Counting and Other Forms of Measurement Dissertation Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Eric Sndyer ∼6 6 Graduate Program in Philosophy The Ohio State University 2016 Dissertation Committee: Professor Stewart Shapiro, Advisor Professor Michael Glanzberg Professor Craige Roberts Professor Kevin Scharp Professor Neil Tennant c Eric Sndyer, 2016 Abstract This thesis is about number expressions, their meanings, and certain puzzles within the philosophy of mathematics arising in their connection. I develop novel solutions to those puzzles based on recent work within linguistic semantics concerned with the meanings of various measurement-related constructions. Along the way, I further develop those analyses in important ways. Some of the more philosophically significant theses I defend are the fol- lowing: though number expressions take on a greater variety of interrelated meanings than philosophers have traditionally recognized, all of these can be derived assuming that e.g. `four' is basically a numeral, thus vindicating a certain form of the Substantival Strategy; there is a natural language distinction between numbers and cardinalities (a kind of degree), and the familiar Fregean cardinality-operator most charitably refers to the latter, not the former, thus casting considerable doubt on whether Hume's Principle can generate numbers in the way assumed by Neo-Fregeans; counting involves measuring the cardinality of collec- 1 1 tions of indivisible objects, or \atoms"; fractions like `2 2 ' in `2 2 oranges' involve a form of 1 counting (at least on one interpretation), so that `Mary ate 2 2 oranges' implies that Mary ate three things, namely two whole oranges and one half-orange; the distinction between sortal and non-sortal concepts, at least when construed linguistically, is not exhausted by the familiar count/mass distinction, but is plausibly exemplified by various other semantic phenomenon previously unrecognized by philosophers; and not all concepts answering `how many'-questions are sortal. Some of the more linguistically significant theses I defend are as follows: though previous accounts have claimed that various uses of `four' can be derived via type-shifting, none of those accounts actually succeeds in deriving meanings appropriate for all of those uses; there are correspondingly distinct referential uses of `four', one referring ii to a number, the other referring to a degree of cardinality; so-called individuation/measure- ambiguities witnessed in container phrases like `four glasses of water' are a special case of the same ambiguity arising in connection with atomic predicates more generally; despite the traditional conceptions of cardinality, all scales (ordered sets of degrees) are dense, includ- ing cardinality scales. My hope is that this thesis not only contributes novel and plausible solutions to important puzzles within the philosophy of mathematics, it also add to our present understanding of the meanings of number expressions and measurement-related phenomena. iii Acknowledgements I would like to sincerely thank the members of this committee for their time, patience, insight, and feedback regarding the material in this thesis. Extra special thanks are due to my advisors, Stewart Shapiro and Craige Roberts, for guiding through this entire process. It is largely due to their help and encouragement that I am in a position to present this material in the first place. And it is probably no accident that this material represents a mixture of their particular specialties and interests { the philosophy of mathematics and linguistic semantics. It has truly been an honor to work with them so closely over the past several years, and any future success I might enjoy as an academic should be attributed chiefly to their excellent guidance. I also owe a large debt of gratitude to the other members of this committee, Michael Glanzberg, Kevin Scharp, and Neil Tennant. I was fortunate to meet Michael at a conference held at Ohio State some years back, and I remember being amazed that someone of his incredible ability and expertise would not only be willing to listen to my ideas, but would also encourage developing them further and stay in touch regarding their development, especially since Michael was not a member of the OSU faculty. That interaction eventually led to my first publication, and I have been very fortunate to maintain a dialog (though from a distance) with him ever since. His seemingly endless encouragement and enthusiasm has been invaluable. My interest in measurement-related phenomena was sparked largely by a seminar I took with Kevin on measurement theory. To his credit, he allowed me to develop a semantics for measurement-related expressions which