Mathematical Objects As Figures of Speech: on Yablo's Figuralism

Mathematical objects as ﬁgures of speech: On Yablo’s ﬁguralism

Ebba Gullberg November 2007

1 Introduction

This paper is intended as a preliminary study for my dissertation, which is centered around Georg Kreisel’s famous dictum: “The problem is not the existence of mathematical objects, but the objectivity of mathematical statements.” Kreisel’s dictum brings up the question whether it is possible to give an account of mathematics that preserves its objectivity without postulating a platonic domain of abstract mathematical objects. I want to explore the relationship between the assumption that mathematics is objective, and the assumption of mathematical objects. In mathematics, it is common to make statements that assert the existence of mathematical objects such as numbers, sets, and functions. From a mathematical point of view it seems obvious that these objects exist. Philosophers, on the other hand, have questioned the existence of numbers. One may ask whether the mathematician and the philosopher are talking about the same thing when the former claims that numbers exist and the latter denies this. If they do, the philosopher’s denial of the existence of numbers comes into conflict with well-established mathematical results. In this context, Carnap [7] introduced a distinction between internal and external statements. The mathematical statement that numbers exist is an internal existence statement within mathematics. Without contesting this mathematical statement, the philosopher asks whether numbers really exist. Should she claim that this is the case, she will have made an external existence statement about mathematical objects. One way of interpreting Carnap’s distinction, which has recently been pro- posed by Stephen Yablo, is to equate the external statements with those that are intended to be literally true and the internal ones with those that are in some sense fictional. Yablo’s approach, figuralism, is the topic of this paper.

1 1.1 Nominalism The main thesis of nominalism can be formulated in the following way: There are no abstract objects. In the case of mathematics this can be narrowed down to the more speciﬁc claim that there are no abstract mathematical objects, such as numbers, functions, or sets.1 One of the attractions of a view like this can be seen by considering what is perhaps the most obvious objection against the opposite position, realism or platonism. The objection has to do with our possibilities of gaining mathematical knowledge. Benacerraf [2, p. 414] writes: “If, for example, numbers are the kinds of entities they are normally taken to be, then the connection between the truth conditions for the statements of number theory and any relevant events connected with the people who are supposed to have mathematical knowledge cannot be made out.” The worry that is expressed here is that the abstractness of the numbers and in particular the impossibility to causally interact with them seem to prevent us from giving a reasonable account of how we can come to know things about them. Since mathematics and mathematical thinking play a huge role in both science and everyday life, the nominalist’s suggestion is that we abandon the idea of abstract objects and try to give an account of mathematics in terms of something else.

1.2 Mathematical realism and ontological commitment However, it is far from clear that nominalism is a winning strategy. Let us for a moment ignore the epistemological difﬁculties with realism and look at what kind of argument that can be given in favor of the existence of abstract mathematical entities, and hence against nominalism. Suppose that T is some mathematical theory. A very inﬂuential line of reason- ing is the following:

1. The theorems of T are literally true. 2. If the theorems of T are literally true, objects of kind P exist. 3. Hence, there exist objects of kind P.

We can get a better picture of the idea of this argument if we look at an example. Let T be the theory of arithmetic and let the objects of kind P be natural numbers. Now take a sentence such as

(*) There are inﬁnitely many prime numbers. 1This deﬁnition of mathematical nominalism presupposes the thesis that if numbers etc. exist, then they are abstract, i.e. not located anywhere in space or time. For some other possible views, see e.g. Balaguer [1].

2 If we agree to treat this sentence no differently than we would treat any other natural language sentence, it does not seem far-fetched to interpret (*) as asserting the existence of a certain kind of entity, namely prime numbers (and, more generally, natural numbers). Another assumption that is unlikely to appear controversial is that this interpretation is the literal interpretation of (*). Combine this with our knowledge that (*) is a theorem of arithmetic and we have accepted Premise 2 of the above argument. All that remains in order for us to reach the conclusion that there actually are numbers is to find an argument convincing us that the literal interpretation is the right one and that, on this interpretation, arithmetic is true. Mathematics, and not only arithmetic, is full of sentences like (*) that, on the face of it, assert the existence of various abstract entities. Some of them can be seen on closer examination to be expressible in terms of other; for instance, talk about numbers can be translated into talk about sets, but at the end of the day there always seem to be some left. We say that mathematics is ontologically committed to abstract objects. At this point it could be objected that our natural language formulations of mathematical theories are misleading (despite the fact that this is how they are usually formulated). Quine [13, p. 242] writes: “The trouble is that at best there is no simple correlation between the outward forms of ordinary affirmations and the existences implied.” Quine’s own solution to this problem is to introduce a canonical notation in the language of first order logic, and a criterion by which to decide what entities exist: “Insofar as we adhere to this notation, the objects we are to be understood to admit are precisely the objects which we reckon to the universe of values over which the bound variables of quantification are to be considered to range.” As it turns out, even with canonical notation and the more formal criterion, the ontological commitment to abstract entities in mathematics apparently cannot be made to disappear.

