Logic, Epistemology, and the Unity of Science 51 Gerhard Heinzmann Gereon Wolters Editors Paul Lorenzen – Mathematician and Logician Logic, Epistemology, and the Unity of Science Volume 51 Series Editor Shahid Rahman, CNRS-UMR: 8163, Université de Lille, Lille, France Managing Editor Nicolas Clerbout, Universidad de Valparaíso, Valparaíso, Chile Founding Editor John Symons, Department of Philosophy, The University of Texas at El Paso, El Paso, TX, USA Editorial Board Jean Paul van Bendegem, Gent, Belgium Hourya Benis Sinaceur, Techniques, CNRS, Institut d’Histoire et Philosophie des Sci, Paris, France Johan van Benthem, Institute for Logic Language & Computation, University of Amsterdam, Amsterdam, Noord-Holland, The Netherlands Karine Chemla, CNRS, Université Paris Diderot, Paris, France Jacques Dubucs, CNRS, IHPST, Université Paris, Paris, France Anne Fagot-Largeault, Philosophy of Life Science, College de France, Paris, France Bas C Van Fraassen, Department of Philosophy, Princeton University, Princeton, NJ, USA Dov M. Gabbay, King’s College, Interest Group, London, UK Paul McNamara, Philosophy Department, University of New Hampshire, Durham, NH, USA Graham Priest, Department of Philosophy, Graduate Center, City University of New York, New York, NY, USA Gabriel Sandu, Department of Philosophy, University of Helsinki, Helsinki, Finland Sonja Smets, Institute of Logic, Language and Computation, University of Amsterdam, Amsterdam, Noord-Holland, The Netherlands Tony Street, Faculty of Divinity, University of Cambridge, Cambridge, UK Göran Sundholm, Philosophy, Leiden University, Leiden, Zuid-Holland, The Netherlands Heinrich Wansing, Department of Philosophy II, Ruhr University Bochum, Bochum, Nordrhein-Westfalen, Germany Timothy Williamson, Department of Philosophy, University of Oxford, New College, Oxford, UK Logic, Epistemology, and the Unity of Science aims to reconsider the question of the unity of science in light of recent developments in logic. At present, no single logical, semantical or methodological framework dominates the philosophy of science. However, the editors of this series believe that formal frameworks, for example, constructive type theory, deontic logics, dialogical logics, epistemic logics, modal logics, and proof-theoretical semantics, have the potential to cast new light on basic issues in the discussion of the unity of science. This series provides a venue where philosophers and logicians can apply specific systematic and historic insights to fundamental philosophical problems. While the series is open to a wide variety of perspectives, including the study and analysis of argumentation and the critical discussion of the relationship between logic and philosophy of science, the aim is to provide an integrated picture of the scientific enterprise in all its diversity. This book series is indexed in SCOPUS. For inquiries and submissions of proposals, authors can contact Christi Lue at [email protected] More information about this series at http://www.springer.com/series/6936 Gerhard Heinzmann • Gereon Wolters Editors Paul Lorenzen – Mathematician and Logician 123 Editors Gerhard Heinzmann Gereon Wolters Archives Henri-Poincaré (UMR 7117) Fachbereich Philosophie Université de Lorraine/Université de Universitat Konstanz Strasbourg/CNRS Konstanz, Baden-Württemberg, Germany Nancy, France ISSN 2214-9775 ISSN 2214-9783 (electronic) Logic, Epistemology, and the Unity of Science ISBN 978-3-030-65823-6 ISBN 978-3-030-65824-3 (eBook) https://doi.org/10.1007/978-3-030-65824-3 © The Editor(s) (if applicable) and The Author(s) 2021. This book is an open access publication. 