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Reconsidering Carnap's Logic of Science Western University Scholarship@Western Electronic Thesis and Dissertation Repository 12-9-2013 12:00 AM The Methodological Roles of Tolerance and Conventionalism in the Philosophy of Mathematics: Reconsidering Carnap's Logic of Science Emerson P. Doyle The University of Western Ontario Supervisor John L. Bell The University of Western Ontario Graduate Program in Philosophy A thesis submitted in partial fulfillment of the equirr ements for the degree in Doctor of Philosophy © Emerson P. Doyle 2013 Follow this and additional works at: https://ir.lib.uwo.ca/etd Part of the Logic and Foundations of Mathematics Commons Recommended Citation Doyle, Emerson P., "The Methodological Roles of Tolerance and Conventionalism in the Philosophy of Mathematics: Reconsidering Carnap's Logic of Science" (2013). Electronic Thesis and Dissertation Repository. 1757. https://ir.lib.uwo.ca/etd/1757 This Dissertation/Thesis is brought to you for free and open access by Scholarship@Western. It has been accepted for inclusion in Electronic Thesis and Dissertation Repository by an authorized administrator of Scholarship@Western. For more information, please contact [email protected]. The Methodological Roles of Tolerance and Conventionalism in the Philosophy of Mathematics: Reconsidering Carnap's Logic of Science (Thesis Format: Monograph) by Emerson P. Doyle Graduate Program in Philosophy A thesis submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy The School of Graduate and Postdoctoral Studies The University of Western Ontario London, Ontario, Canada c Emerson P. Doyle 2013 Abstract This dissertation makes two primary contributions. The first three chapters de- velop an interpretation of Carnap's Meta-Philosophical Program which places stress upon his methodological analysis of the sciences over and above the Principle of Tol- erance. Most importantly, I suggest, is that Carnap sees philosophy as contiguous with science|as a part of the scientific enterprise|so utilizing the very same meth- ods and subject to the same limitations. I argue that the methodological reforms he suggests for philosophy amount to philosophy as the explication of the concepts of science (including mathematics) through the construction and use of suitably robust meta-logical languages. My primary interpretive claim is that Carnap's understand- ing of logic and mathematics as a set of formal auxiliaries is premised upon this prior analysis of the character of logico-mathematical knowledge, his understanding of its role in the language of science, and the methods used by practicing mathematicians. Thus the Principle of Tolerance, and so Carnap's logical pluralism, is licensed and justified by these methodological insights. This interpretation of Carnap's program contrasts with the popular Deflationary reading as proposed in Goldfarb & Ricketts (1992). The leading idea they attribute to Carnap is a Logocentrism: That philosophical assertions are always made relative to some particular language(s), and that our choice of syntactical rules for a language are constitutive of its inferential structure and methods of possible justification. Consequently Tolerance is considered the foundation of Carnap's entire program. My third chapter argues that this reading makes Carnap's program philosophically inert, and I present significant evidence that such a reading is misguided. The final chapter attempts to extend the methodological ideals of Carnap's pro- gram to the analysis of the ongoing debate between category- and set-theoretic foun- dations for mathematics. Recent criticism of category theory as a foundation charges that it is neither autonomous from set theory, nor offers a suitable ontological ground- ing for mathematics. I argue that an analysis of concepts can be foundationally in- formative without requiring the construction of those concepts from first principles, and that ontological worries can be seen as methodologically unfruitful. Keywords: Rudolf Carnap, Explication, Conventionalism, Principle of Tolerance, The Logical Syntax of Language, Foundations of Mathematics, Philosophy of Math- ematics, Rational Reconstruction, Logic of Science, Scientific Philosophy, Logical Pluralism, Logical Empiricism, Logical Positivism, Philosophy of Science. ii Modern science arose from the marriage of mathematics and empiricism; three centuries later the same union is giving birth to a second child, scientific philosophy, which is perhaps destined to as great a career. For it alone can provide the intellectual temper in which it is possible to find a cure for the diseases of the modern world. Bertrand Russell (1936) iii Dedicated to my parents, for their love and support. And to Mark Bronson, for introducing me to philosophy, and for encouragement and sound advice every step of the way. iv Acknowledgements This project would never have come to fruition without the help and support of more than my fair share of remarkable