Neo-Fregean Ontology

Philosophical Perspectives, 20, Metaphysics, 2006

NEO-FREGEAN ONTOLOGY∗

Matti Eklund Cornell University

I. Introduction Neo-Fregeanism in the philosophy of mathematics consists of two main parts: the logicist thesis, that mathematics (or at least branches thereof, like arithmetic) all but reduce to logic, and the platonist thesis, that there are abstract, mathematical objects. I will here focus on the ontological thesis, platonism. NeoFregeanism has been widely discussed in recent years. Mostly the discussion has focused on issues specific to mathematics. I will here single out for special attention the view on ontology which underlies the neo-Fregeans’ claims about mathematical objects, and discuss this view in a broader setting. In neo-Fregean writings—from Michael Dummett’s 1950s writings to Crispin Wright (1983) and Bob Hale (1988) to more recent texts by Wright and Hale, such as those collected in Wright and Hale (2001)—one can find many different considerations relevant to their defense of the platonist thesis that there are abstract, mathematical objects. One can easily single out four. There is the appeal to reconceptualization, with precedent in Frege’s Grundlagen, §64. There is appeal to the idea that, somehow, truth is constitutively prior to reference—this is an idea I will call priority. More recently, see especially Wright and Hale (2000), there is appeal to general considerations concerning when implicit definitions are acceptable. Also more recently, there is appeal to more epistemological considerations: the focus is here not on direct arguments for the existence of mathematical objects but on arguments for taking us to be justified or entitled to believe in the existence of mathematical objects. Here I will focus on the more traditional defenses of neo-Fregean ontology. Moreover, I will focus almost exclusively on the priority idea. The reasons for focusing on priority rather than reconceptualization are two. First, reconceptualization has been more widely discussed in the secondary literature. Second, the priority idea is more readily generalizable so as to provide an acceptable foundation for the ontology of abstract objects in general. Concerning priority, I will then argue, first, that this

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view on ontology has wide ramifications regarding what exists—specifically, it commits the neo-Fregean to a radically promiscuous ontology—and, second, that these commitments threaten to be inconsistent.1

II. Quinean and Neo-Fregean Approaches to Ontology There are several schools within analytic philosophy that can be said to, in different ways, approach ontology via language and thus be linguistic approaches to ontology. Although I will focus on the neo-Fregean school, let me begin by briefly presenting what we may call the Quinean school (although Quine himself, both through his thesis of ontological relativity and through his scientism, is a slightly problematic proponent of that school). According to the Quinean school what we should take there to be is what our best theory of the world is ontologically committed to, which in turn is what it quantifies over. We arrive at our best theory of the world by considering which theory is empirically adequate and the most theoretically virtuous.2 Nothing in the description so far makes it clear just how this approach to ontology is linguistic, except in the limited way that the identification of what a theory is ontologically committed to with what it quantifies over relies on certain views on language. But what makes the approach appropriately called linguistic is this. There are two steps in determining what there is, for our Quinean. The first step is that of determining which the best theory of the world is. The second step is that of determining what that theory quantifies over. It is the second step that characteristically is the province of philosophy. It may not be possible to read off ontological commitments directly from the surface structure of the statements of a theory. Some sort of investigation into the structure of the sentences of the theory is needed—some sort of analysis or regimentation—and it is here that philosophy of language comes in. There are two aspects of this familiar Quinean approach to ontology that I wish to stress. First, there is the emphasis on analysis and regimentation. Second, on this approach, we should approach the question of what exists via the question of which sentences are true, rather than vice versa. This reverses what might perhaps be a more natural thought: that we come to find out which ontologically committing sentences are true via coming to know what there is.3 I mention the Quinean approach to ontology mainly as a foil. What I will focus on is the neo-Fregean approach, championed by theorists like Wright, Hale, and Dummett.4 The neo-Fregean approach can perhaps be best described if we consider the issue the neo-Fregeans have devoted the most attention to, that of the ontology of arithmetic. As Frege in effect showed, the axioms of full second-order Peano arithmetic can be derived from Hume’s Principle (HP), The number of Fs = the number of Gs iff the Fs and the Gs are equinumerous,

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plus definitions.5 Many who believe that numbers exist will also accept that HP is necessarily true.6 But neo-Fregeans like Crispin Wright and Bob Hale have wanted to claim more on behalf of HP. They want to say that HP constitutes an explanation of the concept of number; and that it explains the concept in such a way that nominalist doubts about the existence of numbers are removed. The idea is that, HP being an explanation of number, the existence of numbers in a certain sense demands nothing more of the world than that equinumerosity obtains. It is (something like) a conceptual truth that if equinumerosity obtains then there are numbers. From a different perspective than that of the neo-Fregean, one may hold that what is trivially true, or what is true solely on the grounds of how the concept number is explained, is only that if numbers exist, HP holds. Wright and Hale, however, argue that there is no need for such conditionalization. Here Frege’s context principle—only in the context of a sentence do words have meaning—is supposed to come in.7 The idea, to state it briefly, would be that HP can be shown to endow all sentences in which number words occur with meaning, and to do it in such a way that some such sentences which require for their truth that numbers exist (like the sentences on the left hand side of instances of HP) are true. Any further question about whether there really are numbers asks for the meaning or reference of number words taken in isolation, flouting the context principle. In this way, the context principle is regarded as having significant implications both regarding what exists (there exist abstract objects) and regarding the nature of the relevant existence claims (they can have the status of something like conceptual truths). The neo-Fregean, like the Quinean, approaches the question of what exists via the question of which sentences are true, rather than vice versa. But the neoFregean is somehow more radical than the Quinean. The Quinean does not take her outlook to entail that the existence of numbers is at all trivial: there is a non-trivial question of whether our best theory of the world needs to quantify over numbers. What I will consider is just how the neo-Fregean goes beyond the Quinean, and what is supposed to justify this more radical outlook.

III. The Underlying Metaontological View Appealing to the context principle in this way may seem uncomfortably like trying to pull a rabbit out of a hat. Can so much really be had for so little? The context principle is on its face a purely semantic thesis. Taken as a semantic thesis, its justification is straightforward (whether, in the end, perfectly compelling). When we use language, we use whole sentences. It is in virtue of this use that sentences, and the expressions that go to make them up, get their meaning. The meaning of an individual word then consists entirely in its contribution to the meanings of sentences of which it is part. This semantic thesis can serve to defend the legitimacy of contextual definitions. There is also an epistemological claim in the theoretical vicinity: we do not epistemically access

