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Abstract Jody Azzouni argues that whilst it is indeterminate what the criteria for existence are, there is a criterion that has been collectively adopted to use ‘exist’ that we can employ to argue for positions in ontology. I raise and defend a novel objection to Azzouni: his view has the counterintuitive consequence that the facts regarding what exists can and will change when users of the word ‘exist’ change what criteria they associate with its usage. Considering three responses, I argue Azzouni has best reason to take one that ultimately renders unsuccessful his arguments against mathematical abstracta. INTRODUCTION A familiar question in the philosophy of mathematics is: (Q1) Do mathematical objects, properties, structures, and relations exist?1 (Q1) is a question in first-order ontology, that is, a question about what entities exist. It asks whether mathematical objects like numbers or sets exist, where such entities are understood (unlike concrete objects such as tables or chairs) to be abstract entities that lack both spatio-temporal location and causal powers. Two families of positions answer yes or no respectively to (Q1): Mathematical Platonism: Mathematical objects, as abstract objects, exist. Mathematical Nominalism: Mathematical objects either do not exist simpliciter, or they do not exist as abstract objects.2 Jody Azzouni [2004; 2010a, pp. 96–97; 2010b; 2012a; 2012b; 2012c] has argued for a form of mathematical nominalism. According to Azzouni, mathematical objects do not exist either as abstract objects or as any other kind of (e.g., fictional or concrete) entity.3 Interestingly, and in contrast to other mathematical nominalists, Azzouni’s strategy for establishing his nominalism has been to argue as follows. First, he maintains that whilst it is indeterminate what the criteria for existence are, there is a criterion that has been collectively adopted to use ‘exist’ that we can employ to argue for positions in ontology. Secondly, this criterion, Azzouni argues, entails that mathematical objects — and other abstracta — do not exist: It’s still possible, however, to examine our actual usage practices with respect to ‘exist’, to show that they de facto involve a criterion, and then bring that criterion to bear against candidate items that some take to exist. This has been my strategy with respect to mathematical abstracta. I’ve argued [...] that such are dependent on a mathematical practice, and therefore (according to our criterion for what exists) that there are no such things. [Azzouni, 2010a, p. 96, emphasis original] In this paper, I provide a novel objection to Jody Azzouni’s view that whilst it is indeterminate what the criteria for existence are, there is a criterion that has been collectively adopted to use ‘exist’ that we can employ to argue for positions in ontology. If the objection succeeds, then it provides strong reason to think a significant contender in the debate over what, if any, the criteria for existence are and how, if at all, they can be appealed to in first-order ontological debate — Azzouni’s position — is false. My objection, however, is of broader significance. This is for at least two reasons. First, the objection has an interesting implication for both Azzouni’s mathematical nominalism, and, in turn, the dispute between mathematical platonists and nominalists simpliciter. For I argue that, of the responses to my objection considered, the one Azzouni has best reason to take renders unsound his influential arguments for mathematical nominalism. Secondly and relatedly, my objection highlights the neglected fact that there are ways in which at least certain views defended in meta-ontology can affect the plausibility of arguments for positions in both the ontology of mathematics and first-order ontology more generally. Here, my objection illustrates the under-discussed fact that assumptions made when engaging in first-order ontology regarding what, if any, the criteria for existence are, and how, if at all, such criteria can be appealed to in ontological debates, can affect the plausibility of arguments for positions over what exists. I proceed as follows. In Section 1, I present Azzouni’s view that despite it being indeterminate what the criteria for existence are, there is a collectively adopted criterion to use ‘exist’ that can be employed in ontology. I raise my objection in Section 2: Azzouni’s view has the hitherto unnoticed, counterintuitive consequence that what exists can and will change when users of the word ‘exist’ change what criteria they associate with its usage. In Section 3, I consider three responses. I argue that they are all highly problematic for Azzouni, and suggest he has good reason to take the third response. This response, however, despite avoiding the problems of the others, comes at the high cost of rendering his arguments for mathematical nominalism unsound. I conclude in Section 4. 