Pythagoras’s triangular numbers.
DIVINE PROPERTIES OF NUMBERS
The second-century historian Lucian recounts how Pythagoras connected this property of the natural numbers with the Pythagoreans’ number worship. One day he asked a member of his sect to count. The man began: 1, 2, 3 … When he reached 4, Pythagoras interrupted him and said, “Do you see? What you take for 4 is 10, a perfect triangle, and our oath.”13 Indeed, to the Pythagoreans 10 was a very special number.
Pythagoras and his followers saw the number 1 as the generator of all other numbers and the embodiment of reason. Two, the first even number, was considered female, representing opinion. Three was the first “true male” number, representing harmony because it incorporated both unity (1) and diversity (2). Four represented justice or retribution, since it was associated with the “squaring of accounts.” Being the union of the first true male number (3) and the first female number (2), 5 represented marriage. The number 6 represented creation (and is the first “perfect number,” as we will soon see), and 7 was the number of the Wandering Stars. (In addition to the sun and moon, the Pythagoreans knew of only five planets—Mars, Mercury, Jupiter, Venus, and Saturn—for which the days of the week are named. Sunday, Monday, and Saturday are obvious; for Tuesday through Friday, you can see the correspondence to the planets in their French forms: Mardi, Mercredi, Jeudi, and Vendredi.)
The number 10 was considered the holiest of holies—hence, Pythagoras’s statement in the story above. It even had a special name, tetractys, from the Greek word for four (tetra), in reference to the number of dots to a side in the number’s triangular form. Ten represented the universe as a whole, as well as the sum of the numbers that generate all the possible dimensions of the space we live in. (The number 1 generates all dimensions; 2 generates a line, since a line is created by the joining of two points; 3 generates a plane, since three points not all on a line determine a triangle—i.e., a two-dimensional figure—when joined together; four points, not all on a plane, generate a three-dimensional figure and, hence, three-dimensional space. And 1 + 2 + 3 + 4 = 10.) Pythagoras and his followers called the number 10 their “greatest oath,” as well as “the principle of health.”14 Of course, 10 is also the number of fingers and toes we have, from which fact our entire 10-based number system evolved and eventually superseded the base-60 system of the Babylonians and Assyrians. Equally, vestiges of a base-20 system (presumably emerging from the fact that, together, we have 20 fingers and toes) are still evident in the French language, where the word for 80 is quatre-vingt (four twenties).
Pythagoras was also interested in square numbers—like the triangular numbers, another set of “geometrical” numbers. As triangular numbers form triangles, square numbers can similarly be arranged to form squares. The first square number is 1 (by default, assuming it forms a square rather than a circle; indeed, 12 = 1). The next square number is 4, then 9, then 16, and so on. If we draw these numbers as a two-dimensional figure, as the Pythagoreans did, we see the pattern in the figure.
To proceed from one square number to the next, we add the two sides of a square and add one. For example, to proceed from 42 to 52, we need to add (2 × 4) + 1 to 42 (16). Indeed, 52 (25) is equal to 16 + (2 × 4) + 1. Therefore, we can represent every square number as a sum of odd numbers: (n + 1)2 = 1 + 3 + 5 + … + (2n + 1), where n is an integer.
Pythagoras’s square numbers.
In their search for mystical properties of numbers, the Pythagoreans defined a perfect number as a number that is “equal to [the sum of] its own parts.” In other words, a perfect number is equal to the sum of all its multiplicative factors, excluding itself but including 1. The first perfect number is 6, because 6 = 6 × 1 and 2 × 3. As it happens: 6 = 1 + 2 + 3. The next perfect number is 28, since 28 = 1 + 2 + 4 + 7 + 14. The number 496 is also perfect. How do we know?
