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Home , Barry Mazur, Brian Rotman, Eupatorus, Mathematics and art, Max Dehn, Morris Kline

The Subjects of : , , and the Politics of Value in Twentieth-Century United States

by

Clare Seungyoon Kim

Bachelor of , 2011

Submitted to the Program in , Technology and Society In Partial Fulfillment of the Requirements for the Degree of

Doctor of Philosophy in History, Anthropology, and Science, Technology and Society at the Massachusetts Institute of Technology

September 2019

© 2019 Clare Kim. All Rights Reserved.

The author hereby grants to MIT permission to reproduce and distribute publicly paper and electronic copies of this thesis document in whole or in part in any medium now known or hereafter created.

Signature of Author: Signatureredacted History, Anthr and Sc'nce, Technology and Society Signature redacted- August 23, 2019 Certified by: David Kaiser Germeshausen Professor of the History of the History of Science, STS Professor, Department of Physics Thesis Supervisor

Certified by: Sianature redacted Christopher Capozzola MASSACHUSETTS INSTITUTE OFTECHNOLOGY Professor of History C-) Thesis Committee Member OCT 032019 LIBRARIES

1 Signature redacted Certified by: Stefan Helmreich Elting E. Morison Professor of Anthropology Signature redacted Thesis Committee Member Accepted by: Tanalis Padilla Associate Professor, History Director of Graduate Studies, History, Anthropology, and STS Signature redacted Accepted by: Jennifer S. Light Professor of Science, Technology, and Society Professor of Urban Studies and Planning Department Head, Program in Science, Technology, & Society

2 The Subjects of Modernism Mathematics, Art, and the Politics of Value in Twentieth-Century United States

By Clare Kim

Submitted to the Program in History, Anthropology, and Science, Technology and Society on September 6, 2019 in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy in History, Anthropology, and Science, Technology and Society.

Abstract My dissertation illuminates the status of mathematical knowledge in relation to other intellectual domains and racialized social forms-particularly American Orientalism-in the twentieth-century United States. Observers and practitioners have long engaged in drawing relations between mathematics and . Around the turn of the twentieth century, however, when reconceived mathematics around notions of abstraction, formalism, and "made" theories bearing no necessary relationship to the empirical world, understandings of the relationship of mathematics to the arts changed. Historians of mathematics, mathematicians, and historians of art have since referred to these twentieth-century intellectual changes as a "modernist" transformation. They refer to mathematical modernism in terms of metaphors that reflect shared values of being autonomous, creative, and a form of self-expression. My dissertation recovers an alternative history that upends the assumption that mathematical modernism developed within pre-existing boundaries of its discipline. It tracks a series of collective efforts between mathematicians, , critics, and historians, to use and articulate a place for formally abstract and axiomatically derived mathematical techniques within humanistic and artistic inquiries. Drawing from archival and published sources across four main chapters, I trace four specific efforts that reflect changes and transformations in American higher education and academic institutions between the 1890s and the present. Chapter one chronicles mathematicians', historians', and art collectors' interpretations of Japanese and Chinese mathematical traditions between the 1890s and 1920s. It shows how their conclusions that the resulting "oriental mathematics" was universal but inferior to the current practices were informed by a racialized discourse, treating Japanese and Chinese math as symbols of exotic difference. Chapter two recounts and describes the production of a mathematical theory of aesthetic measure at in the 1930s. It shows how the theory was part of a broader artistic movement to articulate a theory of pure design. Chapter three examines the valuation and of mathematics within the liberal arts setting at in the 1940s and 50s. It recovers how rather than being essential to high art, mathematics was also critical to the resurgence of craft. The final chapter elucidates the contradictions in valuing mathematics as abstract, creative, and autonomous, by examining a copyright dispute between a mathematical designer and a conceptual in the 2000s. The resulting view of US mathematical modernism as embedded within broader intellectual domains illuminates a more nuanced view of changes in what has or has not counted as a mathematical subject.

Thesis Supervisor: David Kaiser Title: Germeshausen Professor of the History of Science and Professor of Physics

3 4 For Rui, Eunice, and Irene

5 Table of Contents

T itle ...... 1

A b stract...... 3

T able of C ontents...... 6

A cknow ledgem ents...... 8

A bbrev iation s...... 11

Introduction When Mathematics Was Modern...... 13

Chapter One Universal Subjects: The Problem of "Oriental"Mathematics...... 31

Chapter Two Introspective Subjects: Efficient Expression andBirkhoff's Theory ofAesthetic Measure...... 50

Chapter Three Formal Subjects: Max Dehn and Black Mountain College...... 75

Chapter Four Creative Subjects: Mathematical Origami and the Limits of Creative Expression...... 83

E p ilo g u e ...... 1 1 1

B ib lio grap h y ...... 1 16

6 7 Acknowledgements

Long in the making with many more rewrites and edits to follow, my work is indebted to the support and generosity of a range of interlocutors: mentors, students, colleagues, research staff, friends, and family. I will try to do justice in these acknowledgements and in my footnotes to the many people who have guided me along the way. My first and biggest thanks goes to my advisor, David Kaiser, whose wit and generosity knows no bounds. I am eternally grateful to Dave for his guidance and support, for his willingness to listen, and for helping me to ground the chaotic, ill-formed thoughts that crowded my mind. His patience and humor have meant more than I can say. To Chris Capozzola, who first introduced me to U.S. intellectual history, and who never fails to ask the questions that get me to think beyond the and STS: thankyou. I would not have been able to move this project in the direction that it did without your generosity. And to Stefan Helmreich, who never tires of my questions and continues to encourage my foray into new strands of literature: thankyou. His belief in this project from its inception made everything that came after possible.

Entering MIT and HASTS six years ago, I was lucky enough to have learned alongside a wonderful of individuals who were as warm and welcoming as MIT's tunnels during the bitter winter months. I especially would like to thank Beth Semel, Lauren Kapsalakis, and Grace Kim for being there in a pinch during the final weeks of writing this dissertation. For their friendship and intellectual wit, I thank Marc Aidinoff, Renee Blackburn, Marie Burks, Ashawari Chaudhuri, Nadia Christidi, Amah Edoh, Richard Fadok, Steven Gonzalez, Kit Heitzman, Shreeharsh Kelkar, Younhun Kim, Rijul Kocchar, Nicole Labruto, Alison Laurence, Lee, Jia-Hui Lee, Lan Li, Srujan Meesala, Lucas Mueller, Peter Oviatt, Canay Ozden-Schilling, Tom Ozden-Schilling, Luisa Reis Castro, Alex Reiss Sorokin, Hilary Robinson, Elena Sobrino, Alex Reiss Sorokin, Mitali Thakor, Theodora Vardouli, Claire Webb, Jamie Wong, and Peter Y. Zhang.

Faculty at HASTS and MIT challenged my thinking in more ways than I could have ever thought. For their questions, humor, and support, I thank Dwai Banerjee, William Broadhead, William Deringer, Mike Fischer, Deborah Fitzgerald, Slava Gerovitch, Caley Horan, Jennifer Light, Anne McCants, Amy Moran-Thomas, Kenda Mutongi, Steven Ostrow, Heather Paxson, Jeffrey Ravel, Harriet Ritvo, Robin Scheffler, Susan Silbey, Emma Teng, Craig Wilder, Roz Williams, William Uricchio, and Yufei Zhao. More significantly, my graduate student life wouldn't have been as smooth as it was without the support of the people who helped make everything run. I am eternally grateful to Karen Gardner, Paree K Pinkney, Gus Zahariadis, Margo F. Collett, Mabel Chin, Megan Pepin, Kathleen Lopes, Ayn Cavicchi, Irene Hartford, Barbara Keller, and Amberly Steward,

Financial support for this project came in the form of the NSF Doctoral Dissertation Research Improvement Grant, the German Historical Institute, and the Notre Dame Institute for Advanced Study Graduate Fellowship. I received enormously generous help from the brilliant staff and archivists at the MIT Libraries, Institute Archives, and Special Collections. They include Michelle Baildon, Myles Crowley, Nora Murphy, Michael Noga, and Ece Turner. Similarly, helpful assistance came from the staff and archivists at the Harvard Art Museum Archives,

8 Harvard University Archives, John Hay Library at Brown University, Western Regional Archives, Archives and Special Collections, Dolph Briscoe Center for American Archives at the University of Texas Austin, and the Rare Book and Manuscript Library at . In particular, I would like to thank Jeff Arnal, Raymond Butti, Sarah Downing, Timothy Engels, Carolyn Grosch, Susan Halpert, Michelle Anna Interrante, Megan Schwenke, Edith Sandler, Alice Sebrell, Holly Snyder, and Heather South.

I have had the pleasure of meeting many scholars who have encouraged this project in the various directions that it went. Of the historians of mathematics, I especially thank Alma Steingart and Stephanie Dick for encouraging me to come to MIT. I am also indebted to the many historical and mathematical conversations I had with Ellen Abrams, Michael Barany, Philip J. Davis, Theodora Dryer, Sam Evens, Bruce Hughes, Barry Mazur, Dave Peifer, Christopher Phillips, Anand Pillay, Mark Schiefsky, and many more. For their feedback and conversations on topics ranging from media studies and art to US history and STS. Many of the ideas introduced in this dissertation coalesced while in residence at the Notre Dame Institute for Advanced Study, with more than a few thinking and writing retreats at HHMI's Janelia Research Campus. I am indebted to the HPS Colloquium at Notre Dame, as well as the Harvard-MIT- Princeton Physical Working Group for hearing out some early ideas. I am forever grateful to Francesca Bordogna, Harvey Brown, John Deak, Erika Doss, Peter Galison, Michael Gordin, Robert Goulding, Fredrik Albritton Jonsson, Rebecca McKenna, Jay Malone, Lisa Mueller, Tom Stapleford, Thomas Tweed, and Johanna Winant for their questions and conversations.

But in as much as these institutions left me with a paper trail to mark their assistance, there were also people moving behind the scenes in bringing this dissertation to life. They include Shoh Asano, Faisal Baqai, Hilary Buxton, Chi-Lun Chang, Tony Cova, Ricky Correira, Bobby Day, Rosa Fry, Larry Hong, David Kang, Jacklyn Lock, Jessica Man, Kim Nguyen, Nancy Niu, Shristi Pandey, Jina Park, Nate Park, Ida Pavlichenko, Emily Robinson, Didem Sarikaya, Anna Shneidman, Samantha Siskind, Eric Shu, Patrick Strotman, Channa Srey, Giannina Schaefer, Ben Stevenson, Gokul Upadhyayula, Robin Wetherill, Rie Yamamoto, and Pam Zhang.

There are two sets of people I would like to thank in conclusion. The first set are three scholars and teachers who inaugurated me into new fields that have led me to pursue the path I'm now on: Geri McCarthy, who in high school introduced me to the world of history and the social sciences; Joan Richards, who as an undergraduate at Brown University introduced met to the history of science and, more importantly, the history of mathematics; and Heather Paxson, whose Arts, Crafts, Science seminar continues to have lasting influence on my thinking.

The last set are the people who helped see me through to the end of writing this dissertation: my family. To Kyung-Suk and Saeja Kim, thank you for sharing with me the struggles of your research life. Your sagely advice on managing time helped me to become a better scholar and person. To Irene Kim and Simon Jenni, thank you for always checking in on me when I needed your humor the most. To Eunice Kim, thank you for being one of my closest readers and for never failing to find a way to relate the classics to mathematics. And to Rui Gao, thank you for the long walks and for your love.

9 10 Abbreviations

AAC Anni Albers Collection, PC.1197, Western Regional Archives, State Archives of North Carolina, Asheville, N.C.

AMS American Mathematical Society

AMSR American Mathematical Society Records 1888-. Call Numbers Ms. 75.1, 75.2, 75.4 John Hay Library, Brown University Archives, Providence, RI. 75.6

BMC Black Mountain College

BMCPC Black Mountain College Project Collection, Western Regional Archives, State Archives of North Carolina, Asheville, N.C.

DESPP David Eugene Smith Professional Papers, 1860-1945. Ms 1167. Rare Book & Manuscript Library, Columbia University, , NY.

GDBP Papers. Harvard University Archives, Cambridge, MA.

MDP Max Dehn Papers, 1899-1979. Archives of American Mathematics, Dolph Briscoe Center for American History, University of Texas at Austin, Austin, TX.

RAP Papers. Call Number M1585. Department of Special Collections and University Archives, Stanford University Libraries, Stanford, CA.

RPMP Richard Peter McKeon Papers, 1918-1985. Special Collections Research Center, University of Chicago Library, Chicago, IL..

II 12 Introduction When Mathematics was Modern

Make it new! -Ezra Pound, Make It New (1935)

Mathematics in the Modern World

In September 1964, published a special issue on "Mathematics in the

Modem World." With contributions from several leading mathematicians in the United States such as Richard Courant, Philip Davis, and , the issue sought to convey the depth and breadth of then current mathematical research. The authors not only elaborated on the concepts of "number," "," ," and "probability" as fields of abstract inquiry within mathematics, but also lauded the introduction of mathematical theories and techniques to other domains of scientific knowledge since World War II, including the physical, biological, and social sciences.' But the widening scope and nature of mathematical knowledge also raised concerns: If mathematics comprised diverse fields of study and increasing areas of application, what, if anything at all, held mathematics together? How should mathematics be characterized as a field or discipline?

As Richard Courant argued in an article introducing the special issue, the problem of defining and describing mathematics stemmed in part from institutional and intellectual transformations in the subject that had unfolded around the turn of the twentieth century.

Professional mathematical organizations in the U.S., for instance, had witnessed unprecedented

' The special issue's table of contents reflects this dual focus and organization. In Scientific American 211 (1964): 3.

13 growth since the 1900s, with membership numbers increasing thirty-fold.2 American colleges

and universities over the same period had experienced a similar increase in the number of

undergraduates majoring in mathematics and graduate students pursuing doctoral studies. For

Courant, such growth came on the heels of renewed interest in the epistemological foundations

of mathematics and the emergence of new fields of research such as and .

These new fields, Courant observed, reflected a tendency towards formalism, abstraction, and

general theory. They also reflected preferences for the treatment of mathematical theories as

worlds (i.e. logically consistent systems of operation) of their own. Despite the inclination

toward "progressive abstraction, logically rigorous axiomatic deduction and ever wider

generalization," Courant maintained that the "essence" of mathematics lay in "[t]he interplay

between generality and individuality, deduction and construction, logic and imagination." 4

2 The New York Mathematical Society was founded in 1888 and had 16 members at its inception. Within 2 years, membership ballooned to 210. Parshall and Rowe, The Emergence of the American Mathematical Research Community, 1876-1900: JJ. Sylvester, Felix Klein, andE. H. Moore (London: London Mathematical Society, 1994),336. 3 Before 1875, American universities had conferred a total of only six degrees in the field. During the next fifteen years, thirty-nine Americans took doctorates in the US, and another fifteen earned their degrees abroad. These figures were dwarfed again by those of the final decade of the century, which witnessed a total of 107 new Ph.Ds. in mathematics, 85 of which were earned in the US. R.G.D. Richardson, "The Ph.D. Degree in Mathematical Research," American Mathematical Monthly 43 (1936): 199-215. 4 Richard Courant, "Mathematics in the Modern World," Scientific American 211 (1964): 42-43.

14 -1

SCIENTIFIC AMERICAN

%14kT1V1 l sI N 1 1il t 01I)UR I\ ~) .1~77 orVVS7N

Jff/V 0A /96

Figure 0.1: Special issue cover. From "Mathematics in the Modern World," Scientific American vol. 24, no. 3 (September 1964).

Given Courant's characterization of mathematics in the mid 1960s, the selection of Rend

Magritte's La Lunette d'approche (Figure 0.1) would seem both a curious and apt choice for the cover of the special issue. Originally painted in 1963, the surrealist depicts a vista that can and cannot be discerned through a window. The image contrasts the bright blue hues of an

15 open sea and sky lying beyond the closed side of the window with the empty, black space revealing itself through the window's opening. While the image makes no overt reference to either mathematics or mathematicians, its elusive subject matter seemingly embodies the abstract nature of mathematics characterized in the special issue. As the editors of Scientific American explained, "[t]he painting symbolizes those aspects of mathematics which make outrageous new assumptions to erect new systems." 5 As art historians have shown, Magritte's surrealist work constituted a new "form of thinking" whose images eschewed representations of nature in favor of showing "nothing except what I have thought." 6 By placing Magritte's work on the cover, the editors intimated a "modem" affinity between mathematics and the arts that saw both as endeavors grounded in human rather than nature.

From the of mathematicians during the Cold War and historians of mathematics in retrospect, the relation of mathematics to the arts demonstrated the dual nature of the subject as a humanistic and scientific discipline.? Between World War II and the 1970s, new fields of mathematical inquiry proliferated in the human sciences through new networks of government funding, seminars, institutes, and conferences that complemented existing university departments, academic journals, and professional societies. , cybernetics, and exemplified the postwar growth of what Peter Galison has called the

5 Ibid., 4. 6 Rend Magritte, quoted in Suzanne Guerlac, "The Useless Image: Bataille, Bergson, Magritte," Representations 97 (2007): 39. Because of the shift away from his earlier work in abstract painting, Magritte has generally been excluded from the modernist canon and accused of producing "dry realism," as argued by Rosalind Krauss in Originalityofthe Avant-Garde and other Modernist Myths (Cambridge: Cambridge University Press, 1985), 91. Critical theorists such as Michel Foucault have analyzed Magritte's as concerned with operations of signification rather than visual . Michel Foucault, Ceci n'est pas une Pipe (Berkeley: University of California Press, 1973). More recent scholarship has sought to resituate Magritte within the greater discursive concerns of artistic modernism and relate his early abstractionist work to his late surrealist paintings. On this, see Roger Rothman, "A Mysterious Modernism: Rend Magritte and Abstraction," Konsthistoriks Tidskrift/Journalof 76 (2007): 224-239; Lisa Lipinski, Rend Magritte and the Art of Thinking (Routledge, 2019). ' Alma Steingart, "Conditional Inequalities: American Pure and , 1940-1975" (PhD diss, MIT 2011).

16 "interdisciplines." 8 On this reading, the apparent tensions and contradictions of mathematics as

both pure and applied, abstract and concrete, were crucial to the institutional remaking of

mathematics following World War II.

Such tensions were also interpreted to be a direct consequence of "mathematical

modernism," in which mathematics was reconceived around notions of abstraction, general

theory, and formalism, and was understood to deal with "made," disembodied concepts. The

nineteenth-century introduction of "non-, [...] bizarre functions, and

transfinite numbers," elaborates Morris Kline, "forced the recognition" of the subject as a

"human, somewhat arbitrary creation, rather than idealization of the realities in nature, derived

solely from nature." 9 Between 1890 and 1930, these understandings emerged out of German

research universities and found their strongest expression among mathematicians in American

universities and academic institutions." The modernist transformation of mathematics was

successful, argues Jeremy Gray, "because it connected fruitfully with what mathematicians were

doing and with the image they were creating for themselves as an autonomous body of

8Peter Galison, "The Americanization of Unity," Daedalus 127 (1998): 45-71. On of funding in the postwar mathematical sciences and their implications for the organization of research, see Michael Barany, "Distributions in Postwar Mathematics" PhD diss., , 2016); Paul Erickson, The World that Game Theorists Made (Chicago: University of Chicago Press, 2015); Hunter Crowther Heyck, HerbertA. Simon: The Bounds ofReason in Modern America (Baltimore, Johns Hopkins Press, 2005); ' Morris Kline, Mathematical Thoughtfrom Ancient to Modern Times, Volume 3 (Oxford University Press, 1999), 1032. 1 Two late-nineteenth century mathematical traditions have characterized German mathematics. One tradition, centered in Berlin, followed a neo-humanist orientation promoting to the almost complete neglect of applied mathematics. The other tradition, based in G~ttingen under the direction of Felix Klein and , studied mathematics with connections to the natural sciences, though it also saw the separation of applied and pure mathematics on an institutional level. See Gert Schbring, "The Conception of Pure Mathematics as an Instrument in the Professionalization of Mathematics," in Social History ofNineteenth-Century Mathematics, ed. Herbert Mehrtens, H.J.M. Bos, and Ivo Schneider (Boston: Birkhauser, 1981), 111-134; Lewis Pyenson, Neohumanism and the PersistenceofPure Mathematics in Wilhelmian (American Philosophical Society, 1983); Eberhard Knoblock, "Mathematics at the Berlin Technische Hochschule/Technische Universitat: Social, Institutional, and Scientific Aspects," in the Historyof Modern Mathematics:Institutions and Applications, ed. David E. Rowe and John McCleary (San Diego, CA: Academic Press, 1989), 109-128; Herbert Mehrtens, Moderne Sprache Mathematik (Suhrkamp, 1990).

17 professionals within, or alongside, the disciplines of philosophy and science."" Here, mathematical modernism emerged as a separate but parallel development to concurrent transformations in the arts and humanities, and was assumed to develop according to an internalist, disciplinary logic.

By the end of the Cold War in the U.S., the dual nature of mathematics seemed to have crystallized by associating the rhetoric of autonomy and the comparison with the creative arts with its "modernist transformation,"-in the words of Jeremy Gray-and by recognizing and advocating the extension of mathematics to broader domains of in the technical, applied, and human sciences.' This telling, however, is incomplete. Seeking to challenge the conceptual dichotomies that structure readings of modern mathematics and "mathematical modernism," this dissertation examines how the meanings and values ascribed to mathematics-as a universal, autonomous, abstract, and creative discourse-proliferated and culturally circulated in US intellectual life throughout the twentieth century. At the turn of the twentieth century, artists and designers encountered new mathematical ideas and techniques as resources for describing, analyzing, and creating new forms of art. Likewise mathematicians joined critics' and art historians' discussions over the meanings of art, craft, and . Such exchanges occurred within interstitial spaces that reflected the shifting institutional organization of American higher education. Providing an account of contemporaneous efforts in mathematics to engage the arts and humanities, I not only show how such exchanges equally contributed to an understanding of

" Jeremy Gray, 's Ghost: The Modernist Transformation of Mathematics, (Princeton University Press, 2008), 3. 12 Ibid., 1. Mathematicians who corroborate this image of mathematics included ", "Matheamtics as a Creative Art," American Scientist 56 (1968): 375-389; Marshall Stone, "The Revolution in Mathematics," American Mathematical Monthly 68 (1961): 715-734. For a historical account of the 1960s emergence of "the mathematical sciences" as a new term of art to reflect this dual nature, see Alma Steingart, "Conditional Inequalities: American Pure and Applied Mathematics, 1940-1975" (PhD diss, MIT 2011), especially 201-262.

18 mathematics as both scientific and humanistic, but also illuminate the concrete connections among the fields in twentieth-century US intellectual life.

By recovering this alternative history, I show that when historians interpret modern mathematics as an intellectual enterprise that is simply separate and parallel to the arts we miss a far more complex historical drama. By remaining within the epistemological frame of disciplinary autonomy, I contend that scholars inadvertently take for granted how mathematicians have rethought the characterization and meanings of their subject in relation to other disciplines. In other words, scholars take as evident mathematicians' claims that the subject matter of each intellectual field as already being known, thereby naturalizing claims about the differences and similarities in their intellectual dynamics. Consequently, they foreclose the possibility of examining those assumptions, practices, and institutional infrastructures that enabled such understandings in the first place. 3 Because mathematics appears at once as historically stable and ever changing, it offers a particularly compelling vantage from which to examine not only the situated nature of mathematical knowledge, but also how its related values and classification as humanistic, artistic, or scientific are locally negotiated.

Unlike humanistic and scientific disciplines such as literature, the study of texts, or physics, which examines the properties and behaviors of matter, the discipline of mathematics has centered around the system of methods it creates and employs." The production of new mathematical theories thereby entailed the adjudication of the contexts and intellectual domains

3 In making this argument, I draw from the arguments in Joan Scott, "The Evidence of Experience," Critical Inquiry 17, no. 4 (Summer, 1991): 773-797. Scott makes a similar argument that historians have taken as evident the category of experience as a foundational epistemology from which its mobilization as a form of historical evidence forecloses understanding of the mechanisms producing such of that experience in the first place. " This peculiar capacity for mathematics to become part of its own subject-matter is what Leo Corry calls the "reflexive character of mathematics." In practices of proof, for instance, work on mathematical equations revolve around ascertaining the existence and uniqueness of solutions. Historians have tended to group this feature under the rubric of meta-mathematics. See Leo Corry, Modern Algebra and the Rise ofMathematical Structures (Basel: Birkhauser, 2004), 8-10.

