The discovery of these ancient conundrums by Zeno the Eleatic and the followers of Pythagoras in the sixth and fifth centuries BCE changed the course of ancient mathematics. From then on, classical mathematicians turned away from the unsettling considerations of the infinitely small and focused instead on the clear, systematic deductions of geometry. Plato (ca. 428–348 BCE) led the way, making geometry the model for correct rational reasoning in his system, and (according to tradition) carving the words “let no one ignorant of geometry enter here” above the entrance to his Academy. His student Aristotle (ca. 384–322 BCE) differed from his master on many issues, but he too agreed that infinitesimals must be avoided. In a detailed and authoritative discussion of the paradoxes of the continuum in book 6 of his Physics, he concluded that the concept of infinitesimals was erroneous, and that continuous magnitudes can be divided ad infinitum.
The turn away from infinitesimals would likely have been final had it not been for the remarkable work of the greatest of all ancient mathematicians, Archimedes of Syracuse (ca. 287–212 BCE). Fully aware of the mathematical risks he was taking, Archimedes nevertheless chose to ignore, at least provisionally, the paradoxes of the infinitely small, thereby showing just how powerful a mathematical tool the concept could be. To calculate the volumes enclosed in circles, cylinders, or spheres, he sliced them up into an infinite number of parallel surfaces and then added up their surface areas to arrive at the correct result. By assuming, for the sake of argument, that continuous magnitudes are, in fact, composed of indivisibles, Archimedes was able to reach results that were well nigh impossible in any other way.
Archimedes was careful not to rely too much on his novel and problematic method. After arriving at his results by means of infinitesimals, he went back and proved every one of them by conventional geometrical means, avoiding any use of the infinitely small. Even so, despite his caution, and his fame as a great sage of the ancient world, Archimedes had no mathematical successors. Future generations of mathematicians steered clear of his novel approach, relying instead on the tried-and-true methods of geometry and its irrefutable truths. For over a millennium and a half, Archimedes’s work on infinitesimals remained an anomaly, a glimpse of a road not taken.
It was not until the 1500s that a new generation of mathematicians picked up the cause of the infinitely small. Simon Stevin in Flanders, Thomas Harriot in England, Galileo Galilei and Bonaventura Cavalieri in Italy, and others rediscovered Archimedes’s experiments with infinitesimals and began once more to examine their possibilities. Like Archimedes, they calculated the areas and volumes enclosed in geometrical figures, then went beyond the ancient master by calculating the speed of bodies in motion and the slopes of curves. Whereas Archimedes was careful to say that his results were only provisional until proven through traditional geometrical means, the new mathematicians were less timid. Defying the well-known paradoxes, they openly treated the continuum as made up of indivisibles and proceeded from there. Their boldness paid off, as the “method of indivisibles” revolutionized the practice of early modern mathematics, making possible calculations of areas, volumes, and slopes that were previously unattainable. A staid field, largely unchanged for centuries, was turned into a dynamic one that was constantly expanding and acquiring new and unprecedented results. Later on, in the late seventeenth century, the method was formalized at the hands of Newton and Leibniz, and became the reliable algorithm that today we call the “calculus,” a precise and elegant mathematical system that can be applied to an unlimited range of problems. In this form, the method of indivisibles, founded on the paradoxical doctrine of the infinitely small, became the foundation of all modern mathematics.
THE LOST DREAM
Yet, useful as it was, and successful as it was, the concept of the infinitely small was challenged at every turn. The Jesuits opposed it; Hobbes and his admirers opposed it; Anglican churchmen opposed it, as did many others. What was it, then, about the infinitely small that inspired such fierce opposition from so many different quarters? The answer is that the infinitely small was a simple idea that punctured a great and beautiful dream: that the world is a perfectly rational place, governed by strict mathematical rules. In such a world, all things, natural and human, have their given and unchanging place in the grand universal order. Everything from a grain of sand to the stars in the sky, from the humblest beggar to kings and emperors, is part of a fixed, eternal hierarchy. Any attempt to revise or topple it is a rebellion against the one unalterable order, a senseless disruption that, in any case, is doomed to failure.