A few centuries later Euclid, the famous Greek mathematician, proved that if the sum of any number of terms of the series 1, 2, 22, 23 … 2n-1 is a prime number, then that sum multiplied by 2n-1 is a perfect number. For example, for n = 3: 20 + 21 + 22 = 1 + 2 + 4 = 7, a prime number. Therefore, 7 multiplied by 2(3-1) must be a perfect number. It is, as 7 × 4 = 28. For n = 4, 1 + 2 + 4 + 8 = 15, which is not prime. For n = 5, however, 1 + 2 + 4 + 8 + 16 = 31, a prime number, so 31 × 16 should equal the next perfect number, 496.
The ancient Greeks also determined the following perfect number, 8,128, and began to notice a pattern: a perfect number always ends either in a 6 or in an 8. But further perfect numbers were beyond their computational ability—the fifth perfect number, 33,350,336, is very large, and the next one, 8,589,869,056, is in the billions. The ninth perfect number has 37 digits!
Pythagoras was once asked by a disciple, “What is a friend?” He replied, “A friend is an alter ego.” This led him to define the concept of friendship for numbers as well, defining two numbers as being “friends” if each one is the sum of the multiplicative factor of the other number. Hence, the numbers 284 and 220 are friends. Why? 284 = 1 + 2 + 4 + 5 + 10 + 11 + 20 + 22 + 44 + 55 + 110, which are all of the factors of 220, while 220 = 1 + 2 + 4 + 71 + 142, which are all of the factors of 284.
Pythagoras and his followers also understood fractions, such as 2/7 or 31/77. We call such numbers rational numbers, perhaps because they make sense to us. A pie can be divided into seven pieces, each being 1/7 of the whole, and you can give someone two pieces—a fraction of the entire pie represented by the number 2/7. But when the Pythagoreans went further in their mathematical and mystical exploration of numbers and their properties, they ran into a conundrum that stunned them and perhaps even brought on their demise. This paradox—which came about in the interface between geometry and arithmetic—would come to a head with the work of Georg Cantor in the nineteenth century, and it continues to haunt us even today.
Raphael’s famous fresco The School of Athens (1510–11) echoes the notion of Plato’s Academy as an intellectual community of learned scholars. Among the thinkers depicted in the painting is Plato himself, who stands at the center of the image, pointing upward.
TWO
PLATO’S ACADEMY
Despite having so little foundational work to build on, the mathematicians of ancient Greece were able to make advances in mathematics that stun us today. Thanks to the pioneering work of Thales and Pythagoras, abstract mathematics became what we know it to be: a science—and, in many ways, an art—based on pure logic and consisting of results we call theorems (and lemmas and corollaries that precede and follow them, respectively), which must be proved. Many of their discoveries were so fundamental that they seem “modern” in their depth of meaning and influence, and as mathematics became interlinked with philosophy, it attracted the attention of the greatest philosopher of the time: Plato.
HIPPASUS OF METAPONTUM
Plato’s dialogues show that he and the members of his Academy in Athens had been stunned by a discovery about the nature of numbers that made their philosophical view that “God is number”—where “number” means a whole number or a fraction made of two integers—shaky. The discovery was made by members of the Pythagorean order sometime before 410 BCE. Some early historians attribute this shocking discovery to the Pythagorean mathematician Hippasus of Metapontum (another name for Crotona, the Pythagoreans’ centuries-long abode). Hippasus was reportedly expelled from the order—or worse—because of what happened next.1
In addition to the study of numbers, the Pythagoreans did much work on pure geometry, as initiated by the work of the great Thales and pursued very actively by Pythagoras himself. Hoping to associate numbers with lengths in geometry, an objective that seemed simple enough, Hippasus drew a square in the sand. As soon as a diagonal was drawn inside the square, however, the question arose: If the length of the side of the square is one unit, what is the length of the diagonal? The result of this elementary investigation would, in fact, change the world of mathematics.
Today nearly every schoolchild knows the famous Pythago
rean theorem, which states that the sums of the squares of the two sides of a triangle that share a right angle between them is equal to the square of the hypotenuse.
So when Hippasus drew a square in the sand and added the diagonal, it was clear to all witnesses that the diagonal was the hypotenuse of a right triangle, with opposite sides equal to one unit each. Based on the Pythagorean theorem, the length of the hypotenuse must be the square root of 2.
