Aristotle simply declared that mathematicians “do not need the infinite, or use it.” Though “potential” infinities could exist in the minds of mathematicians—like the concept of dividing lines into infinite pieces—nobody could actually do it, so the infinite doesn’t exist in reality. Achilles runs smoothly past the tortoise because the infinite points are simply a figment of Zeno’s imagination, rather than a real-world construct. Aristotle just wished infinity away by stating that it is simply a construct of the human mind.
From that concept comes a startling revelation. Based upon the Pythagorean universe, the Aristotelian cosmos (and its later refinement by the astronomer Ptolemy) had the planets moving in crystalline orbs. However, since there is no infinity, there can’t be an endless number of spheres; there must be a last one. This outermost sphere was a midnight blue globe encrusted with tiny glowing points of light—the stars. There was no such thing as “beyond” the outermost sphere; the universe ended abruptly with that outermost layer. The universe was contained in a nutshell, ensconced comfortably within the sphere of fixed stars; the cosmos was finite in extent, and entirely filled with matter. There was no infinite; there was no void. There was no infinity; there was no zero.
This line of reasoning had another consequence—and this is why Aristotle’s philosophy endured for so many years. His system proved the existence of God.
The heavenly spheres are slowly spinning in their places, making a music that suffuses the cosmos. But something must be causing that motion. The stationary earth cannot be the source of that motive power, so the innermost sphere must be moved by the next sphere out. That sphere, in turn, is moved by its larger neighbor, and on and on. However, there is no infinity; there are a finite number of spheres, and a finite number of things that are moving each other. Something must be the ultimate cause of motion. Something must be moving the sphere of fixed stars. This is the prime mover: God. When Christianity swept through the West, it became closely tied to the Aristotelian view of the universe and the proof of God’s existence. Atomism became associated with atheism. Questioning the Aristotelian doctrine was tantamount to questioning God’s existence.
Aristotle’s system was extremely successful. His most famous student, Alexander the Great, spread the doctrine as far east as India before Alexander’s untimely death in 323 BC. The Aristotelian system would outlast Alexander’s empire; it would survive until Elizabethan times, the sixteenth century. With this long-standing acceptance of Aristotle came a rejection of the infinite—and the void, for Aristotle’s denial of the infinite required a denial of the void, because the void implies the existence of the infinite. After all, there were only two logical possibilities for the nature of the void, and both implied that the infinite exists. First, there could be an infinite amount of void—thus infinity exists. Second, there could be a finite amount of void, but since void is simply the lack of matter, there must be an infinite amount of matter to make sure that there is only a finite amount of void—thus infinity exists. In both cases the existence of the void implies the existence of the infinite. Void/zero destroys Aristotle’s neat argument, his refutation of Zeno, and his proof of God. So as Aristotle’s arguments were accepted, the Greeks were forced to reject zero, void, the infinite, and infinity.
There was a problem, though. It is not so easy to reject both infinity and zero. Look back through time. Events have happened throughout history, but if there is no such thing as infinity, there cannot be an infinite number of events. Thus, there must be a first event: creation. But what existed before creation? Void? That was unacceptable to Aristotle. Conversely, if there was not a first event, then the universe must have always existed—and will always exist in the future. You’ve got to have either infinity or zero; a universe without both of them makes no sense.
Aristotle hated the idea of the void so much that he chose the eternal, infinite universe over one that had a vacuum in it; he said that the eternity of time was a “potential” infinity like Zeno’s infinite subdivisions. (It was a stretch, but many scholars bought the argument; some even chose the creation story as further evidence for God. Medieval philosophers and theologians were doomed to battle over this puzzle for several hundred years.)
The Aristotelian view of physics, as wrong as it was, was so influential that for more than a millennium it eclipsed all opposing views, including more realistic ones. Science would never progress until the world discarded Aristotle’s physics—along with Aristotle’s rejection of Zeno’s infinities.