1. AZZOUNI ON CRITERIA FOR EXISTENCE Consider: (Q2) Under what conditions, if any, does any entity $$x$$ exist? (Q2) is a meta-ontological question. That is, it is a question about assumptions made in first-order ontological disputes about what entities exist. Here, (Q2) asks a question about existence: what, if any, are the conditions that are individually necessary and jointly sufficient for any entity to exist. Answers to (Q2) provide criteria for existence (CFE).4 Possible answers to (Q2) include: Alexander’s Dictum (AD): An entity, $$x$$, exists iff $$x$$ has causal powers.5 Neo-Fregean Principle (N-FP): An entity, $$x$$, exists iff $$x$$ is the referent of a singular term in a true proposition.6 Positions like these have been employed to generate certain arguments in first-order ontology. For example, AD has been employed to argue against the existence of allegedly causally impotent entities, on the grounds that they fail to meet a necessary condition of existence. N-FP has been used (in conjunction with semantic theses regarding number terms, such as ‘1’, that treat them as singular referring expressions) to argue for the existence of numbers.7 Pace AD, N-FP, and other answers to (Q2), Jody Azzouni argues for the ‘indeterminacy of a criterion for what exists’ [Azzouni, 2010a, p. 77] and that views which maintain that there are ‘philosophically binding’ [Azzouni, 2004, p. 97] reasons to accept a given CFE are mistaken. Azzouni holds this, he claims, because of ‘the particularities of phrases like “there is”, “exist”, etc.’ [Azzouni, 2010a, p. 77]. However, as noted above, he also maintains there is a criterion for existence that users of ‘exist’ have collectively adopted, and that this criterion can be employed in disputes in ontology over what exists [Azzouni, 2010a, p. 96]. Whilst subject to varying interpretation,8 Azzouni appears committed to at least the following, which I present and explain in turn: Ontic Neutrality (ON): Uses of ‘exist’ in natural language can be ontologically relevant or ontologically irrelevant.9 Indeterminacy: It is indeterminate what the CFE are because ‘exist’ is a criterion-transcendent word.10 Independence Criterion (IC): The criterion that has been collectively adopted by competent users of ‘exist’, when ‘exist’ is used in an ontologically relevant way, is: ‘$$x$$ exists iff $$x$$ is mind- and language-independent’.11 Azzouni maintains ON captures a feature of the use of ‘exist’ in English. By ontologically relevant or irrelevant uses, Azzouni means that ‘exist’ can be used without users being concerned with, or taking themselves to be committing themselves to, the existence of some entity or entities (thereby using ‘exist’ in an ontologically irrelevant way). To illustrate, Azzouni takes (A) to be an ontologically relevant use, and (B) and (C) to be irrelevant uses, of ‘exist’: (A) Chakras really exist. People who think otherwise just are not enlightened. (B) Strategies for circumventing anger exist. Many are found in self-help books. (C) Although waltzes designed to last more than ten hours exist, people rarely dance unexpurgated versions of them. Azzouni is not claiming that the utterers of (B) or (C) will obviously deny the existence of anger-circumventing strategies or waltzes. Rather, that speakers of (B) or (C) do not use ‘exist’ to be making an ontological claim at all. The existence of the former or the latter, he claims, is not of concern when ‘exist’ is used in these sentences — whilst it is to the speaker of (A).12 Next, Indeterminacy. Azzouni restricts his discussion of (Q2) to ontologically relevant uses of ‘exist’. As such (Q2) is understood, he claims, to be asking whether ‘exist’ (when used ontologically relevantly) is co-extensive with any property-attributing phrase — such as ‘possesses causal powers’ or ‘is a referent of a singular term in a true proposition’. Azzouni [2010a, pp. 84–85] draws a distinction between two ways to answer (Q2) so understood: de jure and de facto. The former is to argue for necessary, sufficient, or necessary and sufficient conditions of existence (described in terms of other predicates) by engaging in either ‘conceptual or philosophical analysis’ of ‘exist’, or by arguing for claims about the meaning of ‘exist’ [Azzouni, 2010a, p. 91, emphasis original]. The latter attempt to answer (Q2) is to try to ‘empirically discover’ what properties all entities that exist have.13 Regarding de jure attempts, Azzouni does not clarify what he means by ‘conceptual or philosophical analysis’. Prima facie, he appears to mean analyses of ‘exist’ which purport to show that it has analytic entailments [2010a, pp. 84–88], that is, analyses that demonstrate that a term has an entailment that provides necessary and/or sufficient conditions for its correct usage due to its meaning. Consider ‘sibling’. Azzouni claims that ‘sibling’ intuitively has analytic entailments in virtue of what ordinary speakers call its meaning: something is a sibling iff it is either a brother or a sister.14 To illustrate these analyses, and their relevance to Indeterminacy, Azzouni introduces a distinction between criterion-transcendent and criterion-immanent terms. A criterion-transcendent term is one where (a) it has no analytic entailments, and (b) there are possible situations in which the criterion associated with the term change, whilst users of it judge both that the term has not changed in meaning, and that things that do not meet its new criterion never belonged to its extension [2010a, pp. 84–91]. Take ‘gold’. Azzouni maintains, firstly, that ‘gold’ lacks analytic entailments perceived by ordinary users of the term to be due to its meaning. Secondly, that there are possible situations in which the criterion associated with the use of ‘gold’ changes (e.g., from ‘yellow-coloured metal’ to ‘being composed purely of atoms of atomic number 79’ [Azzouni, 2010a, p. 95]), where users of ‘gold’ judge that it has not changed in meaning and that some objects that no longer fall under its extension (e.g., some yellow-coloured metals) never did. Criterion-immanent terms include ‘sibling’, ‘legal tender’, and ‘refrigerator’ [2010a, pp. 85, 87]. These words, Azzouni holds, do have analytic entailments. Additionally, he maintains that were the criterion associated with their use to change, users of these terms would both perceive the word to have changed in meaning, and insist that things that no longer fall under its extension were once correctly considered to do so (e.g., what was once legal tender would still be described as having been legal tender, and not that it never was). Crucially, Azzouni argues that ‘exist’ is a criterion-transcendent word. As such, on Azzouni’s [2010a, pp. 89–90] view all de jure attempts to answer (Q2) will fail. This is because if there are no analytic entailments from ontologically relevant uses of ‘exist’ (e.g., from ‘tables exist’ to ‘tables possess causal powers’), then no attempt to provide necessary and sufficient conditions as CFE in virtue of the meaning of ‘exist’ could succeed. Indeterminacy, then, appears to be an epistemic claim about attempts to answer (Q2): we cannot establish, through philosophical or conceptual analysis, whether ‘exist’ is co-extensive with other property-attributing phrases (and so what the CFE are) because ‘exist’ is a criterion-transcendent word. Finally, IC. Recall the de jure/de facto distinction, and that a de facto attempt to answer (Q2) is to try to ‘empirically discover’ what properties all entities that exist have. Azzouni maintains that the de facto approach does yield a CFE, although this criterion is not provided by any insight into the meaning of ‘exist’. In his words: [...] there are (what we take to be) robust samples of what (really) exists: people, tables, planets, etc. I’ve claimed that, de facto, the word ‘exist’ [...] is coextensive with [...] ‘not mind- or language-dependent’. This projection from the robust examples of what we (ordinarily) take to really exist provides a criterion for ontically relevant uses of ‘exist’. [2010a, p. 96] Again, consider ‘gold’. There are examples of what ordinary speakers take to be gold. Through empirical observation of these, we discover that they share the salient property of being composed purely of atoms with the atomic number 79. Users of ‘gold’, then, take the criterion associated with its use (upon this discovery) to be ‘being composed purely of atoms of the atomic number 79’ [2010a, pp. 95–96]. Similarly, Azzouni claims that the community of ordinary speakers of ‘exist’ have collectively adopted ‘mind- and language-independence’ as the CFE. Taking robust samples of what such speakers take to exist (people, chairs, planets), he maintains that ‘exist’ appears to be used as co-extensive with ‘mind- and