Below, we make available the speakers’ bios, abstracts, and, where available, the papers. We ask that any reproduction or quotation of any of these papers, is accompanied by the proper citation.
Amir Alexander | John D. Barrow | Pierre Cartier | Gregory Chaitin | Karine Chemla | David Corfield | Leo Corry | Martin Davis | Persi Diaconis | Apostolos Doxiadis | Hans Magnus Enzensberger | Rebecca Goldstein | Timothy Gowers | Dennis Guedj | Martin Krieger | Barry Mazur | Barbara Oliver | Robert Ossermen | Alecos Papadatos | Christos Papadimitriou | John Allen Paulos | Theodore M. Porter | Joan L. Richards | Brian Rotman | Marcus du Sautoy | Mary Terrall | Mark Turner | Robin J. Wilson | M. Norton Wise | Doron Zeilberger
Amir Alexander has been studying the interrelationships between narrative and mathematics for over a decade. His book GEOMETRICAL LANDSCAPES (Stanford UP, 2002) shows how a new and radical narrative equating mathematics to a geographical voyage of discovery helped shape the mathematical techniques that led up to the calculus. He is currently working on a new book (forthcoming Harvard UP, 2007) demonstrating how changing stories about mathematics through the ages went hand in hand with changing mathematical practices.
In addition to his work on mathematics and its history, Alexander also writes popular articles on recent developments in space science. He is a writer and editor for The Planetary Society, and a visiting scholar at the UCLA department of history.
From Heroes to Martyrs: Changing Stories and Changing Practices in Modern Mathematics
One way to appreciate the role of narrative in mathematical thinking is to see what happens when mathematical stories change. Do different narratives about and by mathematicians suggest different mathematical practices? In my talk I will explore a series of such stories, which I provisionally call “the mathematical explorer,” “the mathematical universe,” and “the tragic mathematician.” Each of these narratives was prevalent at a different time in the development of modern mathematics, and each was strongly linked to a specific mathematical approach. Mathematical stories and mathematical practices, in other words, go hand in hand.
John D. Barrow is the author of more than 370 scientific articles in cosmology and astrophysics and 17 books, translated into 28 languages, which explore many of the wider historical, philosophical and cultural ramifications of developments in astronomy, physics and mathematics: these include, The Left Hand of Creation (with Joseph Silk), The Anthropic Cosmological Principle (with Frank Tipler), L’Homme et le Cosmos (with Frank Tipler), The World Within the World, Theories of Everything, Pi in the Sky: counting, thinking and being, Pérche il mondo è matematico?, The Origin of the Universe, The Artful Universe, Impossibility: the limits of science and the science of limits, Between Inner Space and Outer Space, The Book of Nothing, The Constants of Nature: from alpha to omega and, recently The Infinite Book: a short guide to the boundless, timeless and endless . He has written a play, Infinities, which was performed (in Italian) at the Teatro la Scala, Milan, in the Spring of 2002 and again in 2003 under the direction of Luca Ronconi and in Spanish at the Valencia Festival. It was the winner of the Italian Premi Ubu award for best play in 2002 and the Italgas Prize for contributions to Italian scientific culture in 2003.
Infinities and Beyond
We give some preliminary discussion about the expression of science and mathematics through the arts. We will discuss the thinking behind the creation of the play ‘Infinities’ and how it was intended to differ from earlier representations of science and mathematics in the theatre. The five scenes will be described with special emphasis on the novel aspects of the original production at Teatro la Scala in Milan under the direction of Luca Ronconi. See http://www.piccoloteatro.org/infinities/intro.html for video extracts from the scenes and my book, entitled “The Infinite Book”, published by Jonathan Cape, London (2005) for a more detailed discussion of the associated ideas.
Pierre Cartier was born on June 10, 1932 , in the historical city of Sedan, famous for many lost French battles and for the peaceful coexistence of three communities (Catholics, Huguenots and Jews) already in the 16th Century . His own family partakes of all the three, and he learned to read (and sing) German at an early age. This is just to explain why he have been a scientific “Wanderer” and explored pure and applied mathematics, mathematical physics and philosophy. Cartier lives in a family of educators, and is well aware of the teacher’s paradox: to teach the important traditions yet not to respect it too much. By looking at his family, especially his three grandchildren, he is not too pessimistic about human fate. Cartier had also many mathematical siblings, including a mathematical great-granddaughter and he is still enjoying discussing transcendental numbers, or differential equations, with them. Meanwhile, he climbed the canonical ladder of an academic career, taught at Strasbourg, and in various Paris institutions (Ecole Polytechnique, Ecole Normale Superieure, IHES). Cartier is presently “Directeur de Recherches (emerite)” at CNRS, the member of no Academy, and can explain why this fact makes him happy.
