Neo-Fregeans argue that substantial mathematics can be derived from a priori abstraction principles, Hume's Principle connecting numerical identities with one:one correspondences being a prominent example. The embarrassment of riches objection is that there is a plurality of consistent but pairwise inconsistent abstraction principles, thus not all consistent abstractions can be true. This paper considers and criticizes various further criteria on acceptable abstractions proposed by Wright settling on another one—stability—as the best bet for neo-Fregeans. However, an analogue of the embarrassment of (...) riches objection resurfaces in the metatheory and I conclude by arguing that the neo-Fregean program, at least insofar as it includes a platonistic ontology, is fatally wounded by it. (shrink)

While Frege’s own attempt to provide a purely logical foundation for arithmetic failed, Hume’s principle suffices as a foundation for elementary arithmetic. It is known that the resulting system is consistent—or at least if second-order arithmetic is. Some philosophers deny that HP can be regarded as either a truth of logic or as analytic in any reasonable sense. Others—like Crispin Wright and I—take the opposed view. Rather than defend our claim that HP is a conceptual truth about numbers, I explain (...) one way it may be possible to extend our view beyond elementary arithmetic to encompass the theory of real numbers. My approach has affinities to the leading ideas of Frege’s own treatment of the reals, although differing in one fundamental way. I attempt, like the HP approach to elementary arithmetic, to obtain the reals very directly by means of abstraction principles without any essential reliance on a theory of sets. This is the most natural way of extending the neo-Fregean position to the reals. (shrink)

Jonathan Lowe argues that metaphysics should be restored to a central position in philosophy, as the most fundamental form of inquiry, whose findings underpin those of all other disciplines. He portrays metaphysics as charting the possibilities of existence, by identifying the categories of being and the relations between them. He sets out his own original metaphysical system, within which he seeks to answer many of the deepest questions in philosophy. 'a very rich book... deserves to be read carefully by anyone (...) interested in any of the many subjects he discusses.' Katherine Hawley, British Journal of the Philosophy of Science. (shrink)

How do we go about weighing evidence, testing hypotheses, and making inferences? According to the model of _Inference to the Best Explanation_, we work out what to infer from the evidence by thinking about what would actually explain that evidence, and we take the ability of a hypothesis to explain the evidence as a sign that the hypothesis is correct. In _Inference to the Best Explanation_, Peter Lipton gives this important and influential idea the development and assessment it deserves. The (...) second edition has been substantially enlarged and reworked, with a new chapter on the relationship between explanation and Bayesianism, and an extension and defence of the account of contrastive explanation. It also includes an expanded defence of the claims that our inferences really are guided by diverse explanatory considerations, and that this pattern of inference can take us towards the truth. This edition of _Inference to the Best Explanation_ has also been updated throughout and includes a new bibliography. (shrink)

This book is a defense of modal realism; the thesis that our world is but one of a plurality of worlds, and that the individuals that inhabit our world are only a few out of all the inhabitants of all the worlds. Lewis argues that the philosophical utility of modal realism is a good reason for believing that it is true.

Jonathan Lowe argues that metaphysics should be restored to a central position in philosophy, as the most fundamental form of rational inquiry, whose findings underpin those of all other disciplines. He portrays metaphysics as charting the possibilities of existence, by idetifying the categories of being and the relations of ontological dependency between entities of different categories. He proceeds to set out a unified and original metaphysical system: he defends a substance ontology, according to which the existence of the world s (...) one world in time depends upon the existence of persisting things which retain their identity over time and through processes of qualitative change. And he contends that even necessary beings, such as the abstract objects of mathematics, depend ultimately for their existence upon there being a concrete world of enduring substances. Within his system of metaphysics Lowe seeks to answer many of the deepest and most challenging questions in philosophy. (shrink)

'There is much food for thought in McGinn's discussions and each chapter is rich with a series of considerations for thinking that the currently received views on the various topics have some serious difficulties that need confronting... For those interested in metaphysics and the philosophy of logic, this book will stimulate much further thought' -Mind 'The sweep of the book is broad and the pace is brisk... There is much material here to provide the basis for many a deep philosophical (...) discussion' -Mind 'Lucid and provocative little book... clear, direct and well argued' -Times Higher Education SupplementIdentity, existence, predication, necessity, and truth are fundamental philosophical concerns. Colin McGinn treats them both philosophically and logically, aiming for maximum clarity and minimum pointless formalism. He contends that there are real logical properties that challenge naturalistic metaphysical outlooks. These concepts are not definable, though we can say a good deal about how they work. The aim of Logical Properties is to bring philosophy back to philosophical logic. (shrink)

Existence questions have been topics for heated debates in metaphysics, but this book argues that they can often be answered easily, by trivial inferences from uncontroversial premises. This 'easy' approach to ontology leads to realism about disputed entities, and to the view that metaphysical disputes about existence questions are misguided.

