10

10

Make-Believe

Pretence theory construes mathematical sentences as prescribed to be imagined true, so that mathematics is a sort of make-believe. Although the pretence theorist who is also a classical theist will agree with the fictionalist that such sentences, given the neo-Quinean criterion of ontological commitment, are untrue, still he gives a plausible account of why such sentences should not be taken to be true, namely, they are prescribed to be imagined. Therefore, their supposed ontological commitments are just make-believe. Using Kendall Walton’s incisive theory of fiction as a springboard, Mary Leng advocates such a pretence theoretical approach to applied mathematics. This approach is defended against objections based on the obvious truth of mathematics and the disanalogy of mathematics to fiction. A pretence theoretical approach is found to be especially plausible for set theory, which may be taken as a foundation for all of mathematics, in view of the axiomatization of set theory, the creativity of the mathematician within the bounds of consistency, and the incompleteness of mathematical entities.

Closely related to fictionalism and figuralism about abstract discourse is what we may call pretence theory. According to this view, abstract discourse is a kind of fictional discourse and thus analysable in terms of theories of fiction. In that sense, it is more aptly called ‘fictionalism’ than is fictionalism! Like fictionalists and figuralists, pretence theorists do not challenge the neo-Quinean meta-ontological theses that come to expression in premiss (I) of the Indispensability Argument, but instead challenge premiss (II). Pretence theory differs from fictionalism in that it does not claim that sentences involving quantification over or reference to abstract objects are untrue. But neither does it affirm with figuralism that such discourse is often true. Pretence theory is just neutral about the truth value of such discourse.

The pretence theorist’s essential point, rather, is that, whether or not such sentences are true, we are invited

to pretend that they are true. The pretence theorist agrees with the figuralist against the fictionalist that abstract discourse ought not, or need not, be taken literally; but he differs from the figuralist in attributing non-literality, not to the figures of speech employed, but to the fictional nature of such discourse. Thus, abstract discourse, being fictional, does not commit us to the reality of abstract objects. If we have good reason, as the theist does, to think that such objects do not exist, the objects of such discourse may be taken to be mere pretence and nothing more.

Walton’s Theory of Fiction

Contemporary theories of fiction draw much of their inspiration from the brilliant, pioneering work of University of Michigan professor of (p.182) philosophy Kendall Walton.1 Walton takes the practice of ‘make-believe’ to lie at the heart of fiction. ‘The central idea of the make-believe approach’, says Walton, ‘is…that what seem to be commitments, by speakers of theories, to non-existent entities are to be understood in a spirit of pretence or make- believe.’2 He thinks that the activities which produce representational works of art, like literature, painting, film, and so forth, are best seen as continuous with children’s games of make-believe. In support of his viewpoint, Walton makes the interesting observation:

Children devote enormous quantities of time and effort to make-believe activities. And this preoccupation seems to be nearly universal, not peculiar to any particular cultures or social groups. The urge to engage in make-believe and the needs such activities address would seem to be very fundamental ones. If they are, one would not expect children simply to outgrow them when they grow up; it would be surprising if make-believe disappeared without a trace at

the onset of adulthood.3

In Walton’s view, artistic activities such as those mentioned are themselves games of make-believe. Such games are a type of imaginative activity. Walton gives examples to illustrate this familiar activity: Fred imagines himself to be rich, Jennifer imagines herself coming across a bear in the forest, and so on.

Prescribed imagining lies at the heart of Walton’s theory of fiction. Fictional propositions are propositions which in certain contexts we are to imagine to be true.4 Walton recognizes that some propositions which are fictional may not, strictly speaking, be prescribed because they are trivial or distracting (for example, that Hamlet had a heart); nonetheless, they are prescribed to be imagined should the question arise. Games of make-believe involve certain explicit or implicit understandings (what Walton calls principles of generation) as to what someone playing the

game is to imagine to be true. These principles of generation may or may not be explicit. Walton thinks (p.183) that some principles of generation, ‘including most involving works of art, are never explicitly agreed on or even formulated, and imaginers may be unaware of them, at least in the sense of being able to spell them out’.5 But those who play the game correctly have an implicit understanding of what is to be imagined, though in some cases

uncertainty and disagreement may persist.

Walton emphasizes that truth and fictionality are not mutually exclusive. Some of the propositions prescribed to be imagined (by a historical fiction like War and Peace, for example) may be true. Even if all the sentences in a novel about the future like Orwell’s 1984 turned out to be true, that novel still remains fiction—similarly, in the case of a novel whose sentences accurately describe events on some unknown planet in the universe.6What is essential to fictionality is not falsehood, but a prescription to be imagined.

