DOES THE EXISTENCE OF MATHEMATICAL OBJECTS ... - In AAP

Australasian Journal of Philosophy Vol. 81, No. 2, pp. 246–264; June 2003

DOES THE EXISTENCE OF MATHEMATICAL OBJECTS MAKE A DIFFERENCE? Alan Baker In this paper I examine a strategy which aims to bypass the technicalities of the indispensability debate and to offer a direct route to nominalism. The starting-point for this alternative nominalist strategy is the claim that—according to the platonist picture—the existence of mathematical objects makes no difference to the concrete, physical world. My principal goal is to show that the ‘Makes No Difference’ (MND) Argument does not succeed in undermining platonism. The basic reason why not is that the makes-no-difference claim which the argument is based on is problematic. Arguments both for and against this claim can be found in the literature; I examine three such arguments, uncovering flaws in each one. In the second half of the paper, I take a more direct approach and present an analysis of the counterfactual which underpins the makes-no-difference claim. What this analysis reveals is that indispensability considerations are in fact crucial to the proper evaluation of the MND Argument, contrary to the claims of its supporters.

Debates over the pros and cons of the Indispensability Argument have generated substantial literature over the past two decades. The complex and inconclusive nature of these often technical discussions has prompted some philosophers with nominalist sympathies to seek alternative strategies for winning over their platonist opponents. In this paper I shall examine one particular such strategy which its supporters argue bypasses the technicalities of the indispensability debate and offers a direct route to nominalism. The startingpoint for this alternative nominalist strategy is the claim that—according to the platonist picture—the existence of mathematical objects makes no difference to the concrete, physical world. I shall refer to the argument for nominalism based on this claim as the ‘Makes No Difference’ (or MND) Argument. My own sympathies are broadly pro-platonist, and my principal goal in this paper is to show that the ‘Makes No Difference’ Argument does not succeed in undermining platonism. The basic reason why not is that the makes-no-difference claim which the MND Argument is based on is problematic. Arguments both for and against this claim can be found in the literature; I examine three such arguments in Sections III to V of the paper, uncovering flaws in each one. In Sections VI to X, I take a more direct approach and present an analysis of the counterfactual which underpins the makes-no-difference claim. What this analysis reveals is that indispensability considerations are in fact crucial to the proper evaluation of the MND argument, contrary to the claims of its supporters. My conclusions fall into two parts: if mathematics is dispensable for science then the existence of mathematical objects makes no difference to the concrete world. However, if mathematics is indispensable then there is no determinate answer to the question of 246

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Does The Existence Of Mathematical Objects Make A Difference?

whether the existence of mathematical objects makes a difference. Hence there is no clear route to nominalism based on the makes-no-difference claim. I. The ‘Makes No Difference’ Argument The first task is to formulate a specific version of the ‘Makes No Difference’ Argument. Versions of the claim that the existence of mathematical objects makes no difference to the concrete world can be found in many places in the philosophical literature; Since sets are not supposed to be part of the world’s spatio-temporal causal nexus, that nexus would be exactly as it is whether sets existed or not . . . . [Horgan 1987: 281–2] Since mathematical objects are acausal, the existence or non-existence of mathematical objects makes no difference to the actual arrangement of concrete objects. [Cromwell 1992: 80] Perhaps the most prominent recent exponent of this view is Mark Balaguer, in his book Platonism and Anti-Platonism in Mathematics; [E]ven if there were no such things as abstract objects, science could be practiced exactly as it is right now (assumptions of abstracta and all) with exactly the same results. If there were never any such things as abstract objects to begin with, the physical world would be exactly as it is right now, and we would be receiving the very same empirical data we are presently receiving. [Balaguer 1999: 113] From this basic claim, several philosophers—including Balaguer—have been prompted to draw a sceptical and broadly anti-platonist conclusion; The basic reason for resisting abstract [objects] is that the world we can know about would be the same whether or not they existed. [Ellis 1990: 328] [T]his suggests that fictionalism . . . [is] . . . just as plausible as platonism . . . [Balaguer 1998: 137] Taken together, these remarks suggest the following argument against platonism; The Makes No Difference (MND) Argument (1) If there were no mathematical objects then (according to platonism) this would make no difference to the concrete, physical world. (2) Hence (on the platonist picture) we have no reason to believe in the existence of mathematical objects.

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II. The ‘No-Difference’ Hypothesis: Three Arguments Henceforth I shall refer to the premise of the MND argument as ‘(No-Difference)’. At first blush, (No-Difference) seems hard to resist. Surely it is obvious that mathematical objects—if acausal and non-spatiotemporal—make no difference to the arrangement of the concrete world? This attitude is borne out in the literature, (No-Difference)-style claims are often asserted but rarely argued for. Proponents of the MND argument answer the above question in the negative; supporters of the ‘no’ position—as we have seen— include Mark Balaguer, Terence Horgan, and Jodi Azzouni. A few philosophers have also defended the ‘yes’ position, among them Colin Cheyne, Charles Pigden, and Mark Colyvan. Arguments for either position are few. Where such arguments are given, the two sides tend to support their respective intuitions by appealing to quite distinct features of mathematics. Supporters of the ‘no’ answer point to the acausal nature of mathematical objects, arguing that this clearly implies that mathematical objects make no difference to the concrete world. Supporters of the ‘yes’ answer point to the putative indispensability of mathematics for science, arguing that this clearly implies that mathematical objects do make a difference to the concrete world. My own view is that neither side has succeeded in making its case. I have seen no evidence to date of any convincing argument from acausality to the ‘no’ position, nor of any convincing argument from indispensability to the ‘yes’ position. My first task, therefore, will be to examine three arguments that have been offered in the literature (two for the ‘no’ position and one for the ‘yes’ position) and to provide backing for my claim that none of them ultimately is successful.