19 to which such theories could be applied. And decisions about the disciplinary contexts to which

mathematics could operate counted as decisions about the value of mathematics for cultivating that particular subject. In this study, I examine the place of mathematics in the humanistic and

artistic disciplines that have been scrutinized as modernist by scholars in American studies, history of science, and history of art." My dissertation title, "The Subjects of Modernism," is therefore invoked in two senses: first, to refer to the discipline of mathematics and second, to the mathematicians, critics, artists, and intellectual thinkers who thought and tinkered with it.

Modernism in US Intellectual Life

What does it mean for mathematics to be modern? How did mathematicians, artists, designers, and historians think differently with new mathematical techniques and concepts? How did artists, architects, and historians think differently about their subject matters as they crafted a place for mathematics in their work? In accounts of the transformation of the ideas, practices, and institutions that led to the reconceptualization of mathematics as an abstract, autonomous, and creative discipline, mathematical modernism has been a productive concept for both mathematicians and historians. For mathematicians and historians, the term underscores the novelty and emergence of pure mathematics as an independent field of study primarily centered at European institutions. In Moderne-Sprache-Mathematik, for example, Herbert Mehrtens,

" On historians of mathematics and science who write on modernism, see Nina Engelhardt, Modernism, Fiction and Mathematic (Edinburgh: Edinburgh Univesrity Press, 2018); Jose Ferreiros and Jeremy Gray, eds., The of Modern Mathematics (Oxford/New York: Oxford University Press, 2006); Moritz Epple, "An Unusual Career between Cultural and Mathematical Modernism: Felix Hausdorff, 1868-1942," in Jews and Sciences in German Contexts, eds. U. Charpa and U. Deichmann (Tilbingen: Mohr Siebeck, 2007), 77-100. On historians of art and American modernism, see Erika Doss, Benton, Pollock, and the Politics of Modernism: From Regionalism to Abstract Expressionism (Chicago: University of Chicago Press, 1991); Linda D. Henderson, The Fourth Dimension andNon-Euclidean Art (Princeton: Princeton University Press, 1983); Stephen Kern, The Culture of Time & Space 1880-1918: With a New Preface (Cambridge: Harvard University Press, 2003 [1983]); Caroline Jones, Eyesight Alone: 's Modernism and the Bureaucratizationofthe Senses (Chicago: University of Chicago Press, 2006).

20 invokes modernism as a means to characterize various reactions to a wider formalist and axiomatization movement in mathematics, and more specifically to debates surrounding the

foundations of mathematics. On the side of the "modems" stood German David

Hilbert, whose Foundations ofGeometry (1899) treated Euclidean geometry as an abstract, formally deductive system. Terms such as "," "point," or "" no longer relied on intuition but were to be defined by symbolic systems and by their use in the axioms they now rested upon.' 6 German mathematician Felix Klein, French mathematician Henri Poincard, and

Dutch mathematician L.E.J. Brouwer stood in contrast as "countermoderns." While acknowledging the increasing detachment of mathematics from physical reality, each sought to maintain a place for spatial and intuition within mathematics. For Mehrtens, then, mathematical modernism denotes "first, the autonomy of cultural production and second, the departure from the vision of an immediate representation of the world of experience." 17

Building on Mehrtens's understanding of mathematical modernism, Jeremy Gray defines

"mathematical modernism" as an "autonomous body of ideas, having little or no outward reference...and [that] maintain[s] a complicated-indeed anxious, rather than anaive relationship to the day-to-day world." 18 Gray further elaborates that his usage of the term illuminates the "deep modernist interest in the history of its subject, which was often used as a way of legitimizing the new [of mathematics], at least in the eyes of its adherents." 9 Both

Gray and Mehrtens underscore an increasingly insulated mathematical community that no longer defined mathematics as the science of space and magnitude, but rather as the "science of

6 David Hilbert, The Foundationsof Geometry, trans. E.J. Townsend (La Salle, IL: Open Court, 1950). 17 Herbert Mehrtens, "Modernism vs Countermodenrism, Nationalism vs Internationalism: Style and Politics in Mathematics, 1900-1950," in MathematicalEurope: History, Myth, Identity, eds. Catherine Goldstein, Jeremy Gray and Jim Ritter. (Paris: Maison des Sciences de I'Homme, 1996), 521. " Gray, Plato's Ghost, 1. 19Ibid.

21 structures." 2 0 These readings hypostatize an increasingly insulated mathematical community concerned with mathematics qua mathematics, upholding a notion of modernism forwarded by art critic Clement Greenberg as "the use of the characteristic methods of a discipline to criticize the discipline itself, not in order to subvert it but to entrench it more firmly in its area of competence." 2 In so doing, historians of mathematics reify the idea that mathematics developed separately but in parallel to concurrent developments in the arts and architecture that have similarly been described as modernist.2 2 Their interpretations of mathematical modernism complement the turn-of-the-century rise of structural objectivity that, as Lorraine Daston and

Peter Galison describe, "all images, whether they are perceived by the eye of the body or that of the mind."2 3

Likewise, scholarship in the history of art and U.S. intellectual and cultural history has supported a strong divergence between mathematics and the arts, contrasting the unrestrained subjectivity of the artist with the self-discipline of scientist-mathematician.2 4 Writing on the emergence of the "modern system of the arts," for instance, cultural historian Larry Shriner describes a process originating from the European Enlightenment. Against the backdrop of an industrial modernity that embraced efficiency and standardization, "art" emerged by assigning the characteristics of tradition, skill, and function to "craft." In the United States between the

20 Leo Corry, Modern Algebra and the Rise ofMathematicalStructures: Leo Corry, "Mathematical Structures from Hilbert to Bourbaki: The of an Image of Mathematics," in Changing Images in Mathematics: From the French Revolution to the New Millennium, eds. Umberto Bottazzini and Amy Dahan-Dalmedica (London: Routledge, 2001), 167-85. 21 Clement Greenberg, "Modernist Painting," in Forum Lectures (Washington, D. C.: Voice of America, 1960). 22 New art forms, techniques of abstraction, and increased functionality in architectural designs characterized a philosophical and artistic movement that came to be called "modernism". See Matei Calinescu, Five Faces of Modernity: Modernism, Avant-garde, Decadence, , ( Press, 1987); Peter Childs, Modernism (London and New York: Routledge, 2000); Peter Gay, Modernism: The Lure of Heresy, 1st ed (New York, NY: W.W. Norton & Company, 2007). 2 Lorraine Daston and Peter Galison, Objectivity (Cambridge: MIT Press, 2007), 255. 2 Robert Crunden, American Salons: Encounters with European Modernism, 1885-1917 (New York: Oxford University Press, 1993; Wanda Corn, The GreatAmerican Thing: Modern Art and NationalIdentity, 1915-1935 (Berkeley: University of California Press, 1999).

22 1890s and 1930s, this ordering of art over craft gave rise to new and formal artistic practices, which under the label of aesthetic modernism sought to express a subjective introspective

exploration of individual consciousness. 25 In Robert Genter's Late Modernism (2012), we

encounter modernists deeply opposed to the ascendancy of science, suspicious of the

organizational ethos of American life, and committed to human freedom and cultural

autonomy. 26 Modernists, in his telling, were defenders of the humanities who saw totalitarian

implications in mass culture and scientific progress. Only more recently have historians problematized this dichotomy in the European context, revealing how science and the arts were not necessarily the opposing forces they have been made out to be. 27 Generally, however, historians of mathematics, the arts, and US intellectual life have been united in the assumption that while modernism has taken multiple and disciplinary-specific forms, each kind tended towards individuated expression, fragmentation, formalist abstraction, and autonomy. Such features pointed to a historically specific "order of things" that valued novelty over tradition, art over craft, form over function, and abstraction over the real or concrete.2 8

This dissertation develops a conception of modernism that departs from received notions in significant ways. It treats mathematical modernism as an "epistemic space," a concept that emerges from a broader configuration of epistemological commitments, intellectual, material,

2 Larry Shiner, The Invention ofArt: A Cultural History (Chicago: University of Chicago Press, 2001). On literature that relates aesthetic modernism to modernity, see Anson Rabinbach, The Human Motor: Energy, Fatigue, and the Origins of Modernity (University of California Press, 1992); Terry Smith, Making the Modern: Industry, Art, and Design in America (Chicago: University of Chicago Press, 1993). 26 Robert Genter, Late Modernism: Art, Culture, and Politics in Cold War America, (Philadelphia: University of Pennsylvania Press, 2012). 2 Gillian Beer, " Theory and the Rise of Literary Modernism," in Realism and Representation, ed. George Levine (Madison: University of Wisconsin Press, 1993), 193-213; Henning Schmidgen, "1900-The Spectatorium: On Biology's Audiovisual Archive," Grey Room 43 (2011): 42-65; John Tresch, The Romantic Machine: Utopian Science and Technology after Napoleon (Chicago: University of Chicago Press, 2012) Robert Brain, The Pulse of Modernism: PhysiologicalAesthetics in Fin-de-SiecleEurope (Seattle: University of Washington Press, 2015) 21 Michel Foucault, The Orderof Things: An Archaeology ofthe Human Sciences (New York: Vintage Books, 1994).

23 legal, and economic contexts.29 This approach effectively enables a means to mediate between the meanings ascribed to abstract mathematical theories and the concrete practices of mathematicians and artists. In doing so, I trace collective efforts between mathematicians, artists, critics, and historians to use and articulate a place for formally abstract and axiomatically derived mathematical theories and techniques within humanistic and artistic inquiries. The alternative history I recover not only upends the assumption that mathematical modernism developed within preexisting boundaries of its discipline, but also reveals how the meanings and values ascribed to mathematics were equally and concurrently shaped by thinking with and in relation to other intellectual domains.

The Art and Craft of Theory and Practice

In nineteenth-century institutions of higher learning in the United States, mathematics was valued primarily for its role in education and for its presumed exemplification of the highest form of reason. Mathematicians' efforts centered on supplementing the liberal arts curriculum with courses in , geometry, and algebra. 3 0 By the first few decades of the twentieth century, mathematicians and mathematical work occupied a different role. No longer valued just for their ideas and ideals about mathematics in education, mathematicians were to be recognized as members of a research community whose recognition derived from their contributions to the methods and subject matter of mathematics itself.

29 In their study of , Staffan Muller-Wille and Hans-Jdrg Rheinberger use the term "epistemic space" to refer to the "broader realm in which a scientific concept takes shape," but that "nevertheless elude full and final representation." This enables them to locate heredity as an abstraction that emerges from histories of exchange and under multiple contexts. Staffan Millier-Wille and Hans-J6rg Rheinberger, "Heredity-The Formation of an Epistemic Space" in Heredity Produced:At the CrossroadsofBiology, Politics, and Culture, 1500-1870, eds. Staffan Muller-Wille and Hans-Jbrg Rheinberger (Cambridge: MIT Press, 2017), 3-34. See also StaffanMUller- Wille and Hans-J6rg Rheinberger, A Cultural History ofHeredity (Chicago: University of Chicago Press, 2012). " Parshall and Rowe, The Emergence; Karen Parshall, "Defining a Mathematical Research School: The Case of Algebra at the University of Chicago, 1892-1945," HistoriaMathematica 31 (2004): 263-278.

24 Such shifts were embedded within the institutional remaking of American higher education and the emergence of research universities that no longer revolved around a core set of collegiate values. Between 1890 and 1920, institutions such as Harvard, Columbia, Johns

Hopkins, and the University of Chicago increasingly constituted a diverse array of colleges, professional schools, laboratories, liberal arts departments, museums, and observatories.3 1

Academic researchers relied on private industry and philanthropic foundations like the

Rockefeller Foundation and Carnegie Corporation for monetary support.32 American universities also encompassed the development of what Joel Isaac has called the "interstitial academy," an assortment of university seminars, reading groups, and other enclaves through which to discuss or exchange ideas outside of established departments.3 3

In order to account for both the formation of a mathematician or scientist and the theoretical work they produce, historians, sociologists, and anthropologists of science have approached theory as a form of social practice. Historians of physics Andrew Warwick and

David Kaiser, for instance, have focused on the of theory, analyzing how theory is learned through the mastery of theoretical tools and skills. Building on Thomas Kuhn's pedagogical theory of knowledge and drawing from Michel Foucault, they contend that the

"disciplinary regimes" of universities shape the dispositions of scientists-in-training. Whether

31 See Laurence R. Veysey, The Emergenceof the American University (Chicago: University of Chicago Press, 1965); Jonathan R. Cole, The GreatAmerican University: Its Rise to Preeminence, Its IndispensableNational Role, and Why it Must be Protected (New York: Public Affairs, 2009); Julie Reuben, Making of the Modern University: Intellectual Transformationand the Marginalizationof Morality (University of Chicago Press, 1996); Frederick Rudolph, The American College and University A History (Athens: University of Georgia Press, 1990), especially pp. 462-464; Roger L. Geiger, To Advance Knowledge: The Growth ofAmerican Research Universities, 1900-1940 (New York: Oxford University press, 1986), especially pages 1-57. 3 Robert E. Kohler, Partnersin Science: Foundationsand Natural Scientists, 1900-1945 (Chicago: University of Chicago Press, 1991). On the history of relations between mathematics and philanthropic institutions, see Reinhard Siegmund-Schultze, Rockefeller and the Internationalizationof Mathematics Between the Two World Wars (Basel: Birkhauser Verlag, 2001). 3 Joel Isaac, Working Knowledge: Working Knowledge: Making the Human Sciencesfrom Parsonsto Kuhn (Cambridge: Harvard University Press, 2012), 32-36.

25 through classroom demos, paper exams, or supervised drills, students work to constitute themselves as a particular kind of knowing subject.3 4 Other scholarship on the history of the natural and physical sciences has also shown how the production, use, and dissemination of chemical formulae, diagrams, or theories of were materially grounded and mediated by "paper tools." 3 5 Analyzing the cognitive practices involved in calculating or measuring, anthropological and sociological literatures have also explored the diverse ways in which people conceptualize and perform reframe universal procedures. 36 Collectively, these strands of literature move away from descriptions of theories as disembodied ideas or transcendent truths and towards the characterization of theories as a set of transformative techniques or exercises that are performed by and act upon a particular subject.

The treatment of theory as practice was part of a wider "practice turn" in the history, sociology, and anthropology of science that appropriated notions of craft and artisanship in order to describe scientific practices. 37 Historical and sociological accounts of modem laboratory work, for instance, analogically drew on early modem forms of artisanal work in order to

3 Andrew Warwick and David Kaiser, "Kuhn, Foucault and the Power of Pedagogy," in Pedagogy and the Practice of Science: Historicaland ContemporaryPerspectives, ed. D. Kaiser (MIT Press, 2005): 393-409. Kuhn rejected the idea that scientific knowledge involved memorizing a body of formal laws and algorithmic rules for their application. Rather, trainee scientists learned to "puzzle" solve. 3 See Ursula Klein, "Paper Tools in Experimental Cultures," Studies In History and PhilosophyofScience Part A 32, no. 2 (June 2001): 265-302; Andrew Warwick, Masters of Theory: Cambridge and the Rise of Mathematical Physics, (University of Chicago Press, 2003); David Kaiser, Drawing Theories Apart: The Dispersion ofFeynman Diagrams in PostwarPhysics (University of Chicago Press, 2005) Andrew Pickering and Adam Stephanides, "Constructing Quarternions: On the Analysis of Conceptual Practice," in Science as Practiceand Culture, ed. Andrew Pickering (University of Chicago Press, 1992: 139-167. 36 See Jean Lave, Cognition in Practice (Cambridge University Press, 2009); Jean Lave, "The Values of Quantification," The Sociological Review 32 (1984): 88-111; Helen Verran, Science and an African Logic, (University of Chicago Press, 2011); Tim Lenoir, "Practice, Reason, Context: The Dialogue Between Theory and Experiment," Science in Context 2, No. 1 (1988): 3-22; Joan H. Fujimura, Crafting Science: A Sociohistory of the Questfor the Geneticsof Cancer (Cambridge: Harvard University Press, 1997) 3 Treating theories as practice, Andrew Warwick notes, corrects an asymmetry between "culture-specific explanations of experimental practice" and the study of "theorizing...as a transcendent solitary activity." See Andrew Warwick, "Cambridge Mathematics and Cavendish Physics: Cunningham, Campbell and Einstein's Relativity 1905-1911: Part I: The Uses of Theory," Studies in the History and Philosophy ofScience 23 (1992): 632 as quoted in Joel Isaac, "Tangled Loops: Theory, History, and the Human Sciences in Modem America," Modern Intellectual History 6, no. 2 (2009): 404.

26 describe modem laboratory work. 38 Likewise in histories of the physical sciences, attention to

"the crafting and use of paper tools" enabled a means to reveal the "tacit knowledge and craft skill" deployed in theoretical work. 39 The appeal of these notions of craft and artisanship lay in their ability to highlight the physical and refined skills required to pursue scientific inquiry.

Likewise, anthropologists' ethnographic accounts of craft have shown how in learning to craft cheese or confections, artisan apprentices not only embody the dispositions of their particular vocation, but also learn and incorporate social values deriving from a wider context."

My dissertation builds on these previous approaches - though extends and complicates their analyses to examine two additional topics or dynamics. The first is to do with how conceptualizations of race, human embodiment, and economies of exchange informed mathematicians' and artists' valuations of mathematics as formally abstract, creative, and autonomous. The second aim that my alternative history articulates has to do with problematizing the art-craft dyad that many humanistic scholars rely upon in order to treat theories as practice. Historians' reliance on craft as an analytic that deflates theories as lofty abstractions depends, I argue, upon a hierarchical placement of art over craft that is itself a product of a particular historical account of modernism. If craft has been invoked in order to

"lower the tone" about science, then treating mathematical theories as a form of craft practice also requires a more nuanced notion of craft, one that, in treating "math" as a "craft" that can broker connections to "art" can thus itself be used to undo craft/art dichotomies necessitates

" See Steven Shapin, "The Invisible Technician," American Scientist 7 (1989): 554-563; Simon Schaffer, "Babbage's Intelligence: Calculating Engines and the Factory System," CriticalInquiry 21 (Autumn 1994): 203- 227). 3 David Kaiser, Drawing Theories Apart, 11. * See Heather Paxson, The Life of Cheese: Crafting Food and Value in America (University of California Press, 2012); Michael Herzfeld, The Body Impolitic: Artisans and Artifice in the Global Hierarchyof Value (Chicago: University of Chicago Press, 2004); Dorinne K. Kondo, Crafting Selves: Power, Gender and Discourses ofldentity in a Japanese Workplace (Chicago: university of Chicago Press, 1990).

27 examining the place and nature of mathematics beyond self-ascribed disciplinary limits and in broader, interstitial contexts." Neglecting to do so results in an asymmetry and forecloses the analysis and understanding of other intellectual communities who deploy mathematical theories and techniques for ends beyond the creation and maintenance of mathematics itself.

Order of Argument

This study is organized around moments of exchange between mathematicians and other intellectual communities between the 1890s and 2000s, charting the institutional configurations as well as the material and social practices that shaped valuations of mathematics as creative, autonomous, and abstract. Each chapter focuses on a moment within the history of American higher education that inflected a particular institutional organization, as well as extends and complicates historians of mathematics' notions of "mathematical modernism" by showing that this intellectual formation formed in sustained dialogue with practitioners in the arts and the humanities. Collectively, the chapters explore the multiple ways that this alternative history of mathematical modernism raises questions about the effects and limitations of the current historical schemas inherited and employed today.

Chapter One, "Universal Subjects," chronicles mathematicians', historians', and philosophers' interpretations of Japanese and Chinese mathematical traditions between the 1890s and 1920s. Focusing on historian of mathematics David Eugene Smith, a professor at the

Teachers College at Columbia University, it examines how mathematicians and historians scrutinized and compared differences in mathematical inscriptions to assert that "Oriental mathematics" was at once the same as an inferior to the symbolic formalism and axiomatic

" Steve Shapin, "Lowering the Tone in the History of Science," in Never Pure: HistoricalStudies ofScience as ifIt was Produced by People with Bodies, Situated in Time, Space, Culture, and Society, and Strugglingfor Credibility and Authority (Baltimore: Johns Hopkins University Press, 2010), 1-14.

28 mathematical system that had recently emerged. Examining these practices of interpretation

against the emergence of the American research university and a system of international student

exchange, I analyze how these claims were informed by a racialized-and racist-discourse that treated Chinese and as symbols of exotic difference. Scholars, I argue, have unwittingly maintained this view in their accounts of mathematical personae.

Chapter Two, "Introspective Subjects," recounts and describes the production of mathematician George David Birkhoff's theory of aesthetic measure at Harvard between the

1920s and 1930s. Problematizing prevailing accounts of his mathematical theory as an aberration of mathematical work, this chapter reconstructs the theory as a product of an "interstitial academy." As a formal and quantitative technique for assessing artistic forms, Birkhoff's theory of aesthetic measure drew from decorative designs that have been considered "antimodern," developed out of the mathematics classroom, and found new life in design curricula. I also resituate Birkhoff's theory as part and continuation of a broader movement that aimed to ground aesthetic by recourse to the and the phenomenology of experience.

If the previous two chapters examine mathematics against the growth and development of research universities, Chapter Three, "Formal Subjects," examines the valuation and nature of mathematics within the liberal arts during the 1940s and 1950s. Supplementing studies of mathematics within the context of the Cold War university, this chapter examines topologist Max

Dehn's relations to members of the Bauhaus at the liberal arts college of Black Mountain

College. It examines student notes and Dehn's course lectures on "Geometry for Artists."

The final chapter, "Creative Subjects," elucidates the contradictions in valuing mathematics as abstract, creative and autonomous by examining a copyright lawsuit between a mathematical origami designer and conceptual artist in the 2000s. It describes the emergence of

29 mathematical origami in the 1970s and reads the copyright case against the backdrop of intellectual property regimes in university research.

Treating mathematics as craft, the resulting view of mathematical modernism in the

United States as fully embedded within the cultural worlds of which it is a part reveals how racial formations-in particular, Asian and Asian-American identity that answered to the name of American Orientalism-contour what and who counts as a mathematical subject.

4 This dissertation builds on Edward Said's formulation, who in 1978 distinguished Orientalism as a distant geographic and historical imaginary created by European colonialism that was separate, according to Mae Ngai, from the "actual history of real people in the 'Orient."' He also periodized American Orientalism to post-World War II, when the social sciences reworked European Orientalism into area studies. Building on Said's notion and complicating the more uncritical understanding of Orientalism as a shorthand about negative western stereotypes about all Asians, I treat the racialization of mathematical knowledge and the subjects who produced that knowledge as a local and enmeshed production. See Edward Said, Orientalism (New York: Vintage Books, 1978); Mae Ngai, "American Orientalism," Reviews in American History 28 (September 2000): 408-415.

30 Chapter One Universal Subjects: The Problem of "Oriental"Mathematics

What [the Western nations] had failed to take into account was this: that between them and was no common psychological speech. Their thought processes were radically dissimilar. There was no intimate vocabulary...There was no way to communicate Western ideas to the Chinese mind. -Jack London, "The Unparalleled Invasion" (1910)

Young people who continue the study of English after they are fifteen or sixteen, ought to learn something of both historical and comparative grammar, and come to understand how much the work of logicians has done to make of English a language in which it is possible to think clearly and exactly on any subject. The PrincipiaMathematica are perhaps a greater contribution to our language than they are to mathematics. -T.S. Eliot, Criterion(1927)

An unusual thesis title appeared alongside the list of graduate degrees conferred for

Mathematics at Stanford University in June 1916. Produced by Japanese-born Masahachi

Mukaiyama for his Master of Arts degree, "The Value of Oriental Mathematics" specified its

purpose to appraise mathematical theories and practices other to what his classmates had

pursued. Rather than focusing on the "Unicursal Curves" of or the

"Surfaces of the Fourth Order" from , Mukaiyama sought to describe,

compare, and contrast the treatment of mathematics from China and .43 Following a brief

historical overview of mathematical developments in China-such as the implementation of the fangcheng (,i ) procedure to solve linear equations in the 2nd century BCE-he devoted the

" "Graduate Study 1916-1917," Leland Stanford Junior University Bulletin 92 (June 1916): 60.

