Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World (2 page)

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 infinite 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.

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.