Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World

by Alexander, Amir

  Bacon’s warning against the “daintiness and pride” of mathematicians was not lost on his followers, the founders of the Royal Society. Although the Society was officially described as a “Colledge for the Promoting of Physico-Mathematicall Experimentall Learning,” in practice the “mathematicall” studies were strictly subordinated to the “experimentall” ones. For the Society leaders shared Bacon’s concern that mathematics breeds pride and makes it easy to assume that God created the world according to rigid mathematical strictures. Like Bacon, they were worried that mathematical reasoning would lure scientists away from the laborious work of experimentation.

  But the Royal Society founders had other concerns, concerns that went beyond Bacon’s warning half a century before. Mathematics, they believed, was the ally and the tool of the dogmatic philosopher. It was the model for the elaborate systems of the rationalists, and the pride of the mathematicians was the foundation of the pride of Descartes and Hobbes. And just as the dogmatism of those rationalists would lead to intolerance, confrontation, and even civil war, so it was with mathematics. Mathematical results, after all, left no room for competing opinions, discussions, or compromise of the kind cherished by the Royal Society. Mathematical results were produced in private, not in a public demonstration, by a tiny priesthood of professionals who spoke their own language, used their own methods, and accepted no input from laymen. Once introduced, mathematical results imposed themselves with tyrannical power, demanding perfect assent and no opposition. This, of course, was precisely what Hobbes so admired about mathematics, but it was also what Boyle and his fellows feared: mathematics, by its very nature, they believed, leads to claims of absolute truth, dogmatism, threats of tyranny, and, all too easily, civil war.

  Yet, despite the ideological and political dangers, mathematics could not be simply dispensed with. Some of the greatest accomplishments of the New Philosophy were heavily mathematical. Advancements in medicine, such as Harvey’s discovery of the circulation of the blood, were certainly experimental, as were the barometric measurements of atmospheric pressure known as the Torricellian experiment, and William Gilbert’s investigations into the nature of magnetism. But the greatest scientific triumphs of the age were indeed in astronomy, and these were deeply indebted to mathematics.

  What, then, could the Royal Society leaders do? They could not simply ignore the brilliant contributions that mathematics had already made to science, or the strong indications that the former would continue to play a central role in the latter’s advancement. But how could the Society embrace the important contributions of mathematical science and yet avoid its dangerous methodological, philosophical, and political implications? It was a conundrum that left the Society with an ambivalence toward mathematics that characterized its science for many years. And no one felt this conflict more keenly than John Wallis.

  9

  Mathematics for a New World

  AN INFINITY OF LINES

  Wallis was the only mathematician among the Society’s founders, and it therefore fell to him to address the problematic status of mathematics. He fully shared his associates’ abhorrence of dogmatism, and in his autobiography he took pride in his moderation and openness to diverse opinions, even when they conflicted with his own views. “It hath been my endeavour all along,” he wrote by way of a summary, “to act by moderate principles, between the extremities on either hand … without the fierce and violent animosities usual in such cases against all that did not act just as I did, knowing that there were many worthy persons engaged on either side.”

  Yet, as a mathematician and Savilian Professor, Wallis was committed to a field that had traditionally prided itself on its inflexible methodology and the absolute and incontestable truth of its results. How was one to reconcile this with the moderation and flexibility he cherished as a member of the Royal Society? Wallis’s solution was simple, and radical: he created a new kind of mathematics. Unlike traditional mathematics, this new approach would proceed not through rigorous deductive proof but through trial and error; its results would be extremely probable but not irrefutably certain, and they would be validated not through “pure reason” but by consensus, just like the public experiments conducted at the Royal Society. Ultimately, his mathematics would be judged not by its logical perfection, but by its effectiveness at producing new results.

  His mathematics, in other words, was not modeled on Euclidean geometry, the grand logical edifice that had inspired Clavius, Hobbes, and innumerable others over two millennia; it was, rather, designed to emulate the experimental approach practiced at the Royal Society. If Wallis succeeded, he would free mathematics of its association with dogmatism and intolerance and resolve the long-standing objections his fellows at the Society had toward the field. It would be a new “experimental mathematics,” powerful and effective in the service of science, but serving as a model for tolerance and moderation rather than dogmatic rigidity. And at its very core would be the concept of the infinitely small.

  The singular nature of Wallis’s approach is apparent from the very first theorem of the very first work he wrote and published as Savilian Professor, “On Conic Sections” (De sectionibus conicis).

  I suppose, to begin with (according to Bonaventura Cavalieri’s Geometry of Indivisibles) that any plane is made up, so to speak, of infinite parallel lines. Or rather (as I prefer) of an infinite number of parallelograms of equal height, the altitude of each one being of the entire height, or an infinitely small aliquot part (the sign ∞ denoting an infinite number); so that the altitude of all equal the height of the figure.