31 rest of his thesis to recounting the origins and lasting influences of Japanese wasan (Thy)

mathematics initiated by Seki Takazawa (I* I¶) in the 1 7th century. Contrary to what was intimated in his thesis title, Mukaiyama's conclusions stopped short of assigning a total claim of exceptionalism onto mathematics from East Asia. Despite acknowledging Seki's development of a new system of algebra, Mukaiyama affirmed:

But so far as the theory of mathematics is concerned, we do not find any difference between Occidental and Oriental mathematics. The difference is the process of the calculation. Chinese used the Sangi board and Japanese used the Soroban to calculate the numerical and abstract expressions." 44

The attribution of material tools to "Oriental" practices raises questions of how conceptualization of race, and understandings of cultural differences have been mapped onto the figuration of the

'mathematician.'

However, the peculiarity of this statement does not lie so much in Mukaiyama's invocation and then dismissal of material tools, such as the Sangi computing rods or Soroban abacus, to oppose prevailing characterizations of Chinese and Japanese mathematical theory traditions as insufficiently abstract. In a way, Mukaiyama's work itself participated in a common conceptualization of race at the time, which not only categorized Asians as "Orientals," but also understood their material products to be symbols of exotic difference. 45 Rather, the writing piques curiosity on two levels. Contextually, Mukaiyama's status as a graduate student in mathematics marked him as a rare, educated elite at a time when the majority of Chinese and

" Masahachi Mukaiyama, "The Value of Oriental Mathematics," (M.A. thesis, Stanford University, 1916), 19. 41 Robert G. Lee, Orientals:Asian Americans in PopularCulture (Philadelphia, Temple University Press, 1999), especially 58-79; Henry Yu, Thinking Orientals: Migration, Contact, and Exoticism in Modern America (Oxford/New York: Oxford University Press, 2002).

32 Japanese in the US were merchants, laborers, or servants.46 Content-wise, Mukaiyama's

conclusions also give pause precisely because of his refusal to extend his attention to materiality

to thinking about theories of mathematics themselves. That the racialized, cultural and

geographical differences signified by "Occidental" and "Oriental" were negligible for his vision

of mathematical theories as the same across "cultures" presumes a transparency-that is, a

shared use of language-requiring neither explanation nor qualification.

Closer scrutiny of Mukaiyama's thesis, however, underscores the degree to which he took

for granted this assumption of English as a preferred language for writing and representing

mathematical concepts. For whenever Mukaiyama referred to mathematicians, theories, and

techniques from China or Japan, he did so using written English rather than Chinese characters

or Japanese kanji, hiragana, or katakana - and that sometime led to him leaving out key features

of the techniques he hoped to explain and translate. For example, when he reconstructed a

Japanese method for deriving an algebraic curve in the fourth dimension-heimen daisu

kyokusen-he neglected to reproduce or describe the bosho shiki "side notation" method

practiced by wasan mathematicians (see Figure 1.1). This method used a combination of Chinese

ideograms and lines to describe relations between various mathematical entities; Chinese

orthographic practice, in other words, was part of the technique, not a mere commentary to be

either translated or left out. Mukaiyama opted to use a symbolic notational system that

characterized an emerging mathematical tradition: symbolic formalism, which sought to reduce

mathematics to logic using an alphanumeric system of notation (see Figure 1.2).47 This logicist

" According to Judy Yung's work on US census manuscripts, about 7.5% of San Francisco Chinese men in 1900 were professionals. Judy Yung, Chinese American Voices: From the Gold Rush to the Present, eds. Judy Yung, Gordon H. Chang, H. Mark Lai (Berkeley: University of California Press, 2006), 135-136. Information on Mukaiyama's life and career trajectory are scant. Originally hailing from Okinawa, he received his B.A. in Mathematics from the College of the Pacific in 1914, submitting a thesis on "A Model of a of the Fourth Order." He was advised by Robert Edgar Allardice at Stanford. 4 Mukalyama, "The Value of Oriental Mathematics," 10. Part of a larger concern among Euro-American

33 vision aimed to craft formal systems consisting of sets of axioms and rules of inference for

deriving consequences through a symbolic notational system. 48 Joining a formal system

(propositional logic) to a specific way of writing math (symbolic notation system) on a material

medium (paper), this was a dominant form of practice among mathematicians in the U.S. and

Europe in the early twentieth century. Its appeal lay in being a means to ensure abstract cognition

beyond natural languages and numerical values, to the extent that some people claimed it as

universal language. Mukaiyama's delineation of what aspects of mathematics could be defined as

"Oriental," however, not only derived from his recourse to history, but also depended upon

something that "Orientals" did not possess: a (Latin) alphabet. This suggests obliviousness,

characteristic of other mathematicians at the time, to recognize how one mathematical language

and script had been imposed upon another.

mathematicians with foundations from which to build up mathematics. English mathematicians Alfred North Whitehead and Bertrand Russell's PrincipiaMathematica reflects the tradition of symbolic formalism and dream to secure a universal system for representing and working in mathematics. See Alfred North Whitehead and Bertrand Russell, PrincipiaMathematica Vol. I (Cambridge: Cambridge University Press, 1913), 2n edition, 103. 48Mathematical logic has a rich and complex history - intimately bound up with the history of algebra - and it consists of many subfields, each carved out through the definition of the basic logical units in question and the operations that can be applied to them. Gregory Moore offers an account of how first-order logic in "The Emergence of First-Order Logic" in History and Philosophy of Modern Mathematics, Vol. 11 (1988): 95-135. Morris Kline offers a succinct account of the origins of in the nineteenth century, drawing on developments from as far back as Rene Descartes and Gottfried Wilhelm Leibniz. See Morris Kline, "The Rise of Mathematical Logic," in Mathematical Thought From Ancient to Modern Times, Vol. 3 (New York: Oxford University Press, 1972): 1187-1192. Joan Richards offers an account of the cultural and ideological complexity of Augustus de Morgan's contribution to algebra and logic in "Augustus de Morgan, the History of Mathematics, and the Foundations of Algebra," Isis 78, No. 1 (March 1987): 6-30. Richards also explores some relations between the development of mathematical logic and the changing field of geometry in the nineteenth century in Mathematical Visions: The Pursuitof Geometry in Victorian England(Academic Press, 1988). For a more traditional account of the history of logical "ideas and attitudes," see Ernst Nagel, "The Formation of Modern Concepts of Formal Logic in the Development of Geometry" Osiris 7 (1939): 142-224; N. I. Styazhkin, History ofMathematical Logic from Leibniz to Peano (Cambridge: MIT Press, 1969).

34 ;+ 11 7-- Lw A- o C 'k

:k o A 014

05

isilot P 1*t frtjlk Yit '-71

Figure 1.1: 's bosho shiki "side notation" system. On the left page, numerals are written at the side of a vertical lines. Depending on the number of lines and their placement, the "side notation" can describe absolute coefficients, proportions, powers of relations, etc. In Okoshi Moto Kai 2-Kan (tjL2tf* 21), 53. Available at: http://dl.ndl.go.jp/info:ndljp/pid/3508175/15.

I I

-I 471 t)' j~z /e, 0 lit-

Figure 1.2: Masahachi Mukaiyama's description of a curve in the fourth degree using formally symbolic notation. From Mukaiyama, "The Value of Oriental Mathematics," (M.A. thesis, Stanford University, 1916), 16.

35 The view of mathematics as an ideal body of knowledge has been repeatedly challenged and refuted by philosophers, anthropologists, sociologists, and historians of mathematics. 49

Scholarship treating the relationship between how mathematicians think and the tools or notational systems they work with now avoid reproducing notions of universal and disembodied cognition, doing so in relation to a naturalized division between the mind and body.0 Concerned with resurrecting the corporeality of the mathematician from written paper, for instance, scholar

Brian Rotman critiques a view of mathematics that makes no place for the body in favor of disembodied cognition, instead advocating for an analysis of mathematics as embodied knowledge. Positing a semiotic approach to mathematics, according to which a single mathematician comprises three distinct agencies (the person, the subject, and the agent), Rotman analyzes the way a mathematician's body figures within written practice." Alma Steingart has problematized Rotman's semiotic approach of mathematics in order to focus more expansively

49 For the most prominent accounts, see Imre Lakatos, Proofs and Refutations: The Logic of Mathematical Discovery, eds. John Worrall and Elie Zahar (Cambridge: Cambridge University Press, 1976); Philip Kitcher, The Nature of Mathematical Knowledge (New York/Oxford: Oxford University Press, 1985); Sal Restivo, The Social Relations ofPhysics, Mysticism and Mathematics (Boston/Dordrecht: D. Reidel, 1985); David Bloor, Knowledge and Social Imagery, (Chicago: University of Chicago Press, 1991). 5 The relationship between notational systems and cognition has been a subject of particular interest for certain historians of mathematics. Notably, the work of Reviel Netz on the origins of deductive reasoning practices with the lettered diagram and certain linguistic formulations in mathematics. See, for example, Reviel Netz, The Shaping ofDeduction in Greek Mathematics: A Study in Cognitive History (Cambridge: Cambridge University Press, 1999); Netz, "Linguistic Formulae as Cognitive Tools," Pragmaticsand Cognition 7, No. 1 (1999): 147-176. Science studies scholars have also scrutinized inscription practices in relation to other knowledge domains. Ursula Klein, Andrew Warwick, and David Kaiser have explored the role of "paper tools" in the knowledge-practices of nineteenth century organic chemistry, mathematical physics, and . See Andrew Warwick, Masters of Theory: Cambridge and the Rise of MathematicalPhysics (Chicago: University of Chicago Press, 2003); David Kaiser, DrawingTheories Apart: The Dispersion ofFeynman Diagrams in PostwarPhysics (Chicago: University of Chicago Press, 2005); and Ursula Klein, Experiments, Models, Paper Tools: Cultures of Organic Chemistry in the Nineteenth Century (Stanford: Stanford University Press, 2003), Bruno Latour has suggested that a study of the making, circulation, and reproduction of "inscriptions" would go a long way to understanding knowledge- production in for example, Bruno Latour "Visualization and Cognition: Drawing Things Together," in Knowledge and Society: Studies in the Sociology of Culture Past and Present, ed. H Kuklick (Jai Press, 1986), 1-40. 'Brian Rotman has a pointed interest in mathematical notation systems in so far as he wants to reduce mathematics to writing and associated semiotic practices. See especially Brian Rotman, Mathematics as Sign: Writing, Imagining, Counting (Stanford: Stanford University Press, 1993); and Rotman, "Thinking Dia-Grams: Mathematics and Writing" in ed. Mario Biagioli, Science Studies Reader (New York: Routledge,1999), 430.

36 on the "role of bodily in mathematical research beyond symbolic notation," calling

for attention towards "mathematical manifestations." 5 These accounts of theory and practice

have been contextualized within an internationalist framing and narrative of globalization.5

But there is another aspect to the relations between mathematical notation systems and

cognition that scholars, especially within the history of mathematics, have taken for granted, which is in treating their mathematician-subjects monolithically without regard to changes and

differences of racial ideologies. 5 One consequence of this oversight has been that even as attitudes towards conceptualizations of race and the processes of cognition have changed over time, the figure of a mathematician as a fragmented subject has perpetuated.

As such, this chapter offers a historically minded approach to the representation of the academic mathematician by exploring relationships among conceptualizations of racial difference and the formalization of mathematics within the tradition of symbolic formalism, the nascent field of history of mathematics, and the intellectual discourse of American Orientalism during the early twentieth century. In the late nineteenth and early twentieth centuries, American mathematicians took avid interest in what they called the "native" mathematics of China and

Japan. This sense of awe was present in many early histories of mathematics, which promulgated

52Alma Steingart, "Inside: Out," Grey Room Quarterly 59 (Spring 2015): 48. By positing a semiotic approach of mathematics according to which a single mathematician comprises three distinct agencies (the person, the subject, and the agent), Brian Rotman primarily analyzes the way a mathematician's body figures within the written form of mathematical practice (e.g. symbolic notations of numbers and diagrams). Brian Rotman, Ad Infinitum: The Ghost in Turing's Machine (Stanford: Stanford University Press, 1993); Rotman, Mathematics as Sign: Writing, Imagining, Counting (Stanford: Stanford University Press, 1993). 5 See Michael Barany, "Distributions in Postwar Mathematics," PhD diss. (University of Chicago, 2016) for an account of how mathematicians reorganized their discipline around the ideology of internationalism; Karen Hunger Parshall, "'A New Era in the Development of Our Science': The American Mathematical Research Community, 1920-1950," in David E. Rowe and Wann-Sheng Horng, eds., A Delicate Balance: Global Perspectives on Innovation and Tradition in the History of Mathematics, a Festschrift in Honor of Joseph W. Dauben (Basel: Birkhauser, 2015), 275-308. 4 For preliminary literature on race and science, see Stephen Jay Gould, The Mismeasure ofMan (New York: Norton, 1996); Londa Schiebinger, Nature's Body: Gender in the Making of Modern Science (Boston: Beacon, 1993); Donna Haraway, Primate Visions: Gender, Race, and Nature in the World of Modern Science (New York: Routledge, 1989).

37 a narrative of modernization characterized by increased formalization in which the symbols and variables that were crucial to subjects such as algebra no longer stood in for unknown numerical values or quantities, but also algebra-like symbol systems beyond the numerical domain.

But while historians of mathematics have largely dismantled these valuations of the triumph of Western mathematics, few have problematized the presumed status of "Oriental" and

"Occidental" mathematicians as embodying the same kind of subject. This chapter documents the ways in which mathematicians' and historians' debates about "Oriental" mathematics and who could embody them found expression in the publications of American mathematical journals. Partially recounting the ways in which Asians and Asian Americans came to be mathematicians within American academic institutions, this chapter also examines the role of historians of mathematics in mapping categories of racial difference onto mathematical knowledge and practice. In particular, it focuses on the collaboration between historians of mathematics David Eugene Smith and Yoshio Mikami to produce a history of Chinese and

Japanese mathematics for an American readership. I argue that they employed a form of what

Christopher Bush has called "ideographic modernism,"" which forwarded the universal communicability of a particular system of symbolic formal system and thereby treated Japanese and Chinese mathematical cultures [and their notational systems as an inferior system to be consigned to the past.s This ordering paralleled the racial hierarchies in place within the US and higher education.

*Christopher Bush, Ideographic Modernism: China, Writing, Media (New York: Oxford University Press, 2012). 56This is not to say that I fully endorse "ideograph" as the designated descriptor of the written Chinese script. As scholars have shown, the term has been unpleasantly associated with the philological basis of Indo-European tradition, chinoiserie, missionary ethnocentrism, and European colonialism. I invoke "ideograph" in this chapter as a historical category that has never been itself stable, but rather evolved with the different contexts of linguistic and technological standardization under examination-in this case, in mathematics. See John DeFrancis, The Chinese Language: Fact and Fantasy (Honolulu: University of Hawaii Press, 1984) for an especially useful summary of the different and controversial ways of naming Chinese script.

38 This chapter also seeks to contribute to the growing body of literature on Asian American intellectual history. Historians such as Henry Yu, Augusto Espiritu, Diane Fujino, to name a few, have pointed to the contradictions contained within Asian American sociologists and Filipino literary scholars' simultaneous embrace and rejection of the ideals of the US university system.5 7

American Orientalism and Differential Racializations

Reading before the San Francisco Section of the American mathematical Society on April

5, 1919, Yuen Ren Chao proceeded to deliver "A Note on 'Continuous Mathematical

Induction.' 5 8 Reconstructing a seemingly ordinary case of proof by mathematical induction, he concluded: "Like all mathematical theorems, the conclusion is no surer than its hypothesis. In this case, if the argument fails, it is usually because a constantA required in the second hypothesis does not exist...If a wedge is driven with a constant force between two sides which are pushed together by elastic forces, it will be stopped when balanced by the component of the increasing resistance. In this case, the A within which 6 may increase for cp(x + 5) to continue to hold will not be 'uniform for the interval,' so to speak, but will become smaller and smaller as x approaches the dangerous point, beyond which the conclusion ceases to be true." 59

Despite his note on math induction, Chao's conclusions indicate that he was trained and embedded within a particular tradition of theorem proving that emerged in the early twentieth century. That tradition embodied a desire to reduce mathematics to logic: to construct the branches of mathematics as formal axiomatic systems. Proofs in this tradition were meant to

5 See Henry Yu, Thinking Orientals;Augusto Espiritu, Five Faces ofExile: The Nation and FilipinoAmerican Intellectuals (Stanford: Stanford University Press, 2005); Diane Fujino, Heartbeatof Struggle: The Revolutionary Life of Yuri Kochiyama (Minneapolis: Press, 2005); Diane Fujino, Samurai among Panthers:Richard Aoki on Race, Resistance, and a ParadoxicalLife (Minneapolis: University of Minnesota Press, 2012). 5 Yuen Ren Chao, "A Note on 'Continuous Mathematical Induction,"' Bulletin of the American Mathematical Society (1919): 17-18. 59 Ibid., 18.

39 have a particular form: they consisted of the application of deductive rules of inference to the axioms or primitive principles of a formal logical system. In 1895 Giuseppe Peano, an Italian mathematician and early developer of math logic, published Formulairede mathe'matique. It was intended as a catalogue of all mathematical knowledge at the time. Listing theorems, Peano's goal was to collect and circulate established, mathematical results from all branches of mathematics. He wanted to standardize and organize mathematical knowledge in one place. 60 In order to salvage math from what they saw as troubling contradictions and diffusions, certain communities of mathematicians and logicians set out in search of new foundations that could be used to build mathematics from the bottom up, eliminating the possibility of contradiction, and providing justification for mathematical truth claims. They wanted to put math in one place and use the same grounding to justify and present all math proofs. Peano, Chao, logicians, and a growing number of mathematicians at the turn of the twentieth century believed that logic was the answer It was a formal system in which they hoped all of mathematics could be reliably put together and justified.

When Chao presented, he was based at UC Berkeley. In 1918 he had received his doctorate in philosophy, specializing in mathematical logic under the guidance of Henry M.

Sheffer. He had his received his BA in mathematics in 1914 at Cornell under a Boxer Indemnity fellowship. After the Boxer rebellion of 1900, when a group of Chinese revolutionaries attempted to expel Americans and Europeans from China, the indemnity fund paid to the US by

6 Giuseppe Peano, Formulairede mathematique (Bocca, Turin: Rivisita di matematica, 1985). On the history of the emergence of mathematical logic in the late nineteenth century that discusses Peano's life, see Ivor Grattan- Guinness, The Searchfor Mathematical Roots, 1870-1940: Logics, Set Theories and the Foundationsof Mathematicsfrom Cantor through Russell to Gdel (Princeton: Princeton University Press, 2002), especially 219- 267.

40 the Chinese government became the most prominent source of funding for Chinese students traveling abroad.61

Although Orientalism has been discussed primarily within the historical context of European colonialism, the discursive production of an utterly foreign, premodern, alien Oriental in opposition to a rational, modern Western subject has also been operative within the United

States, albeit in different ways.62 In the context of mid-nineteenth-century America,Orientalism constituted an Oriental other through exclusionary U.S. state policies on Asian immigration, and regulated racialized Asian labor through the institution of citizenship. Historian John Tchen also points out that prior to the 1850s, there was another Orientalist formation not organized solely around immigration. Tchen observes that increased trade with China and a growing port culture situated the Chinese as an exotic, curious spectacle for consumption within an emergent industry or urban popular . 63 Broadly, we can understand nineteenth-century American

Orientalisms as a set of discursive formations that are determined by and determining of U.S. economic and political engagements with East Asia and the Pacific, and that provide the ideological structure for domestic processes that produce and manage Asian racial difference within the US.

American Orientalism has structured the manner in which European Americans have dealt with ideas, goods, and immigrants from Asia. As noted by Henry Yu, Asians and Asian

Americans "Have been both valued and denigrated for what was assumed to be different about

61 Madeline Y. Hsu, The Good Immigrants: How the Yellow Peril Became the Model Minority (Princeton: Princeton University Press, 2015), 98-100. 6 On the European colonial context, see Suzanne Marchand, German Orientalism in the Age of Empire: Religion, Race, and Scholarship (New York: Cambridge University Press, 2009). On the specificity of U.S. Orientalism, see Lisa Lowe, Immigrant Acts: On Asian American CulturalPolitics (Durham: Duke University Press, 1996); John Tchen, New York Before Chinatown: Orientalism and the Shaping ofAmerican Culture, 1776-1882 (Baltimore: John Hopkins University Press, 1999). 6 Robert Tchen, New York Before Chinatown, 58-72.

41 them. Further, it has also had a profound effect on Asian American conceptions of themselves." 64

Because white Americans treated them as exotic foreigners, second-generation Chinese

Americans Japanese Americans often tried to "erase any connections to their Asian heritage."6 5

Since the knowledge of white Americans about Asian Americans was structured by the linking together of them as Orientals, the self-understanding of intellectuals who entered American institutions of scholarship reflected this structure. For instance, at the most important institution producing knowledge about Asian Americans during the period between 1920 and 1965, the

University of Chicago's Department of Sociology, Chinese American and Japanese American students were recruited to study what the white sociologists considered the "Oriental problem": the question of whether Orientals could be assimilated into American society. Drawn into academic institutions that were overwhelmingly white and male, these students created knowledge that answered the interests of their colleagues, and at the same time came to understand themselves through the social theories that they learned.

Between 1880 and 1924, large numbers of Japanese immigrants came to the West Coast, but anti-Oriental organizations decrying a "Yellow Peril" and an "Oriental problem' transferred onto them the political rhetoric and cultural representations used to exclude the Chinese. The popularity of pulp fiction novels such as Sax Rohmer's Fu Manchu series used the threat of a

Chinese evil genius to encapsulate and promote a fear of 'Asiatics' in general.6 6 Racial thinking, in the form of biological theories of "Mongolian" inferiority, or arguing the supremacy of

"white" civilization versus the "barbarity of Oriental civilizations," was criticized in the late

6 Henry Yu, 10-11. See also p. 18 65Henry Yu, see in particular 36-54. 6 Karla Rae Fuller, "Masters of the Oriental Detective," Spectator 17 (1996): 54-69; R. John Williams, Chinese Parrot: Techn&Pop Culture and the OrientalDetective Film," Modernism/Modernity 18 (2011): 95-124.

42 1910s and in the 1920s from a number of different points of view. Historians, social scientists, and anthropologists such as Franz Boas began to develop and advocate theories of culture that stressed understanding different communities from the inside, or 'native' perspective. 67

The 1920s marked the point at which a cosmopolitan interest in the exotic unknown became part of the training for becoming a social scientist in the US. The creation of a cosmopolitan ideal was not limited to social science, but it found its most rigorous theorization there in the perspectives of the sociologist or anthropologist as an outsider or stranger, and the social scientific ideal of objectivity. 6 8 The creation in 1920s New York of a certain brand of elite cosmopolitanism, with an attendant fascination with exotic art objects and a connoisseur's appreciation of other cultures, expanded to other elite private universities in the northeastern

US. 6 9 David Eugene Smith, a collector and historian of mathematics housed within the Teachers

College at Columbia University embodied this elite cosmopolitanism.

Historians of Mathematics and Ordering of Oriental Mathematics

Historians of media have documented how many imaginative interpretations about

Chinese and Japanese script systems appeared at the turn of the twentieth century. When analyzing Chinese characters-apparently upside down-for instance, Marshall McLuhan recalled seeing a "vortex that responds to lines of force...a mask of corporate energy."71 Writers

Ezra Pound and Ernest Fenollosa once proposed that Chinese script was "alive and plastic" and

"not only the forms of sentences," but also "literally the parts of speech growing up, budding

67 Franz Boas, The Mind ofPrimitive Man (New York: MacMillian, 1911); Franz Boas, Anthropology and Modern Life (New York: W.W. Norton & Co., 1928). 68 Henry Yu in particular documents this aspect in Thinking Orientals. 69For an interpretation of how a desire for the exotic arose in the United States, see T.J. Jackson Lears, No Place of Grace: Antimodernism and the TransformationofAmerican Thought, 1880-1920 (Chicago: University of Chicago Press, 1994). 70 Marshall McLuhan and Harley Parker Through the : Space in and Painting (Harper & Row, 1968), 38.

