic content is something it “says incidentally” in its effort to say what it really “wants to say”.’7

So fictionalists may regard Platonistic mathematics as either dispensable (Field) or indispensable (Balaguer) to physical science. What unites them is their conviction that mathematical sentences, whether pure or applied, are not true.

The Obvious Truth of Elementary Mathematics

The most evident objection to fictionalism is that some mathematical statements, like ‘2 + 2 = 4,’ are just obviously true. Indeed, they seem to be necessarily true. Therefore, fictionalism is simply ruled out.

Field tries to soften fictionalism’s blow by holding that ‘2 + 2 = 4’ is true according to standard mathematics:

A fictionalist needn’t (and shouldn’t) deny that there is some sense in which ‘2 + 2 = 4’ is true; but granting that it is true in some sense does not commit one to finding any interesting translation procedure that takes acceptable mathematical claims into true claims that don’t postulate mathematical entities. Rather, the fictionalist can say that the sense (p.148) in which ‘2 + 2 = 4’ is true is pretty much the same as the sense in which ‘Oliver Twist lived in

London’ is true: the latter is true only in the sense that it is true according to a certain well-known story, and the former is true only in the sense that it is true according to standard mathematics.8

By appealing to truth according to standard mathematics, Field can differentiate between commonly accepted mathematical falsehoods like ‘2 + 2 = 4’ and outrageous mathematical falsehoods like ‘2 + 2 = 5.’ The latter are obviously not true according to standard mathematics.

But this response fails, I think, to alleviate our misgivings. For the point is surely that the standard model of arithmetic is correct. It is inconceivable, given the meaning of the symbols 0, ′, and + in the logical language of arithmetic, that 0″ + 0″ ≠ 0″″.9 The elementary truths of arithmetic impose themselves upon us, along with the axioms from which these truths are derived.

Field reacts to the claim that a mathematical assertion like ‘2 + 2 = 4’ must be true simply as a consequence of the meaning of its terms by saying that this claim cannot be right because analytic truths cannot have existential implications. He grants that

the claim ‘If there are numbers then 2 + 2 = 4’ has some claims to count as an analytic truth, indeed one so obvious that its denial is unintelligible.… But…it can’t be an analytic or purely conceptual truth that there are objects 1, 2, 3, 4 etc. obeying such laws as that 2 + 2 = 4. An investigation of conceptual linkages can reveal conditions that things must satisfy if they are to fall under our concepts; but it can’t yield that there are things that satisfy those concepts.10

Field rightly contends that we cannot infer the existence of things from merely definitional truths. But it is only the unquestioned presupposition of the neo-Quinean criterion of ontological commitment that leads the fictionalist to think that a statement like ‘2 + 2 = 4’ has existential implications. Deny that criterion and one is not forced into the awkward position of denying that 2 + 2 = 4.

Mary Leng, in response to the obviousness of elementary mathematics, boldly denies that any arithmetical propositions are obviously (p.149) true. For, she points out, if it is obvious that there is, for example, an even prime number, then it is likewise obvious that there are numbers. But the debate over the reality of numbers has been raging for over two thousand years, so ‘surely the mere existence of a debate over this matter speaks against the

claim that the truth of the arithmetic propositions in question is genuinely obvious?’11

Leng’s response assumes that the neo-Quinean criterion of ontological commitment has the same obviousness as the truths of elementary arithmetic, which is absurd. People have not been debating for two thousand years whether there is an even prime number or whether 2 + 2 = 4, but rather the ontological implications of such truths. It has only been within the past century that a criterion of ontological commitment has evolved that takes such truths to be ontologically committing. The millennia-long debate, especially the contemporary debate, over the reality of numbers ought to make us dubious, I think, not of the obviousness of arithmetic, but of the obviousness of the assumed criterion of ontological commitment.

Leng herself confesses that ‘2 + 2 = 4’ is ‘difficult to doubt’, even for fictionalists.12 Remarkably, however, rather than reject the neo-Quinean criterion of ontological commitment, she tries instead to explain away the apparent obviousness of ‘2 + 2 = 4’! She suggests, ‘Perhaps what is obvious to us is just that certain propositions, including the proposition that 2 + 2 = 4, follow from the assumption that there are numbers satisfying the Dedekind-Peano axioms.’13 Such a suggestion seems to me outrageous, since these axioms were formulated only in the late nineteenth century, and yet people have believed since the most primitive times that 2 + 2 = 4. Even today, most people would not have a clue how to derive the truths of elementary arithmetic from the Peano Axioms.

