Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World

by Alexander, Amir

  WALLIS SAVES MATHEMATICS

  If Wallis’s approach was unacceptable to comparatively orthodox mathematicians such as Fermat, for his colleagues at the Royal Society it was the solution to a vexing problem. Boyle, Oldenburg, and their associates had enshrined the experimental method as the proper way to pursue science. To them, it was not only the correct methodology for revealing the secrets of nature, but also a model for the proper workings of the state. Unfortunately, while experimentalism supported the Royal Society’s founders’ vision of both nature and society, it also left mathematics on the wrong side of the methodological and political divide. Mathematics, as commonly understood, left no room for competing points of view, extracting agreement through the irresistible power of its reasoning rather than reaching agreement through freely given consent. It was the exclusive domain of a small number of experts whose work was too technical and esoteric to be competently evaluated by intelligent and educated laymen, and whose pronouncements—absolute and arrogant—had to be accepted as true based on their authority alone. To top it off, mathematics was the cornerstone of a vision of knowledge and of the state that the Royal Society’s grandees viewed with disgust and horror: Hobbes’s authoritarian science and totalitarian commonwealth. As they saw it, whereas experimentalism stood for moderation, tolerance, and peace, mathematics was the tool of the advocates of dogmatism, intolerance, and their inevitable outcome, civil war.

  This left the founders of the Royal Society in a quandary. How could they retain the power and scientific accomplishments of mathematics without also taking on board its unwelcome baggage? Wallis had the answer: his unique brand of mathematics was as powerful as the traditional approach, but also perfectly in line with the Society’s cherished experimentalism. To the founders of the Royal Society it was a godsend: here was a flexible mathematical approach that accommodated dissenting views and was modest in its claims. Precisely the kind of mathematics the Royal Society could endorse, and promote.

  To see just how different Wallis’s mathematics was from the rigid Euclidian approach detested by the Society, it is instructive to compare his practice with the mathematical views of the man the Society feared most, Thomas Hobbes. To begin with, Hobbes insisted that geometrical entities must be constructed by us, from first principles, and were consequently fully known. Not so, retorted Wallis: lines, planes, and geometrical figures were given to us fully formed, and their mysteries should be investigated just as a scientist studies natural objects. Then came the issue of mathematical method, with Hobbes insisting that strict deduction was the only acceptable way to proceed in mathematics, since it alone secured absolute certainty. Wallis, in contrast, advocated induction, which, he argued, was far more effective than deduction in discovering new results. The fact that induction never pretended to achieve the level of certainty that Hobbes so prized was for Wallis a small price to pay. Finally, since Hobbes insisted that his mathematical deductions arrived at absolute truth, he cared nothing for the opinions of others. The proof stood for itself, whether others understood it or not. But Wallis’s inductive proofs are not infallible logical deductions, but rather, strong, persuasive arguments aimed at swaying his audience. Their success depended very much on whether Wallis’s readers believed in the end that the theorems were true for all cases, not just the ones he tried out.

  In almost every aspect, Wallis’s mathematics replicated the experimental practices of his associates at the Royal Society. He investigated external objects, not constructed ones; his mathematics relied on induction, not deduction; it never claimed to arrive at a final truth; and the ultimate arbiter of this truth was the consensus of men. It was precisely the kind of mathematics that one would expect from the only mathematician among the founders of the Royal Society, and it was precisely what the Society grandees were looking for. Instead of being a dangerous rival to experimental practices, mathematics could now join with them to promote proper science and a proper political order.

  Wallis and Hobbes both believed that mathematical order was the foundation of the social and political order, but beyond this common assumption, they could agree on practically nothing else. Hobbes advocated a strict and rigorous deductive mathematical method, which was his model for an absolutist, rigid, and hierarchical state. Wallis advocated a modest, tolerant, and consensus-driven mathematics, which was designed to encourage the same qualities in the body politic as a whole. Across the mathematical and political divide the two faced each other, and the stakes could not have been higher: the nature of truth; the social and political order; the face of modernity.

