Breger and myself, Carlo Cellucci, Donald Gillies, Penelope Maddy, Paolo Mancosu, François De Gandt, Jaakko Hintikka, Madeline Muntersbjorn, Carl

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Marshall Mitchell
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1 Preface This book stems from my engagement with philosophers of mathematics at the University of Rome, the University of Paris, the University of London and King s College, London, and members of the newly formed Association for the Philosophy of Mathematical Practice, as well as mathematics seminars at the Pennsylvania State University, in particular those of the number theorist Wen-Ch ing (Winnie) Li. It also stems from my membership in the Center for Fundamental Theory in the Institute for Gravitation and the Cosmos at Penn State. Recently I went over there to help celebrate the fact that for the past 15 years, the IGC has been one of the main incubators of the spectacular discovery of the gravitational waves that Einstein predicated a century ago. My membership as the resident Humanist began seven years ago, and resulted in a workshop on cosmology and time where philosophers and scientists engaged in fruitful conversation now collected in a special issue of Studies in the History and Philosophy of Modern Physics, Vol. 52, Part A (2015). One of the aspects of research at the IGC that struck me right away was the substantive collaboration between theoretical physicists and highly empirical astronomers, despite the striking disparity in the kinds of texts they generated. Watching their interchanges helped me to develop further ideas about mathematical and scientific research, which I began to formulate during a sabbatical leave supported by a fellowship from the City of Paris and of course also Penn State. So I am indebted to colleagues far away and close to home. The workshop on cosmology and time took place with support from then Head of the Philosophy Department Shannon Sullivan and Dean Susan Welch, warm support from Abhay Ashtekar and Murat Gunaydin at the IGC and Professor of Physics Emeritus Gordon Fleming, as well as John Norton, Director of the Center for History and Philosophy of Science at the University of Pittsburgh. We listened to exchanges between Bryan Roberts and Abhay Ashtekar, William Nelson and Sarah Shandera, Thomas Pashby and Gordon Fleming, David Sloan and Kurt Gibble, Elie During and myself, and Alexis de Saint-Ours and John Norton; later additions included responses or essays by Jeremy Butterfield, Julian Barbour, Klaus Mainzer, and Lee Smolin, to complement the overview essays by Abhay Ashtekar and John Norton that conclude the special issue of SHPMP. v

2 vi Preface The relation of the theoretical physicists to the astronomers in cosmological research seems to me analogous to the relation of the logicians to the number theorists, geometers and topologists I have encountered over the years at Penn State and elsewhere. Philosophy of mathematics for many decades has been dominated by logicians who (I think) misunderstand the role of mathematical logic. It is not an over-discourse that should supplant the others, but one of many, which can be integrated with other mathematical discourses in a variety of fruitful ways. Thus I am happy to note that philosophy of mathematics in the early 21st century is undergoing a long-awaited and long-overdue sea change. Philosophers of mathematics are turning to a serious study of the history of mathematics, logic, and philosophy, as well as interviewing mathematicians and educators, and reading textbooks, to look in detail at mathematical research and instruction. They are trying to give an account of how mathematical knowledge grows and how problems are solved, with the dawning insight that serious problem-solving is by nature ampliative. In addition to the founding of the Association for the Philosophy of Mathematical Practice in 2009, I would point to a series of publications initiated by Carlo Cellucci, and Brendon Larvor s Mathematical Cultures project ( ), as well as Mic Detlefsen s Ideals of Proof project ( ) and Karine Chemla s Mathematical Sciences in the Ancient World project ( ), both of the latter conjured up in the wonderful research cauldron SPHERE (CNRS UMR 7219) at the University of Paris Denis Diderot Paris 7. To this we can now add Springer s SAPERE series under the direction of Lorenzo Magnani. In fact, this change had been in preparation for decades. Imre Lakatos celebrated book Proofs and Refutations: The Logic of Mathematical Discovery was published in 1976, and inspired many young philosophers to think about mathematics in a new way. Donald Gillies wrote his dissertation with Lakatos, and has been active in the philosophy of mathematics since the late 1970s. He published my essay Two Episodes in the Unification of Logic and Topology in the British Journal for Philosophy of Science in 1985, and that same year introduced me to Carlo Cellucci when I was in Rome, who would go on to write three important books on logic and mathematics in Italian, in 1998, 2002, and (His Rethinking Logic: Logic in Relation to Mathematics, Evolution, and Method was published in English in 2013, followed shortly by Danielle Macbeth s similarly motivated but dialectically opposed Realizing Reason: A Narrative of Truth and Knowing in 2014.) In 1995, Donald Gillies edited an important collection of essays, Revolutions in Mathematics, thoroughly Lakatosian in spirit, with contributions by inter alia Paolo Mancosu, Herbert Breger, Caroline Dunmore, Jeremy Gray, Michael Crowe, Herbert Mehrtens, Joe Dauben, and myself. In 1994 and 1995, I organized two conferences at Penn State (with support from a Humboldt Foundation Transatlantic Cooperation grant written with Herbert Breger, then director of the Leibniz Archives), where philosophers and historians of mathematics interacted and wrote pairs of essays together, so that philosophical claims might be supported, countered or complicated by case studies from the history of mathematics, which was precisely what Lakatos had encouraged. This resulted in The Growth of Mathematical Knowledge in 1999, and included essays by Herbert

