DOCSLIB.ORG

0415023351.Pdf

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
0415023351.Pdf PROOF, LOGIC AND FORMALIZATION This book addresses various problems associated with finding a philosophically satisfying account of mathematical proof. It brings together many of the most notable figures currently writing in the philosophy of mathematics in an attempt to explain why it is that mathematical proof is given prominence over other forms of mathematical justification. The difficulties that arise in accounts of proof range from the rightful role of logical inference and formalization to questions concerning the place of experience in proof and the possibility of eliminating impredicative reasoning from proof. The editor, Michael Detlefsen, has brought together an outstanding collection of essays, only two of which have previously appeared. Essential for philosophers and historians of mathematics, it is also recommended for philosophically inclined logicians and philosophers interested in the nature of reasoning and justification. A companion volume entitled Proof and Knowledge in Mathematics, edited by Michael Detlefsen, is also available from Routledge. PROOF, LOGIC AND FORMALIZATION PROOF, LOGIC AND FORMALIZATION Edited by Michael Detlefsen London and New York First published 1992 by Routledge 11 New Fetter Lane, London EC4P 4EE This edition published in the Taylor & Francis e-Library, 2005. “To purchase your own copy of this or any of Taylor & Francis or Routledge’s collection of thousands of eBooks please go to www.eBookstore.tandf.co.uk.” Simultaneously published in the USA and Canada by Routledge a division of Routledge, Chapman and Hall, Inc. 29 West 35th Street, New York, NY 10001 Selection and introductory material © 1992 Michael Detlefsen Individual chapters © 1992 the respective authors. All rights reserved. No part of this book may be reprinted or reproduced or utilized in any form or by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying and recording, or in any information storage or retrieval system, without permission in writing from the publishers. British Library Cataloguing in Publication Data Proof, logic and formalization 1. Mathematics I. Detlefsen, Michael 511.3 Library of Congress Cataloging in Publication Data Proof, logic and formalization/edited by Michael Detlefsen. p. cm. Includes bibliographical references. 1. Proof theory. 2. Logic, Symbolic and mathematical. I. Detlefsen, Michael. QA9.54.P77 1992 511.3–dc20 91–17469 ISBN 0-203-98025-5 Master e-book ISBN ISBN 0-415-02335-1 (Print Edition) In memory of Kenneth Lee Steel, Born 18 March 1947, Lincoln, Nebraska, Died 28 February 1967, Bong Son, South Viet Nam. “Therefore I have uttered what I did not understand, things too wonderful for me, which I did not know. ‘Hear, and I will speak; I will question you, and you declare to me.’” Job 42 CONTENTS Notes on contributors vii Preface viii 1 PROOFS ABOUT PROOFS: A DEFENSE OF CLASSICAL LOGIC. PART I: THE AIMS OF CLASSICAL 1 LOGIC John P.Burgess 2 PROOFS AND EPISTEMIC STRUCTURE 8 Glen Helman 3 WHAT IS A PROOF? 22 Richard Tieszen 4 HOW TO SAY THINGS WITH FORMALISMS 31 David Auerbach 5 SOME CONSIDERATIONS ON ARITHMETICAL TRUTH AND THE ω -RULE 39 Daniel Isaacson 6 THE IMPREDICATIVITY OF INDUCTION 60 Charles Parsons 7 THREE INSUFFICIENTLY ATTENDED TO ASPECTS OF MOST MATHEMATICAL PROOFS: 71 PHENOMENOLOGICAL STUDIES Robert S.Tragesser 8 ON AN ALLEGED REFUTATION OF HILBERT’S PROGRAM USING GÖDEL’S FIRST 88 INCOMPLETENESS THEOREM Michael Detlefsen Index 104 NOTES ON CONTRIBUTORS David Auerbach teaches philosophy and logic at North Carolina State University in Raleigh. John P.Burgess received his doctorate from the Group in Logic and the Methodology of the Deductive Sciences at the University of California at Berkeley, and is now Professor of Philosophy at Princeton University. He is an editor of the Journal of Symbolic Logic and the Notre Dame Journal of Formal Logic, and a frequent contributor to journals and anthologies in mathematical and philosophical logic and the philosophy of mathematics. Michael Detlefsen is Professor of Philosophy at the University of Notre Dame and Editor-in-Chief of the Notre Dame Journal of Formal Logic. His areas of special interest are the philosophy of logic and mathematics. He is the author of Hubert’s Program: An Essay on Mathematical Instrumentalism (Reidel, 1986). Currently, he is working on a book on intuitionism, and coauthoring another with Michael Byrd on Gödel’s theorems. Glen Helman, formerly a member of the Philosophy Department of the Pennsylvania State University, is now Assistant Professor of Philosophy at Wabash College. His publications concern the equivalence of proofs and their relation to the extended lambda calculus. Daniel Isaacson is currently University Lecturer in the Philosophy of Mathematics at Oxford University and Fellow of Wolfson College, Oxford. He has been lecturer in philosophy at the University of Washington in Seattle and visiting professor of philosophy at the University of California, Berkeley, and has held research posts at the Rockefeller University in