Is God a Mathematician? (20 page)

Figure 37

Figure 38

As I have noted several times already, probability and statistics become meaningful when one deals with a large number of events—never individual events. This cardinal realization, known as the
law of large numbers,
is due to Jakob Bernoulli, who formulated it as a theorem in his book
Ars Conjectandi
(
The Art of Conjecturing;
figure 38 shows the frontispiece). In simple terms, the theorem states
that if the probability of an event’s occurrence is
p,
then
p
is the most probable proportion of the event’s occurrences to the total number of trials. In addition, as the number of trials approaches infinity, the proportion of successes becomes
p
with certainty. Here is how Bernoulli introduced the law of large numbers in
Ars Conjectandi:
“What is still to be investigated is whether by increasing the number of observations we thereby also keep increasing the probability that the recorded proportion of favorable to unfavorable instances will approach the true ratio, so that this probability will finally exceed any desired degree of certainty.” He then proceeded to explain the concept with a specific example:

Bernoulli devoted twenty years to the perfection of this theorem, which has since become one of the central pillars of statistics. He concluded with his belief in the ultimate existence of governing laws, even in those instances that appear to be a matter of chance:

The upshot of this tale of the science of uncertainty is very simple: Mathematics is applicable in some ways even in the less “scientific” areas of our lives—including those that appear to be governed by pure chance. So in attempting to explain the “unreasonable effectiveness” of mathematics we cannot limit our discussion only to the laws of physics. Rather, we will eventually have to somehow figure out what it is that makes mathematics so omnipresent.

The incredible powers of mathematics were not lost on the famous playwright and essayist George Bernard Shaw (1856–1950). Definitely not known for his mathematical talents, Shaw once wrote an insightful article about statistics and probability entitled “The Vice of Gambling and the Virtue of Insurance.” In this article, Shaw admits that to him insurance is “founded on facts that are inexplicable and risks that are calculable only by professional mathematicians.” Yet he offers the following perceptive observation:

For someone such as Shaw, who complained that during his education “not a word was said to us about the meaning or utility of mathematics,” this humorous account of the “history” of the mathematics of insurance is quite remarkable.

With the exception of Shaw’s text, we have so far followed the development of some branches of mathematics more or less through the eyes of practicing mathematicians. To these individuals, and indeed to many rationalist philosophers such as Spinoza, Platonism was obvious. There was no question that mathematical truths existed in their own world and that the human mind could access these verities without any observation, solely through the faculty of reason. The first signs of a potential gap between the perception of Euclidean geometry as a collection of universal truths and other branches of mathematics were uncovered by the Irish philosopher George Berkeley, Bishop of Cloyne (1685–1753). In a pamphlet entitled
The Analyst; Or a Discourse Addressed to An Infidel Mathematician
(the latter presumed to be Edmond Halley), Berkeley criticized the very foundations of the fields of calculus and analysis, as introduced by Newton (in
Principia
) and Leibniz. In particular, Berkeley demonstrated that Newton’s concept of “fluxions,” or instantaneous rates of change, was far from being rigorously defined, which in Berkeley’s mind was sufficient to cast doubt on the entire discipline:

Berkeley certainly had a point, and the fact is that a fully consistent theory of analysis was only formulated in the 1960s. But mathematics was about to experience a more dramatic crisis in the nineteenth century.

In his famous book
Future Shock,
author Alvin Toffler defined the term in the title as “the shattering stress and disorientation that we induce in individuals by subjecting them to too much change in too short a time.” In the nineteenth century, mathematicians, scientists, and philosophers experienced precisely such a shock. In fact, the millennia-old belief that mathematics offers eternal and immutable truths was crushed. This unexpected intellectual upheaval was caused by the emergence of new types of geometries, now known as
non-Euclidean geometries.
Even though most nonspecialists may have never even heard of non-Euclidean geometries, the magnitude of the revolution in thought introduced by these new branches of mathematics has been likened by some to that inaugurated by the Darwinian theory of evolution.

To fully appreciate the nature of this sweeping change in worldview, we have first to briefly probe the historical-mathematical backdrop.

