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Carnap, Tarski, and Quine's Year Together: Conversations on Logic

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Carnap, Tarski, and Quine's Year Together: Conversations on Logic, Mathematics, and Science Greg Frost-Arnold September 26, 2008 ii Contents 1 Background and Overview 1 1.1 Introduction: Setting the Historical Stage . 1 1.2 Outline of the Primary Project: A `Finitist-Nominalist' Total Language of Science . 3 1.3 Mathematics in a Finitist-Nominalist Language . 11 1.3.1 Number . 11 1.3.2 Interpreting numerals that are `too large' . 13 1.4 Pre-history of the 1941 Finitist-Nominalist Project . 15 1.4.1 The Poles: Chwistek, Kotarbinski, Lesniewski . 15 1.4.2 Russellian influences . 21 1.4.3 Logical Empiricists skeptical of infinity . 23 2 Justifications 27 2.1 First Justification for the FN Conditions: Verst¨andlichkeit . 28 2.1.1 Assertions meeting the FN criteria are verst¨andlich . 28 2.1.2 What does `understandable' [verst¨andlich] mean in the discussion notes? . 28 2.1.3 What does `understandable' or `intelligible' mean in Car- nap's publications? . 30 2.1.4 What does `understandable' or `intelligible' mean for Quine? 34 2.2 Second Rationale for the FN Conditions: Overcoming Metaphysics 37 2.3 Third Rationale: Inferential Safety, or Taking the Paradoxes Se- riously . 43 2.4 Fourth Rationale for the FN Conditions: Natural Science . 45 2.5 Current Justifications for Nominalist Projects . 47 2.5.1 The positive argument: from a causal theory of knowledge and/or reference . 47 2.5.2 The negative argument: rebutting the indispensibility ar- gument . 48 3 Responses 51 3.1 Why Does Carnap Participate in the FN Project, Given his Reser- vations? . 51 iii iv CONTENTS 3.2 Answering the Finitist-Nominalist's Arguments: Higher Mathe- matics is Meaningful . 53 3.2.1 An analogy between higher mathematics and theoretical physics . 54 3.2.2 Potential infinity to the rescue? . 58 3.3 Are There Any Infinities Compatible with Nominalism? (and a Detour through Finitist Syntax) . 60 3.4 Attacking the FN Conditions . 65 3.4.1 Arithmetic is distorted . 66 3.4.2 Problems with proofs . 70 3.5 An Objection to the FN Project not in the Notes . 71 4 Nominalism and Analyticity 73 4.1 Under a Finitist-Nominalist Regime, Arithmetic becomes Synthetic 74 4.2 Radicalization of Quine's Critique of Analyticity: A Historical Conjecture . 81 5 Analyticity in 1941 89 5.1 What is Analyticity, circa 1940? . 90 5.1.1 Carnap and Tarski on Analyticity . 91 5.1.2 Quine: analysis of language{including analyticity{should be syntactic and extensional . 94 5.2 Tarski's Objections to Analyticity . 98 5.2.1 Tarski's first objection: any sign can be converted into a logical sign . 100 5.2.2 Tarski's second objection: G¨odelsentences . 102 5.3 Quine's Disagreements with Carnap circa 1940 . 107 5.3.1 Analyticity is an empirical concept, not a logical one . 107 5.3.2 Modal and intensional languages are unacceptable . 111 5.3.3 A further step in Quine's radicalization . 113 6 Metaphysics and the Unity of Science 117 6.1 Unity of Science: Unity of Language, not of Laws . 118 6.2 Overcoming Metaphysics . 122 6.2.1 Aufbau .............................123 6.2.2 \Overcoming Metaphysics through Logical Analysis of Lan- guage" . 125 6.2.3 Logical Syntax of Language . 127 6.2.4 \Empiricism, Semantics, and Ontology" . 129 6.2.5 Neurath . 130 6.3 A Difficulty: What Cannot be Incorporated into a Language of Science? . 132 6.4 Conclusion: The Origin of the Term ‘Unified Science' . 135 CONTENTS v Preface Several years ago, I began combing through the Rudolf Carnap Collection at the University of Pittsburgh's Archives of Scientific Philosophy, searching for material about Alfred Tarski's theory of truth. I fortuitously came across a folder full of dictation notes Carnap had taken during conversations with Tarski and others, mostly from 1941. Although I did not fully recognize then what I had stumbled upon, those notes are the originating cause of this book. The first portion of this book consists of my attempts to come to grips with those documents, both in terms of their place in the history of analytic philosophy, and their often surprising philosophical content. As the subtitle of this book suggests, I have grouped this content into three interdependent sections: math- ematical nominalism (mathematics), analytic truth (logic), and the unity of science. This choice of what to focus on in these wide-ranging notes certainly reflects my own, sometimes idiosyncratic interests in and perspective on the his- tory of analytic philosophy; another, almost entirely different book could most likely be written about the same archival material. Because the documents would amply reward such study, an edited version of the archival material itself can be found in the Appendix, along with an English