New York and St John’s College, Oxford. He was an undergraduate at Harvard, and took his doctorate at Oxford. Charles Parsons is Professor of Philosophy at Harvard University. From 1965 to 1989 he was on the faculty of Columbia University. He is the author of Mathematics in Philosophy (Cornell University Press, 1983) and of papers on proof theory, intensional logic, philosophy of logic and mathematics and Kant. Richard Tieszen received his Ph.D. in philosophy at Columbia University and is presently Assistant Professor at San Jose State University, located in the Silicon Valley. His research interests include logic and logic programming, philosophy of mathematics, and cognitive science. Robert S.Tragesser is currently Assistant Professor of Philosophy at Barnard College in Columbia University, and Research Associate at the CUNY Graduate Center. He is the author of two books, Phenomenology and Logic (Cornell University Press, 1977) and Husserl and Realism in Logic and Mathematics (Cambridge University Press, 1984). He is at work on a book, Logical Imagination, which will contain (among other things) an elaboration of the ideas presented in his paper in this volume. PREFACE Among the features of mathematical thought that most invite the philosopher’s attention is its regimentation of justificatory procedures; its canonization of proof as the preferred form of justification. This regimentation is made the more striking by the fact that it is clearly possible to have other kinds of justification (even some of very great strength) for mathematical propositions; a fact which at least suggests that concern for truth and/or certainty alone cannot account for the prominence given to proof within mathematics. This being so, arriving at some understanding of the nature and role of proof becomes one of the primary challenges facing the philosophy of mathematics. It is this challenge which forms the motivating concern of this volume, and to which it is intended to constitute a partial response (on a variety of different fronts). One possibility, of course, is that it signifies nothing distinctive in the epistemic aims of mathematics at all, but is rather the outgrowth of certain purely social or historical forces. However, since regimentation of justificatory practice is the kind of thing that is at all likely to arise only out of consideration for epistemic ideals, to attribute it to nonepistemic, socio-historical factors would require special argumentation, and there is currently little if any reason to suppose such argumentation might be given. Thus, accounts of the ascendancy of proof in mathematics that make appeal to some epistemic ideal(s) would appear to be more promising. One consideration which seems to have weighed heavily in favor of proof as the preferred form of justification is the traditional concern of mathematics for rigor. This drive, in turn, seems to have derived from one of two (in some ways related) impeti. The first of these—an old and persistent one, with a bloodline reaching from Aristotle through Leibniz and on down to Frege —s based on the idea that the aim of a justification is not merely, nor even primarily, to establish the certainty of its conclusion, but rather to identify the metaphysical grounds or origins. This induces a hierarchy of truths at the base of which are located the ur-truths or axioms. The epistemic ideal is to be able to trace a given truth all the way back to the ur- truths which found it, thus obtaining a complete account of why it is true. Rigor is essential to this enterprise because, without it, one cannot get a clear distillation of those truths that are being used (and perhaps even need to be used) to ground a given truth. A historically more influential idea—also associated with Aristotle (in particular, with his conception of a deductive or demonstrative science), as well as with Euclid and, in the modern period, with Descartes—has a less metaphysical, more purely epistemic foundational thrust. It too searches for ultimate truths, but not in the sense of final metaphysical grounds of truth. Rather, it seeks to find premises that are epistemically ultimate—truths, that is, that are (at least in certain respects) epistemically unsurpassed. In this type of scheme, the axioms are not taken to be metaphysically responsible for the truth of the theorems, though there is still a sense in which they may be said to show why they (i.e. the theorems) are true. More recently, these traditional conceptions have given way to less foundationally oriented conceptions of mathematical justification. Axioms are no longer taken as giving the metaphysical grounds for the theorems. Nor are they taken to be insurpassably, or even insurpassedly, evident. Rather, they are seen only as being evident enough (i) to make a search for further evidence seem unnecessary, or perhaps (ii) to remove sufficiently much of whatever rational doubt or indecision it seems possible and/or desirable to remove from the theorems, or maybe just (iii) to be attractively simple and economical, and evident enough to serve the purposes at hand, etc.