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the world by relating individual signs to objects but by relating whole sentences or thoughts to the world. This epistemological claim is something the Quinean subscribes to.8 (When I utter “rabbit” in the presence of a rabbit scurrying by, that is on this view best seen as my uttering a one-word sentence. Insofar as it is ever correct to say that I refer to objects, that is so because of the structure of my sentences.9 ) It is also a claim that the neo-Fregean subscribes to. It is via whole sentences or whole judgments that we respond to the world around us. But then our knowledge of a particular type of entity—say, numbers—is not primarily a matter of direct access between simple symbols in language or thought and those entities but rather of relations between whole judgments and the states of affairs to which they correspond. The semantic and epistemological claims may be problematic enough. There are both problems concerning what exactly they come to, and why we should accept them. But let me set aside whatever problems arise already with respect to these claims. For even if the semantic and the epistemological claims are both true, it does not obviously follow that the ontological claims mentioned above are justified. So how can ontologically radical conclusions be taken to follow fairly immediately from the context principle? Neo-Fregeans have traditionally relied on two separate strands of thought.10 One, appeal to reconceptualization, is what has attracted the most attention in recent literature. Apart from a brief summary I shall not say much about it here. The other, which is what I will call priority, is what I will focus on. First, reconceptualization. Much of the literature on neo-Fregean philosophy of mathematics focuses on HP as an example of reconceptualization. The idea would be that the two sentences flanking the biconditional of a particular instance of HP somehow express the same content: it is only that this content is ‘conceptualized’ differently in the two sentences. If the two sentences have the same content it is clear how HP can guarantee that numbers exist: we arrive at this conclusion just by ‘reconceptualizing’ a sentence of the form “the Fs and the Gs are equinumerous.” One can then, the idea is, go from the truth of a sentence of the form (1) the Fs and the Gs are equinumerous to the truth of a corresponding sentence of the form (2) the number of Fs is identical to the number of Gs by a purely conceptual transformation. (The idea is from Frege’s Grundlagen, §64.) Since the sentences have the same content, the truth of (2) requires nothing

more of the world than the truth of (1) does, despite the fact that (2) but not (1) quantifies over numbers. The neo-Fregean view here has the appearance of inconsistency. Surely, one may think, it cannot be the case that (1) and (2) have the very same content while still only one of them quantifies over numbers. The only way that (1) and (2) can have the very same content is if either they both

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quantify over numbers or neither does: if the surface form of at least one of them is misleading. Naturally, the neo-Fregeans are aware of this challenge, and seek to defuse it. Another issue with respect to appeal to reconceptualization concerns what relation two sentences must stand in to each other for these sentences properly to count as reconceptualizations of each other. Logical equivalence seems to be a necessary condition. But is it sufficient? If the answer is yes, all logically true sentences count as reconceptualizations of each other. If the answer is no, then hard questions arise concerning just what more specific condition might count as sufficient, while not ruling out the neo-Fregeans’ favorite examples of reconceptualizations. I do not have much of substance to add to the already rather extensive literature critically discussing reconceptualization.11 However, one point deserves stressing. If appeal to reconceptualization does the trick when it comes to defending platonism in the case of arithmetic, that is because in this case there is an appropriate content (the Fs and the Gs are equinumerous) to reconceptualize. For neo-Fregeanism to be an acceptable philosophy of mathematics, similarly appropriately reconceptualizable contents must be found across the board. One may legitimately be skeptical, even while finding the neo-Fregean’s points about arithmetic persuasive. Skepticism about generalizability should become more serious still if the neo-Fregean intends to give a general account of the ontology of abstract objects.12 More attention will be devoted to the idea of priority, to which I now turn.13 My main reasons for focusing on priority are two. First, priority has been somewhat neglected in the secondary literature. Second, priority seems to me anyway to be the more fundamental consideration, in part because of the risk that the appeal to reconceptualization is not sufficiently generalizable. In (1956), Dummett says, If a word functions as a proper name, then it is a proper name. If we have fixed the sense of sentences in which it occurs, then we have done all there is to be done towards fixing the sense of the word. If its syntactic function is that of a proper name, then we have fixed the sense, and with it the reference, of a proper name. If we can find a true statement of identity in which the identity sign stands between the name and a phrase of the form ‘the x such that Fx’, then we can determine whether the name has a reference by finding out, in the ordinary way, the truth-value of the corresponding sentence of the form ‘There is one and only one x such that Fx’. There is no further philosophical question whether the name—i.e., every name of that kind—really stands for something or not.14

And in Wright (1983) there are similar passages: . . . when it has been established, by the sort of syntactic criteria sketched, that a given class of terms are functioning as singular terms, and when it has been verified that certain appropriate sentences containing them are, by ordinary criteria, true, then it follows that those terms do genuinely refer.15

100 / Matti Eklund According to [the “thesis of the priority of syntactic over ontological categories,” which Wright presents as implied by the context principle], the question of whether a particular expression is a candidate to refer to an object is entirely a matter of the sort of syntactic role which it plays in whole sentences. If it plays that sort of role, then the truth of appropriate sentences in which it so features will be sufficient to confer on it an objectual reference; and questions concerning the character of its reference should then be addressed by philosophical reflection on the truth-conditions of sentences of the appropriate kind.16

Focus primarily on the emphasized parts. One claim here is that there is an object to which a given singular term refers if certain sentences in which the singular term occurs in the right way (paradigmatically, identity sentences) are true. This by itself, although it can be denied, is not a terribly radical thesis. Most would agree that there is this connection between reference and truth.17 This thesis by itself does not really get the neo-Fregean anywhere. A nominalist opponent can agree on this much, yet insist that as a matter of fact no atomic arithmetical sentence is true, for the reason that no arithmetical term refers. This brings us to the more radical thesis of the neo-Fregeans, which is that truth is, so to speak, constitutively prior to reference. The nominalist’s envisaged point puts the cart before the horse, according to the neo-Fregean, for the nominalist says that atomic arithmetical sentences are never true, for arithmetical terms never refer. The neo-Fregean would rather say that arithmetical terms do refer, for they occur in true sentences of the right kind. And not only is this a point about our knowledge of reference—the point is not simply that we know that arithmetical terms refer because we know some sentences of the right kind to be true—but it is about what it is for a term to refer. As Wright puts it in (1992), discussing the case of abstract objects, “The irresistible metaphor is that pure abstract objects . . . are no more than shadows cast by the syntax of our discourse.”18 The distinction I am here drawing here between two different neoFregean theses is drawn also by Hartry Field (1984). In a discussion of Wright’s views, Field distinguishes, helpfully, between the “weak priority thesis,” to the effect that any expression which syntactically functions as a singular term also semantically functions as a singular term, and the “strong priority thesis,” which adds that “what is true according to ordinary criteria really is true, and any doubts that this is so are vacuous.”19 Consider, to further illustrate the neo-Fregean view, a negative argument from Wright (1992), the target of which is Field’s nominalism, which involves an error theory about mathematics. Wright argues that a theorist who, like Field, holds that all (atomic) sentences of mathematics are untrue is faced with an uncomfortable choice: either to relegate mathematics to bad faith or to say that sentences of mathematics satisfy some norm of correctness distinct from truth. The first is simply implausible; and Field also takes the latter option. But Wright asks, with respect to this option, why should we not say that the truth of a mathematical sentence just consists in its satisfying the relevant norm of

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correctness? Perhaps this is a good question. But Field would appear to have an immediate answer: for (atomic) mathematical sentences to be true there must be mathematical objects for the terms to refer to whereas for mathematical sentences to satisfy the relevant norm of correctness—for Field, conservativeness20 —no such thing is necessary. This brings me to the point I wish to stress. It is that the point that I have envisaged would be Field’s presupposes the natural order of explanation of truth and reference: it is because the relevant objects do not exist that the relevant terms do not refer and hence no atomic mathematical sentences are true. Wright, by contrast, presupposes that this cannot be the proper order of explanation. Rather, mathematical terms refer because some mathematical sentences are true; and some atomic mathematical sentences are true since the relevant norm of correctness is satisfied. According to Wright’s understanding of Frege’s context principle, the import of this principle is that we should conceive of the relation between truth and reference in the way indicated. It is the thoughts underlying Wright’s (1992) argument against Field, and which are spelled out in the above quotations from Dummett and Wright, that I will call priority. No doubt it is still somewhat obscure exactly what priority is supposed to amount to. But I think that despite this obscurity some telling observations can be made. In the next section I will spell out some consequences of the thesis of priority. Later, I will discuss some problems.

