Oxford Logic Guides 47
216 pages | 23 black and white illustrations | 234x156mm
978-0-19-960916-1 | Paperback | April 2011 (estimated)
A clear exposition of independence proofs in set theory presented in its most elegant form--Boolean-valued models
With a foreword by Dana Scott--an illuminating historical account by one of the creators of the subject
Paperback edition of a well-respected, classic monograph
Third edition contains expanded background material on topics including Boolean-valued analysis, Heyting-algebra valued models of intuitionistic set theory, and category theory.
Numerous exercises, along with the enlarged and entirely updated background material, make this an ideal text for students in logic and set theory
This third edition, now available in paperback, is a follow up to the author's classic Boolean-Valued Models and Independence Proofs in Set Theory,. It provides an exposition of some of the most important results in set theory obtained in the 20th century: the independence of the continuum hypothesis and the axiom of choice. Aimed at graduate students and researchers in mathematics, mathematical logic, philosophy, and computer science, the third edition has been extensively updated with expanded introductory material, new chapters, and a new appendix on category theory. It covers recent developments in the field and contains numerous exercises, along with updated and increased coverage of the background material. This new paperback edition includes additional corrections and, for the first time, will make this landmark text accessible to students in logic and set theory.
Readership: Aimed at graduate students and researchers in mathematics, mathematical logic, philosophy, and computer science.
Forward by Dana Scott
Preface
List of Problems
0: Boolean and Heyting Algebras: The Essentials
1: Boolean-Valued Models of Set Theory: First Steps
2: Forcing and Some Independence Proofs
3: Group Actions on V(B) and the Independence of the Axiom of Choice
4: Generic Ultrafilters and Transitive Models of ZFC
5: Cardinal Collapsing, Boolean Isomorphism, and Applications to the Theory of Boolean Algebras
6: Iterated Boolean Extensions, Matrin's Axiom, and Souslin's Hypothesis
7: Boolean-Valued Analysis
8: Intuitionistic Set Theory and Heyting-Algebra-Valued Models
Appendix: Boolean and Heyting Algebra-Valued Models as Categories
Historical Notes
Bibliography
Index of Symbols
Index of Terms
176 pages | 5 line drawings | 234x156mm
978-0-19-960505-7 | Hardback | May 2011 (estimated)
Chapters by world class experts in mathematics, mathematical physics and philosophy
Timothy Gowers - Marcus du Sautoy - Roger Penrose - Peter Lipton - Mary Leng - Michael Detlefsen - Stewart Shapiro - Gideon Rosen - Mark Steiner
Nontechnical presentation is accessible at all levels, both lay reader and expert
Lively, thought-provoking chapter style
Concise and authoritative introduction to a popular subject
Is mathematics a highly sophisticated intellectual game in which the adepts display their skill by tackling invented problems, or are mathematicians engaged in acts of discovery as they explore an independent realm of mathematical reality? Why does this seemingly abstract discipline provide the key to unlocking the deep secrets of the physical universe? How one answers these questions will significantly influence metaphysical thinking about reality.
This book is intended to fill a gap between popular 'wonders of mathematics' books and the technical writings of the philosophers of mathematics. The chapters are written by some of the world's finest mathematicians, mathematical physicists and philosophers of mathematics, each giving their perspective on this fascinating debate. Every chapter is followed by a short response from another member of the author team, reinforcing the main theme and raising further questions.
Accessible to anyone interested in what mathematics really means, and useful for mathematicians and philosophers of science at all levels, Meaning in Mathematics offers deep new insights into a subject many people take for granted.
Readership: Mathematicians and mathematical physicists, philosophers of mathematics and science, and anyone interested in a thought-provoking book that explores a subject many of us take for granted.
John Polkinghorne: Introduction
1: Timothy Gowers: Is Mathematics Discovered or Invented?
2: Marcus du Sautoy: Exploring the Mathematical Library of Babel
3: John Polkinghorne: Mathematical Reality
4: Roger Penrose: Mathematics, the Mind, and the Physical World
5: Peter Lipton: Mathematical Understanding
6: Mary Leng: Creation and Discovery in Mathematics
7: Michael Detlefsen: Discovery, Invention and Realism: Godel and others on the Reality of Concepts
8: Stewart Shapiro: Mathematics and Objectivity
9: Gideon Rosen: The Reality of Mathematical Objects
10: Mark Steiner: Getting More out of Mathematics than What We Put In
Index
378 pages | 234x156mm
978-0-19-969564-5 | Hardback | August 2011 (estimated)
The first book by a leading philosopher
Ground-breaking study of Gottlob Frege, the founder of modern logic and one of the fathers of analytic philosophy
Demonstrates the importance and interest of the topic for current debates in philosophy
Frege's Theorem collects eleven essays by Richard G Heck, Jr, one of the world's leading authorities on Frege's philosophy. The Theorem is the central contribution of Gottlob Frege's formal work on arithmetic. It tells us that the axioms of arithmetic can be derived, purely logically, from a single principle: the number of these things is the same as the number of those things just in case these can be matched up one-to-one with those. But that principle seems so utterly fundamental to thought about number that it might almost count as a definition of number. If so, Frege's Theorem shows that arithmetic follows, purely logically, from a near definition. As Crispin Wright was the first to make clear, that means that Frege's logicism, long thought dead, might yet be viable.