This cursory sketch of Walton’s theory of fiction does not do justice to its fascinating richness, but it does serve to draw our attention to some of the salient features of pretence theory when it is applied to abstract discourse: the centrality of make-believe, the importance of prescribed imagining, the neutrality of fiction with regard to the truth value of propositions which are to be imagined, the absence of any challenge to the neo-Quinean view of quantification and reference—all are elements adapted by pretence theorists in dealing with discourse about abstract objects.

Leng’s Pretence Theoretical Philosophy of Mathematics

Mary Leng is an anti-realist philosopher of mathematics who embraces Walton’s theory of fiction in order to deal with the alleged commitments of our best scientific theories to mathematical objects. She does not appeal to pretence theory when dealing with the sentences of pure mathematics. In Leng’s view, the practice of pure mathematics consists primarily in formulating various mathematical concepts and then inquiring into their

consequences.7 So she treats (p.184) the statements of pure mathematics as disguised conditionals, for example, ‘If

there were numbers, then it would be the case that 2 + 2 = 4.’ This part of Leng’s philosophy of mathematics is thus quasi-deductivist rather than pretence theoretical and forms a relatively tiny part of her defence of anti-realism.

Her central concern, rather, is the applied mathematics found in empirically successful scientific theories. She proposes to treat the applied mathematical sentences of scientific theories as fictional, prescribed to be imagined true. Leng regards the mathematical objects appearing in our scientific theories as merely useful fictions. She explains:

it is reasonable to view the assumption that there are mathematical objects that are related to non-mathematical objects in various ways as generative of a fiction, which provides us with a means, indirectly, to represent non- mathematical objects and their relations. Merely pretending that there are mathematical objects satisfying the axioms of a given mathematical theory provides us with a means to represent non-mathematical objects (as being related to these mathematical objects in certain ways)…all that is confirmed by our empirical successes is that these hypotheses are indeed fictional…, that is, that they are correct in their representation of the non-mathematical objects they concern.8

The echoes of Walton’s theory of fiction are unmistakable: the notions of pretence, principles of generation, fiction, representation, statements’ being fictional all trace back to Walton’s work.

In support of the plausibility of her view, Leng provides two non-mathematical examples of the use of fictions in scientific theorizing: idealizations and theoretical entities. These two cases illustrate the point that, even in our best scientific theories, a reason to speak as if an object existed is not always a reason to believe that that object exists.

With respect to scientific idealizations, Leng notes that scientists sometimes make use of idealizations which are explicitly contradicted by theoretical assumptions made elsewhere.9 For example, textbooks on fluid dynamics treat fluids as continuous (composed of points), in order to be able to say that they have a density, velocity, and temperature defined at a certain point. No real fluid is composed of points. But given the approximation of ideal

fluids to actual fluids, it can be useful to adopt such literally false theories in (p.185)making predictions about real fluids. Obviously, the empirical confirmation of such a theory does not extend to the literal truth of its idealizations. A second example of the scientific use of fictions to which Leng appeals is theoretical entities towards which scientists have an attitude of agnosticism. There are cases in which our theories indispensably posit objects of a certain sort, and yet scientists refrain from saying that these objects exist until they have direct evidence of their reality. Leng has particularly in mind theories of particle physics, which will often posit certain fundamental particles (like the Higgs boson), even as scientists look for evidence of their actual existence. This case differs from the case of scientific idealizations because in this case we have no good reason for taking the fictional content of the theory to be false. The postulated theoretical entities in this case might well exist and will be believed to exist if direct evidence of their existence emerges. This circumstance nicely illustrates the point that fictional statements may well be true. We are to imagine that certain objects exist, and therefore it is fictional that such objects exist, even if it should turn out that they actually do exist.

So in line with Walton’s theory of fiction, Leng does not take the fact that the mathematical sentences in our best scientific theories are fictional to imply their falsity or untruth. She explains:

If Walton is right, then…in viewing our mathematically stated empirical theories as merely fictional, the mathematical fictionalist need take no stand on the question of the actual, metaphysical existence of a realm of objects satisfying the mathematical utterances of our empirical theories. Viewing the mathematical objects posited by our empirical theories as mere fictions requires us only to view our theories as prescribing us to imagine that there are such objects, and as prescribing imaginings concerning their nature (including their relations to real non-

mathematical objects), rather than attempting to answer to the true nature of actual mathematical objects.10

Leng claims that nothing in our current, best scientific world view gives us good reason to believe in mathematical objects as anything more than useful fictions. Hence, we have no reason to think that such objects exist.

(p.186) Objections to Pretence Theory in Mathematics

The Obvious Truth of Mathematics

A common objection to fictionalism, as we have seen, is that some mathematical statements, like ‘2 + 2 = 4,’ are obviously true. This objection to fictionalism might seem irrelevant to pretence theory, since the latter view does not imply that mathematical sentences, in virtue of being fictional, are also false or untrue. Pretence theory, unlike fictionalism, is thus compatible with Platonism and serves merely as a way of defeating the Indispensability Argument for Platonism by showing that quantification and singular reference are not ontologically committing when occurring in a fictional context.

