on the indispensability of the distinctively mathematical

Philosophia Mathematica (III) 20 (2012), 324–338.doi:10.1093/phimat/nkr043 Advance Access publication January 19, 2012

On the Indispensability of the Distinctively Mathematical†

Cory Juhl∗

Indispensability arguments purport to show that empirical data provideevidence for the existence of mathematical entities. In this paper weargue that indispensability arguments fail to show that empirical databears on the mathematical. In order to show this we attempt to clarifywhat it is to be mathematical, and separate the question whether abstractain general exist from whether distinctively mathematical abstracta exist.We introduce the notion of an ‘agnostified’ empirical theory as a heuristicdevice for clarifying the role of mathematical terminology in empiricaltheories. The special case of spaces is then briefly considered.

According to indispensability arguments, empirical evidence supports theexistence of mathematical abstracta. The precise forms of the argumentsvary, but at a level of abstraction sufficient for my purposes here, the argu-ments share the form:

IND: Appeal to1 mathematical objects and/or facts is indispensablewithin empirical science, particularly within physics or within ourmost successful empirical theories.

IND→JME: The fact that appeal to some type of entities/facts is indis-pensable within our most successful empirical theories providesempirical justification for believing that these entities exist/factsobtain.

JME: Therefore, we are empirically justified in believing that mathemat-ical entities exist/facts obtain.

† Thanks to Nora Berenstain, Dan Bonevac, Ray Buchanan, Josh Dever, Austin Gatlin,Rob Koons, Eric Loomis, Bryan Pickel, and David Sosa for helpful discussions. Thanksespecially to Eric Loomis and Todd Stewart for comments on an earlier draft of the paper.Thanks also to two anonymous referees who contributed many helpful objections and sug-gestions for improvement.

∗ Department of Philosophy, The University of Texas at Austin, Austin, Texas 78712,U.S.A. [email protected]

1 Whether such ‘appeal’ requires quantification over the relevant entities is left openfor the purposes of the argument in this paper. Nothing in the argument hinges on thecorrectness or incorrectness of the Quinean ‘criterion of ontological commitment’.

Philosophia Mathematica (III) Vol. 20 No. 3 C© The Author [2012]. Published by Oxford University Press.All rights reserved. For permissions, please e-mail: [email protected]

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THE INDISPENSABILITY OF THE DISTINCTIVELY MATHEMATICAL 325

Many responses have been given to indispensability arguments. Field,for example, denies IND, and tries to show how one could in fact dis-pense with mathematical objects within a complex and successful mathe-matized theory such as Newtonian gravitation theory. Field [1980] tries toshow that one could do Newtonian gravitation theory without appeal to anymathematical abstracta, in part by using ordered structures of space-timepoints as surrogates for real numbers (to simplify an extremely clever andtechnically sophisticated construction). It remains controversial whetherField’s strategy can avoid appeal to mathematical abstracta for a variety ofreasons.2

Another response strategy is to question IND→JME, typically byraising doubts about some form of confirmational holism, which holismis often used to motivate and defend IND→JME. Sober [1993], amongothers, has noted that it is false that given a conjunction that entails someempirical consequence, the obtaining of that consequence empirically sup-ports all of the conjuncts. However, some aspects of Sober’s objections toconfirmational holism (beyond the very straightforward and decisive onejust mentioned) make essential appeal to a still-controversial approachto confirmation that he has developed: ‘contrastive empiricism’. Others,such as Chihara [2004], have pointed out the intuitive falsehood of con-firmational holism, and how such holism, which often fails to distin-guish evidence for truth of a theory from the pragmatic benefits of the-ory acceptance, does not accord well with actual scientific practice. Chi-hara cites examples given by Maddy [1997] and Vineberg [1998] suchas late nineteenth-century disputes concerning the existence of atoms andmolecules, given widespread acceptance of their empirical fruitfulness.One potential difficulty with basing one’s responses on historical cases isthat a purveyor of mathematical indispensability arguments can dig in hisheels and insist that the scientists of the time were being intellectually dis-honest and employing bad empirical methodology (see [Colyvan, 2001]).Real historical situations are complicated, and it is difficult to settle whowas right about the evidence for atoms at some time t , given precisely thecomplicated wealth of data in some sense ‘available’ at that time.

