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

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. Front Matter....Pages i-xiii Nonstandard Construction of Stable Type Euclidean Random Field Measures....Pages 1-18 The Continuum in Smooth Infinitesimal Analysis....Pages 19-24 Constructive Unbounded Operators....Pages 25-37 The Points of (Locally) Compact Regular Formal Topologies....Pages 39-54 Embedding a Linear Subset of B ( H ) in the Dual of Its Predual....Pages 55-61 Nonstandard Analysis by Means of Ideal Values of Sequences....Pages 63-73 Nilpotent Infinitesimals and Synthetic Differential Geometry in Classical Logic....Pages 75-92 On Hyperfinite Approximations of the Field R ....Pages 93-102 Various Continuity Properties in Constructive Analysis....Pages 103-110 Loeb Measures and Borel Algebras....Pages 111-117 On Brouwerian Bar Induction....Pages 119-126 Curt Schmieden’s Approach to Infinitesimals....Pages 127-142 A Sequent Calculus for Constructive Ordered Fields....Pages 143-155 The Puritz Order and Its Relationship to the Rudin-Keisler Order....Pages 157-166 Unifying Constructive and Nonstandard Analysis....Pages 167-183 Positive Lattices....Pages 185-197 Constructive Mathematics without Choice....Pages 199-205 Pointwise Differentiability....Pages 207-210 On Conway Numbers and Generalized Real Numbers....Pages 211-227 The Constructive Content of Nonstandard Measure Existence Proofs—Is There Any?....Pages 229-239 Kruskal’s Tree Theorem in a Constructive Theory of Inductive Definitions....Pages 241-255 Real Numbers and Functions Exhibited in Dialogues....Pages 257-269 Understanding and Using Brouwer’s Continuity Principle....Pages 271-283 Peirce and the Continuum from a Philosophical Point of View....Pages 285-302 Back Matter....Pages 303-316 ....Pages 317-329

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.

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