Is God a Mathematician_ - Mario Livio [12]
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 32, to a 4 × 4 square makes a new, 5 × 5 square: 32 + 42 = 52. 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 52 122 132) 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:
Limit
Unlimited
Odd
Even
One
Plurality
Right
Left
Male
Female
Rest
Motion
Straight
Curved
Light
Darkness
Good
Evil
Square
Oblong
The basic philosophy expressed by the table of opposites was not confined to ancient Greece. The Chinese yin and yang, with the yin representing 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