In this paper I will raise a different objection to indispensability argu-ments for mathematical abstracta. I will argue that even if one grants aversion of the premise IND→JME, and grants that there is a sense inwhich any portion of any empirical theory is empirically defeasible (con-firmable and disconfirmable), one may also maintain that mathematics isnot empirically defeasible, and in particular not empirically supported.Seeing how these can be true, and how one might deny IND in a relevant

2 See [Chihara, 2004, ch. 11] for an illuminating overview of some of Field’s difficulties.

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sense, requires that we first clarify how best to understand the dispute asconcerning the distinctively ‘mathematical’.3

1. On the ‘Mathematical’

There is a further difficulty lurking in the background of the controversyconcerning indispensability arguments. The difficulty is that it is oftentaken for granted that we have an idea what we mean by ‘mathematical’,when we consider whether empirical data can bear on the existence ofmathematical entities. Unfortunately, though, not much has been writtenconcerning the question what makes an entity mathematical.4 One can getthe impression that what it is to be mathematical is simply to be the refer-ent of a mathematical term. Mathematical terms are the sorts of terms usedby mathematicians or scientists when they do math. And so on.

The little that is written usually pertains to the question what ‘abstract’means. It is often taken for granted that the problems for our epis-temic access to mathematical entities arise from the fact that they areabstract, and abstractness entails various features such as not havingspatio-temporal locations or being causally inert. In a later section of thispaper we give reasons for distinguishing the question whether empiricaldata might support the existence of some properties, such as ‘having amass of 2.88 kg’ or ‘having a temperature of 32 degrees Farenheit’, fromthe question whether empirical data support the existence of mathematicalparticulars such as the number 2.88.

Failure to distinguish the distinctively mathematical from other itemsthat are understood to be abstract can make the indispensabilists’ argument

3 It may be helpful to contrast my argument with Azzouni’s as given in his [2004].Azzouni argues that we should take the mathematics used within successful theories to betrue, but that we should nevertheless resist the claim that any abstracta exist, whether phys-ical properties or mathematical entities. In support of this Azzouni argues against Quine’squantificational ‘criterion of ontological commitment’ and proposes a replacement criterionof commitment that requires that we have an intelligible (broadly causal) account of howwe know about the entities in question. What I am concerned to argue in this paper is thatthere are reasons to think that empirical data do not support the existence of distinctivelymathematical entities, whatever the verdict concerning empiricism about physical proper-ties. Given important disanalogies between the two cases, I claim one can take empiricaldata to support the existence of physical properties and yet deny empiricism concerningmathematics.

4 This is somewhat misleading, since many philosophers have taken the main issue tobe what abstractness consists in, and for such philosophers considering the abstract vs.concrete distinction just is to consider what it is to be mathematical, or at least the mostdistinctive feature of the mathematical. Part of the purpose of this paper is to provide rea-sons to reject this assimilation. Someone who, like me, thinks that mathematical entitiesare distinctive in ways besides their purported abstractness, and that these other featuresare central to the epistemic difficulties surrounding them, will not take a discussion ofabstractness to constitute a discussion of what it is to be mathematical.

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for empiricism concerning mathematics appear stronger than it should. Forsome scientific realists it seems quite awkward to deny that empirical datasupport the existence of various physical features, from temperatures tomasses to charges to spin.5 Claims that such properties, if they exist, do notplay any interesting causal role within the physical world are liable to sur-prise most physicists, and many philosophers with realist sympathies. Thisis not the place to settle the question whether there are properties in gen-eral. However, many scientific realists accept their existence on the basisof empirical evidence. If the difficulty for mathematical entities or facts issupposed to arise from their abstractness, which is shared by other entitiesand facts for which we intuitively seem to have empirical evidence, thecase for mathematical empiricism given scientific realism seems strong.

In what follows I explain how it remains possible to deny that empiricalevidence bears on the distinctively mathematical, even for someone whogrants that empirical data can bear on the existence of various physicalproperties or relations, as well as on the existence of physical ‘spaces’ asopposed to mathematical ones. I distinguish the distinctively mathematicalusing features that mathematical entities and facts are widely understoodto have, as well as what would make sense of our epistemic practices con-cerning the mathematical on one hand and the empirical/theoretical on theother. If we do not begin with a reasonably definite idea what we are talk-ing about when we argue about mathematical entities, it can become fairlytrivial to establish their existence. For example, if we grant that empiricaldata support the existence of a continuum of possible temperature values,then why not say that the case for mathematical abstracta is settled? Eitherthe real numbers just are these temperatures, or if there are some othercontinua, then the reals are one of them. Those of us who would recoil atsuch a suggestion owe ourselves and our opponents an account of whatis wrong with identifying the reals with temperatures or any other empiri-cally accessible domain of entities or properties.