  Immediately we are in the highly unorthodox world of Wallis’s infinitesimal mathematics. Like Cavalieri and Torricelli before him, Wallis considered planes as quasi-material objects made up of an infinite number of lines stacked on top of one another, not as the abstract concepts of Euclidean geometry. That this conflicted with the classical paradoxes of Zeno, and with the problem of incommensurability, was obvious to any mathematician reading the tract, and both Hobbes and the French mathematician Pierre de Fermat were quick to point this out. But Wallis was unimpressed with these obvious criticisms. His notion that plane figures are composed of lines was derived from Cavalieri’s famous analogy of a plane to a piece of cloth made of threads, as well as from the Jesuat’s practice of viewing a plane as an aggregate of lines. He therefore simply referred the reader to Cavalieri, who supposedly had already dealt with all objections and moved on. Wallis even invented a sign to mark the number of infinitesimals that make up the plane and their magnitudes, respectively, ∞ and .

  With these basic tools in hand, Wallis then proceeds to demonstrate the power of his approach by proving an actual theorem:

  Since a triangle consists of an infinite number of arithmetically proportionate lines or parallelograms, beginning with a point and continuing to the base (as is clear from the discussion): then the area of the triangle is equal to the base times half the altitude.

  Needless to say, Wallis did not need to provide a complex proof in order to determine that the area of a triangle is half its base times its height. The purpose of the proof was not to prove the result, but quite the opposite: to demonstrate the validity of his unconventional approach, by showing that it led to a correct and familiar result. Once he had established the reliability of his method, he could then use it to resolve more challenging and unfamiliar problems.

  The statement that the lines composing a triangle are “arithmetically proportionate” calls for some explanation. What Wallis means is that if lines are drawn through a triangle parallel to its base, and if those lines are equally spaced along the triangle’s height, then the lengths of the lines form an arithmetic progression. For example, if a line is drawn halfway between the apex of the triangle and the base, its length will be half that of the base, forming the arithmetical series (0, , 1), for the apex, the line, and the base, respectively. If the height is divided into three, and lines are drawn at the one-third and two-third marks, th
eir lengths will form the series (0, , , 1); if the height is divided into ten, the lengths will be (0, , , … , 1), and so on. This holds true regardless of how many parts the height is divided into, as long as those parts are at an equal distance from one another. In his proof, Wallis assumes that this principle holds true even if the height is divided into an infinite number of parts.

  He continues: “It is a well-known rule among mathematicians that the sum of an arithmetical progression, or the aggregate of all the terms, equals the sum of its extremes multiplied by half the total number of terms.” This is a simple rule, familiar today to many a high school student. The sum of all the numbers from 1 to 10, for example, is 11 (that is, 1 + 10) times 5 (half the number of terms in the sequence), that is, 55. Designating the infinitesimal magnitude of a single point by the letter “o,” Wallis then uses the rule to sum up all the indivisible lines that compose the triangle:

  Therefore, if we consider the smallest term “o” (since we suppose that a point equals “o” in magnitude as well as zero in number), the sum of the two ends is the same as the largest term. I substitute the altitude of the figure for the number of terms in the progression, for this reason, that if we suppose the number of terms to be ∞, then the sum of their lengths is × Base (since the base is equal to the sum of the two ends).

  Wallis is after the total length of all the lines that make up the triangle. Since they are infinite in number, and range from zero (or “o”) to the length of the base, their combined length is × Base. He now multiplies this by the thickness of each line:

  But we suppose the thickness or altitude of each (line or parallelogram) to be × the Altitude of the Triangle; by which the sum of the lengths is to be multiplied. Therefore × A multiplied by × Base will give the area of the triangle. That is .

  Figure 9.1. Wallis’s triangles, composed of parallel lines. From De sectionibus conicis, prop. 3. (Oxford, Leon Lichfield, 1655)

  And that is how Wallis calculated the area of the triangle: He summed up the lengths of all the component lines as an arithmetical progression, and then multiplied the sum by the “thickness” of each line. Arriving in this way at an equation that had ∞ in the numerator and ∞ in the denominator, he canceled them against each other and ended up with the familiar formula. QED.

  Now, it is probably an understatement to say that no modern mathematician would follow Wallis in these wild and woolly calculations. Nor would many of his contemporaries, including all the Jesuits and Fermat, among others. Apart from the problematic assumption that a surface is made up of lines with a certain (very small) thickness, Wallis is also assuming without proof that the rules for summing up a finite series also apply to an infinite one. And if these unsubstantiated assumptions are not questionable enough, Wallis then casually divides infinite by infinity, or, to use his own notation, ∞ by ∞. In modern mathematics is undefined, for the simple reason that if = a, then ∞ = a × ∞, and since any number multiplied by ∞ equals ∞, a can be any number. But Wallis treats as an ordinary algebraic expression, and cancels ∞ by ∞. When criticized by Fermat and others, Wallis seemed unconcerned with the logical difficulties of his procedures, refusing to concede any of their points. His approach, after all, was not meant to demonstrate his adherence to strict formal rigor. It was, rather, designed to make mathematics acceptable to his fellows at the Royal Society.