language-independence’. Azzouni does not provide an account of exactly what these kinds of independence amount to. He does, however, provide examples of objects that he takes to fail to meet IC. These are fictional characters, mathematical abstracta, hallucinations, and dreams. He claims that in his sense of these terms, the former two are language-dependent, whilst the latter are examples of mind-dependent phenomena. Importantly for our purposes, finally recall that Azzouni defends these meta-ontological views as part of a strategy of arguing for a position in first-order ontology: mathematical nominalism. His argument against the existence of mathematical objects such as numbers and sets is that they, as abstract objects, are dependent upon mathematical practice, and that given our de facto collectively adopted CFE he defends, they do not exist. 2. AN OBJECTION In this paper, I shall assume ON, Indeterminacy, and IC. I do this to argue that even if these contentious claims are true, the way that Azzouni has claimed a collectively adopted CFE can be appealed to in arguing over what exists gives rise to a forceful objection to his view. This objection arises because of a response Azzouni provides to an objection from Yvonne Raley. She argues that if Azzouni maintains that ‘exist’ is criterion-transcendent, and that attempts to show de jure that there is some CFE fail, then Azzouni must maintain that there is no fact of the matter what exists: [...] it is not just that we do not know the answer to the question of what really exists, but rather that there is nothing to know in the first place. This is because the criterion-transcendence of ‘exists’ forces on us a non-factualism about criteria for what exists. [Raley, 2009, p. 82] Azzouni [2010a, p. 98] responds that Raley illicitly assumes that there is a fact of the matter what a term refers to only if perceived analytic entailments of that term fix its reference, and that this claim is false. Again, take ‘gold’. Azzouni’s response to Raley is that even if ‘gold’ eludes necessary and sufficient conditions associated with it in virtue of its meaning, the term does have a certain criterion associated with it. Users of ‘gold’ call objects that are composed purely of atoms of the atomic number 79 gold, and due to this fact about the use of ‘gold’ there is a fact of the matter what objects are gold [2010a, pp. 97–98]. Azzouni continues: In exactly the same way [that there is a fact of the matter what is gold], there is a fact of the matter about what exists given the criterion we actually use to apply the word. [2010a, p. 98, emphasis original, words in brackets added] Azzouni appears to be suggesting that the (we are assuming) sociological fact that ‘mind- and language-independence’ has been collectively adopted as the CFE is what determines the facts of the matter regarding what exists. This is because in the same way that things are gold, on Azzouni’s view, given the adopted criterion of ‘being composed purely of atoms of atomic number 79’, certain things exist given the collectively adopted criterion of ‘mind- and language-independence’. Importantly, then, Azzouni is not claiming that IC is only a criterion that has been collectively adopted for ‘exist’, but that this criterion somehow provides us with the facts regarding what exists. Because what exists is determined by what criterion is adopted by users of ‘exist’, Azzouni thinks IC can be employed in ontological disputes over what exists to argue, e.g., against the existence of abstracta [2010a, pp. 96–97]. Azzouni’s response has a counterintuitive entailment that the following inference reveals: (1) There is a fact of the matter about what exists given the criterion users of the word ‘exist’ have collectively adopted to use it. (2) If (1), then if users of ‘exist’ changed the criterion associated with its use, then the facts regarding what entities exist can and will change just because the criterion collectively adopted to use ‘exist’ will have changed. (3) If users of ‘exist’ changed the criterion associated with its use, then the facts regarding what entities exist can and will change just because the criterion collectively adopted to use ‘exist’ will have changed. But (3) is strongly counterintuitive: for it is prima facie implausible to think that the facts regarding whether an entity exists (e.g., whether persons exist) can and will change just because users of ‘exist’ have changed how they use the word ‘exist’. A thought experiment illustrates why this is counterintuitive. Suppose members of the community that