Euclidean “Truth”

Until the beginning of the nineteenth century, if there was one branch of knowledge that had been regarded as the apotheosis of truth and certainty, it was Euclidean geometry, the traditional geometry we learn in school. Not surprisingly, therefore, the great Dutch Jewish philosopher Baruch Spinoza (1632–77) entitled his bold attempt to unify science, religion, ethics, and reason
Ethics, Demonstrated in
Geometrical Order.
Moreover, in spite of the clear distinction between the ideal, Platonic world of mathematical forms and physical reality, most scientists regarded the objects of Euclidean geometry simply as the distilled abstractions of their real, physical counterparts. Even staunch empiricists such as David Hume (1711–76), who insisted that the very foundations of science were far less certain than anyone had ever suspected, concluded that Euclidean geometry was as solid as the Rock of Gibraltar. In
An Enquiry Concerning Human Understanding,
Hume identified “truths” of two types:

In other words, while Hume, like all empiricists, maintained that all knowledge stems from observation, geometry and its “truths” continued to enjoy a privileged status.

The preeminent German philosopher Immanuel Kant (1724–1804) did not always agree with Hume, but he also exalted Euclidean geometry to a status of absolute certainty and unquestionable validity. In his memorable
Critique of Pure Reason,
Kant attempted to reverse in some sense the relationship between the mind and the physical world. Instead of impressions of physical reality being imprinted on an oth
erwise entirely passive mind, Kant gave the mind the active function of “constructing” or “processing” the perceived universe. Turning his attention inward, Kant asked not
what
we can know, but
how
we can know what we can know. He explained that while our eyes detect particles of light, these do not form an image in our awareness until the information is processed and organized by our brains. A key role in this construction process was assigned to the human intuitive or synthetic
a priori
grasp of space, which in turn was taken to be based on Euclidean geometry. Kant believed that Euclidean geometry provided the only true path for processing and conceptualizing space, and that this intuitive, universal acquaintance with space was at the heart of our experience of the natural world. In Kant’s words:

To put it simply, according to Kant, if we perceive an object, then necessarily this object is spatial and Euclidean.

Hume’s and Kant’s ideas bring to the forefront the two rather different, but equally important aspects that had been historically associated with Euclidean geometry. The first was the statement that Euclidean geometry represents the only accurate description of physical space. The second was the identification of Euclidean geometry with a firm, decisive, and infallible deductive structure. Taken together, these two presumed properties provided mathematicians, scientists, and philosophers with what they regarded as the strongest
evidence that informative, inescapable truths about the universe do exist. Until the nineteenth century these statements were taken for granted. But were they actually true?

The foundations of Euclidean geometry were laid around 300 BC by the Greek mathematician Euclid of Alexandria. In a monumental thirteen-volume opus entitled
The Elements,
Euclid attempted to erect geometry on a well-defined logical base. He started with ten axioms assumed to be indisputably true and sought to prove a large number of propositions on the basis of those postulates by nothing other than logical deductions.

The first four Euclidean axioms were extremely simple and exquisitely concise. For instance, the first axiom read: “Between any two points a straight line may be drawn.” The fourth one stated: “All right angles are equal.” By contrast, the fifth axiom, known as the “parallel postulate,” was more complicated in its formulation and considerably less self-evident: “If two lines lying in a plane intersect a third line in such a way that the sum of the internal angles on one side is less than the two right angles, then the two lines inevitably will intersect each other if extended sufficiently on that side.” Figure 39 demonstrates graphically the contents of this axiom. While no one doubted the truth of this statement, it lacked the compelling simplicity of the other axioms. All indications are that even Euclid himself was not entirely happy with his fifth postulate—the proofs of the
first twenty-eight propositions in
The Elements
do not make use of it. The equivalent version of the “fifth” most cited today appeared first in commentaries by the Greek mathematician Proclus in the fifth century, but it is generally known as the “Playfair axiom,” after the Scottish mathematician John Playfair (1748–1819). It states: “Given a line and a point not on the line, it is possible to draw exactly one line parallel to the given line through that point” (see figure 40). The two versions of the axiom are equivalent in the sense that Playfair’s axiom (together with the other axioms) necessarily implies Euclid’s original fifth axiom and vice versa.

