In the recent literature there has been some debate between advocates of deflationist and fictionalist positions in metaontology. The purpose of this paper is to advance the debate by reconsidering one objection presented by Amie Thomasson against fictionalist strategies in metaontology. The objection can be reconstructed in the following way. Fictionalists need to distinguish between the literal and the real content of sentences belonging to certain areas of discourse. In order to make that distinction, they need to assign different truth-conditions (...) to the real and the literal content. But it is hard to see what more is required for the literal content to be true than for the real content to be true. So, fictionalism is an unsatisfactory position. Here I offer a novel reply to Thomasson’s challenge. I argue that the literal and the real content need not be distinguished in terms of their truth-conditions; rather, they can be distinguished in terms of their different subject-matters, leaving it open whether their truth-conditions coincide or not. I explain how replying to Thomasson’s objection is crucial for deepening our understanding of fictionalist strategies in metaontology. (shrink)

I here defend a theory consisting of four claims about ‘property’ and properties, and argue that they form a coherent whole that can solve various serious problems. The claims are (1): ‘property’ is defined by the principles (PR): ‘F-ness/Being F/etc. is a property of x iff F’ and (PA): ‘F-ness/Being F/etc. is a property’; (2) the function of ‘property’ is to increase the expressive power of English, roughly by mimicking quantification into predicate position; (3) property talk should be understood at (...) face value: apparent commitments are real and our apparently literal use of ‘property’ is really literal; (4) there are no properties. In virtue of (1)–(2), this is a deflationist theory and in virtue of (3)–(4), it is an error theory. (1) is fleshed out as a claim about understanding conditions, and it is argued at length, and by going through a number of examples, that it satisfies a crucial constraint on meaning claims: all facts about ‘property’ can be explained, together with auxiliary facts, on its basis. Once claim (1) has been expanded upon, I argue that the combination of (1)–(3) provides the means for handling several problems: they help giving a happy-face solution to what I call the paradox of abstraction , they form part of a plausible account of the correctness of committive sentences, and, most importantly, they help respond to various indispensability arguments against nominalism. (shrink)

According to the deflationary theory of truth, to assert that a statement is true is just to assert the statement itself. For example, to say that ‘snow is white’ is true, or that it is true that snow is white, is equivalent to saying simply that snow is white, and this, according to the deflationary theory, is all that can be said significantly about the truth of ‘snow is white’.

The general tendency or attitude that Dreier 2004 calls creeping minimalism is ramping up in contemporary analytic philosophy. Those who entertain this attitude will take for granted a framework of deflationary or minimal notions – principally semantical1 and ontological – by means of which to analyse problems in different philosophical fields – e.g. theory of truth, metaethics, philosophy of language, the debate on realism and antirealism, etc. Let us call sweeping minimalist the philosopher affected by creeping minimalism. The framework of (...) minimal notions that the sweeping minimalist takes for granted encompasses, for instance, the concept of truth, reference, proposition, fact, individual, and property. Minimal notions are characterized in terms of general platitudinous principles expressed by schemata like the following (cf.: 26): ‘S’ is true if and only if S; ‘S’ is true if and only if ‘S’ corresponds to the facts; a has the property of being P if and only if a is P. Where ‘S’ and ‘a is P’ stand for sentences satisfying superficial constraints of truth-aptitude (i.e. sentences in declarative form subject to communally acknowledged standards of proper use), and.. (shrink)

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. Neo-Fregeanism 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. (shrink)

Hartry Field’s epistemological challenge to the mathematical platonist is often cast as an improvement on Paul Benacerraf’s original epistemological challenge. I disagree. While Field’s challenge is more difficult for the platonist to address than Benacerraf’s, I argue that this is because Field’s version is a special case of what I call the ‘sociological challenge’. The sociological challenge applies equally to platonists and fictionalists, and addressing it requires a serious examination of mathematical practice. I argue that the non-sociological part of Field’s (...) challenge amounts to a minor reformulation of Benacerraf’s original challenge. So, I contend, Field’s challenge is not an improvement on Benacerraf’s. What is new to Field’s challenge is as much a problem for the fictionalist as it is for the platonist. (shrink)

In his paper in the same volume, Sider argues that, of maximalism and quantifier variance, the latter promises to let us make better sense of neo-Fregeanism. I argue that neo-Fregeans should, and seemingly do, reject quantifier variance. If they must choose between these two options, they should choose maximalism.