That being said, however, it remains the case that the pretence theorist who is a classical theist is committed to the falsehood or untruth of fictional sentences supposedly quantifying over or involving reference to various abstract objects. For since he denies the existence of uncreated abstract objects, he cannot remain merely agnostic about the truth of sentences taken to be fictional due to their unwanted ontological commitments. Such fictional sentences cannot, unbeknownst to us, be in fact true, for, given the pretence theorist’s assumption of the neo-Quinean criterion of ontological commitment, that would involve us in commitments incompatible with divine aseity. Therefore, given pretence theory’s neo-Quinean assumptions, we cannot find in pretence theory a solution to the challenge facing us unless we are willing to regard fictional sentences involving apparent quantification over or reference to mathematical objects as literally false. So the classical theist who embraces pretence theory must address the challenge posed by the obvious truth of elementary mathematical statements.

In our assessment of fictionalism, we saw that how one assesses the objection from the obviousness of mathematics is apt to hinge upon one’s attitude towards the neo-Quinean criterion of ontological commitment. If we accept the neo-Quinean criterion for ontological commitment, then we shall find upon reflection that the sentences of elementary arithmetic are anything but obvious. On the other hand, if we find that the statements of elementary mathematics force themselves upon us as true, then the classical theist should simply reject the criterion which would saddle us with such strange (p.187) and unwelcome ontological commitments as sets and other mathematical objects.

Even if we are neutralists about ordinary language, we may on occasion be forced to resort to metaphysically heavyweight discourse in order to make our ontological commitments clear. The pretence theorist who is a classical theist will agree with the fictionalist that statements like ‘There are odd as well as even numbers’ are false in a metaphysically heavyweight sense. So what advantage over fictionalism does pretence theory in the end enjoy? The gain to be had from pretence theory, it seems to me, is that, like figuralism, it offers a plausible, positive account of why mathematical and other abstract sentences should not be taken to be ontologically committing to abstract objects. Because mathematical sentences are merely to be imagined as true, mathematical objects are just matters of make-believe.

Mathematics and Fiction

An important objection to interpreting mathematics as make-believe is that mathematics is not of the genre of fiction. In particular, mathematics is not to all appearances a matter of make-believe. Mathematicians and scientists take mathematics to be a body of knowledge and a realm of discovery, not invention. To regard mathematical statements as fictional is to distort the nature of this discipline.

In weighing this objection, we need to keep in mind that the pretence theorist is not defending a hermeneutical claim about how professional mathematicians or scientists in fact understand mathematical sentences. In order to undercut the Indispensability Argument for Platonism, the pretence theorist need show only that mathematical

sentences can be reasonably taken to be fictional. So, we may ask, are the differences between fiction and mathematics so great that mathematical sentences, whether pure or applied, cannot be reasonably regarded as fictional? Remember that the truth of a sentence is not incompatible with that sentence’s being fictional, so the supposed truth of mathematical sentences is not grounds for denying their status as fictional. Rather, the objection must be that mathematics is not reasonably construed to be a matter of make-believe.

(p.188) But why not? Why can we not reasonably take mathematical statements as prescribed to be imagined to be true rather than believed to be true? It might be said that mathematical theories are altogether lacking such prescriptions to imagine. But the absence of such prescriptions is of little significance, since, as Walton notes, representational works of art typically lack such explicit prescriptions.11 Indeed, literary fictions not infrequently

present themselves as genuine historical records, such as a ship’s log or a personal journal. The prescription to imagine them to be historical records is anything but explicit. Mathematics, being a collective enterprise, even lacks any author’s intent to guide us in our interpretation. Perhaps mathematics is in that respect like mythology, which people may take either literally or fictionally.

In fact, there are some features of mathematics that make it seem a prime candidate for a fictional interpretation. I shall mention three such features.

Axiomatizaton

In the first place, axiomatization of mathematical theories naturally invites a pretence theoretical interpretation. For example, the axioms of standard set theory are plausibly taken as prescribed to be imagined. University of California philosopher of mathematics Penelope Maddy distinguishes between believing the axioms of standard set theory

and defending the axioms of standard set theory.12 Although such an apologetic concern seems at first blush to

assume that the axioms are true and that it is the truth of the axioms that is to be defended, what Maddy is in fact talking about is defending the use of the axioms, regardless of their truth. Indeed, her aim is to ‘shift attention away from…elusive matters of truth and existence’.13 In Maddy’s view, what justifies the use of set-theoretical axioms is fruitfulness: axioms (p.189)are properly adopted which are rich in mathematical consequences, or what Maddy calls ‘mathematical depth’.14 In short, we do not have to believe the axioms in order to use the axioms.

