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.

Nonstandard construction of stable type Euclidean random field measures.- The continuum in smooth infinitesimal analysis.- Constructive unbounded operators.- The points of (locally) compact regular formal topologies.- Embedding a linear subset of ß(H) in the dual of its predual.- Nonstandard analysis by means of ideal values of sequences.- Nilpotent infinitesimals and synthetic differential geometry in classical logic.- On hyperfinite approximations of the field R.- Various continuity properties in constructive analysis.- Loeb measures and Borel algebras.- On Brouwerian bar induction.- Curt Schmieden’s approach to infinitesimals. An eye-opener to the historiography of analysis.- A sequent calculus for constructive ordered fields.- The Puritz order and its relationship to the Rudin-Keisler order.- Unifying constructive and nonstandard analysis.- Positive lattices.- Constructive mathematics without choice.- Pointwise differentiability.- On Conway numbers and generalized real numbers.- The constructive content of nonstandard measure existence proofs—is there any?.- Kruskal’s tree theorem in a constructive theory of inductive definitions.- Real numbers and functions exhibited in dialogues.- WIJN/On the quantitative structure of ?20.- Understanding and using Brouwer’s continuity principle.- Peirce and the continuum from a philosophical point of view.