in all honestly probably failed to engage with the central themes of the seminar, but which ultimately grew into much of the work presented in this thesis. His input on much of my published work is evident, and his amazing ability to engage with just about any form of philosophy and to present it to a iv general philosophical audience has proved critical to my success in grad school. Finally, the primary focus of this dissertation grew directly out of a seminar I took with Neil on Neo- Logicism. It was through Neil's own work that I was introduced to the debates mentioned in Chapters 2 and 3, and his excellent work on Neo-Logicism clearly permeates throughout this thesis. There are too many other people to thank, and there is no way I could do justice to everyone who has helped influence the ideas found here. Nevertheless, I want to express my sincere gratitude to the following individuals for helpful discussions, exchanges of ideas, or general enthusiasm concerning the project: Maria Aloni, Chris Barker, Thomas Hofweber, Chris Kennedy, Dan Lassiter, Øystein Linnebo, Carl Pollard, Susan Rothstein, Richard Samuels, Greg Scontras, Robert Stalnaker, William Taschek, and Joost Zwarts. Finally, I would like to thank my good friend and collaborator, Jefferson Barlew. Thanks largely to his friendship and the support of the aforementioned individuals, I can sincerely say that the past several years spent at Ohio State have been the best of life. Thanks again to all of you. v Vita 2004 ............................ B.A. English, Piedmont College 2009 ............................ M.A. Philosophy, University of Georgia 2009-present ................ Presidential Fellow, the Ohio State University Publications Snyder, Eric. 2013. Binding, genericity, and predicates of personal taste. Inquiry 56, pp. 278-306. Shapiro, Stewart & Eric Snyder. 2015. Vagueness and context. Inquiry. Snyder, Eric & Stewart Shapiro. Forthcoming. Frege on the real numbers. To appear in P. Ebert and M. Rossberg (eds.), Essays on Frege's Basic Laws of Arithmetic. Oxford University Press. Snyder, Eric & Jefferson Barlew. Forthcoming. The universal measurer. To appear in Proceedings of Sinn und Bedeutung 20. Snyder, Eric. Forthcoming. Numbers and cardinalities: what's really wrong with the easy argument for numbers? Linguistics and Philosophy. Fields of Study Major Field: Philosophy vi Table of Contents Abstract ............................................................................. ii Acknowledgements .................................................................. iv Vita ................................................................................. vi 1 Introduction ..................................................................... 1 2 Frege's Other Puzzle ............................................................. 16 2.1 Frege's Other Puzzle............................... 16 2.2 Two Strategies of Analysis............................ 21 2.2.1 The Substantival Strategy........................ 21 2.2.2 The Adjectival Strategy......................... 27 2.3 Numerals, Determiners, or Adjectives?..................... 33 2.4 The Adjectival Theory.............................. 39 2.4.1 Different Uses of Number Expressions................. 44 2.4.2 The Extended Adjectival Theory.................... 48 2.5 The Neo-Substantivalist Strategy........................ 51 3 The Easy Argument for Numbers ................................................ 56 3.1 The Easy Argument for Numbers........................ 56 3.1.1 A Parallel Ambiguity?.......................... 58 3.1.2 Various Uses of Number Expressions.................. 59 3.1.3 Specificational and Quantificational Uses............... 62 3.1.4 Diagnosing the Easy Argument..................... 65 3.2 A Brief Review of The Extended Adjectival Theory.............. 68 3.3 Extending the Adjectival Theory........................ 71 3.3.1 The Individual Concept Analysis.................... 74 3.3.2 The Degree-as-Kind Analysis...................... 76 3.3.3 Putting It All Together......................... 80 3.3.4 Cardinality and Number......................... 84 3.4 What's Really Wrong with the Easy Argument?............... 85 3.5 Conclusion.................................... 90 vii 4 The Counting Oranges Puzzle .................................................... 92 4.1 The Counting Oranges Puzzle.......................... 92 4.2 Individuating and Measuring.......................... 99 4.3 The Universal Measurer............................. 107 1 4.4 How to Count 2 2 Oranges............................ 113 4.4.1 Generalized Cumulative Conjunction.................. 115 4.4.2 Fractions................................. 117 4.4.3 Putting it All Together........................