DOCSLIB.ORG

David Hilbert's Lectures of Arithmetic and Logic on the Foundations 19 71

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
David Hilbert's Lectures of Arithmetic and Logic on the Foundations 19 71 David Hilbert’s Lectures on the Foundations of Arithmetic and Logic 1917–1933 David Hilbert’s Lectures on the Foundations of Mathematics and Physics, 1891–1933 General Editors William Ewald, Michael Hallett, Ulrich Majer and Wilfried Sieg Vo lu me 1 David Hilbert’s Lectures on the Foundations of Geometry, 1891–1902 Vo lu me 2 David Hilbert’s Lectures on the Foundations of Arithmetic and Logic, 1894–1917 Vo lu me 3 David Hilbert’s Lectures on the Foundations of Arithmetic and Logic, 1917–1933 Vo lu me 4 David Hilbert’s Lectures on the Foundations of Physics, 1898–1914 Classical, Relativistic and Statistical Mechanics Vo lu me 5 David Hilbert’s Lectures on the Foundations of Physics, 1915–1927 Relativity, Quantum Theory and Epistemology Vo lu me 6 David Hilbert’s Notebooks and General Foundational Lectures William Ewald Wilfried Sieg Editors Michael Hallett Associate Editor David Hilbert’s Lectures on the Foundations of Arithmetic and Logic 1917–1933 in collaboration with Ulrich Majer and Dirk Schlimm 123 Editors William Ewald Wilfried Sieg Law School Department of Philosophy University of Pennsylvania Carnegie Mellon University Philadelphia, Pennsylvania Pittsburgh, Pennsylvania USA USA Ulrich Majer Dirk Schlimm Philosophisches Seminar Department of Philosophy Universität Göttingen McGill University Göttingen Montré, al Québec Germany Canada Michael Hallett Department of Philosophy McGill University Montré, al Québec Canada ISBN 978-3-540 - 20578 - 4 ISBN 978-3-540-69444-1 (eBook) DOI 10.1007/978-3-540-69444-1 Springer Heidelberg New York Dordrecht London Library of Congress Control Number: 2013938947 © Springer-Verlag Berlin Heidelberg 2013 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication 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. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com) This Volume is dedicated to the memory of David Hilbert, on the occasion of his 151st birthday. David Hilbert. Courtesy of the Voit Collection in the Manuscript Division of the Niedersächsische Staats- und Universitätsbibliothek, Göttingen. Preface The present Volume is the third in a series of six presenting a selection from Hilbert’s previously unpublished lecture notes on the foundations of mathe- matics and physics during the period 1890 to 1933. The Hilbert Nachlaß contains approximately eighty lecture notebooks, covering all aspects of his mathematical activity, and spanning almost the entirety of his teaching career; some are in his own hand, others were carefully worked out by his assistants as official protocols of his lectures. Roughly one quarter of these notebooks deal with foundational subjects. Hilbert’s lecture courses represent an enormous fund of learning and invention, and embrace almost every subject common in the mathematical sciences of his time. The notes therefore provide a re- markable record, sometimes almost from day to day, of the development of his ideas, and show, in addition, his engagement with the work of other scientific figures of the first rank. The present Volume treats Hilbert’s lectures on logic, arithmetic and proof theory from the autumn of 1917 on. During this period Hilbert actively resumed his research investigations into the foundations of mathematics, and undertook his intensive collaboration with Paul Bernays, who wrote up many of the lecture notes reproduced in this Volume. The period covered here sees the emergence of modern mathematical logic; the explicit posing of questions of completeness, decidability, and consistency for logical systems; the investigations of the relative strengths of various logical calculi; the formulation of the decision problem; the birth of proof theory and the energetic pursuit by Hilbert and by Bernays of technical work on the Hilbert consistency programme. These developments can here be followed in greater detail than has been possible from the published record alone: one sees the variety of approaches, the shifts in strategy, and obtains a fuller picture of the motivation for Hilbert’s investigations, as well as of his intellectual rela- tionship to the work of such contemporaries as Russell, Whitehead, Brouwer, and Weyl. The widespread