Leng acknowledges that, long before people knew much number theory, it seemed obvious that 2 + 2 = 4, but she entertains sympathetically the claim that the apparent obviousness of basic arithmetic is due to our childhood conditioning, which ‘might create an illusion of obviousness’.14 I should have thought, to the contrary, that the reason we teach elementary arithmetic to our children in the first place is because we find it, upon reflection, to be

evidently true. Leng agrees that we could not have found different number-theoretic (p.150) statements like ‘2 + 2 = 5’ to be obviously true. But she insists that the salient point is that

Our early training in basic arithmetic conditions us to accept claims about numbers as true without regard to the ontological commitments they bring with them, so that by the time we come to consider matters of ontology, we find it hard to take seriously the possibility that there may not be any numbers.15

This claim strikes me as both false and irrelevant. The first clause seems nearly right, though it would have been better to speak of ‘alleged ontological commitments’. Such ontological concerns are doubtless absent from primary school instruction in arithmetic. But the last clause following the word ‘ontology’ should have been ‘we are astonished to learn that some philosophers think that the truths of basic arithmetic commit us to the existence of numbers as mind-independent objects’. It is the disclosure that our childhood belief that 2 + 2 = 4 commits us to the reality of 4 that is hard to take seriously at first. In any case, even if, as Leng thinks, we were conditioned against anti-realism from childhood, this is irrelevant to the obviousness of arithmetic. Statements of elementary arithmetic are still obviously true. So why not call into question the criterion of ontological commitment which has been surreptitiously drilled into us?

Finally, Leng suggests more plausibly that the reason 2 + 2 = 4 seems obvious is due to our experience of counting objects:

When we say that it is obvious that 2 + 2 = 4, it is plausible that we sometimes mean, not that it is obvious

that number theory implies that 2 + 2 = 4, but rather, that it’s obvious that if I correctly count exactly two objects of one sort…and exactly two objects of another sort…, then taking these together I will be able to count exactly four objects that are either of the first sort or the second sort.16

‘But,’ she adds, ‘this just makes adjectival use of the natural numbers, and such uses can be formalized without quantification over natural numbers.’17 Let us suppose that Leng is right, that our experience of counting goes to explain the obviousness of elementary arithmetic and that adjectival use of numbers suffices to capture such experience. Upon being told that nominal use of the number 2 is ontologically committing in a way that adjectival use is not, should we react by (p.151) doubting the truth of elementary arithmetic vouchsafed to us by counting? Should we not rather doubt the criterion of ontological commitment which requires that my belief that ‘Ten is the number of my fingers’ commits me to a mind-independent, abstract object, while ‘I have ten fingers’ does not? To doubt instead the truth of elementary arithmetic would be to display a misplaced confidence in the criterion of ontological commitment. Any criterion of ontological commitment will always be less obvious than the truths of elementary arithmetic.

In short, one’s attitude towards the objection from the obviousness of elementary arithmetic is probably going to depend on one’s attitude towards the neo-Quinean criterion of ontological commitment. If with the fictionalist we are convinced that quantification and singular terms are devices of ontological commitment, then we shall find upon reflection that the sentences of elementary mathematics, if taken literally, are anything but obvious. For we shall come to see that statements which we have unhesitatingly accepted as true since childhood are, in fact, radical ontological assertions about the existence of mind-independent abstract objects. As such, they are not at all obviously true. We come to realize that we have, in fact, misunderstood them all these years; we literally did not understand what we were asserting.

On the other hand, if we find sentences of elementary arithmetic to be obvious because we do not take them to be ontologically committing, then we shall be led to reject the neo-Quinean criterion of ontological commitment which would saddle us with such commitments. This seems to me the obvious course to take. After all, the sentences of elementary mathematics are much more obviously true than any criterion of ontological commitment, and so should be more tenaciously held and less quickly surrendered than the neo-Quinean criterion of ontological commitment.