  What did Wallis accomplish with his unconventional approach? For one, he posited that geometrical objects were objects “out there” in the world, and could be investigated as such, just like any natural object. This is the exact opposite of the traditional view that held that all geometrical objects should be constructed from first principles. It also runs counter to Hobbes’s view that geometrical objects are perfectly known because we construct them. For Wallis, in contrast, the triangle already exists in the world, and the geometer’s job is to decipher its hidden characteristics—just like a scientist trying to understand a geological rock formation or the biological systems of an organism. Drawing on common sense and intuition about the physical world, Wallis concluded that a triangle is composed of parallel lines next to each other, just as a rock formation is made of geological strata, a piece of wood is made of fibers, or (following Cavalieri) a piece of cloth is made of threads.

  Since, according to Wallis, geometrical objects already exist in the world, it follows that mathematical rigor is completely unnecessary. In traditional geometry, in which one constructs geometrical objects from first principles and proves theorems about the relationships between them, logical rigor is indispensable. It is only a strict insistence on correct logical inferences, after all, that guarantees that the results are correct. The case, however, is very different if one is examining an object in the world, because it is external reality that decides whether a result is correct. An overinsistence on strict logical reasoning can be more a hindrance than a help.

  Consider, for example, a geologist investigating a rock formation. He will certainly not throw his results out the window just because someone points out that his grant proposal has a spelling mistake or one of the measurements contains a tiny error. Rather, if the results correctly describe the rock formation—its structure, age, the manner it was formed, etc.—then the geologist will rightly conclude that the overall methodology must be correct as well, regardless of minute inconsistencies. The same is true for Wallis, who studied triangles as external objects no different, fundamentally, from a rock formation. It is all very well to insist on strict rigor, we can imagine Wallis thinking, but not if it gets in the way of our making new discoveries. Some mathematicians could and did gripe about his infinitesimal lines and his division of infinity by infinity, but to Wallis this was mere pedantry. He did, after all, arrive at a correct result.

  Such casual disregard for logical rigor is a strange position for a mathematician to take, but Wallis had signaled his unusual outlook in Truth Tried in 1643. Rejecting the pure reasoning of Euclidean geometry, Wallis instead posited the triangle as an almost material object, one that could be intuited through the senses. For Wallis, the triangle could certainly be seen, its internal structure “felt,” and if it could not quite be “tasted,” then one got the feeling that it almost could be. “Mathematical entities exist,” he wrote confidently in his Mathesis universalis of 1657, “not in the imagination but in reality.”

  Much had happened in Wallis’s life in the years that passed between the publication of Truth Tried and his mathematical publications of the following decade. He had left his Presbyterian roots largely behind, moved from London to Oxford, and become a professional mathematician and Savilian Professor. But when it came to the question of how to acquire true knowledge, Wallis the distinguished mathematics professor was no different from Wallis the young Parliamentarian firebrand: the path lay not through abstract reasoning, but through material intuition, which “it seems not in the power of the Will to reject.”

  EXPERIMENTAL MATHEMATICS

  Wallis’s approach in De sectionibus conicis established geometrical objects as real bodies in the world, but it left open the question of how they should be investigated. In the proof of the area of the triangle, Wallis relied on material intuition to break down the plane into an infinity of parallel lines, and then sum them up. This proves to be effective for the problem at hand, but it is not a “method” applicable to a wide array of mathematical problems. In Truth Tried, Wallis suggested that a broader approach should rely on experiments, but it is far from clear what that means. How should one apply the experimental method, which relies on material scientific instruments and actual physical observations to abstract mathematical bodies such as triangles, circles, and cones? Wallis had an answer, and he gave it in the Arithmetica infinitorum, which was published alongside the De sectionibus conicis in 1656. It is widely considered to be his masterpiece.

  “The simplest method of investigation, in this and various problems that follow, is to exhibit the thing to a certain extent, and to observe the ratios
produced and compare them to each other; so that at length a general proposition may become known by induction.” So wrote Wallis in proposition 1 on the first page of the Arithmetica infinitorum. The critical word here is also the last, induction, and the approach became known both to Wallis and to his critics as his method of induction. Today, mathematical induction is the name given to a perfectly rigorous and widely used method of proof that is taught to high school and college students. It consists of demonstrating a theorem for a particular case, say, n, and then proving that if it is true for n, it is also true for the case n + 1 (or n – 1), and consequently for all n’s. This, however, was developed much later, and is not at all what Wallis had in mind. Because in the seventeenth century, and especially in seventeenth-century England, “induction” was associated with a particular scientific approach and a particular individual: Francis Bacon, Lord Chancellor to James I and prophet and chief advocate of the experimental method.

  Bacon developed his theory of induction in his Novum organum (The New Organon) of 1620, his most systematic work on scientific method. He saw induction as an alternative to deduction, which, according to Aristotle and his followers in European universities, is the strongest form of logical reasoning. Deduction is the type of reasoning employed in Euclidean geometry and also in Aristotelian physics. It moves from the general (“all men are mortal”) to the particular (“Socrates is mortal”) and from causes (“heavy bodies belong in the center of the cosmos”) to effects (“heavy bodies fall to the ground”). But Bacon argued that this kind of reasoning would never lead to new knowledge, because it made no room for the acquisition of new facts through observation and experimentation. Induction, for Bacon, was an alternative form of reasoning that, unlike deduction, could make use of experiments.