Operation and Predicativity: Lorenzen's Approach to Arithmetic
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
Philosophy of Science -----Paulk
PHILOSOPHY OF SCIENCE -----PAULK. FEYERABEND----- However, it has also a quite decisive role in building the new science and in defending new theories against their well-entrenched predecessors. For example, this philosophy plays a most important part in the arguments about the Copernican system, in the development of optics, and in the Philosophy ofScience: A Subject with construction of a new and non-Aristotelian dynamics. Almost every work of Galileo is a mixture of philosophical, mathematical, and physical prin~ a Great Past ciples which collaborate intimately without giving the impression of in coherence. This is the heroic time of the scientific philosophy. The new philosophy is not content just to mirror a science that develops independ ently of it; nor is it so distant as to deal just with alternative philosophies. It plays an essential role in building up the new science that was to replace 1. While it should be possible, in a free society, to introduce, to ex the earlier doctrines.1 pound, to make propaganda for any subject, however absurd and however 3. Now it is interesting to see how this active and critical philosophy is immoral, to publish books and articles, to give lectures on any topic, it gradually replaced by a more conservative creed, how the new creed gener must also be possible to examine what is being expounded by reference, ates technical problems of its own which are in no way related to specific not to the internal standards of the subject (which may be but the method scientific problems (Hurne), and how there arises a special subject that according to which a particular madness is being pursued), but to stan codifies science without acting back on it (Kant). -
Popper, Objectification and the Problem of the Public Sphere
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by The Australian National University Popper, Objectification and the Problem of the Public Sphere Jeremy Shearmur1 1. Introduction: World 3 Popper’s approach to philosophy was anti-psychologistic, at least since he wrote The Two Fundamental Problems of the Theory of Knowledge (cf. Popper 2009 [1979; 1930-3]). In his work, anti-psychologism meant not an opposition to the recognition of psychological activity and its significance (in correspondence with Hayek in the 1950s he identified himself as a dualist interactionist), but, rather, an opposition to the reduction of knowledge to psychology. He stressed – in ways that are familiar from Frege and the early Husserl – the non-psychologistic status of logic. And in The Open Society (Popper 1945), with reference to Marx, he stressed his opposition to attempts to offer psychologistic explanations in the social sciences. His own interpretation of ‘methodological individualism’ viewed human action as taking place in social situations. His short papers on the mind-body problem in the 1950s (included in Popper 1963) developed an argument that the descriptive and argumentative functions of language posed a problem for reductionistic accounts of the mind-body relation. While in his ‘Of Clouds and Clocks’ (in Popper 1972), issues concerning what he was subsequently to call ‘world 3’ played an important role in his more systematic treatment of the mind-body problem. As David Miller suggested, Popper called world 3 in to redress the balance of world 1 and world 2 (see Popper 1976b, note 302). -
Nazi Rule and Teaching of Mathematics in the Third Reich, Particularly School Mathematics
Oral presentations 863 Nazi Rule and Teaching of Mathematics in the Third Reich, Particularly School Mathematics Reinhard SIEGMUND-SCHULTZE University of Agder, Faculty of Engineering and Science, 4604 Kristiansand, Norway [email protected] Abstract The paper investigates the consequences of the Nazi seizure of power in Germany in 1933 for the teaching of mathematics on several levels, particularly for school mathematics. The roles of the Mathematischer Reichsverband and the F¨orderverein during the political coordination will be investigated. Particular emphasis is put on the reactions by school and research mathematicians, in particular by the leading representative of mathematical didactics, Walther Lietzmann, the research mathematician Georg Hamel, and by the would-be didactician of mathematics, Hugo Dingler. It will be shown that in the choice of the subjects of mathematics teaching, the Nazi rule promoted militaristic as well as racist and eugenicist thinking. Some remarks on the effects of the reform of 1938 conclude the paper. Much emphasis is put on basic dates and literature for further study. 