For all his wit, Zeno got himself into serious trouble. Around 435 BC, he conspired to overthrow the tyrant of Elea, Nearchus. He was smuggling arms to support the cause. Unfortunately for Zeno, Nearchus found out about the plot, and Zeno was arrested. Hoping to discover who the coconspirators were, Nearchus had Zeno tortured. Soon Zeno begged the torturers to stop and promised that he would name his colleagues. When Nearchus drew near, Zeno insisted that the tyrant come closer, since it was best to keep the names a secret. Nearchus leaned over, tilting his head toward Zeno. All of a sudden Zeno sank his teeth into Nearchus’s ear. Nearchus screamed, but Zeno refused to let go. The torturers could only force Zeno to let go by stabbing him to death. Thus died the master of the infinite.
Eventually, one ancient Greek surpassed Zeno in matters of the infinite: Archimedes, the eccentric mathematician of Syracuse. He was the only thinker of his day to glimpse the infinite.
Syracuse was the richest city on the island of Sicily, and Archimedes was its most famous resident. Little is known about his youth, but it seems that Archimedes was born around 287 BC in Samos, Pythagoras’s birthplace. He then immigrated to Syracuse, where he solved engineering problems for the king. It was the king of Syracuse who asked Archimedes to determine whether his crown was pure gold or had been mixed with lead, a task beyond the abilities of all the scientists at the time. However, when Archimedes settled into a tub of water, he noticed that the water flowed over the sides, and he suddenly realized that he could measure the density of the crown, and thus its purity, by submerging it in a tub of water and measuring how much water it displaced. Elated by the insight, Archimedes leapt out of the tub and ran through the streets of Syracuse shouting, “Eureka! Eureka!” Of course Archimedes forgot that he was stark naked.
Archimedes’ talents were useful to the Syracusan military as well. In the third century BC the era of Greek hegemony was over. Alexander’s empire had collapsed into bickering states, and a new power was flexing its muscles in the West: Rome. And Rome had set its sights on Syracuse. According to legend, Archimedes armed the Syracusans with miraculous weapons to defend the city from the Romans: stone throwers; huge cranes that grabbed the Roman warships, lifted them up, and dumped them, bow first, into the water; and mirrors made of such quality that they set Roman ships afire at a great distance by reflecting sunlight. The Roman soldiers were so afraid of these war engines that if they saw so much as a bit of rope or wood sticking over the wall they would flee for fear that Archimedes was aiming a weapon at them.
Archimedes first glimpsed the infinite in the polish of his war mirrors. For centuries the Greeks had been fascinated with conic sections. Take a cone and cut it up; you get circles, ellipses, parabolas, and hyperbolas, depending on how you slice it. The parabola has a special property: it takes the rays of light from the sun, or any distant source, and focuses them to a point, concentrating all the light’s energy on a very small area. Any mirror that could set ships afire must be in the shape of a parabola. Archimedes studied the properties of the parabola, and it is here that he first started toying with the infinite.
To understand the parabola, Archimedes had to learn how to measure it; for instance, nobody knew how to determine the area of a section of a parabola. Triangles and circles were easy to measure, but slightly more irregular curves like the parabola were beyond the ken of the Greek mathematicians of the day. However, Archimedes figured out a way to measure the parabola’s area by resorting to the infinite. The first ste
p was to inscribe a triangle inside the parabola. In the two little gaps left, Archimedes inscribed more triangles. This left four gaps, which were filled with more triangles, and so on (Figure 12). It’s like Achilles and the tortoise—an infinite series of steps, each getting smaller and smaller. The areas of the little triangles quickly approach zero. After a long, involved set of calculations, Archimedes summed the areas of the infinite triangles and divined the area of the parabola. However, any mathematician of the day would have scoffed at this line of reasoning; Archimedes used the tools of the infinite, which were so expressly disallowed by his mathematical colleagues. To satisfy them, Archimedes also included a proof, based upon the accepted mathematics of the time, that relied upon the so-called axiom of Archimedes, although Archimedes himself mentioned that earlier mathematicians deserved the credit. As you may recall, this axiom says that any number added to itself over and over again can exceed any other number. Zero, clearly, was not included.
