I argue that constructive nominalism is preferable to scientific realism. Rather than reflecting without distortion the way the mind-independent world is, theories refract. They provide an understanding of the world as modulated by a particular theory. Truth is defined within a theoretical framework rather than outside of it. This does not undermine objectivity, for an assertion contains a reference to the framework in terms of which its truth is claimed.

A metaphysically robust, as opposed to lightweight, Platonism with respect to uncreatable abstract objects is theologically unacceptable because it fatally compromises creatio ex nihilo and divine aseity. The principal argument for Platonism is the so-called Indispensability Argument based on the ontological commitments required by singular terms and existential quantifiers in true sentences. Different varieties of Nominalism challenge each of the argument’s premises. Fictionalism accepts the assumed criterion of ontological commitment but rejects the truth of the relevant sentences. Neutralism accepts (...) the truth of the relevant sentences but denies the assumed criterion of ontological commitment. Both of these perspectives, but especially the last, are plausible routes available for the Christian theist. (shrink)

We observe that Putnam’s model-theoretic argument against determinacy of the concept of second-order quantification or that of the set is harmless to the nominalist. It serves as a good motivation for the nominalist philosophy of mathematics. But in the end it can lead to a serious challenge to the nominalist account of mathematical objectivity if some minimal assumptions about the relation between mathematical objectivity and logical objectivity are made. We consider three strategies the nominalist might take to meet this challenge, (...) and we argue that all these strategies are untenable. (shrink)

Numbers and other mathematical objects are exceptional in having no locations in space or time or relations of cause and effect. This makes it difficult to account for the possibility of the knowledge of such objects, leading many philosophers to embrace nominalism, the doctrine that there are no such objects, and to embark on ambitious projects for interpreting mathematics so as to preserve the subject while eliminating its objects. This book cuts through a host of technicalities that have obscured (...) previous discussions of these projects, and presents clear, concise accounts of a dozen strategies for nominalistic interpretation of mathematics, thus equipping the reader to evaluate each and to compare different ones. The authors also offer critical discussion, rare in the literature, of the aims and claims of nominalistic interpretation, suggesting that it is significant in a very different way from that usually assumed. (shrink)

Theoria, EarlyView. -/- In his recent book Thin Objects, Øystein Linnebo (2018) argues for the existence of a hierarchy of abstract objects, sufficient to model ZFC, via a novel and highly interesting argument that relies on a process called dynamic abstraction. This paper presents a way for a nominalist, someone opposed to the existence of abstract objects, to avoid Linnebo's conclusion by rejecting his claim that certain abstraction principles are sufficient for reference (RBA). Section 1 of the paper explains Linnebo's (...) argument for RBA. It offers a reading of Linnebo's work upon which he has two arguments for RBA: one deductive and one abductive, and argues that whilst the deductive argument is unsound the abductive one is prima facie plausible. The nominalist must therefore find a way to respond to the abductive argument. Section 2 outlines just such a response, by offering an alternative explanation of the cases Linnebo wishes to argue from. Most interestingly, it shows that abstraction in Linnebo's most difficult case (the “reference to ordinary bodies” case) can be achieved using mereological means, rather than relying on RBA. (shrink)

Jody Azzouni argues that whilst it is indeterminate what the criteria for existence are, there is a criterion that has been collectively adopted to use ‘exist’ that we can employ to argue for positions in ontology. I raise and defend a novel objection to Azzouni: his view has the counterintuitive consequence that the facts regarding what exists can and will change when users of the word ‘exist’ change what criteria they associate with its usage. Considering three responses, I argue Azzouni (...) has best reason to take one that ultimately renders unsuccessful his arguments against mathematical abstracta. (shrink)

The Buridanian view of the concrete cognition as the general characteristics of sense perception was adopted by Jodocus Trutfetter and Bartholomaeus Arnoldi of Usingen. This theory was not accepted merely on the basis of authority, but it was argued against the competing view, which appeared as legitimate inside the late medieval school of via moderna.

