Simplicity: Ideals of Practice in Mathematics and the Arts (Mathematics, Culture, and the Arts)

To find "criteria of simplicity" was the goal of David Hilbert's recently discovered twenty-fourth problem on his renowned list of open problems given at the 1900 International Congress of Mathematicians in Paris. At the same time, simplicity and economy of means are powerful impulses in the creation of artworks. This was an inspiration for a conference, titled the same as this volume, that took place at the Graduate Center of the City University of New York in April of 2013. This volume includes selected lectures presented at the conference, and additional contributions offering diverse perspectives from art and architecture, the philosophy and history of mathematics, and current mathematical practice. Preface 7 Quotes 10 Acknowledgments 13 Contents 14 Contributors 16 Inner Simplicity vs. Outer Simplicity 22 References 33 The Complexity of Simplicity: The Inner Structure of the ArtisticImage 36 References 44 Thinking in Four Dimensions 48 References 54 Kant, Co-Production, Actuality, and Pedestrian Space: Remarks on the Philosophical Writings of Fred Sandback 57 Fred Sandback 57 Philosophy at Yale 58 Kantianism, Co-Production, and the Sublime 59 Synthesis 62 Temporality 63 Actuality 64 ``It Is of No Assistance...'' 65 References 66 What Simplicity Is Not 69 1. Simplicity is not outside existence 69 2. Simplicity is not totally subjective 70 3. Simplicity is not necessarily timeless 70 4. Simplicity isn't necessarily functional 71 5. Simplicity isn't necessary 71 6. Simplicity does not equal perfection 72 7. Shallowness is often the wolf in simplicity's clothing 72 8. Shorter does not equal simpler 73 9. Simplicity can always be improved upon 73 10. Simplicity resides in movement 73 Endnote: A mathematical concept of ``Simplicity'' and ``Depth'' 74 References 76 Constructing the Simples 79 The Complexity of Simplicity 79 The Dream 81 The Artworld 84 Mathematics 86 Impurity 87 Residues 88 Creation 91 References 92 The Simplicity Postulate 96 References 100 The Experience of Meaning 103 References 117 Math Currents in the Brain 122 Psychology of Mathematics and Mathematics of Psychology 124 Universality and Evolution 125 Learning Languages and Learning Mathematics 127 Comments and Links 130 References 132 bc, becuz, Because ASCII 140 References 141 ``Abstract, Directly Experienced, Highly Simplified, and Self-Contained'': Discourses of Simplification, Disorientation, and Process in the Arts 144 Primary Structures, Forms, Ideas 144 Object 147 Dissolution of the Object/Form Unitary Object to Processes 149 Chaos/Creative Processes/Multiplicity in the One: La Chair du Visible 150 Fragments of Discourse (Forest Paths) 154 References 155 Remarks on Simple Proofs 159 Introduction 159 Formalization 160 Foundational Theories 161 Purity of Method 162 Brevity 164 Proofs, Short and Simple 164 References 165 The Fluidity of Simplicity: Philosophy, Mathematics, Art 168 Characterization of the Ideal 169 ``Reflecting'' Judgment 173 The Fluidity of Simplicity 176 Simplicity as Artistry 181 Mel Bochner 182 Fred Sandback 184 References 186 ``Mathematical Typography'' (After Donald Knuth, 1978) 191 References 199 Simplicity via Complexity: Sandboxes, Reading Novalis 203 Dual Motion, Sandboxes 203 Simplicity and Complexity, Spiraling 204 Simplicity from Complexity in Mathematics 207 Incompleteness, on the Way to Completeness 207 Therapeutic Completeness? Many Possible Worlds 209 Logic, Geometrizing Away from Language? 210 From Geometry to Logic, Breaking the Language 211 The Price: Loss of Compactness 213 Sandboxes for Simplicity: A Dynamics of Adjunction? 214 Coda: Endless Loop? 214 References 215 On the Alleged Simplicity of Impure Proof 219 The Purity Debate in Overview 219 Simplicity and Impurity in Mathematical Practice 221 A Formal Evaluation of Simplicity of Impure Proof 225 The Theories 225 Impurity 227 Conservativity 230 Speed-Up 230 The Evidence 232 Evaluating the Evidence 233 Conclusions 233 References 235 Minimalism and Foundations 241 I 242 II 244 III 246 References 249 Economy of Thought: A Neglected Principle of MathematicsEducation 253 Introduction 253 What Can Be Simpler Than 3-1=2? 254 Encapsulation and De-Encapsulation 256 De-Encapsulation in Action: ``Questions'' Method 257 Self-Directing Questions 258 Distributed Quantities 260 The Arithmetic/Algebra Boundary 262 Uniform Convergence and ``Likeness'' 264 Order and Equivalence—And Abstraction by Irrelevance 267 Arnold's Principle and the ``Questions Method'' in the Historic and Social Context 270 The Economy of Thought 271 A Bonus of ``Economy of Thought'': ``Reduced Fatigability'' 274 References 276 Simplicity Is the Point 280 References 285 Appendix A: Simplicity, in Mathematics and in Art 288 Not an Absolute Notion 288 Visceral Encounters 290 Contrasting Groups 291 Appendix B: Conference Program 295 Wednesday, April 3 295 Thursday, April 4 298 Friday April 5 302 Art Installation 306 Acknowledgements 306 Index 310 Front Matter ....Pages i-xx Inner Simplicity vs. Outer Simplicity (Étienne Ghys)....Pages 1-14 The Complexity of Simplicity: The Inner Structure of the Artistic Image (Juhani Pallasmaa)....Pages 15-26 Thinking in Four Dimensions (Dusa McDuff)....Pages 27-35 Kant, Co-Production, Actuality, and Pedestrian Space: Remarks on the Philosophical Writings of Fred Sandback (Juliette Kennedy)....Pages 37-48 What Simplicity Is Not (Maryanthe Malliaris, Assaf Peretz)....Pages 49-58 Constructing the Simples (Curtis Franks)....Pages 59-75 The Simplicity Postulate (Marjorie Senechal)....Pages 77-83 The Experience of Meaning (Jan Zwicky)....Pages 85-103 Math Currents in the Brain (Misha Gromov)....Pages 105-118 bc, becuz, Because ASCII (Kate Shepherd)....Pages 119-126 “Abstract, Directly Experienced, Highly Simplified, and Self-Contained”: Discourses of Simplification, Disorientation, and Process in the Arts (Riikka Stewen)....Pages 127-141 Remarks on Simple Proofs (Rosalie Iemhoff)....Pages 143-151 The Fluidity of Simplicity: Philosophy, Mathematics, Art (Juliet Floyd)....Pages 153-175 “Mathematical Typography” (After Donald Knuth, 1978) (Dexter Sinister, D. Reinfurt, S. Bailey)....Pages 177-188 Simplicity via Complexity: Sandboxes, Reading Novalis (Andrés Villaveces)....Pages 189-204 On the Alleged Simplicity of Impure Proof (Andrew Arana)....Pages 205-226 Minimalism and Foundations (Spencer Gerhardt)....Pages 227-238 Economy of Thought: A Neglected Principle of Mathematics Education (Alexandre V. Borovik)....Pages 239-265 Simplicity Is the Point (Dennis Sullivan)....Pages 267-274 Back Matter ....Pages 275-305