Is God a Mathematician?

by Mario Livio

  Plato extended his ideas on “true forms” to other disciplines as well, in particular to astronomy. He argued that in true astronomy “we must leave the heavens alone” and not attempt to account for the arrangements and the apparent motions of the visible stars. Instead, Plato regarded true astronomy as a science dealing with the laws of motion in some ideal, mathematical world, for which the observable heaven is a mere illustration (in the same way that geometrical figures drawn on papyrus only illustrate the true figures).

  Plato’s suggestions for astronomical research are considered controversial even by some of the most devout Platonists. Defenders of his ideas argue that what Plato really means is not that true astronomy should concern itself with some ideal heaven that has nothing to do with the observable one, but that it should deal with the real motions of celestial bodies as opposed to the apparent motions as seen from Earth. Others point out, however, that too literal an adoption of Plato’s dictum would have seriously impeded the development of observational astronomy as a science. Be the interpretation of Plato’s attitude toward astronomy as it may, Platonism has become one of the leading dogmas when it comes to the foundations of mathematics.

  But does the Platonic world of mathematics really exist? And if it does, where exactly is it? And what are these “objectively true” statements that inhabit this world? Or are the mathematicians who adhere to Platonism simply expressing the same type of romantic belief that has been attributed to the great Renaissance artist Michelangelo? According to legend, Michelangelo believed that his magnificent sculptures already existed inside the blocks of marble and that his role was merely to uncover them.

  Modern-day Platonists (yes, they definitely exist, and their views will be described in more detail in later chapters) insist that the Platonic world of mathematical forms is real, and they offer what they regard as concrete examples of objectively true mathematical statements that reside in this world.

  Take the following easy-to-understand proposition: Every even integer greater than 2 can be written as the sum of two primes (numbers divisible only by one and themselves). This simple-sounding statement is known as the Goldbach conjecture, since an equivalent conjecture appeared in a letter written by the Prussian amateur mathematician Christian Goldbach (1690–1764) on June 7, 1742. You can easily verify the validity of the conjecture for the first few even numbers: 4 = 2 + 2; 6 = 3 + 3; 8 = 3 + 5; 10 = 3 + 7 (or 5 + 5); 12 = 5 + 7; 14 = 3 + 11 (or 7 + 7); 16 = 5 + 11 (or 3 + 13); and so on. The statement is so simple that the British mathematician G. H. Hardy declared that “any fool could have guessed it.” In fact, the great French mathematician and philosopher René Descartes had anticipated this conjecture before Goldbach. Proving the conjecture, however, turned out to be quite a different matter. In 1966 the Chinese mathematician Chen Jingrun made a significant step toward a proof. He managed to show that every sufficiently large even integer is the sum of two numbers, one of which is a prime and the other has at most two prime factors. By the end of 2005, the Portuguese researcher Tomás Oliveira e Silva had shown the conjecture to be true for numbers up to 3 1017 (three hundred thousand trillion). Yet, in spite of enormous efforts by many talented mathematicians, a general proof remains elusive at the time of this writing. Even the additional temptation of a $1 million prize offered between March 20, 2000, and March 20, 2002 (to help publicize a novel entitled Uncle Petros and Goldbach’s Conjecture), did not produce the desired result. Here, however, comes the crux of the meaning of “objective truth” in mathematics. Suppose that a rigorous proof will actually be formulated in 2016. Would we then be able to say that the statement was already true when Descartes first thought about it? Most people would agree that this question is silly. Clearly, if the proposition is proven to be true, then it has always been true, even before we knew it to be true. Or, let’s look at another innocent-looking example known as Catalan’s conjecture. The numbers 8 and 9 are consecutive whole numbers, and each of them is equal to a pure power, that is 8 23 and 9 32. In 1844, the Belgian mathematician Eugène Charles Catalan (1814–94) conjectured that among all the possible powers of whole numbers, the only pair of consecutive numbers (excluding 0 and 1) is 8 and 9. In other words, you can spend your life writing down all the pure powers that exist. Other than 8 and 9, you will find no other two numbers that differ by only 1. In 1342, the Jewish-French philosopher and mathematician Levi Ben Gerson (1288–1344) actually proved a small part of the conjecture—that 8 and 9 are the only powers of 2 and 3 differing by 1. A major step forward was taken by the mathematician Robert Tijdeman in 1976. Still, the proof of the general form of Catalan’s conjecture stymied the best mathematical minds for more than 150 years. Finally, on April 18, 2002, the Romanian mathematician Preda Mihailescu presented a complete proof of the conjecture. His proof was published in 2004 and is now fully accepted. Again you may ask: When did Catalan’s conjecture become true? In 1342? In 1844? In 1976? In 2002? In 2004? Isn’t it obvious that the statement was always true, only that we didn’t know it to be true? These are the types of truths Platonists would refer to as “objective truths.”

  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:

  According to the prophets, the last adherent of the Platonic ideal in mathematics will have joined the dinosaurs by the year 2000. Divested of its mythical raiment of eternalism, mathematics will then be recognized for what it has always been, a humanly constructed language devised by human beings for definite ends prescribed by themselves. The last temple of an absolute truth will have vanished with the nothing it enshrined.

  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.

  CHAPTER 3

  MAGICIANS: THE MASTER AND THE HERETIC

  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 o
ther 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:

  Some existing things are natural, while others are due to other causes. Those that are natural are…the simple bodies such as earth, fire, air and water…all these things evidently differ from those that are not naturally constituted, since each of them has within itself a principle of motion and stability in place…A nature is a type of principle and cause of motion and stability within these things to which it primarily belongs…The things that are in accordance with nature include both these and whatever belongs to them in their own right, as traveling upward belongs to fire.

  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:

  Every Greek is a person.

  Every person is mortal.

  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:

  Any list of the three “greatest” mathematicians of all history would include the name of Archimedes. The other two usually associated with him are Newton (1642–1727) and Gauss (1777–1855). Some, considering the relative wealth—or poverty—of mathematics and physical science in the respective ages in which these giants lived, and estimating their achievements against the background of their times, would put Archimedes first.

  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 whic
hever 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:

  Continually bewitched by a Siren who always accompanied him, he forgot to nourish himself and omitted to care for his body; and when, as would often happen, he was urged by force to bathe and anoint himself, he would still be drawing geometrical figures in the ashes or with his fingers would draw lines on his anointed body, being possessed by a great ecstasy and in truth a thrall to the Muses.