Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World

by Alexander, Amir

  Hobbes could never accept this. Muddled foundations, he was convinced, led to muddled thinking, confused knowledge, controversy, and social strife. The only possible reason for Wallis and his fellow divines to be promoting such indefensible stances as magnitudeless points and immaterial spirits, it seemed to Hobbes, was that they were acting out of self-interest, trying to appropriate for themselves the authority that rightfully belonged to the civil sovereign. He was determined to stop them. Leviathan, he explained, was written to expose how much the English ministers, in their efforts to seize as much power as possible, had contributed to the outbreak of the civil war, and his struggle with Wallis was also part of the same struggle against the power-hungry clerics. “My business,” he explained to Sorbière in 1655, was “with all the ecclesiastics of England at once, on whose behalf Wallis wrote against me.” The way to stop the conspiracy and save the commonwealth was to expose the falsehoods of Wallis’s mathematics and drive their author in disgrace from the ranks of mathematicians.

  Hobbes struck directly at the two main pillars of Wallis’s mathematics. First came induction: “Egregious logicians and geometricians,” Hobbes exclaimed in disbelief, “that think an induction … sufficient to infer a conclusion universal and fit to be received for a geometrical demonstration!” The fact that a rule holds for a certain number of cases says nothing about whether it will hold for other cases that were not tested. Fermat had made the same point, as indeed would any classically trained mathematician.

  But it was the second pillar of Wallis’s mathematics that was the focus of Hobbes’s true scorn: the concept of the infinitely small. Wallis had calculated the areas of a triangle by dividing it into an infinite number of parallel lines, each with an infinitesimal width of , and then summing them up. Hobbes, sensing vulnerability, struck with force and precision. “In the first proposition of your Conic Sections you first have this, ‘that a parallelogram whose altitude is infinitely little, that is to say none, is scarce anything else but a line,’” Hobbes began, quoting Wallis’s own words. “Is this the language of geometry?” he thundered. “How do you determine this word scarce?” Intuitively we know quite well what Wallis means by this term, but Hobbes is perfectly correct that scarce is not a mathematical term. This was not a minor issue: The whole point of studying mathematics, for Hobbes, was its rigor, precision, and certainty. Using ambiguous terminology, as Wallis was fond of doing, undermined the entire enterprise.

  But Hobbes had more in store for Wallis. The altitude of the lines/parallelogram that comprise the triangle is either something or nothing, and Wallis’s proof founders either way. If the lines have no breadth, Hobbes charges, “the altitude of your triangle consisteth of an infinite number of no altitudes, that is of an infinite number of nothings, and consequently the area of your triangle has no quantity.” Allowing for the lines to possess a certain width and form minuscule parallelograms is just as disastrous for Wallis’s proof: “If you say that by the parallels you mean infinitely little parallelograms, you are never the better,” Hobbes argues. This is because the opposite sides of the supposed parallelograms in Wallis’s construction form the sides of the triangle. And since, as Hobbes points out, no two sides of a triangle are parallel, neither are the opposite sides of the component parallelograms. This leads inevitably to the conclusion that they are not parallelograms at all.

  On other points in Wallis’s Arithmetica infinitorum, Hobbes was even more scathing. In his proofs, Wallis calculated ratios between an increasing infinite series in the numerator, and an “equal number” of the largest term in the series. But how can an increasing infinite series have a “largest term?” And how can two infinite series, one in the numerator, the other in the denominator, have the “same number” of terms? Acting as if infinite series have a largest term or a certain number of terms, Hobbes charges, is equivalent to treating the infinite as finite, which is a contradiction in terms. “This principle is so absurd,” Hobbes rails, “that I believe it could hardly have been proposed by a sane person.” To Wallis’s casual assertion that Cavalieri showed “that any continuous quantity consists of an infinite number of indivisibles, or of infinitely small parts,” Hobbes responds that although he read Cavalieri’s book (which he suspected rightly Wallis had not), he did not recall that it contained anything of the sort. “For it is false. A continuous magnitude is by its nature always divisible into divisible parts: Nor can there be anything infinitely small.”

  THE BATTLE FOR THE FUTURE

  And that, indeed, was the crux of the matter: Hobbes rejected the concept of the infinitely small and the mathematics that went with it. Mathematics, he insisted, must begin with first principles, and proceed deductively, step by step, to ever-more-complex but equally certain truths. In the process, all geometrical objects must be constructed from simpler ones, using only the simple, self-evident definitions of point, line, surface, and so on. In this manner, Hobbes believed, an entire world could be constructed—perfectly rational, absolutely transparent, and fully known, a world that held no secrets and whose rules were as simple and absolute as the principles of geometry. It was, when all was said and done, the world of the Leviathan, the supreme sovereign whose decrees have the power of indisputable truth. Any attempt to tinker with the perfect rational reasoning of mathematics would undermine the perfect rational order of the state, and lead to discord, factionalism, and civil war.