III. An Argument for the ‘No’ Position: The Blinking-Out Argument Azzouni and Balaguer have offered two formulations of the same quick argument for (NoDifference), based on the acausal nature of mathematical objects. This argument is put in terms of what would (or wouldn’t) happen if mathematical objects suddenly ceased to exist. Azzouni, for example, asks us to [i]magine that mathematical objects ceased to exist sometime in 1968. Mathematical work went on as usual. Why wouldn’t it? [Azzouni 1994: 56]1 Meanwhile, Balaguer writes [I]f all the objects in the mathematical realm suddenly disappeared, nothing would change in the physical world. [Balaguer 1998: 132] 1

It should be noted that in more recent work, Azzouni has explicitly distanced himself from a strict ontological interpretation of this ‘blinking-out’ thought experiment [Azzouni 2000].

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This is less an argument than an appeal to raw intuitions. It is not relied heavily upon by either author; however, it is important because the underlying intuition is, I think, both widely held and potentially misleading. The intuition to which Azzouni and Balaguer appeal is the following; since mathematical objects are by hypothesis acausal, if they were suddenly to blink out of existence then this would have no ‘knock-on’ effects on the concrete, physical world. A moment’s reflection, however, shows that this line of reasoning has no real force. Central to the platonist account is that mathematical objects are non-spatiotemporal; and it is incoherent to hypothesize that an atemporal object suddenly ceases to exist in 1968. The ‘blinking-out’ argument focuses on the acausality of abstract mathematical objects, but in doing so it implicitly attributes temporal properties to them.2 This sort of looseness is symptomatic of a general tendency to view abstract objects as akin to ultra-remote, ultra-inert concrete objects. For there is nothing incoherent about imagining concrete objects blinking out of existence. Nor is it only temporal properties that get inadvertently attributed to mathematical abstracta. A similar phenomenon occurs with spatial properties. It is all too easy, for example, to slip from talking of mathematical objects as being non-spatiotemporal to talking of them as existing outside of space-time. But ‘outside’ is of course a spatial notion, hence it cannot legitimately be applied to abstract mathematical objects. Talking this way encourages the picture of mathematical abstracta existing in some distant realm, a realm which is further from us than even the remotest concrete object.3 An analogous slide occurs with respect to the acausal nature of abstract objects. This feature is often glossed by saying that abstract objects are not causally active, or—more equivocally—that they are ‘causally inert’. However, causal inertness—like remoteness—is a property which can be possessed (to varying degrees) by concrete objects. For example, some gases are classed as inert because they do not easily react with other substances. This again encourages a view of abstract objects that places them at the end of a continuum of cases incorporating successively more inert concrete objects. One moral to be drawn from this is that the (No-Difference) claim must be understood as a timeless counterfactual. The issue, in other words, is not whether the existence—here and now—of mathematical objects makes a difference, but whether their existence—in the unrestricted and timeless sense appropriate to mathematical objects—makes a difference. The blinking-out argument, even if it were coherent, provides no quick route to establishing the ‘no’ position. The principal reason is that it appeals to intuitions which are derived from, and thus only non-question-beggingly have force for, concrete objects.

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One could imagine versions of platonism according to which mathematical objects have temporal properties, and thus can in principle blink out of (and into) existence. Such versions might be open to the Balaguer-Azzouni argument. Nonetheless, the standard platonist position is certainly that mathematical objects are neither spatial nor temporal, and this is a consensus with which both Azzouni and Balaguer explicitly agree. ‘If I assert details of the inhabitants of a distant planet but deny that I have any knowledge of those aliens, then there is no reason why my assertions should be regarded as anything more than idle fancies’ [Cheyne 1998: 34].

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IV. A Second Argument For the ‘No’ Position: The Irrelevance Argument In his 1998 book Platonism and Anti-Platonism in Mathematics, Balaguer presents a detailed formulation of an argument along the lines of the MND argument. In the process he defends a version of (No-Difference). This more careful defence of (No-Difference) runs as follows [Balaguer 1998: 133]; (1) Empirical science ‘knows’, so to speak, that mathematical objects are causally inert. (2) Hence science does not assign any causal role to mathematical objects. (3) Hence science ‘predicts that the behavior of the physical world is not dependent in any way upon the existence of mathematical objects’. (4) ‘[T]his suggests that what empirical science says about the physical world . . . could be true even if there aren’t any mathematical objects’. I have two basic worries about this argument; the first concerns the move from (2) to (3), and the second concerns the force of the conclusion, (4). Steps (1) and (2) conform to the standard platonist picture of mathematical objects as acausal. Balaguer formulates this in terms of what he calls the ‘Principle of Causal Isolation’ (PCI), which says that there are no causal interactions between mathematical and physical objects [ibid.: 110]. However in step (3), Balaguer slides from the claim that the physical world is not causally dependent on the existence of mathematical objects to the stronger claim that it is not dependent ‘in any way’ on their existence. In his initial explication of the argument, the only real motivation Balaguer gives for this stronger claim is the ‘blinking-out’ argument which we examined, and rejected, in the previous section. After reiterating this claim that the sudden disappearance of all mathematical objects would change nothing in the physical world, he writes [T]his suggests that if there never existed any mathematical objects to begin with, the nominalistic content of empirical science could nonetheless be true. [Ibid.: 132] The quoted claim is a timeless conditional, which is a step in the right direction given the remarks at the end of the previous section. However, something other than the ‘blinkingout’ argument is needed to make this claim plausible if it is to be used as support for (NoDifference). Interestingly, Balaguer does explicitly address the worry that he is sliding from mathematical objects having no causal role to their having no role at all. However his remarks in response to this worry do little to advance his argument. He considers the following sentence; (A) The physical system S is 40 degrees Celsius. Balaguer concedes that this sentence expresses a ‘mixed’ mathematical/physical fact, namely that S stands in the Celsius relation to the number 40. However he argues that