According to modalism, modality is primitive. In this paper, we examine the implications of this view for modal epistemology, and articulate a modalist account of modal knowledge. First, we discuss a theoretical utility argument used by David Lewis in support of his claim that there is a plurality of concrete worlds. We reject this argument, and show how to dispense with possible worlds altogether. We proceed to account for modal knowledge in modalist terms.

There is a good deal of disagreement about composition. There is firstorder disagreement: there are radically different answers to the special composition question—the question of under what circumstances the xs compose a y. There is second-order disagreement: there are different answers to the question of whether first-order disagreement is real or merely semantic. Virtually all disputants with respect to both the first-and second-order issues agree that the answer or answers to the special composition question will take the form of a (...) necessary truth or truths even though, as I will argue, such answers do not appear to be good candidates to be necessary truths. This paper provides an analysis of the concept of as it pertains to concrete objects, that fulfils two functions. First, it explicates the sense in which claims about composition are contingent and the sense in which they are necessary, and second, it provides a way of understanding when first-order disputes are substantial and when they are merely semantic. (shrink)

Suppose we are asked to draw up a list of things we take to exist. Certain items seem unproblematic choices, while others (such as God) are likely to spark controversy. The book sets the grand theological theme aside and asks a less dramatic question: should mathematical objects (numbers, sets, functions, etc.) be on this list? In philosophical jargon this is the ‘ontological’ question for mathematics; it asks whether we ought to include mathematicalia in our ontology. The goal of this work (...) is to answer this question in the affirmative, by drawing on considerations on the the applicability of mathematics to natural science. (shrink)

Platonists and nominalists disagree about whether mathematical objects exist. But they almost uniformly agree about one thing: whatever the status of the existence of mathematical objects, that status is modally necessary. Two notable dissenters from this orthodoxy are Hartry Field, who defends contingent nominalism, and Mark Colyvan, who defends contingent Platonism. The source of their dissent is their view that the indispensability argument provides our justification for believing in the existence, or not, of mathematical objects. This paper considers whether commitment (...) to the indispensability argument gives one grounds to be a contingentist about mathematical objects. (shrink)

2014 Reprint of 1954 American Edition. Full facsimile of the original edition, not reproduced with Optical Recognition Software. This two volume classic comprises two titles: "Patterns of Plausible Inference" and "Induction and Analogy in Mathematics." This is a guide to the practical art of plausible reasoning, particularly in mathematics, but also in every field of human activity. Using mathematics as the example par excellence, Polya shows how even the most rigorous deductive discipline is heavily dependent on techniques of guessing, inductive (...) reasoning, and reasoning by analogy. In solving a problem, the answer must be guessed at before a proof can be given, and guesses are usually made from a knowledge of facts, experience, and hunches. The truly creative mathematician must be a good guesser first and a good prover afterward; many important theorems have been guessed but no proved until much later. In the same way, solutions to problems can be guessed, and a god guesser is much more likely to find a correct solution. This work might have been called "How to Become a Good Guesser."-From the Dust Jacket. (shrink)

Develops a structuralist understanding of mathematics, as an alternative to set- or type-theoretic foundations, that respects classical mathematical truth while ...

Logical pluralism is the view according to which there is more than one relation of logical consequence, even within a given language. A recent articulation of this view has been developed in terms of quantification over different cases: classical logic emerges from consistent and complete cases; constructive logic from consistent and incomplete cases, and paraconsistent logic from inconsistent and complete cases. We argue that this formulation causes pluralism to collapse into either logical nihilism or logical universalism. In its place, we (...) propose a modalist account of logical pluralism that is independently well motivated and that avoids these collapses. (shrink)

The Oxford Handbook of Metaphysics offers the most authoritative and compelling guide to this diverse and fertile field of philosophy. Twenty-four of the world's most distinguished specialists provide brand-new essays about 'what there is': what kinds of things there are, and what relations hold among entities falling under various categories. They give the latest word on such topics as identity, modality, time, causation, persons and minds, freedom, and vagueness. The Handbook's unrivaled breadth and depth make it the definitive reference work (...) for students and academics across the philosophical spectrum. (shrink)

Kit Fine develops a Fregean theory of abstraction, and suggests that it may yield a new philosophical foundation for mathematics, one that can account for both our reference to various mathematical objects and our knowledge of various mathematical truths. The Limits ofion breaks new ground both technically and philosophically.