Consider the axioms of standard set theory.15 The following is a representative statement of those axioms:16

(p.190) It is striking that, with the exception of the Axiom of Infinity, these axioms are all universally quantified statements devoid of any existential implications. They explicitly have the conditional logical form attributed to them by a deductivist like Leng. One advantage of pretence theory over deductivism is that we need not reconstrue the Axiom of Infinity to have any other logical form than its apparent form. Rather, one takes the Axiom of Infinity as something we are prescribed to imagine. We are to make-believe that there is an infinity of these things called sets and then are free to explore the fictional world of our imagination. This will certainly be a journey of discovery, which will issue in a great deal of knowledge of the mathematical world determined by the axioms. Such an attitude towards the axioms of set theory is not uncommon among mathematicians and philosophers of mathematics. For example, postulationalism, which treats the axioms of competing theories as postulates whose consequences may be

explored, invites us, in effect, to make-believe that the axioms are true without committing ourselves to their objective truth.17

But is it plausible to take the Axiom of Infinity as something prescribed to be imagined rather than as a straightforward metaphysical assertion? I think the answer is obviously affirmative. It is universally admitted that the Axiom of Infinity is not intuitively obvious. Its lack of intuitive warrant was one of the heavy stones that helped to sink logicism, an early twentieth-century attempt to derive set theory from logic alone. Since the axiom lacks intuitive warrant, the Axiom of Infinity is adopted by contemporary mathematicians for reasons that are variously called ‘pragmatic’ or ‘regressive’ or ‘extrinsic’, reasons which do not justify its truth, but its mathematical utility. Moreover, as Michael Potter points out, there are multiple versions of the Axiom of Infinity which postulate

different sets. A defence of the axiom on pragmatic grounds ‘does not directly give us a ground for preferring one sort of axiom of infinity over another’, he says, (p.191) since any of them will work.18 As a result, says Potter, Platonists ‘have frustratingly little they can say’ by way of justification for their preferred version of the axiom.19 As

a serious assertion of ontology, the Axiom of Infinity is a breathtaking assertion that utterly outstrips our intuitions. If any conception of sets had a claim to be intuitive, it would be the conception of nineteenth-century naïve set theory, which turned out to be riddled with logical paradoxes. The pioneers of set theory, Richard Dedekind, Georg Cantor, and Gottlob Frege, all embraced a principle of universal comprehension, according to which any property serves to determine a set of things having that property. That they were blind to the incoherence implicit in this assumption is testimony to the strength of the intuition motivating the naïve conception of sets.20 Set theorists Abraham Fraenkel and Yehoshua Bar-Hillel believe that, were it not for the paradoxes of naïve set theory, Platonists would most naturally adopt a theory whose main feature is an Axiom Schema of Unrestricted Comprehension, such that for any property there is a set of things having that property.21

Axiomatization was an ad hoc fix aimed at precluding the paradoxes inherent in naïve set theory.22 The Axiom of Infinity postulates the existence of sets governed now by (3) the Axiom Schema of Restricted Comprehension, which is an ad hoc fix of naïve set theory aimed at avoiding the paradoxes. If we take the axioms of standard set theory merely as prescribing what we are to imagine, such (p.192) stipulations are unobjectionable; but if we take them as serious ontological assertions about the existence of mind-independent objects, we have left behind any intuitive support for the existence of such things.23

Indeed, as a result of the axiomatization of set theory and the concomitant abandonment, not just of the naïve conception of what a set is, but of any definition of ‘set’ at all, we really do not know what we are talking about in the theory of sets.24 One might, in line with David Hilbert’s axiomatic approach to geometry in his Foundations of Geometry (1899), take the set-theoretical axioms to offer an implicit definition of ‘set’; but then, as Frege rightly protested, we have sacrificed any intuitive warrant for taking the axioms to be true.25 I could not help but smile at Frege’s indignant comparison of Hilbert’s axiomatic existence assertions with the ontological argument for the existence of God—as if one could simply define things into existence!26 Such a comparison is not altogether fair to Hilbert, it must be said, since he understood mathematical existence in a very lightweight sense, amounting to nothing more than consistency relative to a system of axioms;27 but if we take the Axiom of Infinity to be a piece of

serious metaphysics, then to treat its terms as implicitly (p.193)defined by the other axioms leaves us in perplexity as to whether such things as sets really do exist. On the other hand, if we simply make-believe that they exist, then, Frege notwithstanding, we cannot be accused of trying to define things into existence.