IV. Priority Generalized Given what was said in the previous section concerning priority, there is a crucial point crying to be made. There appears to be nothing special about arithmetic here: the neo-Fregean’s reasoning, such as it is, is perfectly generalizable to other domains. Consider a location where, given that there are cars at all, certainly there is one. (A location where there are “simples arranged carwise,” in Peter van Inwagen’s helpful phrase.) Let the name ‘Herbie’ purport to refer to the car, if any, in this location. Does ‘Herbie’ refer? Is there such an object as Herbie? Some philosophers would deny that; but obviously most would say yes. The neo-Fregean would seem committed to siding with the yes-sayers. For her, the question of whether Herbie exists is a question of whether some sentences in which ‘Herbie’ occurs in the right way are true; and that in turn is a question of whether, by ‘ordinary criteria’ these truth-conditions of these sentences are satisfied. But certainly it seems there can be a successful practice of assertively uttering such sentences. The thought would be this. Any inclination to think that the practice would have to be unsuccessful would seem to have to rely on the belief that there simply is nothing there for the term ‘Herbie’ to refer to. But this consideration, putting reference before truth, so to speak, the neo-Fregean must rule inadmissible.

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In the case just described, the position of the neo-Fregean is the commonsensical one. But consider some more controversial cases. First, consider incars: would-be objects almost like cars except for the difference that they only exist when or insofar as they are inside garages. (“Incar” is supposed to be a substance sortal and not a phase sortal. The claim of an incar believer would not be merely that cars in garages have the property of being incars, but that there is some object that, necessarily, goes out of existence as a car leaves a garage.21 ) Suppose I point toward my car and say, “Let us call the incar here ‘inHerbie’.” Does ‘inHerbie’ refer? Is there an object such as inHerbie? Some philosophers say there is (there are cars inside garages and there is no contradiction involved in supposing that there should be such objects as incars); others would claim that the supposition that inHerbie exists offends their sense of reality. By exactly the same reasoning as above, the neo-Fregean would seem to have to side with the former philosophers. For her, the question of whether inHerbie exists is a question of whether there are sentences containing ‘inHerbie’ in the right place which are true; and that in turn is a question of whether these sentences satisfy the relevant norm of correctness. But certainly it seems that there can be a successful practice of assertively uttering such sentences. Or return to the case of abstract objects. Let ‘q’ purport to refer to a Quinean New Foundations (NF) set (where for our purposes the only important thing about NF is that it is widely agreed to be a counterintuitive and poorly motivated set theory). Does ‘q’ refer? Is there such an object as this set? Provided NF is consistent, it appears there can be a successful practice of assertively uttering sentences where ‘q’ occurs in the right place. But then ‘q’ refers; NF sets exist. Generally, the neo-Fregean must accept into her ontology all sorts of strange objects: she has to accept a radically promiscuous ontology. Appeal to priority appears to commit the neo-Fregean to an ontology where, for a given sortal F, Fs exist just in case (a) the hypothesis that Fs exist is consistent, and (b) Fs do not fail to exist, simply as a matter of contingent empirical fact. (The second condition rules out that the neo-Fregean would be committed to the existence of yetis. The neo-Fregean may be committed to intuitively strange objects; but of course she is not committed to believing in objects that we have empirical reasons not to believe in.) Call this ontological view maximalism. And for simplicity, I will say that according to maximalism, some purported objects, the Fs, exist so long as they satisfy minimal conditions. The reason why the neo-Fregean would be committed to maximalism is that for any object that exists according to maximalism, there can be a singular term occurring in the right way in sentences satisfying the relevant norm of correctness.22,23 An objection to the claim that priority entails maximalism across the board is that successful reference to concrete objects requires more than does successful reference to abstract objects, and that hence existence is not as cheap in the case of concrete objects as in the case of abstract objects. Commonsensically, priority would appear to get things exactly backwards. But maybe priority is not actually so counterintuitive in the special case of mathematics, because of the special

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role singular terms there play. We can, so to speak, verify, e.g., an arithmetical sentence “without first identifying anything as the referents of the numerical singular terms,” to use a nice formulation from Stephen Schiffer.24 But then we have reason to treat discourse about mathematical objects and discourse about ordinary objects differently. Certainly this reasoning has something going for it, both as a reading of Wright and in absolute terms. In principle, substantive discussion of it would be required. Let me just make the following remarks. (Some more remarks will follow in the next section.) (i) Even if this objection should succeed, the point would still stand that the neo-Fregean is committed to a maximalist view with respect to pure abstracta. This restricted maximalism too faces the problems I will go on to discuss. The problems that will be brought up either concern pure abstracta or have clear analogues in the case of pure abstracta. (ii) Consider again Wright’s (1992) argument against Field’s nominalism. The argument is meant to have as a general upshot that an error theory is not likely to succeed in any philosophically interesting case. A main theme in Wright’s (1992) is that traditional antirealist paradigms, like non-factualism and error theory, are bankrupt. The argument against the error theory route is precisely that discussed here. But for this general point to have been established, the reasoning must be properly generalizable. A third objection to taking the neo-Fregean to be committed to maximalism is this. Consider the last two cases discussed, incars and NF sets. In both cases, it is arguably counterintuitive that the objects in question should exist (and the point of the examples is that these are supposed to be counterintuitive entities—so should you not find them counterintuitive just change the examples). When arguing that appeal to priority commits the neo-Fregean to maximalism, I assumed that priority anyway militates in favor of the existence of these objects. The idea was that assertive use of the relevant sentences could be successful; counterintuitiveness was beside the point. But someone might fasten on the neoFregean’s talk of ordinary criteria, and insist that by ordinary criteria sentences requiring for their truth incars and NF sets are not true, as evidenced precisely by the counterintuitiveness of these objects. This would be to ascribe to the neo-Fregean a certain kind of relativism.25 For consider a hypothetical linguistic community that doesn’t find it objectionable to suppose that incars and NF sets should exist. By the lights of the neo-Fregean as we now conceive of her, the names of incars and NF sets that this community uses would refer, even if our names of incars and NF sets do not.26 In a very enlightening overview of the types of problems that face the neoFregean philosophy of mathematics, Fraser MacBride ascribes this relativist view to the neo-Fregean (under the name quietism). MacBride says, The rejectionist [the theorist who holds that abstraction principles cannot guarantee the existence of anything non-linguistic] assumes that the structure of states of affairs is crystalline—fixed quite independently of language. By contrast, the [neo-Fregean] assumes that states of affairs lack an independent structure,

104 / Matti Eklund that states of affairs are somehow plastic and have structure imposed on them by language. As a consequence it is unintelligible for the rejectionist that the method of abstraction might be ensured to disclose additional structure in a state of affairs. By contrast, from a [neo-Fregean] point of view it is inevitable . . . that the method of abstraction will succeed.27