picture of Hilbert as a naïve ‘formalist’ disappears, to be replaced by a much more subtle and nuanced record of the development of Hilbert’s views on the philosophy of mathematics. The structure of this Edition, the nature, location, and condition of the Hilbert lecture notes, their provenance, and what we have been able to re- construct of their history, are all described in the general ‘Introduction to the Edition’, which is to be found at the beginning of Volume 1 (Hallett and Majer 2004 ). That Introduction also explains in detail the criteria for the selection of the texts, the way in which they were edited, and general matters of textual policy. Those matters are uniform for the entire Edition, and we have not repeated the full account here. We do, however, include a description of the textual policies in the section ‘The Editing and Reproduction of the Texts’; this section is intended to provide all the basic information necessary to a reading of the texts. This Volume also reprints from Volume 1 (in slightly revised and expanded form) the list of Hilbert’s lecture courses; see pp. 991ff. That these lectures are finally being published is the result of the efforts, over two decades, of many individuals and institutions. The series as a whole IX X Preface is under the supervision of four General Editors, William Ewald, Michael Hal- lett, Ulrich Majer and Wilfried Sieg, who bear the collective responsibility for editorial policy. For each individual volume, Volume Editors were designated to produce the final selection of texts and to write the scholarly apparatus; this work was carried out in consultation with the General Editors. The des- ignated Editors for this Volume were Ewald and Sieg, in collaboration with Majer and Dirk Schlimm, with Hallett as Associate Editor. As before, the General Editors wish to express their thanks to the Deutsche Forschungsgemeinschaft (DFG) for its generous financial support from 1993 to 2003. To edit even the mere fragment of the voluminous Hilbert Nachlaß that appears in these six volumes required a considerable institutional apparatus located in proximity to the archives in Göttingen. Without the assistance of the DFG, which enabled us to establish a permanent staff in Göttingen, the present Edition could never have been realized. Ulrich Majer, the General Editor who was constantly ‘vor Ort’, supervised the permanent staff and thus had the task of dealing with all the technical problems that an edition of this sort must inevitably face. We again acknowledge the indispensable schol- arly, editorial and technical contributions to the Edition as a whole of Ralf Haubrich, Albert Krayer and Tilman Sauer, all at one time full-time members of the permanent staff. We furthermore thank the Institut für Wissenschaftsgeschichte at the Uni- versity of Göttingen (in particular Lorraine Daston, its former director) for giving the project its first physical home and for recognizing its significance. We are also grateful to the Philosophisches Seminar at the University of Göt- tingen for space and support. Numerous other institutions and individuals provided significant support for the Edition. In Göttingen, from the first, formative stages of the project, we received encouragement and advice from the late Martin Kneser, Samuel Patterson, Günther Patzig and Helmut Rohlfing. The Mathematisches Insti- tut and the Niedersächsische Staats- und Universitätsbibliothek in Göttingen (SUB), the holders of the original Hilbert documents, granted the necessary permission for publication. At an early stage, Martin Kneser was kind enough to place at our disposal his father Hellmuth’s Mitschriften of the lectures ‘Grundlegung der Mathematik’ and ‘Grundlagen der Arithmetik’ given by Hilbert and Bernays in the Winter Semesters 1921/22 and 1922/23 respec- tively, as well as notes for some lectures given by Hilbert in February 1924; Martin Kneser also granted permission for their publication. The manuscripts now form part of an archive of Hellmuth Kneser’s papers in the Handschriften- abteilung of the SUB (signature Cod. Ms. H. Kneser), which has also kindly granted permission for their publication. Warm thanks are also due to Frau Brigitte Peterhans of Chicago for allowing us access to a polished Nachschrift by Walter Peterhans of Hilbert’s 1922/23 lectures. (The Nachschrift is described in some detail below; see pp.