1 Nazi seizure of power, dismissals and “German mathematics” After the Nazis had seized power in Germany in early 1933 there was of course concern among mathematicians and mathematics teachers about the consequences for mathematics both on the university and school levels. The most immediate and visible effect were the dismissals. The purge of school teachers seems to have been on a much lesser scale than the dismissals at the universities: apparently much fewer teachers were affected by the “Aryan paragraph” of the infamous Nazi “Law for the Restoration of the Civil Service” of April 7, 1933.1 One knows of several German-Jewish mathematics teachers who later were murdered in concentration camps,2 and the chances of emigration for teachers were slim. -
Philosophia Scientić, 17-1
Philosophia Scientiæ Travaux d'histoire et de philosophie des sciences 17-1 | 2013 The Epistemological Thought of Otto Hölder Paola Cantù et Oliver Schlaudt (dir.) Édition électronique URL : http://journals.openedition.org/philosophiascientiae/811 DOI : 10.4000/philosophiascientiae.811 ISSN : 1775-4283 Éditeur Éditions Kimé Édition imprimée Date de publication : 1 mars 2013 ISBN : 978-2-84174-620-0 ISSN : 1281-2463 Référence électronique Paola Cantù et Oliver Schlaudt (dir.), Philosophia Scientiæ, 17-1 | 2013, « The Epistemological Thought of Otto Hölder » [En ligne], mis en ligne le 01 mars 2013, consulté le 04 décembre 2020. URL : http:// journals.openedition.org/philosophiascientiae/811 ; DOI : https://doi.org/10.4000/ philosophiascientiae.811 Ce document a été généré automatiquement le 4 décembre 2020. Tous droits réservés 1 SOMMAIRE General Introduction Paola Cantù et Oliver Schlaudt Intuition and Reasoning in Geometry Inaugural Academic Lecture held on July 22, 1899. With supplements and notes Otto Hölder Otto Hölder’s 1892 “Review of Robert Graßmann’s 1891 Theory of Number”. Introductory Note Mircea Radu Review of Graßmann, Robert, Theory of Number or Arithmetic in Strict Scientific Presentation by Strict Use of Formulas (1891) Otto Hölder Between Kantianism and Empiricism: Otto Hölder’s Philosophy of Geometry Francesca Biagioli Hölder, Mach, and the Law of the Lever: A Case of Well-founded Non-controversy Oliver Schlaudt Otto Hölder’s Interpretation of David Hilbert’s Axiomatic Method Mircea Radu Geometry and Measurement in Otto Hölder’s Epistemology Paola Cantù Varia Hat Kurt Gödel Thomas von Aquins Kommentar zu Aristoteles’ De anima rezipiert? Eva-Maria Engelen Philosophia Scientiæ, 17-1 | 2013 2 General Introduction Paola Cantù and Oliver Schlaudt 1 The epistemology of Otto Hölder 1 This special issue is devoted to the philosophical ideas developed by Otto Hölder (1859-1937), a mathematician who made important contributions to analytic functions and group theory. -
Lorenzen's Proof of Consistency for Elementary Number Theory [With An
Lorenzen’s proof of consistency for elementary number theory Thierry Coquand Computer science and engineering department, University of Gothenburg, Sweden, [email protected]. Stefan Neuwirth Laboratoire de mathématiques de Besançon, Université Bourgogne Franche-Comté, France, [email protected]. Abstract We present a manuscript of Paul Lorenzen that provides a proof of con- sistency for elementary number theory as an application of the construc- tion of the free countably complete pseudocomplemented semilattice over a preordered set. This manuscript rests in the Oskar-Becker-Nachlass at the Philosophisches Archiv of Universität Konstanz, file OB 5-3b-5. It has probably been written between March and May 1944. We also com- pare this proof to Gentzen’s and Novikov’s, and provide a translation of the manuscript. arXiv:2006.08996v1 [math.HO] 16 Jun 2020 Keywords: Paul Lorenzen, consistency of elementary number the- ory, free countably complete