Mathematisches Institut

Reuniting the Antipodes

Constructive and Nonstandard Views of the Continuum

- Symposion -

17 - 22 May 1999

Venice International University (VIU), San Servolo, Venice, Italy

Organised by

Ulrich Berger, Horst Osswald, Peter Schuster

Volkswagen-Stiftung (Hannover)

Istituto Italiano per gli Studi Filosofici (Naples and Venice)

Context and targets

The introduction of differentials by Leibniz and Newton might be considered as the first appearance of infinitesimals in the mathematics of modern times. Infinitesimals, i.e. nonvanishing positive `numbers' smaller than any given fraction, were still very useful for Euler and Cauchy until becoming banned by the mathematical analysis founded in the last century. Since, the best known renewal of those venerable concept was carried out in the preceeding four decades by Abraham Robinson and his followers, who designed this nonstandard analysis in order to deal with infinitely small and infinitely large entities in a rigorous manner. Because of making unrestricted use of classical logic and set theory and, in particular, of the axiom of choice, Robinson's theory in its full-fledged form is suspected to be nonconstructive from the outset. In addition, the nonstandard idea of discretising the continuum seems to be even less compatible with the intuitionistic concept of a continuum in the true sense of the word, than with the classical atomistic notion.

The distance between constructive and nonstandard mathematics, however, is actually much smaller than it appears to be. Indications for this are that nonstandard practice often looks rather constructive, and that very small numbers unknown to vanish are indispensable to distinguish constructive mathematics from its traditional counterpart. Moreover, various rather promising attempts to build a constructive nonstandard analysis have been begun quite recently, a progress which cannot be thought of without the liberalisation of constructive mathematics started by Errett Bishop in the 1960es, and without the present flourishing of category theory. By now, the time has come to establish the first meeting of all the people who do research in constructive or nonstandard mathematics, and who are aiming at reuniting these putative antipodes. Besides enabling future collaboration between constructive and nonstandard mathematicians, one of the expected side effects of this event is to exchange the specific tools and methods of these areas. The principal intention of this workshop, however, is to drive forward current research in constructive nonstandard mathematics, thus also inspiring fresh views of the continuum from various standpoints. Beyond pure mathematics, let alone mathematical foundations, all this is of particular interest with regard to the main applications of constructive and nonstandard mathematics in theoretical computer science, mathematical physics, and mathematical economics.

Program

Monday 17 May 1999

9.00	Departure (recommended) from San Zaccaria for San Servolo
9.15	Registration
10.15	Opening
10.45	Dirk van Dalen (University of Utrecht, The Netherlands) Indecomposable subsets of the continuum
11.45	Julia Zink (Pontificia Università Gregoriana, Città del Vaticano) Peirce and the continuum from a philosophical point of view
12.00	John L. Bell (University of Western Ontario, Canada) The continuum in smooth infinitesimal analysis
13.00	Lunch
14.30	Registration (continued)
15.30	Jens Erik Fenstad (University of Oslo, Norway) Computability theory over the nonstandard reals
16.00	Rudolf Taschner (Technische Universität Wien, Austria) Real numbers and functions, exhibited in dialogs
16.45	Break
17.00	Jiang-Lun Wu (Universität Bochum, Germany) On hyperfinite integral representation of Euclidean random field measures
17.30	Erik Palmgren (University of Uppsala, Sweden) Constructive nonstandard analysis
18.30	Dinner

Tuesday 18 May 1999

9.45	Per Martin-Löf (University of Stockholm, Sweden) Nonstandard type theory
10.30	Fred Richman (Florida Atlantic University, Boca Raton) Constructive mathematics without choice
11.15	Break
11.30	Wilhelmus A. J. Luxemburg (California Institute of Technology, Pasadena) A Schauder type theorem for internal linear operators
12.00	Giovanni Sambin (Università di Padova, Italy) Real numbers in formal topology
12.45	Giovanni Curi (Università di Padova, Italy) Points in compact regular formal topologies
13.15	Lunch
15.30	H. Jerome Keisler (University of Wisconsin, Madison) Nonstandard methods in o-minimal structures
16.30	Hajime Ishihara (Japan Advanced Institute of Science and Technology) A note on the Gödel-Gentzen translation
17.00	Break
17.15	Frank Rosemeier (Fernuniversität Hagen, Germany) On Conway numbers and generalized real numbers
17.45	Yeneng Sun (National University of Singapore, Singapore) Asymptotic, hyperfinite and continuum models
18.45	Departure from San Servolo for conference dinner