‘Nominalism’ refers to a family of views about what there is. The objects we are familiar with (e.g. hands, laptops, cookies, and trees) can be characterized as concrete and particular. Nominalists agree that there are such things. But one group of nominalists denies that anything is non-particular and another group denies that anything is non-concrete. These two sorts of nominalism, referred to as ‘nominalism about universals’ and ‘nominalism about abstract objects’, have common motivations in contemporary philosophy.

The subject matter of this thesis is analytic ontology. Chapters II and III deal with two versions of trope theory, or moderate nominalism; these are defined as ontologies which recognise properties and relations but no (real) universals. The key notion of both theories, trope, is characterised as an abstract particular. What the abstractness amounts to differs between the two. Yet another difference is that simplicity is an essential trait of a trope according to one theory, but not according to (...) the other. Though exact similarity is said to play an important role in both theories, as it turns out, this does not seem to be the case. The ontology dealt with in chapter IV is a mixture of moderate nominalism concerning qualities and realism concerning relations. In it, quality instances (moments) and universal relations are the ultimate constituents of the universe. While relations and moments are considered to be constituents of states of affairs, which are characterised as objects of higher orders, complexes that are objects of the first order are made up of moments on their own. Among these complexes one finds the ordinary objects. Paradoxically, although relations are necessary for the existence of complex first order objects, relations are not thought to be among the contents of these objects. The main subject of chapter V is a particular version of moderate realism; it is an ontology which is realistic in its recognition of universals and moderate in its recognition of instances of these universals. Instances combine to form complex networks. A theoretically motivated claim is that although each instance has a predicational aspect as well as a universal one, it is simple in the sense of lacking internal predicative structure; though, this claim can be called into question. Keywords: analytic ontology, moderate nominalism, moderate realism, particular, universal, abstract, concrete, abstract particular, abstract universal, concrete particular, concrete universal, trope, moment, complex unity, collection, instance, unit attribute, intensional aspect, predicational aspect, continuous composite, articulated composite. (shrink)

This chapter is a discussion of Class and Resemblance Nominalism. According to the traditional versions of these theories, properties are classes of particulars. Thus, the property of being red is the class of red particulars, and the property of being square is the class of square particulars. Several objections have been advanced against these theories, and one of the most powerful of such objections is the so-called Coextension Difficulty, according to which Class and Resemblance Nominalism have to wrongly (...) identify distinct but coextensive properties. After discussing the nature of Class and Resemblance Nominalism, some problems of these theories and how they can be deal with, the superiority of Class and Resemblance Nominalism over other theories of properties and the superiority of Resemblance Nominalism over Class Nominalism, I concentrate on the Coextension Difficulty. With respect to this I argue that Class and Resemblance Nominalism can be developed in such a way that they admit properties without identifying them with classes, and this allows them to avoid the Coextension Difficulty, since such versions of these theories are not forced to wrongly identify distinct but coextensive properties. Thus the Coextension Difficulty is not a lethal problem for Class and Resemblance Nominalism. (shrink)

Nominalists, who believe that everything there is is concrete and nothing is abstract, seem to have a problem with mathematics. Mathematics says that there are lots of prime numbers, and prime numbers don’t seem to be concrete. What should a nominalist do with mathematics? In the last few decades several programs in the philosophy of mathematics have been formulated which are, more or less explicitly, accounts of what a nominalist can say about mathematics. These programs, and the criticism of them, (...) make up a substantial part of the literature in the philosophy of mathematics during this time. Usually such a program involves two parts. One is purely philosophical. It deals with the motivation for nominalism, and argues why certain technical results would show that nominalists have an account of mathematics. The other part is to provide the technical results needed in the program. (shrink)

The notion of grounding has gained increasing acceptance among metaphysicians in recent years. In this paper, I argue that this notion can be used to formulate a very attractive version of (property) nominalism, a view that I call ‘grounding nominalism’. Simplifying somewhat, this is the view that all properties are grounded in things. I argue that this view is coherent and has a decisive advantage over competing versions of nominalism: it allows us to accept properties as real, (...) while fully accommodating nominalist intuitions. Finally, I defend grounding nominalism against several seemingly troublesome objections. (shrink)