pseudocomplemented semilattice, inductive definition, ω-rule. We present a manuscript of Paul Lorenzen that arguably dates back to 1944 and provide an edition and a translation, with the kind permission of Lorenzen’s daughter, Jutta Reinhardt. It provides a constructive proof of consistency for elementary number theory by showing that it is a part of a trivially consistent cut-free calculus. The proof resorts only to the inductive definition of formulas and theorems. More precisely, Lorenzen proves the admissibility of cut by double induction, on the complexity of the cut formula and of the derivations, without using any ordinal assignment, contrary to the presentation of cut elimination in most standard texts on proof theory. -
Philosophia Scientiæ, 18-3 | 2014, « Logic and Philosophy of Science in Nancy (I) » [Online], Online Since 01 October 2014, Connection on 05 November 2020
Philosophia Scientiæ Travaux d'histoire et de philosophie des sciences 18-3 | 2014 Logic and Philosophy of Science in Nancy (I) Selected Contributed Papers from the 14th International Congress of Logic, Methodology and Philosophy of Science Pierre Édouard Bour, Gerhard Heinzmann, Wilfrid Hodges and Peter Schroeder-Heister (dir.) Electronic version URL: http://journals.openedition.org/philosophiascientiae/957 DOI: 10.4000/philosophiascientiae.957 ISSN: 1775-4283 Publisher Éditions Kimé Printed version Date of publication: 1 October 2014 ISBN: 978-2-84174-689-7 ISSN: 1281-2463 Electronic reference Pierre Édouard Bour, Gerhard Heinzmann, Wilfrid Hodges and Peter Schroeder-Heister (dir.), Philosophia Scientiæ, 18-3 | 2014, « Logic and Philosophy of Science in Nancy (I) » [Online], Online since 01 October 2014, connection on 05 November 2020. URL : http://journals.openedition.org/ philosophiascientiae/957 ; DOI : https://doi.org/10.4000/philosophiascientiae.957 This text was automatically generated on 5 November 2020. Tous droits réservés 1 This issue collects a selection of contributed papers presented at the 14th International Congress of Logic, Methodology and Philosophy of Science in Nancy, July 2011. These papers were originally presented within three of the main sections of the Congress. They deal with logic, philosophy of mathematics and cognitive science, and philosophy of technology. A second volume of contributed papers, dedicated to general philosophy of science, and other topics in the philosophy of particular sciences, will appear in the next issue of Philosophia Scientiæ (19-1), 2015. Philosophia Scientiæ, 18-3 | 2014 2 TABLE OF CONTENTS Logic and Philosophy of Science in Nancy (I) Preface Pierre Edouard Bour, Gerhard Heinzmann, Wilfrid Hodges and Peter Schroeder-Heister Copies of Classical Logic in Intuitionistic Logic Jaime Gaspar A Critical Remark on the BHK Interpretation of Implication Wagner de Campos Sanz and Thomas Piecha Gödel’s Incompleteness Phenomenon—Computationally Saeed Salehi Meinong and Husserl on Existence. -
What Is Mathematics: Gödel's Theorem and Around. by Karlis
1 Version released: January 25, 2015 What is Mathematics: Gödel's Theorem and Around Hyper-textbook for students by Karlis Podnieks, Professor University of Latvia Institute of Mathematics and Computer Science An extended translation of the 2nd edition of my book "Around Gödel's theorem" published in 1992 in Russian (online copy). Diploma, 2000 Diploma, 1999 This work is licensed under a Creative Commons License and is copyrighted © 1997-2015 by me, Karlis Podnieks. This hyper-textbook contains many links to: Wikipedia, the free encyclopedia; MacTutor History of Mathematics archive of the University of St Andrews; MathWorld of Wolfram Research. Are you a platonist? Test yourself. Tuesday, August 26, 1930: Chronology of a turning point in the human intellectua l history... Visiting Gödel in Vienna... An explanation of “The Incomprehensible Effectiveness of Mathematics in the Natural Sciences" (as put by Eugene Wigner). 2 Table of Contents References..........................................................................................................4 1. Platonism, intuition and the nature of mathematics.......................................6 1.1. Platonism – the Philosophy of Working Mathematicians.......................6 1.2. Investigation of Stable Self-contained Models – the True Nature of the Mathematical Method..................................................................................15 1.3. Intuition and Axioms............................................................................20 1.4. Formal Theories....................................................................................27 -