  As Hobbes saw it, however, the infinitely small, an unwelcome intruder into mathematics, did precisely that: it destroyed the transparent rationality of mathematics, which in turn undermined the social, religious, and political order. For one thing, it did not construct its objects logically and systematically, as mathematics must if it is to be the basis of a universal rational order. Even more critically, the infinitesimal itself was notoriously paradoxical and even self-contradictory, and could as easily be used to produce obvious errors as truths. Such a nonconstructive, paradoxical approach represented to Hobbes everything that mathematics must never be. If the infinitely small were allowed into mathematics, then all order would be at risk, and society and state would descend into ruin. In the vague and ill-defined infinitely small, Hobbes perceived an echo of the unruly Diggers on St. Georges Hill.

  Wallis saw things differently. Practically all the features of the infinitely small that Hobbes considered a disaster in the making Wallis considered clear advantages. Hobbes was convinced that the very existence of dissent led inexorably to chaos and strife, and was determined to quash any hint of it. Mathematics, the only science that (he believed) had succeeded in eliminating dissent served him for this purpose. But Wallis, along with his fellows at the Royal Society, believed that it was dogmatism and intolerance that had led to the disasters of the 1640s and ’50s. Their concern was not that knowledge would be uncertain, but that it would appear to be too certain and dogmatic, excluding competing beliefs. Wallis’s mathematics offered an alternative.

  Unlike classical Euclidean geometry, Wallis’s mathematics did not attempt to construct a mathematical world, but instead to investigate the world as it was. This in itself made it far more palatable to those who feared a strict rational world order. Wallis’s world was still mysterious, unexplored, and ready for new investigations, by mathematical or other means. The ambiguity of the infinitely small, far from disqualifying it as a proper mathematical concept, was also a positive feature: opaque and even paradoxical, it left room for different interpretations and explanations of its nature and workings. Finally, Wallis’s infinitely small proved remarkably successful in revealing new mathematical truths, demonstrating the power of this incompletely understood concept. The way forward, it was clear to its advocates, was to proceed carefully, experimentally, gradually and laboriously, using whatever worked, to reveal the mysteries of the world. Any attempt to construct a perfectly known and rational mathematical world was not only politically dangerous, but also a scientific dead end.

  Epilogue: Two Modernities

  Wallis won
. By the 1660s, only a few years into a war that would last decades, Hobbes was effectively ejected from the ranks of mathematicians, while Wallis remained an honored member of the republic of mathematics. Unlike his associate and fellow Hobbes nemesis Seth Ward, he was not appointed an Anglican bishop, and it is likely that the ardent Presbyterianism of his early years prevented such an appointment. But he remained Savilian Professor and keeper of the archives at Oxford, was regularly employed by the king as a code-breaker for captured correspondence, and was a frequent visitor to court. Among his friends he counted the founders of the Royal Society, including Henry Oldenburg and Robert Boyle, and also their illustrious successors, Isaac Newton and John Locke. He kept publishing throughout his life not only on mathematics but also on mechanics, logic, and English grammar, and he considered himself an expert on teaching the deaf and mute to speak. Two collections of some of the dozens of sermons he delivered over several decades were published during his lifetime. At his death in 1703, at the age of eighty-six, he was mourned as “a man of most admirable fine parts, and great industry, whereby in some years he became so noted for his profound skill in mathematics that he was deservedly accounted the greatest person in that profession of any in his time.”

  But more important than his personal and professional success was that Wallis’s controversial mathematics had prevailed. Whereas Hobbes’s proofs were generally dismissed as the work of an amateur, Wallis’s results in the Arithmetica infinitorum and elsewhere were checked and confirmed by his colleagues. Unquestionably the most important reader of the work was Isaac Newton. When the twenty-three-year-old Newton worked out his own version of infinitesimal mathematics in 1665, the Arithmetica infinitorum, he later reported, was one of his chief sources of inspiration. In the following decades, Newton’s calculus, as well as its rival Leibnizian version, spread far and wide, transforming the practice of mathematics and all the mathematical sciences. Analysis, the new mathematical field that took the calculus as its starting point, became the dominant branch of mathematics in the eighteenth century, and remains one of the chief pillars of the discipline. It made possible the mathematical study of everything from the movement of the planets, to the vibrations of strings, to the workings of steam engines, to electrodynamics—practically every field of mathematical physics from that day to ours. Wallis lived to see only the very early stirrings of this mathematical revolution that transformed the world over the following centuries. But when he died in 1703, the verdict was already in: the infinitely small had won the day.

  From Rome and Florence to London and Oxford, the fight over the infinitely small raged across Western Europe in the middle decades of the seventeenth century. And at the opposite ends of the Continent, the struggle yielded opposite outcomes: in Italy the Jesuits prevailed over the Galileans, whereas in England Wallis prevailed over Hobbes. These outcomes were not a foregone conclusion: if an objective observer in 1630 had been asked to predict the fortunes of mathematics in the two lands, he would almost certainly have predicted the opposite. Italy was home to an illustrious mathematical tradition, whereas England had never produced any geometers of note, with the possible exception of the reclusive Thomas Harriot, who never published. If any land was likely to pioneer a challenging new mathematics, it was Italy, whose art and science had inspired Europe since the Renaissance. England, meanwhile, would likely have remained what it had always been, an intellectual backwater feeding off the scraps of its more cultured continental neighbors.