Moreover, it deserves to be said that, wholly apart from principles of comprehension, sets, even naïvely conceived, are such strange entities that if we take set theory to be a serious piece of ontology, certain intuitive questions arise

that would otherwise be obtuse. Consider, for example, the Axiom of Extensionality. According to George Boolos, the Axiom of Extensionality is the central axiom of standard set theory.28 It requires that sets have their members essentially. But is that true? How do we know that sets do not have their members contingently? Intuitively, the set

of living US Presidents loses and gains members over time. Intuitively, if Ronald Reagan were still alive, he would be a member of the set of living US Presidents. Intuitively, in a possible world in which Jimmy Carter and Bill Clinton lose their respective presidential races, they are not members of the set of living US Presidents, but others are. If I did not eat lunch today, then the Axiom of Extensionality requires that the set of things I ate for lunch is identical to the set of unicorns, namely, the empty set, which is intuitively bizarre. Indeed, the idea of an empty set is itself intuitively strange, for how can there be a collection if there is nothing to be collected? What is it a set of?

Why not think that in such a case there simply is no set rather than that there is a set and it has nothing in it?29 Why

think that the Axiom of Extensionality is true?

Such questions will strike the set theorist as maladroit: sets are simply stipulated to be such that they are identical if they have the same members. Boolos takes the Axiom of Extensionality to be analytically true; any justification for it will look like justification for (p.194) sentences like ‘All bachelors are unmarried.’ Anyone who says that there are distinct sets with the same members is just not talking about sets.30

This sort of stipulation is unobjectionable so long as the Axiom of Extensionality merely prescribes what we are to imagine. But if, as Platonists believe, the axiom is an accurate description of objective reality, then why think that there are any such things as sets?31 If such questions as these seem outrageously inept, I suspect that that is because we are not taking the axioms to be sober descriptions of mind-independent reality, but just stipulations of what sets are like.

And what about the Axiom of Foundation? It is this axiom which precludes any set’s being a member of itself, thus determining the hierarchy of sets. But while such a conception of sets might make good sense on a constructivist view of sets, according to which an ideal mind constructs a hierarchy of sets, it seems an ad hoc and even counter- intuitive manoeuvre given Platonism. For why should there not be sets which are members of themselves? For example, the set of all things mentioned in this book is itself mentioned in this book. Standard axiomatic set theory banishes such sets by sheer fiat.

It is noteworthy that in his classic defence of the so-called ‘iterative concept of set’, Boolos’s approach is undisguised constructivism:

(p.195) a set will include itself as a subset; but how can it contain itself as a member? This idea is paradoxical not in the sense that it is contradictory that some set is a member of itself,…but that if one understands ‘∈’ as meaning ‘is a member of,’ it is very, very peculiar to suppose it is true. For when one is told that a set is a collection into a whole of definite elements of our thought, one thinks: Here are some things. Now we bind them up into a whole. Now we

have a set. We don’t suppose that what we came up with after combining some elements into a whole could have been one of the very things we combined.32

Boolos even describes the formation of the hierarchy of sets as proceeding in a sequence of stages ordered by relations like ‘earlier than’ and ‘immediately after’. At stage 0, we begin by forming collections of individuals. If there are no individuals, only one collection is formed, the null set (how or why this is a collection, rather than there being no collection at all, is not explained). One of the collections formed will be the set of all individuals, no matter how many. Then at stage 1 are formed all sets composed of individuals and stage 0 sets; at stage 2, one forms all sets composed of individuals and stages 0 and 1 sets, and so on through all the natural numbers. ‘Immediately after’ stages 0, 1, 2, 3,…, there is a stage ω, which is succeeded by stage ω + 1, and so on.33 As Charles Parsons observes, the language employed here suggests that the iterative conception of set is analogous to the concepts of constructive

mathematics, according to which an idealized mind forms the sequence of stages over time, a view which is ‘flatly incompatible’ with a Platonistic theory of sets.34 While the iterative conception of set makes good sense on a constructivist view, it is gratuitous on a Platonist view to think that sets obey the Axiom of Foundation so as to form

such a hierarchy.35

(p.196) Or consider the Axiom of Choice. Not included among Zermelo’s original axioms of set theory, it is today widely regarded as intuitively obvious for a finite family of sets but far from intuitively obvious for an infinite family of sets.36 Though once the subject of considerable controversy, the axiom has become virtually universally accepted today—not because of its evident truth, but because of its utility and fecundity. Reflecting on the controversy, Fraenkel and Bar-Hillel muse that mathematicians’ initial acceptance or rejection of the axiom was ‘far

more strongly influenced by emotional or practical reasons than by arguments of principle’.37Use of the Axiom of Choice, they advise, is to be justified by analysing concepts, methods, and proofs actually found in mathematics

rather than in an intuitive manner. Even so, they opine, we should not ask whether the Axiom of Choice is true, but what additional parts of mathematics can be obtained by means of the axiom, while at the same time examining what results can be obtained without it. They recognize that such an attitude is incompatible with Platonism; but it makes perfect sense on the view that we are invited to imagine the Axiom of Choice to be true or false and to explore the respective consequences, just as the pretence theorist advocates.