Heck probes the philosophical significance of the Theorem, using it to launch and then guide a wide-ranging exploration of historical, philosophical, and technical issues in the philosophy of mathematics and logic, and of their connections with metaphysics, epistemology, the philosophy of language and mind, and even developmental psychology. The book begins with an overview that introduces the Theorem and the issues surrounding it, and explores how the essays that follow contribute to our understanding of those issues. There are also new postscripts to five of the essays, which discuss changes of mind, respond to published criticisms, and advance the discussion yet further.
Readership: Scholars and graduate students in philosophy, especially those working on logic, language, or mathematics.
Preface
Editorial Notes
1: Frege's Theorem: An Overview
2: The Development of Arithmetic
3: Die Grundlagen der Arithmetik §§82-83
4: Frege's Principle
5: Julius Caesar and Basic Law V
6: The Julius Caesar Objection
7: Cardinality, Counting, and Equinumerosity
8: Syntactic Reductionism
9: The Existence of Abstract Objects
10: The Consistency of Contextual Definitions
11: Finitude and Hume's Principle
12: A Logic for Frege's Theorem
Index
Series: Studies in the History of Mathematics and Physical Sciences, Vol. 15
1990, XIX, 884 p. 96 illus., Hardcover
ISBN: 978-0-387-97180-3
This scientific biography of the mathematician Joseph Liouville is divided into two parts. The first part is a chronological account of Liouville's career including a description of the institutions he worked in, his relations with his teachers, colleagues and students, and the historical context of his works. It portrays the French scientific community in a period when Germany and England had surpassed France as the leading nations in mathematics and physics. The second part of the book gives a detailed analysis of Liouville's major contributions to mathematics and mechanics. The gradual development of Liouville's ideas, as reflected in his publications and notebooks, are related to the works of his predecessors and his contemporaries as well as to later developments in the field. On the basis of Liouville's unpublished notes the book reconstructs Liouville's hitherto unknown theories of stability of rotating masses of fluid, potential theory, Galois theory and electrodynamics. It also incorporates valuable added information from Liouville's notes regarding his works on differentiation of arbitrary order, integration in finite terms, Sturm-Liouville theory, transcendental numbers, doubly periodic functions, geometry and mechanics.
Content Level ≫ Research
2011, 240 p. 151 illus., 2 in color., Hardcover
ISBN: 978-1-4419-7909-4
Due: June 29, 2011
.Can be used as a basis for a one? or two?semester undergraduate course as well as a supplementary text for a topology or combinatorics class
Covers topics such as fair division, graph coloring problems, evasiveness of graph properties, and embedding problems from discrete geometry
Contains many figures that aid in the understanding of concepts and proofs
Includes an extensive appendix which helps make the book completely self?contained
A Course in Topological Combinatorics is the first undergraduate textbook on the field of topological combinatorics, a subject that has become an active and innovative research area in mathematics over the last thirty years with growing applications in math, computer science, and other applied areas. Topological combinatorics is concerned with solutions to combinatorial problems by applying topological tools. In most cases these solutions are very elegant and the connection between combinatorics and topology often arises as an unexpected surprise.
The textbook covers topics such as fair division, graph coloring problems, evasiveness of graph properties, and embedding problems from discrete geometry. It is written in plain language and many figures support the understanding of concepts and proofs. In many cases several alternative proofs for the same result are given and each chapter ends with a series of exercises. The extensive appendix makes the book completely self?contained.
The textbook is well suited for every undergraduate math major with some basic mathematical experience. Previous knowledge in topology or graph theory is certainly helpful but not necessary. The text may be used as a basis for a one? or two?semester course as well as a supplementary text for a topology or combinatorics class.
Content Level ≫ Upper undergraduate
Keywords ≫ Brouwer's fixed point theorem - Kneser conjecture - Radon theorems - Smith theory - evasiveness of graph properties - homological algebra - partially ordered sets - theoretic topology - topological G-spaces
Preface.- Introduction.- 1 Fair Division Problems.- 2 Graph Coloring Problems.- 3 Evasiveness of Graph Properties.- 4 Embedding and Mapping Problems.- A Basic Concepts from Graph Theory.- B Topology in a Nutshell.- C Partially Ordered Sets, Order Complexes and their Topology.- D Groups and Group Actions.- E Some Results and Applications from Smith Theory.- References.- List of Symbols and Typical Notation.- Index