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Information to Users INFORMATION TO USERS This manuscript has been reproduced from the microfihn master. UMI films the text directly from the original or copy submitted. Thus, some thesis and dissertation copies are in typewriter face, while others may be from any type of computer printer. The quality of this reproduction is dependent upon the quality of the copy submitted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleedthrough, substandard margins, and improper alignment can adversely afreet reproduction. In the unlikely event that the author did not send UMI a complete manuscript and there are missing pages, these will be noted. Also, if unauthorized copyright material had to be removed, a note will indicate the deletion. Oversize materials (e.g., maps, drawings, charts) are reproduced by sectioning the original, beginning at the upper left-hand comer and continuing from left to right in equal sections with small overlaps. Each original is also photographed in one exposure and is included in reduced form at the back of the book. Photographs included in the original manuscript have been reproduced xerographically in this copy. Kgher quality 6” x 9” black and white photographic prints are available for any photographs or illustrations appearing in this copy for an additional charge. Contact UMI directly to order. UMI A Bell & Howell Information Company 300 North Zed) Road, Atm Arbor MI 48106-1346 USA 313/761-4700 800/521-0600 CAN WE INTEND AN INTERPRETATION? DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University by Pierluigi Miraglia, MA. ***** The Ohio State University 1996 Dissertation Committee: Approved by Professor Stewart Shapiro, Adviser Professor Neü Tennant Adviser Professor Mark Wilson Department of Philosophy mix Number: 9710625 UMI Microform 9710625 Copyright 1997, by UMI Company. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. UMI 300 North Zeeb Road Ann Arbor, MI 48103 u ABSTRACT Some mathematical theories are thought to have intended interpretations: they are thought to be about a reasonably well defined subject matter. For such theories, an intended interpretation is also presumed to encompass the intuitive concepts that the theory represents formally. Thus intended interpretations play a semantic, an ontological and an epistemological role: they give the preferred reference of the terms of the theory; they embody a conception of the objects described by the theory; and they are a source of evidence for judging the adequacy of proposed axiomatizations. Two series of questions naturally arise: are there intended interpretations? And if so, how are they determined? In the context of set theory, these questions summarize the “Skolem problem,” for it is most prominently displayed in the considerations relative to the Skolem paradox. According to some, the challenge posed by the Skolem problem is met only by a strong variety of platonism about mathematical objects. It is also thought that one must embrace this variety of platonism in order to account for the epistemological role of intended interpretations. I argue against this conception, with particular regard to the case of set theory. The thesis is that skeptical challenges such as those mentioned can be rejected on the basis of a moderately realist view according to which the intended lU interpretation of this theory is largely dependent upon the logical and inferential practices which the theory codifies. The centerpiece of the argument is the role played by the axiom of Choice in the formation of the notion of set which is at the basis of the modem understanding of set theory. While Choice, as platonists typically argue, is essential to a proper understanding of set-theoretical ontology, it cannot be easily justified in platonist terms, as a feature of that ontology. Rather, I argue that Choice is a logical principle: it is intrinsically tied to the laws of quantification, and the notion of choice function itself responds to a Tarskian criterion for the distinction between logical and non-logical notions. This approach also illuminates some of Skolem’s original contributions to the philosophy of logic. IV ACKNOWLEDGMENTS This work might never have seen completion without the concerted effort and support of the members of my dissertation committee. I grateftilly acknowledge the contribution of my adviser, Stewart Shapiro. Not only did he guide me through the intellectual development that led to this work, he also supported me in more personal ways in the occasions — more frequent than he may wish to remember — when I most needed it. Similarly generous with their advice and philosophical wisdom (not to mention their time) were Neil Tennant and Mark Wilson. For support of a different kind, I am deeply grateful to my wife, Laurie, and my parents, who stayed with me through thick and thin. VITA 1984 .................................. B.A., Philosophy University of Milan 1989 .................................. M.A., Philosophy The Ohio State University 1987 ' 1992 .......................... Graduate Teaching andResearch Associate, Department of Philosophy, The Ohio State University 1993 - present ....................... Research Associate, College of Humanities The Ohio State University PUBLICATIONS 1. “A Note on Truth, Deflation and Irreahsm,” Sorites 3 (Nov. 1995); 48-63. 2. Review of T. Oberdan, “Godel, Camap and the thesis that mathematics is empty,” Jahrb. der Kurt Godel GeseUschaft, 1991. Mathematical Reviews, March 1994. 3. Review of W. Demopoulos and J.L. Bell, “Frege’s theory of concepts and objects and the interpretation of second-order logic,” Philosophia Mathematica 3, 1, 1993. Mathematical Reviews, 1994. 4. Review of R.L. Kirkham, “Tarski’s Physicalism,” Erkenntnis 38, 1993. Mathematical Reviews, August 1994. 5. Review of A. Drago and 0. Vitiello, “La storia dell’introduzione della matematica costruttiva nella flsica teorica: il caso della termodinamica, ” Conferenza sulla Storia della Matematica, 1988. Mathematical Reviews, April 1994. 6. Review of J. Burgess, “Hintikka et Sandu versus Frege in re Arbitrary Functions,” Philosophia Mathematica 1, 1993. Mathematical Reviews, March 1994. VI 7. Review of I. Jane, “A critical appraisal of second-order logic,” Hist, and Philosophy of Logic 14, 1993. Mathematical Reviews, February 1994 . 8. Review of A. Oberschelp, Logik fUr Philosophen, Bibliographisches Institut, Mannheim 1992. Mathematical Reviews, September 1993. 9. Review of T. Koetsier, Imre Lakatos’ Philosophy of Mathematics, North-HoUand, 1992. Mathematical Reviews, January 1993. FIELDS OF STUDY Major Field: Philosophy vu TABLE OF CONTENTS ABSTRACT...........................................................................................................i i i ACKNOWLEDGMENTS..........................................................................................v VITA ..........................................................................................................................v i CHAPTER 1 ............................................................................................................... 1 INTRODUCTION ...................................................................................................1 The Skolem problem ..................................................................................1 Skepticism, through and th rou gh? ................................................... 16 The thesis of moderate anti-skepticism ..................................................22 Use and practice ........................................................................... 24 The role of the axiom of choice ........................................... 26 A twist of epistemology ...............................................................32 An o u tlin e .................................................................................................... 34 CHAPTER 2 .............................................................................................................38 SKOLEMITE R ELATIVISM ........................................................................ 38 The paradox and the skeptic ............................................................ 41 The Lowenheim-Skolem T h e o r e m ..............................................41 The argument for relativism ............................................... 44 Foundations and axiomatic t h e o r ie s ............................................... 58 The big p ictu re................................................................................58 Skolem’s view of axiomatic theories .................................. 68 The role of C hoice ........................................................................... 70 Conclusion: skepticism a g a i n ...................................................................87 CHAPTER 3 ................................................................................................................. 91 SKEPTICISM VS RELATIVITY................................................................................91 Introduction ......................................................................................... 91 Quinean r e la tiv ity ............................................................................ 98 viii How to commit oneself to an ontology ................................................107 Is ontological
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