Foundations without Foundationalism
A Case for Second-Order Logic
By (author) Stewart Shapiro
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Foundations without Foundationalism by Stewart Shapiro
Book DescriptionThe 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.
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Book DetailsISBN: 9780198533917
(242mm x 161mm x 24mm)
Imprint: Clarendon Press
Publisher: Oxford University Press
Publish Date: 19-Sep-1991
Country of Publication: United Kingdom
Books By Author Stewart Shapiro
Varieties of Logic, Hardback (September 2014)
Logical pluralism is the view that different logics are equally appropriate, or equally correct. Logical relativism is a pluralism according to which validity and logical consequence are relative to something. Stewart Shapiro explores various such views. He argues that the question of meaning shift is itself context-sensitive and interest-relative.
Vagueness in Context, Paperback (March 2008)
Stewart Shapiro develops an account of the meaning, function, and logic of vague terms in an idealized version of a natural language. The key feature is that extensions of vague terms vary in the course of conversations and, in some cases, a speaker can go either way without sinning against the meaning of the words or the non-linguistic facts.
Oxford Handbook of Philosophy of Mathematics and Logic, Paperback (June 2007)
Philosophy of mathematics and logic remain central disciplines in contemporary philosophy. This volume covers these disciplines, giving the reader an overview of the major problems, positions, and battle lines. It is aimed at those wishing to learn about the philosophy of mathematics and the philosophy of logic.
Thinking About Mathematics, Paperback (July 2000)» View all books by Stewart Shapiro
'Thinking about Mathematics' covers the range of philosophical issues and positions concerning mathematics. The text describes the questions about mathematics that motivated philosophers throughout history and covers historical figures such as Plato, Aristotle, Kant, and Mill.
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