Load more

Recommended publications
  • Commencement1976.Pdf (4.717Mb)
    1976 Digitized by the Internet Archive in 2012 with funding from LYRASIS Members and Sloan Foundation http://archive.org/details/commencement1976 ORDER OF PROCESSION MARSHALS STANLEY CORRSIN JOHN W. GRYDER MATTHEW A. CRENSON WILLIAM H. HUGGINS ELAINE C. DAVIS ROBERT A. LYSTAD HANS GOEDICKE EVANGELOS N. MOUDRIANAKIS ARCHIE GOLDEN EVERETT SCHILLER GERALD S. GOTTERER JOHN P. YOUNG THE GRADUATES MARSHALS ROBERT B. POND OREST RANUM THE DEANS MEMBERS OF THE SOCIETY OF SCHOLARS OFFICERS OF THE UNIVERSITY THE TRUSTEES * MARSHALS BROWN L. MURR FRANCIS E. ROURKE THE FACULTIES * CHIEF MARSHAL RICHARD A. MACKSEY THE CHAPLAINS THE RECIPIENT OF THE MILTON STOVER EISENHOWER MEDAL FOR DISTINGUISHED SERVICE THE PRESENTOR OF THE RECIPIENT OF THE MILTON STOVER EISENHOWER MEDAL FOR DISTINGUISHED SERVICE THE HONORARY DEGREE CANDIDATES THE PROVOST OF THE UNIVERSITY THE PRESIDENT EMERITUS OF THE UNIVERSITY THE CHAIRMAN OF THE BOARD OF TRUSTEES THE PRESIDENT OF THE UNIVERSITY ORDER OF EVENTS STEVEN MULLER President of the University, presiding * * * FANFARE PROCESSIONAL The audience is requested to stand as the Academic Procession moves into the area and to remain standing after the Invocation. " " Earle of Oxford's Marche William Byrd The Peabody Wind Ensemble Richard Higgins, Director * INVOCATION REV. CHESTER WICKWIRE Chaplain, The Johns Hopkins University THE NATIONAL ANTHEM GREETINGS ROBERT D. H. HARVEY Chairman of the Board of Trustees PRESENTATION OF THE RECIPIENT FOR THE MILTON STOVER EISENHOWER MEDAL FOR DISTINGUISHED SERVICE HELEN B. TAUSSIG PRESENTED BY RICHARD S. ROSS Vice President for the Health Divisions and Dean, School of Medicine * PRESENTATION OF NEW MEMBERS OF THE SOCIETY OF SCHOLARS LEROY E.