  But things turned out differently. Following the war over the infinitely small, advanced mathematics in Italy came to a standstill, whereas English mathematics quickly became one of the dominant national traditions in Europe, rivaled only by the French.

  To appreciate the magnitude of the effect that the acceptance of infinitesimals had on the modern West, consider what the world would have been like without them. If the Jesuits and their allies had had their way, there would be no calculus, no analysis, nor any of the scientific and technological innovations that flowed from these powerful mathematical techniques. As early as the seventeenth century the “method of indivisibles” was applied to problems of mechanics, and proved particularly effective in describing motion. Galileo used it to describe the motion of falling bodies, and his contemporary Johannes Kepler used it to discover and describe the movement of the planets around the sun. Isaac Newton made use of the calculus to create a new physics and mathematically describe the complete “system of the world” bound together by universal gravitation.

  Newton’s work was followed up in the eighteenth century by luminaries such as Daniel Bernoulli, Leonhard Euler, and Jean d’Alembert, who offered general mathematical descriptions of the motion of fluids, the vibrations of strings, and the currents of the air. Their successors Joseph-Louis Lagrange and Pierre-Simon Laplace were able to describe the mechanics of both heaven and earth in a set of elegant “differential equations”—that is, equations that make use of the calculus. Indeed, from their day to ours, analysis—the broader form of the calculus—has remained the fundamental tool used by physicists to explain natural phenomena. And all this had its roots in the “indivisible Line of the Mathematicians” that troubled Sorbière in 1664.

  The impact of the calculus on engineering and technology took a little longer to develop, but when it did, it was just as revolutionary. The mathematical theory of heat diffusion developed by Joseph Fourier and thermodynamics developed by William Thomson in the nineteenth century made possible the design and production of ever-more-efficient steam engines. In the 1860s, James Clerk Maxwell wrote down what became known as the Maxwell equations, a set of differential equations describing the relationship between electricity and magnetism. The subsequent development of electric motors and generators and of radio communications would never have been possible without his work. Add to that the fundamental role of the calculus in aerodynamics (making possible air travel), hydrodynamics (shipping, water collection and distribution), electronics, civil engineering, architecture, business models, and on and on, and the picture becomes clear: the modern world would have been unimaginable without the calculus and the insights it opened up into the workings of the natural world.

  But the story does not end there, for the implications of the struggle over the infinitely small ranged far beyond the mathematical world, or even that of science and technology. The fight was over the face of modernity—and indeed, in the two lands in which the struggle raged fiercest, modernity took starkly different paths. Italy became very much what the Jesuits had made it. Profoundly Catholic, it was imbued with the eternal, unchanging truths of Catholic dogma and dominated by the absolute spiritual authority of Pope and Church hierarchy. This spiritual order supported a secular order that shared many of the same characteristics. While papal supremacy did not allow for the development of a strong state, the petty princelings who ruled Italy as tyrannical autocrats owed their positions to ancient dynastic rights and claimed absolute authority over their domains. Religious dissent was unthinkable, political opposition fiercely repressed, intellectual innovation frowned upon, and social mobility nearly nonexistent. As nations to the north became hotbeds of intellectual debate, technological innovation, political experimentation, and economic progress, Italy remained frozen in time. The land that for centuries had led all Europe in the arts and sciences, whose famous city-states were at one point wealthier and more prosperous than many a great kingdom, became stagnant, backward, and poor.

  In the same years that Italy was falling behind, England was becoming the most dynamic, forward-looking, and fastest-growing land in Europe. Long considered a wild and semibarbarous country at the northern edge of European civilization, it became a leading center of European culture and science and a model for political pluralism and economic success. Here was a vision of modernity opposite in every way to the one that dominated Italy: Instead of dogmatic uniformity, England exhibited a marked and ever-increasing openness to dissent and pluralism. Politically, religiously,
and economically, England became a land of many voices, where rival views and interests competed openly, relatively free from state repression. And it was in this relative freedom that England discovered its path to wealth and power.

  The efforts of the Stuart kings to establish an absolute monarchy were defeated by fierce resistance from Parliament, first during the civil war and finally in the Glorious Revolution of 1688. When the last of the Stuarts, James II, was driven into exile and replaced by the constitutionally minded William of Orange, Parliament became the supreme governing body of the state. To be sure, the English Parliament of the seventeenth and eighteenth centuries was very far from the democratic institution we know today. It was a conservative body that represented the landed and propertied classes, and feared social unrest by the landless far more than it feared royal domination. Nevertheless, it was a deliberative body that allowed for an unprecedented degree of dissent, debate, and the free expression of ideas. Over time, the openness inherent in a parliamentary system triumphed over the class and social loyalties of its members. Throughout the eighteenth and nineteenth centuries the franchise was slowly but irreversibly expanded, and Parliament’s membership expanded to include ever broader segments of the population. The process was not completed until 1928, when all women were given the right to vote.