Then he goes on to suggest that the neo-Fregean should be taken to embrace a kind of ‘quietism’ according to which there is no “intelligible question to be raised about whether there might be some ulterior failure of ‘fit’ between language and reality,” so “we can only submit to the norms of our discourse and record whether according to them the use of language determines the presence of objects.”28 On the view that MacBride discusses, cars exist but incars do not; not because that is what the ultimate structure of reality is like (on this view language and world cannot be compared in this way) but because our language or conceptual scheme so to speak carves out cars but not incars as objects. This is a certain kind of relativist view, for the reason earlier given. I have three remarks to make on the identification of the neo-Fregean as a quietist. First, there is no hint in Wright’s neo-Fregean writings that the existence of numbers should be in any way relative to conceptual schemes. (It might be retorted, though, that there is nothing in the writings of the neo-Fregeans that suggest maximalism.) Second, maximalism does not figure in MacBride’s discussion. That seems to me an important omission. Consider MacBride’s own characterization of the neo-Fregean’s ontological outlook. Maximalism respects central points there. Maximalism respects the idea that states of affairs are somehow “plastic.” (No matter how I attempt to “carve up” a state of affairs into objects I succeed, so long as I do not lapse into inconsistency.) Likewise, it respects the idea that it is inevitable—formal problems to the side—that the method of abstraction will succeed. And maximalism too rejects the notion that there can fail to be a fit between language and reality (problems of consistency aside). The only element in MacBride’s characterization that maximalism does not respect is the relativist-sounding claim about the norms of our discourse. But this just raises the question of how this relativist-sounding talk is supposed enter in to begin with. A general point is that there is an important distinction to be drawn between different ways of approaching ontology via language. First, there is the quietist way: what there is, is what our language ‘carves out’. Second, there is the way suggested by maximalism, as motivated by priority: what there is, is a matter of what can be successfully referred to in some possible language or other.29 Third, more importantly, here is an argument to the effect that the quietist view collapses.30 Suppose Fs are some sort of would-be entity such that by the quietist’s lights, “Fs exist” comes out false in our language, but there is another language L in which “Fs EXIST” (where ‘EXIST’ is the other language’s counterpart of our ‘exist’) is true. According to the norms of their discourse Fs exist, to use MacBride’s formulation. Now let ‘a’ be a term of L which purports to refer to an F. We can suppose, without restriction, that there are true atomic

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sentences of their language of the form ‘P(a)’. That is to say, the quietist should by her own lights be able to conclude that there are such sentences, seeing that the norms of discourse might differ between languages. But if ‘P(a)’ is a true sentence of L, then ‘a’, of L, must refer, for it must do so in order for atomic sentences in which it occurs to be true. But now we have concluded in our language that ‘a’ of L refers. But ‘a’ of L purports to refer to an F. So we can conclude that Fs exist. (The idea behind the argument is simple. The quietist holds that when the existence of Fs s not ruled out by the contingent empirical facts—witness again yetis—there is a possible language with singular terms successfully referring to Fs. But for there to be such terms, Fs must exist. But then quietism is false.) After this longer discussion of the possibility that the neo-Fregean might be committed to quietism, let me turn to a fourth objection that might be raised against the claim that the neo-Fregean relies on an ontological claim committing her to maximalism. HP is a central plank in the neo-Fregean’s philosophy of arithmetic. But given the neo-Fregean’s ontological outlook as it has been laid out here, it is sufficient for the existence of numbers that the hypothesis that there are numbers is consistent (given that the question of the existence of numbers is not hostage to contingent empirical fact). But what then is the relevance of HP? (Can one not then equally well rely on the consistency of the axioms of Peano Arithmetic (PA)?)31 The answer is that HP is not in fact relevant to the neo-Fregean’s platonism. Insofar as HP has a role to play, it is in the defense of the neo-Fregean’s logicism. If HP really can serve as an “explanation of the concept of number”—if it affords some sort of implicit definition of “number”—then the concept of number is explained in terms of equinumerosity, which is a purely logical concept. The axioms of PA do not do anything similar, it can be argued. In fact, the point I am making here should be clear already from the structure of Wright’s (1983). In the first part of the book, Wright is concerned to establish neo-Fregean platonism, and there the context principle is at the center of the discussion but HP hardly mentioned. In the latter part of the book Wright is concerned to establish neoFregean logicism, and it is there that HP has a central role. The point about where and how HP comes in is still worth stressing as, first, this shows that there is still a significant neo-Fregean thesis left very much alive even if the claims about HP cannot be made good on, and second, the attention to HP obscures the general nature of the neo-Fregean’s distinct ontological claim. It deserves to be stressed that insofar as it is regarded as a problem for the interpretation of the neo-Fregean as a de facto maximalist that HP ends up not playing a central role, this should also be regarded as a problem for the interpretation of the neo-Fregean as a de facto quietist. Suppose the quietist view is correct. Then it is because of how we conceptually carve up reality that “there are numbers” is true. Why is there then any need to go via HP? MacBride explicitly mentions Rudolf Carnap as a quietist. But Carnap, seemingly reasonably, took his metaontological outlook entirely to trivialize the question of the existence of numbers.32 Given his view, we can for instance just lay down the axioms of PA

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as analytic truths about numbers, so long as the axioms are consistent. (In other words, if it is regarded as a problem for my interpretation of the neo-Fregean that abstraction principles are accorded no pride of place, MacBride’s interpretation clearly does no better on this score.) In the next section I will further elucidate the priority thesis, by discussing Hilary Putnam and Ludwig Wittgenstein on the philosophy of mathematics. The impatient reader can skip this section. The main thread will be taken up again in section V, where I discuss problems for the maximalism to which the priority thesis commits the neo-Fregean.

V. Interlude: Putnam on Wittgenstein’s Philosophy of Mathematics In order further to elucidate what priority is supposed to come to, let me compare what Putnam, often with reference to Wittgenstein, says about mathematics. Some typical remarks are the following: [Wittgenstein] clearly pooh-poohs the idea that talk of numbers—either ordinary numbers or the so-called “transfinite numbers”—is in any way analogous to talk about objects. If we think that way, then of course we will think that set theory has discovered . . . not just objects, and not just intangible objects, but an enormous— an unprecedentedly large—quantity of intangible objects. And the very vastness of the universe which set theory appears to have opened up for our intellectual gaze will then be part of its charm; but Wittgenstein thinks this reason for being charmed is a bad one. Wittgenstein believes—and I think he is right—that it does not make the slightest sense to think that in pure mathematics we are talking about objects.33 I see the attempt to provide an Ontological explanation of the objectivity of mathematics as, in effect, an attempt to provide reasons which are not part of mathematics for the truth of mathematical statements and the attempt to provide an Ontological explanation for the objectivity of ethics as a similar attempt to provide reasons which are not part of ethics for the truth of ethical statements; and I see both attempts as deeply misguided.34

These two passages point in different directions. Compare the following two theses: (a) Non-existence. Mathematical terms do not refer to objects; there is no special class of objects that mathematical assertions are about; mathematical assertions cannot be said to correspond to reality. (b) Non-explanation. The appearance that we somehow explain the nature of mathematical discourse by appeal to the fact that mathematical terms refer, that mathematical assertions are about mathematical objects, and that true mathematical assertions correspond to reality, is illusory.