In this essay I first outline contemporary Platonism about musical works – the theory that musical works are abstract objects. I then consider reasons to be suspicious of such a view, motivating a consideration of nominalist theories of musical works. I argue for two conclusions: first, that there are no compelling reasons to be a nominalist about musical works in particular, i.e. that nominalism about musical works rests on arguments for thoroughgoing nominalism, and, second, that if Platonism fails, (...) fictionalism about musical works is to be preferred to other nominalist ontologies of musical works. (shrink)

‘Grounding and the indispensability argument’ presents a number of ways in which nominalists can use the notion of grounding to rebut the indispensability argument for the existence of mathematical objects. I will begin by considering the strategy that puts grounding to the service of easy-road nominalists. I will give some support to this strategy by addressing a worry some may have about it. I will then consider a problem for the fast-lane strategy and a problem for easy-road nominalists willing to (...) accept Liggins’ grounding strategy. Both are related to the problem of formulating nominalistic explanations at the right level of generality. I will then consider a problem that Liggins only hints at. This problem has to do with mathematics’ function of providing the sort of covering generalizations we need in scientific explanations. (shrink)

Nominalism is the view that abstract mathematical objects like numbers, functions, and sets do not exist. The chapter articulates and defends a variety of nominalism, based on a reading of mathematical statements in terms of possible linguistic constructions. The chapter responds directly to a recent study of nominalism by Gideon Rosen and John Burgess, and develops a reply to the Quine-Putnam indispensability argument for the existence of mathematical objects.

Curiously, Charles Sanders Peirce and Maurice Merleau-Ponty raise the same objection to British empiricism: its foundational tenet is nominalist. In his 1869 review of a new edition of James Mill's Analysis of the Phenomena of the Human Mind, Peirce traces the foundational tenet of Mill's work back to Hume's Copy Principle—that all of our ideas are fainter copies of our impressions—and then remarks, If I compare a red book and a red cushion, there is, according to them [the "English psychologists," (...) among whom Peirce includes Mill, Hume, and Berkeley], no general sensation red which enters into both these images, nor is there any idea of a general respect... (shrink)

Nominalism is the view that mathematical objects do not exist. This chapter delimits several types of nominalistic projects: revolutionary programs that attempt to change mathematics and hermeneutic programs that attempt to interpret mathematics. Some programs accord with naturalism, and some oppose naturalism. Steven Yablo’s fictionalism is brought into the fold and discussed at some length.

In my book *Resemblance Nominalism* I argued that the truthmakers of ´a and b resemble each other´ are just a and b. In his "Resemblance Nominalism and counterparts" Alexander Bird objects to my claim that the truthmakers of ´a and b resemble each other´ are just a and b. In this paper I respond to Bird´s objections.

Aldo Antonelli offers a novel view on abstraction principles in order to solve a traditional tension between different requirements: that the claims of science be taken at face value, even when involving putative reference to mathematical entities; and that referents of mathematical terms are identified and their possible relations to other objects specified. In his view, abstraction principles provide representatives for equivalence classes of second-order entities that are available provided the first- and second-order domains are in the equilibrium dictated by (...) the abstraction principles, and whose choice is otherwise unconstrained.entities are the referents of abstraction terms: such referents are to an extent indeterminate, but we can still quantify over them, predicate identity or non-identity, etc. Our knowledge of them is limited, but still substantial: we know whatever has to be true no matter how the representatives are chosen, i.e., what is true in all models of the corresponding abstraction principles. This view is backed up by an “austere” conception of universals, according to which these are first-order objects, i.e., ways of collecting first-order objects. Antonelli thus claims that second-order logic does not import any novel ontological commitment beyond ontology of first-order, naturalistically acceptable, objects. Moreover, in the case of arithmetic, even if Hume’s Principle is construed as described above, Frege’s Theorem goes through unaffected, since there is nothing in its proof that depends on an account of the “true nature” of numbers. A viable construal of logicism is then given by the combination of semantic nominalism and a naturalistic conception of abstraction. [Editors note]. (shrink)