    [Show full text]
  • APA Eastern Division 2019 Annual Meeting Program
    The American Philosophical Association EASTERN DIVISION ONE HUNDRED FIFTEENTH ANNUAL MEETING PROGRAM SHERATON NEW YORK TIMES SQUARE NEW YORK, NEW YORK JANUARY 7 – 10, 2019 Visit our table at APA Eastern OFFERING A 20% (PB) / 40% (HC) DISCOUNT WITH FREE SHIPPING TO THE CONTIGUOUS U.S. FOR ORDERS PLACED AT THE CONFERENCE. THE POETRY OF APPROACHING HEGEL’S LOGIC, GEORGES BATAILLE OBLIQUELY Georges Bataille Melville, Molière, Beckett Translated and with an Introduction by Angelica Nuzzo Stuart Kendall THE POLITICS OF PARADIGMS ZHUANGZI AND THE Thomas S. Kuhn, James B. Conant, BECOMING OF NOTHINGNESS and the Cold War “Struggle for David Chai Men’s Minds” George A. Reisch ANOTHER AVAILABLE APRIL 2019 WHITE MAN’S BURDEN Josiah Royce’s Quest for a Philosophy THE REAL METAPHYSICAL CLUB of white Racial Empire The Philosophers, Their Debates, and Tommy J. Curry Selected Writings from 1870 to 1885 Frank X. Ryan, Brian E. Butler, and BOUNDARY LINES James A. Good, editors Philosophy and Postcolonialism Introduction by John R. Shook Emanuela Fornari AVAILABLE MARCH 2019 Translated by Iain Halliday Foreword by Étienne Balibar PRAGMATISM APPLIED William James and the Challenges THE CUDGEL AND THE CARESS of Contemporary Life Reflections on Cruelty and Tenderness Clifford S. Stagoll and David Farrell Krell Michael P. Levine, editors AVAILABLE MARCH 2019 AVAILABLE APRIL 2019 LOVE AND VIOLENCE BUDDHIST FEMINISMS The Vexatious Factors of Civilization AND FEMININITIES Lea Melandri Karma Lekshe Tsomo, editor Translated by Antonio Calcagno www.sunypress.edu II IMPORTANT NOTICES FOR MEETING ATTENDEES SESSION LOCATIONS Please note: this online version of the program does not include session locations.
    [Show full text]
  • SYLLABUS Gödel's Theorems Phil 93902 TR 3:30 143 Debartolo Hall
    SYLLABUS G¨odel's Theorems Phil 93902 TR 3:30 143 Debartolo Hall Michael Detlefsen ([email protected]), 408 Malloy Hall Curtis Franks ([email protected]), 411 Malloy Hall Fall 2007 I. Course Objectives • To acquaint the student with the basic ideas, methods and techniques behind one of the most striking scientific achievements of the twentieth century, the discovery of G¨odel'sincompleteness theorems. • To engage the student in productive thinking concerning the broader significance of these theorems. II. Texts • Kurt G¨odel:Collected Works, vol. I, Feferman et al. (eds.), Oxford, Oxford U. Press, 1986. III. Organization and Projected Schedule The course will divided into the following units, each estimated to run the indicated number of meetings. We plan to cover all of units A through G. We will cover the additional units according to the interests of the seminar participants. A. Historical and conceptual background and basic terminology. Two meetings. B. Basic mathematical background. Proof of the first theorem (G1). Four to five meet- ings. 1 C. Modifications of the proof of the first theorem, esp. Rosser's modification. One to two meetings. D. Proof of the second theorem (G2). Central task is to clarify the proofs of the second and third of the Hilbert-Bernays-L¨obDerivability Conditions (HBL Conditions). Two meetings. E. Alternative proofs of G2. Jeroslow's 1973 proof. Feferman's 1960 proof. Freidman's recent proof. [Note: We may discuss Jeroslow's and Feferman's proofs in inverse historical order because Jeroslow's proof is in some ways the less radical departure from the Hilbert-Bernays-L¨obproof.] Three meetings.