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since 40 isn’t causally relevant to S’s temperature, it follows that if (A) is true, it is true in virtue of facts about S and 40 that are entirely independent of one another. [Ibid.: 133] In other words, Balaguer seems once again to slide from the causal irrelevance of mathematical objects to their irrelevance tout court! My second worry about Balaguer’s argument concerns the strength of its conclusion. Put bluntly, he is not arguing for (No-Difference) but for a claim that is significantly weaker. The final step, (4), of Balaguer’s main argument is not that mathematical objects make no difference, but that they might make no difference. This is reiterated in the second counterfactual claim quoted above; he writes that if there were no mathematical objects then the nominalistic content of science could be true. But ‘could’ is not the same as ‘would.’ The reason for this weakening is tied to the broader, anti-metaphysical project of Balaguer’s book. His ultimate aim is not to argue for the truth of mathematical fictionalism, but merely to show that it is a ‘coherent and sensible’ and a ‘tenable’ position [ibid.: 131, 134]. This weaker goal is obviously less threatening to platonism.4 Moreover, even if it can be established, it does not entail (No-Difference). It is worth examining precisely why not. Balaguer’s strongest formulation of his conclusion is the following claim. [S]ince PCI is true, this nominalistic content [of science] could very easily be true, even if there were no such things as abstract objects. [Ibid.: 134] It is unclear what is the modal force of ‘could very easily be true’. In any case, even if we accept the above claim, it does not follow that the existence of abstract mathematical objects makes no difference to the physical world. Consider the following analogy. Three children are playing with matches in an abandoned building. Child A accidentally lights and drops a match and the building burns to the ground. Clearly the lighting of A’s match caused the building to burn down. And this remains plausible despite our intuition that if A’s match had failed to light then the building ‘could very easily’ have burnt down anyway (say if B or C had accidentally lit their matches). Rephrasing this point, the ‘could very easily’ claim does not undermine the thesis that the lighting of A’s match made a difference to the fate of the building. So why should Balaguer’s analogous claim quoted above undermine the thesis that the existence of abstract mathematical objects makes a difference? To summarize, although there is much of interest in Balaguer’s discussion of mathematical fictionalism, I see no compelling argument for the central claim that the physical and mathematical realms are totally independent. The case for asserting (No-Difference)

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It should be noted that later in the same book, Balaguer strengthens this goal somewhat, from establishing that fictionalism is tenable to establishing that it is ‘quite plausible’ [ibid.: 134]. In this vein, he writes: [E]mpirical science’s picture of the physical world could be accurate, even if there are no such things as mathematical objects. . . . And this is why I think that fictionalism and nominalistic scientific realism are plausible views, why I think they are just as plausible as platonism and ordinary scientific realism [ibid.: 141].

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merely on the basis of the acausal nature of mathematical objects has yet to be convincingly made. V. An Argument for the ‘Yes’ Position: Changing Mixed Mathematical/Physical Facts I turn now to one of the few arguments in the literature directed at undermining the (NoDifference) claim. Colin Cheyne and Charles Pigden have recently offered an argument from the presumption that mathematics is indispensable for science to the claim that mathematical objects make a difference to the concrete world. Their argument proceeds by way of an example: [S]uppose the fact that there are three cigarette butts in the ashtray causes Sherlock to deduce that Moriarty is the murderer, and that if there had been more or fewer butts he would have deduced otherwise. . . . Suppose that the number three is an indispensable constituent of that fact. . . . Its being a constituent of the fact makes a . . . difference. If the number two or the number four were in its place, the effects would differ. [Cheyne and Pigden 1996: 642] 5 The fact that there are three cigarette butts in the ashtray is what might be called a ‘mixed’ mathematical/physical fact: it states that a particular relation holds between the arrangement of cigarette butts in the ashtray (a physical configuration) and the number three (a mathematical entity). Cheyne and Pigden consider a counterfactual situation in which this mixed fact changes, holding instead—for example—between the cigarette butts and the number two. This change clearly involves a change in the physical arrangement of cigarette butts. Does this show that mathematical objects make a difference to the physical world? I claim that it does not. Consider the following threefold classification of facts: first, purely mathematical facts such as, ‘2 + 3 = 5’, or, ‘7 is a prime number’; second, purely physical facts such as, ‘Chlorine reacts with oxygen’, or, ‘Mars orbits closer to the sun than Jupiter’; third, mixed mathematical/physical facts such as the cigarette butt example considered above, or more complex facts such as, ‘The phase-space of particle, p, is an infinite-dimensional Hilbert space’. In the above example, the supposition that there are two cigarette butts in the ashtray rather than three entails a change in the mixed mathematical/physical facts. However purely mathematical facts such as ‘2 + 3 = 5’ remain as they are; no revisions to pure mathematics need to be made in order to accommodate this change in the number of cigarette butts in the ashtray. If Cheyne and Pigden’s argument establishes anything, therefore, it concerns what would be the case if the mixed mathematical/physical facts were different. Thus it is misleading to paraphrase their conclusion by saying that the existence of mathematical objects make a difference; rather it is the relations between these mathematical objects and the concrete world that make a difference. 5

Later in the same paper they state their general conclusion explicitly, viz., ‘The indispensability argument claims that numbers, sets, etc., make a difference (which is why they cannot be dispensed with)’ [Cheyne and Pigden 1996: 644].