employs ‘exist’ collectively change what they take to be samples of existents. Suppose, e.g., that they no longer believe anything other than the entities described in our fundamental physical theories exist. Tables, chairs, and people, then, will plausibly no longer be taken to exist by this community. Asking themselves what property all posits of fundamental physical theories share, they would no longer maintain that the property-ascribing phrase that is co-extensive with ‘exist’ is ‘mind- and language-independence’ but, e.g., ‘being a sub-atomic particle’. By Azzouni’s own lights such a scenario is possible, and since ‘exist’ is a criterion-transcendent word, members of this community will both think that ‘exist’ has not changed its meaning and that it also never applied to some things they previously took it to (such as tables and persons) [2010a, pp. 84–88]. But since Azzouni maintains that facts about what exists are given by the CFE a community has adopted, then he is committed to claiming that the facts regarding what exists both could and indeed have changed in our scenario because this community has changed the criterion they associate with ‘exist’. So whether, inter alia, tables or chairs exist can change (and will, depending upon what criterion is adopted) just by the users of ‘exist’ changing what criteria they associate with it. This is counterintuitive because what exists is intuitively not determined by only whatever criterion users of ‘exist’ employ to apply the word. This entailment, then, burdens Azzouni’s position with a strongly counterintuitive consequence. Before addressing my objection further, I allay a potential concern. One might insist that Azzouni may object to my reading of his quoted claim above. Another way to read his claim is to take him to be asserting that there is a trivial sense in which the things correctly described as existing when one uses the word ‘exist’ depends upon how the word ‘exist’ is used. But this sense neither entails nor suggests that what actually exists is somehow determined by competent users of the word ‘exist’ and what criteria they associate with the correct use of ‘exist’. If so, then Azzouni does not face my objection. In response, however, note that if the relevant passage is read in this alternative way, then it becomes unclear how and why the de facto CFE of mind- and language-independence can be appealed to in ontological disputes (such as those over mathematical objects) that Azzouni employs it in. This is because if the observation that there is a de facto CFE adopted by competent users of ‘exist’ did not play any role in determining what actually exists, then highlighting that there is a de facto CFE becomes only an observation about how the word ‘exist’ is used by competent speakers. How the word is used by competent speakers alone, however, is not sufficient to draw conclusions about what exists — unless how the word is used played some role in determining what actually exists. This alternative reading of Azzouni would undermine his own attempt to appeal to de facto CFE in arguing over what exists, and, in particular, his arguments against mathematical objects and other abstracta. So there is good reason to read Azzouni as I have suggested, lest Azzouni be read as undermining his own arguments against the existence of abstracta.15 3. THREE RESPONSES, AND REJOINDERS I now defend my objection to Azzouni against three responses. 3.1. Biting the Bullet? Azzouni may respond by biting the bullet and accepting that his view entails that what exists can and will change when the CFE associated with ‘exist’ changes. Two reasons render this an unattractive response for Azzouni. First, insisting upon this commitment forces Azzouni to adopt a prima facie implausible, radical anti-realism about what exists. That is, it appears to commit him to the view that (i) the facts regarding what exists are determined by whatever criterion is adopted by a community, and (ii) that what exists can and will change when a community changes the criterion associated with ‘exist’. Absent an argument for such a view, its prima facie implausibility provides reason for Azzouni to avoid this response. Second, maintaining (i) highlights what is, at best, a tension in Azzouni’s view. The tension: it is unclear how Azzouni can consistently maintain that the facts regarding what exists are determined by users of ‘exist’ collectively adopting a criterion to use the word, whilst also maintaining that what exists is ‘mind- and language-independent’ by this adopted criterion. For if the first