Is God a Mathematician? (3 page)

And where do the heavens fit into all of this? Pythagoras and the Pythagoreans played a role in the history of astronomy that, while not critical, was not negligible either. They were among the first to maintain that the Earth was spherical in form (probably because of the perceived mathematico-aesthetic superiority of the sphere). They were also probably the first to state that the planets, the Sun, and the Moon have an independent motion of their own from west to east, in a direction opposite to the daily (apparent) rotation of the sphere of the fixed stars. These enthusiastic observers of the midnight
sky could not have missed the most obvious properties of the stellar constellations—shape and number. Each constellation is recognized by the number of stars that compose it and by the geometrical figure that these stars form. But these two characteristics were precisely the essential ingredients of the Pythagorean doctrine of numbers, as exemplified by the Tetraktys. The Pythagoreans were so enraptured by the dependency of geometrical figures, stellar constellations, and musical harmonies on numbers that numbers became both the building blocks from which the universe was constructed and the principles behind its existence. No wonder then that Pythagoras’s maxim was stated emphatically as “All things accord in number.”

We can find a testament to how seriously the Pythagoreans took this maxim in two of Aristotle’s remarks. In one place in his collected treatise
Metaphysics
he says: “The so-called Pythagoreans applied themselves to mathematics, and were the first to develop this science; and through studying it they came to believe that its principles are the principles of everything.” In another passage, Aristotle vividly describes the veneration of numbers and the special role of the Tetraktys: “Eurytus [a pupil of the Pythagorean Philolaus] settled what is the number of what object (e.g., this is the number of a man, that of a horse) and imitated the shapes of living things by pebbles
after the manner of those who bring numbers into the form of triangle or square.
” The last sentence (“the form of triangle or square”) alludes both to the Tetraktys and to yet another fascinating Pythagorean construction—the gnomon.

The word “gnomon” (a “marker”) originates from the name of a Babylonian astronomical time-measurement device, similar to a sundial. This apparatus was apparently introduced into Greece by Pythagoras’s teacher—the natural philosopher Anaximander (ca. 611–547 BC). There can be no doubt that the pupil was influenced by his tutor’s ideas in geometry and their application to
cosmology
—the study of the universe as a whole. Later, the term “gnomon” was used for an instrument for drawing right angles, similar to a carpenter’s square, or for the right-angled figure that, when added to a square, makes up a larger square (as in figure 2). Note that if you add, say, to a 3 × 3 square, seven pebbles in a shape that forms a right angle (a
gnomon), you obtain a square composed of sixteen (4 × 4) pebbles. This is a figurative representation of the following property: In the sequence of odd integers 1, 3, 5, 7, 9,…, the sum of any number of successive members (starting from 1) always forms a square number. For instance, 1 1
²
; 1 3 4 2
²
; 1 3 5 9 3
²
; 1 3 5 7 16 4
²
; 1 3 5 7 9 25 5
²
, and so on. The Pythagoreans regarded this intimate relation between the gnomon and the square that it “embraces” as a symbol of knowledge in general, where the knowing is “hugging” the known. Numbers were therefore not limited to a description of the physical world, but were supposed to be at the root of mental and emotional processes as well.

Figure 2

The square numbers associated with the gnomons may have also been precursors to the famous
Pythagorean theorem.
This celebrated mathematical statement holds that for any right triangle (figure 3), a square drawn on the hypotenuse is equal in area to the sum of the squares drawn on the sides. The discovery of the theorem was “documented” humorously in a famous
Frank and Ernest
cartoon (figure 4). As the gnomon in figure 2 shows, adding a square gnomon number, 9 3
²
, to a 4 × 4 square makes a new, 5 × 5 square: 3
²
+ 4
²
= 5
²
. The numbers 3, 4, 5 can therefore represent the lengths of the sides of
a right triangle. Integer numbers that have this property (e.g., 5, 12, 13; since 5
²
12
²
13
²
) are called “Pythagorean triples.”

Figure 3

Figure 4

Few mathematical theorems enjoy the same “name recognition” as Pythagoras’s. In 1971, when the Republic of Nicaragua selected the “ten mathematical equations that changed the face of the earth” as a theme for a set of stamps, the Pythagorean theorem appeared on the second stamp (figure 5; the first stamp depicted “1 + 1 = 2”).

