Über die Autorin bzw. den Autor

Apostolos Doxiadis is a writer whose books include Uncle Petros and Goldbach's Conjecture and Logicomix. Barry Mazur is the Gerhard Gade University Professor in the Department of Mathematics at Harvard University. His books include Imagining Numbers and Arithmetic Moduli of Elliptic Curves (Princeton).

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"Circles Disturbed offers a range of possibilities for how narrative can function in mathematics and how narratives themselves show signs of a mathematical structure. An intelligent, exploratory collection of writings by a distinguished group of contributors."--Theodore Porter, University of California, Los Angeles

"This collection is a pioneering effort to trace the hidden connections between mathematics and narrative. It succeeds magnificently, and represents a very significant contribution that will appeal to the professional mathematician as well as the general educated reader. The articles are written by top authorities in their fields."--Doron Zeilberger, Rutgers University

"The idea of a volume devoted to mathematics and narrative is a good one. The strength of the present volume is the breadth of its outlook, and I would imagine a fairly diverse readership from a wide variety of perspectives."--Robert Osserman, professor emeritus, Stanford University

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"Circles Disturbed offers a range of possibilities for how narrative can function in mathematics and how narratives themselves show signs of a mathematical structure. An intelligent, exploratory collection of writings by a distinguished group of contributors."--Theodore Porter, University of California, Los Angeles

"This collection is a pioneering effort to trace the hidden connections between mathematics and narrative. It succeeds magnificently, and represents a very significant contribution that will appeal to the professional mathematician as well as the general educated reader. The articles are written by top authorities in their fields."--Doron Zeilberger, Rutgers University

"The idea of a volume devoted to mathematics and narrative is a good one. The strength of the present volume is the breadth of its outlook, and I would imagine a fairly diverse readership from a wide variety of perspectives."--Robert Osserman, professor emeritus, Stanford University

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CIRCLES DISTURBED

PRINCETON UNIVERSITY PRESS

Contents

Introduction.............................................................................................................................................vii1 From Voyagers to Martyrs: Toward a Storied History of Mathematics AMIR ALEXANDER......................................................................12 Structure of Crystal, Bucket of Dust PETER GALISON....................................................................................................523 Deductive Narrative and the Epistemological Function of Belief in Mathematics: On Bombelli and Imaginary Numbers FEDERICA LA NAVE.....................794 Hilbert on Theology and Its Discontents: The Origin Myth of Modern Mathematics COLIN MCLARTY..........................................................1055 Do Androids Prove Theorems in Their Sleep? MICHAEL HARRIS.............................................................................................1306 Visions, Dreams, and Mathematics BARRY MAZUR..........................................................................................................1837 Vividness in Mathematics and Narrative TIMOTHY GOWERS.................................................................................................2118 Mathematics and Narrative: Why Are Stories and Proofs Interesting? BERNARD TEISSIER...................................................................2329 Narrative and the Rationality of Mathematical Practice DAVID CORFIELD.................................................................................24410 A Streetcar Named (among Other Things) Proof: From Storytelling to Geometry, via Poetry and Rhetoric APOSTOLOS DOXIADIS..............................28111 Mathematics and Narrative: An Aristotelian Perspective G. E. R. LLOYD................................................................................38912 Adventures of the Diagonal: Non-Euclidean Mathematics and Narrative ARKADY PLOTNITSKY................................................................40713 Formal Models in Narrative Analysis DAVID HERMAN.....................................................................................................44714 Mathematics and Narrative: A Narratological Perspective URI MARGOLIN.................................................................................48115 Tales of Contingency, Contingencies of Telling: Toward an Algorithm of Narrative Subjectivity JAN CHRISTOPH MEISTER..................................508Contributors.............................................................................................................................................541Index....................................................................................................................................................545

Chapter One

Toward a Storied History of Mathematics

AMIR ALEXANDER

1. Introduction

Sometime in the fifth century BC, the Pythagorean philosopher Hippasus of Metapontum proved that the side of a square is incommensurable with its diagonal. The discovery was quickly recognized to have far-reaching implications, for it thoroughly challenged the Pythagorean belief that everything in the world could be described by whole numbers and their ratios. Sadly for Hippasus, he did not live long enough to enjoy the fame of his mathematical breakthrough. Shortly after making his discovery, he traveled aboard ship and was lost at sea.

