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Hilbert's Finitism: Historical, Philosophical, And Hilbert’s Finitism: Historical, Philosophical, and Metamathematical Perspectives by Richard Zach Dipl.-Ing. (University of Technology, Vienna, Austria) 1993 M.A. (University of California, Berkeley) 1997 C.Phil. (University of California, Berkeley) 1997 A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Logic and the Methodology of Science in the GRADUATE DIVISION of the UNIVERSITY OF CALIFORNIA, BERKELEY Committee in charge: Professor Paolo Mancosu, Cochair Professor Jack H. Silver, Cochair Professor Barry Stroud Spring 2001 The dissertation of Richard Zach is approved: Cochair Date Cochair Date Date University of California, Berkeley Spring 2001 Hilbert’s Finitism: Historical, Philosophical, and Metamathematical Perspectives Copyright 2001 by Richard Zach 1 Abstract Hilbert’s Finitism: Historical, Philosophical, and Metamathematical Perspectives by Richard Zach Doctor of Philosophy in Logic and the Methodology of Science University of California, Berkeley Professor Paolo Mancosu, Cochair Professor Jack H. Silver, Cochair In the 1920s, David Hilbert proposed a research program with the aim of providing mathe- matics with a secure foundation. This was to be accomplished by first formalizing logic and mathematics in their entirety, and then showing—using only so-called finitistic principles— that these formalizations are free of contradictions. In the area of logic, the Hilbert school accomplished major advances both in introducing new systems of logic, and in developing central metalogical notions, such as completeness and decidability. The analysis of unpublished material presented in Chapter 2 shows that a completeness proof for propositional logic was found by Hilbert and his assistant Paul Bernays already in 1917–18, and that Bernays’s contribution was much greater than is commonly acknowledged. Aside from logic, the main technical contribution of Hilbert’s Program are the development of formal mathematical theories and proof-theoretical inves- tigations thereof, in particular, consistency proofs. In this respect Wilhelm Ackermann’s 1924 dissertation is a milestone both in the development of the Program and in proof theory in general. Ackermann gives a consistency proof for a second-order version of primitive recursive arithmetic which, surprisingly, explicitly uses a finitistic version of transfinite in- ω duction up to ωω . He also gave a faulty consistency proof for a system of second-order arithmetic based on Hilbert’s ε-substitution method. Detailed analyses of both proofs in 2 Chapter 3 shed light on the development of finitism and proof theory in the 1920s as prac- ticed in Hilbert’s school. In a series of papers, Charles Parsons has attempted to map out a notion of mathematical intuition which he also brings to bear on Hilbert’s finitism. According to him, mathemati- cal intuition fails to be able to underwrite the kind of intuitive knowledge Hilbert thought was attainable by the finitist. It is argued in Chapter 4 that the extent of finitistic knowl- edge which intuition can provide is broader than Parsons supposes. According to another influential analysis of finitism due to W. W. Tait, finitist reasoning coincides with primitive recursive reasoning. The acceptance of non-primitive recursive methods in Ackermann’s dissertation presented in Chapter 3, together with additional textual evidence presented in Chapter 4, shows that this identification is untenable as far as Hilbert’s conception of finitism is concerned. Tait’s conception, however, differs from Hilbert’s in important re- spects, yet it is also open to criticisms leading to the conclusion that finitism encompasses more than just primitive recursive reasoning. i Es ist gar nicht richtig, daß der Forscher der Wahrheit nachstellt, sie stellt ihm nach. Er erleidet sie. — Robert Musil ii Contents 1 Introduction 1 1.1 David Hilbert and the Foundations of Mathematics . 1 1.2 Hilbert, Bernays, and Logic . 4 1.3 Hilbert’s Proof-Theoretical Program . 7 1.4 Finitism . 11 2 Completeness before Post: Bernays, Hilbert, and the Development of Proposi- tional Logic 15 2.1 Introduction . 15 2.2 Semantics, Normal Forms, Completeness . 17 2.2.1 Prehistory: Hilbert’s Lectures on Logical Principles of Mathematical Thought 1905 . 17 2.2.2 The Structure of Prinzipien der Mathematik . 21 2.2.3 The Propositional Calculus . 23 2.2.4 Consistency and Completeness . 25 2.2.5 The Contribution of Bernays’s Habilitationsschrift . 27 2.2.6 A Brief Comparison with Post’s thesis . 29 2.3 Hilbert or Bernays? . 31 2.4 Dependence and Independence . 35 2.5 Axioms and Rules . 36 2.6 Lasting Influences . 41 3 The Practice of Finitism: Epsilon Calculus and Consistency Proofs in Hilbert’s Program 56 3.1 Introduction . 56 3.2 Early Consistency Proofs . 58 3.2.1 The Propositional Calculus and the Calculus of Elementary Computation . 61 3.2.2 Elementary Number Theory with Recursion and Induction Rule . 66 3.2.3 The ε-Calculus and the Axiomatization of Mathematics . 