Current versions of nominalism in the philosophy of mathematics have the benefit of avoiding commitment to the existence of mathematical objects. But this comes with the cost of not taking mathematical theories literally. Jody Azzouni's _Deflating Existential Consequence_ has recently challenged this conclusion by formulating a nominalist view that lacks this cost. In this paper, we argue that, as it stands, Azzouni's proposal does not yet succeed. It faces a dilemma to the effect that either the view is not (...) nominalist or it fails to take mathematics literally. After presenting the dilemma, we suggest a possible solution for the nominalist. (shrink)

The aim of the present paper is twofold. One task is to argue that our use of the numerical vocabulary in theory and applications determines the reference of the numerical terms more precisely than up to isomorphism. In particular our use of the numerical vocabulary in modal and counterfactual contexts of application excludes contingent existents as candidate referents for the numerical terms. The second task is to explore the impact of this conclusion on what I call semantic nominalism, which (...) is the view that ordinary physical objects and other contingent existents are eligible to be the referents of numerical terms. I suggest that for semantic nominalism to have a chance we must reform our usage of the numerical vocabulary in modal and counterfactual contexts. I tentatively explore some ways in which this might be done and call attention to some of their costs. (shrink)

The object of this paper is to provide a solution to Nelson Goodman’s Imperfect Community difficulty as it arises for Resemblance Nominalism, the view that properties are classes of resembling particulars. The Imperfect Community difficulty consists in that every two members of a class resembling each other is not sufficient for it to be a class such that there is some property common to all their members, even if ‘x resembles y’ is understood as ‘x and y share (...) some property’. In the paper I explain and criticise several solutions to the difficulty. Then I develop my own solution, which is not subject to the objections I make to the other solutions, and which accords completely with the basic tenets of Resemblance Nominalism. (shrink)

The goal is to sketch a nominalist approach to mathematics which just like neologicism employs abstraction principles, but unlike neologicism is not committed to the idea that mathematical objects exist and does not insist that abstraction principles establish the reference of abstract terms. It is well-known that neologicism runs into certain philosophical problems and faces the technical difficulty of finding appropriate acceptability criteria for abstraction principles. I will argue that a modal and iterative nominalist approach to abstraction principles circumvents those (...) difficulties while still being able to put abstraction principles to a foundational use. (shrink)

It is widely assumed that platonism with respect to a discourse of metaphysical interest, such as fictional or mathematical discourse, affords a better account of the semantic appearances than nominalism, other things being equal. Of course, other things may not be equal. For example, platonism is supposed to come at the cost of a plausible epistemology and ontology. But the hedged claim is often treated as a background assumption. It is motivated by the intuitively stronger one that the platonist (...) can take the semantic appearances at ‘face-value’ while the nominalist must resort to ad hoc and technically problematic machinery in order to explain those appearances away. In this article, I argue that, on any natural construal of ‘face-value’, the platonist, like the nominalist, is not able to take the semantic appearances at face-value. And insofar as the nominalist is forced to resort to ad hoc and technically awkward devices in order to explain those appearances away, the platonist must resort to such devices as well. One moral of the story is that the thesis that platonism affords a better account of the semantic appearances than nominalism – even other things being equal – is not trivial. Another is that we should rethink a widespread methodology in metaphysics. (shrink)

Peter Abelard and William of Ockham represent the two main figures of the nominalism of the Middle Ages. Both share the fundamental thesis of that doctrine, according to which only individual entities exist. The repercussions of nominalism are quite evident in relation to the question of universals, which constitutes a subject that, until now, won the attention of the majority of contemporary studies on the two most important logicians of their time. Nevertheless the nominalism of each of (...) these two protagonists apparently diverges in a significant manner, especially in respect to the role exercised by the mind in the semantic process. Ockham seems to distance himself definitely from his predecessors, including Abelard, when he transfers grammar and semantic functions of conventional language to the level of mental language. In the present article, the chief intention is to expound the theory of sign and signification of each of the authors under consideration. The ultimate objective is to compare their respective semantic positions and indicate the principal points of agreement and disagreement, keeping in mind the hypothesis according to which these two medieval nominalists should have more theoretical elements in common than not. (shrink)