Load more

Recommended publications
  • Weyl's Predicative Classical Mathematics As a Logic-Enriched
    Weyl’s Predicative Classical Mathematics as a Logic-Enriched Type Theory? Robin Adams and Zhaohui Luo Dept of Computer Science, Royal Holloway, Univ of London {robin,zhaohui}@cs.rhul.ac.uk Abstract. In Das Kontinuum, Weyl showed how a large body of clas- sical mathematics could be developed on a purely predicative founda- tion. We present a logic-enriched type theory that corresponds to Weyl’s foundational system. A large part of the mathematics in Weyl’s book — including Weyl’s definition of the cardinality of a set and several re- sults from real analysis — has been formalised, using the proof assistant Plastic that implements a logical framework. This case study shows how type theory can be used to represent a non-constructive foundation for mathematics. Key words: logic-enriched type theory, predicativism, formalisation 1 Introduction Type theories have proven themselves remarkably successful in the formalisation of mathematical proofs. There are several features of type theory that are of particular benefit in such formalisations, including the fact that each object carries a type which gives information about that object, and the fact that the type theory itself has an inbuilt notion of computation. These applications of type theory have proven particularly successful for the formalisation of intuitionistic, or constructive, proofs. The correspondence between terms of a type theory and intuitionistic proofs has been well studied. The degree to which type theory can be used for the formalisation of other notions of proof has been investigated to a much lesser degree. There have been several formalisations of classical proofs by adapting a proof checker intended for intuitionistic mathematics, say by adding the principle of excluded middle as an axiom (such as [Gon05]).
    [Show full text]
  • David Hilbert's Contributions to Logical Theory
    David Hilbert’s contributions to logical theory CURTIS FRANKS 1. A mathematician’s cast of mind Charles Sanders Peirce famously declared that “no two things could be more directly opposite than the cast of mind of the logician and that of the mathematician” (Peirce 1976, p. 595), and one who would take his word for it could only ascribe to David Hilbert that mindset opposed to the thought of his contemporaries, Frege, Gentzen, Godel,¨ Heyting, Łukasiewicz, and Skolem. They were the logicians par excellence of a generation that saw Hilbert seated at the helm of German mathematical research. Of Hilbert’s numerous scientific achievements, not one properly belongs to the domain of logic. In fact several of the great logical discoveries of the 20th century revealed deep errors in Hilbert’s intuitions—exemplifying, one might say, Peirce’s bald generalization. Yet to Peirce’s addendum that “[i]t is almost inconceivable that a man should be great in both ways” (Ibid.), Hilbert stands as perhaps history’s principle counter-example. It is to Hilbert that we owe the fundamental ideas and goals (indeed, even the name) of proof theory, the first systematic development and application of the methods (even if the field would be named only half a century later) of model theory, and the statement of the first definitive problem in recursion theory. And he did more. Beyond giving shape to the various sub-disciplines of modern logic, Hilbert brought them each under the umbrella of mainstream mathematical activity, so that for the first time in history teams of researchers shared a common sense of logic’s open problems, key concepts, and central techniques.
    [Show full text]
  • Einstein and Hilbert: the Creation of General Relativity
    EINSTEIN AND HILBERT: THE CREATION OF GENERAL RELATIVITY ∗ Ivan T. Todorov Institut f¨ur Theoretische Physik, Universit¨at G¨ottingen, Friedrich-Hund-Platz 1 D-37077 G¨ottingen, Germany; e-mail: [email protected] and Institute for Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences Tsarigradsko Chaussee 72, BG-1784 Sofia, Bulgaria;∗∗e-mail: [email protected] ABSTRACT It took eight years after Einstein announced the basic physical ideas behind the relativistic gravity theory before the proper mathematical formulation of general relativity was mastered. The efforts of the greatest physicist and of the greatest mathematician of the time were involved and reached a breathtaking concentration during the last month of the work. Recent controversy, raised by a much publicized 1997 reading of Hilbert’s proof- sheets of his article of November 1915, is also discussed. arXiv:physics/0504179v1 [physics.hist-ph] 25 Apr 2005 ∗ Expanded version of a Colloquium lecture held at the International Centre for Theoretical Physics, Trieste, 9 December 1992 and (updated) at the International University Bremen, 15 March 2005. ∗∗ Permanent address. Introduction Since the supergravity fashion and especially since the birth of superstrings a new science emerged which may be called “high energy mathematical physics”. One fad changes the other each going further away from accessible experiments and into mathe- matical models, ending up, at best, with the solution of an interesting problem in pure mathematics. The realization of the grand original design seems to be, decades later, nowhere in sight. For quite some time, though, the temptation for mathematical physi- cists (including leading mathematicians) was hard to resist.