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The use of general descriptive names, registered names, trademarks, service marks, etc. in this publi- cation does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland For Dr. Brigitte Parakenings, Archivist of the Philosophical Archive at the University of Konstanz (1991–2019), in gratitude for her dedicated work Preface This volume assembles contributions presented at the workshop “Paul Lorenzen: Mathematician and Logician” (8–9 March 2018) at the University of Konstanz. It was organized by the Philosophical Archive of this university, which holds the papers of Lorenzen. Paul Lorenzen (1915–1994) was an outstanding philosopher of the latter half of the twentieth century. His name is associated with the Erlangen School of Methodical Constructivism, whose approach in linguistic philosophy and philoso- phy of science determined philosophical discussions especially in Germany in the 1960s and 1970s. At that time, Lorenzen already had an international reputation as a brilliant mind in mathematics and logic. Focussing at first on abstract algebra, Lorenzen later turned his attention to foundational issues in logic and mathematics. His studies in this field are still highly regarded today and finally led to his concept of operative logic and mathematics, which in turn were the base for his philosophy later on. The contributions in this volume focus on integrating Lorenzen’s original approach into the history of logic and mathematics. Furthermore, they explore the options of how Lorenzen’s systematical ideas can be implemented in today’s debates on the philosophy of mathematics and proof-theoretic semantics. The volume opens with memories of Kuno Lorenz, Paul Lorenzen’s first student. The editors would like to thank Brigitte Parakenings (former Archivist of the Philosophical Archive) for her careful organization of the conference and Christopher von Bülow for his splendid editing of the contributions. We are also very grateful to Paul Lorenzen-Stiftung for their generous financial support for both the workshop and the production of this volume. Nancy, France Gerhard Heinzmann Konstanz, Germany Gereon Wolters May 2020 vii Contents Paul Lorenzens Weg von der Mathematik zur Philosophie – Persönliche Erinnerungen ......................................... 1 Kuno Lorenz Operation and Predicativity: Lorenzen’s Approach to Arithmetic .... 11 Gerhard Heinzmann Conceptions of Infinity and Set in Lorenzen’s Operationist System .. 23 Carolin Antos Lorenzen and Constructive Mathematics ........................... 47 Thierry Coquand Lorenzen between Gentzen and Schütte............................ 63 Reinhard Kahle and Isabel Oitavem Syntax for Semantics: Krull’s Maximal Ideal Theorem .............. 77 Peter Schuster and Daniel Wessel RegularEntailmentRelations...................................... 103 Thierry Coquand, Henri Lombardi and Stefan Neuwirth Connecting Sequent Calculi with Lorenzen-Style Dialogue Games .. 115 Christian G. Fermüller Lorenzen’s Reshaping of Krull’s Fundamentalsatz for Integral Domains(1938–1953) .............................................. 143 Stefan Neuwirth Lorenzen’s Correspondence with Hasse, Krull, and Aubert, Together with Some Relevant Documents .......................... 185 edited by Stefan Neuwirth ix List of Contributors Carolin Antos Department of Philosophy, University of Konstanz, Universitätsstr. 10, DE- 78464 Konstanz, Germany, e-mail: [email protected] Thierry Coquand Computer Science and Engineering Department, University of Gothenburg, Vasaparken, SE-405 30 Göteborg, Sweden, e-mail: [email protected] u.se Christian Fermüller Department of Computer Science, Institute for Logic and Computation, TU Wien, Favoritenstr. 9–11,AT-1040 Wien, Austria, e-mail: chrisf@logic .at Gerhard Heinzmann Archives Henri-Poincaré (UMR 7117), Université de Lorraine/Université de Strasbourg, CNRS-PReST, 1 av. De la Libération, FR-54001 Nancy cedex, France, e-mail: [email protected] Reinhard Kahle Theorie und Geschichte der Wissenschaften, Eberhard-Karls-Universität Tübingen, Keplerstr. 2,D-72074 Tübingen, Germany, and Center for Mathematics and Applications, Faculdade de Ciências e Tecnologia, Universidade Nova de Lisboa, P-2829-516 Caparica, Portugal, e-mail: [email protected] Henri Lombardi Laboratoire de mathématiques de Besançon (UMR 6623), Université