Byrne & Hilbert are right that it might be an objective fact that a particular tomato is unique red, but wrong that it cannot simultaneously be yellowish-red (not only objectively, but from somebody else's point of view). Sensory categorization varies among organisms, slightly among conspecifics, and sharply across taxa. There is no question of truth or falsity concerning choice of categories, only of utility and disutility. The appropriate framework for color categories is Nominalism and Pluralistic Realism.

In this dissertation I consider the merits of certain nominalist accounts of phenomena related to the character of ordinary objects. What these accounts have in common is the fact that none of them is an error theory about standard cases of predication and none of them deploys God or uniquely theistic resources in its explanatory framework. -/- The aim of the dissertation is to answer the following questions: -/- • What is the best nominalist account on offer? • How might (...) it be improved? • Does it ultimately succeed? -/- I will argue that while so-called trope theory is the best account on offer, it can be significantly improved—or replaced—by a novel version of nominalism that is modeled after trope theory. Ultimately, however, I will argue that even the novel version fails. -/- The dissertation unfolds as follows. In Chapter 1, I introduce Austere Nominalism (AN), which is perhaps the most extreme version of nominalism that falls within the scope of the dissertation. AN is often described as the view that there exist only concrete particulars. According to AN, it is unnecessary to posit any entities other than ordinary objects—turkeys, tables, and the like—in order to account for explananda related to the character of those objects. (Such explananda include the phenomenon of attribute agreement, of attribute possession, of true subject-predicate sentences, etc.) In this chapter I argue that AN fails to provide an adequate account of these explananda. In addition, introducing and criticizing AN serves an important heuristic role for the rest of the dissertation. To understand this role, we must distinguish between the basic explanatory strategy deployed by the austere nominalist and the type of explananda for which she deploys that strategy. The austere nominalist deploys the strategy to account for the character of ordinary objects. As I argue in Chapter 1, this deployment is a failure. As I go on to show in Chapter 2, the widespread rejection AN has led to a variety of rival accounts of the character of ordinary objects. In rejecting AN, however, these accounts also tacitly reject its basic explanatory strategy. Thus goes the baby with the bathwater, since, arguably, there are some attractive features of AN’s basic explanatory strategy. Indeed, those who defend the most prominent version of nominalism—trope theory—seem to overlook the advantages of AN’s basic strategy, and by so doing, make an unnecessary concession to the realist. Or so I argue in Chapter 3. And, as I will argue in Chapter 4, the strongest version of nominalism is a novel account, modeled after trope theory, that deploys AN’s basic strategy at a more fundamental level than that of ordinary objects. This novel account—troper theory—is closer in spirit to AN than is traditional trope theory. (Thus, AN serves as a foil for the discussion of other nominalist views.) Finally, in the Afterword I indicate how troper theory is equally vulnerable to some of the traditional objections that plague trope theory. Thus, if you are not convinced that traditional objections to trope theory are conclusive and you want to be a nominalist, then you should abandon trope theory and adopt troper theory. If you take traditional objections against trope theory to have significant force, then you should reject both theories. (shrink)

Biological naturalism claims that all psychological phenomena can be causally, though not ontologically, reduced to neurological processes, where causal reduction is usually understood in terms of supervenience. After presenting John Searle’s version of biological naturalism in some detail, I argue that the particular supervenience relation on which this account depends is dubious. Specifically, the fact that either realism or nominalism is the case implies that there is one fact about belief that does not supervene on neurophysiological processes. Biological naturalism (...) is thereby defeated because it cannot account for belief. Ialso address three likely objections to this argument. (shrink)

