Foundations without Foundationalism: A Case for Second-Order Logic

Paperback | April 30, 1999

byStewart Shapiro

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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 andhigher-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 andhistorical 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

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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 andhigher-order logic, in a way that will be accessible t...

Stewart Shapiro is Professor of Philosophy at Ohio State University.

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Format:PaperbackDimensions:300 pages, 9.21 × 6.14 × 0.71 inPublished:April 30, 1999Publisher:Oxford University PressLanguage:English

The following ISBNs are associated with this title:

ISBN - 10:0198250290

ISBN - 13:9780198250296

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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-orderlanguages; 8. Second-order logic and rule-following; 9. The competition; References; Index

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