Is God a Mathematician?

Home > Other > Is God a Mathematician? > Page 3
Is God a Mathematician? Page 3

by Mario Livio


  There is little doubt that Pythagoras was born in the early sixth century BC on the island of Samos, just off the coast of modern-day Turkey. He may have traveled extensively early in life, especially to Egypt and perhaps Babylon, where he would have received at least part of his mathematical education. Eventually he emigrated to a small Greek colony in Croton, near the southern tip of Italy, where an enthusiastic group of students and followers quickly gathered around him.

  The Greek historian Herodotus (ca. 485–425 BC) referred to Pythagoras as “the most able philosopher among the Greeks,” and the pre-Socratic philosopher and poet Empedocles (ca. 492–432 BC) added in admiration: “But there was among them a man of prodigious knowledge, who acquired the profoundest wealth of understanding and was the greatest master of skilled arts of every kind; for whenever he willed with all his heart, he could with ease discern each and every truth in his ten—nay, twenty men’s lives.” Still, not all were equally impressed. In comments that appear to stem from some personal rivalry, the philosopher Heraclitus of Ephesus (ca. 535–475 BC) acknowledges Pythagoras’s broad knowledge, but he is also quick to add disparagingly: “Much learning does not teach wisdom; otherwise it would have taught Hesiod [a Greek poet who lived around 700 BC] and Pythagoras.”

  Pythagoras and the early Pythagoreans were neither mathematicians nor scientists in the strict sense of these terms. Rather, a metaphysical philosophy of the meaning of numbers lay at the heart of their doctrines. To the Pythagoreans, numbers were both living entities and universal principles, permeating everything from the heavens to human ethics. In other words, numbers had two distinct, complementary aspects. On one hand, they had a tangible physical existence; on the other, they were abstract prescriptions on which everything was founded. For instance, the monad (the number 1) was understood both as the generator of all other numbers, an entity as real as water, air, and fire that participated in the structure of the physical world, and as an idea—the metaphysical unity at the source of all creation. The English historian of philosophy Thomas Stanley (1625–78) described beautifully (if in seventeenth century English) the two meanings that the Pythagoreans associated with numbers:

  Number is of two kinds the Intellectual (or immaterial) and the Sciential. The Intellectual is that eternal substance of Number, which Pythagoras in his Discourse concerning the Gods asserted to be the principle most providential of all Heaven and Earth, and the nature that is betwixt them…This is that which is termed the principle, fountain, and root of all things…Sciential Number is that which Pythagoras defines as the extension and production into act of the seminal reasons which are in the Monad, or a heap of Monads.

  So numbers were not simply tools to denote quantities or amounts. Rather, numbers had to be discovered, and they were the formative agents that are active in nature. Everything in the universe, from material objects such as the Earth to abstract concepts such as justice, was number through and through.

  The fact that someone would find numbers fascinating is perhaps not surprising in itself. After all, even the ordinary numbers encountered in everyday life have interesting properties. Take the number of days in a year—365. You can easily check that 365 is equal to the sums of three consecutive squares: 365=102 + 112 + 122. But this is not all; it is also equal to the sum of the next two squares (365 132 + 142)! Or, examine the number of days in the lunar month—28. This number is the sum of all of its divisors (the numbers that divide it with no remainder): 28= 1 + 2 + 4 + 7 + 14. Numbers with this special property are called perfect numbers (the first four perfect numbers are 6, 28, 496, 8218). Note also that 28 is the sum of the cubes of the first two odd numbers: 28 = 13 + 33. Even a number as widely used in our decimal system as 100 has its own peculiarities: 100 = 13 + 23 + 33 + 43.

  OK, so numbers can be intriguing. Still, one may wonder what was the origin of the Pythagorean doctrine of numbers? How did the idea arise that not only do all things possess number, but that all things are numbers? Since Pythagoras either wrote nothing down or his writings have been destroyed, it is not easy to answer this question. The surviving impression of Pythagoras’s reasoning is based on a small number of pre-Platonic fragments and on much later, less reliable discussions, mostly by Platonic and Aristotelian philosophers. The picture that emerges from assembling the different clues suggests that the explanation of the obsession with numbers may be found in the preoccupation of the Pythagoreans with two apparently unrelated activities: experiments in music and observations of the heavens.

