Language, Truth and Logic in Mathematics / Edition 1

by Jaakko Hintikka

Overview

One can distinguish, roughly speaking, two different approaches to the philosophy of mathematics. On the one hand, some philosophers (and some mathematicians) take the nature and the results of mathematicians' activities as given, and go on to ask what philosophical morals one might perhaps find in their story. On the other hand, some philosophers, logicians and mathematicians have tried or are trying to subject the very concepts which mathematicians are using in their work to critical scrutiny. In practice this usually means scrutinizing the logical and linguistic tools mathematicians wield. Such scrutiny can scarcely help relying on philosophical ideas and principles. In other words it can scarcely help being literally a study of language, truth and logic in mathematics, albeit not necessarily in the spirit of AJ. Ayer. As its title indicates, the essays included in the present volume represent the latter approach. In most of them one of the fundamental concepts in the foundations of mathematics and logic is subjected to a scrutiny from a largely novel point of view. Typically, it turns out that the concept in question is in need of a revision or reconsideration or at least can be given a new twist. The results of such a re-examination are not primarily critical, however, but typically open up new constructive possibilities. The consequences of such deconstructions and reconstructions are often quite sweeping, and are explored in the same paper or in others.

Product Details

ISBN-13:	9789048149230
Publisher:	Springer Netherlands
Publication date:	12/10/2010
Series:	Jaakko Hintikka Selected Papers , #3
Edition description:	Softcover reprint of hardcover 1st ed. 1997
Pages:	250
Product dimensions:	6.10(w) x 9.25(h) x 0.02(d)

About the Author

Jaakko Hintikka is the author or co-author of thirty volumes and of some 300 scholarly articles in mathematical and philosophical logic, epistemology, language theory, philosophy of science, history of ideas and history of philosophy, including Aristotle, Descartes, Leibniz, Kant, Peirce, The Bloomsbury Group, Husserl and Wittgenstein. He has also been active in international scholarly organizations, most recently as the First Vice-President of FISP, Vice-President of IIP and Co-Chair of the American Organizing Committee of the Twentieth World Congress of Philosophy. He has been Editor-in-Chief of the International Journal Synthese and the Managing Editor of Synthese Library since 1965.

Table of Contents

1. What Is Elementary Logic? Independence-Friendly Logic as the True Core Area of Logic.- 2. A Revolution in Logic?.- 3. A Revolution in the Foundations of Mathematics?.- 4. Is There Completeness in Mathematics after Gödel?.- 5. Hilbert Vindicated?.- 6. Standard vs. Nonstandard Distinction: A Watershed in the Foundations of Mathematics.- 7. Standard vs. Nonstandard Logic: Higher-Order, Modal, and First-Order Logics.- 8. The Skeleton in Frege’s Cupboard: The Standard versus Nonstandard Distinction.- 9. An Alternative Concept of Computability.- 10. What is the Logic of Parallel Processing?.- 11. Model Minimization — An Alternative to Circumscription.- 12. New Foundations for Mathematical Theories.

Editorial Reviews

A dozen essays, most previously published, critique the concepts that mathematicians use, particularly their logical and linguistic tools. Typically finds that the concept under scrutiny needs revision or reconsideration. Argues that quantification theory has been only partially and inadequately captured by the received first-order logic, that much of the discussions of the implications of G<:o>del-type incompleteness results are superficial and misleading, that the characterization of Hilbert as a formalist hides rather than reveals the true dynamics of his thought, and that implication of the contrast between standard and non-standard interpretations of higher-order logics have remained largely unacknowledged. Also offers several alternative concepts. No index. Annotation c. by Book News, Inc., Portland, Or.

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