1.3 Indispensability and truth The realist will still need something to support Premise 1 in the argument given in the previous section, since if mathematics is not true, then it hardly matters what its ontological commitments are. Such an argument is given as part of the Quine- Putnam indispensability argument for the existence of mathematical objects. The indispensability argument has been considered by many philosophers, both realists and nominalists, to be the best one available. It is based on one of the most conspicuous features of mathematics, namely its vast applicability in science. If we were deprived of the possibility to use mathematical language and mathematical theories when formulating and practicing science, the argument goes, our scientiﬁc theories would be irreparably damaged and impoverished. Furthermore, since the scientiﬁc theories that we have now are the best we have come up with

3 so far (even though they are not flawless) it can be argued that we are justified in believing that they are true, at least to some extent. So, if mathematics is an indispensable part of these theories, it seems that we are justified in believing that mathematics is true too. We can summarize this in the following way: 1. We are justified in believing that our best scientific theories are true. 2. Mathematics is indispensable for our best scientific theories. That is, mathematics is necessary both for the formulation of scientific theories and for the practice of science. 3. If T and T ′ are theories, T ′ is indispensable for T , and we are justified in believing that T is true, then we are justified in believing that T ′ is true. 4. Hence, we are justified in believing that mathematics is true. By accepting this argument we get half of Premise 1 in the argument for the existence of mathematical objects. The part about literalness seems to follow if we join for instance Quine [13] and Burgess [5] in seeing it as somehow dishonest or insincere to use one language when doing science and another when talking about the same thing in philosophy. The rest of the indispensability argument runs along essentially the same lines as the argument given in the previous section; literal truth and ontological commitment yield the conclusion that there exist mathematical objects.

1.4 Revolutionary and hermeneutic nominalism In order to avoid acceptance of the realist’s account of mathematical existence, there are a few different options for the nominalist to explore. Burgess [5] introduces a distinction between two nominalistic strategies: revolutionary and hermeneutic nominalism. The same distinction, with some slight qualifications, is used in Rosen & Burgess [15]. Revolutionary nominalism involves a normative claim about mathematics. The revolutionary nominalist agrees with the realist that our actual mathematical theories are ontologically committed to abstract entities, but holds that the practice or the interpretation of mathematics can and should be revised to make the commitments go away. Contrary to the adherents of the indispensability argument, the revolutionary nominalist believes that mathematics as it is now is not indispensable for science. Clearly, mathematics is an extremely useful and conve- nient tool for formulating attractive and efficient scientific theories, but, at least in principle, we could manage without it and have theories with the same explanatory power as the ones we have now but that do not refer to abstract objects. This is of course something that needs to be shown by constructing nominalistically and scientifically acceptable alternatives to the existing theories. Of the attempts that

4 have been made in this direction, the one that has probably received the most at- tention is Field’s [10] nominalization of Newton’s theory of gravitation. However, the gravitational theory is just one of many scientiﬁc theories that will need to be reconstructed in order to show that mathematics really is dispensable for science and there does not seem to be a uniform method to handle them all at once.2 The amount of hard work that is needed to carry out a revolutionary nominalistic program has earned its defenders some respect even from those who disagree with the idea; for instance, Rosen & Burgess [15, p. 523] call the revolutionary nominalists “admirably straightforward”. Hermeneutic nominalism, by contrast, does not receive any such positive appraisal. Burgess [5, p. 97] calls it “a desperate device of ‘ostrich nominalism’.” So, what is hermeneutic nominalism? The central idea of hermeneutic nominalism is that mathematics does not really involve us in any “unwanted” ontological commitments, because either the sentences of mathematics do not really mean what they appear to mean, or we do not really believe what they mean.3 This could be understood as a normative claim, but that would bring us back to revolutionary nominalism. Instead, this is supposed to be a description of what actually goes on when we use and talk about mathematics the way we normally do. Thus, the obvious challenge for the hermeneutic nominalist is to give a plausible account of how appearances can be so deceiving: How is it possible that almost no one knows what our mathematical theories are really about, or that so many people go around thinking that they believe something without really believing it? Clearly, the hermeneutic nominalist needs to tell a very good story to make this idea seem at all reasonable. In what follows we will see some examples of what hermeneutic nominalists might say to defend their thesis.