Platonists and fictionalists might try to force us to choose against the obvious truth of mathematical statements by simply stipulating that they are using ‘there is/are’ in a metaphysically heavyweight sense, so that in that sense the anti-realist is forced to say things like ‘There is no number greater than 3’ or ‘There are no prime numbers less than 100’, which sounds absurd.18 Neutralism is thus said to collapse into (p.152)fictionalism once we make clear that we are talking about existence in a metaphysically heavyweight sense.19

Neutralist Jody Azzouni takes a very dim view of this sort of move on the part of the neo-Quinean, characterizing it as a kind of philosophical chicanery:

‘There are fictional mice that talk,’ one says, during a discussion about talking fictional animals. ‘Oh, so you believe fictional mice really exist?’ the philosophical trickster responds. ‘I didn’t say that,’ one responds. ‘Yes, you sure did—you said “there are fictional mice that talk”. What do you think “there is” means?’…But what’s happened is that the philosophical trickster has (implicitly) switched on usage—now one is speaking so that ‘there is’ (at least

for the time being) does convey onticity; and the ordinary person isn’t sophisticated enough to see how he or she has been duped.20

In all fairness, the ‘trickster’ seems to be doing just what Azzouni advises: using tone of voice and qualifying words like ‘really’ to make it clear that he is talking about existence in a metaphysically heavyweight sense. The trick here would be his pretending that he is sticking with ordinary usage of quantificational expressions, with the intention of embarrassing the anti-realist. The trickster puts the ordinary person in an awkward situation only by changing the way in which informal quantifiers are used.

Neo-Quinean Theodore Sider is more candid: ‘Ontological realism should not claim that ordinary quantifiers carve [reality] at the joints, or that disputes using ordinary quantifiers are substantive. All that’s important is that one can introduce a fundamental quantifier, which can then be used to pose substantive ontological questions.’21 We just stipulate explicitly that we are using ‘there is/are’ in a metaphysically heavyweight sense. When we do so, says Sider, we are no longer speaking ordinary English, but ‘a new language—“Ontologese”—whose quantifiers

are stipulated to carve [reality] at the joints’.22 If (p.153) one stipulates that ‘there is/are’ is being used to indicate ontological commitments, then, of course, the anti-realist will regard a statement like ‘There are odd numbers’ as

false. But anti-realism has not thereby been thrust into an embarrassing position, for in a metaphysically heavyweight sense it is anything but obvious that there are numbers.

What this discussion reveals is that the correct logic for Ontologese, or metaphysically heavyweight discourse, should be a positive free logic. For we have already argued for a neutral theory of reference in Chapter 7. Since free logic takes quantification to be ontologically committing but does not take singular terms to be devices of ontological commitment, we may deny the truth of ‘There are numbers,’ while—the fictionalist notwithstanding— affirming the truth of ‘2 + 2 = 4.’ There do not need to be objects which are the denotations of the singular terms ‘2

+ 2’ and ‘4’ in order for ‘2 + 2 = 4’ to be true. So the anti-realist need not and should not be a fictionalist about the statements of arithmetic, even if he is a fictionalist when it comes to statements quantifying over mathematical objects.

Indispensability/Utility of Mathematics

If, as the fictionalist claims, sentences quantifying over or involving reference to mathematical objects are not true, then how can they be so useful and, indeed, indispensable to modern science in describing the physical world? The issue that arises here is why mathematical statements in scientific theories cannot be taken merely instrumentally, just as many other statements in scientific theories are. The philosopher of mathematics Penelope Maddy provides some engaging illustrations:

If we open any physics text with these questions in mind, the first thing we notice is that many of the applications of mathematics occur in the company of assumptions that we know to be literally false. For example, we treat a section of the earth’s surface as flat, rather than curved, when we compute trajectories; we assume the ocean to be infinitely deep when we analyze the waves on its surface; we use continuous functions to (p.154) represent quantities like energy, charge, and angular momentum, which we know to be quantized; we take liquids to be continuous substances in fluid dynamics, despite atomic theory. On the face of it, an indispensability argument based on such an application of mathematics and science would be laughable: should we believe in the infinite because it plays an

indispensable role in our best scientific account of water waves?23

Why should we not take mathematical sentences to be false though useful?