    [Show full text]
  • Notre Dame Report 37:10 (2008-08-03)
    Faculty Notes 387 ������ Honors 387 ������ Activities 389 ������ Publications 390 ������ Correction Administrators’ Notes 390 ������ Activities 390 ������ Publications Documentation 391 ������ Faculty Awards 393 ������ Faculty Recognition and Promotions 395 ������ Senior Fellow Comments 395 ������ Charge to the Class 396 ������ Baccalaureate Mass Homily 397 ������ Invocation 397 ������ Valedictory Remarks 399 ������ Undergraduate School Commencement Address 401 ������ The Laetare Medal Research 404 ������ May 2008 A U G U S T 3 , 2 0 0 8 N U M B E R 1 0 #8-5-387 “Discovery vs� Invention in the History of Mathematics” at the “Mathematics and Its Significance, Templeton Foundation Symposium,” Castel Gandolfo, Italy, Faculty Notes June 21–23; “Arithmetization and the Ideals of Proof” for the Dept� de la Histoire et de Honors Philosophie des Sciences, Univ� of Paris- Diderot and REHSEIS, June 18–19; “Some Michael Detlefsen, the McMahon-Hank Harvey A. Bender, professor of biological Questions Concerning Proof” as an invited Professor of Philosophy, was awarded a sciences, delivered a Hesburgh Lecture and discussant in “Workshop in the History and Senior Chaire d’excellence, by the Agence participated in the annual meeting of the Philosophy of Mathematics,” Dept� de la Nationale de la Recherche, France, for his American College of Medical Genetics in Histoire et de Philosophie des Sciences, U� work on ideals of proof; and is serving Phoenix, Ariz�, March 12–16, 2008� of Paris-Diderot and REHSEIS, June 13–14; as distinguished invited professor
    [Show full text]
  • Counterpossibles in Science: the Case of Relative Computability*
    Counterpossibles in Science: The Case of Relative Computability* MATTHIAS JENNY Massachusetts Institute of Technology forthcoming in Noˆus Abstract I develop a theory of counterfactuals about relative computability, i.e. counterfactuals such as If the validity problem were algorithmically decidable, then the halting problem would also be algorithmically decidable, which is true, and If the validity problem were algorithmically decidable, then arithmetical truth would also be algorithmically decidable, which is false. These counterfactuals are counterpossibles, i.e. they have metaphysically impossible antecedents. They thus pose a challenge to the orthodoxy about counter- factuals, which would treat them as uniformly true. What’s more, I argue that these counterpossibles don’t just appear in the periphery of relative computability theory but instead they play an ineliminable role in the development of the theory. Finally, I present and discuss a model theory for these counterfactuals that is a straightforward extension of the familiar comparative similarity models. *Thanks to Scott Aaronson, Karen Bennett, Matteo Bianchetti, David Boylan, Ben Burgis, Alex Byrne, Michael Detlefsen, Richard Dietz, Kevin Dorst, Nicole Dular, Hartry Field, Melvin Fitting, Juliet Floyd, Cameron Gibbs, Cosmo Grant, Geoffrey Hellman, Jared Henderson, Harold Hodes, Douglas Jesseph, Justin Khoo, Hilary Kornblith, Teresa Kouri, Rose Lenehan, Matt Mandelkern, William Nalls, Eileen Nutting, Alex Paseau, Milo Phillips-Brown, Graham Priest, Agust´ın Rayo, Bernhard Salow, Chris Scambler, Melissa Schu- macher, Kieran Setiya, Stewart Shapiro, Jeremy Shipley, Bradford Skow, Theodore Sider, Robert Stalnaker, Stephen Yablo, Elena Ziliotti, audiences at the 3rd Seoul Philosophy Graduate Student Conference, the 16th Midwest PhilMath Workshop, the 1st Massachusetts-Rhode Island Graduate Workshop in Philosophy, the 2016 Logic & Metaphysics Workshop at the CUNY Graduate Center, the 2016 CSHPM/SCHPM Annual Meeting, and especially Vann McGee for very helpful discussion.