68 3.3 Ackermann’s Dissertation . 73 iii 3.3.1 Second-order Primitive Recursive Arithmetic . 75 3.3.2 The Consistency Proof for the System without Epsilons . 77 3.3.3 Ordinals, Transfinite Induction, and Finitism . 83 3.3.4 The ε-Substitution Method . 85 3.3.5 Assessment and Complications . 91 3.4 Conclusion . 94 4 Finitism and Mathematical Knowledge 109 4.1 Introduction . 109 4.2 The Significance of Finitism . 111 4.3 Finitism in Hilbert . 114 4.3.1 Numbers and Numerals . 114 4.3.2 Statements . 121 4.3.3 Finitistic Functions . 125 4.3.4 Finitism and PRA . 128 4.4 The Status of Finitism . 132 4.5 Parsons’ Criticism of Finitism as Intuitive Knowledge . 134 4.6 Tait’s Analysis of Finitism . 141 4.7 Conclusion . 147 Bibliography 155 iv Acknowledgements The first debt of gratitude I must acknowledge is to my parents Wilhelm and Waltraud and my sister Ruth, without whose support and encouragement I probably would not chosen an academic career, let alone come this far. My teachers at Vienna and Berkeley have shaped my thinking and helped me improve my work more than anyone else. I am indebted in particular to my teachers at Berkeley, foremost among them Charles Chihara, Christos Papadimitriou, Hans Sluga, Barry Stroud, and Hugh Woodin. But above all I must thank my advisors Paolo Mancosu and Jack Silver, without whose support, criticism, praise, and insight this thesis would not be what it is now. Among my teachers at the University of Technology, Vienna, I have to mention Matthias Baaz, Chris Fermuller,¨ Georg Gottlob and Alex Leitsch, who have taught me logic, and that philosophical and methodological reflection are as important as technical sophistication. I am also grateful to Solomon Feferman and Grisha Mints, current colleagues of sorts at Stanford, who have played a not insignificant part in my graduate studies, and my student colleagues at Berkeley and Vienna, specifically Eddie Cushman, Johannes Hafner, Ben Hansen, Christine Heitsch, Georg Moser, Chris Pincock, and Helmut Veith. Special thanks goes to Catalina Cordoba´ Boniffaccini, my mom away from home. Chapter 2 appeared as “Completeness before Post: Bernays, Hilbert, and the develop- ment of propositional logic,” Bulletin of Symbolic Logic 5 (1999) 331–366, and is c 1999 by the Association of Symbolic Logic. It is reprinted here with the kind permission of the ASL. Versions of this paper were presented at a symposium on The Development of Modern Logic in Helsinki, June 1998, at the Townsend Center Working Group on History and Phi- losophy of Logic and Mathematics at the University of California, Berkeley, at the Stanford Logic Lunch, and at the 1999 Spring Meeting of the ASL. I would like to thank in par- ticular Jeremy Avigad, Martin Davis, Solomon Feferman, Georg Kreisel, Grigori Mints, Volker Peckhaus, Chris Pincock, Wilfried Sieg, and the referee for the BSL for their help- ful comments. My thanks also goes to the Wissenschaftshistorische Sammlungen of the ETH Zurich¨ Library, the Handschriftenabteilung of the Niedersachsische¨ Staats- und Uni- versitatsbibliothek,¨ the library of the Institute of Mathematics at the University of Gottin-¨ v gen, and to the curators of the Heinrich Behmann Papers in Erlangen, Christian Thiel and Volker Peckhaus, for their help in my research and permission to quote from unpublished material. Parts of Chapter 3 were presented to the Townsend Center Working Group in History and Philosophy of Logic and Mathematics at the University of Caliornia, Berkeley, Decem- ber 1999, at a workshop on Hilbert at the Institut d’Histoire des Sciences et Techniques of the Universite´ Paris I (partially funded by the Townsend Center through an Interna- tional Working Group grant), May 2000, at the International Symposium on the History of Logic at the University of Helsinki, June 2000, at the Department of Logic and Philosophy of Science of the University of California, Irvine, and at the Philosophy of Mathematics Workshop at UCLA in 2001. I am indebted to Hans Richard Ackermann for providing me with copies of Wilhelm Ackermann’s correspondence, and to Wilfried Sieg for provid- ing me with Kneser’s notes to the 1921–22 and 1922–23 lectures (Hilbert 1922a, Hilbert and Bernays 1923a). I would also like to thank in particular Aldo Antonelli, Solomon Feferman, Ivor Grattan-Guinness, David Kaplan, Penelope Maddy, Tony Martin, Yiannis Moschovakis, and Grigori Mints for valuable suggestions. My understanding of the ε- substitution method in general and Ackermann’s proof in particular has benefitted greatly from conversations with and writings of Grigori Mints, Georg Moser, and W. W. Tait. Any errors in my presentation are, of course, mine. Parts of Chapter 4 appeared as “Numbers and functions in Hilbert’s finitism,” Taiwanese Journal for Philosophy and History of Science 10 (1998) 33–60, and are c 1998 by the Taiwanese Journal for Philosophy and History of Science. They are reprinted here with the kind permission of the editors of the Taiwanese Journal for Philosophy and History of Science.
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