Azzouni acknowledges that scientists often do take an instrumentalist attitude towards their theories, and so he inquires how we can tell when a theory is being treated instrumentally. How does the difference in attitude towards sentences presumed to be true and those which are taken merely instrumentally manifest itself? ‘The answer is that if a theory (or sentence) is one we think true, then we must also include in our body of beliefs (at least

implicitly) all of its implications…if we really do take the theory only as an instrumental device…, then we ignore or quarantine the other implications of the theory from our communal body of beliefs.’24 In Azzouni’s view, this manifestation condition is both a necessary and sufficient condition of a theory’s being construed instrumentally.

In response to Maddy, Azzouni insists, ‘implications of applied mathematical doctrine are never placed in quarantine in the way that (instrumentally construed) implications of empirical doctrine are’.25 He concludes:

the Quinean claim, that an indispensable empirical theory must be regarded as true, has been qualified, because such isn’t the case in ordinary scientific practice if certain implications of such theories are placed in quarantine.…

Nevertheless,…this qualification offers little solace to instrumentally inclined philosophers—those, anyway, who urge us to take applied mathematical doctrine or fundamental scientific laws to be false—because in these cases scientists do not engage in the practice of placing in quarantine (some of) the implications of such theories, and

so (p.155) philosophers have no grounds to regard such theories as false, or even grounds to be agnostic regarding the truth of such theories.26

Now all sorts of questions might be raised concerning the justification for Azzouni’s manifestation condition. But let that pass. There is a more glaring lacuna in the argument, or at least so it seems. Azzouni claims that scientists never place any of the implications of the mathematical doctrine applied in their theories in quarantine. But the fictionalist will contend that such a claim is patently mistaken.27 They will say that scientists most certainly do place into quarantine implications of the mathematical statements used in or presupposed by scientific theories. For example, given the neo-Quinean criterion of ontological commitment, it is an implication of ‘2 + 3 = 5’ that there is a mind- independent object designated by the singular terms ‘2 + 3’ and ‘5’. This is, to be sure, not

a mathematical implication, that is to say, a mathematical theorem; rather, it is, as Maddy says,

an ontological implication. It is with such an ontological conclusion that Maddy is concerned when she protests against believing in the infinite because of the indispensable role it plays in our best theory of ocean waves.

Mathematics can play its indispensable role in scientific theories, even as its ontological implications are ignored, so long as all the mathematical implications hold.

Quarantining, according to Azzouni, involves simply ignoring some of the implications of a theory, that is to say, not including those implications in our (communal) body of beliefs. I think it highly improbable that Platonism is included in the body of beliefs of the scientific community! Even in the absence of a sociological survey, it seems safe to say that most scientists simply ignore any alleged ontological commitments following from the mathematical statements used in their theories.

Even the Platonists John Burgess and Gideon Rosen acknowledge evidence in favour of instrumentalism concerning mathematical objects, namely, the lightheartedness with which novel mathematical entities are introduced,

the indifference when it comes to questions of (p.156) identifying mathematical objects such as the number 2 with other objects such as {{Ø}} or {0, 1}, and the varying reactions of practitioners when pressed by philosophers about the ontological commitments of their theorems.28 Although Burgess and Rosen try to diminish the force of this

evidence, these considerations do, I think, supply strong grounds for suspecting that mathematicians and scientists are not concerned about the ontological implications of mathematical sentences.

If this is correct, the manifestation condition has been met, and the fictionalist is entirely within his rights in construing the indispensable mathematical content of scientific theories instrumentally. In that case, the indispensability of mathematics to empirical science fails to defeat the fictionalist claim that such mathematical statements are false. As Field puts it, mathematics need not be true to be good.

