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The present study, so far, has focused exclusively on vague predicates such as ‘bald’, ‘tall’, and predicates such as ‘determinately bald’ and ‘borderline bald‘. Intuitively, what makes a predicate vague is that there is some indeterminacy over whether it applies to some objects: vague predicates have borderline cases. Some aggregates of sand are borderline heaps, some potential NBA players are borderline tall (for an NBA player), some men are borderline bald, and perhaps some men are borderline-borderline bald. At least prima facie, there also seem to be vague objects. This chapter looks at some variations on this theme, and then extends the model theory to handle them.

Edmund Husserl is one of only a few major philosophers in the last one hundred years or so who holds that it is possible to develop a philosophy of mind in which one can account for the consciousness of abstract objects or ideal objects. This chapter discusses Husserl's ideas in connection with the views of Kurt Gödel and Roger Penrose. It presents an argument that leads from Gödel's incompleteness theorems to recognition of the awareness of abstract or ideal objects. Husserl's view, based on his ideas about intentionality and the phenomenological reduction, shows us how to open up a space for a phenomenology of the consciousness of abstract mathematical objects.

The hypothesis is that space‐time is what there is. W.V. Quine's ‘abstract objects’ are rejected. What exists should play some causal role (Graham Oddie's Eleatic Principle). The nature of space‐time is subject to scientific investigation (Wilfrid Sellars' distinction between the manifest and the scientific image of the world). Is there room for metaphysics? Yes, because a number of topic neutral notions (of which causality is an important instance) are contested by philosophers and scientists. As argued by C.B. Martin, metaphysics seeks a more abstract model of the world than that provided by science.

Frege regarded the foundations of mathematics as self-evident. He maintained that reason could enable one to know of the existence and nature of both mind-independent abstract objects, such as the numbers and thought contents, and mind-independent functions, such as the successor function. There is extensive discussion of evidence for, and the nature of, his Platonism; the view that abstract objects and functions are completely mind-independent of both minds and physical reality. This chapter explains how he squared his rationalism with his Platonism. The key idea is that Frege thought that mind is dependent for being what it is both on functioning to attain truth and to track accurately the basic structures of abstract, mind-independent objects (both logical objects and thought contents) and abstract, mind-independent functions.

There seem to be repeatable artworks. Plays and musical works (like Hamlet or ‘Pictures at an Exhibition’) can be performed again and again; installations (like Sol LeWitt’s Wall Drawing #260) can be installed over and over; culinary dishes (like Jean-George Vongerichten’s tuna tartare) can be prepared many times. This chapter offers an argument against the existence of repeatable artworks. First, the chapter argues that an artwork is repeatable only if it is an abstract object. This jives with standard accounts of repeatable works, which take them to be abstracta. Second, it argues that abstract objects must have all their properties essentially. Finally, since repeatable artworks, if there are any, do not have all their properties essentially, the chapter concludes that there are no repeatable artworks.

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.

Begins by distinguishing different varieties of nominalism. All adherents of nominalism agree in rejecting mathematical and other abstract entities, and many have attempted to develop nominalist interpretations of scientific theories that appear to involve mathematical objects, with some (hermeneutic nominalists) claiming that these interpretations reveal what the theories really meant all along, and others (revolutionary nominalists) admitting that what they are developing are new, replacement theories. Before beginning our examination of these interpretative projects, we stop to examine critically their presuppositions, beginning with the distinction between abstract objects and concrete objects. We devote special attention to epistemological arguments and semantical arguments for nominalism, based on causal theories of knowledge and causal theories of reference. We also consider why more nominalists have not been content simply to adopt instrumentalism, which declares science to be a useful fiction, and offers no reinterpretation to turn theory into fact.