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The first of the two passages (“it makes no sense to think we are talking about objects”) suggests Non-existence. The latter passage, by contrast, suggests Non-explanation. I take it that whatever Putnam means when talking about an “Ontological explanation,” it is compatible with the fact that ontological explanations of the objectivity of mathematics are futile that there should be numbers and other mathematical objects. The same ambiguity that can be found in Putnam can be found also in Wittgenstein’s writings on mathematics. Compare the following passages: Suppose we said first, ‘Mathematical propositions can be true or false.’ The only clear thing about this would be that we affirm some mathematical propositions and deny others. If we then translate the words ‘It is true . . .’ by ‘A reality corresponds to . . .’—then to say a reality corresponds to them would say only that we affirm some mathematical propositions and deny others. We also affirm and deny propositions about physical objects.—But this is plainly not Hardy’s point. If this is all that is meant by saying that a reality corresponds to mathematical propositions, it would come to saying nothing at all, a mere truism: if we leave out the question of how corresponds, or in what sense it corresponds.35 If you have a mathematical proposition about ℵ 0 , and you imagine you are talking about a realm of numbers,—I would reply that you aren’t as yet talking about a realm of anything, in the most important sense of “about.” You are only giving rules for the use of “ℵ 0 .”36

The second passage here quoted suggests Non-existence; the first one suggests Non-explanation. Focus on Non-explanation, mainly because it is this thought that connects with neo-Fregean themes, but also because, some careless formulations to the contrary, it seems to be what Putnam and Wittgenstein chiefly have in mind.37 I will first further motivate and elucidate the thesis. Then I will turn to what are the consequences of the thesis. Let me first turn to what might appear to be a rather different issue: the debate over the correspondence theory of truth. Specifically, consider a correspondence theory that takes seriously the commitment to a special realm of truth-makers—facts—of traditional versions of the theory. On this kind of theory of truth, there is, for each true statement, a fact making this statement true. A well-known line of objection to this kind of correspondence theory runs as follows.38 Whereas this idea of correspondence might seem explanatory of the notion of truth when we are considering certain simple statements, say, “the cat is on the mat,” where we might seem to have an independent conception of the relevant truth-maker—the fact that the cat is on the mat—it fails to be of explanatory value when we consider already somewhat more complex statements, such as negative statements, hypothetical statements, universal statements, as well

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as statements involving concepts less straightforward than those expressed by ‘cat’ and ‘mat’. In these cases, the objection goes, we have no conception of the truth-maker for a statement P save as “the fact that P.” On the basis of this sort of consideration, some might want to dispense with facts altogether. But what this consideration, if sound, most immediately indicates is that the proposed account of truth fails to explain what truth is. The explanans is not explanatorily prior to the explanandum. If there are facts, we cannot explain what truth is by appeal to them. The correspondence theory of truth is a theory of what it takes for a certain class of linguistic expressions to have a particular semantic value. Analogously to the correspondence theory of truth we have what we may call the correspondence theory of reference. Just as, on the correspondence theory of truth, we explain what it is for a statement to be true by appealing to what in the world that it “corresponds to”—in this case, what makes it true—we may, in the case of reference, explain what it is for a singular term to refer to whatever it is it refers to by appealing to what in the world it “corresponds to,” the object that is its referent.39 The correspondence theory of reference might seem so obviously correct as to be entirely trivial. We explain what ‘Fido’ refers to by appealing to what it the world it refers to, Fido. Even the label ‘theory’ may seem a misnomer. But analogous remarks can be made about the correspondence theory of truth. It is sometimes said that on one reasonable understanding of the claim that a proposition is true if and only if it corresponds to the facts, this is a mere truism. On this view, the correspondence theory of truth is only controversial insofar as it seeks to take the right hand side of this equivalence as explanatorily prior and thus to explain the nature of truth by appeal to facts. It is then that the criticism mentioned above has bite: we seem not to have an independent conception of facts, except possibly in a limited range of cases.40 Just as the correspondence theory of truth can be understood either as something much like a truism or as a substantive theory, the correspondence theory of reference can be understood either way. And clearly, insofar as it is supposed to amount to a genuine theory of reference, it must be understood the latter way. And then an objection analogous to that lodged against the correspondence theory of truth can be raised. If one considers only singular terms such as ‘Fido’ one may be led to think that we can, in general, explain the semantics of singular terms by appeal to the fact that names refer to objects, where objects is an independently understood ontological category. But explaining the contribution to truth conditions made by other kinds of singular terms, say, the singular term ‘two’, by saying what they refer to—in this case, that the term refers to the number two—is non-explanatory, because the only grip we have on what the number two is, we have in virtue of our grasp of sentences of which terms purporting to refer to this number are part. To abandon the correspondence theory of reference in light of these considerations is to adopt the thesis of Nonexplanation.

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Suppose now that Non-explanation is true, at least when it comes to mathematical discourse. What should one then say about the question of whether numbers exist? (i) One thing one might say is that numbers then don’t exist: hence adopt also Non-existence. Only if the existence of numbers plays the right sort of explanatory role (whatever it is—and certainly there are good questions regarding what this comes to) should we say there really are numbers. But the stance that Putnam gives voice to in the second passage quoted not only primarily suggests Non-explanation; it is even in tension with Nonexistence. It is in the spirit of these remarks to take the question of whether there are numbers to be one settled by criteria internal to mathematics; but clearly by these criteria there are numbers. Numbers, moreover, are objects in the sense that is at issue: they are in the range of the first-order variables.41 (ii) Putnam, as is well known, defends a thesis of conceptual relativity— a view much like the quietism MacBride ascribes to the neo-Fregean—to the effect that “[T]he logical primitives themselves, and in particular the notions of object and existence, have a multitude of different uses rather than one absolute ‘meaning’”; that “[T]here isn’t one privileged sense of the word ‘object’.” 42 This conceptual relativity thesis meshes well with Non-explanation. If different things can truly be said to ‘exist’ under different conceptual schemes, then it seems natural further to say that facts about what entities there really are cannot serve to explain the nature of our practices. It is harder to reconcile Non-existence with the thesis of conceptual relativity. Suppose the thesis of conceptual relativity is true. Then suppose further that it is common practice to take there to be numbers: in this sense our conceptual scheme licenses us to take numbers to exist. I would have thought that statements to the effect that numbers exist, made ‘within our conceptual scheme’, then are true. The most discussed problems regarding the existence of numbers, like Benacerraf’s problem about our lack of causal contact with would-be mathematical objects, would seem to be beside the point given that we conceive of the existence of numbers this way.43 For numbers to exist, it is, roughly, sufficient that the hypothesis that they do so is coherent, that it is not in conflict with the empirical facts (in the way the hypothesis that there are yetis is), and that numbers exist ‘according to our conceptual scheme’. Naturally, the metaphysical question of what is must be the case for numbers to exist is different from the epistemic question of what knowledge of the existence of numbers requires. But it is clear that the relevant metaphysical issues are directly relevant also to the epistemology of mathematics. (iii) Thirdly, Non-explanation can be used in favor of maximalism with respect to mathematics. The idea would be that if the existence of a type of mathematical object doesn’t, by Non-explanation, have any explanatory role to play but is, so to speak, epiphenomenal, then the question of the truth of the corresponding sentences is not responsible to independently constituted facts about what objects there are. But then, one might think, for these purported

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objects to exist it is sufficient that they satisfy minimal conditions. It is this third option that corresponds to priority, and its supposed relation to maximalism. We get by this argumentative route to maximalism across the board if we think analogues of Non-explanation are true regardless of what kinds of objects we are talking about. Generalizing to the case of material objects may seem hopelessly illadvised. Putnam and Wittgenstein seem clearly to want to contrast mathematical discourse with discourse about material objects. There seems also to be a good philosophical reason for this: we causally interact with material objects but not with mathematical objects. But although it is a truism that we interact causally with material objects, it can be argued that the identities of the objects are irrelevant to the nature of our interactions with them and that therefore regarding reference to objects as fundamentally explanatory is as misguided in the case of material objects as in the case of pure abstracta. Some philosophers like to say that we impose structure on reality when conceptualizing it. To my mind, such talk is objectionable: it makes it sound as if we somehow conceptually help create reality. But maybe there is anyway a kernel of truth to such talk: the structure of our thought—specifically, how our thought carves up the world—is not to be explained by appeal to how the world is independently structured. The claim is not that Non-explanation entails maximalism. There are several ways to go given Non-explanation. My aim in stressing the route from Nonexplanation to maximalism—one possible route to the latter thesis—is rather to elucidate both in general how maximalism can be justified and how priority can be taken to entail maximalism. Non-explanation and priority are evidently related theses.