claim is true, then the facts regarding what exists appear dependent upon either the presence of minded entities who collectively adopt a criterion for using ‘exist’, or a linguistic choice users of ‘exist’ make in associating a criterion with its usage. So what exists appears mind- or language-dependent. But if so, then how can Azzouni maintain that what exists is mind- and language-independent? Azzouni may respond that this is not the kind of mind- or language-dependence he has in mind. As discussed, however, he does not provide an account of precisely what he means by this phrase. At least, then, on this response we are owed an account of mind- or language-dependence that shows how Azzouni can consistently provide this response to my objection. These two problems, in conjunction with the already-noted counterintuitive consequence Azzouni must accept on this response, provide good reason to find it unattractive. 3.2. An Epistemic Criterion? Notice that AD, NF-P, and Azzouni’s proposed de facto CFE can be read in at least two ways. For example, AD can be taken as either of two claims: (a) that an entity exists iff it has causal powers, or (b) one has justified belief in the existence of an entity iff that entity has causal powers. The former is a metaphysical reading of AD, whilst the latter is epistemic.16 This suggests a response to my objection: Azzouni can weaken his claim that the collectively adopted criterion for ‘exist’ determines what exists to the epistemic claim that one has justified belief in the existence of an entity iff it is mind- and language-independent. If such independence is a feature shared by all the entities we (following Azzouni) take to exist, then to claim that we have justified belief in entities existing when they share this feature is prima facie motivated. This allows Azzouni to avoid the counterintuitive consequence noted in Section 2, since what exists will no longer change when the criterion associated with ‘exist’ changes. There are two reasons, however, to find this response problematic for Azzouni. The first: as a tu quoque response, Azzouni cannot make this move. For Azzouni denies that the criterion we have (allegedly) adopted to apply ‘exist’ is epistemic.17 Rather, he maintains that our criterion is a metaphysical claim about under what conditions entities exist and, for at least this reason, thinks we can plausibly appeal to this criterion when arguing over what exists. The second: if Azzouni did claim our collectively adopted CFE is epistemic, then he faces a different problem. Recall that he employs IC to claim that mathematical abstracta do not exist. This appears plausible when IC is read as the discovery of a metaphysical criterion regarding under what conditions users of ‘exist’ take entities to exist. But if Azzouni weakens his claim as suggested, then he is no longer able to argue that it is true that mathematical abstracta and other allegedly mind- or language-dependent objects do not exist. For now our collectively adopted criterion provides only the conditions under which we have justified belief in existence claims. Moreover, suppose that Azzouni decided to maintain that our collectively adopted CFE provided only the conditions under which we have justified belief in existence claims. This might seem sufficient to generate an epistemological problem for those committed to the existence of mathematical abstracta. Notice, however, that Azzouni would now need to provide an account of why these de facto CFE determine the conditions under which we have justified belief in existence claims rather than the conditions under which users of ‘exist’ believe that they have such justified belief. Absent such an account, it is, at best, unclear why we should take the de facto CFE associated with the use of ‘exist’ to reveal the conditions under which we have justified belief in existence claims. This response, then, is unattractive: not only is Azzouni unlikely to take it, but if he did then, firstly, he would be unable to claim — on the basis of this observation about what criterion has been adopted for ‘exist’ alone — that mathematical nominalism is true. And, secondly, Azzouni would owe us an account of why our collectively adopted CFE determines the conditions under which we have justified belief in existence claims, as opposed to when users of ‘exist’ merely believe they are so justified. 