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Reuniting the Antipodes - Constructive and Nonstandard Views of the Continuum : Symposium Proceedings, San Servolo, Venice, Italy, May 16-22, 1999 - Peter Schuster

Reuniting the Antipodes - Constructive and Nonstandard Views of the Continuum

Symposium Proceedings, San Servolo, Venice, Italy, May 16-22, 1999

By: Peter Schuster (Editor), Ulrich Berger (Editor), Horst Osswald (Editor)

Hardcover Published: 31st January 2002
ISBN: 9781402001529
Number Of Pages: 329

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At first glance, Robinson's original form of nonstandard analysis appears nonconstructive in essence, because it makes a rather unrestricted use of classical logic and set theory and, in particular, of the axiom of choice. Recent developments, however, have given rise to the hope that the distance between constructive and nonstandard mathematics is actually much smaller than it appears. So the time was ripe for the first meeting dedicated simultaneously to both ways of doing mathematics - and to the current and future reunion of these seeming opposites. Consisting of peer-reviewed research and survey articles written on the occasion of such an event, this volume offers views of the continuum from various standpoints. Including historical and philosophical issues, the topics of the contributions range from the foundations, the practice, and the applications of constructive and nonstandard mathematics, to the interplay of these areas and the development of a unified theory. This book will be of interest to mathematicians, logicians, and philosophers, as well as theoretical computer scientists, physicists, and economists who are interested in theories of the continuum and in constructive or nonstandard mathematics. The major part is accessible for the non-expert professional reader, from graduate student to academic level.

Forewordp. ix
Nonstandard construction of stable type Euclidean random field measuresp. 1
The continuum in smooth infinitesimal analysisp. 19
Constructive unbounded operatorsp. 25
The points of (locally) compact regular formal topologiesp. 39
Embedding a linear subset of B(H) in the dual of its predualp. 55
Nonstandard analysis by means of ideal values of sequencesp. 63
Nilpotent infinitesimals and synthetic differential geometry in classical logicp. 75
On hyperfinite approximations of the field Rp. 93
Various continuity properties in constructive analysisp. 103
Loeb measures and Borel algebrasp. 111
On Brouwerian bar inductionp. 119
Curt Schmieden's approach to infinitesimals. An eye-opener to the historiography of analysisp. 127
A sequent calculus for constructive ordered fieldsp. 143
The Puritz order and its relationship to the Rudin-Keisler orderp. 157
Unifying constructive and nonstandard analysisp. 167
Positive latticesp. 185
Constructive mathematics without choicep. 199
Pointwise differentiabilityp. 207
On Conway numbers and generalized real numbersp. 211
The constructive content of nonstandard measure existence proofs--is there any?p. 229
Kruskal's tree theorem in a constructive theory of inductive definitionsp. 241
Real numbers and functions exhibited in dialoguesp. 257
On the quantitative structure of [Delta superscript 0 subscript 2]p. 271
Understanding and using Brouwer's continuity principlep. 285
Peirce and the continuum from a philosophical point of viewp. 303
Table of Contents provided by Syndetics. All Rights Reserved.

ISBN: 9781402001529
ISBN-10: 1402001525
Series: Synthese Library
Audience: Professional
Format: Hardcover
Language: English
Number Of Pages: 329
Published: 31st January 2002
Publisher: Springer-Verlag New York Inc.
Country of Publication: US
Dimensions (cm): 23.5 x 15.5  x 2.54
Weight (kg): 1.47