    [Show full text]
  • Hilbert's Program Then And
    Hilbert’s Program Then and Now Richard Zach Abstract Hilbert’s program was an ambitious and wide-ranging project in the philosophy and foundations of mathematics. In order to “dispose of the foundational questions in mathematics once and for all,” Hilbert proposed a two-pronged approach in 1921: first, classical mathematics should be formalized in axiomatic systems; second, using only restricted, “finitary” means, one should give proofs of the consistency of these axiomatic sys- tems. Although G¨odel’s incompleteness theorems show that the program as originally conceived cannot be carried out, it had many partial suc- cesses, and generated important advances in logical theory and meta- theory, both at the time and since. The article discusses the historical background and development of Hilbert’s program, its philosophical un- derpinnings and consequences, and its subsequent development and influ- ences since the 1930s. Keywords: Hilbert’s program, philosophy of mathematics, formal- ism, finitism, proof theory, meta-mathematics, foundations of mathemat- ics, consistency proofs, axiomatic method, instrumentalism. 1 Introduction Hilbert’s program is, in the first instance, a proposal and a research program in the philosophy and foundations of mathematics. It was formulated in the early 1920s by German mathematician David Hilbert (1862–1943), and was pursued by him and his collaborators at the University of G¨ottingen and elsewhere in the 1920s and 1930s. Briefly, Hilbert’s proposal called for a new foundation of mathematics based on two pillars: the axiomatic method, and finitary proof theory. Hilbert thought that by formalizing mathematics in axiomatic systems, and subsequently proving by finitary methods that these systems are consistent (i.e., do not prove contradictions), he could provide a philosophically satisfactory grounding of classical, infinitary mathematics (analysis and set theory).
    [Show full text]
  • CSHPM Bulletin, May 2020
    BULLETIN May/Mai 2020 Number/le num´ero66 WHAT'S INSIDE Articles Announcements ................................................................................................3 Digitized Smithsonian Mathematics Objects [Peggy Aldrich Kidwell] .....................................7 Book Review: Republic of Numbers [Andrew Perry].........................................................8 Books Borrowed Registers [Scott Guthery] ................................................................... 10 New Content from MAA Convergence [Janet Heine Barnett & Amy Ackerberg-Hastings]............... 10 Michael Detlefsen (1948{2019) [Patricia Blanchette, Tim Bays, and Curtis Franks] ...................... 12 Quotations in Context [Mike Molinsky] ...................................................................... 14 Martin Muldoon (1939{2019) [Hardy Grant] ................................................................. 15 Help (Still) Wanted ............................................................................................ 16 Reports President's Message [Maria Zack] .............................................................................2 From the Archives: Councillors ............................................................................... 13 2020 CSHPM Nominating Committee Report [Chris Baltus, Dan Curtin, and Elaine Landry] .......... 13 2019 Financial Statements [Greg Lavers] ..................................................................... 15 New Members .................................................................................................
    [Show full text]
  • Faculty and Index
    304 JOSÉ ANADÓN, Professor of Romance Languages and DAVID E. AUNE, Professor of Theology. B.A., Wheaton University Teaching Literatures. B.A., Albion College, 1968; M.A. Univ. of College, 1961; M.A., ibid., 1963; M.A., Univ. of Minne- Faculty Michigan, 1970; Ph.D., ibid., 1974. (1975) sota, 1965; Ph.D., Univ. of Chicago, 1970. (1999) SILVIA R. ANADÓN, Assistant Professional Specialist and YVES L. AURIOL, Associate Professional Specialist in Concurrent Lecturer in Romance Languages and Literatures. Physical Education. Master’s, Institute National du Sport, The following is the undergraduate teaching faculty roster B.A., Univ. of Michigan, 1970; M.A., ibid., 1972; Ph.D., France, 1968. (1985) for the 2001-02 academic year. This roster includes fac- ibid., 1974. (1976, 1979) ulty members who are on leave during the academic year. REV. NICHOLAS R. AYO C.S.C., Professor of Liberal The date in parentheses at the close of each entry is the THOMAS F. ANDERSON, Assistant Professor of Romance Studies. A.B., Univ. of Notre Dame, 1956; S.T.L., year the individual joined the Notre Dame faculty. Languages and Literatures and Fellow, Helen Kellogg Insti- Gregorian Univ., 1960; M.A., Univ. of Notre Dame, tute for International Studies. B.A., Bowdoin College, 1962; Ph.D., Duke Univ., 1966. (1981) CARL B. ACKERMANN, Assistant Professor of Finance. 1992; M.A., Univ. of Pennsylvania, 1994; Ph.D., ibid., A.B., Amherst College, 1984; Ph.D., Univ. of North 1998. (1998) MARIA PALOMA AYUELA, Associate Professor of Archi- Carolina, 1999. (1998) tecture. M.Arch., Yale Univ., 1989; Ph.D., Universidad PANAGIOTIS J.