Vitae Mathematicae: The Role of Autobiographies from Mathematicians
Most mathematicians are reluctant to speak about themselves and there have been unfortunately very few exceptions to this rule. We are very happy to be able to read the autobiographies of such luminaries as HARDY, KAC, ULAM, WEIL or SCHWARTZ A perusal of these “Vitae Mathematicae” disclose quite different intentions and styles Is the purpose of such autobiographies to make a narrative of some historical period of mathematics, of mathematics themselves, or just to present some outstanding personalities? Some people would like to believe that the inner feelings of mathematicians simply do not exist As in any kind of autobiography, we may also question the sincerity of the author, or the relevance of his own perception I will try to illuminate these issues by refering to two exception-all autobiographies.
In conclusion, we shall advise the prospective writer about the art of mathematical autobiography, in the manner of Rilke, and also question the present trends in the history of mathematics/sociology of science.
Gregory Chaitin is well known for his work on metamathematics and for the celebrated Ω number, which shows that God plays dice in pure mathematics. He has published many books on such topics, including Meta Math! The Quest for Omega. His latest book, Proving Darwin: Making Biology Mathematical, attempts to create a mathematical theory of evolution and biological creativity.
He is a professor at the Federal University of Rio de Janeiro and an honorary professor at the University of Buenos Aires, and has honorary doctorates from the University of Cordoba in Argentina and the University of Maine in the United States. He is also a member of the Académie Internationale de Philosophie des Sciences (Belgium).
Narratives versus formal axiomatic theories: What is mathematics?
In my forthcoming book META MATH! I give case studies showing how math is continually transformed by injecting totally unexpected new ideas. This evolutionary process is incompatible with the notion of math as a static formal axiomatic theory but it fits perfectly into a dynamic math as narrative paradigm.
Karine Chemla, Directrice de recherche (CNRS), studies the history of mathematics in ancient China in the context of a world history of mathematics. She also carries out research about mathematics in 18th and 19th century Europe. More generally, her interests focus on the relationships between mathematics and the cultures within which they are produced, a topic she tackles from the point of view of a historical anthropology. She pays special attention to the texts and inscriptions with which scientific research is conducted. In relation to this, K. Chemla edited History of science, history of text (Kluwer, 2004), a collection of papers devoted to this issue. She recently published with Guo Shuchun (Academia Sinica, Beijing) Les neuf chapitres. Le classique mathématique de la Chine ancienne et ses commentaires, Dunod, 2004. She is chief editor of the journal Extreme-Orient, Extreme Occident, devoted to studying the history scholarly practices in East Asia.
Mathematical Problems as Narratives: Perspectives from Ancient China
The oldest extant mathematical sources from ancient China present mathematical knowledge in the form of problems and algorithms solving them. The paper describes the various narrative forms of the problems and shows how problems were used to carry out mathematical proofs. It hence aims at highlighting an early mode of using narrative to explore mathematical questions.”
David Corfield is a philosopher currently working at the Max Planck Institute for Biological Cybernetics in Tubingen. His main research has been driven by the desire that philosophy engage with mathematics as a vibrant human activity. To this end he has published ‘Towards a Philosophy of Real Mathematics’ (CUP 2003), which has provoked a wide spectrum of responses. Further ideas may be found at http://www-users.york.ac.uk/~dc23/phorem.htm.
He is also the co-author (with Darian Leader) of ‘Why do people get ill?’ to appear with Penguin Books this autumn, part of whose message is to recommend that health professionals pay close attention to the narratives of their patients’ lives.
The Role of Narrative in Mathematical Inquiry
The possibility that narrative might play a crucial role in the practice of mathematics has been paid little attention by philosophers. The majority of Anglophone philosophers of mathematics have followed those working in the logical empiricist tradition of the philosophy of science by carving apart rational enquiry into a ‘context of discovery’ and a ‘context of justification’. In so doing, they have aligned the justification component with the analysis of timeless standards of logical correctness, and the discovery component with the historical study of the contingent, the psychological, and the sociological. The failings of this strategy are by now plain. In this debating arena there can be no discussion of the adequacy of current conceptions of notions such as space, dimension, quantity or symmetry. Such matters become questions purely internal to the practice of mathematics, and no interest is shown in the justificatory narratives mathematicians give for their points of view. In this talk, I would like to outline the views of the moral philosopher, Alasdair MacIntyre, whose descriptions of tradition-constituted forms of enquiry are highly pertinent to the ways in which mathematics can best be conducted, and allow us to discern the rationality of debates concerning, say, the mathematical understanding of space. An essential component of a thriving research tradition is a narrative account of its history, the internal obstacles it has overcome, and its responses to the objections of rival traditions.
Leo Corry is Director of the Cohn Institute of History and Philosophy of Science and Ideas, Tel-Aviv University, and Editor of Science in Context.