1.5 Fictionalism The most serious obstacle to overcome for someone who wishes to be a mathematical nominalist seems to be that mathematics cannot be literally true unless there exist at least some abstract objects. This ontological commitment is taken by realists as a reason to argue in favor of the existence of mathematical objects, it is taken by revolutionary nominalists as a reason to reconstruct scientiﬁc theories, and it is taken by hermeneutic nominalistsas a reason to argue that mathematicians and scientists do not really mean or believe what they say. A positive account of what mathematicians and others could mean when they utter ontologically committing mathematical sentences is given by ﬁctionalists. Their proposal is that we

2Cf. Field [11, p. 65 n.]. 3Rosen & Burgess [15, p. 517] label the former alternative “content-hermeneutic nominalism” and the latter “attitude-hermeneutic nominalism”.

5 can understand such sentences as being “advanced in a fictional or make-believe spirit”.4 There is no general agreement on how fictionalism should be classified in relation to other kinds of nominalism. Rosen & Burgess [15, p. 518] take it to be a version of attitude-hermeneutic nominalism, whereas Burgess [6, p. 19 n.] identi- fies it with instrumentalism. Stanley [17, p. 36] notes that the fictionalist thesis can be put forward both as a normative and as a descriptive claim, and hence makes a distinction between revolutionary and hermeneutic fictionalism. Yablo [22] and Eklund [8] make several other distinctions that reveal a wide spectrum of possible variations on the fictionalist theme. One can read into the term ‘fictionalism’ a suggestion that there is some particular genre of “standard” literary fiction that mathematics is to be compared to, such as novels, short stories, or fables. Burgess [6] argues on the basis of this idea that ‘fictionalism’ is an ill-chosen term since there are many differences (and not very many similarities) between mathematics and any specific fictional genre. However, it is not a common feature of all kinds of fictionalism to try and force mathematical discourse into the frames of literary fiction. The fictionalist view that perhaps comes closest to something like this is meta-fictionalism. Eklund [8] defines meta-fictionalism about Xs in the following way: “the speaker is ‘really’ asserting that according to a certain fiction, the Xs are so and so”. An example of the kind of fiction that a meta-fictionalist could be referring to is “the story of standard mathematics”. So when we assert that 2+2 = 4, what we really assert is that according to the story of standard mathematics 2 + 2 = 4. Yablo [22, p. 76] objects to meta-fictionalism by pointing out that it treats the fiction as a bit too detached from the actual world. For instance, “2 + 2 = 4” is usually taken to be a priori and necessary. However, what this sentence really means on the meta-fictionalist view, namely “According to the story of standard mathematics, 2 + 2 = 4”, does not seem to be a priori or necessary, since the story of standard mathematics could have been different. Moreover, when we use number words to say something like “The number of days in a week is seven”, it is not a plausible claim that what we really mean is “According to the story of standard mathematics, the number of days in a week is seven”. We need a type of fictionalism that takes into account the fact that mathematics is intertwined with the world and not separate from it. Object-fictionalism is a fictionalist view where the focus does not lie primarily on the fiction, but on the world. The fiction is thought to be set up in such a way that what is true in the fiction depends on what is true in the real world.5 So when

4Yablo [22, p. 74]. 5Note, however, that the fiction also is meant to coincide with ordinary mathematics. That is, the statements that the object-fictionalist asserts as being fictionally true are to be exactly the same

6 it looks as if we assert that the number of days in a week is seven, what we really assert, according to the object-fictionalist, is that there are seven days in a week, which is something we can say without referring to numbers as abstract entities. In other words, what makes it true in the number-fiction that the number of days in a week is seven is the fact that it is true in the world that there are seven days in a week, and it is the latter thing—what Yablo [22, p. 77] calls the real content—that we really assert. The idea is that whether or not a statement is in some sense correct can be determined by looking at its real content. One of the things that can make a statement correct is that it is literally true. But sometimes it can be correct to say something that is not literally true. For instance, suppose that we say of someone that he has a heart of gold. Whether or not this is a correct thing to say does not seem to depend primarily on whether or not the person in question actually has a golden heart. What matters is rather if he is a good person or not. So the real content of “He has a heart of gold” can be expressed as “He is a good person”. It is the truth of the real content that makes the statement correct even though it is not literally true. According to the object-fictionalist, something similar holds for mathematics. Even if there are no numbers that can make number statements literally true, there can still be truths underlying them that make them correct. On this view, the real content of 2 + 2 = 4 is that if there are two Fs and two Gs and no Fs are Gs, then there are four (F-or-G)s. Yablo [23, p. 231 f.] generalizes examples like this to show that the real content of any arithmetical truth is a logical truth. In this way the object-fictionalist can explain our impression that mathematics (at least arithmetic) is a priori and necessary. However, even if it withstands objections of the kind directed at meta-fictionalism, object-fictionalism has other weaknesses. Serious problems arise as soon as we start to use numbers to talk about numbers. It is certainly true in the number- fiction that the number of even primes is one. Now what is the real content of this statement that makes it a truth in the number-fiction? According to what has been said about object-fictionalism so far, the real content is supposed to be that there really is an even prime number. But this is something that the fictionalist denies. Instead, what the fictionalist would like to say is that there are no even prime numbers in the world. But if this were true, then it would be true in the number-fiction that the number of even primes is zero.6 statements that the platonist takes to be literally true. 6This objectionis referredto by Eklund[8] as “the Brock-Rosen objection”. Yablo [22, p. 78 f.] uses the more colorful label “the Bomb”.

