# Philosophy and Foundations of Mathematics - Epistemological and Ontological Aspects 2009

A conference dedicated to Per Martin-Löf on the occasion of his retirement. Swedish Collegium for Advanced Study, Uppsala, May 5-8, 2009

Speakers include:

*Peter Aczel ◆ Mark van Atten ◆ Steve Awodey ◆ Thierry Coquand*

*Peter Dybjer ◆ Juliet Floyd ◆ Jean-Yves Girard*

*Sten Lindström ◆ Colin McLarty ◆ Per Martin-Löf*

*Peter Pagin ◆ Erik Palmgren ◆ Jan von Plato*

*Dag Prawitz ◆ Christine Paulin ◆ Aarne Ranta*

*Michael Rathjen ◆ Giovanni Sambin ◆ Anton Setzer*

*Stewart Shapiro ◆ Wilfried Sieg ◆ Jan Smith ◆ Sören Stenlund*

*Göran Sundholm ◆ William Tait ◆ Jouko Väänänen*

**Scope and aim**

The aim of the conference is to bring together philosophers, mathematicians, and logicians to penetrate current and historically important problems in the philosophy and foundations of mathematics. Swedish logicians and philosophers have made important contributions to the foundations and philosophy of mathematics, at least since the end of the 1960s. In philosophy, one has been concerned with the opposition between constructivism and classical mathematics and the different ontological and epistemological views that are reflected in this opposition. A central philosophical question concerns the nature of the abstract entities of mathematics: do they exist independently of our epistemic acts (realism, or Platonism) or are they somehow constituted by these acts (idealism)? Significant contributions have been made to the foundations of mathematics, for example in proof theory, proof-theoretic semantics and constructive type theory. These contributions have had a strong impact on areas of computer science, e.g. through Martin-Löf's type theory.

Two important alternative foundational programmes that are actively pursued today are predicativistic constructivism and category-theoretic foundations. Predicativistic constructivism can be based on Martin-Löf constructive type theory, Aczel's constructive set theory, or similar systems. The practice of the Bishop school of constructive mathematics fits well into this framework. Associated philosophical foundations are meaning theories in the tradition of Wittgenstein, Dummett, Prawitz and Martin-Löf. What is the relation between proof-theoretical semantics in the tradition of Gentzen, Prawitz, and Martin-Löf and Wittgensteinian or other accounts of meaning-as-use? What can proof-theoretical analysis tell us about the scope and limits of constructive and (generalized) predicative mathematics? To what extent is it possible to reduce classical mathematical frameworks to constructive ones? Such reductions often reveal computational content of classical existence proofs. Is computational content enough to solve the epistemological questions?

A central concern for the conference will be to compare the different foundational frameworks - classical set theory, constructive type theory, and category theory - both from a philosophical and a logical point of view. The general theme of the conference, however, will be broader and encompass different areas of philosophy and foundations of mathematics, in particular the interplay between ontological and epistemological considerations.

**Venue**

The conference will take place at the Swedish Collegium for Advanced Study (SCAS), Linneanum, Thunbergsvägen 2, Uppsala, Sweden. Map of Uppsala with a walking path from the Central Station indicated. The street address for arrival by taxi is Villavägen 6.

**Organization and programme committee**

Peter Dybjer, Sten Lindström, Erik Palmgren (Chair), Dag Prawitz, Sören Stenlund, Viggo Stoltenberg-Hansen.

**Programme and Abstracts**

**Attendance**

Attendance is open, and there is no registration fee. However, anyone planning to attend should preregister by emailing **PFM[at]math.uu.se** no later than April 15, 2009. (For reasons of space the number of participants will be limited to 90.)

**Accommodation and Travel**

The main airport serving Uppsala is Stockholm Arlanda Airport. It has direct train and shuttle bus connections to Uppsala Central Station.

**Conference picture**

(Photo by Bjarne Graff, SCAS.)

**List of Participants**

The list of registered participants that attended

**Sponsors**

Swedish Research Council (Vetenskapsrådet) – The Royal Swedish Academy of Sciences – The Royal Swedish Academy of Letters, History and Antiquities – Department of Mathematics, Stockholm University – Department of Philosophy, Stockholm University – Department of Mathematics, Uppsala University – Centre for Interdisciplinary Mathematics, Uppsala University – Department of Philosophy, Uppsala University – Department of Computer Science and Engineering, Chalmers University of Technology and Gothenburg University – The Swedish Collegium for Advanced Study, Uppsala – Swedish National Committee for Logic, Methodology and Philosophy of Science.