PREFACE This special issue contains invited papers from the conference “Found- ations of the Formal Sciences I” which took place at the Humboldt- Universität zu Berlin in May 1999. The conference was part of the conference series “Foundations of the Formal Sciences” which has been successfully bringing together young researchers of the different Formal Sciences not only at its first meeting in Berlin, but in the meantime also at the next meeting in the year 2000 in Bonn. The reader can find more about the general aim of the conference series in the introductory paper “The Formal Sciences: Their Scope, Their Foundations, and Their Unity” of the first editor in this issue. As laid out in that article, we asked the participants to give very special one-hour talks: They should be addressing an interdisciplinary, non-specialist (yet informed) audience, but they should not be mere sur- veys; instead, we expected understandable descriptions of main techniques and lists of open problems with an interdisciplinary character. The conference was a full success: Researchers from fields lying vastly apart as history of mathematics, applied computer science and higher set theory were able to talk to each other, and reached some level of general understanding of each other’s fields. We asked the authors for written versions of their talks that capture this interdisciplinary spirit, and indeed, we received very fine articles that are written for the serious non-specialist reader with a research interest. The papers that the reader can find in this volume are not just compila- tions of interesting results from these areas but also fine explanations of the major concepts and proof techniques as well as descriptions of the important research projects in these fields. Again, we refer the reader to the introduction which contains a couple of remarks on how to utilize this volume. We shall give the list of participants and the conference schedule, but first of all this is the place to mention a couple of notes of gratitude to- wards people without whom we would not have been able to organise the conference: First of all, we have to thank the Studienstiftung des deutschen Volkes and their representatives Dr. Hans-Ottmar Weyand and Dr. Niels Weidtmann. Their financial help was instrumental for the realization of Synthese 133: 1–4, 2002. © 2002 Kluwer Academic Publishers. Printed in the Netherlands. 2 PREFACE the project as a whole. At the Humboldt-Universität zu Berlin, we have to say thanks to the graduate students Mr Thoralf Räsch and Mr Mi- chael Bruening who were of technical assistance during the conference and to the administrative assistant of the Lehrstuhl für Mathematische Logik, Mrs Christa Dobers, who helped with the production of the con- ference programme. The Humboldt-Universität zu Berlin and in particular the Lehrstuhl für Mathematische Logik provided the lecture rooms and some technical equipment for which we would like to thank Ronald Jensen representative for all involved persons. We’d also like to thank Rudolf Rijgersberg of Kluwer Academic Pub- lishers and John Symons, Meredith Enish, and Jaakko Hintikka for their interest and support, and for the rare opportunity to fill a double issue of this journal with the proceedings of a conference. PARTICIPANTS • Susana Balfego, Humboldt-Universität zu Berlin, Lehrstuhl für Mathematische Logik, Unter den Linden 6, D – 10099 Berlin [email protected] • Sebastian Bauer, Charlottenstraße 25, D – 13156 Berlin [email protected] • Christoph Benzmüller, Universität des Saarlandes, Fachbereich In- formatik (FB 14), AG Deduktionssysteme, D – 66041 Saarbrücken [email protected] • Manuel Bodirsky, Rotenbergstraße 12, D – 66111 Saarbrücken [email protected] • Michael Bruening, Humboldt–Universität zu Berlin, Lehrstuhl für Mathematische Logik, Unter den Linden 6, D – 10099 Berlin [email protected] • Wolfgang Burr, Westfälische Wilhelms–Universität Münster, Institut für mathematische Logik und Grundlagenforschung, Einsteinstraße 62, D – 48149 Münster [email protected] • Antje Christensen, Klintholmvej 3, 2. th., 2700 Brønshøj, Dänemark [email protected] • Hartmut Fitz, Wichertstraße 64/III, D – 10439 Berlin [email protected] • Gerhard Fotheringham, Technische Universität Berlin, Sekretariat TIB 4/2–1, Gustav-Meyer-Allee 25, D – 13355 Berlin [email protected] PREFACE 3 • Stefan Geschke, Freie Universität Berlin, Fachbereich Mathematik und Informatik, Arnimallee 2–6, D – 14195 Berlin [email protected] • Kai Hauser, Humboldt–Universität zu Berlin, Lehrstuhl für Mathe- matische Logik, Unter den Linden 6, D – 10099 Berlin [email protected] • Jan Jürjens, The University of