translation. I do not feel that I wrote this book. The final product is very much the result of many people|I just happen to be the person who put the most time into the group's project. My inability to see obvious errors that others spotted easily, and others' insightful and creative suggestions for new directions, truly made this a collaborative effort. John Earman and Nuel Belnap gave me very helpful guidance as this material coalesced slowly into a dissertation. An audience at HOPOS 2004 helped improve the core of what is now the first chapter; Alan Richardson and Chris Pincock, in particular, both provided fruitful suggestions. A more intimate audience at the 2004 PSA helped iron out some of the deficien- cies of the final chapter; Rick Creath provided especially useful feedback both then and later, as the larger project progressed. His Dear Carnap, Dear Van was not only an invaluable research tool for present purposes, but also served as an exemplar after which this book is patterned. Michael Friedman, Don Howard, and Thomas Ricketts each brought their substantial erudition to bear on various ideas presented here; their ability to see the historical `big picture' was a very helpful corrective to my myopia and ignorance. Andre Carus gave me very helpful suggestions about the historical big picture as well, and Chap- ter 1 in particular was thereby much improved. Marion Ledwig, Greg Lavers, and Jon Tsou each undertook the task of reading the entire manuscript when it was close to completion; their fresh eyes provided wonderful new perspective on ideas that had been bouncing around in my head for years. Jon, in partic- ular, engaged my arguments seriously, and (thankfully!) would not let me get away with anything that was too quick. I am certain I am leaving out several people who helped me recognize and rectify deficiencies in the ideas presented here, only because I've been overeager to discuss this material with anyone who would listen for the last five years or so. I apologize to you all. I have saved this book's three greatest intellectual debts for last. First vi CONTENTS is Paolo Mancosu, who found the records of these conversations at roughly the same time I did, and began working on them shortly thereafter. Writing about documents that no one has ever seen before is not something for which the usual historical training on Plato or Descartes really prepares you. I am extremely fortunate that a philosopher as able|and as intellectually generous| as Paolo began studying these works as well: his perspective kept me from being completely lost in the dark as I attempted to wade through Carnap's notes. More than he perhaps realizes, his published work and private comments helped iron out serious distortions and outright misunderstandings in my conception of what is going on in this material, and he helped me to recognize better what is central and what is peripheral. Steve Awodey gave me copious and insightful feedback every step of the way, especially when the project was just beginning: he read very, very rough drafts with more care and respect than they deserved at the time. His ability to comment constructively on ideas that were highly underdeveloped|instead of simply saying `think about this more,' the common and easy response|was a great boon. And Steve has shown excellent editorial touch as well, giving me all the freedom I could want. Plus, he prevented me from saying some very wrong-headed things about Russell's theory of types. Laura Ruetsche's help has been essential from the very beginning of this work. She possesses the enviable knack of seeing to the core of an issue whose surface I'll only scratch for 20 pages or so. Laura has been a true mentor, not merely a very dedicated and astute reader: she struck the perfect balance of encouragement for work I had already done, while showing me how much improvement could yet be done. I am extremely fortunate that she supervised my dissertation. My wife Karen has not only given me a wonderful life, but has been a fantastic philosophical rudder as well. Whenever I get lost in the philosophical trees, she shows me the forest again. More importantly, when I'm confronted with an apparently insuperable obstacle, she shows me that it can be overcome, and when I stumble, she helps me up. Chapter 1 Historical Background and Overview of Harvard Activities 1.1 Introduction: Setting the Historical Stage During the academic year 1940{1941, several giants of analytic philosophy| both established and budding|congregated at Harvard University. The list of philosophers is impressive. Bertrand Russell, who was only at Harvard during the Fall semester of 1940, originally emigrated from Britain in 1938. During 1940, he was embroiled in his infamous legal battles with the City College of New York.
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