Induction was far from a new idea in the early 1600s. It was certainly known to Aristotle and other ancient philosophers, who considered it an inferior form of reasoning compared to deduction. Instead of moving from the general to the particular, induction does the reverse: it requires the gathering of many particulars and drawing from them an overarching rule. It follows that instead of proceeding deductively from causes to effects, induction starts with effects taken from the world around us and then, from them, infers the causes.
The pitfalls of inductive reasoning become clear if we consider the case of the black swan, a favorite cautionary tale of many philosophers. For many centuries Europeans had lived with swans and observed them, and all the swans they had seen were white. Using induction, they reasonably concluded that all swans were white. But when Europeans arrived in Australia in the 1700s, they made an unexpected discovery: black swans. It turns out that despite the innumerable particular observations by Europeans over many centuries, and despite the fact that every single observation was of a white swan, the rule that “all swans are white” was nevertheless false.
Writing in the early 1600s, Bacon knew nothing of black swans, but he was fully aware of the inherent uncertainty of induction. Yet he was undeterred. Aristotelian physics, he believed, was a well-constructed and elegant trap, logically consistent but completely divorced from the world. The only way to expand human knowledge of the world, he argued, was through direct engagement with nature, and that meant systematic observations and experimentation. Since these methods, he conceded, work by induction, they are vulnerable to the weaknesses of induction, and their conclusions are never absolutely certain. But if applied carefully and systematically, Bacon argued, with full awareness of its potential weaknesses, induction can ultimately lead to the advancement of human knowledge. It is the only way, according to Bacon, to study nature and uncover its secrets.
So when Wallis at the beginning of the Arithmetica infinitorum writes that he will proceed through induction, he is associating himself with a very particular philosophical enterprise: the experimental philosophy advocated by the late Sir Francis Bacon and later adopted and promoted by the founders of the Royal Society. Wallis had already demonstrated in De sectionibus conicis that he viewed mathematical objects as existing in the world, just like physical objects. In Arithmetica infinitorum he indicates how he is going to study them: through experiments. He will, in other words, study triangles, circles, and squares using the same method that his friend Robert Boyle used to study the structure of the air and his colleague Robert Hooke to study minute creatures under a microscope. When attempting to establish a mathematical truth, he will begin by trying it out on several particular cases and carefully observing the results of these “experiments.” At length, after he has done this repeatedly for different cases, “a general proposition may become known by induction.” Wallis had found the answer to his colleagues’ suspicion of the mathematical method: he had developed an experimental mathematics to fit the spirit of the Society’s experimental ethos. Rather than deduce universal laws that compelled assent and ruled out dissent, Wallis’s mathematics would gather its evidence gradually, case by case, and slowly, cautiously arrive at general, and provisional, conclusions. For such is the way of the experimenter.
Wallis’s experimental mathematics is the basic tool of the Arithmetica infinitorum, the foundation of his mathematical reputation. The subject of the work is very similar to Hobbes’s most ambitious mathematical venture: determining the area of the circle. There is, however, a crucial difference between their projects. Hobbes tries actually to construct a square equal in area to a circle, using only the traditional Euclidian tools of straightedge and compass. He was doomed to fail, because the side of a square with the area of a circle with radius r is , and π (as it was shown two centuries later) is a transcendental number, which cannot be constructed in this manner. Wallis, of course, does not try to construct anything. He instead tries to arrive at a number that will give the correct ratio between a circle and a square with a side equal to its radius r. Since the area of the square is r2 and the area of the circle is πr2, the number is π. Since π is transcendental, it cannot be described as a regular fraction or a finite decimal fraction. Nevertheless, at the end of the work, Wallis manages to produce an infinite series that allows him to approximate π as closely as he wishes:
Wallis begins his calculation of the area of a circle much as he began his calculation of the area of a triangle: Looking at one quadrant of the circle with radius R, he parses it into parallel lines as seen in Figure 9.2. The longest of these is R, and the others gradually become shorter and shorter until they reach 0 at the circle’s circumference. Let us mark the longest line r0, and the others r1, r2, r3, and so on. Meanwhile, the area of the square enclosing the quadrant is also composed of an infinity of lines, but they are all the same length. Consequently, the ratio between the quadrant and the square is
The more lines we have in the quadrant and in the square—or, as we would say today, as n approaches infinity—the closer this number draws to the ratio between the area of the quadrant and the area of the enclosing square.
Figure 9.2. Parallel lines compose the surface of the quadrant. After John Wallis, Arithmetica infinitorum (Oxford: Leon Lichfield, 1656), p. 108, prop. 135.