    [Show full text]
  • Blanchette Vita for Website
    Patricia A. Blanchette Dept. of Philosophy Office: (574) 631-6439 University of Notre Dame Dept: (574) 631-6471 Notre Dame, IN 46556 e-mail: [email protected] Areas of Specialization: History and Philosophy of logic; Philosophy of mathematics; Foundations of analytic philosophy; Philosophy of Language. Education Ph.D. (Philosophy), Stanford University, 1990. B.A. (Philosophy), University of California, San Diego, 1983. Employment August 1993 – Present: University of Notre Dame, Dept. of Philosophy • June 2017 – Present: Glynn Family Honors Collegiate Chair • May 2012 – Present: Professor • May 2000 – May 2012: Associate Professor o Director of Graduate Studies 2004-7, 2008-11 • August 1993 – May 2000: Assistant Professor July 1990 - July 1993: Yale University. Assistant Professor, Dept. of Philosophy. Publications: Book • Frege’s Conception of Logic, Oxford University Press 2012 Reviewed in: o Philosophia Mathematica 22, 1 (2014) pp 108-120. o Journal of the History of Philosophy 52 (2014) pp 176-177. o Bulletin of Symbolic Logic 19, 2 (2013) pp 219-222. o Notre Dame Philosophical Reviews (2013) online. Articles • “Axioms in Frege,” Essays on Frege’s Basic Laws of Arithmetic, edited by Marcus Rossberg and Philip Ebert, Oxford University Press, forthcoming • “Models in Geometry and Logic: 1870-1920,” Logic, Methodology and Philosophy of Science – Proceedings of the 15th International Congress, edited by Niiniluoto, Seppälä, & Sober, College Publications 2017, 41-61. • “Philosophy of Logic,” Philosophy of Logic: 5 Questions, Automatic Press 2016, 25-35. • “Models and Independence circa 1900,” Logica Yearbook 2015, London: College Publications 2016, 17-38. Much of the material in this paper appears also in the expanded “Models in Geometry and Logic: 1870-1920,” 2017.
    [Show full text]
  • From the Outside Looking In: Can Mathematical Certainty Be Secured Without Being Mathematically Certain That It Has Been?
    From the Outside Looking In: Can mathematical certainty be secured without being mathematically certain that it has been? Dissertation Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University Matthew Souba, MSc, MLitt Graduate Program in Philosophy The Ohio State University 2019 Dissertation Committee: Neil Tennant, Advisor Stewart Shapiro Christopher Pincock Copyright by Matthew Souba 2019 Abstract The primary aim of this dissertation is to discuss the epistemological fallout of Gödel’s Incompleteness Theorems on Hilbert’s Program. In particular our focus will be on the philo- sophical upshot of certain proof-theoretic results in the literature. We begin by sketching the historical development up to, and including, Hilbert’s mature program, discussing Hilbert’s views in both their mathematical and their philosophical guises. Gödel’s Incompleteness Theorems are standardly taken as showing that Hilbert’s Pro- gram, as intended, fails. Michael Detlefsen maintains that they do not. Detlefsen’s argu- ments are the focus of chapter 3. The argument from the first incompleteness theorem, as presented by Detlefsen, takes the form of a dilemma to the effect that either the infini- tistic theory is incomplete with respect to a certain subclass of real sentences or it is not a conservative extension over the finitistic theory. He contends that Hilbert need not be committed to either of these horns, and, as such the argument from the first incompleteness theorem does no damage to Hilbert’s program. His argument against the second incomplete- ness theorem as refuting Hilbert’s Program, what he calls the stability problem, concerns the particular formalization of the consistency statement shown unprovable by Gödel’s theorem, and endorses what are called Rosser systems.