Figure 39

Over the centuries, the increasing discontent with the fifth axiom resulted in a number of unsuccessful attempts to actually prove it from the other nine axioms or to replace it by a more obvious postulate. When those efforts failed, other geometers began trying to answer an intriguing “what if” question—what if the fifth axiom did, in fact, not prove true? Some of those endeavors started to raise nagging doubts on whether Euclid’s axioms were truly self-evident, rather than being based on experience. The final, surprising verdict eventually came in the nineteenth century: One could create new kinds of geometry by
choosing
an axiom different from Euclid’s fifth. Furthermore, these “non-Euclidean” geometries could in principle describe physical space just as accurately as Euclidean geometry did!

Let me pause here for a moment to allow for the meaning of the word “choosing” to sink in. For millennia, Euclidean geometry had been regarded as unique and
inevitable
—the sole true description of space. The fact that one could choose the axioms and obtain an equally valid description turned the entire concept on its ear. The certain, carefully constructed deductive scheme suddenly became more similar to a game, in which the axioms simply played the role of the rules. You could change the axioms and play a different game. The
impact of this realization on the understanding of the nature of mathematics cannot be overemphasized.

Figure 40

Quite a few creative mathematicians prepared the ground for the final assault on Euclidean geometry. Particularly notable among them were the Jesuit priest Girolamo Saccheri (1667–1733), who investigated the consequences of replacing the fifth postulate by a different statement, and the German mathematicians Georg Klügel (1739–1812) and Johann Heinrich Lambert (1728–1777), who were the first to realize that alternative geometries to the Euclidean could exist. Still, somebody had to put the last nail in the coffin of the idea of Euclidean geometry being the one and only representation of space. That honor was shared by three mathematicians, one from Russia, one from Hungary, and one from Germany.

Strange New Worlds

The first to publish an entire treatise on a new type of geometry—one that could be constructed on a surface shaped like a curved saddle (figure 41a)—was the Russian Nikolai Ivanovich Lobachevsky (1792–1856; figure 42). In this kind of geometry (now known as
hyperbolic geometry
), Euclid’s fifth postulate is replaced by the statement that given a line in a plane and a point not on this line, there are at least two lines through the point parallel to the given line. Another important difference between Lobachevskian geometry and Euclidean geometry is that while in the latter the angles in a triangle always add up to 180 degrees (figure 41b), in the former the sum is always
less than 180 degrees. Because Lobachevsky’s work appeared in the rather obscure
Kazan Bulletin,
it went almost entirely unnoticed until French and German translations started to appear in the late 1830s. Unaware of Lobachevsky’s work, a young Hungarian mathematician, János Bolyai (1802–60), formulated a similar geometry during the 1820s. Bursting with youthful enthusiasm, he wrote in 1823 to his father (the mathematician Farkas Bolyai; figure 43): “I have found things so magnificent that I was astounded…I have created a different new world out of nothing.” By 1825, János was already able to present to the elder Bolyai the first draft of his new geometry. The manuscript was entitled
The Science Absolute of Space.
In spite of the young man’s exuberance, the father was not entirely convinced of the soundness of János’s ideas. Nevertheless, he decided to publish the new geometry as an appendix to his own two-volume treatise on the foundations of geometry, algebra, and analysis (the supposedly inviting title of which read
Essay on the Elements of Mathematics for Studious Youths
). A copy of the book was sent in June 1831 to Farkas’s friend Carl Friedrich Gauss (1777–1855; figure 44), who was not only the most prominent mathematician of the time, but who is also considered by many, along with Archimedes and Newton, to be one
of the three greatest of all time. That book was somehow lost in the chaos created by a cholera epidemic, and Farkas had to send a second copy. Gauss sent out a reply on March 6, 1832, and his comments were not exactly what the young János expected: