Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World

by Alexander, Amir

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  To Jordan and Ella

  CONTENTS

  Title Page

  Copyright Notice

  Dedication

  Epigraph

  Introduction

  PART I: THE WAR AGAINST DISORDER: THE JESUITS AGAINST THE INFINITELY SMALL

  1. The Children of Ignatius

  2. Mathematical Order

  3. Mathematical Disorder

  4. “Destroy or Be Destroyed”: The War on the Infinitely Small

  5. The Battle of the Mathematicians

  PART II: LEVIATHAN AND THE INFINITESIMAL

  6. The Coming of Leviathan

  7. Thomas Hobbes, Geometer

  8. Who Was John Wallis?

  9. Mathematics for a New World

  Epilogue: Two Modernities

  Dramatis Personae

  Time Line

  Notes

  Acknowledgments

  Index

  Also by Amir Alexander

  A Note About the Author

  Copyright

  No continuous thing is divisible into things without parts.

  —ARISTOTLE

  Introduction

  A COURTIER ABROAD

  In the winter of 1663 the French courtier Samuel Sorbière was presented at a meeting of the newly founded scientific academy, the Royal Society of London. Sorbière, explained Henry Oldenburg, the Society’s distinguished secretary, was a friend from the dark days of the civil war, when the king was driven out of England and made his court in Paris. Now, three years after Charles II had been restored to his throne in London, Oldenburg was proud to host his old friend in his true home, and to share with him the exciting new investigations taking place under the Royal Society’s roof. For the next three months, Sorbière traveled the land, meeting with political leaders and leading intellectual lights, and even the king himself. Throughout this time, the gregarious Frenchman made the Royal Society his home, attending its meetings and socializing with its fellows. They, for their part, treated him with the greatest respect, and bestowed on him their highest honor: they made him a fellow of the Royal Society.

  Whether Sorbière was worthy of this honor is debatable. Although he was a noted physician in his day, and something of a man of letters, even he did not consider himself an original thinker. By his own testimony he was a “trumpeter” rather than a “soldier” in the “warfare of letters,” one who did not promote his own ideas but who advertised the ingenious inventions of other men through his far-flung web of acquaintances and correspondents. And it was, to be sure, an impressive network, including some of the greatest luminaries in France, and philosophers and scientists in Italy, the Dutch Republic, and England. Sorbière was of a type familiar in intellectual circles from that day to ours, the man whom everybody knows, though not necessarily respects all that much. Of greater concern to his hosts, however, was the fact that Sorbière was a close friend and the French translator of Thomas Hobbes, a man most members of the Society considered a dangerous subversive, and a threat to religion and the state.

  If the powers that be at the Royal Society were willing to overlook these missteps and invite him into their circle, the reason was simple: Sorbière was a man on the rise. In 1650, after years of living in exile in Holland, he’d returned to France and, four years later, abandoned his Protestant faith and converted to Catholicism. At a time when the position of Protestants in France was becoming increasingly precarious, this was a wise choice. Sorbière became a protégé of Cardinal Mazarin, Louis XIV’s chief minister, and was admitted to the king’s inner circle. He was granted a pension and the title of royal historiographer, and tried to use his influence as a high-ranking courtier to establish a scientific academy in France. His journey to England was meant, in part, as a study of the Royal Society, to determine whether it could serve as a useful model for a similar institution back home. To the grandees of the fledgling Royal Society, always on the lookout for patrons and benefactors, Sorbière was an emissary from the brilliant court of Louis XIV, and hence a man to be treated with the utmost consideration.

  If Oldenburg and his fellows hoped to be repaid in kind for the honor they had granted Sorbière, they were quickly disappointed. Mere months after returning home, Sorbière published an account of his experiences in England that showed little appreciation for the country he had so recently visited, stunning his former hosts. In Sorbière’s eyes, England suffered from an excess of religious freedoms and an excess of “Republican spirit,” both of which undermined established religion and royal authority. The official Church of England, Sorbière wrote, was likely the best of the plethora of sects, because its “Hierarchy inspires People with Respect to those who are Supream over them, and is a support to the Monarchy.” But the others—Presbyterians, Independents, Quakers, Socinians, Mennonites, etc.—are the “pernicious” fruit of excessive toleration and have no place in a peaceful realm.

