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Hilbert's tenth problem is one of 23 problems proposed by David Hilbert in 1900 at the International Congress of Mathematicians in Paris. These problems gave focus for the exponential development of mathematical thought over the following century. The tenth problem asked for a general algorithm to determine if a given Diophantine equation has a solution in integers. It was finally resolved in a series of papers written by Julia Robinson, Martin Davis, Hilary Putnam, and finally Yuri Matiyasevich in 1970. They showed that no such algorithm exists. This book is an exposition of this remarkable achievement. Often, the solution to a famous problem involves formidable background. Surprisingly, the solution of Hilbert's tenth problem does not. What is needed is only some elementary number theory and rudimentary logic. In this book, the authors present the complete proof along with the romantic history that goes with it. Along the way, the reader is introduced to Cantor's transfinite numbers, axiomatic set theory, Turing machines, and Gödel's incompleteness theorems. Copious exercises are included at the end of each chapter to guide the student gently on this ascent. For the advanced student, the final chapter highlights recent developments and suggests future directions. The book is suitable for undergraduates and graduate students. It is essentially self-contained.
Provides an essential introduction to classical logic.
Logic is sometimes called the foundation of mathematics: the logician studies the kinds of reasoning used in the individual steps of a proof. Alonzo Church was a pioneer in the field of mathematical logic, whose contributions to number theory and the theories of algorithms and computability laid the theoretical foundations of computer science. His first Princeton book, The Calculi of Lambda-Conversion (1941), established an invaluable tool that computer scientists still use today. Even beyond the accomplishment of that book, however, his second Princeton book, Introduction to Mathematical Logic, defined its subject for a generation. Originally published in Princeton's Annals of Mathematics Studies series, this book was revised in 1956 and reprinted a third time, in 1996, in the Princeton Landmarks in Mathematics series. Although new results in mathematical logic have been developed and other textbooks have been published, it remains, sixty years later, a basic source for understanding formal logic. Church was one of the principal founders of the Association for Symbolic Logic; he founded the Journal of Symbolic Logic in 1936 and remained an editor until 1979 At his death in 1995, Church was still regarded as the greatest mathematical logician in the world.
In the field of philosophy, Plato's view of rhetoric as a potentially treacherous craft has long overshadowed Aristotle's view, which focuses on rhetoric as an independent discipline that relates in complex ways to dialectic and logic and to ethics and moral psychology. This volume, composed of essays by internationally renowned philosophers and classicists, provides the first extensive examination of Aristotle's Rhetoric and its subject matter in many years. One aim is to locate both Aristotle's treatise and its subject within the more general context of his philosophical treatment of other disciplines, including moral and political theory as well as poetics. The contributors also seek to illuminate the structure of Aristotle's own conception of rhetoric as presented in his treatise. The first section of the book, which deals with the arguments of rhetoric, contains essays by M. F. Burnyeat and Jacques Brunschwig. A section treating the status of the art of rhetoric features pieces by Eckart Schütrumpf, Jürgen Sprute, M. M. McCabe, and Glenn W. Most. Essays by John M. Cooper, Stephen Halliwell, and Jean-Louis Labarrière address topics related to rhetoric, ethics, and politics. The final section, on rhetoric and literary art, comprises essays by Alexander Nehamas and André Laks. Originally published in 1994. The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.