Two publications which are relevant to the topic of the meeting appeared in exotic languages:
“Some Scientific Ideas in the Work of Jorge Luis Borges and their Historical Background” Spanish: “Algunas Ideas Científicas en la Obra de Jorge Luis Borges y su Contexto Histórico”, in Borges en Jerusalén, Myrna Solotorevsky and Ruth Fine (eds.), Frankfurt am Main, Vervuert/Iberoamericana(2003), 49-74. “Fermat’s Last Theorem: How does a marginal remark becomes a “Great Mathematical Problem”? – Or does it?” (Hebrew – Galileo. Israeli Journal for Science and Ecology (2004).
Calculating the Limits of Poetic License: Fictional Narrative and the History of Mathematics
“[T]he poet’s function is to describe, not the thing that has happened, but a kind of thing that might happen, i.e. what is possible as being probable or necessary. The distinction between historian and poet is not in the one writing prose and the other verse—you might put the work of Herodotus into verse, and it would still be a species of history; it consists really in this, that the one describes the thing that has been, and the other a kind of thing that might be. Hence poetry is something more philosophic and of graver import than history, since its statements are of the nature rather of universals, whereas those of history are singulars.”
This famous, opening passage of the ninth book of Aristotle Poetics will provide the background of my talk, in which I intend to explore the interrelationship between history of mathematics and narrative about mathematics.
In my lecture, I will address questions of this kind in relation with Mathematics and Narrative. I will ask, on behalf of the public perception of mathematics, about the desired reaction of the historian, professionally committed to the singular, and sometimes boring account of what “has been”, when confronted with the poet’s universal description of the story—perhaps more philosophical and of graver import—of what “might have been”.
Professor Davis was a student of Emil L. Post at City College and his doctorate at Princeton in 1950 was supervised by Alonzo Church. Davis’s book Computability and Unsolvability (1958) has been called “one of the few real classics in computer science.” He is best known for his pioneering work in automated deduction and for his contributions to the solution of Hilbert’s tenth problem for which he was awarded the Chauvenet and Lester R. Ford Prizes by the Mathematical Association of America and the Leroy P. Steele Prize by the American Mathematical Society. His books have been translated into a number of languages including Russian and Japanese. Professor Davis has been on the faculty of the Courant Institute, New York University, since 1965, and was one of the charter members of the Computer Science Department.
Mathematics and Biography
The strategy of using biographies of mathematicians, especially lively anecdotes, to help introduce their discoveries to a general public was pioneered by E.T. Bell. I have tried to use this technique in my book on the role of logicians in the origin of all-purpose computers. It is a challenge to use the biographical material to carry the reader along while gently introducing technical matters. Although one wants to make use of the various juicy anecdotes that have become part of the culture of mathematics, careful investigation will often reveal such anecdotes to be apocryphal, and the scrupulous author is reduced to saying that the very survival of the anecdote, true or not, is itself revealing. Thus there is no evidence that Poincaré said that some day set theory will be regarded as a disease from which one has recovered although he has often been quoted to this effect. It is important not to let the heroic stature of the subjects hide their personal defects in fact some of the most interesting anecdotes arise out of them. A case in point: Gottlob Frege showed almost superhuman honesty and strength of character in his reaction to learning that his just completed two volume master work was deeply flawed. On the other hand, the aftermath of Germany’s defeat in World War I found him joining those on the anti-Semitic far right looking for a strong leader. The writer can not avoid complex controversial social and political matters that often elicit strong feelings. Indeed I have had a long correspondence with a prominent German computational logician who defended Frege from what he insisted was my unfair attack. The challenges of using biography as a narrative tool in mathematical exposition for the general public are very real, but the results can well justify the efforts.
Persi Diaconis is a mathematical statistician who works on down to earth problems such as how many times should a deck of cards be shuffled to mix it (answer seven). He is hard at work on two books for the public. The first ‘on coincidences’ tries to build a theory of surprise. The second mathematics and magic tricks shows that there are good tricks that engage real mathematics. Diaconis has alternated between statistics and mathematics departments at Stanford and Harvard. An early winner of a MacArthur Prize, he is a member of the National Academy of Science. In statistics, Diaconis specializes in Bayesian theory, graphical methods for high dimensional data and probability theory. In mathematics he specializes in combinatorics and group theory. Often, these interests are quite shuffled together.
Mathematical Stories and Stories for Mathematics
I can only work on a mathematical problem when I know it’s story — who cares about this problem? Where does it come from? What might happen if it’s understood? Right now, I’m trying to harness my stories in writing two books for the public: one on coincidences and one on mathematics and magic tricks.
I will illustrate with stories of Jung and Freud on coincidences and a wonderful card trick invented by a Petaluma chicken farmer.