Was Pythagoras truly the first person to have formulated the well-known theorem attributed to him? Some of the early Greek historians certainly thought so. In a commentary on
The Elements
—the massive treatise on geometry and theory of numbers written by Euclid (ca. 325–265 BC)—the Greek philosopher Proclus (ca. AD 411–85) wrote: “If we listen to those who wish to recount ancient history, we may find some who refer this theorem to Pythagoras, and say that he sacrificed an ox in honor of the discovery.” However, Pythagorean triples can already be found in the Babylonian cuneiform tablet known as Plimton 322, which dates back roughly to the time of the dynasty of Hammurabi (ca. 1900–1600 BC). Furthermore, geometrical constructions based on the Pythagorean theorem were found in India,
in relation to the building of altars. These constructions were clearly known to the author of the Satapatha Brahmana (the commentary on ancient Indian scriptural texts), which was probably written at least a few hundred years before Pythagoras. But whether Pythagoras was the originator of the theorem or not, there is no doubt that the recurring connections that were found to weave numbers, shapes, and the universe together took the Pythagoreans one step closer to a detailed metaphysic of order.

Figure 5

Another idea that played a central role in the Pythagorean world was that of
cosmic opposites
. Since the pattern of opposites was the underlying principle of the early Ionian scientific tradition, it was only natural for the order-obsessed Pythagoreans to adopt it. In fact, Aristotle tells us that even a medical doctor named Alcmaeon, who lived in Croton at the same time that Pythagoras had his famous school there, subscribed to the notion that all things are balanced in pairs. The principal pair of opposites consisted of the
limit,
represented by the odd numbers, and the
unlimited,
represented by the even. The limit was the force that introduces order and harmony into the wild, unbridled unlimited. Both the complexities of the universe at large and the intricacies of human life, microcosmically, were thought to consist of and be directed by a series of opposites that somehow fit together. This rather black-and-white vision of the world was summarized in a “table of opposites” that was preserved in Aristotle’s
Metaphysics:

The basic philosophy expressed by the table of opposites was not confined to ancient Greece. The Chinese yin and yang, with the yin rep
resenting negativity and darkness and the yang the bright principle, depict the same picture. Sentiments that are not too different were carried over into Christianity, through the concepts of heaven and hell (and even into American presidential statements such as “You are either with us, or you are with the terrorists”). More generally, it has always been true that the meaning of life has been illuminated by death, and of knowledge by comparing it to ignorance.

Not all the Pythagorean teachings had to do directly with numbers. The lifestyle of the tightly knit Pythagorean society was also based on vegetarianism, a strong belief in metempsychosis—the immortality and transmigration of souls—and a somewhat mysterious ban on eating beans. Several explanations have been suggested for the bean-eating prohibition. They range from the resemblance of beans to genitals to bean eating being compared to eating a living soul. The latter interpretation regarded the wind breaking that often follows the eating of beans as proof of an extinguished breath. The book
Philosophy for Dummies
summarized the Pythagorean doctrine this way: “Everything is made of numbers, and don’t eat beans because they’ll do a number on you.”

The oldest surviving story about Pythagoras is related to the belief in the reincarnation of the soul into other beings. This almost poetic tale comes from the sixth century BC poet Xenophanes of Colophon: “They say that once he [Pythagoras] passed by as a dog was being beaten, and pitying it spoke as follows, ‘Stop, and beat it not; for the soul is that of a friend; I know it, for I heard it speak.’”

Pythagoras’s unmistakable fingerprints can be found not only in the teachings of the Greek philosophers that immediately succeeded him, but all the way into the curricula of the medieval universities. The seven subjects taught in those universities were divided into the
trivium,
which included dialectic, grammar, and rhetoric, and the
quadrivium,
which included the favorite topics of the Pythagoreans—geometry, arithmetic, astronomy, and music. The celestial “harmony of the spheres”—the music supposedly performed by the planets in their orbits, which, according to his disciples, only Pythagoras could hear—has inspired poets and scientists alike. The famous astronomer Johannes Kepler (1571–1630), who discovered
the laws of planetary motion, chose the title of
Harmonice Mundi
(
Harmony of the World
) for one of his most seminal works. In the Pythagorean spirit, he even developed little musical “tunes” for the different planets (as did the composer Gustav Holst three centuries later).