UNIVERSITY OF CALIFORNIA, IRVINE Representing Geometry: Perception, Concepts, and Knowledge DISSERTATION submitted in partial satisfaction of the requirements for the degree of DOCTOR OF PHILOSOPHY in Philosophy by J. Ethan Galebach Dissertation Committee: Professor Jeremy Heis, Chair Professor Penelope Maddy Professor Sean Walsh 2018 c 2018 J. Ethan Galebach TABLE OF CONTENTS Page Acknowledgments iv Curriculum Vitae vii Abstract of the Dissertation viii Introduction 1 1 Does Visual Perception Have a Geometry? 5 1.1 Introduction . 6 1.2 VMS Proponents and Arguments . 8 1.2.1 What is Visual Perception? . 8 1.2.2 Five Arguments for a VMS . 11 1.2.3 Behavioral evidence against a VMS . 26 1.3 Spatial Representation according to Vision Science . 30 1.3.1 Marr's visual perception: demarcation and primary function . 30 1.3.2 Current Views on Shape and Scene Perception . 36 1.4 Conclusion and Open Questions . 43 1.5 Bibliography . 44 2 Are Euclidean Beliefs Grounded in Core Cognitive Systems? 57 2.1 Introduction . 57 2.2 The Core Cognitive Theory of Numerical Concept Acquisition . 58 2.2.1 Two Core Systems of Number . 59 2.2.2 Bootstrapping to Numerical Concepts and Arithmetical Beliefs . 62 2.3 The Core Cognitive Theory of Geometric Concept Acquisition . 65 2.3.1 Two Core Systems of Geometry . 67 2.3.2 Bootstrapping to Geometric Concepts and Beliefs . 70 2.4 Problems for the Core Cognitive Theory of Geometry . 71 2.4.1 Critique of the Two Core Systems of Geometry . 71 2.4.2 Critique of the Proposed Geometric Concepts and Beliefs . 74 2.5 Bibliography . 76 ii 3 Visual Imagination and Mathematical Entitlement 82 3.1 Introduction . 82 3.2 Imagery-based Mathematical Entitlement . 85 3.3 The Geometric Contents of Visual Imagination . 88 3.4 Geometric Knowledge through Visual Imagination . 91 3.5 Knowing the o-minimal IVT through Visual Imagination . 95 3.5.1 The Synthetic IVT . 96 3.5.2 The Analytic IVT . 97 3.5.3 The O-minimal IVT . 100 3.6 O-minimality as a Visual-Epistemological Project . 108 3.6.1 Can o-minimal objects be visualized? . 108 3.6.2 Visual Virtues of O-minimality . 112 3.7 Appendix: Modal Logic of O-minimal Structures . 118 3.8 Bibliography . 122 iii ACKNOWLEDGMENTS This project would not have been possible without the encouragement, guidance, and ex- pertise of dozens of philosophers. I want to thank these people, and especially my advisor Jeremy Heis, for developing my scholarship over many years and for treating my whole self with compassion and respect at every turn. In addition, Pen Maddy and Sean Walsh have dedicated the most time and effort to my academic and personal development. I am deeply grateful for the impact they have had on me. Below, I would like to describe my academic development over the last decade in order to thank the many other charitable souls who have contributed to the successful completion of this project. In my undergraduate studies at the University of Pittsburgh, Mark Wilson was the first pro- fessor to show a sincere interest in my philosophical questions. Without Mark's sustained attention to my senior thesis over countless one-on-one sessions in his office, I doubt I would have developed the confidence and self-worth required to articulate and defend a PhD thesis. I hold a similar appreciation for the advice and generosity of Bob Batterman and Tom Rick- etts. In my first two years of graduate school at UC Irvine, I was fortunate enough to take multiple seminars with Jeremy, Pen, and Sean on a variety of topics within the philosophy of mathematics. Although the subject matters of these seminars varied widely (e.g., Cassirer's philosophy of geometry, Heyting arithmetic, Dedekind's conceptual analysis of continuity, Freges notion of analyticity), each seminar was intensely concerned with documenting how philosophical questions arise naturally out of mathematical practice. I was absorbed by their treatment of mathematics as a science (with its own anomalies, methodological com- mitments, and historical development). These seminars led me to books like Kitcher's The Nature of Mathematical Knowledge, Lakatos' Proofs and Refutations, Mancosu's (ed.) Phi- losophy of Mathematical Practice, Bos' Redefining Geometrical Exactness, and Giaquinto's Visual Thinking in Mathematics, all of which seemed to me to be asking and answering the most interesting and important questions regarding the nature of geometric knowledge. iv In the summer of 2014, I became more confident in the legitimacy and durability of these ap- proaches to the philosophy of mathematics when I participated in the month-long PhilMath Intersem in Paris organized by Mic Detlefsen. While I was there I presented a paper on Giaquinto's theory of mathematical entitlement (which eventually became Chapter 3) and I had dozens of invaluable meal-length conversations with philosophers and historians of mathematics from a handful of countries. In particular, Andrew Arana, Karine Chemla, Mic Detlefsen, Katherine Dunlop, Renaud Chorlay, Paolo Mancosu, David Rabouin, and Vincenzo De Risi all displayed genuine interest in my research and