Nominalists need some account of correctness for sentences committed to the existence of abstract objects. This paper proposes a new statement of such conditions specifically for properties. The account builds on an earlier proposal of mine, but avoids the counter-examples against the latter pointed out by Thomas Schindler, particularly, the sentence ‘There are inexpressible properties’. I argue that the new proposal is independently motivated and more faithful to the spirit of the kind of error-theoretic nominalism that the original proposal (...) was part of. Along the way, I also propose novel treatments of Arthur Pap’s and W.V.O. Quine’s notoriously hard-to-paraphrase sentences about abstracta. (shrink)

Hofweber (Ontology and the ambitions of metaphysics, Oxford University Press, 2016) argues for a thesis he calls “internalism” with respect to natural number discourse: no expressions purporting to refer to natural numbers in fact refer, and no apparent quantification over natural numbers actually involves quantification over natural numbers as objects. He argues that while internalism leaves open the question of whether other kinds of abstracta exist, it precludes the existence of natural numbers, thus establishing what he calls “restricted nominalism” (...) about natural numbers. We argue that Hofweber’s internalism fails to establish restricted nominalism. Not only is his primary argument for restricted nominalism invalid, the analysis of quantification proposed threatens to collapse internalism into either a traditional form of error theory or realism. (shrink)

One major motivation for nominalism, at least according to Hartry Field, is the desirability of intrinsic explanations: explanations that don’t invoke objects that are causally irrelevant to the phenomena being explained. There is something right about the search for such explanations. But that search must be carefully implemented. Nothing is gained if, to avoid a certain class of objects, one only introduces other objects and relations that are just as nominalistically questionable. We will argue that this is the case (...) for two alleged nominalist views: Field’s fictionalism, and Frank Arntzenius and Cian Dorr’s geometricalism. Central to our competing approach to nominalism is a distinction between terms that refer to objects and ones that instead code empirical phenomena while being referentially empty. We next contrast our approach to nominalism, which uses this term-grained distinction between coding and referring, with approaches that instead attempt to make a sentence-grained distinction between mathematical and non-mathematical content. We show the latter approach fails to be responsive to objections raised by van Fraassen. In the end, only one last approach to nominalism is left standing. 1 Introduction2 Troubles for Mathematical Fictionalism2.1 A distinction2.2 Field’s programme2.3 Parts of nominalistically acceptable entities are nominalistically acceptable2.4 Structural assumptions: How far can these be pushed?2.5 Cases of indispensable theories where the mathematical posits are known not to be real2.6 How does science distinguish between the indispensable structural presuppositions taken to be real and those that aren’t?2.7 Do physicists distinguish between mathematical posits and non-mathematical posits?2.8 Mathematical coding versus genuine metaphysics. Case study: Fields2.9 The upshot3 Troubles for Topologism4 Contrasts: Kitcher, Maddy, Sober, and van Fraassen4.1 Sober’s approach4.2 Maddy’s approach4.3 Kitcher’s approach4.4 Where do we go from here?4.5 Van Fraassen’s objection to Maddy’s approach to ontological commitment5 Conclusion. (shrink)

The idea of “material plenitude” has been gaining traction in recent discussions of the metaphysics of material objects. My main goal here is to show that this idea may have important dialectical implications for the metaphysics of properties – more specifically, that it provides nominalists with new resources in their attempt to reject an ontology of universals. I will recapitulate one of the main arguments against nominalism – due to David Armstrong – and show how plenitude helps the nominalist (...) overcome the argument. (shrink)

The platonism/nominalism debate in the philosophy of mathematics concerns the question whether numbers and other mathematical objects exist. Platonists believe the answer to be in the positive, nominalists in the negative. According to non-factualists, the question is ‘moot’, in the sense that it lacks a correct answer. Elaborating on ideas from Stephen Yablo, this article articulates a non-factualist position in the philosophy of mathematics and shows how the case for non-factualism entails that standard arguments for rival positions fail. In (...) particular, showing how and why non-factualists reject nominalism illuminates the originality and interest of their position. (shrink)