3 Preface vii Breger and myself, Carlo Cellucci, Donald Gillies, Penelope Maddy, Paolo Mancosu, François De Gandt, Jaakko Hintikka, Madeline Muntersbjorn, Carl Posy, Mike Resnik, Hourya Sinaceur, Klaus Mainzer, and others. In 2005, Carlo Cellucci edited, with Donald Gillies, Mathematical Reasoning and Heuristics; in 2006, with Paolo Pecere, Demonstrative and Non-demonstrative Reasoning in Mathematics and Natural Science; and in 2011, with myself and Emiliano Ippoliti, Logic and Knowledge, as well as a special issue of the Italian journal Paradigmi. The Festschrift for him edited in 2014 by Emiliano Ippoliti and Cesare Cozzo continues this series. In the Francophone world, the tradition of philosophy of mathematics informed by history of mathematics remained unbroken throughout the 20th century. Jules Vuillemin, for example, who was in a sense the Quine of France, did not share Quine s disdain for history; rather, he was an expert on the history of mathematics, logic and philosophy, as well as the latest texts in analytic philosophy. His book La Philosophie de l algèbre (1962/1993) has been very influential, and so likewise his magisterial Nécessité ou contingence: l aporie de Diodore et les systèmes philosophiques, published in He belonged to the tradition of Henri Poincaré, the great fin de siècle mathematician-physicistphilosopher; indeed, Vuillemin s papers are housed in the Archives Henri Poincaré at the University of Lorraine, under the auspices of Gerhard Heinzmann, who published L Intuition épistémique in Also included in this lineage are Albert Lautman, Jean Cavaillès, and Léon Brunschvicg (and the German Ernst Cassirer), all of whom met their fate in (Emmy Noether worked with Cavaillèson an edition of the Cantor-Dedekind correspondence.) Vuillemin was my philosophical mentor, and the teacher of Claude Imbert (whose Pour une histoire de la logique appeared in 1999) and Hourya Sinaceur (whose book on the theory of fields, Corps et Modèles, appeared in 1991). I met the Newton scholar François De Gandt in 1981 through Rémi Brague, when I attended a study group in Paris on Book I of the Principia; he introduced me to Hourya Sinaceur and Karine Chemla (she translated, with Guo Shuchun, the Chinese mathematical classic Les neuf chapitres), and Amy Dahan and Jeanne Pfeiffer (who together edited Une histoire des mathématiques: routes et dédales, (1986), which was published in English translation in 2009). Almost all the French and Italian philosophers just mentioned were invited to the United States or Canada, and spent some months or a year or so there during the last two decades of the 20th century, interacting with many important philosophers and historians of mathematics. They in turn introduced me to a generation of younger scholars: Marc Parmentier, David Rabouin, Jean-Jacques Szczeciniarz, Christine Proust, Andy Arana, Dirk Schlimm, Agathe Keller, Renaud Chorlay, Valeria Giardino, Marco Panza, Ivahn Smadja, Norma Goethe, Amirouche Moktefi, Justin Smith, Valérie Debuiche, Koen Vermeir, Pierre Cassou-Noguès, Jessica Carter, Giulia Miotti, Roy Wagner, and many others associated with the University of Rome La Sapienza, SPHERE and elsewhere. This vigorous, unbroken European tradition has played a big role in the development of a richer approach to philosophy of mathematics, whose proponents moreover now interact fruitfully with philosophers who continue to be more interested in logic and formalism, as recent

4 viii Preface conferences, lecture series and events attest, organized by Mic Detlefsen, Brendan Larvor, Lorenzo Magnani, José Ferreirós, Jamie Tappenden, Michel Serfati, Marco Panza, Penelope Maddy, Dale Jacquette, Marcus Giaquinto, Paolo Mancosu and others. To this list should be added philosophically inclined mathematicians whose popular expositions of various issues in contemporary mathematics and the history of mathematics raise lively issues about mathematical reason, and add important dimensions to the current conversation: Joseph Mazur, Robyn Arianrhod, Reuben Hersh, William Byers, Edward Frenkel, Donal O Shea, Marjorie Senechal and Chandler Davis. As noted above, I am indebted to the number theorist Winnie Li for allowing me to sit in on her graduate seminars over a period of many years, and to graduate students Travis Morrison, Sonny Arora, Ryan Flynn, William Chen, Ayla Gafni, and Haining Wang for sharing their research with me. And reiterated thanks, in the final preparation of the manuscript of the book, to Gordon Fleming, Winnie Li, Jeremy Gray, and Christelle Vincent. I think it is important to recognize the sea change in philosophy of mathematics (following upon a change in philosophy of science that began with Kuhn), which has taken place over the last forty years, and to bring the scattered philosophers, mathematicians and historians responsible for it into more coherent, though of course still dialectical, conversation. This book is my contribution to the current conversation, which so often focuses on the nature of ampliative reasoning in mathematics and the mathematical sciences, in this case those where we count moments and stars as well as units. Hence the title of this book. University Park, USA Emily Rolfe Grosholz

Introduction: From Practice to Results in Mathematics and Logic

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