3.3. Remaining Silent on What Exists? Azzouni faces the objection of Section 2 because he insists that IC determines the facts regarding what exists and, for that reason, can be appealed to in arguing over the truth of existence claims. Azzouni could relinquish this feature of his view. That is, he could maintain ON, Indeterminacy, and IC but deny that the criterion we have (allegedly) adopted to apply ‘exist’ determines what entities exist. Instead, he could maintain that de facto mind- and language-independence is taken to be the feature in virtue of which extant objects exist, but that this observation at best tells us what users of ‘exist’ take to exist (and not what determines the facts regarding what does exist). Azzouni, then, can remain silent on what entities exist on the basis of IC alone, whilst taking himself to have shown that de facto users of ‘exist’ do take mind- or language-dependent entities not to exist. This latter claim, if true, may be of interest (e.g., in arguments that certain existence claims are in conflict with what CFE users of ‘exist’ employ), but it will no longer be taken to reveal what entities in fact exist. Of the responses considered, I suggest remaining silent on what exists is the most plausible response for Azzouni. For, firstly, he would avoid the form of anti-realism discussed about what exists and the prima facie tension in his view identified in Section 3.1. Secondly, he avoids having both to claim that the criterion that has been adopted by users of ‘exist’ is epistemic, as well as having to provide an account of why this adopted criterion supplies the conditions under which we have justified belief about what exists (rather than the conditions under which we believe we have justified belief about what exists). This response, however, comes at a high cost to Azzouni: if Azzouni remains silent on what exists, surrendering the claim that what exists is given by our collectively adopted CFE, then he can no longer appeal to IC to argue against the existence of abstracta. This undermines Azzouni’s own argument against mathematical abstracta, rendering it unsound. Since in no longer taking collectively adopted CFE to determine the facts about what exists, Azzouni would be forced here to deny himself a crucial feature of his argumentative strategy for mathematical nominalism (namely, his appealing to IC to argue that mathematical abstracta do not exist in virtue of failing to meet it). But as we have seen, it is, at best, unclear how else Azzouni can avoid the counterintuitive consequence of his view raised here. Since Azzouni can retain commitment to ON, Indeterminacy, IC, and avoid the problems with the other responses considered, he appears to have good reason to take this option. But if he does, he pays the heavy price of rendering his arguments in ontology against the existence of mathematical abstracta that appeal to IC unsound. 4. CONCLUSION I conclude that the best response available to Azzouni to block my objection is to remain silent on what exists. As noted, however, this comes at a high cost. For in denying that a collectively adopted criterion to apply ‘exist’ determines the facts of the matter regarding what exists, Azzouni must relinquish appeal to IC in arguing that mathematical abstracta do not exist — thereby accepting that his arguments appealing to IC cannot alone demonstrate their non-existence. Whilst this response allows Azzouni a way out of the objection here considered, and a way to avoid the problems identified with the other responses discussed, it comes at the devastating price of rendering his attempts to show that mathematical abstracta do not exist unsuccessful. I note, in closing, the broader significance of my objection. I have argued that there is good reason to believe that Azzouni’s view that there is a collectively adopted criterion for using the word ‘exist’ that can be appealed to in ontological disputes is false. But, first, in doing so I have shown that the best response considered here to my argument against Azzouni’s view entails that his argument for mathematical nominalism fails. This is a significant result, since if true it undermines an influential argument in the philosophy of mathematics presented by Azzouni in the wider debate between mathematical nominalists and platonists. Secondly, my objection to Azzouni has provided a case that illustrates that the interaction between meta-ontological views about criteria for existence, and appeals to such criteria in arguments for first-order ontological positions, deserves careful attention. Defending a view about what the criteria for existence are, and how such criteria can be appealed to in ontology, appears a prima facie attractive strategy to argue for or against the existence