Foundations without foundationalism : a case for second-order logic
Author(s)
Bibliographic Information
Foundations without foundationalism : a case for second-order logic
（Oxford logic guides, 17）
Clarendon Press , Oxford University Press, 1991
- : pbk
Available at / 34 libraries
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Hokkaido University, Library, Graduate School of Science, Faculty of Science and School of Science図書
dc20:511.3/sh222070226207
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Note
Bibliography: p. [263]-272
Includes index
Description and Table of Contents
- Volume
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: pbk ISBN 9780198250296
Description
Stewart Shapiro presents a distinctive and persuasive view of the foundations of mathematics, arguing controversially that second-order logic has a central role to play in laying these foundations. To support this contention, he first gives a detailed development of second-order and higher-order logic, in a way that will be accessible to graduate students. He then demonstrates that second-order notions are prevalent in mathematics as practised, and that higher-order
logic is needed to codify many contemporary mathematical concepts. Throughout, he emphasizes philosophical and historical issues that the subject raises. Foundations without Foundationalism is a key contribution both to philosophy of mathematics and to mathematical logic.
'In this excellent treatise Shapiro defends the use of second-order languages and logic as framework for mathematics. His coverage of the wide range of logical and philosophical topics required for understanding the controversy over second-order logic is thorough, clear, and persuasive. . . . Shapiro recognizes that it is unlikely that he has had the last word on these controversial philosophical subjects. Nevertheless, his book is certainly an excellent place to start work on them.' Michael
D. Resnik, History and Philosophy of Logic
Table of Contents
- PART I: ORIENTATION
- 1. TERMS AND QUESTIONS
- 2. FOUNDATIONALISM AND FOUNDATIONS OF MATHEMATICS
- PART II: LOGIC AND MATHEMATICS
- 3. THEORY
- 4. METATHEORY
- 5. SECOND-ORDER LOGIC AND MATHEMATICS
- 6. ADVANCED METATHEORY
- PART III: HISTORY AND PHILOSOPHY
- 7. THE HISTORICAL 'TRIUMPH' OF FIRST-ORDER LANGUAGES
- 8. SECOND-ORDER LOGIC AND RULE-FOLLOWING
- 9. THE COMPETITION
- REFERENCES
- INDEX
- Volume
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ISBN 9780198533917
Description
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 development of higher-order logic, including a comprehensive discussion of its semantics.
Professor Shapiro demonstrates the prevalence of second-order notions in mathematics is practised, and also the extent to which mathematical concepts can be formulated in second-order languages . He shows how first-order languages are insufficient to codify many concepts in contemporary mathematics, and thus that higher-order logic is needed to fully reflect current mathematics.
Throughout, the emphasis is on discussing the philosophical and historical issues associated with this subject, and the implications that they have for foundational studies. For the most part, the author assumes little more than a familiarity with logic as might be gained from 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 this subject.
Table of Contents
- PART I: ORIENTATION
- Terms and questions
- Foundationalism and foundations of mathematics
- PART II: LOGIC AND MATHEMATICS
- Theory
- Metatheory
- Second-order logic and mathematics
- Advanced metatheory
- PART III: HISTORY AND PHILOSOPHY
- The historical triumph of first-order languages
- Second-order logic and rule-following
- The competition
- Index.
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