We are thus brought back to the same choice we confronted when dealing with the objection to fictionalism from the obvious truth of elementary arithmetic. It does seem bizarre to deny, as the fictionalist does, that the sentence ‘The number of Mars’ moons is two’ is true. But the only reason the fictionalist denies the truth of this statement is because he takes it to be equivalent to the claim that ‘A mind-independent, abstract object exists which numbers the moons of Mars as two.’ Other anti-Platonists join the fictionalist in rejecting such a claim. The question anti- Platonists face is whether it is more plausible to deny the truth of the neo-Quinean criterion of ontological commitment which gives rise to so extraordinary a claim or to deny the truth of the common-sense assertion about the number of Mars’ moons. It seems to me evident that given the prima facie truth of the common-sense assertion, there is properly a presumption of its truth which can be overturned only by the demonstration that the neo-Quinean criterion of ontological commitment is even more obviously true than the common-sense assertion. If this is right, then fictionalism should be the last resort of the anti-realist, to be entertained only after all attempts to affirm the truth of statements of applied mathematics without commitment to abstract objects have proved themselves futile.

(p.157) It seems to me, therefore, that while the indispensability/utility of mathematics to science might justifiably lead us to call into question the neo-Quinean devices of ontological commitment, the determined fictionalist, like the instrumentalist, can consistently take such mathematical sentences to be literally false. Such an instrumentalist

attitude is manifested by placing some of the implications of the mathematical statements used in science in quarantine.

Is Fictionalism Self-Defeating?

The fictionalist claim that abstract objects such as numbers are merely useful fictions occasions a deeper question: do useful fictions exist? Surely not, for a useful fiction would be an abstract object, and fictionalism denies that there are abstract objects. But then it is strictly false to say that numbers and the like are useful fictions, which seems to make fictionalism self-defeating.

Now at first blush, this problem might seem easy to solve. Strictly speaking, according to fictionalism, there are no fictions. But when fictionalists say that something is a useful fiction, what they mean is merely that the singular terms allegedly referring to such an entity are vacuous but may nevertheless be usefully employed in our scientific theorizing. Thus, we are not committed to the reality of useful fictions.

The problem, however, is not so easily dismissed. For fictionalists want to distinguish between useful falsehoods like ‘2 + 2 = 4’ and outrageous falsehoods like ‘2 + 2 = 5’. They do so, as we have seen, by saying that the former is true according to the standard model or story of arithmetic, while the latter is not. But then the fictionalist seems to have committed himself to the reality of models or stories, which are abstract objects. So the fictionalist has to say that his claim that ‘“2 + 2 = 4” is true in our standard story of arithmetic’ is, strictly speaking, false, which seems to make fictionalism self-defeating. The difficulty is that fictionalism itself, wedded as it is to the neo-Quinean criterion of ontological commitment, seems to be committed to the reality of abstract objects.

At this point, the sensible anti-realist might conclude that such an impasse gives good grounds for jettisoning the criterion of ontological commitment assumed by the fictionalist, which results in his being (p.158)committed to abstract entities like models or stories. But in order to give fictionalism a full run for its money, let us ask what the fictionalist might say in response to the objection.

Paradox and Linguistic Frameworks

The most daring response would be to say that, due to the indispensability of abstract terminology, paradox is inevitable. We inevitably have to recur to abstract terminology in order to express our views, so that we find ourselves trapped by our language into making paradoxical assertions.

Such a response does not strike me as absurd. This fictionalist rejoinder can be made to appear less radical by appealing to Carnap’s distinction between internal and external questions relative to a linguistic framework.29Carnap, it will be remembered, drew a fundamental distinction, which he took to be of paramount

importance, between what he called ‘internal questions’, that is to say, questions posed within a linguistic framework, and ‘external questions’, that is, questions posed from a vantage point outside that framework. Despite the widespread rejection of Carnap’s verificationism, many philosophers find his distinction between external and internal questions relative to a linguistic framework to be intuitive and helpful. Øystein Linnebo puts his finger on Carnap’s fundamental insight when he writes:

In fact, many nominalists endorse truth-value realism, at least about more basic branches of mathematics, such as arithmetic. Nominalists of this type are committed to the slightly odd-sounding view that, although the ordinary mathematical statement

(1) There are prime numbers between 10 and 20.

is true, there are in fact no mathematical objects and thus in particular no numbers. But there is no contradiction here. We must distinguish between the language LM in which mathematicians make their claims and the language LP in which nominalists and other philosophers make theirs. The statement (1) is made in LM. But the

nominalist's assertion that (1) is true but that there are no abstract objects is made in LP. The nominalist's assertion is thus perfectly coherent.30

(p.159) Statements made in the mathematical language LM correspond to Carnap’s internal questions, and statements made in the philosophical language LP correspond to external questions. External questions are meaningful and have objective answers, but those answers may be quite different than the answers to similar questions posed internally. If the claim ‘There are prime numbers between 10 and 20’ is expressed in LM, then the fictionalist will accept the claim as stated in LM while denying, in LP, that there are numbers.