people. My thanks go first to my supervisory committee, Robert DiSalle, Wayne Myrvold, and especially to my supervisor John Bell. Their suggestions, guidance, and encouragement were a great benefit to this project, especially in its beginning stages. To John I must also extend my gratitude for many an enjoyable evening of red wine and great conversation. The enthusiasm displayed by my examiners David DeVidi, Stathis Psillos, Wayne Myrvold, and Robert Mercer has redoubled my own enthusiasm for the project. I must admit that by the time of submission I was looking forward to taking some time away from Carnap, but their encouragement and insightful comments have reminded me of why I decided upon this topic in the first place. In my (not insignificant) time at Western I have benefited from the instruction and example of many exceptional philosophers, but with regard to my own philo- sophical development, none more so than William Demopoulos. It was Bill's courses and our conversations that instilled in me a deep appreciation for early analytic phi- losophy. His own work has inspired much of my thought, and represents an ideal of philosophical analysis and argumentation to which I may only ever strive. With respect to this project, no one has had a greater influence on its immediate ideas and writing than my friend and colleague Steve Bland. My understanding of Carnap and the Deflationary interpretation owes an incredible debt to our conver- sations. Steve also patiently and carefully read through many drafts of most of the chapters, at each stage asking thoughtful questions and offering deep and insightful commentary. In many ways this project would not have been but for Steve's help. In the final stages of writing, and especially in the preparation of my public talk, I owe a great debt of gratitude to Robert Moir and Lori Kantymir. They helped me to see my project in a new light; and have been generous with their kindness, philosophical insight, and friendship for as long as I've known them. My interests and orientation as a philosopher have also been influenced by many other friends and colleagues over the years. I must distinguish for special thanks Alex Beldan, Octavian Busuioc, Sheldon Chow, Nicolas Fillion, Elana Geller, Sona Ghosh, Dave Johnston, Molly Kao, Pamela Malins, Nick Ray, and Ryan Samaroo for their friendship and influence. I am certainly a better and more knowledgeable philosopher as a result of our interactions. And without their friendship and support I surely would not have been able to complete the difficult and often very isolating v journey that is graduate school. More importantly, thanks for making it fun. For getting me into this mess to begin with, I thank my friend Mark Bronson. It was Mark that introduced me to philosophy as a discipline many years ago, and I have benefited from his insight, encouragement, and advice ever since. Finally, I must thank my parents, Brian and Annie Doyle, for their constant love and support. This project would have proven impossible without them. Emerson P. Doyle December 16, 2013 London, ON. vi Contents Abstract ii Epigraph iii Dedication iv Acknowledgements v List of Figures x List of Tables x 1 Introduction: A Bittersweet Renaissance for Logical Empiricism 1 1.1 Carnap's Meta-Philosophy|Overview . .4 1.1.1 Transforming Philosophical Debates . .5 1.1.2 Conventionalism and the Principle of Tolerance . 10 1.1.3 Should We Be Tolerant About Tolerance? . 15 1.2 A Bittersweet Renaissance|Circularity Objections . 18 1.2.1 Carnap's Philosophy of Mathematics . 19 1.2.2 A Possible Carnapian Reply . 20 1.3 Interpreting Carnap|A Deflationary Logic of Science? . 21 1.3.1 The Deflationary Reading . 22 1.3.2 Some Evidence for the Deflationary Reading . 25 1.3.3 Objection: The Priority of Science for Carnap . 26 1.4 Summary of What Follows . 27 1.4.1 Two Final Caveats . 30 Logico-Mathematical Interlude 31 A.1 Logical Syntax|Overview . 32 A.2 Carnap's Motivations and Technical Goals . 34 A.3 Languages I and II . 41 A.3.1 Consequence for LI . 42 A.3.2 Consequence for LII . 44 A.4 General Syntax . 48 A.4.1 Distinguishing Between L- and P-Rules . 49 vii 2 Is Carnap's Meta-Philosophy Viciously Circular? 54 2.1 G¨odel'sCriticisms . 56 2.1.1 G¨odel'sCircularity Objection . 59 2.1.2 Mathematical and Empirical Content . 60 2.2 Other Circularity Objections . 62 2.2.1 Quine on Conventions . 63 2.2.2 The First Incompleteness Theorem . 64 2.2.3 Non-standard Interpretations . 66 2.2.4 The Crux of the Circularity Objections . 68 2.3 Explication and Scientific Methodology . 70 2.3.1 Informality and Material Interpretation . 70 2.3.2 Carnap's Notion of Explication . 75 2.3.3 The Methodological Structure of Science . 83 2.4 Response: Explication as Foundations . 90 3 `Empirical Fact', Tolerance, and Conventionalism: A Reply to Goldfarb and Ricketts 93 3.1 The Deflationary Interpretation . 94 3.2 The Deflationary Response to G¨odel . 97 3.2.1 Review: G¨odel'sCircularity Objection .
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