Now, the precise length of each of the parallel lines r that compose the quadrant is dependent on its distance from the first and longest line, R. If we divide the distance R into n equal parts, and consider each equal part a unit, then the length of the line closest to R is ; the line next to that will be , the one after that , and so on, until we reach the circumference, where the last line is , which is zero. The ratio between the lines dividing the quadrant and the same number of lines dividing the enclosing square will therefore be
Wallis’s goal in the Arithmetica infinitorum is to calculate this ratio as n increases to infinity, and this proves to be no easy task. He approaches the result through a succession of approximations of similar series, that draw ever closer to the desired ratio. But far more significant than Wallis’s calculation of the area of a circle is his method for summing infinite series that ultimately lead to his final result.
Suppose, he suggests at the start of Arithmetica infinitorum, that we have a “series of quantities in arithmetic proportion, continually increasing, beginning from a point or 0 … thus as 0, 1, 2, 3, 4, etc.” What, he asks, is the ratio of the sum of the terms of the series to the sum of an equal number of the largest term? Wallis decides to try it out. He begins with the simplest case, of the two-term series 0, 1. The ratio is, accordingly,
He tests other cases:
Every case yields the same result, and Wallis draws a definite conclusion: “If there is taken a series of quantities in arithmetic proportion (or as the natural sequence of numbers) continually increasing, beginning from a point or 0, either finite or infinite in number (for there will be no reason to distinguish), it will be to a series of the same number of terms equal to the greatest, as 1 to 2.”
Wallis could easily have proven this simple result by giving the general formula for the sum of the sequence of natural numbers beginning with 0, and dividing it by the sum of an equal number of the largest term: divided by n(n + 1), which immediately gives . But his goal was not to calculate the ratio, but to demonstrate the usefulness of the method of induction: try one case, then another, and then another. If the theorem holds in all cases, then to Wallis it is proven and true. “Induction,” he wrote many years later, is “a very good Method of Investigation … which doth very often lead us to the early discovery of a General Rule.” Most important, “it need not … any further Demonstration.”
Once he had established this first theorem, Wallis moved on to do the same for more complex series: what if, instead of adding up a sequence of the natural numbers and dividing the sum by the same number of the largest term, he added up the squares of the natural numbers and divided the sum by an equal number of the la
rgest square? Using his favored method of induction, he tries it out. Starting with the simplest case, he gets:
He then adds more terms, calculating the sum in each case:
Looking at the different cases, Wallis deduces that the more terms there are in the series, the closer the ratio approaches . For an infinite series, he concludes, the difference will vanish entirely. He writes it up in a theorem (proposition 21):
If there is proposed an infinite series, of quantities that are as squares of arithmetic proportionals (or as a sequence of square numbers) continually increasing, beginning from a point of 0, it will be to a series of the same number of terms equal to the greatest as 1 to 3.
Wallis’s proof requires only a single sentence: the result, he writes, is “clear from what has gone before.” Induction needs no further support.
Wallis tries out one more series of this type, looking at the cubes rather than the squares of the natural numbers:
The method of induction proves itself once again. As the number of terms increases, the ratio approaches ever more closely to , which leads to proposition 41:
If there is proposed an infinite series of quantities that are as cubes of arithmetic proportionals (or as a sequence of pure numbers) continually increasing, beginning from a point or 0, it will be to a series of the same number of terms equal to the greatest as 1 to 4.
Like the theorems that preceded it, this, too, requires no proof beyond self-evident induction.
In modern notation, Wallis’s three theorems would look like this:
Wallis considers these ratios important steps on the road to calculating the area of a circle, because each algebraic ratio corresponded for him to a particular geometrical case. The first one shows the ratio between a triangle and its enclosing rectangle, just as Wallis shows in his proof of the area of the triangle in De sectionibus conicis. The series 0, 1, 2, 3,…, n represents the lengths of the parallel lines that make up the triangle, and the series n, n, n, n,…, n represents an equal number of parallel lines composing the enclosing rectangle. The ratio between them, , is indeed the ratio between the areas of the triangle and rectangle (see Figure 9.1). The second case corresponds to the ratio between a half-parabola and its enclosing rectangle or, more precisely, the ratio between the area outside the half-parabola and the rectangle. The parallel lines composing this area increase as squares, that is, 0, 1, 4, 9,…, n2, whereas the rectangle is represented by n2, n2, n2,…, n2. Wallis, in effect, shows that the ratio between the area outside a parabola and the area of its enclosing rectangle is . The third ratio (Figure 9.3) corresponds to a steeper “cubic” parabola, showing that the ratio here is . While Wallis still has a long way to go before calculating the more difficult ratio between the quarter-circle and its enclosing square, his strategy for arriving there is clearly taking shape.
With these results established, Wallis now makes use of induction once again to arrive at an even more general theorem: what is true for natural numbers, their squares, and their cubes, must be true for all powers m of natural numbers:
Wallis does not quite write the results in this form. Lacking our modern notation, he uses a table, where he assigns a ratio, , to the “first power,” another ratio, , to the “second power,” another ratio, , to the “third power,” and so on. The table is open-ended, and the rule is manifest: for any power m, the ratio will be .
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 w
hy 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.
Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World Page 31