In sum, axiomatic set theory seems to be a perfect candidate for a pretence theoretical approach. The philosopher of mathematics Stewart Shapiro observes: ‘The strongest versions of working realism are no more than claims that

mathematics can (or should) be practiced as if the subject matter were a realm of independently existing, abstract, eternal entities.’38 This characterization would make working realists into pretence theorists! We are invited to imagine the axioms are true and then are free to explore the fictional world established by such prescriptions.

Indeed, many mathematicians do, in fact, so understand the axioms.

(p.197) Creative Freedom

A second respect in which set theory seems particularly apt for a fictional interpretation concerns the creative freedom enjoyed by mathematicians. Just as authors of literary fiction are free to shape their characters as they wish, without concern for correspondence with reality, so mathematicians are at liberty to craft and explore different axiomatic systems without worrying that they are misrepresenting reality in doing so. Abraham Fraenkel has spoken eloquently of abstract set theory as showing ‘that possibility of free creation in mathematics which is not equaled in any other science. It is no accident that at the birth of the theory of sets, there was coined the sentence: the very

essence of mathematics is its freedom.’39 The freedom of ‘the creative mathematician’ is, he says, restricted only by ‘the postulate of consistency, of logical non-contradiction’.40 In Fraenkel’s view, sets are ‘logical fictions’, that is to say, ‘abstract concept[s]’ which are the result of ‘an intellectual act’ of collecting things into an aggregate.41 He

notes that taking sets as logical fictions is no impediment to mathematical practice: ‘The logical character of the objects called “sets” is of no importance to the mathematical theory of sets—in the same way as the results of arithmetical calculating are independent of what may be, in the view of the calculator, the logical or psychological meaning of number.’42

So set theorists have felt free to formulate quite a variety of set theories, some with different ontologies. For example, von Neumann-Bernays-Gödel set theory (NBG) includes, in addition to sets, mathematical objects called classes, which are governed by a universal principle of comprehension.43 NBG thus has a fundamentally different ontology (p.198) than standard set theory. Potter lists four variable features of the ‘enormously varied’ potpourri of set theories: (1) whether there are individuals or just pure sets; (2) how many levels there are in the hierarchy of sets;

(3) how rich a conception there is, once a level is given, of the level that follows it in the hierarchy; and (4) whether there are classes as well as sets.44

Theorists working with the axioms of standard set theory are at liberty to bracket certain axioms which are independent of the others and to explore the consequences. Zermelo-Fraenkel set theory without the Axiom of Choice is probably the most familiar example, but one might omit instead the Axiom of Foundation. This will have

the consequence of allowing sets which are not well founded, that is, sets exhibiting an infinite descent or circle of membership rather than ascent.45 Such sets subvert the iterative hierarchy of sets founded on the empty set Ø. According to set theorist Kenneth Kunen, the Axiom of Foundation ‘is never used in mathematics, but it leads to a

much clearer picture of the set theoretic universe’.46 In this case, the Platonist cannot even justify use of the Axiom of Foundation on the grounds of its fruitfulness; its adoption appears to be mere convenience. But then why should we think that the set-theoretical hierarchy really exists?

On the other hand, one might freely add axioms to standard set theory.47 Adding an Axiom of Constructibility will ensure that the universe of constructible sets L is identical to the entire realm of sets V. On the other hand, we can compel moving beyond V = L by replacing the Axiom of Choice with a so-called Axiom of (p.199)Determinacy. In the absence of an Axiom of Constructibility, the mathematician’s imagination may soar boundlessly through inaccessible cardinals, Mahlo cardinals, and indescribable cardinals, to partition cardinals, measurable cardinals, and

so on.48 Pretence theory makes far better sense of this remarkable freedom than does Platonism. For within the bounds of logical consistency, we are free to make-believe that there are mathematical entities of every imaginable sort and to explore the consequences without concern that we are thereby misdescribing reality.

Maddy observes that mathematicians employ ‘maximizing principles of a sort quite unlike anything that turns up in the practice of natural science: crudely, the scientist posits only those entities without which she cannot account for our observations, while the set theorist posits as many entities as she can, short of inconsistency’.49Such a modus operandi seems objectionable given realism. If the set-theoretical universe, like the physical universe, were an

independently existing reality, then caution would be the word of the day, not freedom. Hence, ‘Quine counsels us to economize, like good natural scientists, and thus to prefer V = L, while actual set theorists reject V = L for its miserliness.’50 Why so? Plausibly because the set theorist does not take himself to be ontologically committed to the sets he studies. He can reap the mathematical benefits of his postulates while remaining indifferent to ontological questions.