A feature of agnostic views—views that officially express ignorance about the existence of something —is that they are widely perceived to be epistemically more cautious than views that are committed to the entities in question. This is often seen as giving agnostics a debating advantage: all things being equal, fence-sitters have smaller argumentative burdens. Otávio Bueno argues in this way for what he calls “agnostic nominalism,” the view that we don’t know whether ontologically-independent Platonic objects exist. I show that (...) agnostic nominalism, so called, can be sustained only in ways that don’t give agnostic nominalists debating advantages: the position must either be deduced from antecedently-held broader sceptical assumptions or it requires manufacturing potential referents for one’s terms that artificially generate grounds for scepticism. Neither maneuver leads to an agnostic nominalism with a debating advantage over its nominalist opponents. (shrink)

In his (2002) Gonzalo Rodriguez-Pereyra provides a powerful articulation of the claim that Resemblance Nominalism provides the best answer to the so-called Problem of Universals. Resemblance Nominalism has not been popular for some time, and one influential reason for this is the widespread belief that Resemblance Nominalism cannot dispense with all universals. The realist critics appeal to what is known as Russell’s Regress (cf. Russell 1997). If properties are to be explained in terms of one object’s (...) resembling another, then this seems to leave the relational property of resemblance itself unexplained. The critics’ objection is that this property itself must be explained by a dyadic universal of resemblance. (shrink)

Probably there is no position in Goodman’s corpus that has generated greater perplexity and criticism than Goodman’s “nominalism”. As is abundantly clear from Goodman’s writings, it is not “abstract entities” generally that he questions—indeed, he takes sensory qualia as “basic” in his Carnap-inspired constructional system in Structure—but rather just those abstracta that are so crystal clear in their identity conditions, so fundamental to our thought, so prevalent and seemingly unavoidable in our discourse and theorizing that they have come to (...) form the generally accepted framework for the most time-honored, exact, sophisticated, refined, central, and secure branch of human knowledge yet devised, mathematics itself! Of all the abstracta to question, why sets? Of course, Goodman gave his “reasons”, the unintelligibility of “generating” an infinitude of “constructed objects” automatically from any given object or objects. But critics have been quick to point out that set theory is intended not as a theory of what can be “generated” or “constructed” from given objects in any literal sense but rather as a theory of a certain realm of objects independently existing in their own right. “Construction” is a metaphor. (shrink)

I here investigate whether there is any version of the principle of charity both strong enough to conflict with an error-theoretic version of nominalism (EN) about abstract objects, and supported by the considerations adduced in favour of interpretive charity in the literature. I argue that in order to be strong enough, the principle, which I call (Charity), would have to read, “For all expressions e, an acceptable interpretation must make true a sufficiently high ratio of accepted sentences containing e”. (...) I next consider arguments based on Davidson's intuitive cases for interpretive charity, the reliability of perceptual beliefs, and the reliability of “non-abstractive inference modes”, and conclude that none support (Charity). I then propose a diagnosis of the view that there must be some universal principle of charity ruling out (EN). Finally, I present a reason to think (Charity) is false, namely, that it seems to exclude the possibility of such disagreements as that between nominalists and realists. (shrink)

The present paper will argue that, for too long, many nominalists have concentrated their researches on the question of whether one could make sense of applications of mathematics (especially in science) without presupposing the existence of mathematical objects. This was, no doubt, due to the enormous influence of Quine's "Indispensability Argument", which challenged the nominalist to come up with an explanation of how science could be done without referring to, or quantifying over, mathematical objects. I shall admonish nominalists to enlarge (...) the target of their investigations to include the many uses mathematicians make of concepts such as structures and models to advance pure mathematics. I shall illustrate my reasons for admonishing nominalists to strike out in these new directions by using Hartry Field's nominalistic view of mathematics as a model of a philosophy of mathematics that was developed in just the sort of way I argue one should guard against. I shall support my reasons by providing grounds for rejecting both Field's fictionalism and also his deflationist account of mathematical knowledge—doctrines that were formed largely in response to the Indispensability Argument. I shall then give a refutation of Mark Balaguer's argument for his thesis that fictionalism is "the best version of anti-realistic anti-platonism". (shrink)

