Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World (43 page)

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.

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
r
2
and the area of the circle is
π
r
2
, 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
r
0
, and the others
r
1
,
r
2
,
r
3
, 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