Since that time, different versions of the story have come down to us. In some, Hippasus's "shipwreck" was contrived by his own Pythagorean brothers; in others he was not killed but only expelled from the brotherhood for his indiscretion in revealing its most profound secrets. But whatever version one adopts, it is clear that the story of Hippasus is not meant to be an accurate chronicle of a tragic event that took place 2,500 years ago. Rather, it is a morality tale, intended to convey deeply held truths about the meaning of mathematics and its potential dangers. In this, it was an early example of a "mathematical story," a narrative type that has accompanied the study of mathematics from the very beginning.

Other stories soon followed. Pythagoras himself reportedly sacrificed an ox upon his discovery of what became known as the Pythagorean theorem, and Euclid, according to another popular tale, admonished King Ptolemy that "there is no royal road to mathematics." Archimedes ran naked through the streets of Syracuse shouting "Eureka!" and was killed years later when, oblivious to the sack of the city going on all around him, he asked a Roman soldier to stand aside while he worked out a problem in geometry. In later times stories abounded about mathematicians as heroic explorers or tragic young geniuses. The recent popularity of movies such as A Beautiful Mind and Good Will Hunting strongly suggests that stories remain the constant companions of mathematical studies to this day.

Like all stories, mathematical tales are meant to amuse and entertain. They draw on an existing world that is familiar to their audience, reflecting its historical and cultural realities. Ancient Greek philosophers, for example, were indeed frequent maritime voyagers, and Hippasus's tragic end is far from implausible. Archimedes did die during the fall of Syracuse in 212 BC, many early modern mathematicians were deeply involved in voyages of geographic exploration, and so on. But in addition to mirroring the actual conditions in which mathematical work was carried out, the stories also convey important lessons about what mathematics is and how it should be practiced. Hippasus's tale suggests the dangers of pursuing mathematics to its ultimate conclusions, and Archimedes' death is emblematic of the clash between the pure realm of mathematics and the barbarism of war. Centuries later, the description of mathematicians as enterprising explorers is not only a reflection of their professional affiliations but also a prescription for how mathematics itself should be practiced.

Mathematical tales, in other words, are both descriptive and prescriptive, drawing on the historical conditions of their times while seeking to define the meaning and practice of mathematics itself. On the one hand, like all popular stories, they are firmly anchored in a particular time and place; on the other, they reach out toward the seemingly insular practice of mathematics itself, defining its meaning, its purpose, and how it should be practiced. Anchored firmly in human culture and history while straining toward the ethereal realms of high mathematics, such stories are uniquely positioned to span the great divide that looms between mathematical practices and the cultural realities in which they arose.

This is no small feat. All too often in writing the history of mathematics, mathematical developments are treated as fundamentally separate from their historical culture. Mathematics, in these accounts, has a history only in the sense that different mathematicians in different eras glimpsed different parts of the eternal and unchanging truth that is mathematics. The precise historical circumstances in which these discoveries took place are completely irrelevant to the actual substance of the mathematics, which would be exactly the same no matter when or where it was discovered. When historical circumstances do make their appearance in these accounts, they are not meant to draw connections between earthly history and transcendent mathematics but rather to contrast the senseless contingencies of earthly life with the transcendent perfection of the mathematical world.

Poised as they are between mainstream human culture and high mathematics, mathematical stories open up the possibility of writing a different kind of history. In place of the traditional separation between the mathematical and the historical worlds, mathematical stories make it possible to connect the two in interesting and even surprising ways. History, traditionally treated as background noise to the forward march of mathematical knowledge, can now move to the center of the account, both shaping and in turn being shaped by mathematical developments. The practices of high mathematics can thus be brought into contact with the cultural circumstances that gave birth to them. Mathematics, far from residing on its insular Platonic plane, is shown to be an integral part of the human enterprise.