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The distinction between the Cheyne-Pigden scenario and our discussion of the status of the (No-Difference) hypothesis is that what we are interested in is changes in the purely mathematical facts. The scenario in which there are no mathematical objects is a special case of such a change. To show that the existence of numbers makes a difference to the concrete world, using Cheyne and Pigden’s example, one would need to show that the physical configuration of cigarette butts in the ashtray would be different if there were no numbers at all. This they have not shown, nor do I see any straightforward way of modifying their argument to do so.6 My objection to the Cheyne-Pigden argument can also be formulated in more general terms. Consider any two distinct realms of facts, A and B. However disconnected these two realms are, there will always be some ‘mixed’ facts connecting them (albeit often trivial ones). For example, a mixed fact connecting the realm of astronomical facts with the realm of facts about me might be that the number of coins in my pocket is in 1–1 correspondence with the number of planets orbiting a particular distant star. Now if we stipulate that the pure A-facts are kept fixed, then the following counterfactual is true: if the mixed A/B-facts were different, then the B-facts would be different. Thus, in the above example, if the 1–1 correspondence were not to hold and yet the number of coins in my pocket remained the same, then the number of planets orbiting the distant star would be different. This is true regardless of the genuine connections or lack of connections between A and B. Thus its truth has no bearing on the issue of whether A-facts make a difference to B-facts. The Cheyne-Pigden argument has this structure; thus their conclusion about (pure) mathematical facts making a difference does not follow from the (mixed) counterfactual claim which they establish. Mark Colyvan [1998] has also criticized the Cheyne-Pigden argument. Their argument can be divided into two sub-arguments; the first defending the claim that the existence of mathematical objects makes a difference to the concrete, physical world, and the second proceeding from this claim to the conclusion that mathematical objects have causal powers. My criticisms have been directed at the first of these sub-arguments. Colyvan focuses his attack on the second sub-argument, but he agrees with Cheyne and Pigden on the force of their first sub-argument. Indeed, Colyvan writes that ‘platonists will have no disagreement with [the claim that “mathematical entities make a difference”]!’ [ibid.: 117], and offers an example of his own in support of this thesis. [I]f the angle sum of a triangle is p radians, the space is (locally) Euclidean and so massive bodies experience no net force; if the angle sum is not p radians the space would be non-Euclidean and hence any massive body would be experiencing a net force. Thus, if there were a change in the angle sum of a triangle, the future light cone of the world would be different. . . . [Ibid.: 117]

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Consider the Cheyne-Pigden argument modified for the case where there are no numbers: ‘Suppose that the number three is an indispensable constituent of the fact that there are three cigarette butts in the ashtray. Its being a constituent of the fact makes a difference. If there was no number in its place, the effects would differ.’ Assessing whether this latter conclusion follows seems much less clear than for the original version of the argument. I will return to consider this issue in Section VI.

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Insofar as this example has force, it stems from an equivocation over the term ‘triangle’. The first sentence makes sense only if ‘triangle’ refers to triangles that are spatiotemporally located. Thus when Colyvan entertains the hypothesis that the angle sum of a triangle is p radians, this refers to a mixed mathematical/physical fact—roughly that the geometry of physical space is modelled by a certain pure mathematical theory, namely Euclidean geometry. It is true that if there was a change in this mixed fact, and triangles in space-time did not have an angle sum of p radians, then ‘the future light cone of the world would be different’. However, for the anti-(No-Difference) conclusion to follow, Colyvan needs to establish that changes in the pure facts about what mathematical objects exist make a difference to the physical world. Assume that there are, as postulated by Euclidean geometry, abstract mathematical triangles whose angles sum to p radians. What if no such triangles existed? If we could show that this scenario would result in changes to the physical world, then Colyvan’s argument would go through. But—as it stands— Colyvan’s argument is no more successful than the Cheyne-Pigden sub-argument it is designed to support.7 VI. What If There Were No Mathematical Objects? My goal in Sections III to V was to establish the failure of existing arguments to settle the truth or the falsity of (No-Difference). In this section I attempt a direct analysis of the counterfactual claim implicit in (No-Difference). In so doing, I hope to show how the status of this counterfactual is tightly bound up with issues concerning indispensability. The basic result may be summarized as follows. If mathematics is dispensable for science, then (No-Difference) is true. However, if mathematics is indispensable then the status of (No-Difference) is unclear, perhaps irredeemably so. Hence there is an important sense in which the ‘Makes No Difference’ (MND) Argument cannot be used by nominalists to circumvent the indispensability debate. My original formulation of the MND argument made no mention of the indispensability or otherwise of mathematics for science, nor did the quotes from Balaguer and 7

Stuart Cornwell [1992: 86 n. 13] sketches a second argument against (No-Difference) which focuses on a change in the pure mathematical facts rather than the mixed mathematical/physical facts. Assume that there are exactly 3 apples and 4 oranges on the table. Both of these (mixed) facts can be expressed nominalistically in a familiar fashion using quantifiers and identity. These correspondences allow the platonist to make use of the (pure) mathematical theory of Peano arithmetic to deduce that there are 7 pieces of fruit on the table. Next consider an alternative scenario in which 0 to 4 are the only numbers that exist, and they obey the addition rules of modulo 5 arithmetic. Cornwell argues that applying modulo 5 arithmetic instead of normal Peano arithmetic in the above example yields the consequence that there are exactly 2 apples-or-oranges on the table. Does Cornwell’s argument suffice to show that the existence of mathematical objects does make a difference to the concrete world, i.e., that (No-Difference) is false? I think that the answer manifestly is that it does not. Note that the above application of modulo 5 arithmetic leads not merely to an intuitively peculiar conclusion but to formal inconsistency, for it is possible using logic alone to derive the claim that there are exactly 7 apples-or-oranges on the table. Clearly there cannot be both exactly 2 and exactly 7 apples-or-oranges on the table. Now there is no reason to suspect that modulo 5 arithmetic, considered as a pure mathematical theory, is inconsistent. Hence the inconsistency in the above example must arise from the rules for applying modulo 5 arithmetic to the concrete world.