Neologicists have sought to ground mathematical knowledge in abstraction. One especially obstinate problem for this account is the bad company problem. The leading neologicist strategy for resolving this problem is to attempt to sift the good abstraction principles from the bad. This response faces a dilemma: the system of ‘good’ abstraction principles either falls foul of the Scylla of inconsistency or the Charybdis of being unable to recover a modest portion of Zermelo–Fraenkel set theory with its intended generality. This article (...) argues that the bad company problem is due to the ‘static’ character of abstraction on the neologicist's account and develops a ‘dynamic’ account of abstraction that avoids both horns of the dilemma. 1 Introduction2ion Reconstructed3 From Bad to Worse4 Abstraction Reconceived5 Bad Company Made GoodA AppendixA.1SemanticsA.2LogicA.3ConsistencyA.4Strength. (shrink)

A realist view of numbers often rests on the following thesis: statements like ‘The number of moons of Jupiter is four’ are identity statements in which the copula is flanked by singular terms whose semantic function consists in referring to a number (henceforth: Identity). On the basis of Identity the realists argue that the assertive use of such statements commits us to numbers. Recently, some anti-realists have disputed this argument. According to them, Identity is false, and, thus, we may deny (...) that the relevant statements commit us to numbers. The present paper argues that the correct linguistic analysis of the relevant number statements supports the anti-realist view that Identity is false. However, as will further be shown, pace the anti-realist, this analysis does not establish that such statements do not commit us to numbers after all. (shrink)

Dummettian anti-realism–the refusal to endorse bivalence–is generally thought to be associated with idealism This paper argues that this is only true of the position developed by early Dummett. In a later manifestation Dummettian anti-realism is better thought of as providing the logic for anti-realisms of an error theoretic kind. Early on Dummett distinguished deep from shallow arguments for giving up bivalence: deep arguments followed a strong ‘sufficiency’ reading of Frege’s context principle, and made the sentence the primary vehicle of meaning. (...) Enriched by an account of truth in terms of verifiability, deep arguments implied a form of linguistic idealism. From within a perspective that had already made ontology relative to theory, it was natural for the difference between realism and idealism to hinge on the notion of truth for sentences. Having given up the distinction between deep and shallow arguments against bivalence, Dummett, post 1990, asserts that every rejection of bivalence leads to a form of anti-realism. In the intervening years, he has also come to cast doubt on the sufficiency reading of the context principle, particularly as developed by Crispin Wright. These doubts open the space for some more simple-minded criticisms of this strong version of the context principle, developed in the paper. Once the sufficiency reading of the context principle is given up, arguments for failing to assert bivalence come to hinge on beliefs about ‘genuine’ existence. These forms of anti-realism cannot be taken to imply idealism. Rather, in many cases they will be the expression of views of an error-theoretic kind, that are quite compatible with a thorough-going rejection of idealism. (shrink)

This paper is a reply to George Boolos's three papers (Boolos (1987a, 1987b, 1990a)) concerned with the status of Hume's Principle. Five independent worries of Boolos concerning the status of Hume's Principle as an analytic truth are identified and discussed. Firstly, the ontogical concern about the commitments of Hume's Principle. Secondly, whether Hume's Principle is in fact consistent and whether the commitment to the universal number by adopting Hume's Principle might be problematic. Also the so-called `surplus content' worry is discussed, (...) which points out that the conceptual resources to grasp Hume's Principle vastly outstrip the conceptual resources employed in arithmetical reasoning. And lastly whether Hume's Principle is in bad company with other unsuccessful implicit definitions. In the last section, an account towards our entitlement to Hume&'s Principle is sketched. (shrink)