7 2 Yablo’s ﬁguralism

Yablo [22] presents a modified version of object-fictionalism that avoids the objection given above. He develops his account further by building on ideas of Wal- ton [18] concerning the connection between make-believe games and metaphorical or figurative speech. The resulting view is called figuralism. It is a kind of hermeneutic fictionalism according to which there are many and striking resem- blances between the way we talk about mathematical objects and the way we use metaphors.

2.1 Real content and different kinds of fiction Object-fictionalism allows numbers to be introduced as representational aids. The number-fiction can then help us to describe reality in a simpler and more efficient manner. But it seems that we can only use it to talk about objects that are not themselves part of the fiction, since it makes the arithmetical claim that the number of even primes is one incompatible with the nominalistic claim that the number of even primes is zero. In order to solve this difficulty we must let numbers function not only as representational aids but also as things-represented, i.e. as something we can talk about. As a first step, we need to recognize that we sometimes want to talk about numbers outside the number-fiction without this automatically having consequences for what is true in the fiction. For instance, the nominalist wants to be able to claim that there really are no even prime numbers. But what happens then with the claim made within the number-fiction that the number of even primes is one? What real content can make this claim a true statement in the number-fiction? As Yablo [22, p. 83] observes, a nominalist will typically want to speak about numbers both in a disengaged and an engaged manner. It is the disengaged (from the number-fiction) nominalist that holds that the number of even primes is zero, whereas the nominalist who is doing mathematics and claims that the number of even primes is one is engaged (in the number-fiction). Hence, there is a desire to give a fictionalist account that admits talk of numbers in both these ways; as they really are (on the nominalist’s view: non-existent) and as they are to be imagined. In order to meet this desire Yablo introduces a different kind of fiction, or, in his terminology, game. The kind of fiction that was inherited from the object- fictionalist view is called a basic game. The basic number-game is a game where numbers are used as representational aids to talk about the way things really are. Numbers can also be things-represented in the basic number-game as when the disengaged nominalist says that the number of even primes is zero. The new game is a parasitic game. It can be seen as an extension of the basic game in which numbers also can be used as representational aids to talk about numbers

8 as they are to be imagined within the basic game. This gives the nominalist the opportunity to engage fully in mathematical discourse and say for instance that the number of even primes is one. However, there still seems to be an important piece missing, namely an answer to the question about real content. What is it that is really asserted and that makes it all right to claim within the parasitic number-game that the number of even primes is one? Yablo’s answer comes close to what the meta-fictionalist would say, but without explicitly bringing up the number-fiction: The assertion that is really made when it is claimed that the number of even primes is one is that there is a single even prime number among the numbers as they are to be imagined. Here we see that notion of real content is perhaps more complex than it first ap- peared. According to Yablo, there are two things that can make real content real. It can have objectual reality, i.e. be real in the sense that it is about real objects. This is the kind of real content that is emphasized by the object-fictionalist. But clearly the notion of objectually real content cannot be applied to statements about numbers as they are to be imagined. What makes such a statement’s real content real is that is has assertional reality, i.e. it is something that (unlike the statement’s literal content) is really asserted.7

2.2 Metaphors and make-believe games Yablo wants to argue that his fictionalist account of mathematics is hermeneutic, i.e. that it gives an accurate description of how mathematical discourse works. To make this case a convincing one, Yablo must explain how it can be that nobody seems to be aware of the game playing that supposedly goes on every time we talk about mathematical objects. He does this by claiming that mathematical discourse is analogous to figurative or metaphorical discourse and that metaphors often can be understood in terms of (more or less) unconscious games of make-believe. The possibility of a connection between metaphors and make-believe games is explored by Walton [18]. His suggestion is that the utterance of a metaphorical or figurative statement functions as an introduction to or reminder of some game of make-believe. Such a metaphor game is typically prop oriented, which means that our primary interest lies not in actually playing the game, but rather in how the game can be used to describe and give information about game-independent things. The metaphor lets us imagine something as a prop in a game and invites us to consider what this thing must be like in order to fit into the game. For instance, what we are after when we say that someone is a rock is in general not to initiate a game of make-believe in which this person plays the part of a rock. But maybe we are suggesting that he has properties (of being steady, supportive, etc.) that would

7See Yablo [22, p. 85].