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other authors on which it is based. Indeed Balaguer, writing of his version of (NoDifference), asserts that ‘the whole topic of indispensability is irrelevant to the question of whether [this hypothesis] is true’ [1998: 140]. If the status of the MND argument were in fact independent of the indispensability issue then this would allow its proponents to bypass the technical wrangles over nominalistic alternatives to science which characterize many of the arguments against indispensabilist platonism. I shall argue, however, that the purported independence of the MND argument from indispensability issues is illusory. It is important to get clear about the dialectical structure of this debate. As I am reading it, the MND argument rests on the following two basic claims; (No-Difference) If there were no mathematical objects then (according to platonism) this would make no difference to the concrete, physical world. (Irrelevance) (No-Difference) holds true regardless of the indispensability or otherwise of mathematics for science. Each of these claims affects the assumptions which can be made in evaluating the MND argument. Consider first (No-Difference). Trivially, things which do not exist can make no actual difference. Hence (No-Difference) only makes a substantive claim if it holds true under the platonist assumption that mathematical objects do exist (i.e., if mathematical objects exist then their existence makes no difference). I shall therefore be adopting this assumption for the purposes of the analysis which follows. It is worth emphasizing that in the present context this assumption does not beg the question against the nominalist. Rather the structure of the MND argument is similar to that of a Reductio Ad Absurdum. Its proponents assume platonism, show how (No-Difference) follows from this assumption, and conclude—using the MND argument—that the initial assumption of platonism has no rational basis. What about (Irrelevance)? If this claim is correct then (No-Difference) is true even if mathematics is indispensable for science. Hence it is not question-begging against the supporter of MND to assume indispensability when evaluating the truth of (NoDifference). In any case, the MND argument would lose much of its strategic interest if its soundness were to hinge on indispensability considerations. The chief motivation for formulating MND in the first place was to provide an argument for nominalism that could bypass the inconclusive debate over indispensability. It is important to stress that these two assumptions—that mathematical objects exist, and that mathematics is indispensable for science—are made for quite independent reasons, connected with (No-Difference) and with (Irrelevance) respectively. Furthermore, we are nowhere assuming that indispensability entails existence. Even supporters of the Indispensability Argument concede that it is logically possible for mathematics to be indispensable for describing and explaining the concrete world and yet for no mathematical objects to exist. The mathematical existence assumption is being made in order to give content to (No-Difference), and not because some strong realist interpretation of indispensability is being smuggled in. To summarize, the question I am addressing in the following sections of the paper is the following; if mathematical objects exist, and if mathematics is indispensable for science, does the existence of these mathematical objects make a difference to the concrete, physical world? In other words, given these two assumptions, is (No-Difference) true?

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VII. Counterfactuals Involving Concrete Objects Under the assumption that mathematical objects exist, the (No-Difference) hypothesis makes a counterfactual claim. Mathematical counterfactuals such as this take us into unfamiliar territory, hence raw intuitions are of little help. Nor have the various arguments surveyed in Section II settled the issue one way or the other. But perhaps the counterfactual in (No-Difference) is amenable to direct philosophical analysis. The idea would be to take as a starting-point our procedure for evaluating more familiar kinds of counterfactuals, concerning the non-existence of concrete objects, and then to apply the same general procedure to the evaluation of the more exotic mathematical counterfactual in (No-Difference). The most popular contemporary approach to the analysis of counterfactuals utilizes the apparatus of possible worlds developed by David Lewis. The leading idea is that a counterfactual statement is true if and only if its consequent is true in the closest possible world, or worlds, in which its antecedent is true. For this to have real content much more needs to be said about what factors influence closeness, or similarity, between possible worlds. In his early work on counterfactuals, Lewis distinguishes two basic respects of similarity and difference between worlds: laws and facts. He refined this distinction in later work to produce an analysis in which different respects of similarity receive different ‘weights’ when calculating the overall similarity between worlds. In assessing which relevant possible world is closest to the actual world, Lewis proposes the following system of priorities: (1) It is of the first importance to avoid big, widespread, diverse violations of law. (2) It is of the second importance to maximize the spatiotemporal region throughout which perfect match of particular fact prevails. (3) It is of the third importance to avoid even small, localized, simple violations of law. (4) It is of little or no importance to secure approximate similarity of particular facts, even in matters that concern us greatly. [Lewis 1979: 472] The easiest way to get a feel for Lewis’s analysis is by means of an example. Consider a comparatively straightforward case involving concrete objects. Assume that we postulate the existence of a black hole, B, at some specified point in space, in order to explain certain deviations in the light reaching us from distant galaxies. Consider the following claim; (H)

(According to our current scientific theories) the existence of black hole, B, makes a difference to what we observe.

What grounds our belief in the truth of (H)? As with the mathematical object case (which involved the contrary claim that mathematical objects make no difference), (H) can be rewritten so as to make its counterfactual content explicit; (H*) If black hole, B, did not exist then (according to our current scientific theories) our observational data would be different.

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The natural way to evaluate (H*), I would suggest, is to imagine B suddenly blinking out of existence at some recent time, t. Would our present observational data be different? If we imagine photons leaving the distant galaxies—under the counterfactual supposition that B has blinked out of existence—then their path towards Earth is no longer distorted because there is nothing at the position previously occupied by B to cause such a distortion. Hence (H*) is true; the existence of B does make a difference to what we observe.8 Applying Lewis’s analysis to the above example bears out this informal evaluation. Why, for instance, is the blinking-out possible world in the black-hole example taken to be the closest relevant world? In the counterfactual world the sudden blinking-out of the black hole B at time t involves a ‘small, localized violation of law’. However, this difference is outweighed by there being perfect match of particular fact up to time t, and no widespread violations of law. As Lewis himself puts it, ‘a lot of perfect match of particular fact is worth a little miracle’ [ibid.: 469]. VIII. Counterfactuals Involving Mathematical Objects What happens when we apply Lewis’s analysis to the mathematical counterfactual in (NoDifference)? The antecedent of this counterfactual implicitly narrows the focus to worlds in which there are no mathematical objects. Consider that possible world, call it PW, which matches the actual world in respect of all pure physical facts but contains no mathematical objects. The (No-Difference) counterfactual is true only if PW is the closest of the mathematical-object-free possible worlds to the actual world. Is PW is closest to the actual world according to Lewis’s system of priorities? By stipulation, PW matches the actual world with respect to all purely physical matters of fact, and PW contains no mathematical objects.9 The two most important factors—according to Lewis’s analysis— are the general degree of matching between PW’s laws and the actual laws, and the spatiotemporal extent of perfect match between particular facts in PW and actual facts. It turns out that both these factors depend on whether mathematics is indispensable for science. If mathematics is indispensable for science then our best scientific theories make unavoidable reference to mathematical objects. These references may occur in articulating certain matters of particular concrete fact or in formulating certain scientific laws. Consider first the situation in which unavoidable reference to mathematical objects is made in describing matters of particular concrete fact. Imagine that a particular particle, p, is in a quantum-mechanical state which can only be described using a piece of mathematical apparatus, m*. By assumption, mathematical objects—including m*—exist in the actual world. However in PW the match between p and m* does not obtain, because m* 8

