An Unexplored Aspect of Following a Rule by Rachael Elizabeth Driver Bachelor of Science, University of New South Wales, 2001 Bachelor of Liberal Studies (Honours), University of Sydney, 2006 Submitted to the Graduate Faculty of the Dietrich School of Arts and Sciences in partial fulfillment of the requirements for the degree of Doctor of Philosophy University of Pittsburgh 2020 UNIVERSITY OF PITTSBURGH DIETRICH SCHOOL OF ARTS AND SCIENCES This dissertation was presented by Rachael Elizabeth Driver It was defended on July 27, 2020 and approved by Robert Batterman, Distinguished Professor of Philosophy Warren Goldfarb, Walter Beverly Pearson Professor of Modern Mathematics and Mathematical Logic, Harvard University Erica Shumener, Assistant Professor of Philosophy Dissertation Co-Directors: Thomas Ricketts, Professor of Philosophy Mark Wilson, Distinguished Professor of Philosophy ii Copyright © by Rachael Elizabeth Driver 2020 iii An Unexplored Aspect of Following a Rule Rachael Elizabeth Driver, PhD University of Pittsburgh, 2020 Though Wittgenstein has been most often identified as opposing Platonism in his writings about mathematics, I argue that Wittgenstein’s radical contextualism about mathematics finds its most natural opponent not in Platonism, but in a variety of formalism. One of Wittgenstein’s obvious formalist targets is his colleague the mathematician G. H. Hardy. If we discard this—still influential—picture of mathematics and replace it with a more nuanced account of mathematical activity as exemplified in the metamathematical thinking of the nineteenth century mathematician Augustus De Morgan, the example of the wayward pupil takes on a different significance. Against a more complex background, the wayward pupil can be reinterpreted as representing an exemplar of mathematical discovery. I consider the example of the nineteenth century engineer Oliver Heaviside whose unconventional approach in mathematics, driven by a need to efficiently elicit results from complex formulae for the purposes of aiding his research in electrical engineering, resulted in extraordinary mathematical advances. Yet, his approach to algebraic manipulation has the aspect of a wayward pupil. The wayward pupil, who may be making an error according to our ordinary criteria of rule-following, may be initiating new and fruitful paths. This possibility is largely unexamined in the larger discussion of Wittgenstein’s remarks on following a rule, and it explains Wittgenstein’s hesitation to label the wayward pupil’s actions straightforwardly incorrect. iv Table of Contents Preface ............................................................................................................................................ x Abbreviations ............................................................................................................................... ix 1.0 Introduction ............................................................................................................................. 1 2.0 Following a Rule ...................................................................................................................... 6 2.1 The Wayward Pupil ....................................................................................................... 7 2.2 The Coherence of the Wayward Pupil........................................................................ 12 2.3 The Wayward Pupil as a Grammatical Investigation ............................................... 21 2.3.1 Wittgenstein’s Use of Thought Experiments ...................................................21 2.3.2 The Wayward Pupil as an Episode of a Grammatical Investigation ............28 3.0 Radical Contextualism.......................................................................................................... 32 3.1 The Importance of Context .......................................................................................... 33 3.2 Context and Following a Rule ..................................................................................... 42 4.0 Hardyan Formalism.............................................................................................................. 48 4.1 Conventionalism vs. Platonism .................................................................................... 49 v 4.2 Hardy’s Platonism ........................................................................................................ 52 4.3 Hardy’s Formalism ...................................................................................................... 54 4.4 Divergent Series ............................................................................................................ 60 4.5 Hardy and Divergent Series ........................................................................................ 69 5.0 De Morgan’s Alternative ...................................................................................................... 77 5.1 The British Algebraists ................................................................................................ 79 5.2 Negative and Complex Numbers ................................................................................. 81 5.3 De Morgan’s Conception of Mathematics .................................................................. 87 5.4 Hardy’s Criticism ......................................................................................................... 93 5.5 Hardy and William Frend ........................................................................................... 96 5.6 Asymptotic Series ......................................................................................................... 97 6.0 Reinterpreting Wittgenstein .............................................................................................. 103 6.1 Wittgenstein’s Ambivalence ...................................................................................... 104 6.2 Shifting River-beds ..................................................................................................... 110 6.3 Hardy Revisited .......................................................................................................... 116 vi 6.4 The Wayward Pupil Revisited ................................................................................... 119 6.5 Heaviside ..................................................................................................................... 122 7.0 Conclusion ........................................................................................................................... 127 Bibliography .............................................................................................................................. 131 vii List of Figures Figure 1 The unit square ............................................................................................................ 62 Figure 2 Graph of and ......................................................................................................... 63 1 1 2 x x viii Abbreviations BB WITTGENSTEIN, L., Preliminary studies for the Philosophical Investigations : generally known as The blue and brown books, Blackwell, Oxford 1958. OC WITTGENSTEIN, L., ANSCOMBE, G. E. M. – PAUL, D. (eds.), On Certainty, Harper & Row, New York 1972. PR WITTGENSTEIN, L., RHEES, R. (ed.), Philosophical Remarks, Blackwell, 1989. FM WITTGENSTEIN, L., WRIGHT, G. H. VON – RHEES, R. – ANSCOMBE, G. E. M. (eds.), Remarks on the Foundations of Mathematics, ANSCOMBE, G. E. M. (trad.), Basil Blackwell, Oxford 1978Revised ed. AWL WITTGENSTEIN, L., AMBROSE, A. (ed.), Wittgenstein’s Lectures, Cambridge 1932–1935 From the Notes of Alice Ambrose and Margaret Macdonald, Prometheus Books, Amherst, NY 2001. PI WITTGENSTEIN, L., HACKER, P. M. S. – SCHULTE, J. (eds.), Philosophical Investigations, ANSCOMBE, G. E. M. — HACKER, P. M. S. — SCHULTE, J. (trads.), Wiley-Blackwell, Oxford 2009Revised 4t. Z WITTGENSTEIN, L., ANSCOMBE, G. E. M. – WRIGHT, G. H. VON (eds.), Zettel, ANSCOMBE, G. E. M. (trad.), Basil Blackwell, Oxford 1967. ix Preface This dissertation has taken me a long time to write. Many people have helped me during the years that I have worked on it. I would like to thank my advisor Mark Wilson whose patience and encouragement extended to the entire period it took me to write it. Thanks also go to him for pointing me in the direction of the British Algebraists, G. H. Hardy and Oliver Heaviside. I would also like to express my gratitude to my co-advisor Tom Ricketts and my committee members Bob Batterman, Warren Goldfarb and Erica Shumener. I would also like to thank John McDowell for his help during the early stages of my dissertation. I am grateful to Ken Manders for his constant willingness to discuss my work. I would like to thank those who have made my time in Pittsburgh happy and fruitful. I am grateful for the friendship shown me by my cohort Ori Beck, Jason Carnell, Casey Doyle, Jenn Jhun, Joe Milburn, Eric Palmer and Preston Stovall, and especially my friend and countryman Benjamin Schulz. I would like to thank my fellow graduate students Dan Addison, Mikio Akagi, Sophia Arbeiter, David de Bruijn, Josh Eisenthal, Vera Flocke, Jack Fortune, Kim Frost, Chuck Goldhaber, Samantha Hancox, Leif Hancox-Li, Ulf Hlobil, Brandon Hogan, Kohei Kishida, Gus Law, Kathryn Lindeman, Adam Marushak, Travis McKenna, Katharina Nieswandt, Tyke Nunez, Annalisa Paese, Raja Rosenhagen, Michael Sands, Shawn Standefer, Robert Steel, Jonathan Surovell, Hao Tang and