By distinguishing between internal and external questions concerning abstract objects, the fictionalist could circumscribe the neo-Quinean criterion of ontological commitment. We are at best ontologically committed by quantification and singular terms in LP, the language in which external questions are posed; but use of quantification and singular terms in LM, the language in which internal questions are posed, is not ontologically committing.

Certain statements made internally are to be regarded as fictionally true, even though these statements, when considered externally, are false. Thus, ‘2 + 2 = 4’ is fictionally true, whereas ‘2 + 2 = 5’ is fictionally false, though neither of these statements is true outside the framework.

The pickle in which the fictionalist may find himself is that, for whatever reason, he is able to express certain external claims only by using the language of the framework LM. For example, talk of linguistic frameworks is itself in the language of LM. The fictionalist may explain that he is unfortunately forced by the exigencies of language to express some of his external claims using the language of the framework LM. His fundamental view that no abstract objects exist and therefore no sentence involving putative quantification over or singular terms referring to such objects is true can be expressed externally and coherently in LP; but his views on the difference between statements like ‘2 + 2 = 4’ and ‘2 + 2 = 5’ require him to speak in the language of LM,thereby generating the appearance of self- referential incoherence.

(p.160) It is not clear to me that such a position is untenable; indeed, it seems plausible that we could find ourselves in such an awkward situation. Accordingly, this response provides the fictionalist a not unreasonable way out of self-defeat.

Counterfactual Paraphrases

A second, and different, response to the spectre of self-defeat is to abandon the self-referentially incoherent claims in favour of paraphrases that are consistent with anti-realism about abstract objects. Recall that the present difficulty is the result of the fictionalist’s attempt to distinguish between outrageous falsehoods like ‘2 + 2 = 5’ and falsehoods vital to our getting along in the world like ‘2 + 2 = 4’. The fictionalist says that one is true in our standard story of mathematics, while the other is not, which, given his criterion of ontological commitment, leaves him with abstract objects like stories on his hands.

One way the fictionalist could avoid ontological commitment to things like stories is by employing counterfactual claims to explicate fictional truth. Counterfactual statements are conditionals in the subjunctive mood, such as ‘If there were unicorns, narwhals would not be the only single-horned mammals’ or ‘If a rigid rod were placed in uniform motion through the aether, it would contract in the direction of motion’. Since the neo-Quinean criterion of

ontological commitment applies only to extensional contexts,31 it does not apply when quantification and singular

terms are employed in an intensional context like a counterfactual statement. So, for example, the fictionalist could coherently claim that what is right about ‘2 + 2 = 4’, in contrast to ‘2 + 2 = 5’, is that, if there were numbers, then ‘2

+ 2’ and ‘4’ would refer to the same number. The correctness of our story of mathematics lies in the fact that, if there were numbers, that is what they would be like. The fictionalist can employ counterfactuals in order to meet the challenge of distinguishing outrageous falsehoods from falsehoods which are commonly accepted as mathematical truths: in the one case, one says, ‘If there were numbers, then it would be the case that…’, and in the other case, ‘If there were (p.161) numbers, it would not be the case that…’ We do not need abstract entities like stories, fictions, or whatever, to express the difference between sentences like ‘2 + 2 = 4’ and ‘2 + 2 = 5’.

So in response to the charge of self-defeat, the fictionalist may explain that, in affirming that certain sentences are true in the standard story of mathematics, he means to affirm that, if there were numbers, then things would be as the sentences say. Certain sentences in our best scientific theories, namely, sentences involving quantification and reference only to physical realities, may be taken as true, while sentences involving indispensable quantification over or reference to mathematical objects are to be regarded as false but fictionally true, that is to say, they would be true if the objects in question existed.

