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Foundations of Constructive Mathematics

Metamathematical Studies

  • Michael J. Beeson
Book

Part of the Ergebnisse der Mathematik und ihrer Grenzgebiete book series (MATHE3, volume 6)

Table of contents

  1. Front Matter
    Pages I-XXIII
  2. Practice and Philosophy of Constructive Mathematics

    1. Front Matter
      Pages 1-1
    2. Michael J. Beeson
      Pages 3-32
    3. Michael J. Beeson
      Pages 33-46
  3. Formal Systems of the Seventies

    1. Front Matter
      Pages 93-93
    2. Michael J. Beeson
      Pages 97-147
    3. Michael J. Beeson
      Pages 148-161
    4. Michael J. Beeson
      Pages 162-201
    5. Michael J. Beeson
      Pages 202-215
    6. Michael J. Beeson
      Pages 216-245
    7. Michael J. Beeson
      Pages 246-283
  4. Metamathematical Studies

    1. Front Matter
      Pages 285-285
    2. Michael J. Beeson
      Pages 287-299
    3. Michael J. Beeson
      Pages 300-323
    4. Michael J. Beeson
      Pages 324-345
    5. Michael J. Beeson
      Pages 346-367
    6. Michael J. Beeson
      Pages 368-398
  5. Metaphilosophical Studies

    1. Front Matter
      Pages 399-399
    2. Michael J. Beeson
      Pages 401-416
  6. Back Matter
    Pages 417-466

About this book

Introduction

This book is about some recent work in a subject usually considered part of "logic" and the" foundations of mathematics", but also having close connec­ tions with philosophy and computer science. Namely, the creation and study of "formal systems for constructive mathematics". The general organization of the book is described in the" User's Manual" which follows this introduction, and the contents of the book are described in more detail in the introductions to Part One, Part Two, Part Three, and Part Four. This introduction has a different purpose; it is intended to provide the reader with a general view of the subject. This requires, to begin with, an elucidation of both the concepts mentioned in the phrase, "formal systems for constructive mathematics". "Con­ structive mathematics" refers to mathematics in which, when you prove that l a thing exists (having certain desired properties) you show how to find it. Proof by contradiction is the most common way of proving something exists without showing how to find it - one assumes that nothing exists with the desired properties, and derives a contradiction. It was only in the last two decades of the nineteenth century that mathematicians began to exploit this method of proof in ways that nobody had previously done; that was partly made possible by the creation and development of set theory by Georg Cantor and Richard Dedekind.

Keywords

Cantor Mathematics computability theory computer computer science development eXist forcing model theory organization philosophy proof proof by contradiction proving set theory

Authors and affiliations

  • Michael J. Beeson
    • 1
  1. 1.Department of Mathematics and Computer ScienceSan Jose State UniversitySan JoseUSA

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-642-68952-9
  • Copyright Information Springer-Verlag Berlin Heidelberg 1985
  • Publisher Name Springer, Berlin, Heidelberg
  • eBook Packages Springer Book Archive
  • Print ISBN 978-3-642-68954-3
  • Online ISBN 978-3-642-68952-9
  • Series Print ISSN 0071-1136
  • Buy this book on publisher's site
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