Apostolos Doxiadis, the conference organizer, is a writer interested in the intersection of mathematics and narrative. He studied mathematics at undergraduate and graduate level, first at Columbia University in New York and then at the École Pratique des Hautes Études. He published four novels in Greek and, in 1999, translated into English his Uncle Petros and Goldbach’s Conjecture. Uncle Petros became a bestseller in many of the thirty-plus languages in which it has been published to date. In 2009, Logicomix, a graphic novel on Bertrand Russell’s epic quest for certainty in mathematics was released and became an instant, international bestseller.
The story of the proof is the proof: early stations in a paramathematical odyssey
Most of the talk of “mathematics and narrative” is really about the “narratives of mathematics”, whether historical, anecdotal/biographical, fictional, or combinations thereof; and there has also been interesting work in the opposite direction, of the “mathematics of narratives”, mostly by way of either formalist or AI investigations. But there is yet another way to look at the mathematics/narrative connection, i.e. by trying to locate and describe their common underlying structures. It is mostly in this direction that I shall move in my talk, trying to construct a conceptual context in which such similarities become more natural. Apart from its inherent interest, such a context facilitates journeys in both directions, “narratives of mathematics” and “mathematics of narratives”.
Hans Magnus Enzensberger is a German poet and writer and the recipient of many awards. His works include: The Number Devil, The Consciousness of Industry, and many more.
How our Schools succeed in Blotting out Children’s Natural Gift for Mathematics
Rebecca Goldstein received her doctorate in philosophy from Princeton University. Her award-winning books include the novels, The Mind-Body Problem, Properties of Light, and Mazel and non-fiction studies of Kurt Godel and Baruch Spinoza. She has received a MacArthur Foundation Fellowship and Guggenheim and Radcliffe fellowships and she was elected to the American Academy of Arts and Sciences in 2005.
Mathematicians as characters
Many of my works of fiction, starting with my first novel, The Mind-Body Problem, have featured mathematicians. I plan to discuss why I find mathematicians such compelling characters, as well as to widen my discussion to include other narrative artists who have utilized the special attributes of mathematicians in a serious way. By “a serious way” I mean, first of all, of course, the avoidance of caricatures, centering around such untrue stereotypes as the passionless pasty-faced sort; but I also mean by “a serious way” the attempt to do justice to mathematical thinking itself within fiction. What are the possibilities here? Since both narrative art and mathematics are guided by intuitions of beauty and necessity, one might even argue that there is a natural affinity between appropriating mathematical ideas, the res cogitans of mathematical characters, into narrative art.
Timothy Gowers is the Rouse Ball Professor of Mathematics at Cambridge University. He works in combinatorics, combinatorial number theory, and in the theory of Banach spaces, and has made fundamental contributions to these fields. He has solved many important problems on the structure of Banach spaces, and in combinatorics he has worked on difficult problems involving randomness and regularity in number theory. His exceptional insight and clarity have led to remarkable advances in these theories, arrived at by novel and creative combinations of analytic techniques and combinatorial ingenuity. His achievements were recognised in 1998 by the award of a Fields Medal (in mathematics, the equivalent of a Nobel Prize).
Describing mathematical proofs without losing the plot
It is often remarked that mathematicians have trouble explaining their ideas to a lay audience. What is less widely appreciated is that they have almost as much trouble explaining them to other mathematicians. Although this is partly because their explanations must be rigorous and because what they are explaining is complicated, it also results from certain unfortunate features of what has become the standard style of presentation in articles and textbooks. I shall argue for a different style, one that is much easier to read, involves no loss of rigour, and reflects more closely how we learn and discover mathematics. Moreover, it can be described sufficiently precisely that it might, just conceivably, catch on.
Romancier, mathématicien. Professeur d’Histoire et d’Épistémologie des Sciences à l’université Paris 8.
The Drama of Mathematics
There is a custom to contrast works of fiction with those of reality, the first owing to imagination, the others to rigor, deduction, etc. Emotion on one side, reason on the other. But what stories does reality recount? What dramas are natural to the mathematical field? What emotions does the revelation of a result give birth to? And what principles of internal coherence govern a piece of theatre or a novel answer? The dramas of mathematical notions and concepts can become the material of literature. To a mathematician, mathematical entities have an own existence, they habitate spaces created by their intention.They do things, things happen to them, they relate to one another. We can imagine on their behalf all sorts of stories, providing they don’t contradict what we know of them. The drama of the diagonal, of the square,É We describe certain connections between the tragic and axiomatics, rhetoric and politics.