But if the paradoxes of Zeno and the problem of incommensurability prove anything, it is that the dream of a perfect fit between mathematics and the physical world is untenable. On the scale of the infinitely small, numbers do not correspond to physical objects, and any attempt to force the fit leads to paradoxes and contradictions. Mathematical reasoning, however rigorous and true on its own terms, cannot tell us how the world actually must be. At the heart of creation, it seems, lies a mystery that eludes the grasp of the most rigorous reasoning, and allows the world to diverge from our best mathematical deductions and go its own way—we know not where.
This was deeply troubling to those who believed in a rationally ordered and eternally unchanging world. In science it meant that any mathematical theory of the world was necessarily partial and provisional, because it could not explain everything in the world, and might always be replaced by a better one. Even more troubling were the social and political implications. If there was no rational and unalterable order in society, what was left to guarantee the social order and prevent it from descending into chaos? To groups invested in the existing hierarchy and social stability, infinitesimals seemed to open the way to sedition, strife, and revolution.
Those, however, who welcomed the introduction of the infinitely small into mathematics held far less rigid views about the order of the natural world and society. If the physical world was not ruled by strict mathematical reasoning, there was no way to tell in advance how it was structured and how it operated. Scientists were therefore required to gather information about the world and experiment with it until they arrived at an explanation that best fit the available data. And just as they did for the natural world, infinitesimals also opened up the human world. The existing social, religious, and political order could no longer be seen as the only possible one, because infinitesimals had shown that no such necessary order existed. Just as the opponents of infinitesimals had feared, the infinitely small led the way to a critical evaluation of existing social institutions and to experimentation with new ones. By demonstrating that reality can never be reduced to strict mathematical reasoning, the infinitely small liberated the social and political order from the need for inflexible hierarchies.
The struggle over the infinitely small in the early modern world took different forms in different places, but nowhere was it waged with more determination, or with higher stakes, than in the two poles of Western Europe: Italy in the south and England in the north. In Italy it was the Jesuits who led the charge against infinitesimals, as part of their efforts to reassert the authority of the Catholic Church in the wake of the disastrous years of the Reformation. The story of this fight, from its faint glimmers in the early history of the Society of Jesus to the climactic struggles with Galileo and his followers, is told in part 1 of this book, “The War against Disorder.” In England, too, the struggle over the infinitely small followed in the wake of turbulence and upheaval—the two decades of civil war and revolution in the middle of the seventeenth century during which England was a troubled land without a king. The drawn-out gladiatorial fight over infinitesimals between Thomas Hobbes and John Wallis was a struggle between two competing visions of the future of the English state. The story of this fight—its roots in the terror-filled days of the revolution, its role in the founding of the leading scientific academy in the world, and its effect on the emergence of England as a leading world power—is found in part 2 of this b
ook, “Leviathan and the Infinitesimal.”
From north to south, from England to Italy, the fight over the infinitely small raged across western Europe. The lines in the struggle were clearly drawn. On the one side were the advocates of intellectual freedom, scientific progress, and political reform; on the other, the champions of authority, universal and unchanging knowledge, and fixed political hierarchy. The results of the fight were not everywhere the same, but the stakes were always just as high: the face of the modern world, then coming into being. The statement that “the mathematical continuum is composed of distinct indivisibles” is innocent enough to us, but three and a half centuries ago it had the power to shake the foundations of the early modern world. And so it did: the ultimate victory of the infinitely small helped open the way to a new and dynamic science, to religious toleration, and to political freedoms on a scale unknown in human history.
Part I
The War Against Disorder
THE JESUITS AGAINST THE INFINITELY SMALL
A unity among many cannot be maintained without order, nor order without the due bond of obedience between inferiors and superiors.