  To understand how those mysterious connections among numbers, the heavens, and music materialized, we have to start from the interesting observation that the Pythagoreans had a way of figuring numbers by means of pebbles or dots. For instance, they arranged the natural numbers 1, 2, 3, 4,…as collections of pebbles to form triangles (as in figure 1). In particular, the triangle constructed out of the first four integers (arranged in a triangle of ten pebbles) was called the Tetraktys (meaning quaternary, or “fourness”), and was taken by the Pythagoreans to symbolize perfection and the elements that comprise it. This fact was documented in a story about Pythagoras by the Greek satirical author Lucian (ca. AD 120–80). Pythagoras asks someone to count. As the man counts “1, 2, 3, 4,” Pythagoras interrupts him, “Do you see? What you take for 4 is 10, a perfect triangle and our oath.” The Neoplatonic philosopher Iamblichus (ca. AD 250–325) tells us that the oath of the Pythagoreans was indeed:

  Figure 1

  I swear by the discoverer of the Tetraktys,

  Which is the spring of all our wisdom,

  The perennial root of Nature’s fount.

  Why was the Tetraktys so revered? Because to the eyes of the sixth century BC Pythagoreans, it seemed to outline the entire nature of the universe. In geometry—the springboard to the Greeks’ epochal revolution in thought—the number 1 represented a point •, 2 represented a line , 3 represented a surface , and 4 represented a three-dimensional tetrahedral solid . The Tetraktys therefore appeared to encompass all the perceived dimensions of space.

  But that was only the beginning. The Tetraktys made an unexpected appearance even in the scientific approach to music. Pythagoras and the Pythagoreans are generally credited with the discovery that dividing a string by simple consecutive integers produces harmonious and consonant intervals—a fact figuring in any performance by a string quartet. When two similar strings are plucked simultaneously, the resulting sound is pleasing when the lengths of the strings are in simple proportions. For instance, strings of equal length (1:1 ratio) produce a unison; a ratio of 1:2 produces the octave; 2:3 gives the perfect fifth; and 3:4 the perfect fourth. In addition to its all-embracing spatial attributes, therefore, the Tetraktys could also be seen as representing the mathematical ratios that underlie the harmony of the musical scale. This apparently magical union of space and music generated for the Pythagoreans a powerful symbol and gave them a feeling of harmonia (“fitting together”) of the kosmos (“the beautiful order of things”).

  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 12; 1 3 4 22; 1 3 5 9 32; 1 3 5 7 16 42; 1 3 5 7 9 25 52, 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 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 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).

  From the perspective of the questions that are at the focus of the present book, once we strip the Pythagorean philosophy of its mystical clothing, the skeleton that remains is still a powerful statement about mathematics, its nature, and its relation to both the physical world and the human mind. Pythagoras and the Pythagoreans were the forefathers of the search for cosmic order. They can be regarded as the founders of pure mathematics in that unlike their predecessors—the Babylonians and the Egyptians—they engaged in mathematics as an abstract field, divorced from all practical purposes. The question of whether the Pythagoreans also established mathematics as a tool for science is a trickier one. While the Pythagoreans certainly associated all phenomena with numbers, the numbers themselves—not the phenomena or their causes—became the focus of study. This was not a particularly fruitful direction for scientific research to take. Still, fundamental to the Pythagorean doctrine was the implicit belief in the existence of general, natural laws. This belief, which has become the central pillar of modern science, may have had its roots in the concept of Fate in Greek tragedy. As late as the Renaissance, this bold faith in the reality of a body of laws that can explain all phenomena was still progressing far in advance of any concrete evidence, and only Galileo, Descartes, and Newton turned it into a proposition defendable on inductive grounds.

 

‹ Prev