Martin Krieger, Ph.D., does social-science informed aural and photographic documentation of Los Angeles, including storefront houses of worship and industrial Los Angeles. Professor Krieger has won three consecutive Mellon Mentoring Awards, for mentoring undergraduates, faculty, and graduate students. Professor Krieger has worked in the fields of planning and design theory, ethics and entrepreneurship, mathematical models of urban spatial processes, and has explored the role of the humanities in planning. His eight published books describe how planning, design, and science are actually done. Titles include Advice and Planning (1981); Marginalism and Discontinuity: Tools for the Crafts of Knowledge and Decision (1989); Doing Physics: How Physicists Take Hold of the World (1992); Entrepreneurial Vocations: Learning From the Callings of Augustine, Moses, Mothers, Antigone, Oedipus, and Prospero (1996); Constitutions of Matter: Modeling the Most Everyday of Physical Phenomena (1996); What’s Wrong With Plastic Trees?: Artifice and Authenticity in Design (2000); Doing Mathematics: Convention, Subject, Calculation, Analogy (2003); and, Urban Tomographies (2011). He is currently writing a book on photographing Los Angeles and Paris, and on more general issues of urban documentation. Professor Krieger has been a fellow at the Center for Advanced Study in the Behavioral Sciences and at the National Humanities Center.
The Work of Mathematical Research
What I have looked at: mathematics in mathematical physics, statistics, topology as a subject, long proofs, analogy, tours de force of mathematics or mathematical physics, I have described it in terms of models, conventions, subjects, proof, analogy.
In his recent book Imagining Numbers (particularly the square root of minus fifteen), Barry Mazur invites people who otherwise have never thought about mathematics to make a certain “leap of imagination” that allows them to become comfortable with that wonderful piece of sixteenth century mathematics: “the square root of negative numbers.” Narrative plays a key role in this exercise: it is marvelous how stories shape, and prod, our imagination as we grapple with novel concepts in math. He will tell such a story.
Eureka and Other Stories
I don’t know whether or not Archimedes discovered his hydrostatic principle, the principle of bouyancy, while taking his bath. Nor do I know whether he immediately jumped from the tub shouting “Eureka,” and ran home stark naked, dripping wet. But I do know that we all have our favorite stories that go along with accounts of mathematics; tales that help to explain, to dramatize, to teach, and even to shape in important ways, the mathematical material being recounted. How do they do this? I’ll briefly tell such a story that currently fascinates me and comment on how it does its work.
Barbara Oliver is a director and the founder of the Aurora Theater Company in downtown Berkeley, California.
Mathematics and Narrative – A Happening
I’ll talk about the way PARTITION by Ira Hauptman arrived in my office, and how surprised and pleased I was to read a script that seemed to deal (quite seriously) with important events in the history of mathematics and was, at the same time, good theatre. I was familiar with ARCADIA and COPENHAGEN, but I never expected that I (an “a-mathematical” person) might have the opportunity to produce and direct a script that attracted interest and support from a fairly impressive group of mathematicians as well as the local theatre audience. An important aspect of PARTITION is that much of the action has to do with the inherent conflict between British culture in the early 20th Century and that of South India during the same period. All of that required careful research, thought, and attention to design. Every aspect of the PARTITION project – from first reading through the final performance – was ‘Mathematics and Narrative’ in process.
Download the paper and slide show
Robert Osserman received his PhD in mathematics from Harvard University in 1955. He teaches mathematics at Stanford University.
The Right Spin: Spinning mathematics by spinning a yarn.
“The Right Spin” is the title of a new DVD – my first venture into making a film. Astronaut Michael Foale, the current U.S. record holder for time in space, recounts the harrowing moments on board the Russian space station Mir when it was struck by a cargo ship in the course of an ill-conceived and ill-prepared attempt at a manual docking procedure. The collision punctured the hull of one of the modules on the Mir, and sent the whole ship into an uncontrolled spin that soon deprived them of all power. In the absence of communication with the ground or any ideas coming from the cosmonauts who were ostensibly in charge of the mission, it was left to Michael Foale to devise a means of rescuing the Mir from what could easily have spelled its doom. Needless to say, there is a math angle to the story, and Michael Foale explains it all in this film.
Alecos Papadatos was born in 1959 in Salonica, Greece. He studied Economics at the National Aristotelian University, and received an MA in Marketing from the University of Sorbonne, Paris, all the while learning cartooning and animation. In 1986 he became professionally involved with drawing and design and he has worked ever since in animation, as animation supervisor, cartoon designer and director. In Paris, he worked as animator on animated commercials and cartoon video-clips and he directed and animated cartoon TV series, among others the TV cartoon series “Babar”, produced by Canal Plus, Paris. In 1991 he moved to Athens, where he taught animation and animated film production. In 1993 he established a studio in Athens, with animator and production coordinator Annie Di Donna. Together, they produced of layouts, animation and color backgrounds for several animated series, pilots, feature films and CD-ROMS for the Greek and European market, as well as a great number of TV cartoon commercials in 2D/3D. In addition, Alecos directed and produced cartoon character animation for full-length film credit titles, color background paintings for video projection and animation for theatre plays. In addition, he has created and designed cartoon characters and technical storyboards for TV animated series and illustrated a number of children’s books. He has done commercial work designing comic strips and comic books and was one of the principal cartoonists for the major Athens daily ‘To Vima’. For the past two years he and Annie di Donna have been working full-time on LOGICOMIX.