—IGNATIUS OF LOYOLA
1
The Children of Ignatius
A MEETING IN ROME
On August 10, 1632, five men in flowing black robes came together in a somber Roman palazzo on the left bank of the Tiber River. Their dress marked them as members of the Society of Jesus, the leading religious order of the day, as did their place of meeting—the Collegio Romano, headquarters of the Jesuits’ far-flung empire of learning. The leader of the five was the elderly German father Jacob Bidermann, who had made a name for himself as the producer of elaborate theatrical performances on religious themes. The others are unknown to us, but their names—Rodriguez, Rosco, Alvarado, and (possibly) Fordinus—mark them as Spaniards and Italians, like many of the men who filled the ranks of the Society. In their day these men were nearly as anonymous as they are today, but their high office was not: they were the “Revisors General” of the Society of Jesus, appointed by the general of the order from among the faculty of the Collegio. Their mission: to pass judgment upon the latest scientific and philosophical ideas of the age.
The task was a challenging one. First appointed at the turn of the seventeenth century by General Claudio Acquaviva, the Revisors arrived on the scene just in time to confront the intellectual turmoil that we know as the scientific revolution. It had been over half a century since Nicolaus Copernicus published his treatise proclaiming the novel theory that the Earth revolved around the sun, and the debate on the structure of the heavens had raged ever since. Could it be possible that, contrary to our daily experience, common sense, and established opinion, the Earth was moving? Nor were things simpler in other fields, where new ideas seemed to be cropping up daily—on the structure of matter, on the nature of magnetism, on transforming base metals into gold, on the circulation of the blood. From across the Catholic world, wherever there was a Jesuit school, mission, or residence, a steady stream of questions came flowing to the Revisors General in Rome: Are these new ideas scientifically sound? Can they be squared with what we know of the world, and with the teachings of the great philosophers of antiquity? And most crucially, do they conflict with the sacred doctrines of the Catholic Church? The Revisors took in these questions, considered them in light of the accepted doctrines of the Church and the Society, and pronounced their judgment. Some ideas were found acceptable, but others were rejected, banned, and could no longer be held or taught by any member of the Jesuit order.
In fact, the impact of the Revisors’ decisions was far greater. Given the Society’s prestige as the intellectual leader of the Catholic world, the views held by Jesuits and the doctrines taught in the Society’s institutions carried great weight far beyond the confines of the order. The pronouncements coming from the Society were widely viewed as authoritative, and few Catholic scholars would have dared champion an idea condemned by the Revisors General. As a result, Father Bidermann and his associates could effectively determine the ultimate fate of the novel proposals brought before them. With the stroke of a pen, they could decide which ideas would thrive and be taught in the four corners of the world and which would be consigned to oblivion, forgotten as if they had never been proposed. It was a heavy responsibility, requiring both great learning and sound judgment. Little wonder that only the most experienced and trusted teachers at the Collegio Romano were deemed worthy to serve as Revisors.
But the issue that was brought before the Revisors General that summer day in 1632 appeared far from the great questions that were shaking the intellectual foundations of Europe. While a few short miles away Galileo was being denounced (and would later be condemned) for advocating the motion of the Earth, Father Bidermann and his colleagues were concerning themselves with a technical, even petty question. They had been asked to pronounce on a doctrine, proposed by an unnamed “Professor of Philosophy,” on the subject of “the composition of the continuum by indivisibles.”
Like all the doctrinal proposals presented to the Revisors, the proposition was cast in the obscure philosophical language of the age. But at its core, it was very simple: any continuous magnitude, it stated, whether a line, a surface, or a length of time, was composed of distinct infinitely small atoms. If the doctrine is true, then what appears to us as a smooth line is in fact made up of a very large number of separate and absolutely indivisible points, ranged together side by side like beads on a string. Similarly, a surface is made up of indivisibly thin lines placed next to each other, a time period is made up of minuscule instants that follow each other in succession, and so on.
