Logic/Philosophy of Mathematics

This book contains a collection of twelve metaphysical chapters that address a range of first-order topics, including identity, coincidence, essence, causation, and properties. Some first-order debates are not worth pursuing, the book argues; there is nothing at issue in them. Several of the chapters explore the metaontology of abstract objects, and more generally of objects that are ‘preconceived’, their principal features being settled already by their job-descriptions. The book rejects standard forms of fictionalism, opting ultimately for a view that puts presupposition in the role normally played by pretense.

Towards Non-Being presents an account of the semantics of intentional verbs such as ‘believes’, ‘fears’, ‘seeks’, and ‘imagines’. It tackles problems concerning intentional states which are often brushed under the carpet, such as their failure to be closed under deducibility. Drawing on the noneist work of the late Richard Routley (Sylvan), the book proceeds in terms of objects that may be existent or non-existent, at worlds that may either be possible or impossible. Since Russell, non-existent objects have had a bad press in Western philosophy. The book mounts a full-scale defence, and in the process, offers an account of both fictional and mathematical objects as non-existent.

When ordinary people—mathematicians among them—take something to follow (deductively) from something else, they are exposing the backbone of our self-ascribed ability to reason. This book investigates the connection between that ordinary notion of consequence and the formal analogues invented by logicians. One claim of the book is that, despite our apparent intuitive grasp of consequence, we do not introspect rules by which we reason, nor do we grasp the scope and range of the domain, as it were, of our reasoning. This point is illustrated with a close analysis of a paradigmatic case of ordinary reasoning—mathematical proof.

Is truth objective or relative? What exists independently of our minds? This book is about these two questions. The essays in its pages variously defend and critique answers to each, grapple over the proper methodology for addressing them, and wonder whether either question is worth pursuing. In so doing, they carry on a long and esteemed tradition – for our two questions are among the oldest of philosophical issues, and have vexed almost every major philosopher, from Plato, to Kant to Wittgenstein. Fifteen contributors bring fresh perspectives, renewed energy and original answers to debates that have been the focus of a tremendous amount of interest in the last three decades, both within philosophy and the culture at large.

This book defends an anti-platonist philosophy of mathematics derived from game formalism. Classic formalists claimed implausibly that mathematical utterances are truth-valueless moves in a game. This study aims to develop a more satisfactory successor to game formalism utilizing a widely accepted, broadly neo-Fregean framework, in which the proposition expressed by an utterance is a function of both sense and background circumstance. This framework allows for sentences whose truth-conditions are not representational, which are made true or false by conditions residing in the circumstances of utterances but not transparently in the sense. Applications to projectivism and fiction pave the way for the claim that mathematical utterances are made true or false by the existence of concrete proofs or refutations, though these truth-making conditions form no part of their sense or informational content. The position is compared with rivals, an account of the applicability of mathematics developed, and a new account of the nature of idealization proffered in which it is argued that the finitistic limitations Gödel placed on proofs are without rational justification. Finally a non-classical logical system is provided in which excluded middle fails, yet enough logical power remains to recapture the results of standard mathematics.

This book is a study of ancient logic based upon the John Locke lectures given in Oxford. Its six chapters discuss the following: firstly, certain ancient ideas about truth; secondly, the Aristotelian conception of predication; thirdly, various ideas about connectors which were developed by the ancient logicians and grammarians; fourthly, the notion of logical form, insofar as it may be discovered in the ancient texts; fifthly, the question of the ‘justification of deduction’; and sixthly, the attitude which has been called logical utilitarianism and which restricts the scope of logic to those forms of inference which are or might be useful for scientific proofs.

This book argues that an adequate account of vagueness must involve degrees of truth. The basic idea of degrees of truth is that while some sentences are true and some are false, others possess intermediate truth values: they are truer than the false sentences, but not as true as the true ones. This idea is immediately appealing in the context of vagueness — yet it has fallen on hard times in the philosophical literature, with existing degree-theoretic treatments of vagueness facing apparently insuperable objections. The book seeks to turn the tide in favour of a degree-theoretic treatment of vagueness, by motivating and defending the basic idea that truth can come in degrees, by arguing that no theory of vagueness that does not countenance degrees of truth can be correct, and by developing a new degree-theoretic treatment of vagueness — fuzzy plurivaluationism — that solves the problems plaguing earlier degree theories.

This book develops both a philosophical and a formal, model-theoretic account of the meaning, function, and logic of vague terms in an idealized version of a natural language like English. Extensions of vague terms vary with such contextual factors as the comparison class and paradigm cases: a person can be tall with respect to male accountants and not tall (even short) with respect to professional basketball players. The main feature of the author's account is that the extensions (and anti-extensions) of vague terms also vary in the course of a conversation, even after the external contextual features, such as the comparison class, are fixed. A central thesis is that in some cases, a competent speaker of the language can go either way in the borderline area of a vague predicate without sinning against the meaning of the words and the non-linguistic facts. Shapiro calls thisopen texture, borrowing the term from Friedrich Waismann. The formal model theory has a similar structure to the supervaluationist approach, employing the notion of a sharpening of a base interpretation. In line with the philosophical account, however, the notion of super-truth does not play a central role in the development of validity. The ultimate goal of the technical aspects of the work is to delimit a plausible notion of logical consequence, and to explore what happens with the sorites paradox. Later chapters deal with what passes for higher-order vagueness — vagueness in the notions of ‘determinacy’ and ‘borderline’ — and with vague singular terms, or objects. In each case, the philosophical picture is developed by extending and modifying the original account. This is followed with modifications to the model theory and the central meta-theorems. In this book, vagueness is seen as 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. Vagueness is also due to the kinds of beings we are.

Visual thinking — visual imagination or perception of diagrams and symbol arrays, and mental operations on them — is omnipresent in mathematics. Is this visual thinking merely a psychological aid, facilitating grasp of what is gathered by other means? Or does it also have epistemological functions, as a means of discovery, understanding, and even proof? This book argues that visual thinking in mathematics is rarely just a superfluous aid; it usually has epistemological value, often as a means of discovery. The book explores a major source of our grasp of mathematics, using examples from basic geometry, arithmetic, algebra, and real analysis. It shows how we can discern abstract general truths by means of specific images, how synthetic a priori knowledge is possible, and how visual means can help us grasp abstract structures. This book reopens the investigation of earlier thinkers from Plato to Kant into the nature and epistemology of an individual's basic mathematical beliefs and abilities, in the new light shed by the maturing cognitive sciences.

This book offers a study of Ludwig Wittgenstein's philosophy of mathematics. The fact that he wrote more on this subject than on any other indicates its centrality in his thought. The book traces the development of Wittgenstein's thinking in the context of the mathematical and philosophical work of the times to make sense of ideas that have often been misunderstood. In particular, it illuminates the work of the neglected ‘transitional period’ between the Tractatus and the Investigations. The book argues that a study of Wittgenstein's writings on mathematics is essential to a proper understanding of his philosophy and also that it has much to contribute to current debates about the foundations of mathematics.