Is God a Mathematician? (6 page)

Some mathematicians, philosophers, cognitive scientists, and other “consumers” of mathematics (e.g., computer scientists) regard the Platonic world as a figment of the imagination of too-dreamy minds (I shall describe this perspective and other dogmas in detail later in the book). In fact, in 1940, the famous historian of mathematics Eric Temple Bell (1883–1960) made the following prediction:

Bell’s prophecy proved to be wrong. While dogmas that are diametrically opposed (but in different directions) to Platonism have emerged, those have not fully won the minds (and hearts!) of all mathematicians and philosophers, who remain today as divided as ever.

Suppose, however, that Platonism had won the day, and we had all become wholehearted Platonists. Does Platonism actually explain the “unreasonable effectiveness” of mathematics in describing our world? Not really. Why should physical reality behave according to laws that reside in the abstract Platonic world? This was, after all, one of Penrose’s mysteries, and Penrose is a devout Platonist himself. So for the moment we have to accept the fact that even if we were to embrace Platonism, the puzzle of the powers of mathematics would remain unsolved. In Wigner’s words: “It is difficult to avoid the impression that a miracle confronts us here, comparable in its striking nature to
the miracle that the human mind can string a thousand arguments together without getting itself into contradictions.”

To fully appreciate the magnitude of this miracle, we have to delve into the lives and legacies of some of the miracle workers themselves—the minds behind the discoveries of a few of those incredibly precise mathematical laws of nature.

Unlike the Ten Commandments, science was not handed to humankind on imposing tablets of stone. The history of science is the story of the rise and fall of numerous speculations, hypotheses, and models. Many seemingly clever ideas turned out to be false starts or led down blind alleys. Some theories that were taken to be ironclad at the time later dissolved when put to the fiery test of subsequent experiments and observations, only to become entirely obsolete. Even the extraordinary brainpower of the originators of some conceptions did not make those conceptions immune to being superseded. The great Aristotle, for instance, thought that stones, apples, and other heavy objects fall down because they seek their natural place, which is at the center of Earth. As they approached the ground, Aristotle argued, these bodies increased their speed because they were happy to return home. Air (and fire), on the other hand, moved upward because the air’s natural place was with the heavenly spheres. All objects could be assigned a nature based on their perceived relation to the most basic constituents—earth, fire, air, and water. In Aristotle’s words:

Aristotle even made an attempt to formulate a quantitative law of motion. He asserted that heavier objects fall faster, with the speed being directly proportional to the weight (that is, an object two times heavier than another was supposed to fall at twice the speed). While everyday experience might have made this law seem reasonable enough—a brick was indeed observed to hit the ground earlier than a feather dropped from the same height—Aristotle never examined his quantitative statement more precisely. Somehow, it either never occurred to him, or he did not consider it necessary, to check whether two bricks tied together indeed fall twice as fast as a single brick. Galileo Galilei (1564–1642), who was much more mathematically and experimentally oriented, and who showed little respect for the happiness of falling bricks and apples, was the first to point out that Aristotle got it completely wrong. Using a clever thought experiment, Galileo was able to demonstrate that Aristotle’s law just didn’t make any sense, because it was logically inconsistent. He argued as follows: Suppose you tie together two objects, one heavier than the other. How fast would the combined object fall compared to each of its two constituents? On one hand, according to Aristotle’s law, you might conclude that it would fall at some intermediate speed, because the lighter object would slow down the heavier one. On the other, given that the combined object is actually heavier than its components, it should fall even faster than the heavier of the two, leading to a clear contradiction. The only reason that a feather falls on Earth more gently than a brick is that the feather experiences greater air resistance—if dropped from the same height in a vacuum, they would hit the ground simultaneously. This fact has been demonstrated in numerous experiments, none more dramatic than the one performed by Apollo 15 astronaut David Randolph Scott. Scott—the seventh
person to walk on the Moon—simultaneously dropped a hammer from one hand and a feather from the other. Since the Moon lacks a substantial atmosphere, the hammer and the feather struck the lunar surface at the same time.

The amazing fact about Aristotle’s false law of motion is not that it was wrong, but that it was accepted for almost two thousand years. How could a flawed idea enjoy such a remarkable longevity? This was a case of a “perfect storm”—three different forces combining to create an unassailable doctrine. First, there was the simple fact that in the absence of precise measurements, Aristotle’s law seemed to agree with experience-based common sense—sheets of papyrus did hover about, while lumps of lead did not. It took Galileo’s genius to argue that common sense could be misleading. Second, there was the colossal weight of Aristotle’s almost unmatched reputation and authority as a scholar. After all, this was the man who laid out the foundations for much of Western intellectual culture. Whether it was the investigation of all natural phenomena or the bedrock of ethics, metaphysics, politics, or art, Aristotle literally wrote the book. And that was not all. Aristotle in some sense even taught us
how
to think, by introducing the first formal studies of logic. Today, almost every child at school recognizes Aristotle’s pioneering, virtually complete system of logical inference, known as a
syllogism:

1. Every Greek is a person.
   
2. Every person is mortal.
   
3. Therefore every Greek is mortal.
   

The third reason for the incredible durability of Aristotle’s incorrect theory was the fact that the Christian church adopted this theory as a part of its own official orthodoxy. This acted as a deterrent against most attempts to question Aristotle’s assertions.

