Foundations without Foundationalism: A Case for Second-order Logic

by Stewart Shapiro

Overview

The central contention of this book is that second-order logic has a central role to play in laying the foundations of mathematics. In order to develop the argument fully, the author presents a detailed description of higher-order logic, including a comprehensive discussion of its semantics. He goes on to demonstrate the prevalence of second-order concepts in mathematics and the extent to which mathematical ideas can be formulated in higher-order logic. He also shows how first-order languages are often insufficient to codify many concepts in contemporary mathematics, and thus that both first- and higher-order logics are needed to fully reflect current work. Throughout, the emphasis is on discussing the associated philosophical and historical issues and the implications they have for foundational studies. For the most part, the author assumes little more than a familiarity with logic comparable to that provided in a beginning graduate course which includes the incompleteness of arithmetic and the Lowenheim-Skolem theorems. All those concerned with the foundations of mathematics will find this a thought-provoking discussion of some of the central issues in the field today.

Product Details

ISBN-13:	9780198250296
Publisher:	Oxford University Press
Publication date:	05/18/2000
Series:	Oxford Logic Guides
Edition description:	New Edition
Pages:	300
Product dimensions:	9.21(w) x 6.14(h) x 0.63(d)

About the Author

Ohio State University

Table of Contents

PART I: Orientation1. Terms and Questions2. Foundationalism and Foundations of MathematicsPART II: Logic and Mathematics3. Theory4. Metatheory5. Second-Order Logic and Mathematics6. Advanced MetatheoryPART III: History and Philosophy7. The Historical Triumph of First-Order Languages8. Second-Order Logic and Rule-Following9. The Competition

Editorial Reviews

"Contains more on second-order logic than is readily available in any other textbook or survey. Philosophically, the book also contains many words of wisdom." —Journal of Symbolic Logic

"The most comprehensive all-round account and defense of second-order logic as a vehicle for mathematics known to the reviewer. It is also very comprehensively documented with a large bibliography and a wealth of interesting material in the numerous footnotes." —Mathematical Reviews

"An excellent book, covering all of the main results in second-order logic and its applications to mathematical theories. There is a great wealth of material about second-order logic in this book. . . . it is really an excellent book, and . . . the author is to be commended for a job well done." —Notre Dame Journal of Formal Logic

From the Publisher