Invisible Mathematics in Italo Calvino’s Le città invisibili Ileana Moreno-Viqueira Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Graduate School of Arts and Sciences COLUMBIA UNIVERSITY 2013 © 2013 Ileana Moreno-Viqueira All rights reserved ABSTRACT Invisible Mathematics in Italo Calvino’s Le città invisibili Ileana Moreno-Viqueira This dissertation examines the use of mathematical concepts as an essential structural and thematic element in Italo Calvino’s Le città invisibili. The author‘s conception of literature as a combinatorial art, intrinsically mathematical itself, is the point of departure. Focal to the study is Calvino’s interest in that which is an essential part of the combinatorial game and the key to Gödel Incompleteness Theory, namely, the elements of surprise and the unexpected - the exceptions to the rule. Other critical approaches to Calvino’s work, like semiotic, structuralism and scientific are interrelated to Mathematics, but what this study proposes is a strictly mathematical approach to complement that which has already been pointed out. A mathematical perspective based on an understanding of Mathematics as more than just numbers encompasses the whole analysis. Mathematics is given its proper place as a humanistic discipline. It is an interdisciplinary proposal of literature and science, pertinent to Calvino’s writing. The purpose is to unveil a “hidden math” which from the perspective of this study is an intrinsic tool in Calvino’s writing process of Città. As a versatile writer, Calvino manages to use mathematics in such subtle ways that it may not be perceptible at first sight. Most importantly, within these mathematical concepts and images lies, in part, the potential character of literature for which the author aims: that latent yet invisible possibility, that search for new forms (like the cities). These ideas, particularly related to potential literature, are also analyzed from his interest and involvement in Oulipo (Ouvroir de Littérature Potentielle). The study begins by unfolding what aspects of combinatorial mathematics are present in Le città invisibili; how these concepts as well as other images are used in the construction and design of the cities and the book; and to find out why Calvino finds recourse to mathematics as a narrative and creative strategy. Calvino’s use of mathematical concepts are studied as a “visual instrument” in the organization and construction of his imaginative writing and, furthermore, as a means to achieve “lightness” structurally and thematically through the abstract, aesthetic and, at times, even humorous nature of mathematics. In their own way mathematics and literature attempt to make visible what is invisible, and they both struggle to remove weight from their own “systems” of expression. In conclusion, the investigation intends to demonstrate through Calvino’s Le città invisibili, how mathematics and literature complement each other in the search for new forms, new ideas, new stories. Table of Contents Introduction 1 Chapter One: Genesis of the Book 1.1 Story of a design 14 1.2 From Description to Structure 21 1.3 Design of a Story: The Index 26 1.4 Design of a Story: the Frame 33 Chapter Two: The Book as Space 2.1 Invisible spaces 50 2.1.1 Space as a Combinatorial Construction 54 2.1.2 Combinatorial Game and the Unexpected 62 2.1.3 The Book and the City 71 2. 2 Traveling Through the Cities 80 2.2.1 Mathematical Constructions 82 2.2.2 Doubling doubles 109 Chapter Three: Space in Search of a Form 121 3.1 Motion in Space: The Game of Chess 124 3.2 Spaces in Motion: Complex Models 136 3.3 The Center: Lightness and the Spiral 146 Conclusion 184 Bibliography 190 1 Introduction Se puede comprobar fácilmente que Newton y Kepler modelaron sus órbitas planetarias de una manera esencialmente geométrica. Sin embargo, las elipsis en sí no tienen existencia física en el espacio, son tan sólo senderos invisibles trazados por planetas en órbita. (Richard Mankiewickz. Historia de las matemáticas. Del cálculo al caos.100)1 Not often are literature and mathematics perceived as being intricately linked together, however, through the ages this link between mathematics, poetry and literature is evident. This is precisely the case in Italo Calvino’s Le città invisibili.2 Throughout his writing career, Italo Calvino left many important views regarding the essence of literature. Among these, the definition of “literature as a combinatorial game” stands out. From this perspective of the concept of literature as a “combinatorial game” I intend to focus my investigation of Calvino’s use of mathematical concepts in Le città invisibili. 