Figure 12: Archimedes’ parabola
Archimedes’ proof by triangles was as close as you could come to the idea of limits—and calculus—without actually discovering them. In later works Archimedes figured out the volumes of parabolas and circles, rotated around a line, which any math student knows are early homework problems in a calculus course. But the axiom of Archimedes rejected zero, which is the bridge between the realms of the finite and the infinite, a bridge that is absolutely necessary for calculus and higher mathematics.
Even the brilliant Archimedes occasionally scorned the infinite along with his contemporaries. He believed in the Aristotelian universe; the universe was contained within a giant sphere. On a whim he decided to calculate how many grains of sand could fit in the (spherical) universe. In his “Sand Reckoner,” Archimedes first calculated how many grains of sand would fit across a poppy seed, and then how many poppy seeds would span a finger’s breadth. From finger breadths to the length of a stadium (the standard Greek unit of long distance) and on to the size of the universe, Archimedes figured out that 1051 grains of sand would fill the entire universe, packed full even unto the outermost sphere of fixed stars. (1051 is a really, really big number. Take 1051 molecules of water, for instance. It would take every man, woman, and child now on Earth, each drinking a ton of water a second, over 150,000 years to drink it all.) This number was so large that the Greek system of numeration was unable to handle it; Archimedes had to invent a whole new method of denoting really huge numbers.
The myriad was the biggest grouping in the Greek number system, and by counting myriads the Greeks could denote numbers up to a myriad myriads (100,000,000) and a little further. But Archimedes got beyond this limitation by hitting the reset button. He simply started over at a myriad myriads, setting 100,000,000 equal to one, and counting again, calling these new numbers, numbers of the second order. (Archimedes did not set 100,000,001 equal to one and 100,000,000 equal to zero, which is what a modern mathematician would do. It had never occurred to Archimedes that starting over with zero would make more sense.) The numbers of the second order went from one myriad myriads to a myriad myriad myriad myriads. The numbers of the third order went to one myriad myriad myriad myriad myriad myriads (1,000,000,000,000,000,000,000,000), and so forth, until he reached the numbers of the myriad myriadth order, which he called the numbers of the first period. It was a cumbersome way of doing business, but it got the job done and went far beyond what Archimedes needed to solve his thought experiment. Yet as big as those numbers were, they were finite—and were more than enough to fill the universe to overflowing with sand. The infinite was not needed in the Greek universe.
Perhaps, given more time, Archimedes might have begun to see the lure of the infinite and of zero. But the sand reckoner was destined to meet his fate while reckoning in the sand. The Romans were too powerful for the Syracusans. Taking advantage of a poorly manned watchtower and an easy-to-climb wall, the Romans managed to get some soldiers inside the city. As soon as they realized that Romans were inside the walls of the city, the Syracusans, wild with fear, could not mount a defense. The Romans poured through the city, but Archimedes was deaf to the panic around him. He sat on the ground, drawing circles in the sand, trying to prove a theorem. A Roman soldier saw the bedraggled 75-year-old and demanded that Archimedes follow him. Archimedes refused, since his mathematical proof was not yet finished. The enraged soldier cut him down. Thus died the greatest mind in the ancient world, slaughtered needlessly by the Romans.
Killing Archimedes was one of the biggest Roman contributions to mathematics. The Roman era lasted for about seven centuries. In all that time there were no significant mathematical developments. History marched on: Christianity swept through Europe, the Roman Empire fell, the Library at Alexandria burned, and the Dark Ages began. It would be another seven centuries before zero reappeared in the West. In the meantime two monks created a calendar without zero, damning us to eternal confusion.
Blind Dates
It is a silly, childish discussion and only exposes the want of brains of those who maintain a contrary opinion to that we have stated.
—THE TIMES (LONDON), DECEMBER 26, 1799
This “silly, childish discussion”—whether the new century begins on the year 00 or the year 01—appears and reappears like clockwork every hundred years. If medieval monks had only known of zero, our calendar would not be in such a muddle.