    [Show full text]
  • Arxiv:1504.04798V1 [Math.LO] 19 Apr 2015 of Principia Squarely in an Empiricist Framework
    HEINRICH BEHMANN'S 1921 LECTURE ON THE DECISION PROBLEM AND THE ALGEBRA OF LOGIC PAOLO MANCOSU AND RICHARD ZACH Abstract. Heinrich Behmann (1891{1970) obtained his Habilitation under David Hilbert in G¨ottingenin 1921 with a thesis on the decision problem. In his thesis, he solved|independently of L¨owenheim and Skolem's earlier work|the decision prob- lem for monadic second-order logic in a framework that combined elements of the algebra of logic and the newer axiomatic approach to logic then being developed in G¨ottingen. In a talk given in 1921, he outlined this solution, but also presented important programmatic remarks on the significance of the decision problem and of decision procedures more generally. The text of this talk as well as a partial English translation are included. x1. Behmann's Career. Heinrich Behmann was born January 10, 1891, in Bremen. In 1909 he enrolled at the University of T¨ubingen. There he studied mathematics and physics for two semesters and then moved to Leipzig, where he continued his studies for three semesters. In 1911 he moved to G¨ottingen,at that time the most important center of mathematical activity in Germany. He volunteered for military duty in World War I, was severely wounded in 1915, and returned to G¨ottingenin 1916. In 1918, he obtained his doctorate with a thesis titled The Antin- omy of Transfinite Numbers and its Resolution by the Theory of Russell and Whitehead [Die Antinomie der transfiniten Zahl und ihre Aufl¨osung durch die Theorie von Russell und Whitehead] under the supervision of David Hilbert [Behmann 1918].
    [Show full text]
  • Starting the Dismantling of Classical Mathematics
    Australasian Journal of Logic Starting the Dismantling of Classical Mathematics Ross T. Brady La Trobe University Melbourne, Australia [email protected] Dedicated to Richard Routley/Sylvan, on the occasion of the 20th anniversary of his untimely death. 1 Introduction Richard Sylvan (n´eRoutley) has been the greatest influence on my career in logic. We met at the University of New England in 1966, when I was a Master's student and he was one of my lecturers in the M.A. course in Formal Logic. He was an inspirational leader, who thought his own thoughts and was not afraid to speak his mind. I hold him in the highest regard. He was very critical of the standard Anglo-American way of doing logic, the so-called classical logic, which can be seen in everything he wrote. One of his many critical comments was: “G¨odel's(First) Theorem would not be provable using a decent logic". This contribution, written to honour him and his works, will examine this point among some others. Hilbert referred to non-constructive set theory based on classical logic as \Cantor's paradise". In this historical setting, the constructive logic and mathematics concerned was that of intuitionism. (The Preface of Mendelson [2010] refers to this.) We wish to start the process of dismantling this classi- cal paradise, and more generally classical mathematics. Our starting point will be the various diagonal-style arguments, where we examine whether the Law of Excluded Middle (LEM) is implicitly used in carrying them out. This will include the proof of G¨odel'sFirst Theorem, and also the proof of the undecidability of Turing's Halting Problem.
