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Normativity and Mathematics: a Wittgensteinian Approach to the Study of Number

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Normativity and Mathematics: a Wittgensteinian Approach to the Study of Number ABSTRACT Normativity and Mathematics: A Wittgensteinian Approach to the Study of Number J. Robert Loftis I argue for the Wittgensteinian thesis that mathematical statements are expressions of norms, rather than descriptions of the world. An expression of a norm is a statement like a promise or a New Year’s resolution, which says that someone is committed or entitled to a certain line of action. A expression of a norm is not a mere description of a regularity of human behavior, nor is it merely a descriptive statement which happens to entail a norms. The view can be thought of as a sort of logicism for the logical expressivist—a person who believes that the purpose of logical language (sentential connectives, quantifiers, etc.) is to make explicit commitments and entitlements that are implicit in ordinary practice. The thesis that mathematical statements are expression of norms is a kind of logicism, not because it says that mathematics can be reduced to logic, but because it says that mathematical statements play the same role as logical statements. I contrast my position with two sets of views, an empiricist view, which says that mathematical knowledge is acquired and justified through experience, and a cluster of nativist and apriorist views, which say that mathematical knowledge is either hardwired into the human brain, or justified a priori, or both. To develop the empiricist view, I look at the work of Kitcher and Mill, arguing that although their ideas can withstand the criticisms brought against empiricism by Frege and others, they cannot reply to a version of the critique brought by Wittgenstein in the Remarks on the Foundations of Mathematics. To develop the nativist and apriorist views, I look at the work of contemporary developmental psychologists, like Gelman and Gallistel and Karen Wynn, as well as the work of philosophers who advocate the existence of a mathematical intuition, such as Kant, Husserl, and Parsons. After clarifying the definitions of “innate” and “a priori,” I argue that the mechanisms proposed by the nativists cannot bring knowledge, and the existence of the mechanisms proposed by the apriorists is not supported by the arguments they give. NORTHWESTERN UNIVERSITY Normativity and Mathematics: A Wittgensteinian Approach to the Study of Number A DISSERTATION SUBMITTED TO THE GRADUATE SCHOOL IN PARTIAL FULFILLMENT OF THE REQUIREMENTS for the degree DOCTOR OF PHILOSOPHY Field of Philosophy By J. Robert Loftis EVANSTON, ILLINOIS December 1999 © Copyright by J. Robert Loftis 1999 All Rights Reserved ii ABSTRACT Normativity and Mathematics: A Wittgensteinian Approach to the Study of Number J. Robert Loftis I argue for the Wittgensteinian thesis that mathematical statements are expressions of norms, rather than descriptions of the world. An expression of a norm is a statement like a promise or a New Year’s resolution, which says that someone is committed or entitled to a certain line of action. A expression of a norm is not a mere description of a regularity of human behavior, nor is it merely a descriptive statement which happens to entail a norms. The view can be thought of as a sort of logicism for the logical expressivist—a person who believes that the purpose of logical language (sentential connectives, quantifiers, etc.) is to make explicit commitments and entitlements that are implicit in ordinary practice. The thesis that mathematical statements are expression of norms is a kind of logicism, not because it says that mathematics can be reduced to logic, but because it says that mathematical statements play the same role as logical statements. I contrast my position with two sets of views, an empiricist view, which says that mathematical knowledge is acquired and justified through experience, and a cluster of nativist and apriorist views, which say that mathematical knowledge is either hardwired into the human brain, or justified a priori, or both. To develop the empiricist view, I look at the work of Kitcher and Mill, arguing that although their ideas can withstand the iii criticisms brought against empiricism by Frege and others, they cannot reply to a version of the critique brought by Wittgenstein in the Remarks on the Foundations of Mathematics. To develop the nativist and apriorist views, I look at the work of contemporary developmental psychologists, like Gelman and Gallistel and Karen Wynn, as well as the work of philosophers who advocate the existence of a mathematical intuition, such as Kant, Husserl, and Parsons. After clarifying the definitions of “innate” and “a priori,” I argue that the mechanisms proposed by the nativists cannot bring knowledge, and the existence of