In writing such a history, I propose following the trail of mathematical stories through different historical epochs. In each period I identify one or several dominant narratives that enjoyed wide currency among the broader population as well as among practicing mathematicians. Each of these stories has its roots in a particular cultural context, but each also attempts to define what mathematics is and the place and role of its practitioners within the community. In doing so, the stories might suggest what mathematical questions are relevant or interesting, which methods and approaches are to be considered legitimate, what standards of logical rigor the procedures must adhere to, and what types of solutions would be considered acceptable. Mathematical stories do not determine the contents and details of mathematical proofs, but they did profoundly shape the outlines of mathematical practice in their time.

What follows is a rough outline of just such a story-driven history of mathematics, from the late sixteenth century to the present. The period is divided into three main epochs and a possible fourth, each characterized by a different dominant mathematical story, which in turn is related to a different dominant mathematical style. Unquestionably, this very brief history is far from complete. The dominant tales it identifies by no means exhaust the store of mathematical narratives that existed in each of these periods, as competing stories were always present, sometimes with a message that ran counter to the dominant story's. A full history would necessarily include accounts of such alternative narratives and the mathematical practices with which they were associated.

Nevertheless, I believe the scheme presented here is a significant step toward a storied history of mathematics. It provides the basic outlines of a historical periodization, as well as examples of how narrative can be used to bridge the gap between mathematics and the broader culture. By reading the development of mathematics over time through the lens of mathematical stories this approach provides a unified perspective that combines broad cultural trends and technical mathematical practices. In doing so it reintegrates mathematical practices with their cultural context and makes mathematics once again an inseparable part of human history.

2. Exploration Mathematics

In 1583 the Dutch mathematician and engineer Simon Stevin introduced his Problematim geometricarum with a poem by Luca Belleri extolling the virtues of mathematics:

Truly, then, the ancients called Divine mathesis that which by Its craft enabled to recognize The supreme seat, the ways of the earth and sea And to see in person the hidden places in the dark The secrets of nature. The view of mathematics presented here appears at first sight to be quite unremarkable, and very much in line with the ideas of the great reformers of knowledge of the time. The mathematician is portrayed as an explorer, navigating "the ways of the earth and sea" and viewing "in person" the hidden secrets of nature. Similarly, Francis Bacon in a famous passage in the New Organon challenged the natural philosophers of his time to live up to the example of geographic explorers. "It would be disgraceful" he wrote, "if, while the regions of the material globe—that is, of the earth, of the sea, of the stars—have been in our times laid widely open and revealed, the intellectual globe should remain shut out within the narrow limits of old discoveries." In the years that followed, the great voyages of exploration were repeatedly cited as a model and an inspiration by early modern promoters of the new sciences. The image of the natural philosopher as a Columbus or Magellan, pushing forward the frontiers of knowledge, became a commonplace of scientific treatises and pamphlets of the period. The newly discovered lands and continents seemed both proof of the inadequacy of the traditional canon and a promise of great troves of knowledge waiting to be unveiled.

But while the voyages of exploration served as a powerful trope for promoting the new experimental sciences, the case was very different for mathematics. With its rigorous, formal, deductive structure, mathematics appeared to be a terrain ill-suited for intellectual exploration. No mathematical object, after all, could ever be observed, experienced, or experimented upon. Mathematicians, it seemed, did not seek out new knowledge or uncover hidden truths in the manner of the geographic explorers. Instead, taking Euclidean geometry as their model, they sought to draw true and necessary conclusions from a set of simple assumptions. The strength of mathematics lay in the certainty of its demonstrations and the incontrovertible truth of its claims, not in uncovering new and veiled secrets. Indeed, what could possibly be left hidden and undiscovered in a system where all truths were, in principle, implicit in the initial assumptions?

This view of mathematics was expressed most clearly by Christopher Clavius, the founder of the Jesuit mathematical tradition, in his tract "In Disciplinas Mathematicas Prolegomena," dating from the 1570s. The mathematical sciences, Clavius insisted, "proceed from particular foreknown principles to the conclusions to be demonstrated." He then continues:

The theorems of Euclid and the rest of the mathematicians, still today as for many years past, retain in the schools their true purity, their real certitude, and their strong and firm demonstrations ... and thus so much do the mathematical disciplines desire, esteem, and foster truth, that they reject not only whatever is false, but even anything mere probable, and they admit nothing that does not lend support and corroboration to the most certain demonstrations.