In spite of his impressive contributions to the systemization of deductive logic, Aristotle is not noted for his mathematics. Somewhat surprisingly perhaps, the man who essentially established science as an organized enterprise did not care as much (and certainly not as much as Plato) for mathematics and was rather weak in physics. Even
though Aristotle recognized the importance of numerical and geometrical relationships in the sciences, he still regarded mathematics as an abstract discipline, divorced from physical reality. Consequently, while there is no doubt that he was an intellectual powerhouse, Aristotle does
not
make my list of mathematical “magicians.”

I am using the term “magicians” here for those individuals who could pull rabbits out of literally empty hats; those who discovered never-before-thought-of connections between mathematics and nature; those who were able to observe complex natural phenomena and to distill from them crystal-clear mathematical laws. In some cases, these superior thinkers even used their experiments and observations to advance their mathematics. The question of the unreasonable effectiveness of mathematics in explaining nature would never have arisen were it not for these magicians. This enigma was born directly out of the miraculous insights of these researchers.

No single book can do justice to all the superb scientists and mathematicians who have contributed to our understanding of the universe. In this chapter and the following one I intend to concentrate on only four of those giants of past centuries, about whose status as magicians there can be no doubt—some of the crème de la crème of the scientific world. The first magician on my list is best remembered for a rather unusual event—for dashing stark naked through the streets of his hometown.

Give Me a Place to Stand and I Will Move the Earth

When the historian of mathematics Eric Temple Bell had to decide whom to name as his top three mathematicians, he concluded:

Archimedes (287–212 BC; figure 10 shows a bust claimed to represent Archimedes, but which may in fact be that of a Spartan king) was indeed the Newton or Gauss of his day; a man of such brilliance, imagination, and insight that both his contemporaries and the generations that followed him uttered his name in awe and reverence. Even though he is better known for his ingenious inventions in engineering, Archimedes was primarily a mathematician, and in his mathematics he was centuries ahead of his time. Unfortunately, little is known about Archimedes’ early life or his family. His first biography, written by one Heracleides, has not survived, and the few details that we do know about his life and violent death come primarily from the writings of the Roman historian Plutarch. Plutarch (ca. AD 46–120) was, in fact, more interested in the military accomplishments of the Roman general Marcellus, who conquered Archimedes’ home town of Syracuse in 212 BC. Fortunately for the history of mathematics, Archimedes had given Marcellus such a tremendous headache during the siege of Syracuse that the three major historians of the period, Plutarch, Polybius, and Livy, couldn’t ignore him.

Figure 10

Archimedes was born in Syracuse, then a Greek settlement in Sicily. According to his own testimony, he was the son of the astronomer Phidias, about whom little is known beyond the fact that he had estimated the ratio of the diameters of the Sun and the Moon. Archimedes may have also been related in some way to King Hieron II, himself the illegitimate son of a nobleman (by one of the latter’s female slaves). Irrespective of whichever ties Archimedes might have had with the royal family, both the king and his son, Gelon, always held Archimedes in high regard. As a youth, Archimedes spent some time in Alexandria, where he studied mathematics, before returning to a life of extensive research in Syracuse.

Archimedes was truly a mathematician’s mathematician. According to Plutarch, he regarded as sordid and ignoble “every art directed to use and profit, and he only strove after those things which, in their beauty and excellence, remain beyond all contact with the common needs of life.” Archimedes’ preoccupation with abstract mathematics and the level to which he was consumed by it apparently went much farther even than the enthusiasm commonly exhibited by practitioners of this discipline. Again according to Plutarch:

In spite of his contempt for applied mathematics, and the little importance that Archimedes himself attached to his engineering ideas, his resourceful inventions gained him even more popular fame than his mathematical genius.

The best-known legend about Archimedes further enhances his image as the stereotypical absentminded mathematician. This amusing story was first told by the Roman architect Vitruvius in the first century BC, and it goes like this: King Hieron wanted to consecrate a
gold wreath to the immortal gods. When the wreath was delivered to the king, it was equal in weight to the gold furnished for its creation. The king was nonetheless suspicious that a certain amount of gold had been replaced by silver of the same weight. Not being able to substantiate his distrust, the king turned for advice to the master of mathematicians—Archimedes. One day, the legend continued, Archimedes stepped into a bath, while still engrossed in the problem of how to uncover potential fraud with the wreath. As he immersed himself in the water, however, he realized that his body displaced a certain volume of water, which overflowed the tub’s edge. This immediately triggered a solution in his head. Overwhelmed with joy, Archimedes jumped out of the tub and ran naked in the street shouting “
Eureka, eureka!
” (“I have found it, I have found it!”).

Another famous Archimedean exclamation, “Give me a place to stand and I will move the Earth,” is currently featured (in one version or another) on more than 150,000 Web pages found in a Google search. This bold proclamation, sounding almost like the vision statement of a large corporation, has been cited by Thomas Jefferson, Mark Twain, and John F. Kennedy and it was even featured in a poem by Lord Byron. The phrase was apparently the culmination of Archimedes’ investigations into the problem of moving a given weight with a given force. Plutarch tells us that when King Hieron asked for a practical demonstration of Archimedes’ ability to manipulate a large weight with a small force, Archimedes managed—using a compound pulley—to launch a fully loaded ship into the sea. Plutarch adds in admiration that “he drew the ship along smoothly and safely as if she were moving through the sea.” Slightly modified versions of the same legend appear in other sources. While it is difficult to believe that Archimedes could have actually moved an entire ship with the mechanical devices available to him at the time, the legends leave little room for doubt that he gave some impressive demonstration of an invention that enabled him to maneuver heavy weights.

Archimedes made many other peacetime inventions, such as a hydraulic screw for raising water and a planetarium that demonstrated the motions of the heavenly bodies, but he became most famous in antiquity for his role in the defense of Syracuse against the Romans.