Descartes has often been called the 'father of modern philosophy'. His attempts to find foundations for knowledge, and to reconcile the existence of the soul with the emerging science of his time, are among the most influential and widely studied in the history of philosophy. This is a classic and challenging introduction to Descartes by one of the most distinguished modern philosophers. Bernard Williams not only analyzes Descartes' project of founding knowledge on certainty, but uncovers the philosophical motives for his (...) search. With acute insight, he demonstrates how Descartes' _Meditations _are not merely a description but the very enactment of philosophical thought and discovery. Williams covers all of the key areas of Descartes' thought, including God, the will, the possibility of knowledge, and the mind and its place in nature. He also makes profound contributions to the theory of knowledge, metaphysics and philosophy generally. With a new foreword by John Cottingham. (shrink)

This book is intended to be used as a textbook by students of mathematics, and also within limitations as a reference work.

This book is intended to be used as a textbook by students of mathematics, and also within limitations as a reference work.

This paper is an edited form of a letter written by the two authors (in the name of Tarski) to Wolfram Schwabhäuser around 1978. It contains extended remarks about Tarski's system of foundations for Euclidean geometry, in particular its distinctive features, its historical evolution, the history of specific axioms, the questions of independence of axioms and primitive notions, and versions of the system suitable for the development of 1-dimensional geometry.

This classic undergraduate treatment examines the deductive method in its first part and explores applications of logic and methodology in constructing mathematical theories in its second part. Exercises appear throughout.

Now in its fourth edition, this classic work clearly and concisely introduces the subject of logic and its applications. The first part of the book explains the basic concepts and principles which make up the elements of logic. The author demonstrates that these ideas are found in all branches of mathematics, and that logical laws are constantly applied in mathematical reasoning. The second part of the book shows the applications of logic in mathematical theory building with concrete examples that draw (...) upon the concepts and principles presented in the first section. Numerous exercises and an introduction to the theory of real numbers are also presented. Students, teachers and general readers interested in logic and mathematics will find this book to be an invaluable introduction to the subject. (shrink)

A formal theory of quantity T Q is presented which is realist, Platonist, and syntactically second-order (while logically elementary), in contrast with the existing formal theories of quantity developed within the theory of measurement, which are empiricist, nominalist, and syntactically first-order (while logically non-elementary). T Q is shown to be formally and empirically adequate as a theory of quantity, and is argued to be scientifically superior to the existing first-order theories of quantity in that it does not depend upon empirically (...) unsupported assumptions concerning existence of physical objects (e.g. that any two actual objects have an actual sum). The theory T Q supports and illustrates a form of naturalistic Platonism, for which claims concerning the existence and properties of universals form part of natural science, and the distinction between accidental generalizations and laws of nature has a basis in the second-order structure of the world. (shrink)

I defend Joseph Melia’s nominalist account of mathematics from an objection raised by Mark Colyvan.

For every finite n ≥ 4 there is a logically valid sentence φ n with the following properties: φ n contains only 3 variables (each of which occurs many times); φ n contains exactly one nonlogical binary relation symbol (no function symbols, no constants, and no equality symbol): φ n has a proof in first-order logic with equality that contains exactly n variables, but no proof containing only n - 1 variables. This result was first proved using the machinery of (...) algebraic logic developed in several research monographs and papers. Here we replicate the result and its proof entirely within the realm of (elementary) first-order binary predicate logic with equality. We need the usual syntax, axioms, and rules of inference to show that φ n has a proof with only n variables. To show that φ n has no proof with only n - 1 variables we use alternative semantics in place of the usual, standard, set-theoretical semantics of first-order logic. (shrink)

We naturally think of the material world as being populated by a large number of individuals . These are things, such as my laptop and the particles that compose it, that we describe as being propertied and related in various ways when we describe the material world around us. In this paper I argue that, fundamentally speaking at least, there are no such things as material individuals. I then propose and defend an individual-less view of the material world I call (...) “generalism”. (shrink)