Edinburgh, Laboratory for Foundations of Computer Science, James Clerk Maxwell Building, Room 3311, King’s Buildings, Mayfield Road, Edinburgh EH9 3JZ, Scotland [email protected] • Reinhard Kahle, Wilhelm–Schickard–Institut für Informatik, Eber- hard–Karls–Universität Tübingen, Sand 13, D – 72076 Tübingen [email protected] • Peter Koepke, Rheinische Friedrich–Wilhelms–Universität Bonn, Mathematisches Institut, Beringstraße 6, D – 53115 Bonn [email protected] • Eberhard Knobloch, Technische Universität Berlin, Institut für Philo- sophie, Wissenschaftstheorie, Wissenschafts– und Technikgeschichte, Ernst–Reuter–Platz 7, D – 10587 Berlin [email protected] • Oliver Kutz, Erich-Weinert-Straße 26, D – 10439 Berlin [email protected] • Heiko Mantel, Deutsches Forschungszentrum für Künstliche Intelli- genz GmbH, Stuhlsatzenhausweg 3, D – 66123 Saarbrücken [email protected] • Adrian R. D. Mathias, Institut de Recherche en Mathématiques et Informatique Appliquées, Université de la Réunion, 15 avenue René Cassin, BP 7151, F – 97715 Saint-Denis [email protected] • Ralph Matthes, Ludwig–Maximilians–Universität München, Institut für Informatik, Lehrstuhl für Theoretische Informatik, Oettingenstraße 67, D – 80538 München [email protected] • Guy Merlin Mbakop, Humboldt-Universität zu Berlin, Institut für Mathematik, Unter den Linden 6, D – 10099 Berlin [email protected] • Stephan Merz, Ludwig–Maximilians–Universität München, Institut für Informatik, Lehr– und Forschungseinheit Programmierung und Softwaretechnik, Oettingenstraße 67, D – 80538 München [email protected] 4 PREFACE • Hans Jürgen Prömel, Humboldt–Universität zu Berlin, Institut für In- formatik, Lehr– und Forschungsgebiet Algorithmen und Komplexität, Unter den Linden 6, D – 10099 Berlin [email protected] • Thoralf Räsch, Humboldt–Universität zu Berlin, Lehrstuhl für Mathe- matische Logik, Unter den Linden 6, D – 10099 Berlin [email protected] • Michael Stolz, Eberhard–Karls–Universität Tübingen, Mathemati- sches Institut, Auf der Morgenstelle 10, D-72076 Tübingen [email protected] • Christian Tapp, Hensenstraße 168, D – 48161 Münster [email protected] • Andreas Weiermann, Westfälische Wilhelms–Universität Münster, Institut für mathematische Logik und Grundlagenforschung, Einstein- straße 62, D – 48149 Münster [email protected] SCHEDULE Friday, May 7, 1999 Saturday, May 8, 1999 Sunday, May 9, 1999 COMPUTER SCIENCE I MATHEMATICS II Chair: Kahle Chair: Weiermann 830 – 920 Prömel 830 – 920 Kahle 930 – 1020 Matthes 930 – 1020 Burr Coffee Break Coffee Break 1100 – 1150 Jürjens 1100 – 1150 Koepke 1200 – 1215 Closing LUNCH BREAK 1300 – 1320 Opening MATHEMATICS I HISTORY &PHILOSOPHY II Chair: Prömel Chair: Rudolph 1330 – 1420 Weiermann 1330 – 1420 Christensen 1430 – 1520 Geschke 1425 – 1515 Hauser Coffee Break Coffee Break HISTORY &PHILOSOPHY I COMPUTER SCIENCE II Chair: Koepke Chair: Koepke 1600 – 1650 Stolz 1530 – 1620 Mantel 1700 – 1750 Knobloch 1630 – 1720 Merz Coffee Break Coffee Break 1815 – 1840 Mathias 1730 – 1820 Benzmüller B.L. F.R. Bonn, February 2001 BENEDIKT LÖWE THE FORMAL SCIENCES: THEIR SCOPE, THEIR FOUNDATIONS, AND THEIR UNITY Organizing a conference series with the title “Foundations of the Formal Sciences” obliges us to fill the terms “Formal Sciences” and, in particular, “Foundations of the Formal Sciences” with meaning. There are two very natural answers to the question “What are the Formal Sciences?”: • Answer 1: ‘There is a profound duality in the classification of sciences according to their scientific approaches: some sciences are empirical, some are formal. The former deal with predictions and their falsifica- tion, the latter with the understanding of systems without empirical component, be it man-made systems (literary systems, the arts or social systems) or formal systems”. • Answer 2: “Formal sciences are those that deal with the deductive analysis of formal systems (i.e., systems independent of direct human influence)”. These two answers differ more from each other than it seems at first sight. Answer 2 is much stricter about what it allows to be called a formal science: for example, literary sciences classify as formal in the sense of Answer 1. Nonetheless, the do not (mainly) deal with formal systems, so they are not formal sciences in the sense of Answer 2. Answer 1 rests on the traditional dichotomy between natural sciences and liberal arts. Stressing this dichotomy leads to ignoring those formal sciences that we want to talk about; even worse, the fact that anything computational is often seen as the handmaiden of the empirical sciences, shifts mathematics and computer science towards the borderline between formal and empirical sciences