of some entity. For example, if causal efficacy is a necessary condition of existence, then arguing that some purported entity does not meet this condition provides a strong argument against the existence of that entity. Appealing to views about criteria for existence, however, provides a potential target for one’s opponents. And the strategy can backfire: one’s meta-ontological views about criteria for existence can, in fact, render one’s argument(s) for a view in ontology unsound. I have argued this is so in the case of Azzouni’s argument for mathematical nominalism. Clearly, there is much more to be said about the relationship between criteria for existence and appeals to them in arguments in ontology. But, to be sure, the dangers of this argumentative strategy in ontology are faced by any who adopt it. Footnotes 1 Henceforth, I use ‘mathematical objects’ to refer to mathematical objects, properties, structures, and relations. 2 This formulation correctly allows that there are versions of mathematical nominalism, such as certain forms of fictionalism, that maintain that mathematical objects do exist but not as abstracta. Here, I am concerned only with a form of nominalism that denies that mathematical objects exist simpliciter. 3 Mathematical nominalists of various kinds include, inter alia, Field [1980; 1989]; Hellman ; Yablo ; Bueno ; Leng . For an excellent overview of nominalism in the philosophy of mathematics, including detailed discussion of Azzouni’s form of nominalism, see [Bueno, 2014]. 4 [Azzouni, 1998, pp. 2–4]. There, Azzouni distinguishes between CFE and criteria for what a discourse ontologically commits one to. Here, I am concerned only with the former. 5 [Cotnoir and Edwards, 2015, p. 119]. Sometimes called the Elaetic Principle (see [Armstrong, 1978]). The name ‘Alexander’s Dictum’, popularized by Jaegwon Kim [2003, p. 348], is due to the work of Samuel Alexander . 6 [Cotnoir and Edwards, 2015, p. 120]. Commonly associated with the work of Bob Hale & Crispin Wright (see, e.g., their ). 7 See [Sider, 2007, pp. 201–232] and [Hawley, 2007, pp. 233–249] for discussion. 8 See [Raley, 2009, pp. 73–83]; Rosen, 2006; Burgess, 2004; Melia, 2005; Maddy, 2007, pp. 397–400]. 9 [Azzouni, 2010a, pp. 74–101]. Azzouni uses the phrase ‘ontic relevance/irrelevance’. In what follows, I use ‘ontologically relevant/irrelevant’ to describe the alleged phenomenon captured by ON. 10 [Azzouni, 2010a, pp. 74–101; Raley, 2009, pp. 77–79]. 11 [Azzouni, 2004, pp. 82, 91–99, 112–113; 2010a, pp. 95–100]. Azzouni labels mind- and language-independence ‘ontological independence’. 12 [Azzouni, 2010a, pp. 81–84]. ON is famously denied by W.V.O. Quine  and his followers (such as Peter van Inwagen [2009, 2014]). 13 [Azzouni, 2010a, pp. 84–85]. I shall return to de facto attempts to answer (Q2) when explaining IC below. 14 Assuming the species in question has exactly two sexes. Azzouni’s use of ‘meaning’ is explicitly taken to be whatever ordinary users of words intuitively judge to be meant when they claim that a word like ‘sibling’ ‘means brother or sister’ [2010a, p. 85]. In this way, analyses of analytic entailments can provide us with criterion for being a sibling. 15 I am extremely grateful to Jody Azzouni for discussion on this point, and to an anonymous referee for raising this concern. Azzouni agrees that I have not misread his position, though he has indicated that there may be a case to be made for his being able to endorse the reading I have argued against without facing the objection raised to it. I hope to learn of this response in his possible future work, and to engage with it thereafter. 16Azzouni [2004, pp. 81–87] draws this distinction between forms of CFE. 17 In [2004, pp. 81–87], Azzouni considers the possibility that the CFE collectively adopted by users of ‘exist’ are epistemic in the sense here outlined. However, he concludes that we should deny this is so, declaring instead: There seems no way around it. We’re forced to an excursion into metaphysics — specifically into a study of necessary and sufficient conditions for what exists — because only then can we raise (and perhaps answer) the question of whether things that exist can be the sorts of things that we can learn about in these various [...] ways. [p. 87, emphasis original] I thank an anonymous reviewer for prompting me to clarify this. REFERENCES Alexander, Samuel [ 1920 ]: Space, Time, and Deity . Vol. 2 . London : Macmillan . 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Published: Apr 3, 2018
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