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

When I first began to take an interest in the debate over nominalism in philosophy of mathematics, some twenty-odd years ago, the issue had already been under discussion for about a half-century. The terms of the debate had been set: W. V. Quine and others had given “abstract,” “nominalism,” “ontology,” and “Platonism” their modern meanings. Nelson Goodman had launched the project of the nominalistic reconstruction of science, or of the mathematics used in science, in which Quine for a time had (...) joined him before turning against him. William Alston, Rudolf Carnap, and Michael Dummett had raised doubts about what the point of Goodman’s exercise could be, and though they had unfortunately been largely ignored, Quine’s contention that the exercise cannot be successfully completed had gained wide publicity as the so-called “indispensability” argument against nominalism. By contrast, two subtle discussions of Paul Benacerraf had been appropriated by nominalists and turned into the so-called “multiple reductions” and “epistemological” arguments for nominalism. (shrink)

v. 1. Nominalism and realism.--v. 2. A theory of universals.

Descartes has often been called the 'father of modern philosophy'. His attempts to find foundations for knowledge, and to reconcile the existence of the soul with the emerging science of his time, are among the most influential and widely studied in the history of philosophy. This is a classic and challenging introduction to Descartes by one of the most distinguished modern philosophers. Bernard Williams not only analyzes Descartes' project of founding knowledge on certainty, but uncovers the philosophical motives for his (...) search. With acute insight, he demonstrates how Descartes' _Meditations _are not merely a description but the very enactment of philosophical thought and discovery. Williams covers all of the key areas of Descartes' thought, including God, the will, the possibility of knowledge, and the mind and its place in nature. He also makes profound contributions to the theory of knowledge, metaphysics and philosophy generally. With a new foreword by John Cottingham. (shrink)

Descartes has often been called the 'father of modern philosophy'. His attempts to find foundations for knowledge, and to reconcile the existence of the soul with the emerging science of his time, are among the most influential and widely studied in the history of philosophy. This is a classic and challenging introduction to Descartes by one of the most distinguished modern philosophers. Bernard Williams not only analyzes Descartes' project of founding knowledge on certainty, but uncovers the philosophical motives for his (...) search. With acute insight, he demonstrates how Descartes' _Meditations _are not merely a description but the very enactment of philosophical thought and discovery. Williams covers all of the key areas of Descartes' thought, including God, the will, the possibility of knowledge, and the mind and its place in nature. He also makes profound contributions to the theory of knowledge, metaphysics and philosophy generally. With a new foreword by John Cottingham. (shrink)

In order to perfectly describe the world, it is not enough to speak truly. One must also use the right concepts - including the right logical concepts. One must use concepts that "carve at the joints", that give the world's "structure". There is an objectively correct way to "write the book of the world". Much of metaphysics, as traditionally conceived, is about the fundamental nature of reality; in the present terms, this is about the world's structure. Metametaphysics - inquiry into (...) the status of metaphysical questions - turns on structure. The question of whether ontological, causal, or modal questions are "substantive" is in large part a question of whether the world has ontological, causal, and modal structure - whether quantifiers, causal relations, and modal operators carve at the joints. (shrink)

Science Without Numbers caused a stir in 1980, with its bold nominalist approach to the philosophy of mathematics and science. It has been unavailable for twenty years and is now reissued in a revised edition with a substantial new preface presenting the author's current views and responses to the issues raised in subsequent debate.

On the now dominant Quinean view, metaphysics is about what there is. Metaphysics so conceived is concerned with such questions as whether properties exist, whether meanings exist, and whether numbers exist. I will argue for the revival of a more traditional Aristotelian view, on which metaphysics is about what grounds what. Metaphysics so revived does not bother asking whether properties, meanings, and numbers exist (of course they do!) The question is whether or not they are fundamental.

This is an obituary of Bernard Williams.

This paper distinguishes two kinds of realist issue -- the issue of whether the propositions of a given domain are factual and the issue of whether they are fundamental. It criticizes previous accounts of what these issues come to and suggests that they are to be understood in terms of a basic metaphysical concept of reality. This leaves open the question of how such issues are to be resolved; and it is argued that this may be done through consideration of (...) what grounds the facts of a given domain, when fundamentality is in question, and what grounds our engagement with the putative facts, when factuality is in question. (shrink)