  GOLIATH AGAINST THE BACKBITERS

  The first volley in the war between the Savilian Professor of Geometry and the courtly political philosopher was fired in the summer of 1655, when Wallis published the Elenchus geometriae Hobbianae, a scathing critique of Hobbes’s geometrical efforts in De corpore. The last volley was fired twenty-three years later, when the ninety-year-old Hobbes published Decameron physiologicum, which included a discussion of “the proportion of a straight line to half the arc of a quadrant.” It was Hobbes’s last effort to defend his mathematics and undermine his rival, but the back-and-forth would likely have continued indefinitely had Hobbes not died the very next year. In between, Wallis published an additional ten books and essays aimed directly at Hobbes, while Hobbes published at least thirteen tracts aimed specifically at Wallis. To these may be added innumerable other insults, slurs, accusations, and (occasionally) serious critiques, which constituted asides in other works by these two very prolific authors. When the battle was at its height, accusations were flying back and forth at a ferocious pace, with each man charging the other with not only mathematical incompetence, but also political subversion, religious heresy, and personal villainy.

  When the war began, the two had very likely never met. Wallis undoubtedly knew of the celebrated author of Leviathan, and his friend and colleague Seth Ward had traveled to London to meet Hobbes when he returned to England in 1651. If Hobbes knew of Wallis at all, it was only as the new (and apparently unqualified) Savilian Professor at Oxford, appointed by Parliament for overtly political reasons. Even in later years, when the two spent a great deal of their time trying to demolish each other’s reputations, there is no record that they ever met, although it is hard to imagine that they did not run into each other in the tight social circles of the English intellectual elite. The conflict between them was rooted in their opposing political, religious, and methodological views, not in personal animosity. But it did not take long for the exchanges to turn personal, and vicious.

  Wallis set the tone early on: “No-one can doubt how puffed up with pride and arrogance is this man,” he wrote in his dedication of Elenchus to John Owen, dean of Christ Church College and vice chancellor of the University of Oxford. “When I look at him, the equal of ‘Leviathan’ (which made his name for him) or rather ‘Goliath,’ parading with such arrogance, I decided that he should be thoroughly attacked, so that he may see he cannot do anything he likes without being called to task…” A puffed-up and arrogant Goliath parading around as if he had sole possession of the truth became Wallis’s favored caricature for Hobbes, and he vowed to “burst the balloon” of “that man, so full of airy talk.” As for Hobbes, he seemed unfazed by Wallis’s abusive language, and while occasionally complaining about his opponent’s uncivil tone, he joined in the war with relish and gave as good as he got.

  The condescending title of Hobbes’s first response to Wallis’s invective foreshadowed much that was to follow: Six Lessons to the Professors of Mathematics, One of Geometry, the Other of Astronomy. If Wallis considered Hobbes arrogant before, then this tract surely confirmed it. Here was Hobbes, a house intellectual for the Cavendish clan, with no credentials or position, presuming to teach geometry to Wallis and Ward, holders of two of the most distinguished mathematical chairs in Europe. Nor did Hobbes stop there, for in his dedication to Lord Pierrepont, he went on to argue that he was in fact far more deserving of their
positions than they: by setting forth the true foundations of geometry in De corpore, he argued, “I have done that business for which Dr. Wallis receives the wages.”

  In the work itself, Hobbes moves from defending his own mathematical work to deriding Wallis, and answers contempt with contempt: “I verily believe,” he wrote of the Arithmetica infinitorum and Wallis’s work on the angle of contact, “that since the beginning of the world there has not been, nor shall be, so much absurdity written in geometry as is to be found in those books.” Of Wallis’s use of algebraic symbols, some of them (such as ∞) of his own invention, he opines that “symbols are poor, unhandsome, though necessary, scaffolds of demonstration; and ought no more to appear in public, than the most deformed necessary business which you do in your chambers.” Wallis’s “On Conic Sections,” according to Hobbes, “is so covered with the scab of symbols, that I had not the patience to examine whether it be well or ill demonstrated.” It may be that Wallis had these witticisms in mind when he complained, years later, of those who deride symbols and insist on classical proofs decked in “the pompous ostentation of Lines and Figures.” When Wallis tried to respond to some of Hobbes’s more substantial criticisms, Hobbes brushed him off like an imperious schoolmaster disciplining an unruly child: “You do shift and wiggle,” he wrote impatiently, “and throw out ink, that I cannot perceive which way you go, nor need I.” But it is all to no avail: “your book of the Arithmetica Infinitorum is all nought, from the beginning to the end.”