Otávio Bueno gives a positive and accurate summary of my defense of Platonism, with special emphasis on the epistemological issues. He criticizes “skeptical nominalism”, and proposes instead an “agnostic nominalism”, which treats mathematical objects as “objects of thought”, and neither rejects nor accepts abstract entities. In my response I argue that the main problem for nominalism is to account for abstract properties and relations, and that treating mathematical objects as objects of thought does not provide a satisfactory (...) solution to that end.Otávio Bueno apresenta um resumo positivo e correto de minha defesa do Platonismo, com ênfase especial nos aspectos epistemológicos. Ele critica o “nominalismo cético” e propõe um “nominalismo agnóstico” que trata os objetos matemáticos como “objetos de pensamento”, sem rejeitar ou aceitar entes abstratos. Em minha réplica argumento que o problema principal para o nominalismo é dar conta das propriedades e relações abstratas, e que tratar os objetos matemáticos como objetos de pensamento não é uma solução satisfatória para esse fim. (shrink)

Mathematical Nominalism Mathematical nominalism can be described as the view that mathematical entities—entities such as numbers, sets, functions, and groups—do not exist. However, stating the view requires some care. Though the opposing view (that mathematical objects do exist) may seem like a somewhat exotic metaphysical claim, it is usually motivated by the thought that mathematical … Continue reading Mathematical Nominalism →.

Nominalism faces the task of explaining away the ontological commitments of applied mathematical statements. This paper reviews an argument from the philosophy of logic that focuses on this task and which has been used as an objection to certain specific formulations of nominalism. The argument as it is developed in this paper aims to show that nominalism in general does not have the epistemological advantages its defendants claim it has. I distinguish between two strategies that are available (...) to the nominalist: The Evaluation Programme, which tries to preserve the common truth‐values of mathematical statements even if there are no mathematical objects, and Fictionalism, which denies that mathematical sentences have significant truth‐values. It is argued that the tenability of both strategies depends on the nominalist’s ability to account for the notion of consequence. This is a problem because the usual meta‐logical explications of consequence do themselves quantify over mathematical entities. While nominalists of both varieties may try to appeal to a primitive notion of consequence, or, alternatively, to primitive notions of logical or structural possibilities, such measures are objectionable. Even if we are equipped with a notion of either consequence or possibility that is primitive in the relevant sense, it will not be strong enough to account for the consequence relation required in classical mathematics. These examinations are also useful in assessing the possible counter‐intuitive appeal of the argument from the philosophy of logic. (shrink)

What was the basis for the adoption of mathematics as the primary mode of discourse for describing natural events by a large segment of the philosophical community in the seventeenth century? In answering this question, this book demonstrates that a significant group of philosophers shared the belief that there is no necessary correspondence between external reality and objects of human understanding, which they held to include the objects of mathematical and linguistic discourse. The result is a scholarly reliable, but accessible, (...) account of the role of mathematics in the works of Galileo, Kepler, Descartes, Newton, Leibniz, and Berkeley. This impressive volume will benefit scholars interested in the history of philosophy, mathematical philosophy and the history of mathematics. (shrink)

The goal is to sketch a nominalist approach to mathematics which just like neologicism employs abstraction principles, but unlike neologicism is not committed to the idea that mathematical objects exist and does not insist that abstraction principles establish the reference of abstract terms. It is well-known that neologicism runs into certain philosophical problems and faces the technical difficulty of finding appropriate acceptability criteria for abstraction principles. I will argue that a modal and iterative nominalist approach to abstraction principles circumvents those (...) difficulties while still being able to put abstraction principles to a foundational use. (shrink)

In a series of works, Jody Azzouni has defended deflationary nominalism, the view that certain sentences quantifying over mathematical objects are literally true, although such objects do not exist. One alleged attraction of this view is that it avoids various philosophical puzzles about mathematical objects. I argue that this thought is misguided. I first develop an ontologically neutral counterpart of Field’s reliability challenge and argue that deflationary nominalism offers no distinctive answer to it. I then show how this (...) reasoning generalizes to other philosophically problematic entities. The moral is that puzzle avoidance fails to motivate deflationary nominalism. (shrink)