On what basis might we deny that temperatures are numbers? Gener-ally, we might appeal to a variety of differences between the two sorts ofentities, such as modal features or subjunctive relations that we take tohold between facts or events involving uncontroversially physical featuressuch as temperatures or mass, and other facts or events within the twodomains. We might also note differences concerning what we take to bereliable methods for accessing information about them. In order to see how

5 It remains controversial whether fruitful applications of predicates such as ‘has elec-trical charge q’ provide evidence for the existence of the property of having electricalcharge q . Whatever our ultimate verdict on these questions, though, the point emphasizedin this paper is that there are additional and independent reasons, analogous to those presentin the Santa theory example below, for doubting the evidential relevance of empirical obser-vations to distinctively mathematical entities and facts.

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a general reply to indispensability arguments for mathematical empiricismmight proceed from such considerations, let us consider a structurally anal-ogous example.

2. Santa Theory

Let us imagine that a popularization of the Santa Claus story at some timein the near future is more specific about the details of the story than thevague version with which we are now familiar. For example, in the futureversion of the story there are precisely 117 elf-types, elves are related invarious sibling or rivalry relations, and other details are specified. How-ever, the basic story line continues in the standard way. Santa lives at theNorth Pole, and is the boss of the elves, and drives a reindeer-drawn sleigh.Suppose that some clever physicists of the time notice that the structureof superstrings can be mapped, at some level of abstraction, to relationsbetween Santa Claus and his elves. They name the central attracting stringposited by the theory ‘Santa’, and give other important superstrings (ofwhich there are 117 types) labels of elf-types, and so on. The details actu-ally matter very little for the main moral of our imagined scenario. Themain moral is this: no one should take the empirical data that would sup-port the string theory in the envisaged situation to show that, contrary topopular opinion, Santa Claus really exists, along with his elves.6 Let usconsider what reasons there might be for remaining agnostic or antirealistsconcerning ‘distinctively Xmasy’ entities while being convinced ‘Santatheorists’ in physics.

First of all, as many have noted, the fact that some aspects of the storyare empirically supported fails to show that other ‘unused’ statements ofthe standard story are true. If Santa theory, as the string theory comes tobe called, makes various predictions that turn out to be true, one needs tolook at what statements of Santa theory were essential to the predictions.True predictions would not, in general, support statements of Santa theorythat had no bearing on the predictions, i.e., statements that were irrele-vant to deducing the predictions. For example, if the claim that Santa islocated at the North Pole of planet Earth most of the time is not employedin Santa theory to predict masses of the ‘elves’ generated in particle accel-erators, then that accelerator data does not support that claim. Similarly, ifthe elf-types that are quantified over within Santa theory are distinguishedin terms of ear-shapes, magical powers, and gift-wrapping abilities, clearlyaccelerator evidence does not bear on the existence of those elf-types.

6 There are some ways of filling out the story such that we should then go on to say thatphysicists did after all discover that the Santa story is true, and that Santa turns out to be asuperstring as well as riding in a sleigh. Nevertheless, there are many obvious ways to fillout the story such that it would be absurd to make such claims.

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By adopting the view that predictions are evidentially relevant only tothe parts of the theory actually used7 to make them I am in agreement withmany holists such as, e.g., Quine, where he states

[T]he falsity of the observation categorical does not conclusivelyrefute the hypothesis. What it refutes is the conjunction of sentencesthat was needed to imply the observation categorical. . . . This is theimportant insight called holism. [Quine, 1992, pp. 13–14]

In this passage Quine distinguishes the evidential bearing of correctprediction on parts of a theory that are actually used to derive a predictionfrom the rest of the theory. One would think that the most natural correlateof this idea to confirmation would be to claim that the part of a theory thatis confirmed by an observation is also ‘the conjunction of sentences thatwas needed to imply the observation categorical’. It is important to contrastthis view, what I will call the Natural View, with some more ambitious andcontroversial claims, suggested sometimes by Quine and other Quineans,to the effect that correct predictions support some entire theoretical system,if not our world-view as a whole, rather than those parts of our world-viewemployed in generating the predictions.8 For the purposes of this paper, Iwill assume that the Natural View is correct, and furthermore I will takefor granted the uncontroversial claim that frequently premises that employmathematical language are used to derive true predictions.