Deflationism and Mathematical Utility

Hartry Field’s deflationary view of truth suggests yet a third way of escape for the fictionalist from self-defeat. To begin with, Field maintains that what distinguishes commonly accepted mathematical falsehoods from outrageous falsehoods is simply the utility of the former. He explains:

What makes the mathematical theories we accept better than…alternatives to them is not that they are true and the modifications not true, but rather that they are more useful: they are more of an aid to us in drawing consequences from those nominalistic theories that we are interested in. If the world were different, we would be interested in different nominalistic theories, and in that case some of the alternatives to some of our favorite mathematical theories might be of more use than the theories we now accept.32

Unfortunately, by appealing to the utility of our mathematical theories to distinguish commonly accepted falsehoods from unhelpful falsehoods, Field appears to be ontologically committed to the reality of theories, which are abstract objects. Gideon Rosen asks pointedly:

(p.162) When Field says that set theory is false but useful,…what precisely does he mean by ‘set theory’? A scattered mass of ink and chalk? A theory shaped region of spacetime? The general worry is that on a wide range of restrictive ontological views, theories turn out to be among the entities the theorist professes not to believe in. And whenever this is the case, the fictionalist way out is simply not available.33

Given that a theory, if it existed, would be an abstract object, the fictionalist must say that it is false that accepted mathematical theories are more useful than alternatives, which seems to be self-defeating.

Field’s deflationary view of truth, however, enables him to evade this objection. On a deflationary view, the truth predicate ‘is true’ is merely a device of semantic ascent, whereby we talk about a claim rather than simply make the claim itself. For example, rather than asserting, ‘Hitler was an evil man’, we may ascend semantically and say, ‘It is true that Hitler was an evil man.’ The first assertion is about Hitler; the second assertion is about the first assertion, namely, that it is true.

Similarly, instead of saying that a mathematical theory is useful, the fictionalist may make use of a deflationary schema like

(CONC) The concrete world is so structured that it is useful to assume that

where the ‘ ’ is to be filled by the axioms of the mathematical theory in question. Since the completed sentence does not mention or refer to the theoretical axioms, but is merely completed by inserting the relevant words into the blank, such a sentence would make no ontologically committing reference to theories. The fictionalist will assert such a sentence when the blank is filled by the axioms of a standard mathematical theory like Peano arithmetic, but he will not make such an assertion when the axioms of some non-standard theory fill the blank.

Such a deflationary approach to theories enables the fictionalist to avoid self-referential incoherence in the statement of his position. Contrary to Rosen, the fictionalist will not assert that set theory—or any other mathematical theory—is useful though false. On a deflationary view, one will not ascend semantically to talk of the truth

or (p.163) utility or consistency of scientific or mathematical theories. One will simply affirm of the concrete world that it is so structured that it is useful to assume that [stick in the axioms here].

Of course, that raises the question as to why the physical world is so structured as to make mathematical sentences useful in describing the world—what physicist Eugene Wigner famously referred to as ‘the unreasonable effectiveness of mathematics’.34 This is the question, not of mathematics’ indispensability, but of its applicability.

Neither the Platonist nor the fictionalist as such has much to contribute towards the explanation of mathematics’ applicability.

As Leng points out, for the Platonist, the fact that physical reality behaves in line with the dictates of acausal mathematical entities existing beyond space and time is ‘a happy coincidence’.35 This is simply to reiterate that abstract objects are causally inert. The idea that Platonism somehow accounts for the applicability of mathematics ‘is actually very counterintuitive’, muses Balaguer. ‘The idea here is that in order to believe that the physical world

has the nature that empirical science assigns to it, I have to believe that there are causally inert mathematical objects, existing outside of spacetime’, an idea which is inherently implausible.36

But the fictionalist is no better off. Philosopher of science Tim Maudlin muses, ‘The deep question of why a given mathematical object should be an effective tool for representing physical structure admits of at least one clear answer: because the physical world literally has the mathematical structure; the physical world is, in a certain sense, a mathematical object.’37 Leng says that there is no happy coincidence on anti-realism, since mathematical relations just mirror the relations among things in the physical world. This explanation of mathematics’ applicability in fact

falls short, as Mark Steiner emphasizes in his groundbreaking The Applicability of Mathematics as a Philosophical Problem;38 but never mind. Even in cases where (p.164)mathematical structure does mirror the structure of the physical world, what remains wanting is an explanation why the physical world exhibits so complex and stunning a

mathematical structure in the first place. Balaguer admits that he has no explanation why, on fictionalism, mathematics is applicable to the physical world or why it is indispensable in empirical science. He just observes that the Platonist cannot answer such ‘why’ questions either. That answer must be sought elsewhere.

