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SEP 0 111993 C IMPREDICATIVITY AND TURN OF THE CENTURY FOUNDATIONS OF MATHEMATICS: PRESUPPOSITION IN POINCARE AND RUSSELL by Joseph Romeo William Michael Picard B.A. Honors, Philosophy, University of Calgary, 1986 M.Sc., Philosophy, Massachusetts Institute of Technology, 1990 Submitted to the Department of Linguistics and Philosophy in Partial Fulfillment of the Requirements for the Degree of DOCTOR OF PHILOSOPHY IN PHILOSOPHY at the Massachusetts Institute of Technology July 1993 0 J.R.W.M. Picard, 1993. All rights reserved. The author hereby grants to MIT permission to reproduce and to distribute publicly paper and electronic copies of this thesis document in whole or in part. Signature of Author Department off lnguistics and Philosophy July 8, 1993 Certified by- Professor Richard Caltwright Thesis Supervisor, Department of Linguistics and Philosophy Accepted by Arofessor George Boolos Chairman, Departmental Committee on Graduate Studies MASSACHUSETTS INSTITUTE OFrT•CHNOLOGY 1 SEP 0 111993 -. ·.na uiCO c ABSTRACT The primary purpose of this dissertation is to state a modal account of impredicativity. A (formal or explicit) definition (under a p -ticular interpretation) is impredicative if the object defined )n that interpretation is a value of a bound variable occurring in the detinition. An object may also be called impredicative (with respect to a given language), namely just in case it can be defined in that language but only by means of an impredicative definition. Poincar6 charged in (1906) that impredicative definitions are viciously circular and that impredicative objects dc not exist. Russell agreed with these charges and went on to formulate a logic (ramified type theory) on the basis of a principle which banned imprediwativity (vicious circle principle). The main purpose in this dissertation is to show how certain modal-semantic considerations can be used to mtke sense of the subject of impredicativity, and give an interesting account of what it is for an object to be impredicative. A secondary purpose is to rebut in amore direct manner the charge of vicious circularity. Chapters 1 and 3 are on Russell. In Chapter 1, I examine Russell's early idealist work (1895-1898) in the foundations of geometry. Although Russell increasingly disassociated himself from this work, as indeed from Kant and Hegel, an examination of Russell's idealist foundations can shed light on Russell's later ban on impredicativity. Russell's idealist metaphysical views (especially regarding the continuum) make extensive appeal to modal notions such as essentiality and presupposition (ontologic&l dependency). It was largely his change in attitude toward just these modal notions that lead him to reject idealism and adopt in its place logical atomism and an analytic philosophical methodology. The modal account of impredicativity I give in Chapter 3 will rely chiefly on modal notions Russell rejected when he abandoned his idealist philosophy. Thus the purpose of the first chapter is largely historical: to sketch Russell's views regarding essentiality and ontological presupposition as they were applied in foundations of mathematics. Chapter 2 concerns Poincard. I present Poincar6's views in the foundations of arithmetic and geometry prior to his rejection of impredicativity in 1906. I then try to highlight certain tensions in his thought which the rejection of impredicativity created. These tensions arise from Poincar4's use of Kant's claim that mathematical knowledge is based upon synthetic a priori intuition. The principles Poincar4 held such intuition to justify require, for their proof, the use of impredicative definitions or the postulation of impredicative objects. Poincar6 took his ban on impredicativity to show that explicit proofs of these principles were not possible, and that therefore these principles presupposed a role for synthetic a priori intuition. I argue that this conclusion is misguided, and that Poincar4 does not successfully avoid impredicativity in the foundations. In Chapter 3 I discuss Russell's ramified type theory and argue first that Russell's motivations for introducing this theory can be expressed as certain modal prejudices Russell held. I then extend the modal notions used to express Russell's motivations to define a notion of mutual presupposition or reciprocal ontological dependency, which can be seen to constitute the impredicativity of objects in the context of ramified type theory. One outcome of my modal account of impredicativity is an explanation why Russell thought his restriction to predicative definition could be justified on strictly logical grounds. Russell's later atomistic metaphysics simply ruled out the use of the modal notions required to make sense of impredicative objects. Thesis Supervisor: Professor