Stewart Shapiro

According to a familiar, simple argument, numbers exist because sentences like ‘Two is an even number’ are true. Whereas realists accept the argument as sound, anti-realists either reject that number words function referentially in such sentences (non-referentialism) or else that such sentences are true (fictionalism). We argue that this dialectic, though familiar, drastically underestimates the extent to which natural language supports realism. Indeed, if dominant accounts of number and measurement-related expressions within linguistic semantics are correct, then far more than just our overt talk about numbers as objects would not be true if numbers do not exist. The purpose of this paper is to sketch this strengthened argument for realism and to survey its consequences for extant forms of anti-realism.

This paper is concerned with Gottlob Frege’s theory of the real numbers as sketched in the second volume of his masterpiece _Grundgesetze der Arithmetik_. It is perhaps unsurprising that Frege’s theory of the real numbers is intimately intertwined with and largely motivated by his metaphysics. The account raises interesting, and surprisingly underexplored, questions about Frege’s metaphysics: Can this metaphysics even accommodate mass quantities like water, gold, light intensity, or charge? Frege’s main complaint with his contemporaries Cantor and Dedekind is that their theories of the real numbers do not build the applicability of the real numbers directly into the construction. In taking Cantor and Dedekind’s Arithmetic theories to be insufficient, clearly Frege takes it to be a desideratum on a theory of the real numbers that their applicability be essential to their construction. We begin with a detailed review of Frege’s theory, one that mirrors Frege’s exposition in structure. This is followed by a critique, outlining Frege’s linguistic motivation for ontologically distinguishing the cardinal numbers from the real numbers. We briefly consider how Frege’s metaphysics might need to be developed, or amended, to accommodate some of the problems. Finally, we offer a detailed examination of Frege’s Application Constraint – that the reals ought to have their applicability built directly into their characterization. It bears on deeper questions concerning the relationship between sophisticated mathematical theories and their applications.

Hale has articulated and defended a thesis that properties are tied to predicates in possible languages. The same goes for functions. A necessary and sufficient condition for the existence of a property or function is that it is expressed by a predicate in a possible language that can be understood by finite beings, like us humans. The purpose of this chapter is to assess this view and determine how it fares against the output of standard mathematics. In order to interpret mathematics, Hale must defend some prima facie implausible claims about what sorts of languages are possible, for us finite beings. The only alternative is for Hale to reject, on purely philosophical grounds, large chunks of contemporary mathematics. The relevant mathematics is whatever relies on impredicative definitions and the axiom of choice. This includes his, and Crispin Wright’s, abstraction-based neologicism, which has been shown to rely indispensably on impredicative definitions.

This chapter investigates the question of when it is reasonable to replace an inconsistent concept. After surveying a number of proposals for how one might understand constitutive principles, it goes on to endorse Burgess’s (2004) account of being pragmatically analytic, as a possible source of insight into constitutive principles. The chapter then raises a question: If truth is an inconsistent concept, does it need to be replaced? According to the argument in the chapter, when an inconsistent concept paralyzes valuable projects, it is time to replace it. And if we are to replace a concept, our replacement should be able to do the work that the inconsistency-yielding one did. This, of course, raises a fundamental question concerning what work the notion of truth does for us. The chapter mounts a case for the claim that inflationists, but not obvious deflationists, about truth should offer a replacement for the concept of truth.

The purpose of this article is to assess the prospects for a Scottish neo-logicist foundation for a set theory. We show how to reformulate a key aspect of our set theory as a neo-logicist abstraction principle. That puts the enterprise on the neo-logicist map, and allows us to assess its prospects, both as a mathematical theory in its own right and in terms of the foundational role that has been advertised for set theory. On the positive side, we show that our abstraction based theory can be modified to yield much of ordinary mathematics, indeed everything needed to recapture all branches of mathematics short of set theory itself. However, our conclusions are mostly negative. The theory will fall far short of the power of ordinary Zermelo-Fraenkel set theory. It is consistent that our set theory has models that are relatively small, smaller than the first cardinal with an uncountable index. More important, there is a strong tension between the idea that the iterative hierarchy is somehow ineffable, or indefinitely extensible, and the neo-logicist theme of capturing mathematical theories with abstraction principles.