    [Show full text]
  • Henkin's Method and the Completeness Theorem
    Henkin's Method and the Completeness Theorem Guram Bezhanishvili∗ 1 Introduction Let L be a first-order logic. For a sentence ' of L, we will use the standard notation \` '" for ' is provable in L (that is, ' is derivable from the axioms of L by the use of the inference rules of L); and \j= '" for ' is valid (that is, ' is satisfied in every interpretation of L). The soundness theorem for L states that if ` ', then j= '; and the completeness theorem for L states that if j= ', then ` '. Put together, the soundness and completeness theorems yield the correctness theorem for L: a sentence is derivable in L iff it is valid. Thus, they establish a crucial feature of L; namely, that syntax and semantics of L go hand-in-hand: every theorem of L is a logical law (can not be refuted in any interpretation of L), and every logical law can actually be derived in L. In fact, a stronger version of this result is also true. For each first-order theory T and a sentence ' (in the language of T ), we have that T ` ' iff T j= '. Thus, each first-order theory T (pick your favorite one!) is sound and complete in the sense that everything that we can derive from T is true in all models of T , and everything that is true in all models of T is in fact derivable from T . This is a very strong result indeed. One possible reading of it is that the first-order formalization of a given mathematical theory is adequate in the sense that every true statement about T that can be formalized in the first-order language of T is derivable from the axioms of T .
    [Show full text]
  • A Quasi-Classical Logic for Classical Mathematics
    Theses Honors College 11-2014 A Quasi-Classical Logic for Classical Mathematics Henry Nikogosyan University of Nevada, Las Vegas Follow this and additional works at: https://digitalscholarship.unlv.edu/honors_theses Part of the Logic and Foundations Commons Repository Citation Nikogosyan, Henry, "A Quasi-Classical Logic for Classical Mathematics" (2014). Theses. 21. https://digitalscholarship.unlv.edu/honors_theses/21 This Honors Thesis is protected by copyright and/or related rights. It has been brought to you by Digital Scholarship@UNLV with permission from the rights-holder(s). You are free to use this Honors Thesis in any way that is permitted by the copyright and related rights legislation that applies to your use. For other uses you need to obtain permission from the rights-holder(s) directly, unless additional rights are indicated by a Creative Commons license in the record and/or on the work itself. This Honors Thesis has been accepted for inclusion in Theses by an authorized administrator of Digital Scholarship@UNLV. For more information, please contact [email protected]. A QUASI-CLASSICAL LOGIC FOR CLASSICAL MATHEMATICS By Henry Nikogosyan Honors Thesis submitted in partial fulfillment for the designation of Departmental Honors Department of Philosophy Ian Dove James Woodbridge Marta Meana College of Liberal Arts University of Nevada, Las Vegas November, 2014 ABSTRACT Classical mathematics is a form of mathematics that has a large range of application; however, its application has boundaries. In this paper, I show that Sperber and Wilson’s concept of relevance can demarcate classical mathematics’ range of applicability by demarcating classical logic’s range of applicability.
    [Show full text]
  • Old and New Results in the Foundations of Elementary Plane Euclidean and Non-Euclidean Geometries Marvin Jay Greenberg
    Old and New Results in the Foundations of Elementary Plane Euclidean and Non-Euclidean Geometries Marvin Jay Greenberg By “elementary” plane geometry I mean the geometry of lines and circles—straight- edge and compass constructions—in both Euclidean and non-Euclidean planes. An axiomatic description of it is in Sections 1.1, 1.2, and 1.6. This survey highlights some foundational history and some interesting recent discoveries that deserve to be better known, such as the hierarchies of axiom systems, Aristotle’s axiom as a “missing link,” Bolyai’s discovery—proved and generalized by William Jagy—of the relationship of “circle-squaring” in a hyperbolic plane to Fermat primes, the undecidability, incom- pleteness, and consistency of elementary Euclidean geometry, and much more. A main theme is what Hilbert called “the purity of methods of proof,” exemplified in his and his early twentieth century successors’ works on foundations of geometry. 1. AXIOMATIC DEVELOPMENT 1.0. Viewpoint. Euclid’s Elements was the first axiomatic presentation of mathemat- ics, based on his five postulates plus his “common notions.” It wasn’t until the end of the nineteenth century that rigorous revisions of Euclid’s axiomatics were presented, filling in the many gaps in his definitions and proofs. The revision with the great- est influence was that by David Hilbert starting in 1899, which will be discussed below. Hilbert not only made Euclid’s geometry rigorous, he investigated the min- imal assumptions needed to prove Euclid’s results, he showed the independence of some of his own axioms from the others, he presented unusual models to show certain statements unprovable from others, and in subsequent editions he explored in his ap- pendices many other interesting topics, including his foundation for plane hyperbolic geometry without bringing in real numbers.