But what is the square root of 2? We know that the square root of a square number is an integer (e.g., the square root of 4 is 2, the square root of 9 is 3, and so on). Two is not a square number—so what is its square root? Until that point, the Pythagoreans knew of only (positive) integers and what we call rational numbers: fractions made of two integers—a numerator and a denominator. The square root of 2 is clearly not an integer, since 2 is not a square number, so the Pythagoreans assumed that it was a rational number. They searched in vain for the two integers that made up the numerator and denominator of the fraction but came up empty-handed. It can be proved mathematically that such integers do not exist.2
It is believed that Hippasus broke the Pythagoreans’ code of secrecy by revealing to the outside world the existence of numbers that could not be written as fractions of integers. One story has him simply expelled from the order for this crime, but according to another story, the Pythagoreans erected a tombstone with his name on it and presumably killed him. According to yet another version, he was forced on a sea voyage from which he did not return. All this for revealing the existence of irrational numbers, which comprise many other square roots, higher-order roots (e.g., the cube root of 2), the natural numbers π and e, and infinitely many others. We will learn about these numbers in detail when we arrive at the nineteenth century.
ANAXAGORAS OF CLAZOMENAE
Anaxagoras of Clazomenae lived in the fifth century BCE. At the beginning of this century, the Persian invaders of Greece were defeated, and at the end of the century, Athens was defeated by Sparta. The period between these two key events is called the Age of Pericles. Known for its great achievements in art and literature—including the amazing statues and plays that form much of the foundation of Western composition—it was an age of peace and prosperity, as well as flourishing intellectual activity.
Athens attracted the greatest mathematicians of its vast dominion. Zeno of Elea (in southern Italy) gave us the famous paradoxes about time, space, and infinity, including the notable one about Achilles and the tortoise, in which Achilles races against the tortoise but cannot win because each time he covers half the distance to the tortoise, the tortoise has already advanced farther, and when Achilles has covered half that distance, the tortoise has advanced again, and so on. This period was also the era of Democritus, who proposed that the universe is made of atoms—something that would be proved by scientists almost two and a half millennia later.
A deep thinker who concerned himself with uncovering the structure of the cosmos, Anaxagoras could well be called one of the earliest natural philosophers in history. His birth date is unknown, but we know that he died in the year 428 BCE and that he came from Clazomenae in Ionia, a region of western Greece that includes the islands of Corfu and Ithaca. After acquiring a degree of fame for his novel views about nature and the universe, Anaxagoras became a tutor to the great leader Pericles, but his ideas were so ahead of his time that he was frequently imprisoned for heresy.
Anaxagoras, perhaps the first mathematician to tackle one of the three classical problems of antiquity, is portrayed in this detail from the 1888 fresco Philosophers of Athens.
Anaxagoras shocked many Athenians by publicizing his view that the sun was not a god but rather a flaming rock in the sky that was as large as the entire Peloponnesian peninsula. He contended that the moon was another planet, like Earth, and that it had its own population and reflected light from the sun. Many people took offense at his unprecedented theories, and accusations of heresy led to his imprisonment. Anaxagoras’s friend and protector Pericles eventually forced his release from prison, but the more he spouted his ideas, the longer he found himself behind bars.
During his imprisonment, Anaxagoras used to while away the time by working on a mathematical problem that no one could solve. According to Plutarch, he was trying to square the circle. Plutarch’s account is the first mention in history of a problem that would consume the time and effort of many a mathematician until proved impossible in the nineteenth century. What is so beautiful about this problem is that it is a purely intellectual one—as was most of Greek mathematics.
THE THREE CLASSICAL
PROBLEMS OF ANTIQUITY
Squaring the circle is the first of the celebrated “three classical problems of antiquity.” All three problems are now known to be impossible, thanks to the work of Évariste Galois (1811–32), whose genius and tragic life are discussed later in this book. The problem of squaring the circle presented the challenge of constructing a square with the same area as the area of a given circle. Later sources than Plutarch tell us that only a straightedge and a compass—the two tools that Greek construction employed—were allowed to be used in this effort. Although Anaxagoras is best known for his work in natural philosophy, he is credited with the first known attempt to solve this problem.