Maddy imagines a set theorist recommending an extension of standard set theory, according to which V≠ L on the grounds of its mathematical attractiveness. She then imagines how the realist might respond.

From the realist’s point of view, it would seem fully rational to respond, ‘Yes, I agree with you that [it] is a very nice theory, but if all sets are (p.200) constructible, this theory is false, despite its niceness.’ From this point of view, the set theorist owes evidence, not that the theory has many virtues, but that it is actually true in the real world of sets…Yet—and here’s the problem—this reply to the set theorist’s argument…seemed to me out of place, out of

step with the actual practice of set theory.51

The significance of Maddy’s point needs to be appreciated. Realism, if taken seriously, would be a positive impediment to mathematical practice and progress. The maximizing principles she mentions are unobjectionable on a pretence theoretical approach, since we are at liberty to make-believe as we choose. But such freedom is hard to reconcile with Platonic realism. Platonism cannot accept such principles, since we have not only an abundance of nice, rival theories, but more fundamentally, no warrant to think that ‘niceness’ coincides with reality in the first place.

Set theory is typically regarded as foundational for the rest of mathematics, since the whole of mathematics can be reductively analysed in terms of pure sets. Maddy comments:

The astounding achievement of the foundational studies of the late nineteenth and early twentieth centuries was the discovery that these fundamental assumptions could themselves be proved from a standpoint more fundamental still, that of the theory of sets. The idea is simple: the objects of any branch of classical mathematics—numbers, functions, spaces, algebraic structures—can be modeled as sets, and resulting versions of the standard theorems can be proved in set theory. So the most fundamental of the fundamental assumptions of mathematics, the only such assumptions that truly cannot be proved, are the axioms of the theory of sets itself.

In this sense, then, our much-valued mathematical knowledge rests on two supports: inexorable deductive logic, the stuff of proof, and the set theoretic axioms.52

Such a reductive analysis makes sense on a pretence theoretical approach. We imagine that there are objects such as are characterized by the set-theoretical axioms and allow them to stand in for all other mathematical objects. As Maddy explains, the mathematical benefits of such a procedure are impressive in terms of exposing inconsistencies, organizing knowledge, leading to theories of greater power and (p.201) fruitfulness, and so on. But these benefits do not depend on set-theoretical foundations’ providing any sort of metaphysical insight.

A realist ought to be sceptical of such reductive analyses, since the ability to represent, for example, numbers as sets in no way proves that numbers are sets. Indeed, such a claim seems intuitively bizarre, for we do not normally take numbers, for example, to have members, as sets do. Moreover, different set-theoretical reductions of mathematical objects are available and equally tenable. Is the number 3 to be identified with {{{Ø}}}, as Zermelo suggested, or

with{Ø, {Ø}, {Ø, {Ø}}}, as von Neumann held?53 The Platonist must hold that there is an objective truth about this

matter, yet any answer seems equally good. The Platonist might rightly say that our inability to discern the truth about the ontology of numbers is not a good argument for thinking that there is no such objective truth. But our concern here is not to offer a rebutting defeater of Platonism, but simply to inquire whether a pretence theoretical perspective is not just as plausible as Platonism. The answer to that question seems evidently, ‘Yes.’ We can make-

believe that numbers are the sets defined by Zermelo or we can make-believe that numbers are von Neumann’s sets, and it will make no difference to mathematical practice.54

Incompleteness of Mathematical Entities

Finally, a third respect in virtue of which set theory is especially amenable to a pretence theoretical interpretation is the incompleteness of mathematical entities. One of the most striking features of fictional characters is their incompleteness. It is not merely unknown whether, for example, Hamlet was left-handed or Sherlock

Holmes (p.202) wore a size nine shoe; rather, there is no fact of the matter at all about such things because there are no fictional truths concerning such matters. It is neither fictional that Hamlet was left-handed nor fictional that Hamlet was right-handed. It is for that reason that many philosophers think that Hamlet could not possibly exist. For real objects are complete, at least if they are concrete, and there is nothing that would make a left-handed ‘Hamlet’ rather than a right-handed ‘Hamlet’ really be Hamlet.

The mathematical objects of infinite set theory are incomplete in a number of significant ways. The most celebrated example is the Continuum Hypothesis (CH), that the number of real numbers is ℵ1, the successor to ℵ0, which is the number of natural numbers. Paul Cohen demonstrated in 1963 that CH is independent of the standard set-theoretical axioms.55 The pretence theorist plausibly maintains that CH is neither fictional nor not fictional in the game prescribed by those axioms. Of course, the mathematician is free to play a different game using different principles

of generation, and in some of these CH may be fictional. For example, if we imagine certain inaccessible cardinals to exist, then CH follows. The Platonist will say that some such system of higher mathematics is true, so that CH has an absolute truth value. But none of these stronger systems, which permit virtually any answer to the question of the size of the continuum, has any intuitive warrant. The pretence theorist may surely be excused if he is sceptical about the objectivity of such conjectural systems.