43 forth one from another." 7 1 Collectively, both McLuhan and Pound belonged to a line of commentators who referred to Chinese writing as a kind of ideal alterity.7 2

Historians of mathematics, an emergent profession, similarly did so. One of the first

English of the history of Japanese mathematics, produced by David Eugene Smith and Yoshio Mikami in 1910, differentiated the exoticness of Japanese mathematics with recourse to its different inscription systems. Writing about an eighteenth-century mathematician named

Ajima, they noted:

Thus we at last find in Ajima's work, the calculus established in the native Japanese mathematics, although possibly with considerable European influence. With him the use of a double series again appears, and by him the significance of double integration seems first to have been realized. He lacked the simple of the West, but he had the spirit of the theory, and although his contemporaries failed to realize his genius in this respect, it is now possible to look back upon his work and evaluate it properly.

Looked at from the standpoint of the West, and weighing the evidence as carefully and as impartially as human imperfections will allow, this seems to be a fair estimate of the ancient wasan:--The Japanese, beginning in the 1 7th century, produced a succession of worthy mathematicians...But the mathematics of Japan was like her art, exquisite rather than grand. She never developed a great theory that in any way compares with the calculus as it existed when Cauchy, for example, had finished with it.

From the standpoint of opportunity Japan did remarkable work; from the standpoint of mathematical discovery this work was in every way inferior to that of the West. 73

Smith and Mikami's valuations paralleled a moment in which Chinese script shouldered blame for hampering civilizational advancement. This argument resurfaced among Greek classicists in the mid-twentieth century, when the questioned relationship between orality and literacy invited speculations on whether a writing system like the alphabet was responsible for the advancement

71 Ernest Fenollosa, The Chinese Written Characteras a Mediumfor Poetry: A Critical Edition, ed. Haun Saussy (New York: Fordham University Press, 2008). Originally published in 1919 and heavily edited by Ezra Pound. 72 See R. John Williams, The Buddha in the Machine: Art, Technology, and the Meeting ofEast and West (New Haven: Yale University Press, 2014) for an elaboration of what he calls the "techne-zen" aesthetic among early twentieth-century writings in the United States. 73 David Eugene Smith and Yoshio Mikami, A History ofJapanese Mathematics (Chicago: Open Court Publishing Company, 1914), 280.

44 of philosophy and science in ancient Western civilization.74 Mathematician William F. Osgood and others have similarly argued that the advent of the Greek alphabet, superseding its

Phoenician origins, was the first writing system to successfully reduce ambiguity between physically similar words. 7 5 According to this formalization of the history of mathematics, this system of adaptation spurred the into developing higher and higher levels of abstraction that formed, in short, the prerequisite mental framework for science.

David Eugene Smith's immersions in these discussions were conditioned by his training.

Beginning his higher education at the State Normal School in Cortland and then continuing onto

Syracuse University, Smith initially pursued studies on art, classical languages, and Hebrew. 7 6

He received his doctorate degree in aesthetics and the history of art. Following a brief career in law, he turned his attention to teaching mathematics in 1884. In 1891 he became the chair of the

Department of Mathematics at the Michigan State Normal College in Ypsilanti, where he began to amass a collection of over 700 volumes of books related to mathematics. In 1898 Smith relocated to New York and became principal of the State Normal School at Brockport, where he became acquainted with the publisher George Arthur Plimpton. In 1901, he accepted the position of Chair in mathematics at the Teachers College at Columbia University, where he remained until his retirement in 1926.

Between 1880-1920, the period historians have marked off as the decades in which consumer developed, a market for art and other collectibles from the Far East

?4 Caroline Winterer, The Culture of : Ancient Greece and Rome in American Intellectual Life 1780-1910 (Baltimore: Johns Hopkins University Press, 2002), 204-220. 75 Discussions surrounding this played out on the pages of American MathematicalMonthly. See William F. Osgood, "Discussions: Is There a Student Standard of Truth? A Reply," American Mathematical Monthly 34 (Aug- Sep 1927): 365-366; George W. Evans, "The Greek Idea of Proportion," American Mathematical Monthly 34 (Aug- Sep 1927): 354-357; Florian Cajori, "On the Chinese Origin of the Symbol for Zero," American Mathematical Monthly 10, no. 2 (1903): 35. 76 Joseph W. Dauben, Writing the History of Mathematics: Its HistoricalDevelopment, eds. Joseph W. Dauben and Christoph J. Scriba (Boston/Basel: Birkhauser, 2002), 269.

45 intensified in the United States. 77 In 1893, Ernest Fenollosa observed that a "craze" for Japanese art had emerged and that the prevalent exhibition of Japanese objects at world exhibitions were

"already working a revolution in our theories of decoration." 78 Avid collectors toured Japan,

China, and other parts of Asia to catalog and collect books, furniture, and other decorative objects David Eugene Smith was part of that milieu. While on leave from Teachers College in

1907, Smith traveled abroad to South and East Asia for the first time, visiting Burma, , Sri

Lanka, Japan, and China. In November, Smith began to amass a collection related to the

"Orient." While viewing a collection of books related to Buddhism while visiting Burma, Smith noted:

One [of the books] was written on gilded copper, and the other on thick paper, also gilded. The one on copper had beautiful lettering in sepia lacquer, and the other was written in India ink. I looked at all of the books very carefully and then I said to my interpreter: 'David, those are the only two books in that lot that I want.79

Smith's trips to other parts of Asia focused exclusively on stocking Columbia's libraries with

Oriental art, books and artifacts about mathematics. Later visiting Lahore, India, Smith negotiated with a dealer over the price of a particular manuscript written by Timurid sultan and astronomer Ulugh Beg. Securing the text at the price of $84.25, Smith expressed enthusiasm over having acquired the "finest mathematical manuscript that I have ever bought in the East.""

In 1909, following his return from travels abroad, Smith became Director of the

Educational Museum of Teachers College. It was established in 1899 by James Earl Russell, then the Dean of Teachers College, in order to centralize material relating to the "manual training, art,

77 Dawn Jacobson, Chinoiserie(London: Phaidon 1999); See also Madeleine Jarry, Chinoiserie. Chinese Influence on EuropeanDecorative Art 17th and 18th Centuries (Vendome Press, 1981). 7' Ernest Fenollosa, "Contemporary Japanese Art," The Century 46 (August 1893): 478. 79David Eugene Smith, 1936 notebook, 3-4, DESPP, Box 32. 8 Ibid., 8.

46 domestic science, domestic art, and natural science."" From 1901, the Museum occupied Room

215 on the second floor of Main Hall-now Zankel Hall of Teachers College. Smith incorporated the books and materials he had collected into the exhibits, displaying rare mathematical artifacts he had also brought back from his travels to China and Japan. The collection incorporated major Japanese and Chinese mathematical texts, such as the "Chinese encyclopedia of mathematics published by the Jesuit influence in the seventeenth century; the first Chinese edition of Vlacq's table of ; an early Chinese edition of ; numerous

Japanese manuscripts and printed works, and an early Manchu treatise on mathematical astronomy." 82 Smith's collection served as a primary resource for A History ofJapanese

Mathematics (1914).

Contemporaries and visitors were struck by the "degree of wealth" exhibited in Smith's collections of Chinese and Japanese mathematical materials.83 Despite their rarity and remoteness, the artifacts were rendered familiar and contemporary by repeated analogies with modem mathematical materials and objects. Chinese counting rods and abaci were compared to contemporary instruments. The compatibility of "ancient" Chinese and Japanese mathematical artifacts and scripts were emphasized by the display of the various notational scripts (Figure 2.3) alongside each other. At the same time, they were also displayed in chronological order of their

"discovery." 84 The history of Smith's public display of his "Oriental materials" point to how such artifacts could be framed and transformed from embodying.the "character of an age" to a physical artifact that was part of a wider public enthusiasm for the Oriental aesthetic.

81Benjamin Richard Andrews, Museums ofEducation:Their History and Use, PhD Diss. (Columbia University, 1908),15. 82"Educational Museum of Teachers College," 1909, 7, DESPP, Box 19, Folder "Educational Materials." 83"A Mathematical Exhibit of Interest to Teachers," Science 25 (1907): 232-234. " David Eugene Smith, "Chinese Inventory," 1925, DESPP, Box 19, Folder "Collections Inventory." See also E.F. Donoghue, "In Search of Mathematical Treasures: David Eugene Smith and George Arthur Plimpton," Historia Mathematica 25 (1998): 359-365.

47 U ____

Figure 2.3: Educational Museum exercise celebrating the "equivalence" among varied scripts of numeracy at the Educational Museum in New York. 1913. From DESPP, Box 45, Folder "Teaching Photographs."

The Problem of Origins

Debates over the problem of "origins" in mathematics played out in mathematical journals and publications. Historian and mathematician Florian Cajori took part in a series of exchanges in American MathematicalMonthly in 1903 on the "Chinese Origin of the Symbol for

Zero." Reporting on behalf of Yoshio Mikami of Tokyo, Cajori noted how "Until recently, the symbol for zero and the principle of local value in our notation of numbers were supposed to be of Hindu origin. A few years ago, our attention was called to the early work of the Japanese, and now the priority appears to be passing to the Chinese." 85 Mikami corroborated this observation,

" Florian Cajori, "On the Chinese Origin of the Symbol for Zero," The American Mathematical Monthly, 10, no. 2 (1930): 35.

48 noting that "I have found very important relations between the mathematics of India and of

China." 86

Conclusion

In this chapter, I have argued that the subject of the discipline of mathematics was one whose formal aesthetics were informed by US American visions of the "Oriental" as insufficiently linear and abstract. Meanwhile, I have described mathematicians, critics, and historians of mathematics as subjects who both undermined and shaped these visions.

86 Yoshio Mikami, quoted in George Bruce Halsted, "Our Symbol for Zero," American MathematicalMonthly 10, no. 4 (1903): 90.

49 Chapter Two Introspective Subjects: QuantifiedExpressions and G.D. Birkhoff's Theory of Aesthetic Measure

"The major abstraction is the commonal. The inanimate, difficult visage. Who is it?" -Wallace Stevens, Notes Toward a Supreme Fiction (1942)

In the winter of 1928, Harvard mathematician George David Birkhoff set sail for Japan to begin research on a project that formed part of a larger project. His first aim was to examine the

"Internationalization of the Mathematical Bases of the Art," and in turn to explore whether aesthetic judgments about art could be made through objective and mathematical means." 87

Throughout the rest of that year, he traveled across various parts of Europe and Asia-including

India, Siam, China, Egypt, and Hungary-to collect, catalogue, and analyze the whole range of artistic forms that were specific to a particular culture. From one country to the next, Birkhoff planned to dissect the literatures that described how "past and present practices and theories in art" had helped form new kinds of artistic creations, whether paintings, , musical compositions, or literary texts. Knowing that background was important, but only up to a certain point. A more crucial and essential task was to derive the particular order, arrangement, and set of elements undergirding a specific piece of art in the first place. Birkhoff set out to mathematically define and "give a more mature formulation" of art forms in systematic terms.8 8

87 George D. Birkhoff, "Proposed Research: The Internationalization of the Mathematical Bases of Art as Shown in Form, Color and Sound," December 1926, GDBP, HUG 4213.2, Box 9, Folder "Correspondence: 1927, R-Z #2, 1928-29, A-Q." 88 Ibid.

50 His larger ambition, as he later explained in retrospect, was to demonstrate and, if possible, to prove, the necessity of "purely mathematical thought forms" for grasping those qualities of everyday life that remained elusive and therefore indescribable: "the subjective." 89

Extracting the inner thoughts and uncovering the mechanisms behind feelings of "aesthetic pleasure" within individuals posed a problem that Birkhoff wanted to resolve. Approaching this task, however, required answering several questions first: how could one draw on mathematicians' "skillful work" on the "modification of abstractions" to elucidate "aesthetic pleasure?" 9 If so, in what ways could the "formal principles of mathematics be used to analyze artistic elements and determine their subjective, aesthetic values?"9 1 Birkhoff's mathematical work broadly questioned the place of mathematics within and across all domains of inquiry.

Following a series of activities, including giving regular presentations at mathematics conferences, and offering courses to explore the "mathematical elements of the arts," Birkhoff consolidated his thoughts and findings into a book published by Harvard University Press in

1934. Entitled, Aesthetic Measure, Birkhoff introduced a theoretical and quantitative approach to turn the private, individuated, and inexpressible act of judging art into a consistent, reliable, and objective assessment of what a viewer's response to art was.92

A "longing for a common world" that could be "communicated," as Lorraine Daston and

Peter Galison have noted, animated prior efforts to systematically secure and generalize abstract knowledge in early twentieth-century mathematics. 93 With recourse to the symbolic language of logical proof, for instance, studies of differential equations increasingly focused on deriving the

89 George D. Birkhoff, "Mathematics: Quantity and Order" Science Today (1934): 293-317. 911 Ibid. 91 Ibid. 92 George D. Birkhoff, Aesthetic Measure (Cambridge: Harvard University Press, 1934). 93 Daston and Galison, Objectivity, 301.

51 properties of their solutions or on verifying proofs of their existence and uniqueness. Less attention was placed on computing actual solutions or developing rigorous methods of approximation. Similarly, analyses of geometry or the "modem algebra" of groups and rings revolved around locating "mathematical structures" or "invariants under transformation." 94 Such universal endeavors were neither new nor unique in the early twentieth century. In the 1920s, and 1930s. philosophers of the Vienna constituted another-if not the-quintessential example of the ambition to establish a unified system of knowledge through logical constructions of elementary propositions and simple observation reports. However, G.D. Birkhoff's interest in this pursuit took an alternative form. Contrasting the Vienna Circle's aim to dispose of metaphysical thoughts and empirically ground a new system of knowledge, Birkhoff's ambition to produce a mathematics of aesthetics hinged on a new confidence in mathematics enabled by disciplinary developments; that is, the ability to render and articulate the most subjective and elusive aspects of everyday life. Because mathematical knowledge was no longer necessarily tied to the empirical world, its value lay in its ability to express and codify all forms of reasoning.

Birkhoff's theory was part of a larger movement in the United States that brought the arts and the human sciences together from the 1890s through the 1930s. During this period, a diverse group of participants from the art world forwarded a new, formalist approach to art that historians have labeled as American and aesthetic modernism. 95 They sought to break away from a tradition of producing representational imagery, and instead, under the mantra of abstraction, capture the qualities of everyday life and experience. As Robert Brain and Robin Veder deftly

94 On the qualitative theory of differential equations, see Henri Poincard, "Les m6thodes nouvelles de la m6canique celeste," Il Nuovo Cimento 10 (July 1899): 128-130; George D. Birkhoff, "Surface transformations and their dynamical applications," Acta Mathematica 43 (1920: 1-119. For a historical discussion on invariants and structures, see Mehrtens, Moderne-Sprache-Mathematik,315-326; Leo Corry, Modern Algebra and the Rise of Mathematical Structures, 21-65. 95 For an historical overview, see Daniel Joseph Singal, "Towards a Definition of American Modernism," American Quarterly39, no. 1 (Spring 1987): 7-26.

52 illustrate in the cases of fin-de-siecle Europe and 1920s United States, artists were stimulated by the field of physiological aesthetics. 9 6 They consistently drew from the theories and experiments of physiological psychologists in order to describe modem art in terms of the body. Likewise, psychologists working on physiological aesthetics sought to understand how viewers responded to art through the body. Collectively, the arts and human sciences were united in their aim to discern traces of subjectivity itself. In this chapter, I describe and reconstruct the means by which

Birkhoff arrived at a quantitative theory and approach to evaluating art. In turn, it examines

Birkhoff's participation in these conversations, particularly with art educators, and his explication of mathematics as a useful means to express internalized responses to art, or more broadly to render visible the inner feelings of the mind.

Although mathematicians and historians to date have grappled with many aspects of

Birkhoff's mathematical career and broader social life, his theory of aesthetic measure is frequently mentioned as idiosyncratic and conducted separately from the field, whether in his research or in his efforts to promote the internationalization of mathematics. Much of the literature frequently documents mathematicians' impatience with Birkhoff's writings on aesthetics. As British mathematician G.H. Hardy quipped in response to the publication of

Aesthetic Measure, "Good, now he can get back to doing real mathematics." 97 However, scholarship takes for granted two aspects that shaped his theory: first, the role of institutional and disciplinary developments in mathematics, and second, the wider, non-disciplinary social contexts in which Birkhoff operated. In this case, there is an asymmetry between how

96 See Robert Brain, The Pulse of Modernism: PhysiologicalAesthetics at Fin-de-Siecle Europe (Seattle: University of Washington Press, 2015); Robin Veder, The Living Line: Modern Art and the Economy ofEnergy (Hanover: Dartmouth College Press, 2015); Susan Lanzoni," Practicing psychology in the art gallery: Vernon Lee's aesthetics of empathy," Journal of the History of the BehavioralSciences 45, no. 4 (2009): 330-54. '"Garrett Birkhoff recalls this quip from a conversation he had with Hardy about the senior Birkhoff's work in Donald Albers and Gerald L. Alexanderson, eds. MathematicalPeople: Profiles and Interviews (Boston: Birkhauser, 1985), 14.

53 mathematical practices are contextualized within their disciplinary, social contexts as scientific, and how mathematical practices operating at the edges of-or within-a humanistic domain get contextualized. Stated alternatively, analyses of intellectual practices or inquiries in areas that seem aberrant should equally be assessed as a site in which mathematical thinking or practices take place. To do so otherwise risks performing another kind of separation between technical works and the social contexts in which they operate.98

This chapter proceeds by focusing on how Birkhoff's development of aesthetic measure as both a theory and quantitative technique to evaluate aesthetic judgments emerged and related to concurrent developments in physiological aesthetics and design theory. The point is not to argue that Birkhoff's theory of aesthetic measure was right or wrong, but to show what made it possible, how it inflected Birkhoff's ideas about mathematics, and how other values or forms of thought underwrote it.

Theory and Technique of Aesthetic Measure

George David Birkhoff seemingly exemplified the image of a modern mathematician and the independence of American mathematics. He was part of a new generation of American

9 By forwarding this point, I mean to directly engage the wider literature concerning the history of American intellectual life. Intellectual historians of the United States such as David Hollinger, Thomas Haskell, and Bruce Kuklick have examined various facets of disciplinarity, as well as the access to and consequences of professionalization for American intellectual life. Hollinger's focus on the modem American academy concerns its significance on the scientists, social scientists, philosophers, and historians who fought the Christian biases that had kept Jews from fully participating in American intellectual life. In order to discuss how objective knowledge claims operated in the natural and social sciences, Haskell considers academic professions as institutions. Lastly, Kuklick emphasizes professionalization as a leading factor for supplanting "vision" with "technique." Each illustrate how seemingly technical or academic matters have broader cultural resonances, but always with respect to some form of the sciences. See David Hollinger, Science, Jews, and Secular Culture: Studies in Mid-Twentieth-CenturyAmerican Intellectual History (Princeton: Princeton University Press, 1996); Thomas Haskell, Objectivity Is Not Neutrality: ExplanatorySchemes in History (Baltimore: Johns Hopkins University Press, 1998); Bruce Kuklick, A History of Philosophy in America, 1720-2000 (Oxford: Oxford University Press, 2001). For a treatment of the cultural and social significances of the academic professions in postwar America see David Hollinger, ed., The Humanities and the Dynamics ofInclusion Since World War II (Baltimore: Johns Hopkins University Press, 2006).

54 mathematicians who chose to remain in the US for graduate studies rather than to go abroad.9 9 In

1907, he received a doctorate from the University of Chicago under the supervision of Eliakim

Hastings Moore, whose own work reflected ongoing changes within the field. Moore's

Introduction to a Form of General Analysis (1910), for instance, expanded on David Hilbert's axiomatic approach to turn the analysis of axiomatic systems into a subject of intrinsic mathematical interest in its own right. 100 Following positions at the University of Wisconsin-

Madison and Princeton, Birkhoff assumed an assistant professorship at Harvard and later served as president of the American Mathematical Society and American Association for the

Advancement of Science.

In his work, Birkhoff frequently tacked back and forth between the abstract and the concrete. In 1913, he achieved international recognition for his proof of Henri Poincard's "last geometric theorem."' 0 1 During World I, his work on mathematical problems related to ballistics and submarine detection reflected the growing relations drawn between formal mathematics and the physical sciences. And Birkhoff's later work on mathematical theories of dynamics has often been interpreted by historians of mathematics as Birkhoff's attempt to accommodate the "strive towards purity" and the "acknowledgement, reinforced by the war, that mathematics ought to be concerned with applications." 0 2

99 In the early twentieth century, a majority of American students had traveled to Europe, mostly Germany for training. For a larger discussion of these migrating dynamics, see Karen Hunger Parshall, "Perspectives on American Mathematics," Bulletin of the American Mathematical Society 37, no. 4 (2000): 381-406; John W. Servos, "Mathematics and the Physical Sciences in America, 1880-1930," Isis 77 (1986): 611-629. 10o David Aubin, "George David Birkhoff, Dynamical Systems (1927)," in Landmark Writings in Western Mathematics 1640-1940, ed. Ivor Grattan-Guinness (Elsevier Science, 2005), 180. 10' The problem asks whether an infinite number of periodic solutions exist for three bodies. For an elaboration of Poincards influence on Birkhoff, especially on his theory of dynamical systems, see June Barrow-Green, Poincar and the Three Body Problem (Providence: American Mathematical Society, 1997). 102 David Aubin writes that while Birkhoff always emphasized applications in the studies of the physical sciences, he also "never computed an orbit." In Aubin, "George David Birkhoff, Dynamical Systems (1927)," 178.

55 Yet Birkhoff's conviction in the value of mathematics for applications were also predicated on disciplinary developments within mathematics, including those concerning

Euclidean geometry. Previously understood to refer to incontrovertible spatial truths and a circumscribed form of reasoning inherited from ancient Greece, Euclid's theorems and its assumptions were increasingly challenged by the proliferation of non-Euclidean geometry in the nineteenth century. When American mathematician Oswald Veblen introduced geometric axioms of his own in 1903, for instance, he could not take for granted that his system "codified in a definite way our spatial judgments."' 03 After all, a few years prior David Hilbert had radically questioned geometry's direct relations to spatial experience. If a geometric statement consists of something like "[tiwo points define a line," Hilbert stated, then terms such as "tables," "chairs," and "mugs" could easily substitute for "points," "lines," and "planes."'04 That geometric postulates could be abstracted or substituted for concepts or symbols taken from a variety of other subjects implied that representations, or inscriptions, of mathematics entailed choice. As such, mathematics instantiated a mode of thought of a more general nature, one that in turn could be employed to explore and codify other systems of thought. Birkhoff emphasized this value in the presentation of his own axioms as the simplest possible system of Euclidean geometry" in

1926, stating that the "prime concern" was "to make the students articulate about the sort of thing that hitherto [they have] been doing quite unconsciously.' 05 Geometry-and more broadly

" Oswald Veblen, "A System of Axioms for Geometry" Transactionsofthe American Mathematical Society 5 (1903),3. 104David Hilbert, Foundationsof Geometry, trans. Leo Unger (La Salle, Ill.: Open Court, 1971/1899), 2-4. 1 Oswald Veblen, "A System of Axioms for Geometry," Transactionsof the American Mathematical Society 5 (1904): 343-384, on p. 343. On the history of non-Euclidean geometry's reception, see Joan Richards, Mathematical Visions (cit. n. 13); Jeremy J. Gray, Janos Bolyai, Non-Euclidean Geometry, and the Nature of Space (Cambridge: MIT Press, 2004); George D. Birkhoff, "A Set of Postulates for Plane Geometry, Based on Scale and Protractor," Annals of Mathematics (1932): 329-345. See Christopher Phillips, The New Math: A PoliticalHistory (Chicago: University of Chicago Press, 2015) for an account of George Birkhoff's axioms in the Cold War classroom, and the co-authorship of an experimental textbook between George D. Birkhoff and Ralph Beatley in Basic Geometry (Chicago: Scott Foresman, 1941).