Richard Cartwright Acknowledgements Many people and institutions have helped me in the long labor of completing this dissertation. I would like above all to thank Professor Richard Cartwright, who read and reread many drafts and tirelessly commented upon them. I would also like to thank Professor Robert Stalnaker, especially for long discussions which have considerably improved the form of my ideas. Special thanks are reserved for Professors James Higginbotham and George Boolos, who were on my thesis committee until near the end. Much of the present work shows their influence, and I am very appreciative of the time and work they have put into helping me over the years. I must also thank MIT, which has provided funding and employment during my graduate studies, not to mention an exciting and remarkably supportive community, being part of which will long be a fond memory for me. Other sources if funding were the Social Sciences and Humanities Research Council of Canada and the Deutscher Akademischer Austaushdienst. I am especially grateful for their generous support. Professor Charles Parsons of Harvard is also deserving of special mention, though his influence on me has been exerted almost entirely through his writings. My debt to him is evident chiefly in the conclusions I reach. It seems proper to mention this debt here, since it is not fairly recorded either in the text or in the references. Finally, I would like to thank Catherine, Hugh, Shirley and ZoltAn for their various efforts in my support. I dedicate my dissertation to Patricia and Keith Harriell, who have been to me family for the last two years. It is still incredible to me that such a chance arrangement as ours should have proven so deep a source of strength and contentment for me. I will miss it, and I will miss them. Table of Contents 0. Introduction . .... ....... ... .. ... 6 0.1 Opening Remarks . 6 0.1.1 Outline . ............. .... 8 0.2 Two Informal Examples .................. 10 0.3 Definitions .. .......... ......... .. 15 Chapter Is Early Russell ..................... 21 1.0 Introduction ...................... 21 1.1 The Philosophy of the Continuum. ............ 22 1.2 Russell's Early Foundational Program: Contrasts with Kant . ......................... 31 1.3 Difficulties with Difference. .............. 37 1.4 Relativity ................... .... 42 1.5 Essential Relativity and the Notion of Logical Subject . 45 1.6 Getting to Monism ................... 49 1.7 From Monism to Analytic Philosophy ........... 52 1.7.1 Russell'" Dialectic Revisited .......... 53 1.7.1.1 Ontological Dependency and Internal Relation ................. 57 1.7.2 Revolution and the Rejection of Idealism . .. 59 Chapter 2s Poincar6: Two Barda Choices . .............. 63 2.0 Introduction ....................... 63 2.1 Poincard's Program: The Primacy of Arithmetic . ..... 64 2.1.1 Pure Arithmetical Intuition and Mathematical Induction ................... 66 2.1.2 Poincard's Kantian Credentials .. .......... 74 2.1.3 The Hard Choice in Arithmetic . ......... 84 2.2 The Changing Place of Continuity ... ............ 94 2.2.1 Convention and the Continuity of Actual Space . 94 2.2.1.1 Arithmetization: Early Optimism ..... 97 2.2.1.2 Arithmetization: Late Doubts ...... 102 2.2.1.3 The Kantian Credentials of Geometric Intuition ................. 108 2.2.2 Poincar6's Construction of Mathematical Continuum .................... 112 2.2.3 The Hard Choice in Analysis . ........... 120 Chapter 3: Modality and Impredicativity .............. 126 3.0 Introduction ...................... 126 3.0.1 Possible Worlds Semantics ............ 127 3.0.2 Meta-mathematical Results concerning Ramified Type Theory . .................. 128 3.1 Necessity and Logical Priority . ............ 130 3.1.1 Serious Difficulty for Russell . ......... 134 3.2 The Vicious Circle Principle and Russellian Bound Variables . ...................... 136 3.3 Pluralities, Existence, and Ontological Dependency . 139 Bibliography of Ia/predicativity ................. 145 Reference .............................. 154 5 O. Introduction 0.1 Opening Remarks The subject of this dissertation is impredicativity. The term "impredicative" can significantly be applied to a variety of things. I will begin by saying how the term "impredicative" is to be used in this work, and then go on to say why impredicativity has been thought to be problematic. The word "impredicative" was applied originally to definitions, which one can think of for present purposes as stated in a formal or symbolic language. A definition or, more precisely, a definition under a particular interpretation, is called impredicative if the object defined on that interpretation is a value of a bound variable occurring in the definition. By extension, the term "impredicative"
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