The second of the three famous problems is called doubling the cube. Also known as the Delian problem, the story of its genesis is both stirring and sinister. In 429 BCE a great plague raged in Athens, killing a quarter of the city’s population. Persistent and recurring, the scourge is believed to have claimed the life of Pericles, and as we will see, it formed the basis from which the famous second classical problem of antiquity emerged.
The island of Delos lies in the Aegean Sea, a body of water in the eastern Mediterranean that stretches from mainland Greece down to Crete. A small, now arid island, it is the navel of the Cyclades, a chain of islands that appears to form a “cycle” around it. Because of its special position in the center of the islands, the ancient Greeks considered Delos to be a holy place, and each of the Greek city-states built temples dedicated to their gods on the tiny landmass. Ruins of the many temples built on this island can still be seen today. (The island is easily accessible by boat from nearby Mykonos.)
When the plague first began to spread in Athens around 430 BCE, the Athenians scrambled to try to save themselves from what they viewed as the wrath of the gods. They sent a delegation to Delphi, located on the Greek mainland not far from Athens, to ask the oracle to intercede with the gods on their behalf. The oracle came back with the pronouncement: “Apollo wants you to double his temple on Delos.”
The Athenians frantically set to work. They doubled the length, width, and height of the Athenian temple to Apollo on Delos. But the plague continued to rage. So again the Athenians sent a delegation to the Delphic oracle to find out why Apollo was still angry with them and to plead for an end to the plague—after all, they had done exactly what he had asked them to do. “No,” replied the oracle. “You haven’t done as the god had instructed you to do. Go back to Delos and do as he has commanded you!”
The Athenian masons and engineers soon realized why they had failed: Apollo wanted the volume of his cubic temple doubled—not all its dimensions. By doubling each side of this cube-shaped temple, what they had inadvertently done was to increase the volume eightfold (2 × 2 × 2 = 23 = 8). What they needed to do in order to double the volume of the temple while maintaining its cubic dimensions was to increase the length, width, and height of the cube by a factor equal to the third root of 2. Only this way could a cubic volume be doubled, since (21/3)3 = 2. The engineers, masons, and builders realized that they needed to go back to the original cubic temple and, using the only tools of their trade—a straightedge and a compass—expand each of the three dimensions so that its length would increase by a factor of exactly the cube root of 2. But it couldn’t be done, and the plague raged on. As we will see, the work of Évariste Galois in the nineteenth century also proved that
the Delian problem inspired by this story—the problem of doubling the volume of a given cube using only a straightedge and a compass; i.e., to construct, geometrically, a length that is the cube root of 2 times as large as a given length—is impossible.
The third of the so-called classical problems of antiquity is a problem that, like the two discussed above, was circulating in intellectual circles in Athens at that time. It is the problem of trisecting an arbitrary angle using only a straightedge and a compass. With these tools, some angles can be trisected, meaning that given an angle, one can sometimes construct an angle that is a third as large. But the problem is to be able to do this trisection with any given angle. Archimedes, discussed later in this chapter, was able to use the famous Archimedean spiral to trisect angles, but his method required more than just a straightedge and compass, so he failed to solve the original problem.
PLATO
The Greek philosopher Plato (428–348 BCE) was born just a year after the infamous plague. He was not a mathematician, but he believed that mathematics was the study of truth. To emphasize this view, Plato placed a sign over the gate of his Academy in Athens: LET NO ONE IGNORANT OF GEOMETRY ENTER HERE! Plato thus became known as the maker of mathematicians, and he encouraged many geometers and algebraists to join what would later be considered the first center of philosophy and knowledge in the world.
Despite Hippasus’s discovery that not all numbers were expressible as ratios of natural numbers, Plato was undeterred in his love of numbers and in his belief in their divinity, and he remained faithful to the Pythagorean ideal of number until death.3
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