the mechanisms proposed by the apriorists is not supported by the arguments they give. iv Acknowledgements Molly Hinshaw, who normally gets paid good money to edit academic manuscripts, agreed to look this one over for free. For this and many other generosities, I thank her. I would also like to thank my committee, Arthur Fine, Meredith Williams, and Michael Williams, for their helpful comments and for working with me. I owe a debt of gratitude to the support staff at Northwestern, especially Donna Chocol, for making the years spent in the philosophy department there much more bearable. Finally, I thank my parents for their love and support. v Bibliographic Conventions Most works are referred to by the author’s name, date of publication, and page number. Reprinted works will have two publication dates; the page number is taken from the later edition. Mulitvolume works include the volume number as a Roman numeral and the page number as an Arabic numeral. Most locations in Wittgenstein’s works are identified by the section number. Citations of middle period works, which often feature long sections, include the number of the paragraph within the section enclosed in square brackets after the section number. When a page number is used in a citation of Wittgenstein’s work, I have explicitly included the abbreviation “p.” or “pp.” Wittgenstein’s works are referred to using the common abbreviations for their titles: AWL Wittgenstein’s Lectures, Cambridge 1932–1935 (Wittgenstein 1979c) BB The Blue and Brown Books (Wittgenstein 1969b) CV Culture and Value (Wittgenstein 1980) NB Notebooks, 1914–1916 (Wittgenstein 1961) OC On Certainty (Wittgenstein 1969a) PI Philosophical Investigations (Wittgenstein 1967a) PG Philosophical Grammar (Wittgenstein 1978) PR Philosophical Remarks (Wittgenstein 1975) RFM Remarks on the Foundations of Mathematics (Wittgenstein 1979a) WWK Wittgenstein and the Vienna Circle (Wittgenstein 1979b) Z Zettel (Wittgenstein 1967b) Classical and very famous works are also referred to by their abbreviations TT Boethius, The Theological Tractates (c.510/1936) CPR Immanuel Kant, Kritik der Reinen Vernunft (1781 and 1787/1969) Ethics Baruch Spinoza, Ethica Ordine Geometrico Demonstrata (1677/1992) vi Table of Contents Abstract……………………………………………………………………………………. iii Acknowledgements………………………………………………………………………... v Bibliographic Conventions………………………………………………………………… vi Introduction…………………………………………………………………………..……. 1 Chapter 1: Empiricism……………………………………………………………………... 8 Introduction……………………………………………………………………………... 8 Psychologism…………………………………………………………………………… 10 A Priori versus Empirical ……………………………………………………………... 28 Analytic versus Synthetic………………………………………………………………... 31 How Statements of Basic of Basic Arithmetic and Set Theory Are Known……………… 45 Ontology………………………………………………………………………………… 53 History………………………………………………………………………………….. 58 Chapter 2: Critique of Empiricism………………………………………………….……… 65 Introduction……………………………………………………………………………... 65 The Middle Period Argument…………………………………………………………… 71 The Late Period Argument……………………………………………………………… 82 The Historical Objection………………………………………………………………... 112 The Application Problem………………………………………………………………... 119 Conclusion……………………………………………………………………………… 127 Chapter 3: Nativism and Apriorism………………………………………………………... 129 Introduction……………………………………………………………………………... 129 Two Unworkable Definitions of ‘Innate’…………………………………………………134 The Representationalist Nativist Position……………………………………………….. 144 The Intuitionist Apriorist Position……………………………………………….……….174 The Combined Nativist-Apriorist Position………………………………………………. 197 Chapter 4: Critique of Nativism and Apriorism…………………………………….……… 221 Critique of Representationalist Nativism………………………………………………... 221 Critique of Intuitionist Apriorism………………………………………………………... 234 Critique of Dispositionalist Nativism and Negative Apriorism………………………….. 243 Chapter 5: Numbers as Normative Facts……………………………………………………258 Introduction……………………………………………………………………………... 258 Are Norms Inherently Social?……………………………………………………………260 The Objectivity of the Expression of Norms……………………………………………... 285 The Universality of Mathematical Norms……………………………………….………. 299 Reference List………………………………………………………………………………316 vii Introduction In what follows I will argue for the Wittgensteinian thesis that mathematical statements are expressions of norms rather than descriptions of the world. An expression of a norm is a claim that someone is committed or entitled to a certain line of action. Promises and New Year’s resolutions, statements where the speaker commits herself to an action or behavior, are one important class of expressions of norms. Another is statements like “You ought to be quiet during the movie,” statements that acknowledge commitments or entitlements that the speaker already believes to be in force. An expression of a norm is not a mere description of a regularity in human
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