  And so it went, back and forth, for nearly a quarter century. Hobbes was the better stylist and sharper wit, but Wallis held his own thanks to the sheer fervor and volume of his denunciations. He was also by far the better connected of the two, using his positions at Oxford and the Royal Society to gradually isolate Hobbes and discredit him among the English literati. If in the 1650s Hobbes was widely considered a formidable scientist and mathematician, by the 1670s he was seen as a political philosopher who had unwisely strayed beyond his field of expertise, only to be exposed as an incompetent amateur. Even Charles II, Hobbes’s former student, joined the general fun of tormenting the aging philosopher: “Here comes the beare to be bayted!” the king would announce when Hobbes made one of his frequent visits to court. Consequently, despite being one of the most celebrated scholars in England, and despite having many acquaintances in the ranks of the Royal Society, the aging philosopher was never elected a fellow. Hobbes attributed this to the implacable hostility of his powerful enemies in the Society, Wallis and Boyle, and there is undoubtedly some truth to this. But there is also the fact that his bruising decades-long fight with Wallis (and a shorter fight with Boyle) left his scientific reputation in tatters, so that he could legitimately be dismissed as unworthy of being a fellow.

  But beneath the sound and fury generated when two intellectuals engage in public combat, much was in fact at stake. Wallis said as much when he explained why he had launched his first assault on Hobbes’s mathematics. Why, he asked, “should I undertake to refute his Geometry, leaving out Theology and other Philosophies, when there are other things in which he has made far more dangerous errors?” The reason, he explains, is that Hobbes had “set such store by geometry that without it there is hardly anything sound that could be expected in philosophy.” Hobbes is so sure of the mathematical grounding of his system that “if he sees anyone disagreeing with him in Theology or Philosophy he thinks they should be sent away with the supercilious reply that since they are unlearned in geometry they do not understand those things.” The one sure way to topple Hobbes’s entire philosophical edifice is to show that he is in fact a mathematical ignoramus. Then the man “so full of airy talk” will be “quite deflated,” and people will know “… that there is no more to be feared of this Leviathan on this account, since his armour (in which he had the greatest confidence) is easily pierced.”

  Wallis repeated this explanation, after enduring several punishing rounds with Hobbes, in a 1659 letter to the Dutch polymath Christiaan Huygens. The “very harsh diatribe” against Hobbes, he explained, was not caused by his lack of manners, but by the “necessity of the case.” It was “provoked by our Leviathan, when he attacks with all his might, and destroys our universities … and especially the clergy, and all institutions and all religion.” Since this “Leviathan” relied so much on mathematics, Wallis continued, “it seemed necessary that now some mathematician … should show how little he understands this mathematics (from which he takes his courage).” Destroying Hobbes’s mathematical credibility would discredit his teachings and preserve the institutions threatened by his destructive philosophy.

  Fortunately for Wallis, Hobbes’s unconventional brand of geometry offered many openings for attack by a skilled mathematician. If Hobbes had stuck close to the classical geometry that had captivated him years before, when he stumbled upon an open volume of Euclid in a continental gentleman’s library, he would have been on surer ground. But for Hobbes this was not enough: in order to support his political system, his geometry must be a perfect science, capable of resolving all outstanding problems. And in his quest to transform geometry into this ideal, Hobbes foundered. This is not because he was a mathematical ignoramus, since his attempted proofs show a powerful mathematical mind at work. It was because it is simply impossible to resolve the ancient problems by classical means, and the project he had embarked upon was doomed from the start.