9 make him suitable for the part if the game was to be played. As Yablo [24, p. 98] puts it: “A metaphor on [Walton’s] view is an utterance that represents its objects as being like so: the way that they need to be to make the utterance ‘correct’ in a game that it itself suggests.” Walton also discusses to what extent we are or need to be aware of the make- believe game that is implied by a metaphorical utterance. He argues that some metaphors may correspond to very simple games generated by only one or a few principles. But quite often it seems that we are unable to specify exactly how a metaphor game is built up. Walton [18, p. 81] writes: “What metaphors do, in many cases, is to activate relevant dispositions or abilities, rather than to make us aware of the principles of generation.” This kind of account indicates that we can understand and appreciate metaphors without at all participating in the games they imply. Perhaps it is not even necessary to recognize that there is a game that could be played.

2.3 Mathematics as prop oriented make-believe The figuralist view of mathematics that Yablo proposes incorporates large parts of Walton’s theory of metaphors as prop oriented make-believe games. The reason that we never think of mathematics as a game is that it is a prop oriented game; we tend to look away from the game itself and focus instead on what it tells us about the world (in the case of applied mathematics), or about mathematical objects as they are to be imagined (in the case of pure mathematics). Another, related, reason might be that we often confuse “literal” meaning with “ordinary” or “standard” meaning. When someone says that 7 + 5 = 12 in an everyday context, the most natural way to think about this is probably not as a statement about the properties of some abstract entities, but as a statement of the fact that if we have seven things and another five things and put them together, then we have twelve things. The real/metaphorical content comes to mind so quickly that we never look for any other content, and so we become convinced that the truths of arithmetic are literally true.8 Our tendency to take the metaphorical content of mathematical statements to be their literal content is reinforced by the circumstance that we sometimes cannot express literally the things that we say metaphorically using mathematics. Math- ematical objects are sometimes representationally essential metaphors. As an example, Yablo [23, p. 231] takes a case where we want to express that there are more sheep than cows in a field. The easy way to do this is to say that the number of sheep is greater than the number of cows. But suppose now that we want to express the real content of this literally without any reference to mathematical

8Cf. Yablo [21, p. 223 f.].

10 objects. It turns out that this requires an inﬁnite disjunction, i.e. a sentence that we cannot complete: “There is one sheep and no cows or there are two sheep and no cows or there are two sheep and one cow or ...” This can also be part of an explanation of why we came up with the idea of mathematical objects in the ﬁrst place. Of course, as Yablo [20, p. 301] notes, to argue that mathematics could be interpreted as metaphorical make-believe is not the same as showing that it is metaphorical make-believe. A conclusive argument for this is not given by Yablo, but he does present quite a long list of, as he calls them, “suggestive similarities” between objects that are obviously metaphorical, and “platonic” objects such as numbers or functions.9 Here are a few examples:

Insubstantiality: All properties of both metaphorical and mathematical objects seem to be captured by what follows from our conception of them. We do not make investigations into what stomach-butterflies eat or what color they are, because the only thing there is to know about stomach-butterflies is that they are part of what it feels like to be nervous. Similarly, we do not attribute properties to natural numbers unless they follow from Peano’s axioms. Translucency: Reference to both metaphorical and mathematical objects tends to go unnoticed; we “see through” it. If we are told that someone has a heart of gold, we are unlikely to stop and think about this strange metal organ—we simply hear that this is a good person. In the same way, if we are told that the number of polar bears is decreasing, we are unlikely to stop and think about this strange number that is getting smaller—we simply hear that there are fewer polar bears now than before. Irrelevance: The phenomena that we use metaphorical or mathematical objects to explain would occur even if the objects were not available. That there really are no butterflies in my stomach does not stop me from feeling nervous. And even if there are no one-to-one functions, there are still as many knives as there are forks on my dinner table.

2.4 Figuralism and the existence of mathematical objects Fictionalism and ﬁguralism have been presented here as something a nominalist might consider as a way of avoiding ontological commitment to abstract objects. However, it is not the traditional ontological issues that Yablo focuses on in his arguments for ﬁguralism: “It is not that Xs are intolerable, but that when we

9Other examples of platonic objects are possible worlds, models, and properties. Cf. Yablo [20, p. 277].