VI. Incompatible Objects I have argued that one metaontological consideration underlying the claims of the neo-Fregeans is the priority thesis, and that the priority thesis leads to maximalism. I will now turn to a serious problem with the maximalist view. One of the most prominent objections to neo-Fregeanism in the philosophy of mathematics is what has become known as the bad company objection. Suppose HP really is something like a conceptual truth. Then, it is natural to think, it is so in virtue of its form. If HP is something like a conceptual truth then all (second-level) abstraction principles, principles of the form (ABS) f(F) = f(G) iff R(F,G), where f is a function from concepts to objects and R is a relation between concepts, are conceptual truths.44 But this general claim is false. Frege’s Basic Law V,

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The value-range of F = the value-range of G iff F and G are coextensive, is of this form but is inconsistent, and hence cannot be conceptually true. A revised idea might be that all consistent principles of the form (ABS) are conceptual truths. But this revision does not get around the basic problem. Consider—to take the original example used to press this objection, from George Boolos—the parity principle, The parity of F = the parity of G iff F and G differ evenly [where two concepts differ evenly if an even number of things fall under one but not the other]. If HP succeeds in implicitly defining number, this principle should be regarded as succeeding in implicitly defining parity. Both principles are satisfiable. But HP is satisfiable only in infinite domains and the parity principle is satisfiable only in finite domains, so these principles are not co-satisfiable. The problem for the neo-Fregean is that it would seem that whatever can be claimed on HP’s behalf can also be claimed on behalf of the parity principle. After all, they are of the same form, and they are both consistent.45 Since Boolos first pressed the objection, there has been much discussion of it. Wright has a reasonable reply to the objection in its original form. He can stress that HP, but not the parity principle, is conservative, and this is what decides things in HP’s favor.46 This appeal to conservativeness is a potential tiebreaker. However, even if Wright’s reply is reasonable there are problems looming: there are more serious versions of the bad company objection. Here is one from Stewart Shapiro and Alan Weir (1999). Consider two principles of the form ι(F) =ι(G) iff [(φF ∧ φG) ∨ ∀x(Fx ↔ Gx)].

(Where ı is the function from concepts to objects being defined.) For the first, let φ = ‘has the size of a limit in the series of inaccessibles’ and for the second, let φ = ‘has the size of a successor in the series of inaccessibles’. Then, given the axiom of strong inaccessibles (that for every cardinal κ there is a larger strong inaccessible), both principles are satisfiable, but the first principle is satisfiable only at limits in the series of inaccessibles and the second principle is satisfiable only at successors in this series.47 Appeal to conservativeness cannot serve as a tiebreaker here, even if it can in the case of HP versus the parity principle. The revised objection does not directly involve HP, but raises a theoretical problem regarding the exact conditions under which principles of HP’s form succeed in defining concepts. What I would like to stress in regards to the bad company objection, and the literature on it, is that if I am right about the neo-Fregean’s reliance on priority, and about priority being best seen as entailing maximalism, this

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objection is merely one special case of a very general problem. The maximalism to which the neo-Fregean is committed is false if there are Fs and Gs such that both Fs and Gs exist, given maximalism, but the hypothesis that Fs and Gs coexist leads to contradiction. (Let us say that the Fs and Gs in a case like this are incompatible objects.) Numbers, conceived of as characterized by HP, and parities, conceived of as characterized by the parity principle, threaten to be incompatible objects. In this case, one might perhaps argue that since HP is more liberal than the parity principle—it is true in the domains of the higher cardinality—it should on maximalist grounds be preferred. Already this example shows that maximalism as formulated is false. But the tweak sufficient to get around the parities example might be straightforward.48 The problem presented by Shapiro-Weir example is more serious: there is no obvious tiebreaker in that case.49 There is a more drastic response to the problems thus far introduced. The neo-Fregean maximalist faced with these examples can get around the problem by deeming implicit definition by abstraction principles illegitimate. Given the centrality of abstraction principles to neo-Fregeanism, this response might hurt. But there is nothing, either in maximalism as such or in the neo-Fregean justification of it, that requires that implicit definition by abstraction principles should be admissible. It is possible, at least in principle, for the neo-Fregean maximalist to take the route indicated (whether or not there are non-ad hoc reasons for her to take this route). But however that may be, abandoning abstraction principles to their fate would not in fact get around the problem of incompatible objects. The bad company objection is, as already urged, just an instance of a more general problem. Here is another problem of incompatible objects, slightly modified from Schiffer (2003). Certainly numbers exist, on a maximalist view. But then consider anti-numbers, where the concept of an anti-number is the concept of an abstract object whose existence both supervenes on anything one likes and rules out the existence of numbers.50 Anti-numbers too would seem to satisfy maximalism’s conditions for existence. But trivially, numbers and anti-numbers cannot coexist. Corresponding problems arise with respect to material objects. Consider strong atoms: objects without parts which, necessarily, exist only if no complex objects do. The worry is that maximalism entails both that complex objects exist and that strong atoms do.51 Schiffer’s way out of the problem of anti-numbers exactly parallels Wright’s way out of the parities problem: he argues that numbers but not anti-numbers satisfy a conservativeness requirement which can be independently motivated. But here too it is doubtful whether appeal to conservativeness, even on the assumption that it works well as far as it goes, can deal with all variants of the relevant problem. Consider a problem similar to that of anti-numbers. Let us define xhearts to be almost like hearts except that they exist only if xlivers do not, and let us define xlivers to be almost like livers except that they exist only if xhearts do not. Then it seems that xhearts exist according to maximalism and so

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do xlivers, but xhearts and xlivers cannot coexist.52 Perhaps there is a different way around this problem: rule against both xhearts and xlivers on the ground of the circularity in the definitions. Someone taking this route must defend it on two related scores. It is clear that some circular definitions really succeed.53 She must defend her view that there is something specific that is bad with these ones. Moreover, maximalism is a radically promiscuous ontological view. Someone taking the route described must also show that her strictures against xhearts and xlivers are consistent with maximalism. Clearly, the purported examples of incompatible objects can be multiplied.54 One may think accordingly that it is more appropriate to seek a general response to the problem than to deal with purported counterexamples individually. But what might a general response look like? Here is a suggestion: what we have here is a special kind of indeterminacy. Suppose we have a purported case of incompatible objects to which the maximalist has no other satisfactory answer. For concreteness suppose the xhearts/xlivers case to be such. Then the maximalist can say that it is simply indeterminate whether it is xhearts or xlivers that exist. She can even insist that she has independent reason to say this. Can xhearts consistently exist, given the empirical facts? That is, do xhearts satisfy the minimal conditions that some purported objects must satisfy to exist, given maximalism? It can be said: they do only if xlivers do not exist. Mutatis mutandis for xlivers. There is nothing that determines whether it is xhearts or xlivers that exist. So it is indeterminate which exist. But indeterminacy the maximalist can live with. The problem was that it seemed that the maximalist would have to say that both xlivers and xhearts exist, which is impossible. There are some problems regarding exactly what the nature of the indeterminacy would be. This indeterminacy cannot be semantic, on any natural understanding of semantic indeterminacy: for in order for the indeterminacy to be semantic, objects of both kinds will have to exist. (Consider an ordinary kind of semantic indeterminacy. We want to say that it is semantically indeterminate whether ‘mass’, as used in Newtonian physics, denotes rest mass or relativistic mass. For this to be so, both rest mass and relativistic mass must exist. If either failed to exist, it would not be a candidate referent for ‘mass’.55 ) So the indeterminacy would seem to have to be ontological: and ontological indeterminacy is widely regarded with suspicion. How can the world itself be indeterminate? What does it even mean to say that it is? However, the kind of ontological indeterminacy that would be at issue here would have a special source. For the neo-Fregean, the source would be that indicated by the considerations regarding the nature of questions about what there is: the ontological indeterminacy would arise because the question of the existence of objects is secondary to the question of truth, and in the cases under consideration where the questions of truth are left indeterminate by the facts to which the neo-Fregean allows reference, the questions of what objects there are likewise left indeterminate. So we have some kind of broadly linguistic explanation of the ontological indeterminacy that arises.

