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DISTRIBUTIONS IN POSTWAR MATHEMATICS Michael Jeremy Barany A DISSERTATION PRESENTED TO THE FACULTY OF PRINCETON UNIVERSITY IN CANDIDACY FOR THE DEGREE OF DOCTOR OF PHILOSOPHY RECOMMENDED FOR ACCEPTANCE BY THE PROGRAM IN HISTORY OF SCIENCE Adviser: Michael D. Gordin September 2016 © Copyright by Michael Jeremy Barany, 2016. All rights reserved. Abstract Distributions in Postwar Mathematics examines the intertwined histories of French mathematician Laurent Schwartz’s theory of distributions and the American-hosted 1950 International Congress of Mathematicians in order to explain how mathematicians in the Second World War’s wake rebuilt a discipline newly capable of and dependent on intercontinental exchanges of people and texts. Schwartz’s theory, first formulated in 1944-1945, offered a new way to study differential equations for non-differentiable functions using topological vector spaces and an analogy to the basic calculus technique of integration by parts. Drawing on archives from three continents, Distributions in Postwar Mathematics shows how elite mathematicians joined with agents of government, philanthropic, and other institutions to foster a theory and a discipline across what would come to be called the First, Second, and Third Worlds. This account connects abstract theories to their communities of users, technologies of investigation and exposition, and social, political, and institutional contexts, casting mid-century mathematics as a heterogeneous enterprise sustained through a range of resource-intensive and partially integrated means of coordination. Chapter 1 details European and North American mathematicians’ efforts from the turn of the century through World War II to organize their discipline across national and regional scales, refracted through the Americans’ two failed attempts to host interwar International Congresses of Mathematicians. These developments set the institutional and geopolitical background for Schwartz’s endeavors, discussed in chapter 2, to promote distributions between 1945 and 1949. Chapters 3 and 4 offer a close analysis of American mathematicians’ postwar struggles to organize the 1950 International Congress, tying their challenges to postwar reconstruction and the emerging Cold War. Both Schwartz and the ICM’s organizers, in their separate contexts, iii used shifting and ambivalent formulations of, respectively, techniques of mathematical analysis and ideals of internationalism, to coordinate a range of interests and entities across great distances. Chapter 5 then develops this account of coordination through polysemy and ambiguity into an explanation of distributions’ intercontinental presence following the 1950 Congress, linking the theory’s adoption and adaptation to postwar institutional configurations by tracing the theory to new venues in and beyond South America, North America, and Europe. iv Table of Contents Abstract. iii Table of Contents. v Acknowledgements. viii Notes on Sources. xvi Introduction: The Promised Land. Out of the Wilderness. 1 International and Intercontinental Science. 5 Modern Mathematics and the Nature of Mathematical Theory. 14 The Dissertation’s Parts. 19 Chapter 1: Disintegration by Parts. At a Glance. 25 A Series Interrupted. 31 A Peace Apart. 39 Mathematics With Borders. 47 A Bid Deferred. 58 A Shifting Center. 64 A War of Mathematics. 73 Table 1: Attendance at pre-World War I and interwar International Congresses. 83 v Chapter 2: A Widely Anticipated Invention. A Derivative Innovation. 87 Distinctions Among Differences. 92 Anticipation and Priority. 95 The Propaganda Tract. 105 Making Senses of Distributions. 117 A Ritual Phrase. 125 International Recognition. 130 A Theory from Zero. 139 Chapter 3: The Closed Circuit of Open Ideals. Reconciling Internationalisms. 146 A Few Discreet Inquiries. 150 Money, Politics, and Scheduling. 156 Reframing the Soviets and the Union. 161 International Mathematical Disunion. 168 Moving Money, Moving Mathematicians. 179 Colonial Mathematics. 183 Chapter 4: Internationalism by Committee. Foreign Intelligence. 195 Ranks and Files. 200 Diplomatic Preparations. 210 Political Tests. 215 vi The Schwartz Affair I: A Definitive Showdown. 228 The Schwartz Affair II: Hopes and Promises. 236 Valiant Efforts. 245 Chapter 5: Integration by Parts. Considerable Excitement. 250 A Synthesis and a Simplification. 255 A Definitive Textbook. 265 Expository Translations, West and East. 272 Classrooms and Seminars. 286 The Distributions Road Show. 291 Conclusion: Intercontinental Mathematics. 305 Works Cited. 