    [Show full text]
  • Mathematical Languages Shape Our Understanding of Time in Physics Physics Is Formulated in Terms of Timeless, Axiomatic Mathematics
    comment Corrected: Publisher Correction Mathematical languages shape our understanding of time in physics Physics is formulated in terms of timeless, axiomatic mathematics. A formulation on the basis of intuitionist mathematics, built on time-evolving processes, would ofer a perspective that is closer to our experience of physical reality. Nicolas Gisin n 1922 Albert Einstein, the physicist, met in Paris Henri Bergson, the philosopher. IThe two giants debated publicly about time and Einstein concluded with his famous statement: “There is no such thing as the time of the philosopher”. Around the same time, and equally dramatically, mathematicians were debating how to describe the continuum (Fig. 1). The famous German mathematician David Hilbert was promoting formalized mathematics, in which every real number with its infinite series of digits is a completed individual object. On the other side the Dutch mathematician, Luitzen Egbertus Jan Brouwer, was defending the view that each point on the line should be represented as a never-ending process that develops in time, a view known as intuitionistic mathematics (Box 1). Although Brouwer was backed-up by a few well-known figures, like Hermann Weyl 1 and Kurt Gödel2, Hilbert and his supporters clearly won that second debate. Hence, time was expulsed from mathematics and mathematical objects Fig. 1 | Debating mathematicians. David Hilbert (left), supporter of axiomatic mathematics. L. E. J. came to be seen as existing in some Brouwer (right), proposer of intuitionist mathematics. Credit: Left: INTERFOTO / Alamy Stock Photo; idealized Platonistic world. right: reprinted with permission from ref. 18, Springer These two debates had a huge impact on physics.
    [Show full text]
  • Foundations of Geometry
    California State University, San Bernardino CSUSB ScholarWorks Theses Digitization Project John M. Pfau Library 2008 Foundations of geometry Lawrence Michael Clarke Follow this and additional works at: https://scholarworks.lib.csusb.edu/etd-project Part of the Geometry and Topology Commons Recommended Citation Clarke, Lawrence Michael, "Foundations of geometry" (2008). Theses Digitization Project. 3419. https://scholarworks.lib.csusb.edu/etd-project/3419 This Thesis is brought to you for free and open access by the John M. Pfau Library at CSUSB ScholarWorks. It has been accepted for inclusion in Theses Digitization Project by an authorized administrator of CSUSB ScholarWorks. For more information, please contact [email protected]. Foundations of Geometry A Thesis Presented to the Faculty of California State University, San Bernardino In Partial Fulfillment of the Requirements for the Degree Master of Arts in Mathematics by Lawrence Michael Clarke March 2008 Foundations of Geometry A Thesis Presented to the Faculty of California State University, San Bernardino by Lawrence Michael Clarke March 2008 Approved by: 3)?/08 Murran, Committee Chair Date _ ommi^yee Member Susan Addington, Committee Member 1 Peter Williams, Chair, Department of Mathematics Department of Mathematics iii Abstract In this paper, a brief introduction to the history, and development, of Euclidean Geometry will be followed by a biographical background of David Hilbert, highlighting significant events in his educational and professional life. In an attempt to add rigor to the presentation of Geometry, Hilbert defined concepts and presented five groups of axioms that were mutually independent yet compatible, including introducing axioms of congruence in order to present displacement.
