Toward a Kripkean Concept of Number

City University of New York (CUNY) CUNY Academic Works All Dissertations, Theses, and Capstone Projects Dissertations, Theses, and Capstone Projects 2-2016 Toward a Kripkean Concept of Number Oliver R. Marshall Graduate Center, City University of New York How does access to this work benefit ou?y Let us know! More information about this work at: https://academicworks.cuny.edu/gc_etds/703 Discover additional works at: https://academicworks.cuny.edu This work is made publicly available by the City University of New York (CUNY). Contact: [email protected] ! TOWARD A KRIPKEAN CONCEPT OF NUMBER by Oliver R. Marshall A dissertation submitted to the Graduate Faculty in Philosophy in partial fulfillment of the requirements of the degree of Doctor of Philosophy, The City University of New York. 2016. ! ii! © 2016 Oliver R. Marshall All Rights Reserved ! iii! This manuscript has been read and accepted for the Graduate Faculty in Philosophy to satisfy the dissertation requirement for the degree of Doctor of Philosophy. Jesse Prinz ______________ ___________________________________________________ Date Chair of Examining Committee Iakovos Vasiliou ______________ ___________________________________________________ Date Executive Officer Supervisory Committee: Gary Ostertag, Advisor Saul Kripke Nathan Salmon Jesse Prinz Michael Levin THE CITY UNIVERSITY OF NEW YORK ! iv! Abstract TOWARD A KRIPKEAN CONCEPT OF NUMBER By Oliver R. Marshall Saul Kripke once remarked to me that natural numbers cannot be posits inferred from their indispensability to science, since we’ve always had them. This left me wondering whether numbers are objects of Russellian acquaintance, or accessible by analysis, being implied by known general principles about how to reason correctly, or both. To answer this question, I discuss some recent (and not so recent) work on our concepts of number and of particular numbers, by leading psychologists and philosophers. Special attention is paid to Kripke’s theory that numbers possess structural features of the numerical systems that stand for them, and to the relation between his proposal about numbers and his doctrine that there are contingent truths known a priori. My own proposal, to which Kripke is sympathetic, is that numbers are properties of sets. I argue for this by showing the extent to which it can avoid the problems that plague the various views under discussion, including the problems raised by Kripke against Frege. I also argue that while the terms ‘the number of F’s’, ‘natural number’ and ‘0’, ‘1’, ‘2’ etc. are partially understood by the folk, they can only be fully understood by reflection and analysis, including reflection on how to reason correctly. In this last respect my thesis is a retreat position from logicism. I also show how it dovetails with an account of how numbers are actually grasped in practice, via numerical systems, and in virtue of a certain structural affinity between a geometric pattern that we grasp intuitively, and our fully analyzed concepts of numbers. I argue that none of this involves acquaintance with numbers. ! v! Acknowledgements Firstly, I would like to thank those not formally associated with my project, who nevertheless provided invaluable assistance. In particular, I thank Jody Azzouni and Haim Gaifman for discussion of the material in chapter 1, and the participants in the 2013 Columbia University Frege Reading Group, especially Robert May and Haim once again. I am also grateful to Haim for introducing me to Babylonian mathematics, and for his feedback on my proposal in chapter 4, which also benefitted from correspondence with Stephen Chrisomalis. Finally, I thank Bob Fiengo for useful discussion of hedging. My biggest debt is to my readers and mentors. I would like to thank Saul Kripke, whose teaching and forthcoming Whitehead and Notre Dame Lectures inspired me to write this dissertation and provided much of its subject matter. I will always be grateful to my supervisor, Gary Ostertag, for his limitless patience and generosity, and for his incisive and detailed feedback on every aspect of this project. I also thank Nathan Salmon for his brilliant teaching, for sharing his work in progress with me, and for corresponding at such great length about so many issues. Further, I am grateful to Jesse Prinz for the feedback he provided during his fantastic dissertation seminar, and for chairing my examination committee. My most longstanding reader is Michael Levin, whom I thank for getting me into graduate school, introducing me to the philosophy of mathematics, and providing years of help and encouragement. I would also like to thank Romina Padro and Dehn Gilmore for their support, both professional and moral. Further, I am extremely grateful to my family —my parents, my brother, and my parents in law— for their love, patience and confidence. Finally, I thank my wonderful wife, Emelie Gevalt, for tolerating and enabling my workalholism. ! vi! Table of Contents Abstract iv Acknowledgements v Introduction 1 – 8 Chapter 1 Our intuitive grasp of number 9 – 59 1. Introduction 