11 examine X-language in a calm and unprejudiced way, it turns out to have a whole lot in common with language that is fictional on its face.”10 In other words, Yablo does not rule out the possibility that numbers and other mathematical objects actually exist. Rather than being a convinced nominalist, he takes an agnostic standpoint concerning the existence of abstract objects. The interpretation of a mathematical metaphor (or any metaphor) is not fixed in advance. Yablo [20, p. 299 f.] describes the utterance of a metaphorical statement as a kind of “invitation” to look for a suitable game of make-believe where the metaphor makes sense. Should it be the case that mathematical objects really exist, then the most suitable “game” to play with the mathematical metaphors might be the “null game”, i.e. the one where we interpret metaphors as meaning what they literally say. Thus, how a metaphor is best understood is not something that the speaker can or needs to decide. Exactly what is said is to some extent indeterminate. This view puts Yablo in a position to question the Quinean view that our ontological commitments can be determined by looking at how our best scientific theories are formulated. The idea is that Quine’s criterion of ontological commitment cannot be used as long as we lack another criterion, namely one that tells us how to distinguish figurative speech from literal. A natural way to draw such a distinction would be to use ontological facts. The literal interpretation is correct if the things we talk about really exist. Otherwise, what we say should be interpreted figuratively. But since Quine himself admits that figurative speech is not ontologically committing,11 it seems that we are trapped: we cannot tell what there is unless we know which statements are to be interpreted literally, and we cannot tell which statements are to be interpreted literally unless we know what there is.12 This failure of Quine’s criterion to resolve the question about the existence or non-existence of mathematical objects leads Yablo to the conclusion that no ultimate answer can be given. However, he does not take this to have any serious consequences for mathematical discourse. Talk of numbers etc. is “maybe- metaphorical”, i.e. to be taken literally if numbers etc. exist, and to be interpreted in accordance with some suitable game of make-believe otherwise. In either case, mathematical language is fine just the way it is; there is no need to make any

10Yablo [22, p. 87]. 11See e.g. Quine [13, p. 249 f.]: “The doctrine of ideal objects in physics is ‘symbolic’ [...] It is a deliberate myth, useful for the vividness, beauty, and substantial correctness with which it portrays certain aspects of nature even while, on a literal reading, it falsiﬁes nature in other respects.” Quine seems to take it for granted that these “myths” always can be recognized and paraphrased away when needed. Yablo’s point is that this may be impossible. 12See Yablo [19, p. 255 ff.]. The argument rests, of course, on the intuition that there are no promising alternatives to Quine’s criterion of ontological commitment and no ontology- independent way of drawing the distinction between literal and ﬁgurative speech.

12 changes or qualifications. It seems that when we do mathematics it is not very important to us if the objects we talk about exist or not. Yablo [20, p. 301] takes this to support his thesis that the metaphorical interpretation of mathematics is the correct one: “[T]he main way these metaphors reveal themselves to us is through our—otherwise very peculiar!—insouciance about the existence or not of their apparent objects.” A fictionalist view based on nominalistic concerns about the problematic nature of abstract objects could in principle be refuted by a conclusive argument for the existence of such objects. Yablo’s figuralism with its agnostic starting point appears much less vulnerable to this kind of argument since it does not explicitly deny that abstract objects exist. It seems that even someone who believes in the existence of mathematical entities could decide that figuralism gives the best account of how we talk about them. According to Yablo [22, p. 88] other kinds of fictionalism have chosen the wrong strategy for arguing against platonism/realism. Instead of emphasizing the advantages of their own theory, they have spent most of their energy elaborating on the negative sides of platonism, thereby making it seem as if the only way to argue convincingly for fictionalism is to make the alternatives look worse. Yablo’s own strategy is to argue that we can do just as much with the pretended mathematical objects of figuralism as we can with platonism’sreal ones. The only prima facie advantage of platonism over figuralism that he sees has to do with the relation between objects and objectivity. For instance, the existence of numbers seems to guarantee that there is an objective fact of the matter regarding which arithmetical statements are true and which are false. It is not at all clear that we can get that kind of stability without the actual numbers—how can we be certain that our mathematical work is correct if there is no external norm to compare it to? Yablo [22, p. 88 f.] argues that this supposed power of numbers to ensure the objectivity of arithmetic is an illusion. What matters is whether we have a determinate enough conception of the numbers. If our conception is such that anything that answers to it settles the truth-value of a given arithmetical statement in the same way, then the pretended numbers are quite sufficient. If, on the other hand, we have a conception of numbers that leaves some questions open in the sense that there can be one structure answering to our conception such that A and another such that not-A, then it seems we will have trouble recognizing the actual numbers even if they exist. In any case, the existence of the numbers does not contribute to the objectivity of arithmetic the way the platonist assumes.

13 3 Some objections

Fictionalism, and in particular the hermeneutic variety, is a controversial view and many objections have been directed at it. This section mentions just a few of them together with some possible ﬁctionalist/ﬁguralist responses.

