Experiencing Mathematics : What Do We Do, When We Do Mathematics?

Most mathematicians, when asked about the nature and meaning of mathematics, vacillate between the two unrealistic poles of Platonism and formalism. By looking carefully at what mathematicians really do when they are doing mathematics, Reuben Hersh offers an escape from this trap. This book of selected articles and essays provides an honest, coherent, and clearly understandable account of mathematicians' proof as it really is, and of the existence and reality of mathematical entities. It follows in the footsteps of Poincaré, Hadamard, and Polya. The pragmatism of John Dewey is a better fit for mathematical practice than the dominant "analytic philosophy". Dialogue, satire, and fantasy enliven the philosophical and methodological analysis. Reuben Hersh has written extensively on mathematics, often from the point of view of a philosopher of science. His book with Philip Davis, The Mathematical Experience, won the National Book Award in science. Hersh is emeritus professor of mathematics at the University of New Mexico. Experiencing Mathematics What do we do, when we do mathematics......................................... 1 Contents........................................................................................... 8 Preface............................................................................................ 12 Credits............................................................................................ 14 Acknowledgments.................................................................................... 18 Overture........................................................................................... 20 The Ideal Mathematician (with Philip J. Davis)..................................................... 24 Manifesto.......................................................................................... 32 Self-introduction.................................................................................. 36 Mathematics Has a Front and a Back................................................................. 54 Part 1 “Mostly for the right hand”................................................................. 60 Introduction................................................................................... 62 True Facts About Imaginary Objects............................................................. 66 Mathematical Intuition (Poincare, Polya, Dewey)................................................ 70 Mathematical Intuition..................................................................... 78 Polya...................................................................................... 80 Mental Models.............................................................................. 82 Mental Models Subject to Social Control.................................................... 86 Dewey and Pragmatism....................................................................... 87 References................................................................................. 89 Acknowledgments............................................................................ 91 To Establish New Mathematics, We Use Our Mental Models And Build On Established Mathematics.... 92 Introduction............................................................................... 92 Wiles’ proof of FLT isn’t an axiomatic proof, it’s a “Mathematicians’ Proof”............... 93 Established mathematics.................................................................... 94 Mathematicians’ proof vs. axiomatic proof.................................................. 96 Mathematicians’ proof is semantic, not syntactic........................................... 97 Established mathematics is fallible........................................................ 99 Published vs. private, rigorous vs. plausible..............................................101 Established mathematics is not controversial...............................................102 Acknowledgments............................................................................104 How Mathematicians Convince Each Other or “The Kingdom of Math is Within You”..................108 Introduction and abstract..................................................................108 A quote from Hardy.........................................................................109 What some mathematicians say they are doing................................................110 The Materialist versus the Platonist: Changeux and Connes..................................115 What, then, is a mathematicians’ proof ?...................................................120 Relation between formal proof and mathematicians’ proof....................................121 Aristotle, Kant, and Locke.................................................................123 Is this mere Platonism?....................................................................123 Heron’s area theorem.......................................................................124 Conclusions................................................................................127 Acknowledgments............................................................................127 Appendix...................................................................................127 On the interdisciplinary study of mathematical practice, with a real live case study...........134 Wings, not foundations!....................................................................144 1. What foundations?...................................................................144 2. Lived experience as “foundation”....................................................147 Inner Vision, Outer Truth......................................................................150 Mathematical Practice as a Scientific Problem..................................................156 Atiyah’s pleasant surprise.................................................................156 Does “existence” matter?...................................................................156 For a multi-disciplined study of mathematical practice.....................................158 The basic problem..........................................................................160 Timely or timeless?........................................................................161 Conclusion.................................................................................163 Educational implications...................................................................163 Proving is Convincing and Explaining...........................................................166 I. What is proof?..........................................................................166 II. Proof among professional mathematicians................................................166 III. Three meanings of “proof”.............................................................168 IV. Variation in proof standards...........................................................169 V. The four-color theorem..................................................................169 VI. Proof in our classrooms................................................................172 VII. Coda..................................................................................174 Fresh Breezes in the Philosophy of Mathematics.................................................176 Foundations lost...........................................................................176 Phil / m and pliii / sci...................................................................177 Taking the test............................................................................179 Definition of mathematics......................................................................182 Introduction to “18 