  Up to a point, Hobbes was following in the footsteps of Clavius and the Jesuits, who tried to use geometry as a model for the proper order of knowledge, society, and the state. The rigid state they advocated, in which the word of the sovereign (for Hobbes) or the Pope (for the Jesuits) had the force of law, and all opposition was deemed absurd, mirrored the rational geometrical order. But Hobbes went a step further than the Jesuits: rather than settling on treating geometry as a model and an ideal, he tried logically and systematically to derive his philosophy from his modified geometrical principles. This required that he show that all things in the world could be constructed from geometrical principles, and in De corpore, he set out to do just that.

  But the world, as it turns out, cannot be derived from mathematics. The Pythagoreans learned this more than two thousand years earlier, when the existence of incommensurable magnitudes upended their belief that everything in the world could be described in terms of the ratio of whole numbers.

  Hobbes tried to defend himself. When Wallis ridiculed his repeated failures at squaring the circle, he protested about Wallis’s regurgitation of discarded results: “Seeing you knew I had rejected that Proposition,” Hobbes wrote, “it was but poor Ambition to take wing as you thought to do, like Beetles from my egestions.” He admitted his mistakes in his first two attempts in De corpore, but insisted that the cause of the error was mere negligence, and not any problem with his method.

  Hobbes would not, and probably could not, concede that his fundamental approach was flawed. His entire philosophical system was at stake, he believed, and to admit that his geometry was hopeless was, to him, the same as admitting that everything he had ever written down and argued for was worthless. So, repeatedly, for more than two decades, Hobbes kept producing new “proofs,” and Wallis, who well understood how critical they were for Hobbes, kept demolishing them. Cornered and isolated, and facing a rising tide of mathematical criticism, Hobbes hunkered down. “I do not wish to change, confirm, or argue anymore about the demonstration that is in the press,” he wrote to his friend Sorbière in 1664 about yet another of his quadratures of the circle. “It is correct; and if people burdened with prejudice fail to read it carefully enough, that is their fault, not mine.” He went to his grave fully convinced that he had succeeded in squaring the circle.

  WHICH MATHEMATICS?

  Wallis had his fun demolishing each of Hobbes’s solutions to the three classical problems whenever the old philosopher proposed a new one. But he also had methodological criticisms of his rival’s mathematical approach, and these too made their way into hi
s denunciations of Hobbes’s mathematics. In particular, Wallis objected to Hobbes’s attempt to construct mathematics from physical material principles. “Who ever, before you, defined a point to be a body? Who ever seriously asserted that mathematical points have any magnitude?” If points have size, Wallis continued, then adding two, three, or a hundred points together will increase their size two, three, or a hundred times, which is absurd. The same critique applied to Hobbes’s use of other physical properties in his definition of geometrical concepts such as his “conatus” and “impetus.” Hobbes needed these terms because he believed that all proofs must be driven by material causes. But Wallis, seeing an opening, adopted the classical Euclidean position that drew a strict distinction between perfect geometrical objects and their flawed material counterparts: “To what end,” Wallis asked, “need there be a consideration of time or weight or any other such quantity” in geometrical definitions? Such physical attributes, he argued, have no place in the world of geometry.

  Wallis’s criticism of Hobbes on the grounds that his mathematics incorporated material notions was, to put it mildly, disingenuous. After all, Wallis’s own mathematical approach did much the same thing: positing geometrical objects as existing “out there” in the world, dividing them up into their indivisible components, and studying them experimentally. Indeed, Hobbes was happy to return the favor by condemning Wallis for precisely that. Nevertheless, Wallis’s mathematics was, in the end, far less vulnerable to methodological attacks than was Hobbes’s, because Hobbes wanted to use mathematics as a bulwark of certainty that would buttress his political philosophy. Consequently, any criticism that raised questions about the logical soundness of his approach struck at the very core of his enterprise. Wallis, in contrast, cared little for methodological certainties. His purpose, he explained years later, “was not so much to shew a Method of Demonstrating things already known … as to shew a way of Investigation of finding out of things yet unknown.” It was the effectiveness of his approach in establishing new results that mattered to Wallis. Using a perfect and irreproachable method of proof counted for very little indeed.