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The formulation of this black-hole example differs from that of the mathematical objects example in talking about changes in our observational data rather than changes in the concrete world. This is to avoid rendering claims about concrete objects making a difference trivial; if a particular concrete object did not exist then by stipulation the concrete world would be different. I take it that what we are saying in these sorts of cases is that the concrete object in question is making a difference to our actual or potential observations. Note that we are continuing to operate—for the purposes of this discussion—under the hypothesis that platonism is true.

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does not exist in PW. This is a particular mixed mathematical/physical fact which holds in the actual world but not in PW; occurrences of facts of this sort limit the spatiotemporal area of match of particular fact between PW and the actual world, cited by Lewis as priority (2) in assessing the closeness of PW. The second situation, in which mathematics is indispensable for formulating scientific laws, is more interesting. Assume, for example, that the basic laws of general relativity cannot be formulated without quantifying over mathematical objects. In this case, PW will differ from the actual world not only with respect to certain particular mixed mathematical/physical facts, but also with respect to certain laws. Any essentially mathematized law that holds in the actual world will fail in PW; moreover, any spatiotemporally restricted version of the law will also fail because of the non-spatiotemporality of mathematical objects. But this means that PW differs from the actual world in the respect that is most important according to Lewis’s analysis: PW will contain ‘big, widespread, diverse violations of laws’ that hold in the actual world. If, on the other hand, mathematics is dispensable for science then any reference to mixed mathematical/physical facts or laws can in principle be eliminated from our best descriptions of the world. Hence the best scientific theories in PW match our best scientific theories. Thus PW matches the actual world in all matters of fact that are crucial to science, and the laws of PW are the same as the actual laws of nature. Colyvan discusses this scenario with respect to a ‘Newtonian world’ [1998: 118]. Suppose that Hartry Field has succeeded in nominalizing Newtonian gravitational theory. If we lived in a world which was Newtonian rather than relativistic, then mathematics would be dispensable for science and PW would be closely similar to the actual world. The upshot, therefore, is that dispensability implies that (No-Difference) is true. We have seen how the indispensability or otherwise of mathematics for science affects the degree of similarity between the mathematical-object-free world, PW, and the actual world. But this is not yet enough to show that the indispensability issue is crucial to the evaluation of (No-Difference). Recall that we are proceeding under the assumption that mathematical objects actually exist. Even if mathematics is indispensable, and thus PW is remote from the actual world, it may nonetheless be the closest possible world that satisfies the antecedent of (No-Difference). If so, then (No-Difference) would be true regardless of whether mathematics is indispensable for science or not. IX. Essentially Mathematized Laws Can PW be shown to be the closest mathematical-object-free world to the actual world, even under the assumption that mathematics is indispensable for science? This is what is required to establish the truth of (No-Difference). Consider the following argument supporting a positive answer to this question. If mathematics is indispensable for science, and if mathematical objects actually exist, then some of the laws governing the actual world are essentially mathematized. Hence every possible world that satisfies the antecedent of (No-Difference) contains violations of actual world laws, since all such worlds lack the mathematical objects required to underpin the truth of these laws. Thus PW is no worse than any other mathematical-object-free worlds in this respect. However, PW matches the pure physical facts exactly, hence it is at least as close to the actual world as any of these other worlds. Call this the Closeness Argument.

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The Closeness Argument can be broken down into two key claims which I shall examine separately in this section and the next. (i) All essentially mathematized laws are violated in each mathematical-object-free world. (ii) Given possible worlds where actual laws are violated, the one with the most perfect match of pure physical facts is closest to the actual world. How plausible is the first of these two claims? A mathematized scientific law typically postulates a lawlike connection between some type of physical configuration and some mathematical object or structure. For example a law of quantum mechanics might postulate an isomorphism between the phase-space of some type of physical particle, p, and an infinite-dimensional Hilbert space. For some such mathematized law, L, let P be the physical configuration type, M be the mathematical structure, and R be the relation which L postulates to hold between them. If there are configurations of type P in the (mathematical-object-free) world, PW, then clearly L is violated in PW. After all, P is instantiated in PW but M is not, hence there cannot be a lawlike relation between them. What about possible worlds in which there are no P-configurations (i.e. L is physically uninstantiated)? Recall that the supporter of (No-Difference) claims that any possible world that contains no mathematical objects will be a world in which mathematized laws such as L are violated. This claim is true if L has the following logical form; L1 $x[Mx Ù "y(Py É Ryx)] L1 entails the bare existence claim, $xMx.10 Hence if there are no mathematical objects then L1 is false, and it is false even in those possible worlds with no P-configurations. However L1 is not the only option for paraphrasing L. Another way to represent the lawlike connection between M and P is as follows; L2 "x[Px É $y(My Ù Rxy)] Unlike the first paraphrase, L2 does not entail $xMx.11 Indeed the universal generalization in L2 would be true in a world with no mathematical objects provided that there were no P-configurations of concrete objects either. The assessment of the key claim that mathematized laws are violated in every mathematical-object-free possible world depends crucially on whether L1 or L2 is the correct approach to analysing such laws. If L1 is correct then the claim is true, while if L2 is correct then the claim is false. My own view is that L1 is intuitively more natural than L2 as a paraphrase of L, because it better reflects the role played by mathematical objects in formulating scientific laws. L1 incorporates the mathematical existence claim as a presupposition of the law, L, while L2 forges a subjunctive conditional link between the mathematical and physical components of the law. This latter approach has certain counterintuitive consequences; L2 implies, for example that, while there may exist no 10 11

What L1 asserts, roughly speaking, is that there exists a mathematical structure, M, and that any physical configuration of type P bears relation R to M. In L2 the mathematical existence claim is inside the conditional. What L2 asserts, roughly speaking, is that for any physical configuration of type P, there exists a mathematical structure, M, which bears relation R to P.

