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Leibniz's Laboratory of Concepts LEIBNIZ’S LABORATORY OF CONCEPTS: THE STATUS AND STRUCTURE OF INFINITESIMALS AS METAPHYSICAL LABORATORY by TZUCHIEN THO (Under the Direction of O. BRADLEY BASSLER) ABSTRACT In this dissertation I take up a reading of Leibniz’s metaphysics from the perspective of his mathematical reflections on the nature of infinitesimals. A longstanding tradition of interpreting Leibniz’s treatment of infinitesimals has focused on the problem of its logical consistency as the key to unlocking its relation to the rest of his philosophy. Scholarly inquiry around the status of “infinitesimals” throughout Leibniz’s work has thus tended toward logical reduction. By pointing to some key moments where this approach runs into difficulty, I develop an alternative where the speculative consequences of Leibniz’s infinitesimals can be viewed through the lens of the mathematical, rather than logical, concepts produced through his mathematical work. The issue of the status of infinitesimals thus provides a means by which to understand important aspects of his metaphysics which is often misunderstood. In turn, this exploration allows a rereading of important developments in Leibniz’s treatment of the structure of reality as an actual infinitely divided one. This rereading provides a different means to understand the role that the reality of bodies and their motion plays in his mature metaphysics. As such, I suggest that Leibniz’s mathematical reflections serve as a laboratory of concepts which not only have profound consequences for his own means of thinking through problems but can also guide us in understanding aspects of his method and scope of his work. The first chapter sets the stage for the rest of the dissertation by a cursory examination of the short-comings of logical reduction in order to underline the nature of the speculations afforded by Leibniz’s lifelong engagement with the infinite and infinitesimal. The second and third chapters treat the problem of the status of infinitesimals in Leibniz’s work through a critique of contemporary interpretations and underlining the importance of his mathematical reflections through epistemological and metaphysical lenses. The fourth and fifth chapters treat the development of Leibniz’s commitment to the reality of corporeal bodies and motion in the light of his mathematically embedded assertions about the actual infinite dividedness of reality. INDEX WORDS: Leibniz, infinite, infinitesimal, indivisible, syncategorematic, metaphysics of motion, corporeal substance, labyrinth of the continuum. LEIBNIZ’S LABORATORY OF CONCEPTS: THE STATUS AND STRUCTURE OF INFINITESIMALS AS METAPHYSICAL LABORATORY by TZUCHIEN THO BA., LOYOLA UNIVERSITY CHICAGO, 2002 A Dissertation Submitted to the Graduate Faculty of The University of Georgia in Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY ATHENS, GEORGIA 2011 © 2011 Tzuchien Tho All Rights Reserved LEIBNIZ’S LABORATORY OF CONCEPTS: THE STATUS AND STRUCTURE OF INFINITESIMALS AS METAPHYSICAL LABORATORY by TZUCHIEN THO Major Professor: O. Bradley Bassler Committee: Elizabeth Brient Richard D. Winfield Thomas Cerbu Electronic Version Approved: Maureen Grasso Dean of the Graduate School The University of Georgia May 2011 DEDICATION To my grandmothers 涂陳璧人 and 杜劉月嬌 iv ACKNOWLEDGEMENTS Thanks, above all, goes to Amy M. Anderson, my loving wife who tolerated my intolerable behavior during this period of writing and research and who continues daily to encourage me in this prolonged paradoxical situation. She reminds me daily that truth is sought in actions. I also mention the memory of the late Dr. Hao-Yuan (Joseph) Yeh who first allowed me to understand that there is something eternal in our work. Dr. O. Bradley Bassler’s depth of friendship and breath of thought provided the model for the stakes of unceasing philosophical struggle. My brother Tho Tzusing has always been a mirror for my thoughts and actions, what is reflected is neither of us but something that continually proves the allure of unfettered honesty and the value of a true conversation. My parents, Mr. Tho Kee Ping and Mrs. Tho-Tu Hsiu Chen, have constantly provided the reminder to overcome myself. To Alain Badiou, I owe gratitude for the time he took to allow me to challenge him on his claims and in return, challenge me. The memory of my late paternal grandfather, Tho Nai Ping, accompanies me daily. My maternal grandfather, Tu Chiang Shiang, taught me how to cherish texts. I extend gratitude to my childhood English teacher, Ms. Luan, who with great care and precision first taught me how to use an English sentence. I thank my teachers in higher education who have left an indelible mark on my thinking and in my life: David Ingram, Andrew Cutrofello, Patricia Huntington, the late Hans Siegfried, Thomas Cerbu, Edward Halper, Richard Winfield, Elizabeth Brient and Clarke Wolfe v This process would have hardly been as meaningful without my friends who have stayed with me, encouraged me and criticized me through the long conversations that thought requires of us. I cannot name all of you here but special mention is necessary: Colin Mcquillan, Nathaniel Holdren, Zachary Luke Fraser, Pietro Bianchi, Giuseppe Bianco, Oliver Feltham, Bruno Besana, Ryan Anderson, Daniel Kugler, Benjamin Bishop, Patricia Goldsworthy, Krisha McCune, François Farellacci, Brian Howell, Nathan Vacher. vi TABLE OF CONTENTS Page ACKNOWLEDGEMENTS.............................................................................................................v LIST OF FIGURES ....................................................................................................................... ix CHAPTER 1 MOTIVATION AND INTRODUCTION.....................................................................1 1. Some mathematical and philosophical motivations.............................................1 2. Entering the laboratory: A comparison of arguments concerning the infinite...12 3. The ignorance of mathematics: Russell and the priority of logic ......................25 4. The immanence of the mathematical as a laboratory of concepts .....................39 PREFACE TO CHAPTERS 2 AND 3: THE STATUS OF THE INFINITE AND INFINITESIMALS......................................................................................................42 2 THE STATUS OF FICTION.......................................................................................48 1. The rhetoric of “fictions”...................................................................................48 2. The contemporary status of the status problem: Robinson and Ishiguro...........50 3. Infinite/simal fiction and the false dichotomy of the sensible and the ideal......65 4. What is the syncategorematic infinite/simal? A closer look..............................78 5. The syncategorematic reframing of the discrete and the continuous.................87 3 THE STATUS OF THE WELL-FOUNDED..............................................................95 1. Against reduction and dissolution: Preserving the power of fiction..................95 2. Ideal, actual and imaginary................................................................................98 vii 3. Nothing without motion: Leibniz’s indivisible as infinitesimal ......................118 4. From indivisibles to actual fictions..................................................................136 5. Consequences of the syncategorematic infinitesimal ......................................149 PREFACE TO CHAPTERS 4 AND 5: METAPHYSICAL STRUCTURE AND THE INFINITE AND INFINITESIMAL...........................................................................158 4 THE STRUCTURE AND REALITY OF BODY.....................................................164 1. What is structure?............................................................................................164 2. The problem of body........................................................................................176 3. The correspondence with Arnauld and the new models of substance .............193 4. From well-founded phenomena to well-founded reality: Body and life..........209 5 MOTION AND METAPHYSICAL STRUCTURE..................................................223 1. Force and substance.........................................................................................223 2. Return to the reality of infinitesimals: From force to motion (and back)........249 3. Mathematical and metaphysical structure: From phenomenon to reality .......263 REFERENCES ............................................................................................................................267 APPENDICIES............................................................................................................................271 A Note on Abbreviations...............................................................................................271 viii LIST OF FIGURES Page Figure 1:.........................................................................................................................................22 Figure 2:.........................................................................................................................................60 Figure 3:.........................................................................................................................................82 Figure 4:.......................................................................................................................................125 Figure 5:.......................................................................................................................................138
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