Much revised since its first appearance in 1941, Willard Van Orman Quine’s Elementary Logic, despite its brevity, is notable for its scope and rigor. It provides a single strand of simple techniques for the central business of modern logic. Basic formal concepts are explained, the paraphrasing of words into symbols is treated at some length, and a testing procedure is given for truth-function logic along with a complete proof procedure for the logic of quantifiers.

Fully one third of this revised edition is new, and presents a nearly complete turnover in crucial techniques of testing and proving, some change of notation, and some updating of terminology. The study is intended primarily as a convenient encapsulation of minimum essentials, but concludes by giving brief glimpses of further matters.

The mysteries of the physical world speak to us through equations--compact statements about the way nature works, expressed in nature's language, mathematics. In this book by the renowned Dutch physicist Sander Bais, the equations that govern our world unfold in all their formal grace--and their deeper meaning as core symbols of our civilization.

Trying to explain science without equations is like trying to explain art without illustrations. Consequently Bais has produced a book that, unlike any other aimed at nonscientists, delves into the details--historical, biographical, practical, philosophical, and mathematical--of seventeen equations that form the very basis of what we know of the universe today. A mathematical objet d'art in its own right, the book conveys the transcendent excitement and beauty of these icons of knowledge as they reveal and embody the fundamental truths of physical reality.

These are the seventeen equations that represent radical turning points in our understanding--from mechanics to electrodynamics, hydrodynamics to relativity, quantum mechanics to string theory--their meanings revealed through the careful and critical observation of patterns and motions in nature. Mercifully short on dry theoretical elaborations, the book presents these equations as they are--with the information about their variables, history, and applications that allows us to chart their critical function, and their crucial place, in the complex web of modern science.

Reading The Equations, we can hear nature speaking to us in its native language.

No one has figured more prominently in the study of the German philosopher Gottlob Frege than Michael Dummett. His magisterial Frege: Philosophy of Language is a sustained, systematic analysis of Frege's thought, omitting only the issues in philosophy of mathematics. In this work Dummett discusses, section by section, Frege's masterpiece The Foundations of Arithmetic and Frege's treatment of real numbers in the second volume of Basic Laws of Arithmetic, establishing what parts of the philosopher's views can be salvaged and employed in new theorizing, and what must be abandoned, either as incorrectly argued or as untenable in the light of technical developments.

Gottlob Frege (1848-1925) was a logician, mathematician, and philosopher whose work had enormous impact on Bertrand Russell and later on the young Ludwig Wittgenstein, making Frege one of the central influences on twentieth-century Anglo-American philosophy; he is considered the founder of analytic philosophy. His philosophy of mathematics contains deep insights and remains a useful and necessary point of departure for anyone seriously studying or working in the field.

For many philosophers, modern philosophy begins in 1879 with the publication of Gottlob Frege's Begriffsschrift, in which Frege presents the first truly modern logic in his symbolic language, Begriffsschrift, or concept-script. Danielle Macbeth's book, the first full-length study of this language, offers a highly original new reading of Frege's logic based directly on Frege's own two-dimensional notation and his various writings about logic.

Setting out to explain the nature of Frege's logical notation, Macbeth brings clarity not only to Frege's symbolism and its motivation, but also to many other topics central to his philosophy. She develops a uniquely compelling account of Frege's Sinn/Bedeutung distinction, a distinction central to an adequate logical language; and she articulates a novel understanding of concepts, both of what they are and of how their contents are expressed in properly logical language. In her reading, Frege's Begriffsschrift emerges as a powerful and deeply illuminating alternative to the quantificational logic it would later inspire.

The most enlightening examination to date of the developments of Frege's thinking about his logic, this book introduces a new kind of logical language, one that promises surprising insight into a range of issues in metaphysics and epistemology, as well as in the philosophy of logic.

Widespread interest in Frege’s general philosophical writings is, relatively speaking, a fairly recent phenomenon. But it is only very recently that his philosophy of mathematics has begun to attract the attention it now enjoys. This interest has been elicited by the discovery of the remarkable mathematical properties of Frege’s contextual definition of number and of the unique character of his proposals for a theory of the real numbers.

This collection of essays addresses three main developments in recent work on Frege’s philosophy of mathematics: the emerging interest in the intellectual background to his logicism; the rediscovery of Frege’s theorem; and the reevaluation of the mathematical content of The Basic Laws of Arithmetic. Each essay attempts a sympathetic, if not uncritical, reconstruction, evaluation, or extension of a facet of Frege’s theory of arithmetic. Together they form an accessible and authoritative introduction to aspects of Frege’s thought that have, until now, been largely missed by the philosophical community.

In this illuminating collection, Charles Parsons surveys the contributions of philosophers and mathematicians who shaped the philosophy of mathematics over the course of the past century.

Parsons begins with a discussion of the Kantian legacy in the work of L. E. J. Brouwer, David Hilbert, and Paul Bernays, shedding light on how Bernays revised his philosophy after his collaboration with Hilbert. He considers Hermann Weyl’s idea of a “vicious circle” in the foundations of mathematics, a radical claim that elicited many challenges. Turning to Kurt Gödel, whose incompleteness theorem transformed debate on the foundations of mathematics and brought mathematical logic to maturity, Parsons discusses his essay on Bertrand Russell’s mathematical logic—Gödel’s first mature philosophical statement and an avowal of his Platonistic view.

Philosophy of Mathematics in the Twentieth Century insightfully treats the contributions of figures the author knew personally: W. V. Quine, Hilary Putnam, Hao Wang, and William Tait. Quine’s early work on ontology is explored, as is his nominalistic view of predication and his use of the genetic method of explanation in the late work The Roots of Reference. Parsons attempts to tease out Putnam’s views on existence and ontology, especially in relation to logic and mathematics. Wang’s contributions to subjects ranging from the concept of set, minds, and machines to the interpretation of Gödel are examined, as are Tait’s axiomatic conception of mathematics, his minimalist realism, and his thoughts on historical figures.