Unconventional Essays on the Nature of Mathematics”........................186 Part 2 “Mostly for the left hand”..................................................................192 Introduction...................................................................................194 Rhetoric and Mathematics (with Philip J. Davis)................................................196 Part I: Mathematics as Rhetoric............................................................197 Part 2: Rhetoric in Mathematics............................................................201 Closure....................................................................................207 Math Lingo vs. Plain English: Double Entendre..................................................210 Independent Thinking...........................................................................214 The “Origin” of Geometry.......................................................................218 The Wedding....................................................................................224 Mathematics and Ethics.........................................................................226 Ethics for Mathematicians......................................................................232 Under-represented Then Over-represented: A Memoir of Jews in American Mathematics..............236 Paul Cohen and Forcing in 1963.................................................................246 Part 3 Selected book reviews.......................................................................252 Introduction...................................................................................254 Review of How Mathematicians Think by William Byers............................................260 Review of The Mathematician’s Drain by David Ruelle............................................266 Review of Perfect Rigor by Masha Gessen........................................................270 Review of Letters to a Young Mathematician by Ian Stewart......................................274 Review of Number and Numbers by Alain Badiou...................................................276 Part 4 About the Author............................................................................282 An amusing elementary example..................................................................284 Annotated research bibliography................................................................286 Curriculum Vitae...............................................................................290 List of articles...............................................................................292 Index..............................................................................................298 Back Cover.........................................................................................311 The question “What am I doing?” haunts many creative people, researchers, and teachers. Mathematics, poetry, and philosophy can look from the outside sometimes as ballet en pointe, and at other times as the flight of the bumblebee. Reuben Hersh looks at mathematics from the inside; he collects his papers written over several decades, their edited versions, and new chapters in his book Experiencing Mathematics, which is practical, philosophical, and in some places as intensely personal as Swann's madeleine. —Yuri Manin, Max Planck Institute, Bonn, Germany What happens when mid-career a mathematician unexpectedly becomes philosophical? These lively and eloquent essays address the questions that arise from a crisis of reflectiveness: What is a mathematical proof and why does it come after, not before, mathematical revelation? Can mathematics be both real and a human artifact? Do mathematicians produce eternal truths, or are the judgments of the mathematical community quasi-empirical and historically framed? How can we be sure that an infinite series that seems to converge really does converge? This collection of essays by Reuben Hersh makes an important contribution. His lively and eloquent essays bring the reality of mathematical research to the page. He argues that the search for foundations is misleading, and that philosophers should shift from focusing narrowly on the deductive structure of proof, to tracing the broader forms of quasi-empirical reasoning that star the history of mathematics, as well as examining the nature of mathematical communities and how and why their collective judgments evolve from one generation to the next. If these questions keep you up at night, then you should read this book. And if they don't, then you should read this book anyway, because afterwards, they will! —Emily Grosholz, Department of Philosophy, Penn State, Pennsylvania, USA Most mathematicians, when asked about the nature and meaning of mathematics, vacillate between the two unrealistic poles of Platonism and formalism. By looking carefully at what mathematicians really do when they are doing mathematics, Reuben Hersh offers an escape from this trap. This book of selected articles and essays provides an honest, coherent, and clearly understandable account of mathematicians'proof as it really is, and of the existence and reality of mathematical entities. It follows in the footsteps of Poincaré, Hadamard, and Polya. The pragmatism of John Dewey is a better fit for mathematical practice than the dominant “analytic philosophy”. Dialogue, satire, and fantasy enliven the philosophical and methodological analysis. Reuben Hersh has written extensively on mathematics, often from the point of view of a philosopher of science. His book with Philip Davis, The Mathematical Experience, won the National Book Award in science. Hersh is emeritus professor of mathematics at the University of New Mexico. "Most mathematicians, when asked about the nature and meaning of mathematics, vacillate between the two unrealistic poles of Platonism and formalism. By looking carefully at what mathematicians really do when they are doing mathematics, Reuben Hersh offers an escape from this trap. This book of selected articles and essays provides an honest, coherent, and clearly understandable account of mathematicians' proof as it really is, and of the existence and reality of mathematical entities. It follows in the footsteps of Poincaré, Hadamard, and Polya. The pragmatism of John Dewey is a better fit for mathematical practice than the dominant 'analytic philosophy'. Dialogue, satire, and fantasy enliven the philosophical and methodological analysis."--Provided by publisher "Most mathematicians, when asked about the nature and meaning of mathematics, vacillate between the two unrealistic poles of Platonism and formalism. By looking carefully at what mathematicians really do when they are doing mathematics, Reuben Hersh offers an escape from this trap. This book of selected articles and essays provides an honest, coherent, and clearly understandable account of mathematicians' proof as it really is, and of the existence and reality of mathematical entities. It follows in the footsteps of Poincaré, Hadamard, and Polya. The pragmatism of John Dewey is a better fit for mathematical practice than the dominant 'analytic philosophy'. Dialogue, satire, and fantasy enliven the philosophical and methodological analysis."--page [4] of cover. Cover -- Title page -- Contents -- Preface -- Permissions and acknowledgments -- Acknowledgments -- Overture -- The ideal mathematician (with Philip J. Davis) -- Manifesto -- Self-introduction -- Chronology -- Mathematics has a front and a back -- Part I. Mostly for the right hand -- Introduction to part 1 -- True facts about imaginary objects -- Mathematical intuition (Poincaré, Polya, Dewey) -- To establish new mathematics, we use our mental models and build on established mathematics -- How mathematicians convince each other or "The kingdom of math is within you