This chapter examines the dispute between Quine and Carnap about how to understand ontological commitment and what ontology to adopt. The central dispute is over Carnap’s acceptance of abstract objects, including numbers, properties, and propositions, which Quine characterizes in “On What There Is” (1948) as a form of Platonism. Carnap vigorously disagrees, responding in “Empiricism, Semantics, and Ontology” (1950, 1956). For him, commitments to these things are unproblematic consequences of accepting an optimal theoretical framework for doing science. Philosophers haven’t seen this because, he believes, they have approached ontology in an unscientific way.

This paper starts by offering a brief reconstruction of the Neo‐Fregean approach as suggested in Frege's Conception of Numbers as Objects and distinguishes various challenges against the method of Abstraction. It then focuses on one line of criticism––Rejectionism––which is endorsed by Field in his Review of the previously mentioned book. The thought is to grant that the method of abstraction provides singular terms, however questions its ability to produce true statements. Furthermore, Field draws an analogy between the stipulation of Hume's Principle, which commits one to the existence of numbers and the ontological argument, which commits one to the existence of God. It is then shown that this analogy is amiss and that there is no real point of affinity with the Fregean platonist's ontological strategy and the ontological arguments. A further objection concerning the tacit ontological commitments on the right hand side of Abstraction Principle is discussed. The paper concludes considering ‘the onus of proof’ – issue for Nominalism‐Platonism debates.

This chapter presents some objections to the argument contending that if a conclusion is an immediate consequence of the data, the burden of proof lies on those who would deny it. The main objection posits that each claim appearing to refer to a type is merely a manner of speaking for a claim that does not actually refer to a type. Because such apparent references can be “paraphrased away,” they are harmless, and we need not suppose that types exist. In denying the existence of types, the main motivation is epistemological in nature: knowledge of types, which are abstract objects, is not possible since knowledge requires some sort of relationship, and abstract objects do not stand in causal relations with humans.

McGinn's aim is two‐fold: to undermine both descriptive and causal theories of reference, and to argue for his preferred, ‘contextual’ theory of reference. McGinn is moved to this position by emphasizing indexicals—which he takes to be the primary referential devices—rather than proper names. Linguistic reference, for McGinn, is a conventional activity governed by rules that prescribe the spatio‐temporal conditions of correct use; the semantic referent of a speaker's term is given by combining its linguistic meaning with the spatio‐temporal context in which the speaker is located. McGinn concludes his defence of this theory by demonstrating the plausibility of its implications for such topics as abstract objects, self‐reference, attribution, the language of thought hypothesis, truth, and the reducibility of reference.

Ontological theories of musical works fall into two broad classes, according to whether they take musical works to be abstract objects of some sort. The chapter uses the terms ‘Platonism’ and ‘nominalism’ to refer to these two kinds of theory. It first considers recent debates within Platonism about musical works—the theory that musical works are abstract objects. It then considers reasons to be suspicious of such a view, motivating a consideration of nominalist theories of musical works. The chapter argues 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.

Mathematics poses a difficult problem for methodological naturalists, those who embrace scientific method, and also for ontological naturalists who eschew non-physical entities such as Cartesian souls. Mathematics seems both essential to science but also committed to abstract non-physical entities while methodologically it seems to have no place for experiment or empirical confirmation. The chapter critically reviews a number of responses naturalists have made including logicism, Quinean radical empiricism, and Penelope Maddy’s variant thereof and suggests some further problems both for ontological and for methodological naturalists.

This chapter sets out an account of intuitive knowledge that is analogous to the account developed for perceptual knowledge in the Chapter 6. According to it, if an intuition experience representing that p puts you in a position to know that p, then it does so in part because it makes you aware of a truth-maker for the proposition that p. Since intuitions are generally about abstract matters, this account raises the question: how can we stand in awareness relations to the portions of reality that make propositions about abstract matters true? The bulk of the chapter is dedicated to addressing this question. Sections 7.2 through 7.5 develop an account according to which intuition experiences can make their subjects aware of abstract subject matter in part because they stand in a distinctive type of non-causal dependence relation to that subject matter