Most centrally for the purposes of this paper, however, is the fact thateven those sentences that are used to make predictions within Santa the-ory plausibly have different referents in Santa theory versus ordinary Xmasdiscourse. To a ‘Xmasian empiricist’ who claimed otherwise, asserting thatthe physical evidence cited in journals supports ordinary Santa claims suchas that he lives at the North Pole, we might note that as physicists use ‘theNorth Pole’ in their Santa theory, their methods of determining whetheran item is there suggests that ‘the North Pole’ is located near the centerof some faraway supercluster of galaxies. Their detection methods wouldseem completely inappropriate for checking whether something is at theEarth’s North Pole. To check whether ‘Santa is now coming to distributepresents’ physicists use microscopic particle accelerator data that wouldseem inappropriate if Santa is a humanoid flying through the air. Pressed

7 This is not strictly correct, at least if we move beyond a fairly simple form ofhypothetico-deductivism, since other parts of the theory might also have conditional prob-abilities on the data. But I think that ultimately Quinean indispensability arguments arein a worse position rather than a better one if we move to a more general probabilisticframework such as Bayesianism, and further complications arise; so for the purposes ofthis paper I will adopt the simpler account.

8 To quote one of many statements in this spirit, ‘As Pierre Duhem urged, it is thesystem as a whole that is keyed to experience.’ [Quine, 1953, p. 222], cited in [Colyvan,2001].

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on why the two are not compatible, we might note that different subjunc-tives seem true if Santa were flying through the atmosphere as opposed toif a superstring were close to the center of a supercluster. In other words,we take different subjunctives involving the word ‘Santa’ to be true whenwe are doing Santa theory versus when we are discussing ordinary Xmas.And so on. This is all fairly trivial and obvious in the Santa case envisaged.The point of the Santa example is that thinking that there is empirical evi-dence for distinctively mathematical entities and facts seems similar tomany of us to the ‘Xmasian empiricist’ claim that in the envisaged sce-nario there would be empirical evidence for the existence of the referentof ‘Santa’ and an elaborate Xmas story, as standardly interpreted. In bothcases, there are good reasons to deny the identity of the two classes of pur-ported entities and facts, since physicists’ epistemic behaviors would seemcompletely irrational otherwise.

The moral of the Santa theory situation is that we should not be tooquick to take successful prediction/explanation to support parts of the-ories as standardly interpreted. When we speak of correct predictionssupporting a ‘theory’ or parts thereof, the Santa case shows that whendetermining the bearing of evidence on fact, we must determine whetherthe standard referents of the terms of the theory are essential to predic-tive/explanatory success, or whether we should think that some other enti-ties or facts, distinct from the ones given by standard interpretations ofthe terms involved, are the entities and facts genuinely supported by thedata.9 In the imagined Santa Theory case, scientists would be well-advisedto notice, as they surely would, that not only is the fact that Santa isa human being who lives at the North Pole not essential to the predic-tions, but also that accelerator experiments do not bear on the existenceof a humanoid entity or its geographical location. The accelerator data donot support the existence of distinctively Xmasy entities and facts. WhatI hope to make clearer in the following is that mathematical theory andlanguage, when employed in empirical application, can be understood tofunction in a way analogous to the way Santa theory and language areemployed in the imaginary scenario. There is something empirically con-firmed in both the Santa theory and the mathematical physics cases, but theexistence of Santa, understood as distinctively human, and geographicallylocated, is not confirmed, and neither are mathematical objects and facts,understood as distinctively mathematical. I will explain what one mighttake to characterize the ‘distinctively mathematical’ in the next section.

9 Note that there is no need to take a definite stand for present purposes on the ques-tion whether Xmasy or mathematical terms ‘really have’ Xmasy or mathematical referentswhen applied within physics. The point is that, whatever semantic theory we give for physi-cists’ language, the relevant empirical evidence does not seem to bear on the distinctivelyXmasy or mathematical entities and facts.

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The problem that these observations generate for indispensability argu-ments of the standard contemporary form arises from the fact that:

DIST: In applications of physics and other empirical theories, the claimthat the explanatorily relevant entities and facts are distinctivelymathematical is highly implausible.

3. The Distinctively Mathematical

Many authors have noted that there is not a great deal of clarity con-cerning what it is to be abstract as opposed to concrete. Abstractness,in particular mathematical abstractness, has been taken to entail at leastnon-spatiality and acausality. A third widely (although not universally)accepted feature of mathematical facts (including existence claims) is theirnecessity. A closely related feature to both necessity and acausality, andarguably more fundamental, is the subjunctive or counterfactual irrele-vance of mathematical facts to empirical ones. With respect to time, math-ematical abstracta have been taken to exist at all times, or ‘outside of time’,and either of these could be taken to be either a consequence of their neces-sity or what we might replace necessity with. Similarly, mathematical factscould be taken to be eternal, atemporal, or necessary. Finally, mathematicalfacts are taken to be knowable a priori, and in addition, often our mathe-matical knowledge is taken to be empirically indefeasible. A priori knowa-bility and/or empirical indefeasibility are crucial to what many philoso-phers and scientists would be inclined to count as the distinctively math-ematical. There is a case to be made that empirical indefeasibility and/orsubjunctive irrelevance to the empirical is the core distinctive feature ofthe distinctively mathematical, which explains intuitions favoring the otherfeatures. The main point of this paper is to show how someone who thinksthat mathematics is distinctive in these core respects can continue to main-tain this, and on that basis reject indispensability arguments as showingthat mathematics is empirically supported, even while granting that anypart of a physical theory, properly understood, is empirically revisable.

