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In the area of mathematical logic, a great deal of attention is now being devoted to the study of nonclassical logics. This book intends to present the most important methods of proof theory in intuitionistic logic and to acquaint the reader with the principal axiomatic theories based on intuitionistic logic.
This book examines the role of acts of choice in classical and intuitionistic mathematics. Featuring fifteen papers – both new and previously published – it offers a fresh analysis of concepts developed by the mathematician and philosopher L.E.J. Brouwer, the founder of intuitionism. The author explores Brouwer’s idealization of the creative subject as the basis for intuitionistic truth, and in the process he also discusses an important, related question: to what extent does the intuitionistic perspective succeed in avoiding the classical realistic notion of truth? The papers detail realistic aspects in the idealization of the creative subject and investigate the hidden role of choice even in classical logic and mathematics, covering such topics as bar theorem, type theory, inductive evidence, Beth models, fallible models, and more. In addition, the author offers a critical analysis of the response of key mathematicians and philosophers to Brouwer’s work. These figures include Michael Dummett, Saul Kripke, Per Martin-Löf, and Arend Heyting. This book appeals to researchers and graduate students with an interest in philosophy of mathematics, linguistics, and mathematics.
Intuitionism and Proof Theory: Proceedings of the Summer Conference at Buffalo N.Y. 1968
L. E. J. Brouwer, the founder of mathematical intuitionism, believed that mathematics and its objects must be humanly graspable. He initiated a program rebuilding modern mathematics according to that principle. This book introduces the reader to the mathematical core of intuitionism – from elementary number theory through to Brouwer's uniform continuity theorem – and to the two central topics of 'formalized intuitionism': formal intuitionistic logic, and formal systems for intuitionistic analysis. Building on that, the book proposes a systematic, philosophical foundation for intuitionism that weaves together doctrines about human grasp, mathematical objects and mathematical truth.
This book represents the most comprehensive account to date of an important but widely contested approach to ethics--intuitionism, the view that there is a plurality of moral principles, each of which we can know directly. Robert Audi casts intuitionism in a form that provides a major alternative to the more familiar ethical perspectives (utilitarian, Kantian, and Aristotelian). He introduces intuitionism in its historical context and clarifies--and improves and defends--W. D. Ross's influential formulation. Bringing Ross out from under the shadow of G. E. Moore, he puts a reconstructed version of Rossian intuitionism on the map as a full-scale, plausible contemporary theory. A major contribution of the book is its integration of Rossian intuitionism with Kantian ethics; this yields a view with advantages over other intuitionist theories (including Ross's) and over Kantian ethics taken alone. Audi proceeds to anchor Kantian intuitionism in a pluralistic theory of value, leading to an account of the perennially debated relation between the right and the good. Finally, he sets out the standards of conduct the theory affirms and shows how the theory can help guide concrete moral judgment. The Good in the Right is a self-contained original contribution, but readers interested in ethics or its history will find numerous connections with classical and contemporary literature. Written with clarity and concreteness, and with examples for every major point, it provides an ethical theory that is both intellectually cogent and plausible in application to moral problems.
This anthology reviews the programmes in the foundations of mathematics from the classical period and assesses their possible relevance for contemporary philosophy of mathematics. A special section is concerned with constructive mathematics.
An Introduction to Proof Theory provides an accessible introduction to the theory of proofs, with details of proofs worked out and examples and exercises to aid the reader's understanding. It also serves as a companion to reading the original pathbreaking articles by Gerhard Gentzen. The first half covers topics in structural proof theory, including the Gödel-Gentzen translation of classical into intuitionistic logic (and arithmetic), natural deduction and the normalization theorems (for both NJ and NK), the sequent calculus, including cut-elimination and mid-sequent theorems, and various applications of these results. The second half examines ordinal proof theory, specifically Gentzen's consistency proof for first-order Peano Arithmetic. The theory of ordinal notations and other elements of ordinal theory are developed from scratch, and no knowledge of set theory is presumed. The proof methods needed to establish proof-theoretic results, especially proof by induction, are introduced in stages throughout the text. Mancosu, Galvan, and Zach's introduction will provide a solid foundation for those looking to understand this central area of mathematical logic and the philosophy of mathematics.
This is a long-awaited new edition of one of the best known Oxford Logic Guides. The book gives an informal but thorough introduction to intuitionistic mathematics, leading the reader gently through the fundamental mathematical and philosophical concepts. The treatment of various topics has been completely revised for this second edition. Brouwer's proof of the Bar Theorem has been reworked, the account of valuation systems simplified, and the treatment of generalized Beth Trees and the completeness of intuitionistic first-order logic rewritten. Readers are assumed to have some knowledge of classical formal logic and a general awareness of the history of intuitionism.