    [Show full text]
  • Nebraska), Mort Le 21 Octobre 2019 À L’Hôpital Universitaire De Chicago
    In memoriam Michael Detlefsen1 Né le 20 octobre 1948 à Scottsbluff (Nebraska), mort le 21 octobre 2019 à l’hôpital universitaire de Chicago. Michael Detlefsen (Mic, pour la plupart de ceux qui le connaissent) était un ami et un maître pour nombre d'entre nous au sein de la petite communauté des philosophes et historiens des mathématiques en France. Mic racontait volontiers l'histoire incroyable de son arrière-grand-père et de son arrière-arrière-grand oncle qui ont décidé, à l'âge de 9 et 11 ans, de quitter le Danemark et leur famille pour se lancer à la conquête du « rêve américain ». La longue et éprouvante aventure qui s'ensuivit les conduisit, en bateau, en train, puis à pieds, jusqu'à la ville de Dannebrog - Nebraska. Le gouvernement leur a donné des terres à condition qu'ils les cultivent pendant au moins cinq ans. Le rêve est devenu réalité. Et Mic se demande alors : Pourquoi ces enfants ont-ils décidé de quitter leur famille ? La vérité, telle qu'elle lui a été racontée par son frère aîné, est- elle la vérité ? Une seule vérité existe-t-elle jamais, ou la vérité est-elle affectée par notre propre histoire ? Et qu'en est-il de notre propre histoire par rapport à l'histoire ? Au Wheaton College dans la grande banlieue de Chicago, où il a commencé ses études, Mic s'est tourné vers les sciences, les mathématiques, la physique, ainsi que vers la philosophie. Après une thèse sur le second théorème de l'incomplétude de Gödel en 1976 à l'Université Johns Hopkins, il a d'abord été Associate Professor à l'Université du Minnesota-Duluth (1975-1983), puis Associate Professor à l'Université Notre Dame (1984-1989), avant d'être nommé professeur sur la chaire McMahon-Hank à cette même université.
    [Show full text]
  • Outline Winter 2005
    The University of Calgary Department of Philosophy Philosophy 567.03/667.16 PHILOSOPHY OF MATHEMATICS Winter 2005 — Richard Zach Course Outline Instructor: Richard Zach Office: 1254 Social Sciences Office Hours: Tu 3:30-4:30 (tentatively) Phone: 220–3170 email: [email protected] Lectures: W 3–6 SS 105 Course Description The philosophy of mathematics deals with, as its name suggests, philosophical issues that are raised by mathematics. Perhaps the most important difference between mathematics and the other sciences is that mathematics deals with entirely abstract concepts (such as number, set, function). Mathematicians do not conduct experiments to confirm their hypotheses, they find proofs. One important issue then, for instance, is clarifying the role of proofs in mathematics, and with it, the nature of mathematical knowledge. Another issue is the question of the status of mathematical objects—do they inhabit some Platonic realm of mathematical objects, are they constructions of the mind, or are there perhaps, strictly speaking, no mathematical objects, are they fictitious? In the course of discussing these issues, we will first focus on three important schools of thought: logicism, intuitionism, and formalism. These schools are associated with, respectively, Gottlob Frege and Bertrand Russell, Immanuel Kant and L. E. J. Brouwer, and with David Hilbert. In the second half of the course, we will study some of the more recent developments in the philosophy of mathematics (after 1950). This includes the debate about the metaphysics of mathematical objects, in particular, whether there are any (realism) or not (nominalism), and the status of mathematical truths; structuralism in mathematics; mathematical explanation; and empiricist and naturalist ap- proaches to mathematics.
    [Show full text]