    [Show full text]
  • Mechanism, Mentalism, and Metamathematics Synthese Library
    MECHANISM, MENTALISM, AND METAMATHEMATICS SYNTHESE LIBRARY STUDIES IN EPISTEMOLOGY, LOGIC, METHODOLOGY, AND PHILOSOPHY OF SCIENCE Managing Editor: JAAKKO HINTIKKA, Florida State University Editors: ROBER T S. COHEN, Boston University DONALD DAVIDSON, University o/Chicago GABRIEL NUCHELMANS, University 0/ Leyden WESLEY C. SALMON, University 0/ Arizona VOLUME 137 JUDSON CHAMBERS WEBB Boston University. Dept. 0/ Philosophy. Boston. Mass .• U.S.A. MECHANISM, MENT ALISM, AND MET AMA THEMA TICS An Essay on Finitism i Springer-Science+Business Media, B.V. Library of Congress Cataloging in Publication Data Webb, Judson Chambers, 1936- CII:J Mechanism, mentalism, and metamathematics. (Synthese library; v. 137) Bibliography: p. Includes indexes. 1. Metamathematics. I. Title. QA9.8.w4 510: 1 79-27819 ISBN 978-90-481-8357-9 ISBN 978-94-015-7653-6 (eBook) DOl 10.1007/978-94-015-7653-6 All Rights Reserved Copyright © 1980 by Springer Science+Business Media Dordrecht Originally published by D. Reidel Publishing Company, Dordrecht, Holland in 1980. Softcover reprint of the hardcover 1st edition 1980 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any informational storage and retrieval system, without written permission from the copyright owner TABLE OF CONTENTS PREFACE vii INTRODUCTION ix CHAPTER I / MECHANISM: SOME HISTORICAL NOTES I. Machines and Demons 2. Machines and Men 17 3. Machines, Arithmetic, and Logic 22 CHAPTER II / MIND, NUMBER, AND THE INFINITE 33 I. The Obligations of Infinity 33 2. Mind and Philosophy of Number 40 3. Dedekind's Theory of Arithmetic 46 4.
    [Show full text]
  • Georg Cantor English Version
    GEORG CANTOR (March 3, 1845 – January 6, 1918) by HEINZ KLAUS STRICK, Germany There is hardly another mathematician whose reputation among his contemporary colleagues reflected such a wide disparity of opinion: for some, GEORG FERDINAND LUDWIG PHILIPP CANTOR was a corruptor of youth (KRONECKER), while for others, he was an exceptionally gifted mathematical researcher (DAVID HILBERT 1925: Let no one be allowed to drive us from the paradise that CANTOR created for us.) GEORG CANTOR’s father was a successful merchant and stockbroker in St. Petersburg, where he lived with his family, which included six children, in the large German colony until he was forced by ill health to move to the milder climate of Germany. In Russia, GEORG was instructed by private tutors. He then attended secondary schools in Wiesbaden and Darmstadt. After he had completed his schooling with excellent grades, particularly in mathematics, his father acceded to his son’s request to pursue mathematical studies in Zurich. GEORG CANTOR could equally well have chosen a career as a violinist, in which case he would have continued the tradition of his two grandmothers, both of whom were active as respected professional musicians in St. Petersburg. When in 1863 his father died, CANTOR transferred to Berlin, where he attended lectures by KARL WEIERSTRASS, ERNST EDUARD KUMMER, and LEOPOLD KRONECKER. On completing his doctorate in 1867 with a dissertation on a topic in number theory, CANTOR did not obtain a permanent academic position. He taught for a while at a girls’ school and at an institution for training teachers, all the while working on his habilitation thesis, which led to a teaching position at the university in Halle.
    [Show full text]