3.1 Literalness and make-believe One large family of objections is centered around the hermeneutic fictionalist’s claim that the most common usage of many statements is such that they are not intended to be interpreted literally. This goes against the intuitions of e.g. Burgess [6, p. 26] who argues that to mean something non-literally takes a special effort and that it is not a plausible standpoint that there are words or sentences that are more often used non-literally than literally: “[T]he ‘literal’ interpretation is not just one interpretation among others. It is the default interpretation.” Similar remarks are made by Rosen & Burgess [15, p. 532 f.] when they claim that if a sentence is intended to be interpreted non-literally, this is something that is always clearly recognizable and can be explicitly pointed out if needed. If we for instance say about some very reliable person that he is a rock, it is possible (albeit very unlikely) that someone misinterprets us and objects that this must be wrong since rocks are really hard and cannot move. This kind of misunderstanding is easily resolved if we make it clear that we were not intending the statement to be taken literally. Now, the point that Rosen & Burgess want to make is that mathematical statements cannot be misunderstood in this way. As an example they take the statement “There are two abstract groups of order four” and ask what a “literalistic misconstrual” of it would look like. They imagine that someone might react to the statement by asking where these groups are to be found, what they are like intrinsically, and how they can be known to exist. But when it comes to what a mathematician would answer if confronted with questions like these, Rosen & Burgess find it implausible that there would be any mention of figurative speech or non-literal interpretations in the reply. More likely, it would run along the following lines: “Groups don’t exist in some special location. They’re abstract. [. . . ] I know abstract groups exist because I can describe concrete instances of them, for instance, the integers modulo 4 [...]”.13 The upshot of this argument is that there cannot be any literalistic misconstruals of mathematical sentences and that this in turn means that mathematical language is not figurative. It could be argued that the example with abstract groups is not quite represen- tative since abstract groups are abstractions made within mathematics. Questions

13Rosen & Burgess [15, p. 533].

14 about their nature and existence could therefore very well be interpreted as purely mathematical questions to which purely mathematical answers are given. For a practicing mathematician this would probably be the most natural interpretation and a reply like the one sketched by Rosen & Burgess would seem appropriate. But if the questions were not meant as purely mathematical, the reply can give rise to more questions, this time about the “concrete instances” of the abstract groups. Where are they located? How do we know that they exist? What the mathematician will do with questions like these that are no longer open to a purely mathematical interpretation seems much less obvious. Another objection to the claim that mathematical language is figurative is brought up by Stanley [17, p. 48 f.] who argues that this is empirically falsified by research in psychology concerning how persons with autism handle figurative language and games of make-believe. Such research indicates that the same mech- anisms are involved in playing games of make-believe as in understanding figurative language and that autistic persons have difficulties with both these things. However, they do not in general have problems with mathematical discourse to the extent that could be expected if there really was such a close relation between mathematical and figurative language as the figuralist claims. This means, according to Stanley, that a make-believe account of mathematics must be very carefully balanced; mathematics cannot be completely like make-believe since this apparently comes into conflict with empirical data, but the differences cannot be too big either or the comparison loses its point. Stanley [17, p. 50 ff.] also questions the motivation for hermeneutic fictionalism. On his interpretation, the hermeneutic fictionalist wants to “account for the ontological commitments the speaker believes she incurs when she endorses the truth of an utterance”. One of Stanley’s worries about this is that it seems to assume a very strong connection between a speaker’s believed ontological commitments and her actual ontological commitments. But if there is such a connection, then how can the fictionalist maintain that there is no similar connection between what we believe and what actually is the case when it comes to whether or not we are engaged in make-believe when we do mathematics? If anything, it should be the other way around: Whether or not we are pretending should be clear to us. What our ontological commitments are, on the other hand, might not be something that we are fully aware of. Yablo [22, p. 99 ff.] agrees with this intuition, but argues that it is compatible with fictionalism. To begin with, a distinction has to be made between commitments that we are not aware of having, and commitments that we are aware of not having. According to Yablo, it is only for the latter that Stanley’s formulation of the motivation for fictionalism is relevant. The idea is that someone can make a statement that seems to commit him to some kind of entity, but on careful reflection conclude that he does not believe in entities of this kind. A fictionalist

15 analysis can then explain how it is possible to make the statement and deny the entities without being inconsistent. However, this does not mean that the fictionalist account automatically applies to any commitment that we are unaware of having. We may have commitments that are consequences of other commitments but that we have not reflected on. Something similar to this holds for make-believe. For instance, we could make a statement and not think very much about whether we meant it literally or not, but if we find on reflection that we are not committed to what the statement literally means, then we might conclude that we were not speaking literally. Again, a fictionalist analysis in terms of make-believe could explain what we really meant, even if we were not deliberately engaging in make-believe when we made the statement.