they each offered mean- ingful advice and encouragement for my academic future. My exposure to this constellation of ideas and well-rounded academics solidified my commitment to studying the history of geometry and the nature of spatial representation. Three months later, I presented a logic- focused variant of my paper on Giaquinto at another interdisciplinary conference called Spatial Cognition in Riga, Latvia. This is where I met Nora Newcombe, Barbara Landau, and many other psychologists and linguists of spatial representation. They introduced me to an adjacent world of concepts, theorists, and debates that inform all of my chapters. The Spatial Cognition conference made me aware of the close alignment between philosophical and psychological questions about the nature of spatial representation. Giaquinto's book was merely the tip of an iceberg. When I returned, Pen Maddy introduced me to Stephen Palmers Vision Science and David Marr's Vision, both of which became my portals into the debates and assumptions driving various traditions within cognitive science. The most general question of my dissertation had taken shape: what are the prospects for using experimental and theoretical psychology to answer common philosophical questions about geometric representation? In terms of isolating questions within the philosophical dis- course on spatial perception, Pen's seminar on vision theory in the Winter of 2016 introduced me to the work of Gary Hatfield, and a reading group with Jeremy and Alysha Kassam in Summer of 2016 introduced me to the work of Tyler Burge. The conversations about the border of perception and whether its content had a metric relational structure arose out of v these two experiences. Chapter 1 was born. Regarding concept acquisition, a handful of LPS colloquia on arithmetical knowledge (as well as the works of Newcombe, Landau, and Maddy) pointed me toward to the theory of core number cognition as a useful framework for asking metaphysical, semantic, and epistemological questions about mathematics. The more recent literature on core cognitive theory of geometric concept acquisition seemed to offer parallel hopes for understanding geometric knowledge. In the Spring of 2016, Pen organized a reading group that allowed me to explore and discuss the claims core geometric cognition. The disanalogies I discovered during this reading group gave rise to Chapter 2. The third and final chapter on mathematical entitlement has a longer history. In my first year at UC Irvine, after reading Giaquinto's book (mentioned above), I started discussing the philosophical significance of the nineteenth-century “flight from intuition" with Sean and Jeremy. Initially, I was searching for alternatives to set-theoretic reductionism in the philosophy of geometry, but this metaphysical concern was eventually overshadowed by my growing interest in the visual imagery content that Giaquinto appeals to in his theory of geometric knowledge. While trying to make this content logically precise, Sean Walsh intro- duced me to the model-theoretic field of o-minimality. After reading some of the early texts of this field, it became clear that many of these logicians developed o-minimality with some intention to make precise the spatial content of human mental representation. The content and restrictions of o-minimal structures even seemed to complement the cognitive science models of visual imagery that Giaquinto was using in his theory of mathematical entitle- ment. These elements combined to form the main concern of my third chapter: delimiting the scope of mathematical theorems that can be known through imagery-based mathematical entitlement. In addition to the philosophers mentioned above, I would not have been able to finish writing this dissertation without the loving support of my family (Elliott, Harold, Jean, Karen, Mark), my partner (Elaine), and my closest friends (Sam, Alysha, Valerie, and Tyler). vi CURRICULUM VITAE J. Ethan Galebach EDUCATION Doctor of Philosophy in Philosophy 2018 University of California, Irvine Irvine, CA Masters of Arts in Philosophy 2015 University of California, Irvine Irvine, CA Bachelor of Science in Physics and Philosophy 2012 University of Pittsburgh Pittsburgh, PA TEACHING EXPERIENCE Instructor (PHIL 110) 2018 Santa Ana College Santa Ana, CA Instructor (LPS 29, 60, 105, 205) 2015{2018 University of California, Irvine Irvine, CA Teaching Assistant 2012{2015 University of California, Irvine Irvine, CA vii ABSTRACT OF THE DISSERTATION Representing Geometry: Perception, Concepts, and Knowledge By J. Ethan Galebach Doctor of Philosophy in Philosophy University of California, Irvine, 2018 Professor Jeremy Heis, Chair In this dissertation, I investigate how humans represent space and other geometric enti- ties. The topics of my three chapters are delimited by three kinds of spatial representation: perception, conception, and propositional knowledge. In chapter 1, argue against the widely- held philosophical view that the content of visual perception includes a \geometry," or more precisely, a metric space.