  Sorbière, to be fair, did lavish praise on the Royal Society, and spoke with admiration of the experiments conducted in its halls and of the civility of the debates among its members. He even predicted that “if the advance-project of the Royal Society be not some way or other blasted,” then “we shall find a world of people fall into Admiration of so excellent a learned Body.” The details of Sorbière’s account, however, were far less flattering. He claimed that the Society was divided between the followers of two French philosophers, Descartes and Gassendi, an assertion that offended the English on both patriotic and principled grounds; the Royal Society prided itself on following nature alone, eschewing any systematic philosophy. Sorbière insulted the Society’s patron the Earl of Clarendon, Charles II’s Lord High Chancellor, by writing that he understood the formalities of the law, but little else, and had “no knowledge of literature.” Of the Oxford mathematician John Wallis (1616–1703), one of the Society’s founders and leading lights, Sorbière wrote that his appearance inclined one to laughter and that he suffered from bad breath that was “noxious in conversation.” Wallis’s only hope, according to Sorbière, was to be purified by the “Air of the court in London.”

  For the Society’s nemesis Thomas Hobbes, however, who was also Wallis’s personal enemy, Sorbière had only praise. Hobbes, he wrote, was a courtly and “gallant” man, and a friend of “crowned heads,” despite his upbringing as a Protestant. Furthermore, Sorbière claimed, Hobbes was the true heir of the illustrious Sir Francis Bacon, the late Lord Chancellor of England and prophet of the new science. This last was the most egregious of Sorbière’s offenses in the eyes of the Royal Society grandees. Bacon was venerated in the Society as its guiding spirit and, effectively, patron saint. To have his mantle bestowed on Hobbes was intolerable. As Thomas Sprat, the Society’s historian, wrote in a thorough rebuttal of Sorbière’s account, there was no more likeness between Hobbes and Bacon than there was “between St. George and the Waggoner.”

  Sorbière, as it turned out, paid dearly for his perceived ingratitude to his English hosts. He may have cared little for Sprat’s insults, hurled from faraway London, but he could not ignore the unpleasant ramifications in the royal court in Paris. France at the time sided with England in its war against the Dutch Republic, and Louis XIV was not pleased that one of his own court
iers was causing diplomatic friction with a useful ally. He promptly stripped Sorbière of his status as royal historiographer and banished him from court. Though the banishment was lifted several months later, things were never the same for Sorbière. He repeatedly tried to ingratiate himself with the king, and when that failed, he traveled to Rome to seek the Pope’s patronage instead. He died in 1670, never having regained the status and prestige he had enjoyed on the eve of his journey to England.

  Though disastrously ill-timed as far as his career was concerned, Sorbière’s A Voyage to England expresses views that in many ways were what one might expect from a man in his position. He was, after all, a courtier to Louis XIV, the king most responsible for establishing royal absolutism in France, and whose philosophy of governance was well encapsulated in his (possibly apocryphal) saying “L’état c’est moi.” In the 1660s Louis was rapidly concentrating state power in royal hands, and was well on his way to creating a single-faith state, a process completed with the expulsion of the Protestant Huguenots in 1685. If the French court’s ambition was to create a nation of “one king, one law, one faith” (“un roi, une loi, une foi”), then certainly Sorbière saw little evidence of that in England. Not only had the English effectively suppressed the true Catholic faith, but they had not even succeeded in replacing it with a single religion of their own. A plethora of sects was undermining the established state religion, and thereby the authority of the king. Personages whose actions during the civil war had suggested dangerous republican tendencies now occupied respectable positions in both church and state, whereas Hobbes, a steadfast royalist whose philosophy supported “crowned heads,” was marginalized.