3.2 The Irrelevance Problem As we have mentioned (see e.g. Section 2.4), Yablo’s own motivation for fictionalism is not based on ontological concerns about the nature of mathematical objects. What he finds problematic is rather that many sentences seem to commit us to their existence without really being about them. An example would be the sentence “The number of cats in the yard is two”. The fictionalist’s intuition is that this sentence is about cats. But on a literal reading it also commits us to the existence of the number two, despite the fact that we think there would be two cats in the yard even if there is no such thing as numbers. Rayo [14] calls this the Irrelevance Problem. Rosen & Burgess [15, p. 531 f.] reject the Irrelevance Problem altogether and argue that “The number of cats in the yard is two” actually is about the number two as well as about cats. Rayo [14], on the other hand, admits that there is a problem. However, he has an objection to Yablo’s solution. The fictionalist idea is that we can imagine that there are numbers and then use these imagined objects as representational aids that make it easier to express certain facts about the real world. Assertions that mention numbers do not commit us to the actual existence of numbers, since what we assert is not the literal content of the assertion but the real content. Yablo [23, p. 230 ff.] specifies how the real content of an arithmetical sentence corresponds to its literal content by characterizing a function that maps arithmetical sentences to sentences that express their real content. For instance, the real content of n = m is given by ∃n!x(Fx) → ∃m!x(Fx), i.e. if there are exactly n Fs, then there are exactly m Fs. Now, Rayo [14, p. 3] argues that a natural re- quirement is that the real content of any arithmetical sentence should be true if and only if it is true according to the standard, platonistic, interpretation. Further- more, we do not want any mathematical objects in our domain of quantification,

16 since this would commit us to their existence. Rayo [14, p. 8] notes that these conditions can be satisfied by Yablo’s function, but only if we have an infinite domain of quantification. We can see that this is necessary if we consider a situ- ation where there are only finitely many, say n, objects in the domain. Then for any F, ∃n+1!x(Fx) is false. Consequently, ∃n+1!x(Fx) → ∃n!x(Fx) is vacuously true. But this is supposed to be the real content of n + 1 = n, which is false under the platonistic interpretation. Thus it seems that the correspondence between real and literal content depends on whether or not there are infinitely many objects to quantify over. Rayo therefore claims that the fictionalist makes no real progress concerning the Irrelevance Problem: “[W]hether or not non-philosophers take an assertion of ‘∃n(n + 1 = n)’ to be false does not depend on whether they believe that the world contains infinitely many individuals, any more than it depends on whether or not they believe that the world contains numbers. So the Irrelevance Problem has not been solved; it has simply been relocated.” Yablo is clearly aware of this problem, but it appears that he is less strict than Rayo when it comes to which objects are let into the domain of quantification. He allows “imagined quantification” over imagined objects and sees no problem with a rule like the following (where *S* means that it is to be imagined that S, and F is a predicate that applies to imagined and/or concrete objects):14

(N) If*∃n!x(Fx)* then *there is a thing n=the number of Fs*. If we have access to a rule like (N), we no longer have to worry about running out of concrete objects to quantify over. As Yablo notes: “[(N)] gets us ‘all’ the numbers even if there are only ﬁnitely many concreta. 0 is the number of non- self-identical things, and k + 1 is the number of numbers ≤ k.” However, perhaps the method of “imagined quantiﬁcation” can be questioned.

4 Concluding remarks

There are many aspects of Yablo’s ﬁguralism that makes it an interesting position in the philosophy of mathematics. In particular, it addresses several important questions concerning the objectivity and usefulness of mathematics and suggests answers to them that do not presuppose the existence of objects that seemingly have nothing to do with the phenomena that we use mathematics to explain. Fig- uralism also has the advantage of being non-revisionary, i.e. it builds on an acceptance of actual mathematics and attempts to give an account of what actually takes place when we get involved in mathematical discourse. But, as the previous section shows, the view is not unproblematic. For instance, it is unclear how far the comparison between mathematical and ﬁgurative

14See Yablo [23, p. 231].

17 language can be stretched without being falsiﬁed by empirical data. There is also a question about the ontological status of ﬁctional objects and what makes them different from the “traditional” abstract objects that platonists assume that mathematics is about. These are some of the issues that need to be confronted in the future.

References

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18 [16] Shapiro, S. (ed.), 2005. The Oxford Handbook of Philosophy of Mathematics and Logic, New York: Oxford University Press. [17] Stanley, J, 2001. “Hermeneutic Fictionalism”, Midwest Studies in Philoso- phy 25 (1), 36–71. [18] Walton, K., 1993. “Metaphor and Prop Oriented Make-Believe”, in [12], 63–87. [19] Yablo, S., 1998. “Does Ontology Rest on a Mistake?”, Proceedings of The Aristotelian Society, Supplementary Volume, 72 (1), 229–262. [20] Yablo, S., 2000. “A Paradox of Existence”, in [9], 275–312. [21] Yablo, S., 2000. “Apriority and Existence”, in [4], 197–228. [22] Yablo, S., 2001. “Go Figure: A Path through Fictionalism”, Midwest Studies in Philosophy 25 (1), 72–102. [23] Yablo, S., 2002. “Abstract Objects: A Case Study”, Philosophical Issues 12, 220–240. [24] Yablo, S., 2005. “The Myth of the Seven”, in [12], 88–115.