In order to support DIST further, let us introduce as a heuristic devicethe notion of a (mathematically) agnostified theory. An agnostified the-ory is the result of substituting new, uninterpreted constants for all basicterms in the theory. Individual constants, function symbols, and predicatesymbols all count as ‘constants’ in this extended sense. A version of DISTcan be expressed by the claim that if we take any mathematized physicaltheory and agnostify it, including agnostified correlates of the purely math-ematical theories employed in conjunction with the mathematical physicaltheory, the resulting theory will be exactly as empirically adequate as thepre-agnostified version. The point is to substitute a theory with the sameform (individual constants for objectual constants, predicate or function

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constants of some type for predicate or function terms of that type) for theold mathematized theory, but where the resulting theory is not assumed tocontain terms with distinctively mathematical referents.

Agnostifying a theory can help us see how to resist the temptationto think that when the numeral ‘100’ appears in a physical theory, sincethe term denotes a mathematical entity, appeal to that number is essen-tial to the success of the applications of the theory. Similarly, we leaveopen the question whether the real numbers are quantified over as opposedto, say, the totality of physical spatial locations or possible energies, tem-peratures, or other physical magnitudes. The procedure allows us to seethat the use of mathematical language, understood as having distinctivelymathematical referents, is inessential for empirical applications by show-ing more clearly that whatever empirical success the theory achieves canbe achieved without any explicit or implicit (presupposed in our interpre-tations of terms, say) premise as to the distinctively mathematical naturesof the entities seemingly appealed to in standard theory formulations.In other words, agnostification makes it easier to see that mathematicalterms are akin to ‘Santa’ in the Santa theory case. The fact that physicistsuse the same expression-types when doing physics that mathematiciansuse when doing mathematics can appear to lead inexorably to the con-clusion that since the same entities are referred to in both settings, thesame entities that the mathematician reasons about a priori are the onesthat the physicist learns about empirically. The agnostification procedureis a heuristic device for seeing a way to resist this line of thought. Themain point to be made is to note that the identification of the entities andfacts that are explanatorily relevant in physical processes with distinctivelymathematical entities and facts is highly implausible, as implausible asidentifying the field-theoretic Santa with ordinary Santa or elf-types withsuperstring-types. As in the Santa case, the very methodological presup-positions concerning the best way to access facts about, say, the energytensor (i.e., empirically) versus finding out mathematical facts about math-ematical tensors, and causal/subjunctive consequences of the energy ten-sor being such and such versus causal consequences of a mathematicaltensor having such-and-such mathematical features, makes it very difficultto understand how we could take the two ‘tensors’ or the relevant classesof facts (physical and mathematical) to be identical. These reasons con-cerning epistemic coherence for resisting the bearing of empirical data onthe distinctively mathematical do not in general transfer to abstracta suchas properties, as we discuss in the next section. For that reason even ascientific realist who believes in physical properties on the basis of empir-ical evidence can coherently resist empiricism concerning the distinctivelymathematical.

Agnostification can yield other benefits as well. If we restrict ourselvesto suitably agnostified physical theories, we may permit any part thereof

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to be empirically revised, in conformity with Quinean ‘global empiricaldefeasibility’.

The agnostification procedure also reveals new questions typically notraised by physicists, at least not directly. When we add two Hilbert-spacevectors, for example, and a student asks, ‘How do we know that Hilbert-space vectors add that way?’, one might be tempted to answer, ‘Hilbertspaces are mathematical objects. That is what makes something aHilbert space (in part), the fact that there are vectors within the space thatadd in accord with these rules.’ On the other hand, Hilbert-space vectorsare thought to represent physical states of quantum systems. We can surelyask whether quantum states ‘physically add’ in a particular way. It is anempirical question whether there are states that ‘add’ in that particular wayor not. Although often it may not matter for purposes of doing calculations,it can lead to confusions when objects such as Hilbert spaces, conceivedof as distinctively mathematical, are identified with physical ‘spaces’ ofpossible states.10 We can also begin to consider whether certain generalclaims within an agnostified theory are lawlike or just materially true. It isdifficult to raise the question of nomological versus material generalizationconcerning, say, Hilbert-space vector addition. It is easy simply to presup-pose that all mathematical truths are necessary, and then slip into thinkingthat the way physical states ‘add’ is nomologically necessary. Moving toan agnostified counterpart theory can help us see more clearly some salientempirical questions concerning the nomological (more broadly, modal orsubjunctive) status of various parts of the theory.