3 The present study will examine how in this particular book mathematical concepts constitute an essential element both structurally as well as thematically.4 Although some attention has been paid to the more explicit aspects of combinatorial structures of his texts (particularly, in Il castello dei destini incrociati) further complexities and subtleties of his combinatorial mathematics have been, if not completely neglected, at least not fully perceived. Among his many writings, Le città invisibili, the book which the author claims that he “managed to say the most”, presents itself as ideal to pursue the 1“One could easily prove that Newton and Kepler modeled their planetary orbits in an 2“Symmetries and arithmetic have always tempted Calvino’s imagination to grow flirtatious and to begin its fantastic displays” (Seamus Heaney 77). 3Another contemporary critical study regarding mathematics and literature is Guillermo Martínez “Borges y la matemática” (2012). 2 hypothesis that his use of mathematical concepts, such as combinatorial art, goes beyond what has been analyzed so far. Calvino’s work has been the object of a vast amount of criticism (McLaughlin 1998) from a wide range of approaches. The main focus has been in aspects other than mathematical, which is fundamental to fields of study such as science; also linguistics, structuralism, semiotics, post-modernism. The critical research from these perspectives shows links to their mathematical basis. Studies referring to the scientific interests of the author have placed much emphasis, for instance, in Cosmicomics, but also in his essays and their repercussions in other writings. Massimo Bucciantini in Italo Calvino e la scienza (2007) analyzes Calvino’s relationship with science as well as his scientific images and language. Concerning mathematics, Gabrielle Lolli’s Discorso sulla matematica (2011) proposes reading Italo Calvino’s Six Memos for the Next Millennium “come una parabola sulla matematica”. Furthermore, Calvino’s books have been used with a didactic purpose to illustrate mathematics, particularly, Città. Recently, the visual feature of his work has had a significant impact on critical studies (Ricci 2001, Belpoliti 2006, Ragusa 1983). The theme of the city –a constant topic throughout most of Calvino’s literary career – has been the object of critical works with an architectural approach, for example, Modena (2011). All in all, recent criticism keeps pointing to Calvino’s interdisciplinary view on literature, while the overall structure of Città, as designed by Calvino has been commented by the author and the critic. This study intends to complement the research accomplished at present, particularly on Calvino’s visual, scientific, architectural and other interdisciplinary interests. Mathematical concepts are at the basis of these approaches, yet they are merely 3 mentioned and, generally, in other contexts. The purpose of this dissertation is to demonstrate a mathematical perspective, which has been pointed out, but not particularly studied in extension or depth. From my perspective, this “hidden math” is an intrinsic tool in his creative process of writing. As a versatile writer, he managed to use mathematics in such subtle ways that may not be perceptible at first sight. Most importantly, within these concepts and images lies, in part, the “potential” character of literature that the author was aiming for. The term ars combinatoria was inspired by Leibniz and his famous Dissertatio de Arte Combinatoria (1666), following Giordano Bruno’s admiration of the Medieval Catalan monk Ramón Lull ‘Ars Magna (1305). In essence, this ‘art’ consists in nothing but arranging a limited number of elements subject to certain rules. By altering the sequences, different arrangements are formed. The goal becomes to create as many combinations as possible. For Calvino, narrative is a combinatorial game “[which] plays with possibilities intrinsic to its own material”. This ars combinatoria, he adds, is “a game that at a certain point is invested with an unexpected meaning.” (“Cybernetics and Ghosts” 22; “Cibernetica e fantasmi” 210) Combinatorial systems are by definition formal systems. Thus, according to Gödel’s Incompleteness Theorem, they cannot escape a fundamental paradox. Unavoidably, within any system, something must escape, something that bypassing its own rules becomes, in every sense, surprisingly exceptional: “In every system, something emerges which is not truly part of the system” (D. Hofstadter 3). 4 Even if these statements may seem, at first, potentially contradictory, they are, as will be here examined, complementary. In fact, this paradoxical character of the author’s writing will be essential in the present discussion. Precisely this controversial element of surprise, the exceptions to the rule, will be the argumentative focus of