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The world’s indeterminacy would not, so to speak, be a brute metaphysical fact. Maybe this is sufficient to make the postulated ontological indeterminacy prima facie palatable. But there are other problems. The indeterminacy move is decidedly unattractive as a response to some of the examples discussed. Consider the strong atoms example. The indeterminacy move, in response to this example, would yield almost complete indeterminacy with respect to which material objects exist. But then again, there may well be other ways around the strong atoms example. Moreover, there are other problems with the indeterminacy move. Consider strengthenings of examples like the one concerning xhearts/xlivers: let dhearts be almost like hearts, except they exist only if dlivers determinately do not exist, and let dlivers be almost like livers, except they exist only if dhearts determinately do not exist. Saying that it is indeterminate whether dhearts exist and indeterminate whether dlivers exist leads straight to contradiction.56 (The dhearts/dlivers objection is not immediately decisive against the proposal. It is modeled on parallel examples from the literature on the liar paradox. What to say about the dhearts/dlivers objection in the end depends on how those examples are best accounted for.) I have here argued that the neo-Fregean via the doctrine of priority is led to maximalism, and that maximalism faces problems with what I have called incompatible objects—a kind of generalization of the bad company objection. Others before me have argued that the neo-Fregean’s philosophical considerations in favor of the existence of numbers generalize uncomfortably. Field (1984) asks whether, if the existence of numbers flows from the explanation of the concept of number, the existence of God might not equally well from the explanation of the concept of God as the concept of the perfect being, along the lines of the ontological argument for the existence of God. Similarly, he asks whether, for the ancient Greeks, the truth of “Zeus is throwing thunderbolts” by Wright’s reasoning followed from “It thunders.” Divers and Miller (1995)—in a discussion focusing mostly on Wright (1992)—argue that neo-Fregean arguments would establish the existence of fictional characters. These arguments may themselves be compelling. But they allow the neoFregean more wiggle room than does an appeal to incompatible objects. The general neo-Fregean strategy with respect to these examples, whether in the end successful or not, is best illustrated by consideration of the most simple-minded of these examples, that of Zeus. The neo-Fregean might say that some possible term ‘Zeus’, perhaps even one in some respects like the Greeks’ term ‘Zeus’ can be guaranteed to refer by the fact that sometimes it thunders. But this does not mean that the Greeks’ term ‘Zeus’ refers: for they would not have conceived of Zeus’s existence in consisting simply in there being thundering. The reason the ‘Zeus’-example seems problematic, insofar as it does, is that we are conflating the artificial term ‘Zeus’ with the actual term ‘Zeus’. A similar “divide and conquer” strategy can be applied to other cases. Maybe a suitably well-crafted definition of “God” would yield that “God,” thus defined, by the neo-Fregean’s lights refers: but the strategy is then to indicate how differently this “God” behaves

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from our “God.”57 As for fictional characters, the neo-Fregean could say of our discourse ostensibly committing us to the existence of fictional characters that we do not there actually aspire to literal truth. Perhaps, the neo-Fregean could say, a discourse which is like our fiction discourse but aspires to literal truth would, by the argument of Divers and Miller, be successful and the relevant terms would refer: so in that sense there are fictional characters. But what we must keep in mind is that these entities need not thereby be referents of our actual fictional terms.58

Notes ∗

1. 2. 3.

4.

5. 6.

7. 8. 9. 10. 11.

I have presented earlier versions of this paper, or closely related material, at University of Oslo, Gothenburg University, MIT, University of Stockholm, University of Rochester and Syracuse University. Thanks to the audiences for helpful comments. Moreover, thanks to David Liebesman, Øystein Linnebo, Ted Sider and Zolt´an Szabo´ for helpful discussions of these matters. Separate discussion would be needed of the newer neo-Fregean ideas. See e.g. Rayo (2003) for some such discussion. See further also footnote 12. See e.g. Quine (1948) and (1951). The claim that we should approach existence via truth is trivial if among the relevant sentences we count sentences of the form “Fs exist.” But as stressed in the text, the Quineans focus on what we must quantify over. This makes the point less trivial. See especially Dummett (1956), Wright (1983), Hale (1988), and Wright and Hale (2001). There are qualifications to be made here. Dummett has come to distance himself somewhat from what I will call neo-Fregeanism: he is more of a paradigmatic neo-Fregean in early papers like his (1956) piece “Nominalism,” and ch. 14 of his (1981). There are also distinctions to be made between early and late versions of Wright and Hale. Wright and Hale are most obviously neoFregean in their early books, Wright (1983) and Hale (1988), respectively. Their statements in the essays collected in Wright and Hale (2001) are somewhat more guarded. Though I will focus mostly on Wright’s and similar views, it is also worth comparing the neo-Fregean views of Reck (1997) and Linnebo (2005, forthcoming, ms). Compare too the discussion in Rosen (1993). For discussion, see Parsons (1965) and Wright (1983). Why just ‘many’? Why is not HP obviously true, once doubts about the existence of numbers are bracketed? Because HP actually has consequences we might be doubtful about; primarily, perhaps, that there is a number of all the numbers there are. See Boolos (1997), p. 313f. See Frege (1884/1980), pp. x, 71, 73, 116. See especially Hylton (2000) and (2004). See Hylton, op. cit. As remarked above, recent writings focus also on other considerations. For further discussion, see footnote 12. For some important criticisms of the neo-Fregeans’ ideas here, see Potter and Smiley (2001), Potter and Smiley (2002) and Fine (2002).