For Clavius, as for many of his contemporaries, all forms of mathematics proceeded by deducing undisputed truths from generally known and accepted first principles. They had little to say of the exploration and discovery of hidden and unknown realms of knowledge.

Belleri's account of Stevin's work as a voyage of exploration was a sharp departure from this long-standing tradition. The mathematician is seen here as an explorer, navigating "the ways of the earth and sea" and viewing "in person" the hidden secrets of nature. This, of course, is precisely what reformers such Bacon and Giordano Bruno were trying to achieve in their new models of knowledge. Dismissing the authority of the older canon, they sought to gain knowledge of the world as explorers do, through direct personal experience. It was clearly not what mathematicians themselves had sought to achieve over the centuries. Indeed, as Clavius had argued, the strength of mathematics lay precisely in the fact that it was not dependent on personal experience or sense perception but was based strictly on pure and rigorous reasoning from first principles. In speaking of mathematics as a voyage of exploration, Belleri's poem is proposing a shift in the understanding of the very nature of the field.

Stevin returned to this metaphor in his dedication to Dime, his best-selling treatise on decimal notation, where he compared himself to a "mariner, having by hap found a certain unknown island," who reports his rich discovery to his prince. "Even so we may speak freely of the value of this invention," he concludes. Here Stevin is again an explorer in the uncharted lands of mathematics. Rather than promote his treatise as resulting from rigorous mathematical deduction, he chooses to describe it as an "invention" (equivalent to our modern "discovery"), the happy result of his mathematical travels. Like an unknown land, Stevin's "invention" is discovered through chance wanderings, and like it, it holds the promise of great riches. Stevin, it should be noted, was not an academic mathematician: he was a practicing engineer and high-level official in the court of Prince Maurice of Nassau, responsible for digging canals, building dams, and constructing fortifications. We should not, perhaps, be surprised to find that he did not share Clavius's lofty insistence on the strictly deductive nature of mathematics. But the imagery of mathematical exploration did not long remain the exclusive domain of practical men like Stevin. It soon found its way into more academic settings.

In the 1630s and 1640s, leading members of Galileo's circle in Italy began referring to their mathematical studies in terms of travel and exploration. Unlike their Dutch and English counterparts, Italian mathematicians and natural philosophers were usually far removed from actual maritime ventures, but the rhetoric and imagery of the voyages nevertheless flourished among them. Maritime voyages and physical experiments on board ship figured prominently in Galileo's Dialogue Concerning the Two Chief World Systems. He often referred to scientific work as unveiling the hidden secrets of nature, and applied this vision to the study of mathematics as well.

At the end of the first day of the Dialogue, for instance, Galileo explained that mathematical truths are "clouded with deep and thick mists, which become partly dispersed and clarified when we master some conclusions." Similarly, in congratulating his disciple, the mathematician Evangelista Torricelli, for his achievements, he wrote that by using his "marvellous concept," he "demonstrates with such easiness and grace what Archimedes showed through inhospitable and tormented roads ... a road which always seemed to me obstruse and hidden." The language is indeed suggestive. As before, we are in a land of marvels and secrets, clouded by thick mists, with only difficult and convoluted passages leading through to them. Torricelli is praised for breaking through to his "marvellous concept" and blazing a trail for others to follow. He is indeed a mathematical explorer.

Torricelli himself uses the travel imagery more explicitly in a lecture on the nature of geometric reasoning given in the 1640s. In "the books of human knowledge," he writes, the truth is "so much entangled in the mist of falsities" that it is impossible to separate "the shadows of fog from the images of truth." But "in geometry books you will see in every page, nay, in every line, the truth is laid bare, there to discover among geometrical figures the richness of nature and the theatres of marvels." Much like Clavius, Torricelli is here intent on preserving mathematics' traditional claim to clarity and certainty as against the confused and contested nature of other fields of knowledge. But the source of these unique features of mathematics is radically different for Torricelli than it was for his predecessor. Geometry's superiority is not derived from its rigorous logical structure but resides instead in its ability to reveal the riches and marvels that are hidden among geometric figures. For Torricelli, the geometer is one who explores and seeks out those hidden secrets and brings them to light—a very different image of mathematics indeed than Clavius's systematic elaboration of deductive truths.

(Continues...)

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