Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World (47 page)

Figure 9.3. Half a cubic parabola and its enclosing rectangle. Wallis’s ratios show that the ratio between the area
AOT
outside the cubic parabola and the area of the enclosing rectangle is
. After Wallis,
Arithmetica Infinitorum
, prop. 42.

Wallis viewed geometrical figures as material things, and therefore believed that, just like any object, they were composed of fundamental parts. Plane figures were made up of indivisible lines arranged next to each other, and solids of planes stacked on top of each other, just as they were for Cavalieri and Torricelli before him. But unlike the Italian masters, Wallis’s preferred method for investigating mathematical objects was Baconian induction, which made his methodology resemble that of an experimentalist in his laboratory rather than a mathematician at his desk. Material, infinitesimal, and experimental, Wallis’s method was one of the most unorthodox ventures in the history of Western mathematics.

It should come as no surprise, then, that not everyone was impressed with Wallis’s accomplishment. Pierre de Fermat is remembered today mostly as the author of Fermat’s Last Theorem, one of the longest-standing unsolved problems in mathematics, until it was proven by British mathematician Andrew Wiles in 1994. But in his day, the Frenchman was one of the most renowned and respected mathematicians in Europe. He read the
Arithmetica infinitorum
shortly after its publication in 1656, and by the following year he was engaged in a lively debate with Wallis. Fermat was skeptical, and his critiques went straight to the unconventional heart of Wallis’s approach. First, he went after Wallis’s infinitesimalism, which uncritically assumed that one can sum up the lines in a plane figure to calculate its area. Wallis, Fermat argued, had it backward: one cannot sum up the lines of a figure unless one already knows the area of the figure, arrived at by traditional means. If Fermat was right, then Wallis’s entire project was pointless, since it pretended to demonstrate what in fact it already assumed.

If Fermat was unhappy about Wallis’s casual use of infinitesimals, he was no happier about Wallis’s unusual method of proof. Initially, commenting on Wallis’s work to the Catholic English courtier Kenelm Digby, he was at least superficially chivalrous: “I have received a copy of the letter of Mr. Wallis, whom I esteem as I must,” he began, leaving open the question of exactly how much esteem that is. Judging by what follows, it may not have been much: “But his method of demonstration, which is founded on induction rather than on reasoning in the style of Archimedes, may be somewhat difficult for novices who want demonstrative syllogisms from beginning to end.” You and I, he suggested rather patronizingly, surely understand Wallis’s unusual method, but mathematical “novices” might have trouble with it, and perhaps Wallis would be so good as to accommodate them. But politeness and condescension aside, it immediately became clear that Fermat’s concern wasn’t really the needs of the mathematically unlettered, but Wallis’s method itself: it is much better, he wrote, “to prove things by the ordinary, legitimate, and Archimidean way.” Wallis’s method, one is left to conclude, was neither ordinary nor legitimate.

Fermat made clear the problem with induction in a separate letter, which he penned shortly thereafter. One must be extremely careful using this method, he warned, because it allows one to propose a rule that “will be good for several particulars, and nevertheless will be false in effect and not universal.” The method can be useful in some circumstances, he continued, if used with care. It must not, however, “be used for the foundation of a science, that from which it is deduced, as does Mr. Wallis: for that, one must settle for nothing less than a demonstration.” The unavoidable implication that Wallis’s inductions were
not
demonstrations is left unsaid.

Wallis was unmoved. His mathematics of infinites, he replied, was founded on Cavalieri’s method of indivisibles, and Fermat’s criticism regarding the composition of geometrical figures was therefore fully answered in Cavalieri’s books. Far from being a radical departure from traditional practices, his method was simply a shorthand for the irreproachable method of exhaustion used by the ancient masters Eudoxus and Archimedes. If Fermat nevertheless wished to reconstruct all the proofs in the classical form, Wallis wrote, “it was free for him to do it.” But “he might spare himself the labour, because it was already done to his hand by Cavallerius.”

Wallis artfully deflected Fermat’s valid criticism of infinitesimals without answering it directly. The claim that “there is nothing new here” sounds disingenuous coming from someone who loudly proclaimed the novelty of his work. “You may find this work new (if I judge rightly),” he wrote in his dedication of the
Arithmetica infinitorum
to William Oughtred, adding that “I see no reason why I should not proclaim it.” His claim that Cavalieri had already answered all objections was an effective strategy also used by Torricelli, Angeli, and other promoters of the infinitely small. It ignored the withering attacks on Cavalieri by the Jesuits and others, thereby presenting indivisibles as far more widely accepted than they actually were. It also relied on the high likelihood that Fermat had never actually read Cavalieri’s tomes, whose notorious unreadability provided cover to many seventeenth-century indivisiblists.

Wallis was equally unimpressed by Fermat’s critique of his method of induction. Proofs by induction, Wallis claimed, “are plain, obvious and easy,” and require no additional demonstration. “If any think them less valuable,” he wrote, “because not set forth with the pompous ostentation of Lines and Figures, I am quite of another mind.” Any competent mathematician who put in the time, Wallis argued, could convert his proofs by induction into traditional geometric proofs, but to do so would be mere fussiness: “I do not find that
Euclide
was wont to be so pedantick,” he wrote, and “I am sure
Archimedes
was not.” Pedants such as Fermat, according to Wallis, were a distinct minority: “[M]ost mathematicians that I have seen, after such induction continued for some steps … are satisfied (from such evidence) to conclude universally
and so in like manner for the consequent powers
. And such Induction hath been hitherto thought … a conclusive argument.” With these brief and contemptuous remarks, Wallis dismissed thousands of years of tradition.

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.