This paper defends the view that ontological questions (properly understood) are easy—too easy, in fact, to be subjects of substantive and distinctively philosophical debates. They are easy, roughly, in the sense that they may be resolved straightforwardly—generally by a combination of conceptual and empirical enquiries. After briefly outlining the view and some of its virtues, I turn to examine two central lines of objection. The first is that this ‘easy’ approach is itself committed to substantive ontological views, including an implausibly (...) permissive ontology. The second is that it, like neo-Fregean views, relies on transformation rules that are questionable on both logical and ontological grounds. Ultimately, I will argue, the easy view is not easily assailed by either of these routes, and so remains (thus far) a tenable and attractive approach. (shrink)

NeoFregeanism is an intriguing but elusive philosophy of mathematical existence. At crucial points, it goes cryptic and metaphorical. I want to put forward an interpretation of neoFregeanism—perhaps not one that actual neoFregeans will embrace—that makes sense of much of what they say. NeoFregeans should embrace quantifier variance.

European Journal of Philosophy, Volume 30, Issue 2, Page 578-600, June 2022.

One recent `neologicist' claim is that what has come to be known as "Frege's Theorem"–the result that Hume's Principle, plus second-order logic, suffices for a proof of the Dedekind-Peano postulate–reinstates Frege's contention that arithmetic is analytic. This claim naturally depends upon the analyticity of Hume's Principle itself. The present paper reviews five misgivings that developed in various of George Boolos's writings. It observes that each of them really concerns not `analyticity' but either the truth of Hume's Principle or our entitlement (...) to accept it and reviews possible neologicist replies. A two-part Appendix explores recent developments of the fifth of Boolos's objections–the problem of Bad Company–and outlines a proof of the principle $N^q$, an important part of the defense of the claim that what follows from Hume's Principle is not merely a theory which allows of interpretation as arithmetic but arithmetic itself. (shrink)

The syntactic priority thesis (henceforth SP) asserts that the truth of appropriate sentential contexts containing what are, by syntactic criteria, singular terms, is sufficient to justify the attribution of objectual reference to such terms (Wright, 1983, 24). One consequence that the neo-Fregean draws from SP is that it is through an analysis of the syntactic structure of true statements that 'ontological questions are to be understood and settled' (Wright, 1983, 25). Despite the significant literature on SP, little consideration has been (...) given to this bold meta-ontological claim.1 My concern here is accordingly not with specific applications of SP to debates in the philosophy of mathematics, but rather with the .. (shrink)

The prevailing approach to the problem of the ontological status of mathematical entities such as numbers and sets is to ask in what sense it is legitimate to ascribe a reference to abstract singular terms; those expressions of our language which, taken at face value, denote abstract objects. On the basis of this approach, neo‐Fregean Abstractionists such as Hale and Wright have argued that abstract singular terms may be taken to effect genuine reference towards objects, whereas nominalists such as Field (...) have asserted that these apparent ontological commitments should not be taken at face value. In this article I argue for an intermediate position which upholds the legitimacy of ascribing a reference to abstract singular terms in an attenuated sense relative to the more robust ascription of reference applicable to names denoting concrete entities. In so doing I seek to clear up some confusions regarding the ramifications of such a thin notion of reference for ontological claims about mathematical objects. (shrink)

Can there be objects that are ‘thin’ in the sense that very little is required for their existence? A number of philosophers have thought so. For instance, many Fregeans believe it suffices for the existence of directions that there be lines standing in the relation of parallelism; other philosophers believe it suffices for a mathematical theory to have a model that the theory be coherent. This article explains the appeal of thin objects, discusses the three most important strategies for articulating (...) and defending the idea of such objects, and outlines some problems that these strategies face. (shrink)

Frege proposed that sentences like ‘The number of planets is eight’ be analysed as identity statements in which the number words refer to numbers. Recently, Friederike Moltmann argued that, pace Frege, such sentences be analysed as so-called specificational sentences in which the number words have the same non-referring semantic function as the number word ‘eight’ in ‘There are eight planets’. The aim of this paper is two-fold. First, I argue that Moltmann fails to show that such sentences should be analysed (...) as specificational sentences. Second, I show that even if they are to be analysed in this way, Moltmann’s proposed specificational analysis is unsatisfactory. (shrink)