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mathematical objects M, such objects would exist if there were physical P-configurations. I conclude that L1 is the better of the two methods for paraphrasing mathematized laws. As we have seen, L1 implies that all essentially mathematized laws are violated in each mathematical-object-free world. Hence the first step of the Closeness Argument is justified. X. Counterlegals The second step of the Closeness Argument is the claim that, given possible worlds where actual laws are violated, the one with the most perfect match of pure physical facts is closest to the actual world. In Section VIII, we discussed Colyvan’s example of a Newtonian world in which mathematics is dispensable for science and hence the existence of mathematical objects makes no difference to the concrete world. One might try to argue, on this basis, that the closest mathematical-object-free world to the actual world (given our indispensability assumption) is a Newtonian world, call it NW. This would undermine the second step of the Closeness Argument and with it (No-Difference). The existence of mathematical objects makes a difference because if there were no mathematical objects then our world would be Newtonian rather than relativistic! The force of this argument clearly depends on whether NW is indeed closer to the actual world than the physically matching, mathematical-object-free-world, PW. There is approximate matching of pure physical facts between NW and the actual world, at least for medium scales of space and time. The laws of NW also hold to a high level of approximation in many actual-world situations. By contrast, PW exactly matches the pure physical facts of the actual world. However, any essentially mathematized actual-world laws are violated in PW. How do these factors play off against one another? The comparison of the relative closeness to the actual world of NW and PW stretches both our intuitions and Lewis’s framework of analysis beyond the point where we can expect useful guidance. For example, ‘perfect match of particular fact’ is cited by Lewis as the second most important factor in determining similarity between possible worlds. Is perfect match of the pure physical facts enough here? If the actual world contains essentially mixed mathematical/physical facts, presumably a possible world which matches these facts also is closer than one which does not. And what about pure mathematical facts? Does matching of these facts matter in determining closeness? Consider a possible world, MW, which contains only mathematical objects and which matches all the pure mathematical facts of the actual world. Is MW, which matches all the pure mathematical facts but none of the pure physical facts, closer or further from the actual world than PW, which matches all of the pure physical facts but none of the pure mathematical facts? We could stipulate answers to these questions, but on what basis? Faced with these imponderable questions, it is tempting to look beyond Lewis’s framework to see if some other analysis can be given of the counterfactual in (NoDifference). The twin assumptions of mathematical existence and of indispensability entail that the antecedent of (No-Difference) conflicts with the actual laws of nature. Igal Kvart [1996] refers to this type of law-breaking counterfactual as a counterlegal.12 Central 12

Kvart defines counterlegals as ‘counterfactuals whose antecedents, alone, are incompatible with the laws of nature (i.e. apart from any description of factual circumstances)’ [208].

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to Kvart’s analysis of counterlegals is a distinction between what he terms ‘weak counterlegals’ and ‘strong counterlegals’. A weak counterlegal is one whose antecedent implies that the actual laws change at some time, t, but does not imply that the actual laws have always been different. An example is, (1) If the gravity constant had dropped by a third yesterday then various satellites would have escaped to outer space. Kvart argues, convincingly I think, that weak counterlegals can be evaluated in much the same way as ordinary counterfactuals. The closest possible world in which the antecedent of (1) is true is a world, call it W*, that perfectly matches the actual world until yesterday, at which point the gravitational constant in W* drops by a third. In W*, various satellites escape into outer space following this change. Hence (1) is true. By contrast, a strong counterlegal—according to Kvart’s definition—is one which implies that certain (actual) laws ‘have been different from time immemorial’. For example, (2) If the gravity constant had been a third of its actual value, always and everywhere, then . . . . The key point about strong counterlegals is that they cannot be interpreted in terms of a local divergence from the actual world because there is no spatial or temporal sub-region of any of the possible worlds they describe which exactly matches a corresponding subregion of the actual world. For this reason, strong counterlegals must be evaluated qualitatively differently from weak counterlegals. In particular, Kvart argues that weak counterlegals are ‘world de re’, that is they are about the particular possible world at which they are evaluated. Thus the analysis of a weak counterlegal can bring in factual features specific to the actual world which are not entailed or presupposed by the counterlegal. In evaluating (1), for example, we can legitimately assume that the mass of the Earth in the counterfactual world is the same as its actual mass. Ordinary counterfactuals share this feature of being world de re. Kvart goes on to argue that strong counterlegals cannot be world de re; instead they are ‘world de dicto’. This is because there is no way to interpret strong counterlegals as matching the actual world up to some point in time, and then diverging from it. For example, any world in which the antecedent of (2) is true has a history that does not overlap with the history of the actual world. It is for this reason that specific background facts about the actual world cannot be assumed when evaluating strong counterlegals. Kvart draws the following conclusion: Strong counterlegals thus come out as fairly impoverished . . . in that the counterfactual construction collapses in these cases to the relation of logical consequence (plus accommodation for noncounterfactual presuppositions). Their interpretation is accordingly quite formal, and not literal like that of weak counterlegals. [Kvart 1996: 253] If this analysis is correct then it has important ramifications for our evaluation of (NoDifference). Under our assumption that mathematics is indispensable for science, the (NoDifference) claim,