311 vii Acknowledgements My very first class as a freshman at Cornell University was Bob Strichartz’s introductory graduate course in analysis. On the morning of that first meeting at the start of the Fall 2005 semester, Bob introduced the plan for his two-semester course by talking about a historian and philosopher of science named Thomas Kuhn who proposed that the sciences advance through paradigm shifts. The course would be structured around two paradigm shifts in modern analysis, Bob explained, relating to the two fundamental concepts in analysis of integration and differentiation: Henri Lebesgue’s theory of the integral from the turn of the twentieth century and Laurent Schwartz’s theory of the derivative from the mid-twentieth century. This dissertation on Schwartz’s theory grew from a seed Bob planted that morning, which he and many others have cultivated over the past decade. That seed took hold in soil prepared by even more teachers, mentors, family, and friends. Though I cannot name you all here, please know that I am grateful for the lessons of many kinds that you have taught me. My advisor Michael Gordin has shaped my scholarship, writing, and professional development in more ways than I dare enumerate. The depths of his care, insight, patience, and generosity continue to astonish me after six years under his wing. Every passage of this dissertation has benefited many times over, both directly and indirectly, from his efforts. I thank Angela Creager and Phil Nord for being part of my committee, for teaching and mentoring me in their courses, seminars, and beyond, and for sharing with me not just their collegiality but their genuine joy in doing all that they do. David Aubin, my external examiner, has been a vital advocate and critic at multiple junctures. I thank him for his insights and provocations, and especially for supervising my research in Paris during a pivotal early stage of my research. Beyond my committee, I owe special thanks to Charlie Fefferman, Joan Richards, Donald viii MacKenzie, Matt Jones, Chris Skinner, Karen Parshall, David Kaiser, and Liliane Beaulieu for their generous mentorship and support at different points in my project and graduate education. For launching me on this academic trajectory in the first place, I blame (with extreme gratitude) Suman Seth, Mike Lynch, and Kevin Lambert, and I thank them for their continued presence in my life. I have leaned heavily on the scholarship and advice of two others who have written dissertations about Laurent Schwartz and the theory of distributions. Jesper Lützen kindly shared his recollections, willingly compared notes, and graciously hosted me for a very fruitful visit to Copenhagen. Anne-Sandrine Paumier has been an invaluable guide (particularly during my time in Paris) and collaborator, and I have found myself returning again and again to her insights and observations in attempting to understand my own materials and research questions. This dissertation is quite different from what I proposed at the start of my research, and I have been able to pursue the topics I did with the detail and emphases I did only because I could rely for foundation and counterpoint on Jesper and Anne-Sandrine’s own expansive investigations. I could not have asked for a finer group of friends and classmates as interlocutors and companions. They have offered support, distraction, mirth, and meaning, have sharpened my thinking, broadened my perspectives, lightened my burdens, and added heft to my foundations. I am especially glad to count in this number a remarkable generational cohort of historians of mathematics who fill me with optimism for our field and excitement for many collaborations and debates ahead. Some names I associate with this project by way of anything from conversations shared to couches surfed (with apologies to those I have elsewhere compartmentalized): Ellen Abrams, Anthony Acciavatti, Josh Bauchner, Tom Beck, Natalie Berkman, Edna Bonhomme, Matthew Chan, Hannah-Louise Clark, Gerardo Con Diaz, Henry Cowles, Teresa Davis, Will ix Deringer, Steph Dick, Theo Dryer, David Dunning, Catherine Evans, Margarita Fajardo Hernandez, Christina Florea, Chris Florio, Yael Geller, Ben Gross, Cynthia Houng, Jiří Hudeček, Abram Kaplan, Emily Kern, Clare Kim, Miri Kim, Kyrill Kunakhovich, Molly Lester, Jemma Lorenat, Heather Macbeth, Martín Marimón, Sarah Milov, David Moak, Wangui Muigai, Chris Phillips, Flori Pierri, Rebecca Davis O’Brien, Ingrid Ockert, Felix Rietmann, Padraic Scanlan, Robin Scheffler, Margaret Schotte, Brittany Shields, Richard Spiegel, Alma Steingart, Craig Sylvester, Ksenia Tatarchenko, Melissa Teixeira, Laura Turner, Janet Vertesi, Iain Watts, Caitlin Wylie, and Adrian Young. In his own recently-completed dissertation acknowledgements Evan Hepler-Smith termed me his “scholarly fraternal twin”; as with so much else over the last six years, I can do no better than to borrow Evan’s apt and perceptive rendering. If you read both of our dissertations
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