After Math: (Re)configuring Minds, Proof, and Computing in the Postwar United States The Harvard community has made this article openly available. Please share how this access benefits you. Your story matters Citation Dick, Stephanie Aleen. 2015. After Math: (Re)configuring Minds, Proof, and Computing in the Postwar United States. Doctoral dissertation, Harvard University, Graduate School of Arts & Sciences. 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Dissertation Advisor: Professor Peter Galison Stephanie Aleen Dick After Math (Re)configuring Minds, Proof, and Computing in the Postwar United States Abstract This dissertation examines the history of three early computer programs that were designed to prove mathematical theorems: The Logic Theory Machine, the Program P, and the Automated Reasoning Assistant, all developed between 1955 and 1975. I use these programs as an opportunity to explore ways in which mathematical knowledge and practice were transformed by the introduction of modern computing. The prospect of automation generated disagreement about the character of human mathematical faculties like intuition, reasoning, and understanding and whether computers could be made to possess or participate in them. It also prompted novel discourse concerning the character of mathematical knowledge and how it should be produced. I track how the architects of each program built their beliefs about minds, computation, and proof into their theorem-proving programs and, in so doing, crafted new tools and techniques for the work of mathematics. The practitioners featured in this dissertation were interested in whether or not computers could “think.” I, on the other hand, am interested in how people think differently when they work with computers. And in particular how they thought dif- ferently about mathematics as they crafted a place for computers in the work of proof. Ilookfortracesoftheirnewwaysofthinkinginhowtheyimplementedtheirsoftware. This is a new historiographical approach from existing history of computing that, for the most part, does not engage software at all or engages high-level descriptions or models of software. iii I argue this: what were for my actors implementation concerns are in fact signif- icant epistemological issues for the history of mathematics. Especially in the early decades, actually getting programs to run on computers was no small feat. In imple- menting programs practitioners had to craft many new tools, both formal and material -fromprogramminglanguagesanddatastructurestopunchedcardencodingsanduser interfaces. The work of implementation spans multiple media - from paper to transis- tor - and involves many practices - from diagramming to coding - demonstrating that the media of the “digital age” are multiple indeed. Moreover, implementation is the site where we see practitioners rethinking their objects of interest, their disciplines, their theories, through the lens of computation - what is possible and impossible for computers to do. Implementation is the practice of automation. In implementing their theorem-proving software, the actors in this dissertation gave new formulations and had new experiences of mathematical intuition, logical rules of inference, and other key tenets of twentieth-century notions of proof. The communi- ties explored here were among the most influential and celebrated early contributors to automated theorem-proving. Each had quite a different relationship to the postwar American academic landscape relative to their disciplinary, institutional, and political makeup. Most importantly for they fundamentally and explicitly disagreed with each other about how the automation of proof should be done. Because of this, they devel- oped very different automated theorem-proving programs – one seeking to simulate the human mind, another seeking to surpass it, and another seeking to develop theorem- proving software that would collaborate with a human user. Each of these projects intervened in the history of mathematics by introducing new forms – both social and technical – of proof. iv Acknowledgements My first debt of gratitude is to my advisor, Peter Galison. It would be impossible to thank him sufficiently here for his incisive reflections about my work and his unfailing advocacy and encouragement. I am also grateful to Peter for creating many opportu- nities for me to explore museum and theater projects related to my research. I share his belief that “making” and artistic collaboration can be truly enriching to academic work, and for me this was certainly the case. Peter invested in me both as a scholar and as a person and I relied on his brilliance, his kindness, and his guidance at every stage. I also owe special thanks to Stefan Helmreich who was one of my most valued interlocutors and a tireless and amazing (and very close) reader of my work. Stefan’s generosity and (lightning fast) insightfulness are unparalleled. My research was also deeply informed by a generals field I did with him in cultural and social anthropology. I can only hope that the dissertation adequately reflects the impact that anthropological theory, as wielded by Stefan, has had on my thinking. I am extremely grateful as well to Jimena Canales, also a member of my committee, and also a gracious and close reader of my work. Jimena has an incredible knowledge of vast literatures and directed me toward many conversations that were important for this project but that I would not have discovered without her guidance. I also greatly valued her always honest, generous, and pragmatic advice about the dissertation itself and about the landscape of early academic careers. I was honored to also have mathematician Barry Mazur on my committee. The late John Murdoch once said to me that Barry just “sees things,” referring I think to the simultaneous breadth, depth, and seeming ease of his knowledge and understanding across many fields. I came to share this view, marveling at the subtlety and incisiveness of Barry’s thinking in conversation, in seminars, and in his always thoughtful reading v of my work. To Mario Biagioli and Joan Richards I also owe special thanks. Each supervised one of my generals fields and my thinking was shaped in significant ways through engagement with them. Both continue to be great mentors, readers, and colleagues. IthankaswelltheotherfacultyintheDepartmentofHistoryofScienceatHarvard, including especially Katy Park who was Director of Graduate Studies while I was a student as well as Janet Browne, Everett Mehndelson, and Charles Rosenberg for their advice, generous advocacy, and enthusiasm for graduate scholarship. I am very lucky to have had many other mentors whose seminars, conversations, collaborations, feedback, and interest were invaluable for my intellectual development - especially David Kaiser, Michael Gordin, Heidi Voskhul, Mark Schiefsky, Vincent Lepinay, Sophia Roosth, Sam Schweber, Nathan Ensmenger, and Amir Alexander. I am especially grateful to Nathan and Amir for affording me opportunities to publish pieces of this material. I must also thank Garth McCavana, Margot Gill and GSAS at Harvard for their unwavering support for graduate students of which I was a frequent beneficiary. My graduate experience was, at every turn, shaped by the friendship, insights, and kindness of my fellow graduate students. My cohort was a singular example of col- leagues who are also friends - I relied on them for advice, happy hour, feedback and, when needed, commiseration. The Modern Sciences Working Group, Phunday, and the Dissertation and Writers Group were central venues from which I received amaz- ing feedback on many versions of this material and I am enormously grateful for them. Special thanks to Lukas Rieppel, Alex Czsisar, Alex Wellerstein, Dan Volmar, He Bian, James Bergman, Michael Rossi, Hallam Stevens, Henry Cowles, Ion Milhailescu, Colin Webster, Nassar Zakariya, and Evan Hepler-Smith for countless important conversa- tions, wonderful feedback, collaborations, guidance, and many invaluable insights. I would also like to thank my cohort and mentors from the Harvard Horizons program for reflecting so thoughtfully on my work. Among my graduate student colleagues, I must thank especially Meg Shields- Formato and Jenna Tonn. Both are dear friends and each of them read and thoughtfully engaged many pieces and versions of the material presented here. I have also, since the first day, truly looked up to Chris Phillips and Alma Steingart - I feel tremendously grateful to have them as like-minded colleagues, friends, and collaborators in the his- tory of mathematics. Vive la révolution! Finally, I thank Myrna Perez Sheldon and Lisa Crystal who are among my best friends and closest colleagues and without whom vi Iwouldhavebeenquitelostatmanyjunctures. IamgratefultothefacultyandgraduatestudentsattheInstitutefortheHis- tory and Philosophy