116 / Matti Eklund 12. This may be the place to register some worries concerning the newer neo-Fregean ideas, the appeal to implicit definition and the turn to ideas in epistemology. When, as in Wright and Hale (2000), the neo-Fregeans focus on a general theory of implicit definitions, they consider the question of when an ontologically committing implicit definition may be acceptable and provide the thinker with apriori justification or entitlement to believe the ontological commitments incurred. They argue that HP satisfies the rather strict requirements on such a definition. But again there is a worry concerning generalizability: unless similarly satisfactory principles can be found in the cases of all other abstract objects we would wish to accept, the resulting philosophy of abstract objects is radically incomplete. As for appeal to ideas in epistemology, the worry is that the conclusion of such considerations is too weak to be of much interest. For even if a thinker may be apriori justified in believing in numbers on the basis of, e.g., HP, there may for all that be powerful arguments for nominalism such that these arguments defeat the justification for belief in numbers. What the neo-Fregean who takes this general route needs, over and above a convincing argument for prima facie apriori justification, is either (a) an argument that this justification is indefeasible, or at any rate (b) that the arguments for nominalism are unpersuasive. Consider in this connection an argument from Wright (1990). Wright there charges the nominalist who rejects HP because of its consequence that there are abstract objects with a non-sequitur: “it cannot always be true that the consequences of a statement must be verified independently before that statement may be regarded as known—if it were, advancement of knowledge by inference would be impossible” (161). But what the nominalist is likely to hold is not merely that the claim that there are abstract objects has not been independently verified, but more strongly that we have strong reasons not to believe it. 13. Other critical discussions of the priority idea can be found in Field (1984), Divers and Miller (1995), and MacBride (2003). I remark on these discussions below. 14. Dummett (1956), p. 40f; my emphasis. Compare too Dummett (1981), ch. 14, especially pp. 497, 504 and 509. 15. Wright (1983), p. 13f; my emphasis. 16. Wright (1983), p. 51f; my emphasis. 17. Although some theorists may disagree already about this. Moreover, there can be significant disagreement about which sentences are of the right kind. For example, some theorists may hold that “Vulcan = Vulcan” is true even if Vulcan is an empty name. 18. Wright (1992), p. 181f. 19. Field (1984), pp. 153ff. As Field notes, much of the discussion in Wright serves only to support the “weak priority thesis”—to the extent that it is unclear how the strong priority thesis is meant to be supported. Similar remarks are appropriate concerning Hale (1988). 20. Where, roughly, a mathematical theory T is conservative over nominalist theories iff for any collection of nominalistically acceptable statements N, T+ N has no consequence statable in purely nominalistic language that is not a consequence of N itself. 21. The example is from Hirsch (1976), p. 362. A note on the “almost like” in the characterization. The noted difference between incars and cars arguably entails that there are other differences as well. For example, does an incar

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22. 23.

24.

25. 26.

27. 28. 29.

30. 31.

32. 33. 34. 35.

36. 37. 38.

39.

cost as much as the corresponding car costs? The important thing about the example is that incars are supposed to exist only when or insofar as the corresponding cars are inside garages; and that they then are ontologically controversial. Compare what Field (1984) and Divers and Miller (1995) say about Wright. I will discuss these authors further at the very end. There are significant problems concerning the appropriate formulation of maximalism. For example: What does ‘consistency’ amount to? How exactly is the reference to empirical facts in condition (b) to be understood? Here I set these problems aside, serious though they may be, and rely on an informal understanding. Schiffer (2003), p. 78. In the relevant passage, Schiffer discusses precisely the difference between mathematical discourse and discourse about ordinary objects from the perspective of neo-Fregeanism. A kind of relativism specifically about ontological questions, that is. A kindred idea would be Putnam’s doctrine of “conceptual relativity,” discussed later. Here is a difficulty I slide over in the main text: If this other community uses ‘incar’ differently from us in the way indicated—do they really believe in incars (as opposed to some incar-like entities they call ‘incars’)? MacBride (2003), p. 127. Ibid., p. 127f. Some of the passages on Frege’s views on ontology in Dummett (1981) sound distinctly relativist. For example, Dummett says, “we should say that, for Frege, the world does not come to us articulated in any way; it is we who, by the use of our language (or by grasping the thoughts expressed in that language), impose a structure on it” (p. 504). It is hard not to see a focus on “our” in this passage. However, as also is clear from the passage, Dummett is here expounding Frege’s thought: and it would be odd, to say the least, if Dummett attributed a relativist claim to Frege without even remarking on it (which he does not). This is a much abbreviated version of an argument discussed at more length in Eklund (forthcoming a) and (forthcoming b). Compare too Hawthorne (2006). ¨ Godel’s theorem of course suggests that there are problems with respect to our knowledge of the consistency of PA. But the same goes for the consistency of HP, so there is no disanalogy here. Carnap (1950). Putnam (2001), p. 150. Putnam (2004), p. 3. Wittgenstein (1975), p. 239. The mathematician Hardy apparently wanted to say that mathematical statements are about mathematical objects in a a way analogous to how ordinary empirical statements are about material objects. Wittgenstein (1975), p. 251. See my (forthcoming c) for further discussion of Putnam. This is for instance a theme in Strawson (1950). See also Blackburn (1984), p. 233f. The discussion of correspondence in Wright (1992), especially ch. 3, is also relevant here. Compare Eli Hirsch’s discussion of the “referential correspondence theory” in (2002a), p. 57. It is not clear that Hirsch has in mind exactly what I here dub the correspondence theory of reference, but there certainly are similarities.

118 / Matti Eklund 40. See e.g. the discussion of the “correspondence platitude” in Wright (1992). It deserves to be mentioned that even with the claim of explanatory priority set to the side, it is not clear that the correspondence theory has the status of a truism. Slingshot arguments may still be brought to bear against it. See e.g. Neale (2001) for discussion. 41. Compare here the following passage from Putnam (1992): “Wittgenstein is not puzzled, as many philosophers are, about how we can ‘refer’ to abstract entities . . . .For Wittgenstein the fact is that the use of number words is simply a different use from the use of words like cow. Stop calling three an ‘object’ or an ‘abstract entity’ and look at the way number words are used, is his advice” (p. 168). 42. The quotes are from Putnam (1987), p. 71 and Putnam (1994), p. 30, respectively. 43. See Benacerraf (1973). 44. There are also abstraction principles like The direction of line a = the direction of line b iff a and b are parallel.

45. 46. 47. 48.

49.

50. 51. 52.

These are not of the form (ABS), since the function referred to on the left hand side is a function from objects to objects, not from concepts to objects. This is what might be called a first-level abstraction principle. The abstraction principles discussed in the main text are all second-level. The function f is one from concepts to objects. See Boolos (1990). Wright (1997), p. 297. Shapiro and Weir (1999), p. 319f. “Might be.” I am not saying that it is in fact straightforward. I am setting aside whatever problems there may be concerning formulation just because there are anyway worse problems awaiting. There are further issues along these lines. It might be suggested that the maximalist should look to Kit Fine’s (2002) theory of abstraction principles. Let cp(κ) be the number of cardinals less than κ. Let a cardinal κ be unsurpassable if 2cp(κ) ≤ κ. If we assume further that abstracts generated by different abstraction principles are identical only if the associated equivalence classes are the same. Then Fine shows that all non-inflationary and predominantly logical abstraction principles are jointly satisfiable in a model of unsurpassable cardinality. (Where an abstraction principle is non-inflationary if the number of equivalence classes under the equivalence relation used on the principle’s right hand side does not exceed the number of objects in the domain; and the logicality of an abstraction principle is a matter of the equivalence relation being invariant under permutations of objects.) However, one potential problem concerning making use of this idea is that the principles in the class of abstraction principles that Fine singles out cannot be jointly satisfied in V, the set-theoretic universe. For discussion, see Shapiro (2005) and Fine (2005). Cf. Schiffer (2003), p. 55. Schiffer’s own example (which he in turn attributes to Kit Fine and Joshua Schechter) is that of fictional and anti-fictional entities. Many thanks to Ted Sider for the strong atoms example. On the “almost like,” compare footnote 21 on incars. The important thing is that xhearts, if they exist, are colocated but not identical with hearts; and mutatis mutandis for xlivers and livers.

Neo-Fregean Ontology / 119 53. 54. 55. 56. 57. 58.

See e.g. Yablo (1993). The liar paradox suggests further examples. See my forthcoming (d). See Field (1973). Many thanks to Stephen Yablo for the dhearts/dlivers example. See, too, Wright’s own remarks on the ontological argument, in (1990), p. 163f. The way I read Wright’s response to the problem of fictional entities—see Wright (1994)—this is essentially the response he gives (though in other terms).

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