This book not only outlines the indispensability argument in considerable detail but also defends it against various challenges.

This book expounds a system of ideas about the nature of mathematics which Michael Resnik has been elaborating for a number of years. In calling mathematics a science he implies that it has a factual subject-matter and that mathematical knowledge is on a par with other scientific knowledge; in calling it a science of patterns he expresses his commitment to a structuralist philosophy of mathematics. He links this to a defense of realism about the metaphysics of mathematics--the view that mathematics (...) is about things that really exist. (shrink)

Here the author of How to Solve It explains how to become a "good guesser." Marked by G. Polya's simple, energetic prose and use of clever examples from a wide range of human activities, this two-volume work explores techniques of guessing, inductive reasoning, and reasoning by analogy, and the role they play in the most rigorous of deductive disciplines.

Metaphysicians speak of laws of nature in terms of necessity and universality; scientists, in terms of symmetry and invariance. In this book van Fraassen argues that no metaphysical account of laws can succeed. He analyzes and rejects the arguments that there are laws of nature, or that we must believe there are, and argues that we should disregard the idea of law as an adequate clue to science. After exploring what this means for general epistemology, the author develops the empiricist (...) view of science as a construction of models to represent the phenomena. (shrink)

David Lewis (1941-2001) was Class of 1943 University Professor of Philosophy at Princeton University. His contributions spanned philosophical logic, philosophy of language, philosophy of mind, philosophy of science, metaphysics, and epistemology. In On the Plurality of Worlds, he defended his challenging metaphysical position, "modal realism." He was also the author of the books Convention, Counterfactuals, Parts of Classes, and several volumes of collected papers.

David Miller is the foremost exponent of the purist critical rationalist doctrine and here presents his mature views, discussing the role that logic and argument play in the growth of knowledge, criticizing the common understanding of argument as an instrument of justification, persuasion or discovery and instead advocating the critical rationalist view that only criticism matters. Miller patiently and thoroughly undoes the damage done by those writers who attack critical rationalism by invoking the sterile mythology of induction and justification that (...) it seeks to sweep away. In addition his new material on the debate on verisimilitude is essential reading for all working in this field. (shrink)

First published in 1971, Professor Putnam's essay concerns itself with the ontological problem in the philosophy of logic and mathematics - that is, the issue of whether the abstract entities spoken of in logic and mathematics really exist. He also deals with the question of whether or not reference to these abstract entities is really indispensible in logic and whether it is necessary in physical science in general.

When ordinary people - mathematicians among them - take something to follow from something else, they are exposing the backbone of our self-ascribed ability to reason. Jody Azzouni 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. (shrink)

Selection of thirty six articles by Putnam, mostly written after 2000.

Metaphysics is supposed to tell us about the metaphysical nature of our world: under what conditions composition occurs; how objects persist through time; whether properties are universals or tropes. It is near orthodoxy that whichever of these sorts of metaphysical claims is true is necessarily true. This paper looks at the debate between that orthodox view and a recently emerging view that claims like these are contingent, by focusing on the metaphysical debate between monists and pluralists about concrete particulars. This (...) paper argues that we should be contingentists about monism and pluralism, and it defends contingentism against some necessitarian objections by offering an epistemology of contingent metaphysical claims. (shrink)

Here, Bob Hale and Crispin Wright assemble the key writings that lead to their distinctive neo-Fregean approach to the philosophy of mathematics. In addition to fourteen previously published papers, the volume features a new paper on the Julius Caesar problem; a substantial new introduction mapping out the program and the contributions made to it by the various papers; a section explaining which issues most require further attention; and bibliographies of references and further useful sources. It will be recognized as the (...) most powerful presentation yet of a neo-Fregean program. (shrink)

Neo-Fregean logicism attempts to base mathematics on abstraction principles. Since not all abstraction principles are acceptable, the neo-Fregeans need an account of which ones are. One of the most promising accounts is in terms of the notion of stability; roughly, that an abstraction principle is acceptable just in case it is satisfiable in all domains of sufficiently large cardinality. We present two counterexamples to stability as a sufficient condition for acceptability and argue that these counterexamples can be avoided only by (...) major departures from the existing neo-Fregean programme. (shrink)