Wednesday 19 May 1999

9.45	Detlef Laugwitz (Technische Universität Darmstadt, Germany) Curt Schmieden's approach to infinitesimals---an eye-opener to the historiography of analysis
10.45	Nigel J. Cutland (University of Hull, England) Constructive aspects of nonstandard methods in fluid mechanics
11.30	Break
11.45	Douglas S. Bridges (University of Canterbury, Christchurch, New Zealand) Constructive investigations of functions of bounded variation
12.30	Luminita Simona Dediu (University of Canterbury, Christchurch, New Zealand) Embedding a linear subset of B(H) in its predual
13.00	Lunch
15.30	Chris Impens (University of Gent, Belgium) Some thoughts on nonstandard geometry
16.00	Hans Vernaeve (University of Gent, Belgium) Reducing distributions to hyperreal functions
16.30	Petr V. Andreyev (Nizhnii Novgorod State University, Russia) Definable standardness predicates in internal set theory
17.00	Break
17.15	James Henle (Smith College, Northhampton, Massachusetts) Non-nonstandard analysis: category, measure, and integration
18.00	Peter Schuster (Universität München, Germany) Elementary choiceless constructive analysis
18.30	Dinner
20.00	Public discussion Reuniting the Antipodes---a task for the next millenium?
21.45	Departure (recommended) from San Servolo

Thursday 20 May 1999

9.45	Manfred Wolff (Universität Tübingen, Germany) On the approximation of operators and their spectra
10.45	Wim Veldman (Catholic University of Nijmegen, The Netherlands) On some sets that are not positively Borel
11.30	Break
11.45	Sara Negri (University of Helsinki, Finland) On the proof theory of constructive ordered fields
12.30	Juha Oikkonen (University of Helsinki, Finland) Some geometric ideas related to Brownian motion
13.00	Lunch
14.45	Departure (recommended) from San Servolo

Friday 21 May 1999

9.45	Joan R. Moschovakis ((Occidental College, Los Angeles) (University of Athens, Greece)) The intuitionistic continuum as an extension of the classical one
10.15	P. Ekkehard Kopp (University of Hull, England) Hyperfinite discretisations and convergence in option pricing models
10.45	M. Ali Khan (Johns Hopkins University, Baltimore, Maryland) Modelling `negligibility' in mathematical economics: an application of Loeb spaces
11.15	Break
11.30	Mohamed Khalouani (Université de Franche-Comté, Besançon, France) Étude constructive de problemes de topologie pour les reels irrationnels
12.00	David Ross (University of Hawaii) The constructive content of nonstandard measure existence proofs: is there any?
13.00	Lunch
15.30	Herrmann Render (Universität Duisburg, Germany) Borel measure extensions of measures defined on sub-sigma-algebras
16.00	Paolo Giordano (Università di Novara, Italy) Nilpotent infinitesimals and synthetic differential geometry in classical logic
16.30	Break
17.00	Jan von Plato (University of Helsinki, Finland) The lattice of open intervals of the intuitionistic continuum
17.30	Michael Reeken (Universität Wuppertal, Germany) Discretising the continuum
18.00	Mauro Di Nasso(Università di Pisa, Italy) Hyperreals as ideal limits: an elementary axiomatic presentation of nonstandard analysis
18.30	Dinner
20.30	Hervé Perdry (Université de Franche-Comté, Besançon, France) Computing in the constructive henselisation of a valued field
21.00	Sebastiaan Terwijn (Universität München, Germany) Effective measure theory
21.45	Departure (recommended) from San Servolo

Saturday 22 May 1999

9.45	Xiaoai Lin (National University of Singapore) On the almost independence of correspondences on Loeb space
10.15	Bas Spitters (Catholic University of Nijmegen, The Netherlands) A constructive converse of the mean value theorem
10.45	Break
11.00	Frank A. Waaldijk (Catholic University of Nijmegen, The Netherlands) The hidden presence of the fan theorem in the constructive definition of `continuous function on the continuum'
11.30	Monika Seisenberger (Universität München, Germany) Kruskal's tree theorem in a constructive theory of inductive definitions
12.00	Evgenii I. Gordon (Nizhnii Novgorod State University, Russia) Mathematics in hyperfinite sets theory
13.15	Departure (recommended) from San Servolo

Further attending scientists

Stefano Baratella (Università di Trento, Italy)
Josef Berger (Universität München, Germany)
Ulrich Berger (Universität München, Germany)
Jens Blanck (Högskolan i Gävle, Sweden)
Jonas Eliasson (University of Uppsala, Sweden)
Ferruccio Guidi (Università di Padova, Italy)
Göran Hamrin (University of Uppsala, Sweden)
Horst Osswald (Universität München, Germany)
Hans Ploss (Universität Wien, Austria)
Juha Ruokolainen (University of Helsinki, Finland)

We would like to express our particular gratitude to Dr. Raffaela Toffolo and to Prof. Umberto Curi (both Istituto Italiano per gli Studi Filosofici) for their extremely helpful personal assistance.

The official language of this symposion is English. Visitors are welcome as many seats are available; application is required by Thursday 13 May 1999.

Symposion Antipodes
c/o Peter Schuster
Mathematisches Institut der Universität
Theresienstr. 39, 80333 München, Germany
phone ++49-(0)89-2394-4401, fax ++49-(0)89-2805248
antipode@rz.mathematik.uni-muenchen.de
http://www.mathematik.uni-muenchen.de/~antipode