Logicomix: A Graphic Novel of Logic and Mathematics
I will describe work on Logicomix, a work still in progress, written by Apostolos Doxiadis and Christos Papadimitriou, designed by me and colored by Annie di Donna. Logicomix is a long work in two parts, called “Greek Exercises” and “Computers for Poets”, each about 400 pages long. It is a ‘graphic novel’, i.e. a fully illustrated, full-length comic book, partly modern autobiographical tale and partly historical novel of ideas. The autobiographical part is set in present-day Athens, and is mostly self-referential, concerning the creation of the work itself. The historical novel of ideas, which covers at least 75% of the length, is about the history of mathematical logic, the foundational crisis of mathematics, the attempt to formalize mathematics and how this influenced the creation of computers. A lot of emphasis is also given on the historical setting of the action, and the way the characters interact with it. The story is narrated by Bertrand Russell, almost in the form of a personal mathematical-logical Bildungsroman, in which, apart from Russell and his family, the main characters are Frege, Cantor, Hilbert, Whitehead, Wittgenstein, Godel, Turing — and a few more! I will briefly describe the history of the project and then expand on the work to visualize some of the concepts. Also, I shall talk about our visual approach to the period, our visit with Annie di Donna to many of the original settings of the story, our preparatory work on the characters. I will present a slide show of this preparatory work, as well as a portion of the finished work.
Christos H. Papadimitriou was born and grew up in Greece. He studied electrical engineering at the National Technical University, Athens, and then was awarded a Ph.D. in computer science, from Princeton. After teaching at Harvard, MIT and Stanford, he now holds the Lester C. Hogan Chair at the University of California at Berkeley. Christos research work is in the theory of algorithms, computational complexity and game theory, fields in which he is one of the leading international experts. He has published over three hundred original articles in leading scientific journals, which have received, to date, over twenty-five thousand citations. His books, Elements of the Theory of Computation, Computational Complexity and Combinatorial Optimization: Algorithms and Complexity, are the standard textbooks in their fields, while his first novel, Turing, was published in 2003 by MIT Press, he is also the co-author, with Apostolos Doxiadis, of Logicomix. Christos is a member of the American Academy of Arts and Sciences and the National Academy of Engineering of the USA, and has been awarded numerous honorary doctorates and other distinctions, among them the prestigious Charles Babbage Prize.
On Narrative and Computer Programming
As with the relationship between narrative and mathematics, the connections between storytelling and computer programming go both ways:
At one end we have very few recent examples of novels about coding, such as Ellen Ullman’s “The Bug,” and a small but expanding body of fascinating stories about computer programs. At the other end, some recent trends and experiments in literature have roots and parallels in algorithms: self-conscious/self-referential books and interactive novels are obvious examples. And certain kinds of programming activities, such as the authoring of computer games, aim at the creation, if not of narratives, certainly of diegeses, of storied worlds. I argue that this two-way traffic is not coincidental, but the consequence of deep structural and functional affinities and parallels between the two crafts.
John Allen Paulos is an extensively kudized author, popular public speaker, and monthly columnist for ABCNews.com (archived or current, copyright JAP) and the Guardian. Professor of mathematics at Temple University in Philadelphia, he earned his Ph.D. in the subject from the University of Wisconsin.
His writings include Innumeracy (NY Times bestseller for 18 weeks), A Mathematician Reads the Newspaper (on the readers’ list of the Random House Modern Library’s compilation of the 100 best nonfiction books of the century **), Once Upon a Number (chosen by the LA Times as one of the best books of 1998), and the just released A Mathematician Plays the Stock Market (reviews at left, a brief tenant on the BusinessWeek bestsellers list). He’s also written scholarly papers on probability, logic, and the philosophy of science as well as scores of OpEds, book reviews, and articles in publications such as the NY Times, the Wall Street Journal, Forbes, the Nation, Discover, the American Scholar, and the London Review of Books.
Stories and Statistics, Numbers and Narratives
Nearly everyone has seen those urbanocentric posters of New York or of some other city with the region’s attractions in the foreground and the rest of the world sketched in the receding background. Our psychological worlds are similarly egocentric, other people forming the background for our lives (and, most annoyingly, we for theirs). How can all these parochial posters and self-conceptions be reconciled with accurate maps, external complexities, and the disembodied view from nowhere?
Put differently: To what extent can the logical and psychological gap between stories and statistics – and the related ones between subjective viewpoint and objective probability, between informal discourse and formal logic, and between meaning and information – be closed or at least clarified?