56 mathematics-enabled a means of introspection, a way to render explicit the intuitive and internalized reasoning they were already doing.

It was due to this new attitude towards mathematics and its virtue as a general form of reasoning that Birkhoff believed an objective analysis of aesthetic experience was possible. By aesthetic experience, Birkhoff's interest lay in the problem of reliably measuring a viewer's emotional response to an art form. Aesthetic judgements of this kind had previously seemed impossible to collect, either because viewers struggled to verbally express their inner responses or because viewers' descriptions of their responses varied to a degree that they could not be reasonably compared. In Birkhoff's view, a reliable measure of aesthetic value (M) could be made by joining knowledge about human perception drawn from physiological aesthetics to a logical and formal mode of reasoning drawn from mathematics. In his words:

The typical aesthetic experience may be regarded as compounded of three successive phases: (I) a preliminary effort of attention, which is necessary for the act of perception, and which increases in proportion to what we shall complexity (C) of the object; (2) the feeling of value or aesthetic measure (M) which rewards this effort; and finally (3) a realization that the object is characterized by a certain , , or order (0), more or less concealed, which seems necessary to the aesthetic effect.1 06

Stated alternatively, Birkhoff contended that in the resulting equation M = O/C, the measure of aesthetic -or a viewer's emotional response-depended upon the density of order relations in an aesthetic object. Aesthetic quality was understood to be inversely proportional to the amount of attention required to wholly perceive an object.

The reliability of this measure relied on two assumptions. The first was that there were limits to the kinds of artistic forms to which Birkhoff's measure could apply. As he clarified,

"We shall endeavor at all times to choose formal elements of order" that are "logically

11 Birkhoff, Aesthetic Measure, 3-4.

57 independent." 7That is, only objects that were "made" and "original" acts of creation, rather than "representational of nature" or exhibiting "connotative" features would be considered.' 08

Birkhoff wanted to include only the properties specific to an aesthetic object as constituting its order, a term borrowed from group theory. He deemed "connotative" aspects such as references to an aesthetic object's history or inclusion as part of a tradition as unnecessary elements to include. In this sense, Birkhoff's image of the art viewer was premised on a particular model of the mathematician, one embodying the new attitudes of the field.

Second, the claim that an aesthetic object's complexity, or form, was directly related to a viewer's "efforts of attention" depended upon an understanding of bodily perception drawn from physiological aesthetics. Citing the late nineteenth-century psychophysiological experiments of

Wilhelm Wundt, Gustav Theodor Fechner, and , Birkhoff appealed to the

''muscles,"' "nerve currents" and "automatic motor adjustments" of sensory organs bodily and cognitive means by which aesthetic experience occurred (See Figure 2.1).109 Birkhoff's references to "motor nerve currents" as the sources of movement within the body connected to a larger discourse on energy and resources in the economy. In the context of growing industry and human-machine relations, metaphors such as "human motors" became a means to rethink the conditions of human labor and standards of production." 0 The curious effect of these assumptions was that Birkhoff's aesthetic measure revolved around a labor of perception, treating the reception of an aesthetic object as an economy of optical stimulation and emotive cognition.

107 Ibid., 15. " Ibid. 109Ibid.,14. " This sort of thinking spanned various effects. For an account of how Hermann von Helmholtz's nineteenth- century studies of thermodynamics provided industrial capitalism with the "human motor" metaphor, see Anson Rabinbach's The Human Motor (Berkeley: University of California Press, 1992).

58 DIAGRAM OF THE AESTHETIC FORMULA

Field of attention in cerebrum {Field of aesthetic Field of Field of tension associations sensation)

Motor nerve current of Sensory nerve current automatic adjustment

Sense organs Muscles (eye, ear) (eye,throat)

C (complexity)is measured by weighted automatic motor adjustments. 0 (order) is measured by weighted aesthetic associations. M = 0,'C (aesthetic measure) indicates comparative aesthetic value.

Figure 2.1: George D. Birkhoff's "Diagram of the Aesthetic Formula." From Aesthetic Measure (Cambridge: Harvard University Press, 1934), 14.

In practice, the ways of calculating aesthetic measure varied with the kind of object being examined, but all were predicated on the idea that specific "elements" of the object were independent, could be isolated, and assigned numerical grades to establish its order. Judgment, or the amount of "pleasure felt," was understood to be rendered explicit and precise because each measure would be calibrated. To calculate the aesthetic measure of a polygonal form, for example, the viewer began with reference to the object's shape and then determined its essential properties. The ability of a viewer to discern those elements within a polygonal form also required that the viewer know, or at least be trained, to pick out the form's "structural relations"

59 through mathematical reasoning." Other features such as color and the material upon which the form was expressed could be dismissed on account of being "connotative."I 2 From there, the viewer could begin to quantify a polygonal form's overall aesthetic measure.

Birkhoff wanted the viewer to evaluate an object's order by scoring specific elements and then adding up the values. He constructed a set of values for polygonal forms drawn from a geometrical perspective, resulting in a specific formula (see Fig. 2.2):

0 V+E+R+HV-F M -= C C

Elements included in the calculation of a polygon's order mostly concerned its symmetry permutations, or "group of motions," across a plane. 1 13 The greatest weight (3) was given to a polygonal form's rotational , the least (-1) to the absence of a symmetry along a vertical axis or some other pre-defined value. To calculate complexity (C), the viewer counted

"the number of distinct straight lines containing at least one side." "4 In the case of a square, its complexity amounted to 4, and its order to 6, resulting in an aesthetic measure of 1.5. In

Birkhoff's theory, the resulting quotient value favored a particular form of compactness, in which an aesthetically pleasing object exhibited as much order and as little complexity as possible. In his words: "The beautiful is that which gives us the greatest number of ideas in the shortest period of time." 1 5 The higher the number calculated, the more "pleasing" a polygon was considered to be.

" Birkhoff, Aesthetic Measure, 203. 112 Ibid., 15 Birkhoff's quantitative approach requires that the viewer abstract the form away from the material upon which it gets expressed. This assumption, of course, takes for granted what many scholars in the history of science and STS scholars now are careful to attend to. As noted by Stefan Helmreich, "When abstractions are realized in particular media, the media make a difference to how the abstractions are understood." See Stefan Helmreich, Sounding the Limits ofLife: Essays in the Anthropology of Biology and Beyond (Princeton: Princeton University Press, 2016): 184. S13Birkhoff, Aesthetic Measure, 36. 114 Ibid., 42 "5 Ibid., 4 and 199

60 C: complexity (-), V: vertical symmetry(+), Fq: equilibrium (+), R: (+), HV: relation to a horizontal-vertical network(+), F: unsatisfactory form (-) involving some of the following factors: too small distances fromvertices to vertices or to sides, or between parallel sides; angles too near o° or 1800; other ambiguities; unsupported re- entrant sides; diversity of niches; diversity of directions; lack of sym. metry.

Figure 2.2: Birkhoff's set of prescribed values for calculating the complexity, elements of order, and overall aesthetic measure of polygonal forms. From George. D. Birkhoff, Aesthetic Measure (Cambridge: Harvard University Press, 1934), 34.

Birkhoff admitted that there were limitations to applying his quantitative method. As a measure, the resulting numerical value could never yield a specificfact or give a "real" and absolute measure of a viewer's aesthetic response. "Even in the most favorable cases," he confessed, "the precise rules adopted for the determination of 0, C, and thence of the aesthetic measure M, are necessarily empirical." The symbolic variables 0 and C, in this case, merely represent "social values, and share in the uncertainty common to such values."" i6 But for

Birkhoff, that did not detract from why a mathematical approach to aesthetic response mattered.

His frequent descriptions of aesthetic measure as a "theorem" to "indicate its completeness" point to another aspect. Instead, the theory of aesthetic measure offered a systematic, "purely mathematical (logical) reasoning" to answer aesthetic questions" and offer a "method of direct

" 6 Ibid., 46.

61 introspection" that heretofore had remained unwieldy to organize or collect.""I7 For Birkhoff, aesthetic measure enabled the individuals to more rigorously express their innermost, elusive

feelings, as well as indicate a qualitative index about which aesthetic forms individuals preferred.

Aesthetic Measure as Reason and Design

To pursue his research and travels in 1928, Birkhoff drew on an extensive network of mathematicians, philanthropic institutions, and new funding bodies centered around efforts towards internationalization."I 8 He received partial support from the Rockefeller Foundation's

International Education Board. In the years following World War I, the IEB led an effort to chart, understand, and reproduce the most successful aspects of European mathematics. It aimed to promote communication and disciplinary growth between both sides of the Atlantic. Financial support for Birkhoff's research came on the heels of his participation in 1926-and first time abroad-surveying the structure and institutions of European mathematics.'19 His largest source of funding came from the Bureau of International Research, which granted "$8,400 for travels to the Far East. 2 0 Established in 1924 through a large donation from the Laura Spelman

Rockefeller Memorial, the Bureau was jointly administered by a Harvard-Radcliffe Committee to "develop research of an international character and of an advanced nature, such as might not

17Ibid. '" A wide range of literature in the history of mathematics explores the ideology of internationalism in the recasting of mathematics post WWII. See especially Michael Barany, "Distributions in Postwar Mathematics" PhD diss., Princeton University, 2016); Michael Barany, Anne-Sandrine Paumier, Jesper Lutzen, "From Nancy to Copenhagen to the World: The Internationalization of Laurent Schwarts and His Theory of Distributions," HistoriaMathematica 44 (April 2017): 367-394. "' Reinhard Siegmund-Schultze, Rockefeller and the Internationalizationof Mathematics Between the Two World Wars: Documents and Studiesfor the Social History of Mathematics in the 2yCentury (Basel: Birkhduser, 2001). 12 George Grafton Wilson to George D. Birkhoff, 14 December 1927, GDBP, HUG 4213.2, Box 9, Folder "Correspondence: 1927, R-Z #2, 1928-29, A-Q." Wilson further noted, "The Committee in charge of the Bureau of International Research considered your request, contained in your letter of 8 December, and provisionally approved the arrangement which you suggest."

62 otherwise be undertaken." 21 The Bureau mostly promoted studies of contemporary problems in law, and international relations. But it also offered an enclave for mathematicians like Birkhoff to conduct inquiries beyond established disciplines and departments.

Perhaps the largest aid for the project came through his student Marshall Stone and

Stone's father Harlan F. Stone, who at the time served as an Associate Justice member of the

Supreme Court. With both Stones serving as intermediaries, Birkhoff quickly secured easy access to the "diplomatic and consular office of the United States" through a letter signed by the

Secretary of State, Frank B. Kellogg. 122 "To the American Diplomatic and Consular Officers,"

Kellogg's letter of February 2 started, "I take pleasure in introducing to you George D.

Birkhoff...who is about to proceed abroad. I cordially bespeak for Doctor Birkhoff such courtesies and assistance as you may be able to render, consistently with your official duties."123

The letter may have proven especially helpful a little less than a month earlier when the IEB returned Birkhoff's passport to him. While the IEB had successfully "procured the Japanese visa" at his behest, they could not do so for Turkey since "no one ever seemed to be in their office."1 2 4

Following his research travels, Birkhoff's confidence in the development of a reliable and general theory of aesthetic measure was buttressed not only by inquiry, but also pedagogy.

Towards the end of his travels abroad in 1928, mathematician William Benjamin Fite invited

Birkhoff to join Columbia's mathematics department as a visiting lecturer the following summer:

21 "Harvard Gets Money for International Research: Professor Wilson Chairman of Fund Committee," The Harvard Crimson, October 19, 1925, 8. 12 Harlan Stone to Marshall Stone, 3 February 1928, GDBP, HUG 4213.2, Box 10, Folder "Correspondence: 1928- 1929, R-Z, 1930, A-Z." 2 Frank B. Kellogg to George D. Birkhoff, 2 February 1928, GDBP, HUG 4213.2, Box 10, Folder "Correspondence: 1928-1929, R-Z, 1930, A-Z." 124Hazel Hauahare to George D. Birkhoff, 9 January 1928, GDBP, HUG 4213.2, Box 10, Folder "Correspondence: 1928-1929, R-Z, 1930, A-Z."

63 "The department is unanimous in its desire to have you lecture at Columbia in the Summer

Session of 1929," Fite wrote on October 24, "We should like to have you give two courses, each

five hours a week" and with a salary of"$1,5000 for the session."1 25 Birkhoff eagerly accepted the offer before the end of the month had passed, with "[e]verybody in the [Columbia] department...greatly pleased" that he would be in residence.' 2 6 In the months leading up to the

summer session, Columbia's course catalog advertised Birkhoff's course as an elective for 3 credits directed at graduate students and advanced undergraduates. Students enrolled in the course attended daily lectures from 10:30 am to 12:30 pm during the summer session. 127

Birkhoff retaught the course the following summer at Harvard as a graduate elective.

When in the summers of 1929 and 1930 Birkhoff taught "Mathematical Elements of the

Arts," he was not so much interested in inculcating the theory of aesthetic measure into his students as in demonstrating the efficacy of it. The course was a way of testing whether students' exercises calculating aesthetic measures were also exercises in introspective analysis, whether their exercises matched their aesthetic responses and therefore corroborated his theory. As the course notebook of Barnard undergraduate student Ruth Ellen Rablen reveals, "The purpose of the course is to study the elements of art and determine their mathematical significance."1 28

Students were also tasked with cultivating the skill to objectively compare two artistic forms.

"When these forms are both members of a sufficiently restricted class," Birkhoff stated in his

12 William Benjamin Fite to George D. Birkhoff, 24 October 1928, GDBP, HUG 4213.2, Box 9, Folder "Correspondence: 1927, R-Z #2, 1928-29, A-Q." " William Benjamin Fite to George D. Birkhoff, 11 January 1928, GDBP, HUG 4213.2, Box 9, Folder "Correspondence: 1927, R-Z #2, 1928-29, A-Q." 127 "Columbia University in the City of New York: Catalogue Number for the Sessions of 1929-1930," (New York: Columbia University, 1928), 236. 12 Ruth Ellen Rablen, "Mathematical Elements of Art" student notebook, Summer 1929, RPMP, Box 71, Folder "Mathematical Elements of Art."

64 introductory lecture, "such intuitive comparison [between two objects] is possible and we can say which form is preferable." 129

In his lectures to students, Birkhoff did not treat the mathematical elements of art as a formula or theory to be memorized and then applied. Neither did he treat them as a completely

"known" or "mapped" subject. Instead, he presented the course as an ongoing field of research, in which significant problems remained to be understood: "What is the role of these mathematical elements?" How much does an artistic form's "value depends [sic] on their mathematical nature?" In introducing the course in this manner to an array of graduate students from mathematics and the arts-some of whom had had limited previous exposure to the field itself-Birkhoff emphasized the breadth of questions that the mathematics had yet to answer and map out beyond the disciplinary boundaries of the field. To compensate for the wide range of students whose backgrounds in mathematics varied, Birkhoff centered the course on the study of polygonal forms, from which the of "tile work and design, curvilinear form, vases, and the musical quality of poetry" could proceed. 3 0

Through paper-based exercises, Birkhoff's course engaged students in the appreciation of polygonal forms in two ways; they had to learn to look and to think, or reason, with mathematics.

But not just any specific system of mathematics. Birkhoff repeatedly maintained throughout his lectures that "bringing connotations into opinions as to artistic value" should be repeatedly avoided. 3 ' Consequently, his lectures made little reference to the historical contexts and styles of art in which polygonal forms or geometric designs were often found. He avoided structuring his lectures around the history of aesthetics in favor of promoting an object-oriented pedagogy.

129 Ibid. 13 Ibid. "1 Ibid.

65 To manage these undesirable incursions, Birkhoff disciplined students' bodies to repeatedly engage a polygon's visual geometric form and elements through a variety of mathematical means. One was geometrically, by having students compare the lengths and angles of the various polygons to consider their overall shape, proportions, and complexity. Another was algebraically and numerically, by assigning symbolic variables or numbers to a polygon's sides and manipulating equations of how one polygonal form's area varied from another's form. A final exercise, deemed the most important to Birkhoff, was analyzing the polygon's "group of motions." In a series of exercises, Ruth Ellen Rablen practiced calculating the possible set of spatial orientations-or symmetries-of a square, a triangle, a pentagon, and an octagon (see

Fig. 2.3). Birkhoff's mathematical exercises implied that the "mathematical elements of art" constituted numerous kinds of computational practice.

66 / --

OF

00-- ~ I

0+ArA *~Th

Figure 2.3: Ruth Ellen Rablen's visual and symbolic exercises computing the various spatial orientations of a polygonal form. From Mathematical Elements of Art" student notebook, Summer 1929, RPMP, Box 71, Folder "Mathematical Elements of Art."

In lecturing students and assigning exercises on a polygon's "groups of motions,"

Birkhoff had a larger aim. His interest in developing an objective, quantitative approach to aesthetic response was closely tied to a larger project of organizing and classifying different sets of comparable aesthetic forms. By formulating classifications, determining objects' aesthetic measures, and then correlating them with numerical values, the collected data could enable a

67 I means to objectively make comparative aesthetic judgements among objects in each class. To show what a practice of classifying aesthetic objects would look like look like with his theory, he drew on the problem of ordering 90 polygonal forms. For each polygon, he had students score the elements of order for each shape (i.e. its vertical symmetry, achievement of equilibrium of proportion, rotational symmetries) and calculate the resulting measure.

According to Birkhoff's theory, once completed, the aesthetic measure of a polygonal form served as a record of students' assessments towards that shape. Because the resulting measure had been arrived at through repeated exercises of scoring the weighted elements of a restricted class of objects-in this case polygons-each student documented the reliability of

Birkhoff's theory by arriving at the same numerical measure for each shape. Hence Birkhoff's confidence in his theory when he later wrote, "I obtained the consensus of aesthetic judgment as to the arrangement of these polygons. The results so obtained were found to be in substantial agreement with the arrangement obtained by the formula."' The practice of flattening analyses of a polygon's geometric elements into numbers instantiates, as scholarship in science and technology studies repeatedly show, how a wide array of shapes or other kinds of values can be scrutinized independently and compared to other objects deemed similar through calculation.3 3

The resulting table of polygons inscribed aesthetic judgements as measurement (See Fig. 2.4a and 2.4B)

132 Birkhoff, Aesthetic Measure, 45-46. m3 For example, see Christopher Phillips, "The Machine: Sense, Subjectivity, and in the California Wine-World," Social Studies ofScience 46 (2016): 461-481; William Deringer, Calculated Values: Finance, Politics, and the QuantitativeAge (Cambridge: Harvard University Press, 2018); Theodore Porter, Trust in Numbers: The Pursuit of Objectivity in Science and Public Life (Princeton: Princeton University Press, 1995).

68 r

A~

4+ LZ ~'jj +

Figure 2.4a and 2.4b: On the left, Ruth Ellen Rablen's calculations of each polygon's elements of order. From "Mathematical Elements of Art" student notebook, Summer 1929, RPMP, Box 71, Folder "Mathematical Elements of Art." On the left, the ranked list of the first 15 polygonal forms according to their "," with the square as the most aesthetically pleasing. From George D. Birkhoff, Aesthetic Measure (Cambridge: Harvard University Press, 1934), 33.

In a sense, Birkhoff's exercises and lectures can be read as a means to "discipline" students' judgements. On the other, given their focus on measuring the internalized and inexpressible feelings of pleasure for the individual, they were also an expression of an individuated response to an aesthetic form. But more significantly, Birkhoff's theory of aesthetic measure instantiated a particular vision of what mathematics was and what it could do.

The Modernism of Ornament and Craft

Birkhoff's inquiry into aesthetics as an object of inquiry developed as an outgrowth of disciplinary concerns, pedagogy, and scientific-political networks, but his pursuit also formed in

69 a milieu of fascination for the decorative arts, Asian aesthetics, and revival of the Arts and Crafts movement underway in New England. The production of hand-crafted objects like ceramics had increasingly been displaced with the advent of industrial techniques during the late nineteenth century. Drawing from the craft ideals inaugurated by British writers and designers such as John

Ruskin and William Morris, turn-of-the-century craft enthusiasts re-valorized the handcrafted appearance of a work.' 34 These developments dovetailed with a growing influx of Japanese woodblock prints to the United States that began in the late nineteenth century, as well as a renewed attention within the Boston arts community on the "geometric simplicity" and "bold lines" of Japanese ukiyo-e prints.'35

Against this backdrop, the design and implementation of Birkhoff's research on mathematical aesthetics can more clearly be understood. When clarifying his interests "in the role of relations of order in art," for instance, Birkhoff's 1926 research proposal emphasized that his plans to examine the "mathematical principles" of art forms had developed around exposure to various "tileworks," "vases," and other handicraft objects.1 36 And when validating his treatment of geometric forms such as polygons as aesthetic objects, Birkhoff averred that their

"aesthetic appeal" have "always been recognized, and is borne out by their wide use for decorative purposes in East and West."1 37 His 1928 travels abroad focused attention on collecting

134 For a discussion of the Arts and Crafts movement and its revival in the United States, see William Morris, Selected Writings, edited by G.D.H. Cole (New York: Random House, 1934); Eileen Borris, Art and Labor: Ruskin Morris and the Craftsman Ideal in America (Philadelphia: Temple University Press, 1986), and especially T.J. Jackson Lears, "The Figure of the Artisan: Arts and Crafts Ideology," in No Place of Grace: Antimodernism and the Transformation ofAmerican Culture, 1880-1920 (New York: Pantheon, 1981), 59-96. 13 Arthur Wesley Dow, "The Responsibility of the Artist as an Educator," Lotos 8 (February 1896): 610-611. Dow further commented in this same article, "Oriental art has never felt the touch of Leonardo's system," offering a way of "learning to feel beauty and to create it in simple ways." For a larger discussion of engagement with Japanese prints during this period, see R. John Williams, The Buddha in the Machine: Art, Technology, and the Meeeting of East and West (New Haven: Yale University Press, 2014), especially 24-58. 136 George D. Birkhoff, "Proposed Research: The Internationalization of the Mathematical Bases of Art as Shown in Form, Color and Sound," December 1926, GDBP, HUG 4213.2, Box 9, Folder "Correspondence: 1927, R-Z #2, 1928-29, A-Q." 17 Birkhoff, Aesthetic Measure, 16.

70 a variety of "Oriental objects," including " records of Chinese , "wood-cuts," and Japanese poems. 138 These collected objects became the material sources that Birkhoff analyzed in his later publication on Aesthetic Measure.

Within the decorative arts, practices of ornamentation occupied a special place in

Birkhoff's treatment of aesthetic forms. Ornaments consisted of "[a]ny figure, traced, painted, embossed, or otherwise executed on a surface," provided that there was "at least one possible movement which moves the figure, as a whole, but not point for point, back to its initial position."' 39 Birkhoff prioritized ornament as a form whose aesthetic effectiveness could easily be analyzed by casting aside historical interpretations or abstracting the ornament from other

"connotative" interpretations. Otherwise:

When the theories of an artist overbalance his aesthetic judgment and experience, he is likely to produce what may be termed 'puzzle-art.' This kind of art has been exemplified in many experiments of recent decades. Any novel artistic form which cannot be appreciated without advance knowledge of the theory underlying it may be suspected of falling into the category of puzzle-art.4

Birkhoff never clarified the artists or artistic pieces that he categorized as "puzzle-art," but he constantly maintained that his focus on craftwork and ornament drew from dissatisfaction with this category.1 4 '

Birkhoff's ideas stood in stark contrast to those of European architect and contemporary

Adolf Loos, who in 1929 disparaged ornament as "criminal" since "it "is no longer an expression

" Harlan Stone to Marshall Stone, 3 February 1928, GDBP, HUG 4213.2, Box 10, Folder "Correspondence: 1928- 1929, R-Z, 1930, A-Z."; E.H. Cressy to George D. Birkhoff, 5 October 1928, GDBP, HUG 4213.2, Box 9, Folder "Correspondence: 1927, R-Z #2, 1928-29, A-Q"; Reverend H.G.C. Hallock to George D. Birkhoff, 12 August 1929, GDBP, HUG 4213.2, Box 9, Folder "Correspondence: 1927, R-Z #2,1928-29, A-Q." 13 Birkhoff, Aesthetic Measure, 49 140 Ibid., 216. 14 Historian of art Jeannette Redensek has interpreted this term to be his explicit reaction against "artistic modernism," as decrying artists who occupy a realm divorced from the authority of history, public taste, and political authority. See Jeannette Redensek, "Art is Good for Nothing," Art as a Way ofKnowing (San Francisco: Exploratorium, 2011), 1-7.