2. The number sense hypothesis 3. The use of number sense by numerate human adults 4. The triple code model 5. Why the triple code model cannot explain our intuitive grasp of numbers 6. Digital models 7. Philosophical applications of cognitive science 8. Acquaintance 9. Giaquinto on acquaintance 10. Burge on acquaintance 11. Burge on de re thoughts about numbers 12. Burge on de re understanding 13. Problems for Burge’s account 14. Giaquinto on our intuitive grasp of the number structure 15. A methodological moral Chapter 2 Frege’s Logicism 60 – 95 1. Introduction 2. Frege’s philosophical goals and their precedents ! vii! 3. Frege’s methodology 4. Frege’s doctrine of the primitive truths 5. Frege’s definitions 6. Frege’s views on analysis 7. Assessing Frege’s definition of number 8. The need for a non-logical axiom Chapter 3 Frege’s Theorem 96 – 126 1. Introduction 2. Is HP a primitive truth? 3. Just as many 4. Comprehension axioms 5. Does Frege’s Theorem require predicative or impredicative comprehension? 6. Are impredicative comprehension axioms primitive truths? Chapter 4 Set-theoretic logicism 127 – 186 1. Introduction 2. The derivation of arithmetic in ZF 3. What are the motivations for representing arithmetic in ZF? 4. Benacerraf’s dilemma again 5. Wittgenstein’s objections and Steiner’s response 6. Problems with Steiner’s proposal 7. Kripke’s counterexample 8. Kripke’s proposal ! viii! 9. Kripke’s acquaintance theory 10. The representation of Kripke’s proposal in ZF 11. Assessing Kripke’s definition of number 12. Babylonian notation 13. Decimal notation Chapter 5 Another application of Kripke’s theory 187 – 217 1. Introduction 2. The contingent a priori 3. Kripke’s proposal 4. Scylla again 5. A contextualist response to Kripke’s proposal 6. Invariantist and ambiguity theories 7. Hedging 8. Interest-relativity and hedging Chapter 6 A proposal 218 – 251 1. Introduction 2. Count and mass nouns 3. Salmon’s puzzle about count nouns 4. A contextualist response to Salmon’s puzzle 5. Equinumerosity properties 6. Church’s type-theory 7. Equinumerosity properties again ! ix! 8. Assessing the above definitions 9. Are the axioms of STT primitive truths? References 253 – 260 ! x! List of Tables Table 1: Kripke’s alphabetic base-26 notation 252 ! 1! Introduction Saul Kripke once remarked to me that natural numbers cannot be posits inferred from their indispensability to science, since we’ve always had them. This left me wondering whether numbers are objects of Russellian acquaintance, or grasped by conceptual analysis (being implied by known general principles such as those of logic or set theory), or both, or indeed neither, perhaps being grasped by an understanding of numerical concepts that is not achieved by analysis. The question is pressing, since a full account of human knowledge must include an account of mathematical knowledge, and here we should begin with numbers, since these are a prerequisite of other mathematics. Happily, my interest in this topic has coincided with a resurgence of work in the area, with leading psychologists and philosophers, including Kripke himself, staking out positions corresponding to the aforementioned options. Given this resurgence, a dissertation on the topic seems overdue. I begin, in chapter 1, by discussing recent attempts to explain how we actually grasp numbers, using the resources of cognitive science. First I consider the influential view that we are acquainted with numbers via the innate sense of quantity known in the psychological literature as our “number sense.” Having rejected this proposal, I turn to attempts by Tyler Burge and others to use it —as well as other proposals from cognitive science— as part of an account that purports to explain our grasp of numbers in terms of our acquisition of numerical concepts. In essence, my criticism is that such accounts of acquisition are either insufficient to explain our grasp of numbers, or presuppose too much about what they purport to explain, with the result that they are uninformative. I’ll now say a little more about this, so that the reader can appreciate the dialectic that unfolds in the following chapters. ! 2! According to Burge, numbers are not implied by our knowledge of known general principles, since propositions about numbers are “underived from general principles” (2000: 40). Further, while Burge claims that such propositions are “irreducibly singular” (ibid), he also claims that this is not because numbers are grasped by acquaintance, but because they are grasped by an immediate, non-discursive and elementary kind of understanding that he calls “comprehension.” I’ll now say a bit about comprehension of propositions about the smallest natural numbers. According to Burge, learning to count with numerals gives us the ability to deploy the corresponding numerical concepts, in the context of applied arithmetical propositions like there are 2 houses of Congress. Further, we can immediately perceptually apply these concepts to small concrete pluralities without counting (perhaps using our number sense).

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