To summarize, then, consideration of agnostified theories makes clearerhow one could take any part of an agnostified theory to be empiricallyrevisable, and thereby grant some truth to part of the Quinean account,while resisting the view that mathematics is empirically defeasible. It alsohelps us to see more clearly that both the truth and the nomological statusof various parts of the theory are to be empirically ascertained, which canbe obscured particularly in the pre-agnostified theory parts that employmathematical terminology. And most centrally for the purposes of thispaper, we see how to get all of this while preserving the very naturaland widespread sense that mathematics is empirically indefeasible anddistinctive.

10 There is an issue here as to whether a physical system can ‘be’ a Hilbert space,whether we should count some physical systems as also mathematical ones. I am taking‘Hilbert space’ as a distinctively mathematical entity, but there is no reason to object tospeaking of ‘physical Hilbert spaces’, meaning spaces with a structure in common withmathematical Hilbert spaces. The point is that we should not illicitly assume that we haveempirical evidence for the existence of a distinctively mathematical Hilbert space simplybecause we have evidence for the existence of a physical system or space of states that hasa structure that can be correlated with the structure of the mathematical space.

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4. A Dubious Conflation of ‘Abstracta’

A very common way to carve up the dialectical space treats all ‘abstracta’,whether mathematical particulars or properties/relations or physical prop-erties/relations, as equally problematic. This paper focuses on the distinc-tion between distinctively mathematical entities (whether particulars orproperty/relation terms) on the one hand, and empirical entities (whetherparticulars or properties/relations) on the other. Many philosophers havethought that there is a general ‘problem of abstracta’, where these includeall properties and relations, in addition to mathematical particulars. Wor-ries about our epistemic access to mathematical particulars have beenthought to extend unproblematically to all ‘abstracta’, and vice versa. Thispoint of view, however, does not appear to capture the distinctively puz-zling nature of mathematics and our epistemic access to it, as opposed toempirical hypotheses and our epistemic access to their truth. We ought todistinguish varieties of abstracta and the nature of the epistemic difficultiesthat they generate. It is widely (although not universally) accepted that theproperty of being electrically charged, and the property of having a tem-perature of 32 degrees Fahrenheit, and the property of having a volume of3 cubic meters, are abstract. This is taken to be true in virtue of the pur-ported fact that all properties are abstract. Unfortunately, in the absence ofclarity concerning what abstractness comes to, it can be difficult to decidewhether these physical properties are abstract. The most plausible sense inwhich all properties are abstract is that they do not obviously have spatiallocations. But nonspatiality does not entail acausality, necessity, or aprior-ity, and so is a long way from the distinctively mathematical. Furthermore,it remains controversial whether physical properties play essential roles inthe causal framework.

The question concerning how scientists come to know the truth of state-ments involving empirical predicates such as ‘has a temperature T ’ or‘has a mass M’ would be answered by everyone involved in the debatein the same way: they come to know such truths by appeal to experience.But many philosophers will resist the inference to the existence of phys-ical properties simply on the basis of successful applications of empiri-cal predicates. For example, if someone accepts the Quinean ‘criterion ofontological commitment’ according to which theorists are only ontologi-cally committed to entities over which they take quantifiers to range (i.e.,given existential generalizations that they take to be true, they are onlycommitted to the items within the domain of the quantifiers), then in orderto be convinced that the existence of properties is empirically supportedthey will want to be shown that scientists need to quantify over propertieswithin the best theory.

Whether it is appropriate to adopt a more permissive criterion thanQuine’s, and take true applications of predicates to support the existence