71 of our culture." Because architectural designs now tended towards "everyday use," insisted Loos, ornaments needed to be discarded from exterior surfaces and buildings facades as incongruities of the past.1 42 The resulting architectural style tended towards a stripped down, clean aesthetic derived from the need to preserve "the essential values of modem individuality" and a modernist dichotomy in which "form follows function."1 43

Read in this context, Birkhoff's early twentieth-century valorization of ornaments and craft wares can seem "antimodern" and place him as a participant within a turn-of-the-century movement seeking to return to an authentic and organic experience. His lauding of Chinese and

Japanese craft aesthetics and dismissal of "puzzle-art" certainly echoes a more reactionary-and racist-agenda. Such is the story described in T.J. Jackson Lears No Place of Grace. 44 But

Birkhoff related ornament to the subject differently than Loos, with Loos' understanding between buildings and their viewers founded on distant viewer. Birkhoff, however, interpreted the viewing subject's relations to ornament and other aesthetic forms as an affective .

His valuation of mathematics as a logical, formalist, and abstract tool to approach problems of aesthetic judgment, and, more specifically, subjectivity, was also premised on a particular kind of perceiving viewer, one drawn from the theories and experiments of physiological aesthetics.

In this frame, the references to the body's "motor and nerve currents" enabled the perception of art through the eye.

14 Adolf Loos, "Ornament and Crime," in Adolf Loos, trans. Michael Mitchell, Ornament and Crime: Selected Essays (London: Ariadne Press 1997 [1913]): 167. On the circumstances surrounding "Ornament and Crime," see Christopher Long, "The Origins and Context of Adolf Loos's 'Ornament and Crime'," Journal of the Society of Architectural Historians68, no. 2 (June 2009): 200-23. 43 Antoine Picon, Ornament: The PoliticsofArchitecture and Subjectivity (West Sussex: John Wiley & Sons Ltd., 2013),8. 144 T.J. Jackson Lears, No Place of Grace: Antimodernism and the Transformation ofAmerican Culture, 1880-1920 (University of Chicago Press, 1981).

72 In a way, Birkhoff's theory of aesthetic measure and assessing art was an alternative articulation of approaching an image through the logicism and structural modes of analysis of mathematics, but one not premised on the disembodied viewer. His theory offered an alternative approach to articulating how art was consumed, even when the viewer was vehemently opposed to the images traditionally associated with modernism, or when the art itself was not associated with craft. For Birkhoff, transformations within mathematics offered new tools towards a

"science of subjectivity," of extracting and expressing the inner elusive feelings of the individual by bringing different systems of mathematics to bear on the analysis of aesthetics forms.

Conclusion

The seeming incompatibility that a system of measurements and mathematical rules could strengthen the creativity and designs of architects became central to Birkhoff's theory and a touchstone of his appeal to some contemporary artists, particularly within Harvard's Graduate

School of Design. Birkhoff found strong supporters in Denman Ross and Arthur Pope, design theorists and educators who included Birkhoff's work within their teaching. For Arthur Pope,

Birkhoff's aesthetic measure encouraged the designer to "contemplate the overall composition of the work, rather than get sidetracked by the representation of individual details." 4 5 For Pope, having a system with which to clarify compositional structure strengthened the designer's conviction in their creations.

Birkhoff's theory of aesthetic measure therefore helps give a firmer grasp on an elusive yet critical moment in the history of the in the United States and on modernism.

Actively primarily in the 1910s-1930s, Birkhoff's theories preceded the Bauhaus, and follow in

145Arthur Pope to George D. Birkhoff, to George D. Birkhoff, 21 December 1932, GDBP, HUG 4213.2, Box 11, Folder "Correspondence: 1931-1933, A-Z." For a larger discussion on the work of Denman Ross and Arthur Pope, see Marie Frank, Denman Ross and American Design Theory (Hanover/London: University Press of New England, 2011).

73 lock-step with design theorists such as Denman Ross who have typically been associated as a continuation of British critic John Ruskin and the Arts and Crafts Movement. 14 6 The next chapter looks to reexamining the legacy of the Bauhaus within the history of mathematics and the history of science.

146 Historians of science have paid close attention to the parallels between the Bauhaus and the logical positivists in constructing a vision of high modernism in the mid-twentieth century. For an overview of this relationship, see Peter Galison, "Aufbau/Bauhaus: Logical Positivism and Architectural Modernism," CriticalInquiry (Summer 1990): 709-752.

74 Chapter Three Formal Subjects: Max Dehn and Black Mountain College

"Topography displays no favorites; North's as near as West. More delicate than the historians' are the map-makers' colors." -Elizabeth Bishop, "The Map" (1946)

Mathematics in the Liberal Arts

The phrase "Cold War rationality" calls to mind coldly calculating, mathematically oriented strategists cultivating habits of logical reasoning, wielding the tools of , and mulling over the doctrine of mutual assured destruction in the immediate postwar years. But historians of science and politics have lately characterized Cold War rationality not by its signature celebration of logic, but rather by its "expansion of the domain of rationality at the expense of that of reason,"14 7 an expansion forged out of a war-time pragmatism that left little space for informal discussions of philosophy or foundations in the boardroom or classroom.

Instead, crunching numbers and calculations eclipsed discussions that smacked of

"interpretation," or worse, "philosophy." Historians of science and mathematics today are not the first to recognize the irony of a rationally centered style of thought. A community of artists and progressive educators working in the US during and after World War II saw such contradictions as well. Establishing Black Mountain College in rural North Carolina, their project was nothing short of the reassertion of reason and the aim to "educate as a person and as a citizen."1 48

147Paul Erickson, Judy L. Klein, Lorraine Daston, Rebecca Lemov, Thomas Sturm, and Michael D. Gordin, How Reason Almost Lost Its Mind: The Strange Careerof Cold War Rationality (Chicago: University of Chicago Press, 2013),2. 14' Kenneth Kurtz, "Black Mountain College, Its Aims and Methods," Black Mountain College Bulletin 8 (1944): 3. Reprinted in HaverfordReview 3 (Winter 1944): 18.

75 Contrasting with histories of mathematics that have focused on the Cold War research university dynamics of mathematics, this chapter describes the formalist values embedded within the mathematics curriculum that developed at Black Mountain College. If the dominant image of

"Cold War rationality" includes large, lavishly funded research projects, a stance of political and moral detachment despite increasing militarization, and slavish imitation of the natural sciences, then Black Mountain College and its educators instantiated its negative: institutionally precarious and underfunded, suffused with pacifist values, and deeply philosophical.

An unaccredited college in rural Appalachia founded in 1933, Black Mountain College is remembered as the site of a crucial transatlantic dialogue between Euro-American modernist aesthetics and pedagogy in the postwar United States. Inspired by the work of philosopher John

Dewey, who later joined the College advisory board, its pedagogy emphasized arts training, and its founders hoped to loosen or abolish altogether the types of separations between students and faculty, and the faculty and administration. BMC assumes a prominent place in the genealogies of widely disparate fields of thought. 149 It has been heralded as one of the influential points of contact for European exiles emigrating from Nazi Germany; as a standard-bearer of the legacy of intentional or planned communities; as the bellwether campus of Southern racial integration; as an important testing ground for proponents of progressive education; and as a seminal site for postwar practices of the arts in the US. The number of famous participants-in addition to Josef

149 On the topic of European emigres at Black Mountain, particularly those coming from the Bauhaus, see Hal Foster,"The Bauhaus Idea in America," in Albers and Moholy-Nagy: From the Bauhaus to the New World, ed. Achim Borchardt-Hume (London: Tate, 2006), 92-102; Carl Goldstein, "Teaching Modernism: What Albers Learned in the Bauhaus and Taught to Rauschenberg, Noland and Hesse," Arts Magazine 54 (December 1979): 108- 16. For a history of the College within a legacy of American communitarian experiments, see Martin Duberman, Black Mountain: An Exploration in Community (New York: W.W. Norton, 1972). For information regarding racial integration and Black Mountain and its larger legacy in progressive education in the United States, see John Katherine Chaddock Reynolds, Visions and Vanities: John Andrew Rice ofBlack Mountain College (Baton Rouge: Louisiana State University Press, 1998); Camille Clark, "Black Mountain College: A Pioneer in Southern Racial Integration," JournalofBlacks in Higher Education 51 (Spring 2006): 46-48.

76 Albers, John Cage, , faculty included Merce Cunningham, art critic Clement

Greenberg, and former members of the Bauhaus Josef and Anni Albers.

The faculty also included an unlikely but well-known figure within the mathematics community at the time: topologist Max Dehn. An exile from Nazi Germany, Dehn served as

Professor of mathematics at BMC from 1945-1952, having emigrated in 1940 through an arduous route from Scandinavia/USSR/Japan. Today mathematicians might recognize his name as an adjective (for Dehn surgery and Dehn invariants). As a student of David Hilbert's, he is also remembered as the first mathematician to solve one of Hilbert's famous problems (the third), as well as for his pioneering work in theory and combinatorial topology. Since Dehn emerged as a mathematician from an emerging tradition that emphasized a formalistic and axiomatic approach, we might have expected Dehn to eschew the use of illustrations and models in math pedagogy, instead espousing generalization and unification above all. Yet attention to

Dehn's lecture notes and courses with the students with BMC, I suggest, reveals that visualization practices remained crucial to his mathematical teaching. With his systematic exploration of subtle variations of form in his mathematics classes, he attempted to instill techniques pushing visual perception beyond habit in his students. In this sense, he participated in mid-century modernist project in the arts to emphasize process over outcomes, promoting forms of experimentation and learning in action that could dynamically change routine habits of seeing.

Pedagogy - Craft and Perception Between Science and Intuition in the Liberal Arts

Despite Dehn's mathematical background and the emphasis of art education at the college, he was a good fit for Black Mountain. After all, the goal was not to produce professional artists but to consider all individuals as possible creators and to offer training for what Bauhaus

77 artist Josef Albers, a prominent figure and educator at the time, termed a "flexible and productive mind that wants to do something with the world...we are on the way to the researcher, discoverer, to the inventor, in short, to the worker who produces or understands revelations." 5 0

Dehn had expressed similar views about the mentality of the mathematician in a speech delivered in 1928:

We must realize that the progress of science depends not on comfortable plodding but on perceiving and forming new ideas. The progress of our discipline depends not on mass efforts, not on a flood of papers filled with investigations of insignificant special cases of generalizations, but on individual creative achievements.' 5

Dehn viewed mathematical activity and artistic practice not just as activities aimed at representing the world, but also as inherently acts of creation.

Max Dehn was not completely cut off from the mathematical community in the United

States. He taught two students who went on to pursue mathematical careers, including Peter

Nemenyi and Trueman McCarthy. Dehn arranged for leading mathematician and German 6migre

Emil Artin to travel to Black Mountain College in 1948 and administer a matheamtics exam for

Peter Nemenyi. 5 2 In the summer of 1948 and 1949, he also taught at the University of

Wisconsin-Madison.

Between 1945 and 1952, Dehn offered an array of courses in mathematics that offered, for the artists, alternative ways to articulate form and possibility in rearticulating it creatively.

Dehn's course-"Geometry for Artists"-concentrated on shape through the exact observation and transcription of form in space. Drawing was conceived as a "test of seeing" that graphically reported visual data honed by exercises in foreshortening, overlapping distance, and nearness.

15 Josef Albers, "Speech at Black Mountain College luncheon at the Faculty Club," 29 April 1938, AAC, Box 1, Folder "Speeches." 151 Max Dehn, "The Mentality of the Mathematician. A Characterization," Mathematical Intelligencer 5:2 (1983[1928]), 25. 15 Peter Nemenyi, "Recollections," 18 March 1968, BMCPC, Box 18, Folder "Nemenyi, Peter."

78 Dehn encouraged students to observe the disposition of line in various contexts; in one study, the depiction of repeated bent and scrolled planes tested the precise spatial of two dimensions into three (Figure 3.1). His teaching exercises employed uncomplicated geometric forms such as squares, triangles, and , as well as simple figures such as letters and numbers, to perform changes in perspective and to create anamorphic effects that demonstrated a mastery of spatial representation.

V I '1.'~ A.

Figure 3.1: Lorna Blaine's notes from Max Dehn's Course, 1945. From BMCPC, Box 4, Folder "Blaine, (Howard)(Halper), Loma - BMC Project Correspondence."

What did it mean for a mathematics course to be called "Geometry for Artists?" Did course materials and student exercises take a different from than what would have been taught in a Geometry course at a technical institution? In the archival record, the course stands out from a slew of other technical courses that were offered at Black Mountain College, including physics, chemistry, and . In other words, Dehn was emphasizing the humanistic and

79 philosophical nature of mathematics. In all likelihood, the classification of the course was due to factors both practical and theoretical. To be sure, Dehn was trying to encourage students who did not have a technical background to enroll. But in a broader sense, it was an attempt to acton a shared commonality with arts education to create and produce knowledge, drawing not just from the tools of a specific discipline but using them in other practices.

Dehn's course on , however, was more technical, dealing with

"points in space and pure logical proofs."' Well attended by students like Ruth Asawa and

Elaine Schmitt, both of whom as proponents of geometric abstraction in the arts, recalled bringing to class only a straight edge and pencil. "We used no measurements, only lines and proofs." Once, standing on a chair (as his example required the reach), his chalked line wrapped neatly around a black painted pipe that crossed the board. Proud of his proof, he asked, "can you see the music? To him, the harmony of mathematical relationships and those of music were of the same order." 1The course involved lot of perspective or . These were well-attended classes, but often times they proved to be obscure for many of the students, who didn't have much previous academic training. He taught more advanced courses in mathematics on an individual basis. The effects on his students were numerous. They included Dorothea

Rockburne, Ruth Asawa, and Elaine Schmitt.

1Max Dehn, "Lecture 1," Spring 1948, MDP, Box 4RM 132, Folder "Lectures-BMC 1948." 1 Ruth Asawa and Elaine Schmitt to Toni Dehn, 24 November 1951, RAP, Box 17, Folder 2 "Correspondence: 1949-1953."

80 U

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Figure 3.2: Ruth Asawa and some of her crafted wire art sculptures around 1954. From RAP, Box 217, Folder 3.

Dehn's lecture notes are revealing of his understanding of art and mathematics both as forms of an epistemological project, as aform ofknowledge in which the better "vision" that attentive perception provokes can in fact increase awareness about routinely assigned meanings, and thus can encourage people to transform their customary patterns of comprehension. This view contrasted with other emergent forms of formalism in the arts and Cold War emphasis on rationality, such as those of Russian formalist and art critic Viktor Shklovsky, who privileged the

"special perception" of art over knowledge, elevating art to a category of direct experience

81 surpassing epistemology, and even attempted to negate the "ends" of meaning production in

favor of an "'ends" of "means."1 55 The structure of perception was related to the growth and transformation of cognitively assigned meanings in art and in the world at large. The diverse

forms of modernity are themselves always changing, yet habit-driven behaviors reinforce accustomed understandings of forms and their existing, known relations to one another.

Maintaining an alert attention to the appearance and constitution of form short-circuits assumptions that corroborate preexisting categories.

As historians of art have noted, the legacy of BMC is precisely bound up with contradictory visions of modernism as inextricably interwoven with the logic of experimentation and skepticism towards rationality. BMC is considered a site of what historian of art Eva Diaz has called "the crucial midcentury modernist practice-experimentation." BMC participants' ambitions to transform habits of perceptions, systems of intention, and patterns of tradition have essential implications for understanding not only modernist but subsequent art practices.s56

Instructors such as Josef Albers, abandoned an attentive examination of the structure and serial organization of form, but also continued to pursue a familiarly modernist goal of changing audiences' relationships to established patterns of perception. Architect and designer

Buckminster Fuller, who taught at Black Mountain College in the summers of 1948, 1949 and

1954 and emphasized a return to structure and form, nevertheless also sought to change students' relationships to their designs.1 57

15Viktor Shkolovsky, "Art as Technique," in Art in Theory 1900-2000: An Anthology of ChangingIdeas, ed. Charles Harrison and Paul Wood (Oxford: Blackwell, 2003), 279-80. 156Eva Diaz, The Experimenters: Chance and Design at Black Mountain College (Chicago: University of Chicago Press, 2015), 13. 157 See Buckminster Fuller, "Emergent Humanity: Its Environment and Education",[1965]. In R. Buckminster Fuller on Education, edited by Peter H. Wayschal and Robert D. Kahn, 86-133 (Amherst: University of Massachusetts Press, 1979.

82 Chapter Four Creative Subjects: Mathematical Origami and the Limits of Expression

"Realistic, naturalistic art had dissembled the medium, using art to conceal art; Modernism used art to call attention to art." -Clement Greenberg, "Modernist Painting" (1960)

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Figure 4.1a and 4.1b: On the left, crease of the five-horned rhinoceros . Eupatorus gracilicornis,Opus 476, 2008. On the right, the completed beetle when folded from the crease pattern. Originally designed in 2003-2004. From Libby Hruska, Peter Hall, Paolo Antonelli, et. al., eds., Design and the Elastic Mind (New York: Museum of Modern Art, 2008), 58.

In 2008, a new and temporary exhibition called "Design and the Elastic Mind" opened at the Museum of Modern Art (MoMA) in New York. The exhibit featured a range of objects and installations highlighting the entanglements between design practices and technoscientific research. Of the pieces selected for inclusion by curator Paolo Antonelli, one image depicted a

83 blush-colored assemblage of dispersed in varying parts and sizes (Figure 4.1a). It also featured a patterned array of black, intersecting lines that, when folded along by hand on an uncut square piece of paper in a particular way, would transform into a full-bodied model of a five-homed rhinoceros beetle, replete with an enlarged abdomen, jointed legs, and five horns, one of which extended and protruded upwards from its head (Figure 4.1b). Entitled "Eupatorus gracilicornis, Opus 476," the patterned image was produced on 30.5cm x 30.5cm origamido paper, a textile of blended fibers incorporating hemp, flax, cotton, and linen. The design also constituted a "crease pattern," a diagram whose lines designate the paper folds, or creases, necessary to produce a specified final form.1 58

The work of origami designer and consultant Robert J. Lang, the crease pattern belonged to a new form of the paperfolding craft that emerged during the last quarter of the twentieth century and continues to grow in popularity today: mathematical and computational origami.

Previously, origami sekkei, the art of technical folding, proceeded through a more exploratory, hands-on approach in which the folder intuited new folding techniques through trial-and-error. In the 1970s, however, mathematicians and mathematically-inclined origami artists turned their attention towards examining paperfolding from a geometrical and algebraic perspective. With the aid of computing technologies, they sought to increase the range of origami folds and forms possible, proceeding to convert paper creases into mathematical expressions amenable to formalization and calculation. Their efforts spurred the emergence of a new approach to origami design motivated by mathematical pursuit.

' In Libby Hruska, Peter Hall, Paolo Antonelli, et. al., Design and the Elastic Mind (New York: Museum of Modem Art, 2008), 58-60. The crease patterns were one of more than fifteen origami-related design objects featured on loan from several origami designers. This portion of the exhibit explored the effects of mathematization of origami not just as an art form motivated by a geometrical pursuit, but also as a field of research spanning pedagogical and technical applications. Designers applied their knowledge to practical considerations ranging from how to fold a car's airbag most efficiently to how to send a100-meter in diameter telescope into space.

84 Concurrent with MoMA's exhibit in 2008, another image debuted in the lower-level exhibition spaces of the Fondation Beyeler in Switzerland. It bore a striking to Lang's crease pattern designs but also exhibited clear differences in its material manifestation and discursive purposes. Entitled "Black Beetle," the large-scale mural featured an array of white lines and bright graphic colors that had been painted onto a 23.7m x 3.8m wall using high-gloss household paint (Figure 4.2a). The mural, along with other pieces such as "Rhino Beetle,"

(Figure 4.2b) formed a larger series of paintings called Origami and were produced by New

York artist and filmmaker Sarah Morris between 2007 and 2009. As a craft that "originated in

China with the advent of paper," origami intrigued Morris as subject matter for an even larger art project about contemporary . For Morris, the art form's deep past and production of evermore "complex forms" served as a symbol to "intro-duce [sic] a revised mapping of power, desire, urbanism and design" in contemporary China. 15 9 Each Origamipainting engendered

"architectural motifs" and "visual surfaces" that allegorically analyzed the "multiple interpretations" and "structures of control.""' In the wake of the 2008 Olympics, the Origami series also functioned to "signify" Beijing as an "impending event." 1 6' In doing so, her work treats origami as an Orientalized craft practice that is reminiscent of interpretations from the early twentieth century.

'"Sarah Morris' Black Beetle on View at Fondation Beyeler," Art Daily, May 31, 2008, http://artdaily.com/news/24483/Sarah-Morris--Black-Beetle-on-View-at-Fondation-Beyeer. 16 Museo d'Arte Moderna di Bologna, "Sarah Morris: China 9, Liberty 37," MAMbo Bologna press release, May 26, 2009, Press Release. http://www.mambo- bologna.org/files/documenti/archiviocomunicatiENG/Sarah%20MorrisMAMbo%20may%2026%20- %20july%2026,%202009.pdf. 161 "Sara Morris' Black Beetle on View at Fondation Beyeler," Art Daily.

85 -w

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Figure 4.2a and 4.2b: On the left, installation view of "Black Beetle" at Fondation Beyeler, 2008. Photo by Todd Eberle, Todd Eberle Photography. On the right, Sarah Morris's Rhino Beetle [Origami], 2008.

Although Morris's Origami series claimed to be based on "traditional Eastern paper compositions," the perceived similarities of its lines-depicted on paintings like "Rhino

Beetle"-to several of Lang's and other origami designers' crease patterns-such as "Opus

476"-became the basis of a copyright infringement lawsuit in 2011.162 Several questions arose over the course of the dispute: who authors and owns a work or design? Who, in turn, has the right to use or transform it? Such questions are ever present and constantly negotiated in intellectual property law. In the domain of U.S. copyright law, however, and depending on the nature of the work, those same questions are tethered to a particular kind of author operating within a specific intellectual domain. Questions over copyright in mathematics, for instance, are typically circumscribed around discussions of credit and scientific authorship.' 6 3 If by the late

162 Ibid. 163See, for example, Bruno Latour and Steve Woolgar, LaboratoiyLife: The ConstructionofScientific Facts (Princeton: Princeton University Press, 1986); Mario Biagioli and Peter Galison, eds. Scientific Authorship: Credit and Intellectual Property in Science (New York: Routledge, 2003); Adrian Johns, Piracy: The Intellectual Property Warsfrom Gutenberg to Gates (Chicago: University of Chicago Press, 2010); Mario Biagioli, Peter Jaszi, Martha

86( twentieth century modem mathematics became valued for being a creative art that was at once abstract, "made," and autonomous, how has mathematical knowledge been construed as a form of property?

The arguments and interpretations surrounding the Lang vs. Morris case speak not only to intellectual, but also to legalistic tensions inherent in the narrow discursive framings drawn around art, mathematics, and the professions who think or work with them. As a means to delineate the limits of the modernist values drawn around mathematics as a creative art in the present, this chapter examines the dispute between Lang and Morris over the authorship and ownership of the crease patterns. The case indicates a juncture in which mathematics and art operate beyond the context of American universities and are embedded within a craft that cannot so easily be disaggregated along disciplinary lines.