  Nor were things better when it came to the personal manners of Englishmen. In France, membership in court society was the highest social as well as political aspiration of any man or woman eager to leave his or her mark. The members of this exclusive society were distinguished by their fashionable dress and refined manners, all designed to set them apart from outsiders and establish their social superiority. Sorbière’s English hosts, however, showed little inclination to follow the French example. While some of them—including Royal Society president Lord Brouncker and the noble-born Robert Boyle—were members of the high aristocracy, whose upbringing equaled that of any French courtier, others were not. And as was clear in the case of Wallis, a lack of courtly grace did not disqualify one from a place of honor in the highest intellectual circles. Hobbes, in contrast, had spent a lifetime as a member of aristocratic houses, had adopted their manners, and was therefore a man after Sorbière’s own heart. By ridiculing Wallis and praising Hobbes, Sorbière was doing more than merely expressing his personal sensibilities; he was criticizing the lack of courtly refinement in English society and lamenting the fact that, in England, the court did not set the cultural tone of the land as it did in France. When the low mixed with the high, and boors such as Wallis were allowed in high society, what hope was there for court and king to establish their authority? Such mixing would never have been allowed at the court of the Sun King, and only confirmed Sorbière’s opinion that a dangerous “Spirit of Republicanism” was lurking beneath the surface of English society.

  Hobbes, in Sorbière’s view, was everything a cultured man should be: courtly in his manners, a friend and companion to the great men of the realm, a steadfast loyal subject, and a philosopher whose teachings (in Sorbière’s opinion) supported the rule of kings. Wallis was quite the opposite: uncourtly and uncouth, and a former Parliamentarian who had made war on his king and who had undeservedly been granted a place of honor by the restored monarch. Little wonder that in the long feud between Wallis and Hobbes, the French monarchist sided with Hobbes. But in his account of their dispute, Sorbière does not dwell on the two men’s political or religious differences, but concentrates on something else entirely: “The Argument,” he explains, “was about the indivisible Line of the Mathematicians, which is a mere Chimera, of which we can have no idea.” For Sorbière, it all boiled down to this: Wallis accepted the concept of mathematical indivisibles; Hobbes (and Sorbière with him) did not. Therein lay the difference.

  The notion that a political essayist reviewing the institutions of a foreign land would focus on an obscure mathematical concept seems not only surprising to us today, but outright bizarre. The concepts of higher mathematics appear to us so abstract and universal that they cannot be relevant to cultural or political life. They are the domain of highly trained specialists, and do not even register with modern-day cultural critics, not to mention politicians. But this was not the case in the early modern world, for Sorbière was far from the only nonmathematician to be concerned about the infinitely small. In fact, in Sorbière’s day, European thinkers and intellectuals of widely divergent religious and political affiliations campaigned tirelessly to stamp out the doctrine of indivisibles and to eliminate it from philosophical and scientific consideration. In the very years that Hobbes was fighting Wallis over the indivisible line in England, the Society of Jesus was leading its own campaign against the infinitely small in Catholic lands. In France, Hobbes’s acquaintance René Descartes, who had initially shown considerable interest in infinitesimals, changed his mind and ultimately banned the concept from his all-encompassing philosophy. Even as late as the 1730s the High Church Anglican bishop George Berkeley mocked mathematicians for their use of infinitesimals, calling these mathematical objects “the ghosts of departed quantities.” Lined up against these naysayers were some of the most prominent mathematicians and philosophers of that era, who championed the use of the infinitely small. These included, in addition to Wallis: Galileo and his followers, Bernard Le Bovier de Fontenelle, and Isaac Newton.