The monks can’t be faulted for their ignorance. Indeed, during the Middle Ages the only Westerners who studied math were the Christian monks. They were the only learned ones left. Monks needed math for two things: prayer and money. To count money, they needed to know how to…well…count. To do this, they used an abacus or a counting board, a device similar to an abacus, where stones or other counters get moved around on a table. It was not a very demanding task, but by ancient standards it was the state of the art. To pray, monks needed to know the time and the date. As a result, timekeeping was vital to the monks’ rituals. They had different prayers to be said at different hours, day in and day out. (Our word noon comes from the word nones, the midday prayer service of medieval clergy.) How else would the night watchman know when to roust his fellows out of their comfortable straw beds to begin the day’s devotions? And if you didn’t have an accurate calendar, you couldn’t know when to celebrate Easter. This was a big problem.
Calculating the date of Easter was no mean feat, thanks to a clash of calendars. The seat of the church was Rome, and Christians used the Roman solar calendar that was 365 days (and change) long. But Jesus was a Jew, and he used the Jewish lunar calendar that was only 354 days (and change) long. The big events in Jesus’ life were marked with reference to the moon, while everyday life was ruled by the sun. The two calendars drifted with respect to each other, making it very difficult to predict when a holiday was due. Easter was just such a drifting holiday, so every few generations a monk was drafted to calculate the dates when Easter would fall for the next few hundred years.
Dionysius Exiguus was one of these monks. In the sixth century the pope, John I, asked him to extend the Easter tables. While translating and recalculating the tables, Dionysius did a little research on the side; he realized that he could figure out just when Jesus Christ was born. After chugging through a bit of math, he decided that the current year was the 525th year since the birth of Christ. Dionysius decided that the year of Christ’s birth should, thenceforth, be the year 1 anno Domini, or the first year of Our Lord. (Technically, Dionysius said that Christ’s birth happened on December 25 the year before, and he started his calendar on January 1 to match the Roman year.) The next year after that was 2 AD, and the next 3 AD, and so forth, replacing the two dating systems then most commonly in use.* But there was a problem. Make that two.
For one thing, Dionysius got the date of Christ’s birth wrong. The sources agree that Mary and Joseph fled the wrath of King Herod, since Herod had heard a prophecy about a newborn Messiah. But Herod died in 3 BC, years before the supposed birth of Christ. Dionysi
us was clearly wrong; today most scholars believe that the birth of Christ was in 4 BC. Dionysius was a few years off.
In truth, this mistake was not so terrible. When choosing the first year of a calendar, it really doesn’t matter which year is chosen, so long as everything is consistent after that. A four-year error is inconsequential if everyone agrees to make the same mistake, as, indeed, we have. But there was a more serious problem with Dionysius’s calendar: zero.
There was no year zero. Normally this would be no big deal; most calendars of that day started with the year one, not the year zero. Dionysius didn’t even have a choice; he didn’t know about zero. He was brought up after the decline of the Roman Empire. Even during the heyday of Rome, the Romans were not exactly math whizzes. In the year 525, at the start of the Dark Ages, all Westerners clung to the clunky Roman style of numbers, and there was no zero in that counting system. To Dionysius, the first year of Our Lord was, naturally, the year I. The next year was year II, and Dionysius came to this conclusion in the year DXXV. In most circumstances this would not have caused any trouble, especially since Dionysius’s calendar did not catch on immediately. In 525 there was serious trouble for the intellectuals in the Roman court. Pope John died, and in the ensuing power shift all the philosophers and mathematicians like Dionysius were kicked out of office. They were lucky to escape with their lives. (Others were not so lucky. Anicius Boethius was a powerful courtier who was among the finest medieval Western mathematicians, which makes him worth noting. At about the same time that Dionysius was kicked out of office, Boethius, too, fell from power and was imprisoned. Boethius is not remembered for his math but for his Consolation of Philosophy, a tract in which he comforts himself with Aristotelian-style philosophy. He was clubbed to death soon afterward.) In any case, the new calendar languished for years.
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