Typically, the US legal system treats mathematical work such as proofs as an original creation and expression of abstract ideas that are protected under copyright law. In 1981, however, the scope of protection expanded such that mathematical knowledge and practices could be folded into patent claims. In Diamond v. Diehr, the Supreme Court decided on whether a mathematical formula and programmed digital computer central to a rubber-curing process could be patented.' 6 4 The Court ruled in favor of granting the patent, opening the door to software patenting. 6 5 In writing the majority opinion, Justice William Rehnquist laid out the

Woodmansee, eds. Making and Unmaking IntellectualProperty: Creative Productionin Legal and Cultural Perspective (Chicago: University of Chicago Press, 2011). 64 Diamond v. Diehr 450 U.S. 175 (1981). Engineers James Diehr and Theodore Lutton's patent application included the use of the Arrhenius equation of the form In(v)= CZ + x to examine the temperature dependence of reaction rates. 165 Legal scholars and historians of science and technology in the past two decades have more recently focused on patent law. See Mario Biagioli, "Between Knowledge and Technology: Patenting Methods, Rethinking Materiality," AnthropologicalForum 22 (2012): 285-300; Mario Biagioli, "Patent Republic: Specifying Inventions, Constructing Authors and Rights," Social Research 73 (2006): 1129-1172.; Daniel Kevles, "Inventions, yes; Nature, No: The Products-of-Nature Doctrine from the American Colonies to the U.S. Courts," Perspectives on Science 23 (2015): 13-34.

87 legal precedents for establishing mathematical work as property by parsing knowledge from practical applications, arguing that while the mathematical formula and concept undergirding it were "not patentable in isolation," the equation's facilitation of a "more efficient solution" of a

"process" warranted patent protection.' 66 That is, while a mathematical formula could not be patented in and of itself, its embodiment as a crucial component of a computer program did not preclude the process from being patent eligible.' 67 The case marked a moment when the tensions

of newness, abstraction, and autonomy over mathematics were publicly contested and revealed.

Anyone committed to mathematics as a creative field, however, would have been dissatisfied with the Court's reasoning that patentability could be extended to "an application of a law of nature or mathematical formula to a known structure or process." 168 In other words, the legal threshold for establishing proprietary claims seemed to favor a utilitarian understanding of mathematics rather than preserve the notion of mathematics as a product of creative endeavor.

The shifting status of mathematics raised by these issues and legal disputes are not new,

forming parts of long-standing debates over the nature of mathematical knowledge and narratives of creativity, invention, and discovery.1 69 Taking into account the increasingly interdisciplinary nature of mathematical work advances another set of issues that exceeds the concerns and

'66Diamond v. Diehr 450 U.S. 175 (1981), 187-188. Justice Rehnquist further notes: "It is now commonplace that an application of a law of nature or mathematical formula to a known structure or process may well be deserving of patent protection...Arrhenius' equation is not patentable in isolation, but when a process for curing rubber is devised which incorporates in it a more efficient solution of the equation, that process is at the least not barred at the threshold by 101." 167 For a historical discussion on the elision between software and computer programs, see Gerardo Con Diaz, "The Text in the Machine: American Copyright Law and the Many of Computer Programs, 1974-1978," Technology & Culture 57 (October 2016): 763-779; Gerardo Con Diaz, "Contested Ontologies of Software: The Story of Gottschalk v. Benson, 1963-1972," IEEE Annals qfthe History of Computing 38 (January-March 2016): 23-33. 168 Diamond v. Diehr 450 U.S. 175 (1981), 187. 169 Mathematicians have continued to frame such debates along these terms. See, for example, Paul R. Halmos, "Mathematics as a Creative Art," American Scientist 56 (1968): 375-389; Philip J. Davis and , The MathematicalExperience (Boston/Basel: Birkhauser, 1981); Michael Harris, Mathematics without Apologies: Portraitofa Problematic Vocation (Princeton: Princeton University Press, 2015).

88 questions addressed in discipline- and mathematics-centric histories. In what follows, I first describe the mathematical and computational techniques brought to bear on Lang's origami design practices, which engaged crease patterns as guides to fold new origami forms and also treated them as original creative expressions in themselves. I then examine the artistic practices of Morris, who appropriated, or borrowed, elements of Lang's crease patterns as an artistic strategy in conceptual art to repurpose source material for critical artistic commentary. In sketching out a comparative account of Morris's art and Lang's design, I then directly relate their practices to each other, contextualizing their respective uses of creative expressions or creative art. The expansion of the mathematics-as-creative-art narrative into legal domains, I argue, coincides with the shift in knowledge management beyond the university setting and, in the words of Gabriella Coleman, the "neoliberal drive to make property out of almost anything."170

In analyzing the processes by which mathematics gets articulated in an interdisciplinary and legal frame, this chapter accounts for the aesthetic, economic, and ethical reach of the modernist values ascribed to mathematics.

Origami as Mathematical Art and Functional Design

Origami enthusiasts trace the long history of the art form to several different traditions involving ceremonial wedding practices and gift exchange in China and Japan.171 In the

17" Gabriella Coleman, Coding Freedom: The Ethics andAesthetics of Hacking (Princeton: Princeton University Press, 2012), 4. 171 It is beyond the scope of this chapter to give a full overview, but origami enthusiasts have constructed a history through stitching together references from extant sources from Korea, China, and Japan. For examples, see David Lister, "A Short History of Paper Folding," in The Origami Bible, ed. Nick Robinson (Cincinnati: North Light Books, 2004): 10-21; Eric Keenway, Complete Origami:An A-Z Facts and Folds, with Step-by-Step Instructionsfor Over 100 Projects (New York: St. Martin's Griffin Books, 1987); Koya Ohashi, "The Roots of Origami and Its Cultural Background," in Origami Science and Art: Proceedingsof the Second InternationalMeeting of Origami Science and Scientific Origami, eds. Koryo Miura, Tomoke Fuse, Toshikazu Kawasaki, and Jun Maekawa (Otsu: Seian Univesrity of Art and Design, 1994): 503-510.

89 nineteenth century, however, paperfolding became more widespread in Europe and Japan for educational purposes. Following the pedagogical philosophy of Friedrich Fr6ebel, for instance, many kindergarten educators adopted paperfolding exercises to teach children to intuit mathematics and the laws of geometry through play. 172 Such uses were part of a larger trend in late-nineteenth century trend in mathematics in which, as Herbert Mehrtens notes,

''mathematicians took models as imperfect representations of geometrical entities that could be used as an aid in communication about mathematics." 7 3

In the years following World War II, origami underwent a midcentury revitalization through the development of new notational practices and the emergence of an international network comprising professional societies, publications, and conferences. Historians and origami enthusiasts often attribute the midcentury resurgence of origami to the influence of origami artist

Akira Yoshizawa, who devised a standardized approach to creating and recording origami designs. Published in 1954, his New OrigamiArt (AtarashiOrigami Geuutsu) introduced a diagramming notation that employed dotted lines and dashed lines to represent the creases left on paper after performing a mountain fold (folding paper away from oneself) or a valley fold

(folding a paper towards oneself). American origami artist Samuel Randlett and British magician

Robert Harbin expanded on this notation in 1961, producing the Yoshizawa-Randlett system. 7 4

The standardization of this notational system and the popularization of origami in the U.S. was facilitated by Lillian Oppenheimer, who in 1958 established the Origami Center and ran

172 Frebel referred to the use of geometrical paperfolding exercises to impart mathematical understanding and knowledge as "folds of knowledge." Frbebel's educational system was adopted by the Japanese government during the 1870s. In Michael Friedman, "'Falling Into Disuse': The Rise and Fall of Froebelian Mathematical Folding within British Kindergartens," PaedagogicaHistorica: International Journal of the History ofEducation 54 (2018): 464-587. For a broader historical discussion on the legacy of Fr6ebel, see Norman Brosterman, Inventing Kindergarten(New York: Harry N. Abrams, 1997). 17 Herbert Mehrtens, "Mathematical Models," in Models: The Third Dimension ofScience, eds. Nick Hopwood and Soraya de Chadarevian (Stanford: Stanford University Press), 301. 1' Samuel Randlett, The Art of Origami: Paper Folding, Traditionaland Modern (Boston: E.P. Dutton, 1961).

90 paperfolding sessions out of her apartment in New York. Oppenheimer networked with many origami enthusiasts and artists, including Yoshizawa. In May 1959, she helped organize an exhibition on paperfolding at the Cooper Union Museum of New York called "Plane Geometry and Fancy Figures."1 7 5 In the 1980s, she helped establish the Friends of The Origami Center of

America, which was renamed OrigamiUSA in 1994 and became the largest origami organization in the United States.

Mathematical origami developed out of the art form's midcentury revitalization, with the standardization of a diagramming system enabling more systematic studies of origami crease patterns. Much of the early work in the field centered around analyzing the properties of paper and crease patterns. In the 1970s, researchers and mathematically inclined origami artists aimed to establish and codify the necessary conditions for producing paper folds. They began enumerating the possible combinations of folds and studying the types of distances that were constructible by combining them in various ways.1 7 6 At the 1989 International Meeting of

Origami Science and Technology, the first conference devoted to mathematical aspects of paper folding, mathematicians and scientists gathered in Ferrara, to share their early findings.

Participants from Holland, Italy, , Japan, and the United States gathered to listen to a variety of talks ranging from "high dimensional flat origami," the problem of lying an origami modelflat without introducing additional folds, to the "algebra of paper-folding" and the elucidation of the kinds of geometric constructions possible in origami." 7 7

175 Lillian Oppenheimer, ed., Plane Geometry and Fancy Figures: The Art and Technique ofPaperFolding (New York: Cooper Union Museum for the Arts of Decoration, 1959). 176Jacques Justin, "Mathematics of Origami, Part 9" British Origami 107 (June 1986): 28-30. 177 Toshikazu Kawasaki, "On high dimensional flat origamis," Proceedings of the FirstInternational Meeting of Origami Science and Technology, ed. Humiaki Huzita (Ferrara: 1989), 131-141; Humiaki Huzita and Benedetto Scimemi, "The Algebra of Paper-Folding," in Proceedingsof the First InternationalMeeting of Origami Science and Technology, ed. Humiaki Huzita (Ferrara: 1989), 215-222; Jacques Justin, "Aspects mathematiques du pliage de papier," Proceedingsof the FirstInternational Meeting of Origami Science and Technology, ed. Humiaki Huzita (Ferrara: 1989), 263-277.

91 The organization of a research conference around mathematical origami and the subsequent publication of its proceedings were indicative of broader efforts to organize mathematical origami into a cohesive research field in its own right. Several of the mathematicians who first participated at the conference included Emma Frigerio and Benedetto

Scimemi from Italy, Jacques Justin and Michel Mendes from France, and Toshikazu Kawasaki,

Koryu Miura, and Jun Maekawa from Japan. For Italian mathematician and origami artist

Humiaki Huzita, the conference enabled a means to gather publications and research that had heretofore been "scattered" or "only found with difficulty by people who [were] interested in the

subject."178

One of the larger insights to come out of the first international meeting was that paper folding was amenable to axiomatization or description according to a set of mathematical principles. In a paper entitled "An Axiomatic Development of the Origami Geometry," Humiaki

Huzita described six possible ways in which a fold could be made on paper. He expressed the

axioms in terms of alignments that could be drawn between existing points, finite lines, and the

fold itself on a flat plane or piece of paper. 7 9 Given a set of points and lines on a sheet of paper,

Huzita's axioms offered a formulaic account of the new geometric lines that could be constructed

with new, additional points being defined from the intersections among old and new lines. In

2003, Japanese paper folder Koshiro Hatori discovered an additional fold that could be made,

thereby increasing the number of axioms to seven. While French mathematician Jacques Justin

had also expressed these same results in 1989, the resulting axioms are known as the Huzita-

Hatori Axioms. A couple of years after Hatori formally expressed the seventh axiom, Robert

1Humiaki Huzita, Proceedings of the FirstInternational Meeting of OrigamiScience and Technology, 1. 17The talk was further elaborated on at the Second International Conference on Origami in Education and Therapy in Humiaki Huzita, "Understanding Geometry through Origami Axioms," Proceedingsof the International Conference on Origami in Education and Therapy, ed. J. Smith (London: British Origami Society, 1995) 37-70.

92 Lang proved that the set was complete, meaning that no additional new folds could be defined or found. 80

Since the first International Meeting of Origami Science and Technology, researchers and paper folders have continued regularly to meet at the annual Origami in Science, Mathematics, and Education conference. While not considered mainstream or cutting-edge research in mathematics, mathematical origami has sustained an active research front and continued to produce new results. By treating a crease pattern as a mathematical graph, for instance, mathematicians were able to establish that every crease pattern in flat origami is two- colorable.' 81 Moreover, mathematical problems concerning the properties of origami and paperfolding do not necessarily need to result in practical knowledge aimed at physical folding.

180Robert Lang, "Axiomatic Origami," in Origami and Geometric Constructions(Robert J. Lang: 2010), 40-53. Lang's proof of completeness relies in part on counting degrees of freedom in a system of operations. In order to enumerate the degrees of freedom, his proof draws on an algebraic description of points, lines, and operations. "8 This is referred to as Maekawa's Theorem. In Thomas Hull, "On the Mathematics of Flat Origamis," Proceedingsof the Twenty-fifth Southeastern InternationalConference on , , and Computing (Boca Raton: Congressus Numerantium, 1994), 215-224.

93 /X

Figure 4.3a and 4.3b: On the left, crease pattern for a deer and on the right, the completed form of the deer. From Robert Lang, "The Math and Magic of Origami," filmed February 2008 in Monterey, California, TED video, 15:50, https://www.ted.com/talks/robert-lang_foldswaynew-origami?language=en.

At the TED2008 Conference held in Monterey, California, Robert Lang spoke about the affordances for design and research enabled by the convergence of origami and mathematics.

Describing the mathematical techniques undergirding his practice, Lang presented the crease pattern and folded form of a deer that resulted from his approach (Figure 4.3a and 4.3b).

Underlying his design was an understanding of paper as governed by geometric and mathematical principles, and therefore as amenable to formal description and analysis. In contrast to previous origami models that utilized a dozen steps or folds, Lang's designs required the use of more than two-hundred folds. The significance of this approach was not just that the crease pattern enabled the construction of a more complex and ornate model. Instead, it was that the range of origami structures possible broadened to the point that one could "do an elk. Or you could do a moose. Or, really, any other kind of deer." "And what this has allowed," Lang

94 continued, "is the creation of origami-on-demand."is 2 Lang's presentation gave form to the sense that mathematizing origami design and transcribing paperfolding into mathematical expressions had given rise to a creative and generative method of endless production.

Lang's crease pattern of a deer is one of over 500 original designs that he has produced by drawing on mathematical and geometrical ideas from his scientific and mathematical training.

He engaged paperfolding at an early age, expressing enthusiasm for the craft while also being

"hooked on 's recreational math column in Scientific American."I83 He received a bachelor's degree in electrical engineering from Stanford, and a doctorate in applied physics from Caltech. While completing his dissertation on "Semiconductor Lasers: New and Spectral Properties," Lang developed origami designs for a "hermit crab, a mouse in a mousetrap, an ant, a skunk, and more than fifty other pieces."'8 4 Following graduation he pursued a career as a research scientist, first as a laser physicist at the NASA Jet Propulsion

Laboratory, and then at Spectra Diode Libraries and JDS Uniphase before deciding to pursue origami design and consulting work fulltime in 2001.

In the , Lang rose to prominence within origami circles for his participation in the

"Bug Wars," a series of friendly competitions among origami artists to produce extreme designs of assigned subjects. If twentieth-century expressionist art had veered away from the real towards the abstract, then the craft of technical and mathematical origami in the 1990s signaled a return. Members of the Japanese-based Origami Tanteidan, or Origami Detectives, and other participants of the Bug Wars, recalled Lang, endeavored to imbue their designs with

182 Robert Lang, "The Math and Magic of Origami," filmed February 2008 in Monterey, California, TED video, 15:50, https://www.ted.com/talks/robert-langfolds-way-neworigami?language=en. 183 Susan Orlean, "Robert Lang and the Global Reach of Origami," The Asia-Pacific Journal5 (2007): 3. 184Robert Lang, quoted in ibid., 4.

95 "complexity, realism, and life."1 8 5 The emphasis on the liveliness and complexity of new origami forms are qualities that have characterized the field's development since the 1950s. When Akira

Yoshizawa thought back on his role in the modernization of origami in a 1997 interview, for instance, he insisted: "[W]henever I fold from nature, I think about the structural lines, how the object grows and develops...if you can't feel the liveliness in the model it means nothing." 86

Lang participated in these events, annually presenting a new model of a spider or to showcase before other artists. Although Lang's crease patterns folded into complex and individually distinct origami structures, his crease patterns always rely on a geometric as a common starting point for their design.

Subject Tree Base Model

easy Hard easy

I -r V IN.

Figure 4.4: Process of producing a new crease pattern and model in mathematical origami. From Robert Lang, "The Math and Magic of Origami," filmed February 2008 in Monterey, California, TED video, 15:50, https://www.ted.com/talks/robert-lang_foldsway-neworigami?language=en.

To produce an original crease pattern, Lang begins by converting a subject that is to be modeled in origami form into a "stick figure" called a "tree graph" (Figure 4.4). Mathematically,

85 Robert Lang, "A Different Sort of Competition," LangOrigami.com, November 3, 2007, http://1angorigami.com/art/challenge/challenge.php4. ' Akira Yoshizawa, quoted in Peter Engel, Origamifrom Angelfish to Zen (Mineola: Dover Publications, 1994), 37.

96 the tree serves as a diagrammatic shorthand to describe the topology of a desired folded model, with each line or "edge" representing a subject's appendage or body segment, each edge corresponding to a flap of paper in the final origami structure, and each edge length denoting the length of the associated flap of paper. Collectively, the flaps come together to form an origami

"base," an intermediate configuration of folds whose arrangement of flaps correspond to the orientation of edges on the tree. When flattened, the resulting creases on the pattern reflect the basic crease pattern for a desired structure. Creating the base poses the most difficult challenge for origami designers. Before the mid-twentieth century, only four origami bases were known and utilized: the fish, the bird, the kite, and the frog. But with the treatment of origami bases as tree graphs to be solved for, the number of possible bases possible appears infinite. Any added folds made to the base are subsequently considered extra details to the final origami form.

At the same time, Lang manages the constraints posed by the materiality of origami paper by drawing on the mathematical technique of circle packing to design new crease patterns. The problem of arranging flaps onto a square piece of paper can be reformulated as a problem of circle packing. To return to the stick figure, draw a circle around each node with a radius that is half the distance to another node. For Lang, the problem of mathematical origami is finding a way of positioning these nodes such that the paper can be folded in such a way that each node represents a vertex in the final shape, is then equivalent to finding an optimal way of packing the circles. In this framing, a flap requires an area of circle whose center is at the tip of it and whose radius equals to its length. The flap requires only a half or a quarter of the circle if the center is on the edge of paper. "The reason to make a distinction between the three different types of flaps-corner, edge, middle-is that for the same length flap, each of the three types of flaps

97 consumes a different amount of paper."1 87 While circle packing is a step in the creation of a

folded figure, its purpose is not for elucidating a specific 3-dimensional form. For Lang, it

produces a shape that has "enough material in the right places."1 88

These formal and analytical design techniques were implemented onto a computer

program called Treemaker, a tool intended to automate crease pattern designs. Using computer

code called CFSQP, the program converts the weighted tree into a set of algebraic equations that

are then solved. The program then color-codes the creases according to their structural role and

indicates to the designer where in the folded form a given crease lies (See Figure 4.3a for an

image of what a crease pattern produced using TreeMaker looks like). As Lang elaborates:

TreeMaker computes the 'stacking order' of all of the layers of paper, which is encoded by a graph on the facets that defines how the layers stack. Once the we know the stacking order, we can determine whether each fold is a mountain fold (solid black), valley fold (dashed colored), or is unfolded, or flat (solid grey).'89

This culminates in the fully computed crease pattern. "What TreeMaker does is come up with

something that's anywhere from 5 to 50 times more efficient in its use of paper," Lang

explains.190 There are two sides to the optimization problem that the program helps solve. On

one hand, TreeMaker helps to create a base for a given stick figure with the small possible square piece of paper. On the other hand, it also helps make the largest, scaled-up base possible from a

given square piece of paper. "It finds a local maximum," Lang Says." And with a little bit of

human intelligence, you can convince yourself that you've found the global optimum." 9 '

' Robert Lang, Origami Design Secrets: Mathematical Methodsfor an Ancient Art (Natick: A K Peters, 2003), 292. 188Robert Lang, "Mathematical Methods in Origami Design," in Bridges 2009: Mathematics, Music, Art, Architecture, Culture (Banff: Bridges, 2009), 12. 19 Ibid., 15. 9' Robert Lang, quoted in Barry A. Cipro, "In the Fold: Origami Meets Mathematics," SIAMNews vol. 34 (2001): 1-2. 191 Ibid.

98 In his various activities related to mathematical origami, then, Lang seemingly bridges the academic and industrial, as well as the scientific and artistic. He has displayed his crease patterns in major art events such as the Venice Biennale, but also displayed the folded forms and patterns in educational settings and museums. His work on mathematical origami has also brought him high-profile commissions-origami characters and landscapes for Mitsubishi and

McDonald's TV ads-and consulting work solving high-tech engineering problems using folding techniques.' 9 2 In 2004 at the invitation of mathematician , who himself is a world-renowned mathematical origami artist, Lang was an artist-in-residence at MIT. Outside the art world, Lang is known to the scientists and mathematicians not only as an artist, but also as a computational designer, consultant, and active contributor to scientific research.'9 3 Between

2007-2010, for instance, he served as Editor-in-Chief of the IEEE Journalof Quantum

Electronics. He has authored more than 80 technical publications and holds more than 50 patents on devices related to optics, lasers, and computational origami. And more recently in 2013, he became a Fellow of the American Mathematical Society.1 94 Because Lang and his crease patterns crisscrossed the art world and the public sphere in both distribution, visual, and material strategies, they can be read in many registers-mathematical, consumerist, informational, and artistic. His role as a designer folding multiple discursive practices can be distinguished from the singular discursive practice that defines a scientist.

Lang once described his practice of producing crease patterns as something akin to producing a "proof certificate," an indication that an object that had never been folded before

192Robert Lang, "Making of Mitsubishi," LangOrigami.com, September 29, 2015, https://Iangorigami.com/article/making-of-mitsubishi/. 193More recently he has worked on developing new folding telescopes for NASA as well as the problem of folding airbags in automobiles. '9 Allyn Jackson, "Fellows of the AMS: Inaugural Class," Notices ofthe American AMS 60 (May 2013): 631.

99 was mathematically possible. 195 In other words, the crease pattern was a representation of a desired object to be folded. It was also a referent, a material instantiation of geometrical principles. At the same time, crease patterns constituted a form of art:

Crease patterns as folding guides have a long history in the world of origami, but once their texture and patterns reached a certain level of richness and structure, they began to stand alone, not simply as internal tools of the creative folder, but as artworks worthy of independent contemplation, with diminishing, tenuous, or even nonexistent connections to their associated folded form.' 96

By articulating crease patterns as a , Lang also insisted that crease patterns were referents of another kind and that instantiated expressive, aesthetic ideals. For Lang, the production of crease patterns entailed aesthetic choices concerning the coloring and shading of lines that reflected the "creativity" of the origami designer. As he elaborated, "[fjor those designs based on circle packings, inclusion of the circles...in the background creates an echo of the mental processes that led to the original design...and it can be combined with the use of background colors to lend richness and texture to crease patterns."1 97 Lang's descriptions of crease patterns along these lines sought to delineate the personal, expressive components of his mathematical origami designs.

Sarah Morris, Conceptual Art, and Practices of Appropriation

In contrast to Robert Lang and other paper folders in the field of mathematical origami,

Sarah Morris is neither an origami designer nor extensively trained in mathematics. Influenced primarily by Pop, Minimalism, Conceptual Art, and architecture, her works are often characterized as featuring "hard-edge geometric abstraction" that explore the "physicality and

"9 Robert Lang, "Interview with Robert J. Lang on Origami, Sarah Morris Lawsuit," interview by Cat Weaver Hyperallergic, June 7, 2011, https://hyperallergic.com/25741/lang-art-origami-science/. 196Robert Lang, "Crease Patterns as Art," LangOrigami.com, September 28. 2015, https://Iangorigami.com/article/crease-patterns-as-art/. 197 Ibid.