Ted Porter teaches history of science in the Department of History at UCLA. Much of his work has concerned the history of measurement, quantification, and statistics, both as techniques of calculation and ways of thinking, in relation to the sciences and to public life. He has also worked extensively on the history of the social sciences and their reciprocal interactions with natural science. The Rise of Statistical Thinking (1986) is about the rise of statistics as a science of mass regularities in relation to social and scientific ideas and practices in the nineteenth century. Trust in Numbers (1995) is about calculation as a project to standardize reasoning, and hence about the effort to make impersonal standards of objectivity. His most recent book, Karl Pearson: The Scientific Life in a Statistical Age (2004) shows how the wide-ranging ambitions of a restless intellectual came to be focused on a new program of statistical mathematics.
Karl Pearson’s Statistical Lives
Karl Pearson’s biometric program of statistics, worked out mainly in the period from about 1890 to 1905, formed the chief basis for statistics a field of applied mathematics. In contrast to “pure” mathematics in the era of Hilbert and mathematical modernism, biometrics emphasized empirical strategies of measurement and quantification as well as analysis and deduction, and a mission to advance solutions to scientific and social problems, not just mathematical ones. Such work created different roles those of pure mathematics, and hence required a different sort of mathematician. (I would go further and say a different kind of person.) Pearson thought so too, and in narratives of his own life as well as those of other statistical heroes such as Francis Galton and W. F. R. Weldon, he told of wide-ranging interests and ambitions as well as scientific self-sacrifice for the good of society. These stories were intended also to defend an engaged statistics against the mathematicians who would perfect the formalism at the cost of its scientific soul.
Joan Richards teaches the history of science as a professor in the History Department of Brown University. Her first book — Mathematical Visions: Non-Euclidean Geometry in Victorian England — focused on the reception of a geometrical theory in the wider culture of nineteenth century England. Her second book — Angles of Reflection— was at once a memoir and an exploration of the logical work and family life of Augustus De Morgan. She is currently writing a two-generational family history focusing on changing views of rationality in the Frend/De Morgan family. All of these projects—as well as her many mathematical historical articles—are linked by an abiding interest in the ways that mathematics has served as a model of thinking that has developed in interaction with other approaches to the human mind, be they psychological, spiritual, physical, or phrenological.
Historical narrative and enlightened mathematics
One of the oft-repeated stories in the history of mathematics is of the way at the beginning of the nineteenth century Cauchy established the foundation of the calculus on a rigorous definition of the limit. Implicit and/or explicit in this story is the assumption that this breakthrough was the culmination of a long eighteenth century search for a rigorous calculus.
This paper challenges that assumption. It begins in 1749 when, in the “Preliminary Discourse” to his Histoire naturelle, Buffon characterized mathematics as rigorous, and therefore solipsistic and unenlightening. Two years later, in the “Preliminary Discourse” to the Encyclopedie D’Alembert answered Buffon’s critique by developing a “natural”, rather than rigorous, view of mathematics. One of the defining features of natural mathematics was that it was intimately entwined with the external world in the subjects of mixed mathematics, including optics and mechanics. Equally important was the way mathematical development was bound into the progressive sweep of history.
Brian Rotman’s primary interest in mathematics has been its functioning as a system of signs between embodied individuals. Besides “Mathematics as Sign: Writing, Imagining, Counting” his books include two dedicated to particular sites of mathematical meaning-making – “Signifying Nothing: the Semiotics of Zero” and “Ad Infinitum … the ghost in Turing’s Machine”. He is currently interested in ghosts – mathematical and otherwise.
Gesture in the Head
I explore the idea that gesture (‘disciplined distribution of mobility’) might be a source, essential and inexhaustible, of meaning and narrative novelty in mathematics. If so, this would run counter to any conception of mathematics as a structure of truths that transcended or pre-existed human corporeality.
Marcus du Sautoy is Professor of Mathematics at the University of Oxford and a Fellow of All Souls College. He is currently a Research Fellow at the Royal Society. His best-selling popular mathematics book “The Music of the Primes” is published by Fourth Estate and is being translated into 9 languages. He writes for the Times, Daily Telegraph and the Guardian and is frequently asked for comment on radio and TV. In September 2004 he presented his own series “5 Shapes” on Radio 4. Currently he is presenting the popular TV game show “Mind Games” for BBC4.
The Music of the Primes
Why did Beckham choose the number 23 shirt? How is 17 the key to the evolutionary survival of a strange species of cicada? Prime numbers are the atoms of arithmetic – the hydrogen and oxygen of the world of numbers. Despite their fundamental importance to mathematics, they represent one of the most tantalising enigmas in the pursuit of human knowledge. In 1859,the German mathematician Bernhard Riemann put forward an idea – a hypothesis – that seemed to reveal a magical harmony at work in the numerical landscape. A million dollars now awaits the person who can unravel the mystery of the hidden music that might explain the cacophony of the primes. Like an unsolved murder mystery, the story of the primes illustrates the strong narrative drive that underpins mathematics.