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(No-Difference) If there were no mathematical objects then (according to platonism) this would make no difference to the concrete, physical world, turns out not only to be a counterlegal, but to be a strong counterlegal. It is a counterlegal because the natural way of paraphrasing essentially mathematized scientific laws implies that they are violated in any world with no mathematical objects. And it is a strong counterlegal because the non-existence of mathematical objects cannot be temporally or spatially restricted, hence mathematized laws are violated ‘always and everywhere’ in any mathematical-object-free world. This analysis has the following crucial consequence. Since (No-Difference) is a strong counterlegal, and since strong counterlegals are world de dicto, the only statements which can be truthfully asserted in the consequent of (No-Difference) are those which follow logically from its antecedent. It does not follow logically from the non-existence of mathematical objects that the physical properties of the world are different from their actual properties, nor does it follow logically that these physical properties are all the same as their actual properties. Hence (No-Difference) is neither true nor false. XI. Conclusions The conclusion of the previous section—that if mathematics is indispensable then the (No-Difference) claim lacks a truth-value because its antecedent is underspecified—helps explain philosophers’ conflicting intuitions concerning the truth of (No-Difference), and also the lack of convincing arguments on either side of the debate. For it entails that the MND argument may ultimately rest on a premise whose truth is impossible to establish. Thus the dispute over (No-Difference) has turned into a dispute over whether mathematics is indispensable for science. And this precisely bears out the principal claim of this paper, namely that the MND argument does not allow nominalists to circumvent the indispensability debate and thereby offer a direct route to undermining platonism. What makes (No-Difference) initially so plausible? Our intuitions are misleading in this context, I have argued, because we tend to fall back on implicit analogies between abstract objects and inert, remote concrete objects. However, parallels between the concrete and abstract cases are superficial at best. The theoretical contribution of the concrete objects postulated by science is paradigmatically via their causal role. Thus if a theory postulates the existence of a concrete object which is causally inert then this provides good grounds for thinking that the object in question is dispensable from that theory. No such chain of reasoning holds for abstract mathematical objects, because the contribution which they make to theories, whatever it is, is not in virtue of their causal role. In the abstract case, causal activity and indispensability come apart. At first blush, the MND argument seems closely similar in structure to the traditional sceptical argument against the existence of the external world. This latter argument draws its sceptical strength from the claim that, for all we know, we could be being systematically deceived by an evil demon as to the existence of the material objects we see around us. In other words, whether there is a material world or an evil demon makes no difference to the totality of our experiences. The MND argument can be formulated in analogous fashion: for all we know, we could be in a mathematical-object free world; hence we have no reason to believe in the existence of abstract mathematical objects. However, to

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interpret the MND argument in this way is to underplay its force. The (No-Difference) claim does not imply merely that there exists some mathematical-object-free world which is identical in all concrete respects to the actual world. It also implies that a concretely matching world is the closest mathematical-object-free world to the actual world.13 It is this stronger claim which gives the MND argument its extra force compared with the traditional sceptical argument, but it also presents the main stumbling-block to establishing the truth of (No-Difference). It may be that at the end of the day the dispute over whether the existence of mathematical objects make a difference stems not from any explicit thesis of platonism but from a certain background picture. This picture, inherited from traditional platonism, portrays two disconnected realms—the concrete realm of physical objects and the abstract realm of mathematical objects. It is a picture that has been challenged for example by the St. Andrews school of neo-Fregean platonism. Once this picture is in place it gives the nominalist opponent the resources to describe independent variation in the two realms, and to question whether the goings-on in one realm make any difference to the other. My aim in this paper has been to show that, while such questions are easy enough to raise, turning intuitions based on this two-realm picture into a convincing argument against platonism is far from straightforward. The ‘Makes No Difference’ argument has been advertised by its proponents as an effective argument against platonism, and one which is successful regardless of whether mathematics is indispensable for science. I have sought to challenge both of these claims. This is not to say that the MND argument is unsound. It is rather that its central claim—that the existence of abstract mathematical objects makes no difference to the concrete, physical world—is one whose proper evaluation remains unclear. If platonism is indeed self-undermining, then further argument will be needed to show this.14 Xavier University Received: January 2002 Revised: August 2002

REFERENCES Azzouni, Jodi 1994. Metaphysical Myths, Mathematical Practice, Cambridge: Cambridge University Press Azzouni, Jodi 2000. Stipulation Logic and Ontological Independence, Philosophia Mathematica 8: 225–43. Balaguer, Mark 1998. Platonism and Anti-Platonism in Mathematics, New York: Oxford University Press. 13

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Azzouni makes a related point about closeness in comparing MND-style arguments to traditional sceptical arguments. He talks of ‘the epistemic story being left intact’ by the (No-Difference) claim: ‘It is compatible with mathematical practice as it is (we are really using computers, using pens, carrying out proofs) that, despite this, there are no mathematical objects’ [Azzouni 2000: 226]. Thanks to Alexander Bird, Mark Colyvan, and Seahwa Kim for constructive criticism and comments, and to audience members of the Moral Sciences Club at the University of Cambridge for helpful feedback.

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Balaguer Mark 1999. Book Review: Mathematics as a Science of Patterns, by Michael Resnik, Philosophia Mathematica 3: 108–26. Cheyne, Colin and Charles Pigden 1996. Pythagorean Powers, or A Challenge to Platonism, Australasian Journal of Philosophy 74: 639–45. Colyvan, Mark 1998. Is Platonism a Bad Bet?, Australasian Journal of Philosophy 76: 115–19. Cornwell, Stuart 1992. Counterfactuals and the Applications of Mathematics, Philosophical Studies 66: 73–87. Cheyne, Colin 1998. Existence Claims and Causality, Australasian Journal of Philosophy 76: 34–47. Ellis, Brian 1990. Truth and Objectivity, Oxford: Basil Blackwell. Horgan, Terence 1987. Discussion: Science Nominalized Properly, Philosophy of Science 54: 281–2. Kvart, Igal 1996. A Theory of Counterfactuals, Indianapolis: Hackett. Lewis, David 1979. Counterfactual Dependence and Time’s Arrow, Noûs, 13: 455–76.