Conjectures and Refutations
CONJECTURES AND REFUTATIONS -i- By the same Author The Open Society and Its Enemies Vol. I: The Spell of Plato Vol. II: The High Tide of Prophecy: Hegel, Marx, and the Aftermath The Poverty of Historicism The Logic of Scientific Discovery -ii- CONJECTURES AND REFUTATIONS The Growth of Scientific Knowledge by KARL R. POPPER BASIC BOOKS, Publishers NEW YORK LONDON -iii- © Karl R. Popper 1962 Manufactured in the United States of America -iv- TO F. A. VON HAYEK -v- Experience is the name every one gives to their mistakes. OSCAR WILDE -vi- PREFACE THE ESSAYS and lectures of which this book is composed are variations upon one very simple theme--the thesis that we can learn from our mistakes. They develop a theory of knowledge and of its growth. It is a theory of reason that assigns to rational arguments the modest and yet important role of criticizing our often mistaken attempts to solve our problems. And it is a theory of experience that assigns to our observations the equally modest and almost equally important role of tests which may help us in the discovery of our mistakes. Though it stresses our fallibility it does not resign itself to scepticism, for it also stresses the fact that knowledge can grow, and that science can progress--just because we can learn from our mistakes. The way in which knowledge progresses, and especially our scientific knowledge, is by unjustified (and unjustifiable) anticipations, by guesses, by tentative solutions to our problems, by conjectures. These conjectures are controlled by criticism; that is, by attempted refutations, which include severely critical tests. -
The Law of Causality and Its Limits Vienna Circle Collection
THE LAW OF CAUSALITY AND ITS LIMITS VIENNA CIRCLE COLLECTION lIENK L. MULDER, University ofAmsterdam, Amsterdam, The Netherlands ROBERT S. COHEN, Boston University, Boston, Mass., U.SA. BRIAN MCGUINNESS, University of Siena, Siena, Italy RUDOLF IlALLER, Charles Francis University, Graz, Austria Editorial Advisory Board ALBERT E. BLUMBERG, Rutgers University, New Brunswick, N.J., U.SA. ERWIN N. HIEBERT, Harvard University, Cambridge, Mass., U.SA JAAKKO HiNTIKKA, Boston University, Boston, Mass., U.S.A. A. J. Kox, University ofAmsterdam, Amsterdam, The Netherlands GABRIEL NUCHELMANS, University ofLeyden, Leyden, The Netherlands ANTH:ONY M. QUINTON, All Souls College, Oxford, England J. F. STAAL, University of California, Berkeley, Calif., U.SA. FRIEDRICH STADLER, Institute for Science and Art, Vienna, Austria VOLUME 22 VOLUME EDITOR: ROBERT S. COHEN PHILIPP FRANK PHILIPP FRANK THELAWOF CAUSALITY AND ITS LIMITS Edited by ROBERT s. COHEN Boston University Translated by MARIE NEURATH and ROBERT S. COHEN 1Ii.. ... ,~ SPRINGER SCIENCE+BUSINESS MEDIA, B.V. Library of Congress Cataloging-in-Publication data Frank, Philipp, 1884-1966. [Kausalgesetz und seine Grenzen. Englishl The law of causality and its limits / Philipp Frank; edited by Robert S. Cohen ; translation by Marie Neurath and Robert S. Cohen. p. cm. -- (Vienna Circle collection ; v. 22) Inc I udes index. ISBN 978-94-010-6323-4 ISBN 978-94-011-5516-8 (eBook) DOI 10.1007/978-94-011-5516-8 1. Causation. 2. Science--Phi losophy. I. Cohen, R. S. (Robert Sonne) 11. Title. 111. Series. BD543.F7313 -
Reconsidering Carnap's Logic of Science