There are quite a number of other mathematical hypotheses which are independent of the axioms of standard set theory, notably the Axiom of Constructibility that V = L. While the Platonist may insist that there is some objective truth about such matters, it seems equally plausible to maintain that the mathematical entities imagined to exist in such cases are, like fictional characters, simply incomplete in the standard set-theoretical story. Other stories can be told, for example, those postulating certain inaccessible cardinals, in which some of these hypotheses are fictional; but there is no reason to think that those stories are any less prescriptive of what we are to imagine.

A prescient von Neumann, discussing the question of the categoricity of set theory, that is to say, whether the axioms uniquely determine the system they describe, voiced a pessimistic conclusion:

(p.203) The consequence of all this is that no categorical axiomatization of set theory seems to exist at all; for probably no axiomatization will be able to avoid the difficulty connected with the axiom of restriction and the

‘higher’ systems. And since there is no axiom system for mathematics, geometry, and so forth that does not presuppose set theory, there probably cannot be any categorically axiomatized infinite systems at all.56

‘This circumstance’, he mused, ‘seems to me to be an argument for intuitionism.’57 That is to say, the Law of Excluded Middle fails for statements such as the Continuum Hypothesis: neither CH nor not-CH is true. Such a view fits well with the intuitionists’ view of mathematics as the product of human intellectual construction; but it sits ill with Platonism. Intuitionism is not one’s only choice, however: it is equally plausible to take set theory to be a realm of the human imagination, a fictional world of make-believe entities. In such a world, it is neither fictional that CH nor fictional that not-CH.

In summary, we have seen three features of set theory which make it especially amenable to a pretence theoretical interpretation: (i) the axiomatization of conflicting set theories that outstrip any intuitions that may have undergirded naïve set theory, (ii) the creative freedom of the mathematician to extend the mathematical realm as far as consistency allows, and (iii) the incompleteness of the mathematical objects referred to in mathematical theories.

These seem to make a pretence theoretical approach to such objects at least as plausible as Platonism.

Concluding Remarks

Since mathematics is the last bastion of Platonism, the viability of a pretence theoretical approach to set-theoretical discourse is a devastating blow to realism. Without challenging the Platonist’s assumed neo-Quinean criterion of

ontological commitment, the pretence theorist undercuts the Indispensability Argument by giving a plausible account of mathematics that is not committed to its literal truth.

(p.204) It is this positive account of mathematical discourse that makes pretence theory superior to the flat assertion of standard fictionalism that mathematical sentences involving quantification over or reference to abstract objects are untrue. Although the pretence theorist who is also a classical theist will agree with the fictionalist that such statements, given the neo-Quinean criterion of ontological commitment, are untrue, still, he gives a plausible account of why such sentences should not be taken to be true, namely, they are prescribed to be imagined.

Therefore, their supposed ontological commitments are just make-believe.

Pretence theory’s treatment of axiomatic set theory as fictional also makes pretence theory even more credible, I think, than figuralism. For while figuralism plausibly claims that the real content of many mathematical truths is purely logical, the same cannot be said of mathematical existence axioms such as the Axiom of Infinity. The lesson of logicism’s failure is that infinite set theory cannot be reduced to purely logical truths about collectings, for the Axiom of Infinity is an existence assertion that has non-logical content.58 Better to view this axiom as prescribed to be imagined, and then the consequences logically deduced.

Figuralist Stephen Yablo in fact sometimes characterizes figuralism as a sort of pretence theoretical fictionalism,59 which shows just how blurred the lines of demarcation between these three views can become. His

mature view seems to be that figurative language is not a matter of make-believe (which he, unlike Walton, takes to be a conscious activity), but of simulating belief (which he takes to be unconscious). Someone is simulating belief that S if, although things are as if that person believed that S, he finds upon reflection that he really either disbelieves or is agnostic about S taken literally or feels that the propriety of his stance does not depend upon

believing (p.205) S taken literally. Such a view of figurative discourse remains pretence theoretical at heart.

In short, pretence theory is a tenable and, I should say, attractive option for theists seeking to meet Platonism’s challenge to God’s status as the sole ultimate reality. For that status is not threatened by the spectre of make-believe bogeymen.

Notes:

(6) Ibid., 74.

(7) Mary Leng, Mathematics and Reality (Oxford: OUP, 2010), 83. (8) Ibid., 9–10.

(9) Ibid., 111.

(10) Ibid., 206–7.

(14) Ibid., 82.

(19) Ibid., 72.

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