Mary Terrall has written and published extensively about the cultural history of science in the 18th century. Her book The Man Who Flattened the Earth: Maupertuis and the Sciences of the Enlightenment (University of Chicago Press, 2002) was awarded the 2003 Pfizer Prize for the best book of the year by the History of Science Society. She teaches the history of science at UCLA.
Mathematics in Narrative: 18th Century Scientific Expeditions
In 18th-century Europe, mathematicians engaged vigorously with non-specialist audiences for their work. Although mathematical work might well require a measure of temporary isolation from the exigencies of social engagements and obligations, they did not (for the most part) operate in a rarefied sphere of “pure mathematics” abstracted from the world. Indeed, there was a great deal of public interest in mathematics in the middle decades of the century, and it became a fashionable pursuit. In this context, the geodetic expeditions to Lapland and Peru appealed to the public imagination in part because of their exotic destinations, and in part because of the mathematical and technical rigors of the undertaking. This paper examines the way mathematics functioned in the narrative accounts of the expeditions, and in the disputes they engendered. How was mathematics displayed in these narrative contexts? What place did mathematics occupy in the stories of scientific heroism in distant lands? And how did this reflect on mathematics as an intellectual pursuit?
Mark Turner is Institute Professor and Professor of Cognitive Science at Case Western University. He is Founding Director of the Cognitive Science Network; Founding President of the Myrifield Institute for Cognition and the Arts; Fellow of the Institute for Advanced Study, the Center for Advanced Study in the Behavioral Sciences, the National Humanities Center, the John Simon Guggenheim Memorial Foundation, the Institute of Advanced Study at Durham University, the New England Institute for Cognitive Science and Evolutionary Psychology, the National Endowment for the Humanities, and the Institute for the Science of Origins; Extraordinary Member of the Humanwissenschaftliches Zentrum der Ludwig-Maximilians-Universität; External Research Professor of the Krasnow Institute for Advanced Study. For 2011-2012, he is a fellow of the Centre for Advanced Study at the Norwegian Academy of Science and Letters.
The Role of Narrative Imagining in Blended Mathematical Concepts
The Way We Think (Gilles Fauconnier and Mark Turner; Basic Books, 2002) presents a theory of conceptual integration, or “blending,” as a basic mental operation. See http://blending.stanford.edu. This talk will explore someways in which narrative imagining plays a role in blended mathematical concepts.
Robin J. Wilson teaches mathematics at the Open University.
Writing Popular Mathematics Books: How and Why?
There has recently been an explosion in the number of popular books on mathematics — on topics ranging from Fermat’s last theorem and the Riemann hypothesis to codes, mathematical stamps and the four-colour problem. In particular, the history of mathematics has witnessed a growth of popular interest. What makes a good popular mathematics book, and how should we go about writing one? Has this popularization led to a lowering of standards in mathematical and historical accuracy — and if so, does this matter?
Norton Wise is professor of history at UCLA. He has worked primarily in the history of nineteenth and twentieth century physics, with attention to how physicists participated in the broader culture, drawing resources from it while helping to shape it. Such cultural histories of science are most effective when concrete. Thus Wise has focused on technologies, of both material and mental kinds (e.g., engines and curves), that provided a space of mediation between the natural and the social worlds, suggesting ways to think about new kinds of problems in new ways. This approach appears in the book he is completing on Bourgeois Berlin and Laboratory Science, as well as in earlier works like Energy and Empire: A Biographical Study of Lord Kelvin (co-authored with Crosbie Smith).
How Fourier Analysis Become Rigorous, or How Functions became Curves: A Dirichlet Narrative
The method of representing functions as sums of harmonic components that Fourier presented in various forms from 1809 to 1824 ran up against the opposition of Lagrange, Laplace, and Poisson, among other mathematicians. It was not until Dirichlet produced a quite general proof of validity, first in 1828 and then more accessibly in 1837, that Fourier analysis became mathematically respectable as well as widely used in physics. This paper is concerned not so much with the details of Dirichlet’s proof as with how he was positioned in Berlin when he produced it and with what significance that had for his interest in the problem and for his approach to solving it. This is a story of curves as a new language for mathematical physics.
Doron Zeilberger is a mathematician and programmer with strong opinions who also dabbles in paramathematics.
Sipur and Mispar in Ibn Ezra’s Sefer HaMispar
Rabbi Abraham Ibn Ezra (1089-1164) was not only one of the greatests biblical commentators, grammarians, philosophers, poets, and (most importantly (to him)) astrologers, of all times, he was also a great story-teller. Hence his mathematical textbook “Sefer HaMispar” (Book of Number) is a fascinating and gripping read. But why shouldn’t it be? Indeed, the first sentence of his treatise quotes Sefer Yetsira’s dictum that the paths of knowledge consist of sfar (counting), sefer (book), and sipur (story). These three words all stem from the same Hebrew root (SFR), as does mispar (number).
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