One much discussed issue concerns the metaphysical nature, or the source or cause, of vagueness. Is vagueness a purely linguistic matter, concerned (merely) with how the world gets represented via language, or is there a sense in which the world itself is vague? This chapter argues against the intelligibility of the question of metaphysical vagueness, at least if it is put in such a straightforward and perhaps naive manner. Reflection on the origins and nature of vagueness in language suggests that the phenomenon is due to both the way the world is and the way that we, the users of language, and our faculties of representation are, and that there is no sense to separating those factors and attributing vagueness to one of them exclusively.

This paper argues that the dialetheist is subject to a criticism much like one that Priest levels against consistent theories of truth: there are certain notions and concepts that the dialetheist invokes (informally), but which he or she cannot adequately express, unless the meta-theory is (completely) consistent. Such notions include disagreement, simple truth, simple falsity, non-diatheia, and even consistency. The insistence on a consistent meta-theory would undermine the key aspect of dialetheism; namely, the uniformity of object language and meta-language.

The philosophy of mathematics articulated and defended in this book goes by the name of “structuralism”, and its slogan is that mathematics is the science of structure. The subject matter of arithmetic, for example, is the natural number structure, the pattern common to any countably infinite system of objects with a distinguished initial object and a successor relation that satisfies the induction principle. The essence of each natural number is its relation to the other natural numbers. One way to understand structuralism is to reify structures as ante rem universals. This would be a platonism concerning mathematical objects, which are the places within such structures. Alternatively, one can take an eliminative, in re approach, and understand talk of structures as shorthand for talk of systems of objects or, invoking modality, talk of possible systems of objects. Shapiro argues that although the realist, ante rem approach is the most perspicuous, in a sense, the various accounts are equivalent. Along the way, the ontological and epistemological aspects of the structuralist philosophies are assessed. One key aspect is to show how each philosophy deals with reference to mathematical objects. The view is tentatively extended to objects generally: to science and ordinary discourse.

Number words seemingly function both as adjectives attributing cardinality properties to collections, as in Frege’s ‘Jupiter has four moons’, and as names referring to numbers, as in Frege’s ‘The number of Jupiter’s moons is four’. This leads to what Thomas Hofweber calls Frege’s Other Puzzle: How can number words function as modifiers and as singular terms if neither adjectives nor names can serve multiple semantic functions? Whereas most philosophers deny that one of these uses is genuine, we instead argue that number words, like many related expressions, are polymorphic, having multiple uses whose meanings are systematically related via type shifting.

This chapter explores the ramifications of a relativist view of logic with respect to the meanings of logical terms. It is a twist on the old question of whether classicists and intuitionists have any substantial disagreement, or whether they are merely talking past each other as they attach different meanings to the crucial logical terminology. The author argues that the very question of whether the meanings are the same or different is itself a context sensitive matter: it depends on what aspects of the situation are salient.

Stewart Shapiro's ambition in Vagueness in Context is to develop a comprehensive account of the meaning, function, and logic of vague terms in an idealized version of a natural language like English. It is a commonplace that the extensions of vague terms vary according to their context: a person can be tall with respect to male accountants and not tall (even short) with respect to professional basketball players. The key feature of Shapiro's account is that the extensions of vague terms also vary in the course of conversations and that, in some cases, a competent speaker can go either way without sinning against the meaning of the words or the non-linguistic facts. As Shapiro sees it, vagueness is a linguistic phenomenon, due to the kinds of languages that humans speak; but vagueness is also due to the world we find ourselves in, as we try to communicate features of it to each other.