Western University Scholarship@Western Electronic Thesis and Dissertation Repository 12-9-2013 12:00 AM The Methodological Roles of Tolerance and Conventionalism in the Philosophy of Mathematics: Reconsidering Carnap's Logic of Science Emerson P. Doyle The University of Western Ontario Supervisor John L. Bell The University of Western Ontario Graduate Program in Philosophy A thesis submitted in partial fulfillment of the equirr ements for the degree in Doctor of Philosophy © Emerson P. Doyle 2013 Follow this and additional works at: https://ir.lib.uwo.ca/etd Part of the Logic and Foundations of Mathematics Commons Recommended Citation Doyle, Emerson P., "The Methodological Roles of Tolerance and Conventionalism in the Philosophy of Mathematics: Reconsidering Carnap's Logic of Science" (2013). Electronic Thesis and Dissertation Repository. 1757. https://ir.lib.uwo.ca/etd/1757 This Dissertation/Thesis is brought to you for free and open access by Scholarship@Western. It has been accepted for inclusion in Electronic Thesis and Dissertation Repository by an authorized administrator of Scholarship@Western. For more information, please contact [email protected]. The Methodological Roles of Tolerance and Conventionalism in the Philosophy of Mathematics: Reconsidering Carnap's Logic of Science (Thesis Format: Monograph) by Emerson P. Doyle Graduate Program in Philosophy A thesis submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy The School of Graduate and Postdoctoral Studies The University of Western Ontario London, Ontario, Canada c Emerson P. Doyle 2013 Abstract This dissertation makes two primary contributions. The first three chapters de- velop an interpretation of Carnap's Meta-Philosophical Program which places stress upon his methodological analysis of the sciences over and above the Principle of Tol- erance. -
Karl Popper for and Against Bertrand Russell
KARL POPPER FORAND· AGAINST BERTRAND RUSSELL I. GRATTAN-GUINNESS Mathematics / Middlesex University Enfield, Middlesex EN3 4SF, England E-mail: [email protected] I. IMPACTS he day that Popper died. early in September 1994, I was in Basel, Switzerland; next day the local Sunday newspaper carried an T article about him as the first page of the arts section. The day that he was cremated I was in Zaragoza, Spain; the local newspaper carried a substantial piece, probably syndicated, written by a Spaniard. Since Popper had had no special links with either city, then presumably I had randomly sampled the (non-stupid end ofthe) international press, and I concluded that it had given him world-wide acknowledgement. Such tribute to a philosopher had probably not happened since the death of Bertrand Russell in February 1970: then the reaction was still greater, though driven more by his general social concerns, especially the political activities ofhis final decade, than by his philosophical achieve ments. Russell's activities crossed my path in the mid-1960s, when I was taking a Master's degree at the London School ofEconomics in Popper's department. In addition to the compulsory courses on philosophy, I specialized in mathematical logic and the philosophy ofmathematics (as did David Miller and my colleague Allan Findlay). During the first year some of the instruction in logic was given by Popper himself, in the form ofa late afternoon discourse. On one day-to be precise, Monday, 15 February 1965--<>ur discussion was cut short, so that we three could go downstairs to hear a speech in the Old Theatre by Russell. -
History of Mathematics Clio Mathematicæ the Muse of Mathematical Historiography Craig Smorynski´
History of Mathematics Clio Mathematicæ The Muse of Mathematical Historiography Craig Smorynski´ History of Mathematics A Supplement 123 Craig Smorynski´ 429 S. Warwick Westmont, IL 60559 USA [email protected] ISBN 978-0-387-75480-2 e-ISBN 978-0-387-75481-9 Library of Congress Control Number: 2007939561 Mathematics Subject Classification (2000): 01A05 51-Axx, 15-xx c 2008 Springer Science+Business Media, LLC All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper. 987654321 springer.com Contents 1 Introduction ............................................... 1 1 An Initial Assignment . 1 2 AboutThisBook........................................ 7 2 Annotated Bibliography ................................... 11 1 General Remarks . 11 2 General Reference Works . 18 3 General Biography . 19 4 General History of Mathematics . 21 5 History of Elementary Mathematics . 23 6 SourceBooks........................................... 25 7 Multiculturalism . 27 8 Arithmetic ............................................. 28 9 Geometry .............................................. 28 10 Calculus................................................ 29 11 Women in Science . 30 12 MiscellaneousTopics..................................... 35 13 Special Mention . 36 14 Philately .