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of corresponding physical properties or to adopt some other ‘criterion ofontological commitment’ is unnecessary to settle for the purposes of thispaper. The main point of the considerations adduced here is that thereare reasons, independent of how that debate about ontological commit-ment plays out, for resisting empiricism concerning mathematics. This isbecause there are reasons for resisting the bearing of experience on distinc-tively mathematical entities and facts that do not straightforwardly apply tophysical properties and facts.11 Unlike distinctively mathematical entities,properties, and facts, statements involving physical predicates are under-stood to be essential to predictions and causal explanations of measure-ment results, as enmeshed as appeals to physical entities themselves. Howone discovers many truths involving physical predicates, including sub-junctive truths, is by performing experiments and making empirical obser-vations. In particular, everyone accepts that there are subjunctive truthssuch as ‘If there were no charged objects in the system, the causal conse-quences would be different . . . ’. In that sense, everyone accepts that expe-riences subjunctively depend (perhaps only probabilistically) on whethersome empirical predicate applies to some physical entity, and in that senseeveryone takes experience to depend on facts involving physical entitiesand predicates, whether or not predicates are ontologically committing.There would be no question of ignoring empirical data when attempt-ing to determine the truth of contingent statements involving tempera-ture predicates, for example. In contrast, mathematicians ignore empiricalevidence when attempting to make mathematical discoveries, discoveriesconcerning distinctively mathematical entities, relations,12 and facts, andmost non-Quineans take them to be justified in doing so. Physicists by andlarge agree, and do not take their experimental results to bear on the purelymathematical. The sharp methodological contrast between the proper wayto discover facts about physical ‘abstracta’ as opposed to mathemati-cal entities, relations, and facts is widely accepted among (non-Quinean)

11 The coarse-grained conception of ‘abstractness’ has often led the debate astray, in myview, leading some who are suspicious of mathematical entities to claim that there are noproperties of any sort, and the other side to pretend that, if we have a good story aboutsome properties (and therefore, some abstracta), there is no further problem concerningthe distinctively mathematical. There is room (more comfortable and more tastefully deco-rated) for views that deny empiricism concerning mathematics for reasons that do not settlequestions concerning our knowledge of, or the existence of, physical properties.

12 There is a complication with mathematical properties and relations. It seems naturalto say that some geometrical relations, for example, are instantiated both in empirical sys-tems and in mathematical systems. Similarly for other ‘structural properties’, which mightbe shared by both mathematical and non-mathematical structures. In order to sidestep thiscomplication, I sometimes focus on mathematical abstract particulars and facts. Whetherwe should distinguish distinctively mathematical properties from empirical ones is an inter-esting question that will require a lengthy further discussion, and I do not think that theanswer matters much for the arguments of this paper.

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philosophers, practicing empirical scientists, and mathematicians, and ouracceptance of this difference often remains robust even after many years ofreflection on arguments on behalf of empiricist approaches to mathemat-ics.13 The widely accepted methodological differences between mathemat-ics and empirical science make sense given the differences in their pre-suppositions concerning the subjunctive connections (or absence of sub-junctive connections) between facts in the different domains, just as Santatheorists’ ignoring accelerator data when telling Xmas stories is compati-ble with realism and empiricism about Santa theory thought of as a theoryabout superstrings rather than distinctively Xmasy entities and facts.

5. Empirical Accessibility Without Causation:The Enigma of Spaces

Mark Colyvan (e.g., in [Colyvan, 2001]) has recently defended indis-pensability arguments at great length, and has raised a number of veryserious objections to views according to which we cannot know aboutanything that is not causally related to us in some suitable way. Such an‘Eleatic principle’ has been used to motivate rejection of mathematicalabstracta and to undermine indispensability arguments. I cannot addressall of his and others’ very interesting examples and arguments againstEleatic principles and for indispensability arguments. I will only brieflyremark on one key class of examples. Many of the key examples thatColyvan relies on are geometrical or spatial. We can explain lengthcontraction, for example, by appeal to facts concerning Minkowskispacetime. Shall we say that the spacetime caused the contraction? Thatdoes not seem like a natural thing to say. And yet it also seems natural tothink that empirical data such as length contraction empirically supportor provide evidence for the existence and structure of such a relativisticspacetime. Relatedly, Resnik [1997] has argued that there is no naturalway to divide the distinctively mathematical from the theoretical, andhis best example, arguably, is the case of space and points of space. Itis nontrivial to characterize points as physical entities, and they seemto be in some sense essentially unobservable. Spatial examples havecontinued to play a central role in disputes concerning indispensabilityarguments and the possibility of empirical evidence for mathematical

13 Another helpful analogy here is provided by Field’s ‘isolated Nepalese village’. (See[Field, 1989, pp. 26–27].) It would be a very puzzling combination of views that, on theone hand, a Nepalese village is permanently causally isolated from us, and on the otherhand, that we have reliable belief-forming mechanisms concerning the goings-on withinthe village. In contrast, if the village is clearly visible in a valley below us, and we arewatching it with a telescope, if there is any puzzlement concerning how we know that,say, redness or triangularity are instantiated somewhere within the village, it is of a verydifferent character.