100 psychology of cities."' 98 As an undergraduate student at Brown University in the 1980s, she majored in -later renamed modem culture and media-and pursued architectural studies her junior year abroad at Cambridge University. Following graduation, she entered the

Independent Study Program at the Whitney Museum of American Art in New York and worked as an assistant for American artist Jeff Koons. In 1995 she opened her own studio in Times

Square called Parallax Studios before eventually relocating to Brooklyn. ElP111111 -- m------m

--- "W. U. IUI ------U U. Ii ---- I IEEE EElI m------I --- I I IEEE. I U ------*------U

Figure 4.5: Sarah Morris, Midtown - Madison Square Garden (Stairwell), 1998. Household gloss on canvas.

'9 Ken Johnson, "Art in Review: Sarah Morris-'Crystal,"' New York Times, December 21, 2001, 44; Martin Street, "Reviews/Sarah Morris," Frieze, November 1, 2008, https://frieze.com/article/sarah-morris-1. See also Anthony Byrt, "Sarah Morris-Capitain Petzel," Artforum (September 2011): 358-9.

101 Since the inception of her career in the early 1990s, Sarah Morris has exhibited extensively throughout the world in various institutions of high art. In the late 1990s, she was widely praised within the art world for the production of stark, grid-like paintings entitled the

"Midtown" series, which were based on photographs of glass-curtain building facades around

Times Square (See Figure 4.5).199 The series highlighted the architectural qualities of skyscrapers through the use of single-point perspective and color saturated grid work. The graphical qualities and contrasting hues have since become a signature of hers. She regularly employs bright graphic works on a large scale, having created installations for public spaces such as the Palais de Tokyo in Paris or the Gloucester Road Underground Station in London. She has also received numerous awards for her work, including a Berlin Prize Fellowship from the

American Academy in Berlin in 1999-2000, and the Joan Mitchell Foundation Award in 2001.

Since the late 1990s, Morris has also engaged work as a filmmaker, producing more than ten films that each focus on a major urban center. Examples have included New York City,

Beijing, Las Vegas, and Washington, D.C. Morris typically develops these films in conjunction with a new series of geometrical and abstract paintings reflective of that work. Despite the abstract forms of the paintings, Morris's location-based titles for her art pieces have contributed to a reading of her paintings as a form of portraiture, based not on specific individuals but on the

idea of a particular city. 20 0 If her paintings on canvases favor obscurity or lack details about the

specificity of the places she paints, then her films convey information with photographic precision. Her films present the urban settings of a city at a fast-paced, disjunct sequence to

signal a city's accelerated growth. Her films also convey the urban and commercial growth of the

'99 Roberta Smith, "Art in Review: Sarah Morris-'Midtown,"' , November 26, 1999, 41. 201 Ibid.; Hal Foster, The FirstPop Age: Paintingand Subjectivity in the Art of Hamilton, Lichtenstein, Warhol, Richter, and Ruscha (Princeton: Princeton University Press, 2014), especially pp. 280-285.

102 cities she examines by "appropriating the visual vocabulary of Hollywood" as a form of commentary.201

Appropriation is an artistic strategy in contemporary art that Morris regularly employs in order to work in the mode of a conceptual artist, an approach characterized by institutional critique, dematerialization of the artistic object, and linguistic play. 20 2 In art-historical discourse, appropriation has been framed by the critical engagement of American artists with U.S. copyright law since the 1980s, in which the artist's agency is expressed by appropriating a

"readymade" or other form of property into an aesthetically meaningful act.203 The repurposing, use, or even abuse of commercial objects and imagery for artistic performances are typically read as productive and creative forms of consumption that "performs resistance."204As Winnie Wong has more recently shown in her historical and ethnographic analyses of appropriative practices of activist artists and copyist painters in China trouble the universalist assumptions inherent in aesthetic theories underlying intellectual property law. 205

21 Ania Siwanowicz, "Follow Up," in Sarah Morris: Los Angeles, ed. Cay Sophie Rabinowitz (Cologne: Galerie Aurel Schreibler, 2005), 21. 202 Conceptual art has been historicized as an artistic movement engaged with the political, economic, and social conditions of Europe and the United States from the 1960s onwards. For a historical overview and critical commentary, see Jon Bird and Michael Newman, Rewriting Conceptual Art (London: Reaktion Books, 1999) and Alexander Alberro and Blake Stimson, Conceptual Art: A CriticalAnthology (Cambridge: MIT Press, 1999). 203 Readymade was first used by French artist to describe works of art he made form manufactured objects. It has since been applied more generally to artworks by other artists made in this way. On practices of the readymade and artistic appropriation, see Douglas Crimp, On the Museum's Ruins (Cambridge: MIT Press, 1993); Craig Owens, "The Allegorical Impulse: Toward a Theory of Postmodernism," October 12 (Spring 1980): 67-86; John C. Welchman, Art After Appropriation: Essays on Art in the 1990s (Amsterdam/London: G + B Arts International/Routledge, 2001). 204 On the use and abuse of commercial objects and imagery for countercultural resistance, see Garcia Canclini, Consumers and Citizens: Globalization and Multicultural Conflicts (Minneapolis: University of Minnesota Press, 2001); Rosemary Coombe, The CulturalLife ofIntellectual Properties:Authorship, Appropriation, and Intellectual Property Law (Durham: Duke University Press, 1998), especially pp. 57-58; Dick Hebdige, Hiding in the Light (New York: Routledge, 1989); Hal Foster, Recodings: Art, Spectacle, Cultural Politics (New York: The New Press, 1985). 201 Winnie Wong, "The Panda Man and the Anti-Counterfeiting Hero: Art, Activism, and Appropriation in Contemporary China," Journal of Visual Culture I I(April 2012): 20-37; Winnie Wong, Van Gogh on Demand: China and the Readymade (Chicago: University of Chicago Press, 2013).

103 In order to produce her Origami paintings in the mid-2000s, Sarah Morris searched the internet for images of origami designs that she understood to be "objet trouv6," or "found" or

"readymade" images. 20 6 She came across Robert Lang's OrigamiDesign Secrets. Mathematical

Methodsfor an Ancient Art (2003) online, and proceeded to purchase the book in order to source the crease patterns for her respective project on Beijing. As she notes on the process:

I think that's something specific to my generation of artists, going cross-media and also cross-disciplines, being able to use everything from industrial design to architecture to politics to the entertainment system to maybe commercial strategies, and not only using those ideas as subject matter.2 07

A documentary entitled Sarah Morris in the Studio helps elucidate the practices by which she produced her paintings. She relied on a time-consuming process of masking and layering paint on stretched canvas or wall. For the design of "Black Beetle" (Figure 4.2B), for instance, masking tape was used to map out the straight lines dividing up the canvas and crease pattern.

After masking off each section of the crease pattern, she proceeded to plan which colors to apply to a specific area. Rather than mix individual colors with traditional oil paint, Morris applied household gloss paint to achieve a shiny surface on the canvas. "I'm obsessive and particular about color choice and quality," she recalled about her approach to production.20 8 In order to achieve a glossy sheen on the surface of the colors for print pieces, she applied a final wash of acrylic varnish over the printing ink. The final process of the painting involved peeling and removing the masking tape to reveal the crisp, white lines that were left behind (Figure 4.6).

206 "Sarah Morris, "Morris Deposition," Bay Oak Law, May 16, 2012, 113-125. 207Sarah Morris, "Sarah Morris: An Interview," interview by Paul Laster and Cay Sophie Rabinowitz, Art in America, February 25, 2009, https://www.artinamericamagazine.com/news-features/magazines/interview-sarah- morris/. 20 Sharon L. Butler, "Artistic Industrial Complex," Brown Alumni Magazine (July/August 2010), 58.

104 U

Figure 4.6: The unmasking of one of Sarah Morris's completed paintings. Still from Sarah Morris in the Studio: PartII (New York: White Cube, 2011).

In appropriating Lang's complex crease patterns and producing her high-gloss Origami paintings in this manner, Morris sought to preserve a rhetoric in which the skill of origami design, or craft, was separated from her conceptual work, or art. To state otherwise, if artists deploy certain materials with the intention of highlighting a specific texture, they do so in order to signal the "trace of the author" and signal a sense of movement or spatial depth on a painting's surface. Morris's Origamipaintings, however, consciously sought to eliminate all "traces" of her hand by utilizing materials such as house-gloss paint, which lent themselves to this aim. As historian of art John Roberts comments on such practices, "at no point in looking at the picture is handcraft ever identified with the maker of the picture, but neither is the hand ever absent from

105 the picture's making." 209 In appropriating Lang's complex crease patterns, then, Morris sought to preserve a rhetoric by which the skill or origami design, or craft, was separated from her conceptual work, or art. The "originality" of her work would become the source of contention in her lawsuit with Robert Lang.

Fair Use and the Politics of Creativity Wallpaper

L '0

O1amsrymns

Figure 4.7a and 4.7b: On the left, the April 2009 magazine cover of Wallpaper*, featuring Sarah Morris's Angel, 2009. On the right, crease pattern of Jason Ku's "Harpy," 2004.

209John Roberts, The IntangibilitiesofForm: Skill and Deskilling in Art After the Readymade (London: Verso Books, 2007), 76.

106 Robert Lang learned about Sarah Morris's works in April 2009, when the South African origami artist Sipho Mabona drew his attention to the cover of that month's Wallpaper* magazine, which featured a painting from Morris's Origami series entitled Angel (2009) (Figure

4.7a). Mabona and Lang noticed that the lines on the painting bore a striking similarity to a two- dimensional origami crease pattern by origami artist and computational geometry researcher

Jason Ku called "Harpy," which when folded into its origami form produced a human figure with wings (Figure 4.7b). Less than two years later, Lang, Ku, and four other mathematical origami artists filed a federal lawsuit against Morris for copyright infringement, alleging that Morris had

"created confusion as to the authorship" of the crease patterns depicted in her paintings2 10

Represented by Bay Oak Law Firm in California, Lang and the other origami artists alleged that twenty-four of Morris's paintings from her Origami series violated their copyrights.

Nine of Lang's crease patterns alone were identified to have been infringed upon. In the suit, the attorneys for the origami artists described their copyrights in the following way:

The lines of a crease pattern represent the folds needed to create a three-dimensional origami model from a sheet of paper, but the intricacy of these geometric diagrams gives crease patterns their own aesthetic appeal. Crease patterns thus lend themselves to derivative works, such as colorized versions. 211

By claiming that crease patterns were exclusively works of creative art and original designs, the origami artists effectively sought to downplay the mathematical nature of the crease patterns.

Instead, they alleged that Morris had threatened the their "professional reputation" as artists by

"by making repeated affirmative representations about the origins of the crease patterns she copied." 2 12 Morris, for her part, sought to defend herself on grounds of fair use, affirming her

21 Origami artists that were part of the initial lawsuit included: Robert J. Lang, Noboru Miyajima, Manuel Sirgo, Nicola Bandoini, Toshikazu Kawasaki, and Jason Ku. Bay Oak Law Firm, "Complaint for Damages and Injunctive Relief for Copyright Infringement: Demand for Jury Trial," December 5, 2011, 5. 211 Ibid., 3. 212 Ibid., 5.

107 claim that her paintings had been based on "'found diagrams,' 'found designs,' and 'traditional origami diagrams." 2 1 3Her use of "found" and "traditional" operate as disingenuous modes with the effect of erasing authorship and credit.

The principles of copyright law, of which fair use is a subset, rely on a modernist conceptual binary that disaggregates the expressive or ornamental features of a work from abstract ideas or a work's utilitarian aspects. In order to "promote the Progress of Science and useful Arts," copyright law enforces the principle that, while the "function" of works can and should be copied (as promoted through U.S. patent law), its "decoration" or "expressive" aspects should be protected and not copied (to prevent confusion).2 1 4 In American practice, the validity of "fair use" of a copyrighted work is delineated according to a four-fold test: (1) the transformative factor, or purpose and character of the use; (2) the nature of the copyrighted work;

(3) the amount and substantiality of the portion taken; and (4) the effect of the use upon the potential market.2"5

Under legal representation from Julia A. Ahrens, Director of Copyright and Fair Use at

Stanford Law School's Center for Internet and Society, and Don Zaretsky, Morris began to mount a case for fair use. While endeavoring to show how her practices of conceptual art and the labor behind the paintings' production revealed enough of a "transformative" factor to constitute fair use, her defense's arguments focused on the nature of the crease patterns. Notably, Lang's crease patterns were not considered art objects on insurance documents when the Museum of

Modern Art exhibited them alongside folded origami forms in its 2008 exhibition, "Design and

213 Ibid. 214 The Copyright Act states: "A 'useful article' is an article having an intrinsic utilitarian function that is not merely to portray the appearance of the article or to convey information....the design of a useful article, as defined in this section, shall be considered a pictorial, graphic, or sculptural work only if, and only to the extent that, such design incorporates pictorial, graphic, or sculptural features that can be identified separately from, and are capable of existing independent of, the utilitarian aspects of the article." In Copyright Act of 1976, 17 U.S.C. §101 (2012). 215 Ibid., §§106-107.

108 the Elastic Mind." In a "Memorandum of Law in Support of Motion for Summary Judgment" on

January 29, 2012, her lawyers argued:

Crease patterns provide a map of the folds required to transform a single square sheet of paper into a three-dimensional object. As such, they are functional and educational diagrams that reveal the structural relationship between the paper and the origami model...and as a proof regarding geometric structure, crease patterns are analogous to mathematical ideas. Many of the crease patterns at issue in this case from Lang's book, Origami Design Secrets: Mathematics Methodsfor an Ancient Art.2 6

Morris and her defense team sought to not only treat crease patterns as an intermediary, and therefore non-artistic work in the design process of new origami forms, but to constrain the intellectual domain of crease patterns to mathematical and functional ideas.

By September 2012, several origami artists voluntarily dismissed their initial lawsuit.

They included: Nicola Bandoni of Italy, Manuel Sirgo of Space, and Toshikazu Kawasaki of

Japan. They removed themselves from the lawsuit due to the "financial burden of continuing the case." 2 17Proceeding to refute Sarah Morris's fair use arguments that crease patterns were more functional and therefore subject to fair use, Lang claimed that in practice. "[p]aradoxically, the more information we put into a decorated crease pattern, the less useful it becomes as a folding guide, as visually, the various levels compete with one another in perception." 218 Notably, this runs exactly in opposition to the high modernist principle of "form follows function." That is, the more formally unrelated a work is from the work's function, the more distinctive and hence legally protectable it would be as copyrightable property.

2 1Robert W. Clarida, Julia A. Ahrens, and Donn Zaretsky, "Memorandum of Law in Support of Motion for Summary Judgment," June 29, 2012, 6. Available at: http://cyberlaw.stanford.edu/files/publication/files/2012.06.29%20Memorandum%20f/`2OLaw%20iso%20MSJ_0. pdf. 21Andrew K. Jacobsen and Caroline N. Valentino, "Reply Memorandum of Law in Support of Plaintiff's Motion for Summary ," September 24, 2012, 3. Available at: http://cyberlaw.stanford.edu/files/publication/files/2012.09.24%20Pltfs%27%2OReply%20Memorandum%20iso%2 OMSJ.pdf. 218 Ibid., 9.

109 No landmark decision regarding the lawsuit was ever reached. On February 1, 2013,

Judge Katherine B. Forrest, District Judge of the Southern District of New York, ruled that the case should move to trial-by-jury. As she reasoned when she made the decision, "[t]he question of fair use necessarily requires a fact finder to balance a number of factors." 219 In doing so, she limited the scope of the case and the ability for Morris to set a precedent for appropriative practices in conceptual art as they relate to mathematically inclined crafts. Morris settled the case with Lang, agreeing to include credits to the creators of the crease patterns on future wall labels of her Origami series.

Conclusion

This chapter explored another facet of notions of exchange and credit in science, particularly between mathematics and the arts. Sarah Morris, as conceptual artist, and her legal disputes over intellectual property with mathematical origami designers did not fall neatly along disciplinary lines. In the contemporary moment, the case indicates the contradictions of some long-standing historical dynamics regarding the values attached to mathematics and the arts. It is indicative of the present condition in US intellectual life towards the commercialization and increasing privatization of knowledge.

219Lang v. Morris, 1 Civ. 8821, (S.D. NY 2013), http://cyberlaw.stanford.edu/files/publication/files/2013.02.01%200rder%20on%20MSJ.pdf.

110 Epilogue

In spirit we mathematicians at the Institute [for Advanced Study] would cast our lot in with the humanists. Mathematicians are the freest and most fiercely individualistic of artists. - to Frank Aydelotte (1941)220

In the past decade, efforts to reaffirm the image of mathematics as a creative, abstract, and artistic endeavor have arisen anew. In 2013, Russian 6migre Edward Frenkel, a professor of mathematics at U.C. Berkeley, published a New York Times bestseller titled Love and Math, in which he sought to correct the image of mathematicians as "unworldly creatures who were so deeply involved in their work that they had no interest whatsoever in other aspects of life such as

Arts and Humanities." 22 'As part of this project, he recounted the making of his collaborative

2010 short film entitled "Rites of Love and Math," a 26-minute homage to a 1966 Japanese film

"The Rite of Love and Death" by a writer and filmmaker named Yukio Mishima. Mishima's original production is a short, black and white silent film, acted on a set that resembles a

Japanese Noh theater stage. 222The film takes place in the 1930s and features two characters, an army lieutenant torn between being loyal to the emperor or staging a coup d'etat, as well as his devoted wife. The film ends with the protagonist committing ritual suicide, seppuku, with his wife in a show of love and devotion to her.

Whereas Mishima's film is a commentary on loyalty of the subject to the Japanese

Empire, Frenkel's film seeks to embody what he perceives to be a mathematician. Frenkel not

22 Marston Morse quoted in George Dyson, Turing's Cathedral: The Origins of the Digital Universe (New York: Pantheon, 2012), 333. Many thanks to Dave Kaiser for pointing me to this quotation. 22 Edward Frenkel, Love and Math: The HeartofHidden Reality (New York: Basic Books, 2013), 10. 22See Ibid., 229-241; Edward Frenkel and Reine Grave, dir., "Rites of Love and Math" (Paris/Berkeley: Fondation Sciences Math6matiques de Paris/Mathematical Science Research Institute, 2010), DVD; Yukio Mishima and Masaki Domoto, dir., "Rite of Love and Death" (New York: The Criterion Collection, 2008[1966]), DVD.

IH1 only co-wrote, co-directed, and co-produced the film with French filmmaker Reine Graves, he

also starred as the nameless protagonist, "the mathematician," who has conceived a mathematical

formula for love. Recreating Mishima's set in color, the film centers around two characters, the mathematician and his lover, Mariko, which means "truth" in Japanese and who is intended to be the embodiment of truth and beauty. In Frenkel's film, rather than being torn between loyalty

and disloyalty to an emperor, the mathematician is tom between the purity of his mathematical formula for love and the potential harm that it could cause if used by others. The film ends with

the mathematician deciding to preserve his work by literally tattooing the formula onto Mariko's

torso, before ultimately ending his life. For Frenkel, "Mishima created the aesthetic language we

needed" to "create a different image of a mathematician as someone who is fighting for ideas"

rather than a "reclusive, social misfit." 2 23 However, what Frenkel does not address is that the

creativity that is embodied in the film and its production of a mathematician as creative and a

humanist is at the expense of reinscribing an image of the exoticized, feminine Asian who is

ultimately decorative.224

Through this crafted narrative of mathematics and the arts, Frenkel reprises two

significant framings discussed in this dissertation. First is the image of mathematical knowledge

as abstract and made that has historically been equated with values of creativity and autonomy.

Second is the social production of mathematicians and their discipline that was informed by

racialized logics. My dissertation shows how different social forms are pervasive in the

development of modern mathematics and how the overreliance on notions of autonomy

22Edward Frenkel, quoted in Carol Ness, "The true language of love? It's math, says Berkeley professor Edward Frenkel, whose steamy new film touches a nerve," Berkeley News, November 30, 2010: https://news.berkeley.edu/2010/11/30/rites/. 22 Anne Anlin Cheng labels this racialized and feminized logic as a form of "ornamentalism." See Anne Anlin Cheng, "Ornamentalism: A Feminist Theory for the Yellow Woman," Critical Inquiry 44 (Spring 2018): 415-446 and Ornamentalism (New York/London: Oxford University Press, 2019).

112 permitted people to ignore or forget that fact. In particular, my dissertation raises new questions

about what it means for mathematics to have been described as modem.

Mathematics being described as modem has always been tied to how it became

disciplined, i.e. how the field consolidated around a particular set of values. In the case of

modem mathematics, these values encompassed autonomy and creativity. It is an ongoing form

of rhetoric that mathematicians use to defend themselves against the pervasiveness of what

mathematics can do or be applied to (i.e. the more functionalist image), especially in the wake of

World War II. To some extent, my dissertation reaffirms this view by following the areas of

mathematics between 1890 to the present day in which these values were seemingly reaffirmed.

However, it also shows what the limits are of abiding by that rhetoric. Because this dissertation

recovers an alternative history that examines the inter-disciplinary exchanges between

mathematics and the arts and humanities, it gives us another lens to examine how ideas about

what a discipline is or should be is inseparable from how a discipline relates to other intellectual

domains.

Chapter one affirms how the axiomatic approach of modem mathematics was coupled to assimilation of historical documents of Japanese and in the early twentieth century, while at the same time playing down their sophistication with recourse to language.

Chapter two explores how Harvard mathematician G.D. Birkhoff's fascination with the Far East animated his mathematical theory of aesthetic measure. Birkoff's was motivated by his attempt to "quantify" the exoticism of oriental art. Chapter three discusses the mathematical pedagogy developed by mathematician Max Dehn at Black Mountain College in the 1940s and

1950s. His course, "Geometry for Artists", enrolled artists as students, including Asian-

American artist Ruth Asawa, who went on in turn to elevate craft into a form of high art. Here

113 the interplay between Max Dehn, the mathematician, and his students, the artists, seemingly complicated the hierarchy of art over craft, and by extension also the alignment of modern mathematics to fine art versus functional craft. Chapter four looks at the copyright dispute between a mathematical origami artist and a conceptual artist that further elucidates the problems that arise when the values of autonomy and self-expression are ascribed to an academic field. It outlines an attempt by the conceptual artist to deny the artistry of mathematical origamists' works, rooted in a very dated form of orientalist discourse that equates origami to a practical (i.e. non-artistic) craft. In this case study, which remained ultimately unresolved legally, again the alignment of modern mathematics to high art was blurred since the boundary between high art and what can be considered craft was called into question. And here, not only do mathematicians and artists play a role in defining what is modem mathematics, but so does the American legal system.

This work runs in parallel to more recent discussions of the "over-representation" of

Asians and Asian-Americans in the fields of science and mathematics."' But those discussions always frame Asians and Asian-Americans in the field of mathematics as being mechanical and automatons, denying their creativity. My history brings these two threads together to show how throughout the past century, Asian/Oriental contributions to mathematics have been downplayed by aligning their contributions to craft rather than to the concept of modem mathematics as a creative and artistic endeavor.

225An earlier instance of this framing is in David Brand, "Those Asian-American Whiz Kids: Why Asian Americans Are Doing So Well, and What It Costs Them," Time Magazine, August 31, 1987, 51. More recently, see Pew Research Center, Women and Men in STEM Often at Odds Over Workplace Equity: A Survey (Washington: Pew Research Center, 2018), 14-16; Joan C. Williams, Marina Multhaup, and Rachel Korn, "The Problem With 'Asians Are Good at Science,"' The Atlantic, January 31, 2018: https://www.theatlantic.com/science/archive/2018/01/asian- americans-science-math-bias/551903/; Lien, Pei-te, M. Margaret Conway, and Janelle Wong, The Politics ofAsian America: Diversity and Community (New York: Routledge, 2004).

114 The case studies in my dissertation draw attention, which needs to be paid, to craft as an analytic category (i.e. craft and how it attaches to specific social forms, specifically race and belonging), to the particular moment in time in which these cases occur, and to the notions of autonomy as occluding more than they reveal, i.e. that modem mathematics has never been an entirely autonomous discipline. By encompassing a large historical scope of the "modem" era, from 1890 to present, this dissertation shows the re-perpetuation, continuation, and reverberations of the same phenomena as reflected in the case studies. Taken together, these episodes compel us to reconsider who has had access to shaping the elite view of modem mathematics, who has been excluded, and at what cost.

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