It seems to me that, whether one is a realist or an anti-realist about mathematical objects, the theist enjoys a considerable advantage over the naturalist in explaining the uncanny success of mathematics. On the one hand, the theistic realist can argue that God has fashioned the world on the structure of the mathematical objects. This is essentially the view that Plato defended in his dialogue Timaeus. God looks to the realm of mathematical objects and models the world on it. The world has mathematical structure as a result. Thus, the realist who is a theist has a considerable advantage over the naturalistic realist in explaining why mathematics is so effective in describing the physical world.

On the other hand, the theistic anti-realist also has a ready explanation of the applicability of mathematics to the physical world: God has created the world according to a certain blueprint which He had in mind. He might have chosen any number of blueprints. The world exhibits the mathematical structure it does because God has chosen to create it according to the model He had in mind. This was the view of Philo of Alexandria, who maintained in his treatise On the Creation of the World that God created the physical world on the mental model in His mind. Thus, the theist—whether he be a realist or an anti-realist about mathematical objects—has the explanatory resources to account for the mathematical structure of the physical world and, hence, for the otherwise unreasonable effectiveness of mathematics—resources which the naturalist lacks.

To return, then, to the objection that fictionalism is self-defeating, it seems to me that fictionalists can escape this charge. At worst, even (p.165) if fictionalism cannot be thoroughly explicated without use of abstract object talk of stories, theories, models, and the like, the fictionalist can, by distinguishing between internal and external questions, coherently hold that, due to the indispensability of abstract object talk, we are stuck with explicating at least some external questions only by using vocabulary belonging to the abstract linguistic framework, thus giving the appearance of incoherence. But it is far from clear that the fictionalist is, in fact, stuck in such an awkward position. For even if abstract object talk is indispensable to scientific theorizing and, indeed, life in general, still it seems that the central claims of fictionalism can be coherently explicated by dispensing with reference to stories, theories, models, and the like in favour of either counterfactual claims about what mathematical entities would be like were they to exist or else deflationary claims about what is useful to assume.

Concluding Remarks

I have not in this chapter argued that theists should embrace fictionalism, but have simply examined its tenability as an option for the classical theist. The three objections to fictionalism which I have discussed—namely, the obvious truth of elementary arithmetic, the indispensability/utility of mathematics, and the self-referential incoherence of fictionalism—are, I think, the principal objections that have been raised against it in the literature. We have seen that, insofar as these objections cast doubt on the fictionalist’s claim that mathematical statements are false, they cast doubt, not on anti-realism, but on the criterion of ontological commitment assumed by the fictionalist, which would force on us Platonistic ontological commitments as a result of seemingly innocuous truths. Nevertheless, granted the neo-Quinean criterion, fictionalist claims about the falsehood of such statements seem not only coherent, but even plausible.

This conclusion is important because, even if we adopt a neutralist perspective on quantification and reference, as I have advocated, or even if abstract object talk is taken to be figurative or merely prescribed to be imagined, as figuralists and pretence theorists maintain, still there are occasions on which we shall want to speak Ontologese in order to make metaphysically heavyweight assertions like ‘There (p.166) are no abstract objects’ or ‘There are no mathematical objects and, hence, no numbers’. We shall then adopt a fictionalist take on such quantificational

discourse. But, contrary to the fictionalist, we still want to be able to assert truly that 2 + 2 = 4 or that 3 < 5. That implies that what we want is a positive free logic for our metaphysically heavyweight assertions. A positive free logic will enable us to use singular mathematical terms in sentences we take to be true without ontological commitment to real-world denotations of those terms, and at the same time, to deny with the fictionalist that there are mathematical and other abstract objects.

Notes:

(7) Ibid., 141.

(12) Ibid., 92.

(15) Ibid., 93.

(25) Ibid., 46.

(26) Ibid., 47.

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