This simple notion is far from implausible. In fact, it seems commonsensical, and fits very well with our daily experience of the world: Aren’t all objects made up of smaller parts? Is not a piece of wood made of fibers; a cloth, of threads; an hour, of minutes? In much the same way, we might expect that a line will be composed of points; a surface, of lines; and even time itself, of separate instants. Nevertheless, the judgment of the black-robed fathers who met at the Collegio Romano that day was swift and decisive: “We consider this proposition to be not only repugnant to the common doctrine of Aristotle, but that it is by itself improbable, and … is disapproved and forbidden in our Society.”
So ruled the holy fathers, and in the vast network of Jesuit colleges, their word became law: the doctrine that the continuum is composed of infinitely small atoms was ruled out, and could not be pursued or taught. With this, the holy fathers had every reason to believe, the matter was closed. The doctrine of the infinitely small was now forbidden to all Jesuits, and other intellectual centers would no doubt follow the order’s example. Advocates of the banned doctrine would be excluded and marginalized, crushed by the authority and prestige of the Jesuits. Such had been the case with numerous other pronouncements coming out of the Collegio, and Father Bidermann and his colleagues had no reason to think that this time would be any different. As far as they were concerned, the question of the composition of the continuum had been settled.
Looking back from the vantage point of the twenty-first century, one cannot help but be struck, and perhaps a bit startled, by the Jesuit fathers’ swift and unequivocal condemnation of “the doctrine of indivisibles.” What, after all, is so wrong with the plausible notion that continuous magnitudes, like all smooth objects, are made of tiny atomic particles? And even supposing that the doctrine is in some way incorrect, why would the learned professors of the Collegio Romano go out of their way to condemn it? At a time when the struggle over Copernicus’s theory raged most fiercely; when the fate of Galileo, Copernicus’s ardent advocate and the most famous scientist in Europe, hung in the balance; when novel theories on the heaven and the earth seemed to pop up regularly, didn’t the illustrious Revisors General of the Society of Jesus have greater concerns than whether a line was composed of separate points? To put it bluntly, didn’t they have more important things to worry about?
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Apparently not. For, strange as it might seem to us, the condemnation of indivisibles in 1632 was not an isolated incident in the chronicles of the Jesuit Revisors, but merely a single volley in an ongoing campaign. In fact, the records of the meetings of the Revisors, which are kept to this day in the Society’s archives in the Vatican, reveal that the structure of the continuum was one of the main and most persistent of this body’s concerns. The matter had first come up in 1606, just a few years after General Acquaviva created the office, when an early generation of Revisors was asked to weigh in on the question of whether “the continuum is composed of a finite number of indivisibles.” The same question, with slight variations, was proposed again two years later, and then again in 1613 and 1615. Each and every time, the Revisors rejected the doctrine unequivocally, declaring it to be “false and erroneous in philosophy … which all agree must not be taught.”
Yet the problem would not go away. In an effort to keep abreast of the most recent developments in mathematics, teachers from all corners of the Jesuit educational system kept proposing different variations on the doctrine in the hope that one would be tolerated: Perhaps a division into an infinite number of atoms was allowable, even if a finite number was not? Maybe it was permitted to teach the doctrine not as truth but as an unlikely hypothesis? And if fixed indivisibles were banned, what about indivisibles that expanded and contracted as needed? The Revisors rejected all of these. In the summer of 1632, as we have seen, they once again ruled against indivisibles, and Father Bidermann’s successors (including Father Rodriguez), when called to pass judgment on it in January 1641, again declared the doctrine “repugnant.” In a sign that these decrees had no more lasting effect than their predecessors, the Revisors felt the need to denounce indivisibles again in 1643 and 1649. By 1651 they had had enough: determined to put an end to unauthorized opinions in their ranks, the leaders of the Society produced a permanent list of banned doctrines that could never be taught or advocated by members of the order. Among the forbidden teachings, featured repeatedly in various guises, was the doctrine of indivisibles.
Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World Page 2