  Why did the best minds of the early modern world fight so fiercely over the infinitely small? The reason was that much more was at stake than an obscure mathematical concept: The fight was over the face of the modern world. Two camps confronted each other over the infinitesimal. On the one side were ranged the forces of hierarchy and order—Jesuits, Hobbesians, French royal courtiers, and High Church Anglicans. They believed in a unified and fixed order in the world, both natural and human, and were fiercely opposed to infinitesimals. On the other side were comparative “liberalizers” such as Galileo, Wallis, and the Newtonians. They believed in a more pluralistic and flexible order, one that might accommodate a range of views and diverse centers of power, and championed infinitesimals and their use in mathematics. The lines were drawn, and a victory for one side or the other would leave its imprint on the world for centuries to come.

  THE TROUBLE WITH INFINITESIMALS

  To understand why the struggle over indivisibles became so critical, we need to take a close look at the concept itself, which appears deceptively simple but is in fact deeply problematic. In its simplest form the doctrine states that every line is composed of a string of points, or “indivisibles,” which are the line’s building blocks, and which cannot themselves be divided. This seems intuitively plausible, but it also leaves much unanswered. For instance, if a line is composed of indivisibles, how many and how big are they? One possibility is that there is a very large number of such points in a line, say a billion billion indivisibles. In that case, the size of each indivisible is a billion-billionth of the original line, which is indeed a very small magnitude. The problem is that any positive magnitude, even a very small one, can always be divided. We could, for instance, divide the original line into two equal parts, then divide each of them into a billion billion parts, which would result in segments that were half the size of our original “indivisibles.” That means that our supposed indivisibles are, in fact, divisible after all, and our initial supposition that they are the irreducible atoms of the continuous line is false.

  The other possibility is that there is not a “very large number” of indivisibles in a line, but actually an infinite number of them. But if each of these indivisibles has a positive magnitude, then an infinite number of them arranged side by side would be infini
te in length, which goes against our assumption that the original line is finite. So we must conclude that the indivisibles have no positive magnitude, or, in other words, that their size is zero. Unfortunately, as we know, 0 + 0 = 0, which means that no matter how many indivisibles of size zero we add up, the combined magnitude will still be zero and will never add up to the length of the original line. So, once again, our supposition that the continuous line is composed of indivisibles leads to a contradiction.

  The ancient Greeks were well aware of these problems, and the philosopher Zeno the Eleatic (fifth century BCE) codified them in a series of paradoxes with colorful names. “Achilles and the Tortoise,” for example, demonstrates that swift Achilles will never catch up with the tortoise, be the latter ever so slow, if Achilles first has to pass one-half of the distance between them, then one-quarter, one-eighth, and so on. Yet we know from experience that Achilles will catch up with his slower rival, leading to a paradox. Zeno’s “Arrow” paradox asserts that an object that fills a space equal to itself is at rest. This, however, is true of an arrow at every instant of its flight, which leads to the paradoxical conclusion that the arrow does not move. Though seemingly simple, Zeno’s mind-benders prove extremely difficult to resolve, based as they are on the inherent contradictions posed by indivisibles.

  But the problems do not end there, for the doctrine of indivisibles also runs up against the fact that some magnitudes are incommensurable with others. Consider, for example, two lines with lengths given as 3 and 5. Obviously the length 1 is included three full times in the shorter line, and five full times in the longer. Because it is included a whole number of times in each, we call the length 1 a common measure of the line with length 3 and the line with length 5. Similarly, consider lines with lengths of 3½ and 4½. Here the common measure is ½, which is included 7 times in 3½ and 9 times in 4½. But things break down if you consider the side of a square and its diagonal. In modern terms we would say that the ratio between the two lines is , which is an irrational number. The ancients put it differently, effectively proving that the two lines have no common measure, or are “incommensurable.” This means that no matter how many times you divide each of the lines, or how thinly you slice them, you will never arrive at a magnitude that is their common measure. Why are incommensurables a problem for indivisibles? Because if lines were composed of indivisibles, then the magnitude of these mathematical atoms would be a common measure for any two lines. But if two lines are incommensurable, then there is no common component that they both share, and hence there are no mathematical atoms, no indivisibles.