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entities; so a ‘mathematical distinctivist’ who objects to indispensabilityarguments in the way that I have in this paper should explain what shethinks concerning the status of space as either distinctively mathematicalon one hand or physical/theoretical on the other.

When we try to understand physical systems, part of this involvesdetermining a space of possible states of these systems. What the spacesof states are, how they are structured, seems an empirical question. Buthow can we empirically access state spaces, especially if we come tothink that the spaces themselves, as well as the ‘points’ in such spaces,are not causally efficacious? To abbreviate what is surely a long andcomplicated story, we determine possibilities by investigating actualconfigurations, and then we conjecture as to what other (nomologically)possible configurations there are. On what empirical basis might we arguethat our spacetime is special-relativistic? It seems that we would say thingslike, ‘If spacetime were different, when we moved that stick really fastit would have yielded different measurement results than were observed,and if spacetime is special-relativistic, we should get the measurementresults that we in fact observe’. If someone asks how we know whethersome location is the possible location of a physical object, we might notethat an object has actually been observed there, or alternatively we mightappeal to random selections of predicted locations, and note how all suchselections have yielded genuine possible locations, in that things weresuccessfully placed there. In both of these appeals, it seems that thereis taken to be a counterfactual or subjunctive connection between thestructure of the space and empirical data. We say things like, ‘if the spaceof possible states/properties/relations/magnitudes were not like the onehypothesized, we would expect that the data would (alternatively, wouldprobably) have been different’. The existence and structure of a statespace explain phenomena by partially explaining why the data look theway they do rather than the way they would have looked if the state spacehad been different. Such a subjunctive relationship between facts aboutspaces and empirical data does not entail that we should think that spatiallocations, possible locations, or the structure of a space overall shouldbe said to ‘cause’ the data, even if such spatial entities and facts help toexplain the data. Thus one might agree generally with Colyvan and otherswhen they claim that scientists posit some entities and structures evenwhen these entities or structures do not cause any phenomena. But onemight nevertheless demur from their further claim that empirical evidencefor such a theory provides evidence that distinctively mathematicalentities exist or facts obtain. As noted earlier, mathematical entities andfacts are widely understood as necessary, and also as counterfactually orsubjunctively irrelevant to the empirical, physical world, in the strongsense that even the probabilities or likelihoods of empirical data areindependent of the distinctively mathematical.

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338 JUHL

In contrast, for physical spaces, there is understood to be acounterfactual or subjunctive connection between the data and spatialentities and facts, at least in the sense that the likelihoods of data areunderstood to be different given some spatial hypothesis S than if not-S.If there were thought to be no such subjunctive (even probabilistic) con-nections between these theoretical posits and empirical data, it becomesvery difficult to see how the structure of the space could explain anything,and relatedly, why we would think that we have empirical evidence for itsexistence.

6. Saving the Trans-Phenomena

If a structure of some type is appealed to in a successful physical theory,there can be evidence for the existence of a system with correlatedstructure out there in the physical world. For example, if an enormouslysuccessful theory such as thermodynamics appeals to a collection of tem-peratures that is at some level of abstraction isomorphic to the reals, thenthat is reason to think that there is some real-number-like structure that iscaptured by the theory. The Santa example and its elaboration is supposedto show nevertheless how to resist the thought that distinctively mathemat-ical entities or facts are thereby empirically supported.

References

Azzouni, Jody [2004]: Deflating Existential Consequence: A Case for Nominal-ism. Oxford: Oxford University Press.

Chihara, Charles [2004]: A Structural Account of Mathematics. Oxford:Oxford University Press.

Colyvan, Mark [2001]: The Indispensability of Mathematics. Oxford: OxfordUniversity Press.

Field, Hartry [1980]: Science Without Numbers: A Defense of Nominalism.Princeton: Princeton University Press.

——— [1989]: Realism, Mathematics, and Modality. Oxford: Blackwell.Maddy, Penelope [1997]: Naturalism in Mathematics. Oxford: Oxford Univer-

sity Press.Quine, W.V.O. [1953]: ‘On mental entities’, in The Ways of Paradox and Other

Essays. Rev. ed., pp. 221–227. Harvard University Press, 1976.——— [1992]: Pursuit of Truth. Cambridge, Mass.: Harvard University Press.Resnik, Michael D. [1997]: Mathematics as a Science of Patterns. Oxford:

Oxford University Press.Sober, Elliott [1993]: ‘Mathematics and indispensability’, Philosophical

Revieww 102, 35–57.Vineberg, Susan [1998]: ‘Indispensability arguments and scientific reasoning’,

Taiwanese Journal for History and Philosophy of Science 10, 117–140.

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on the indispensability of the distinctively mathematical

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