Is God a Mathematician?

by Mario Livio

  After La Flèche, Descartes graduated from the University of Poitiers as a lawyer, but he never actually practiced law. Restless and eager to see the world, Descartes decided to join the army of Prince Maurice of Orange, which was then stationed at Breda in the United Provinces (The Netherlands). An accidental encounter in Breda was to become very significant in Descartes’ intellectual development. According to the traditional story, while wandering in the streets, he suddenly saw a billboard that appeared to present a challenging problem in mathematics. Descartes asked the first passer-by to translate the text for him from Dutch into either Latin or French. A few hours later, Descartes succeeded in solving the problem, thus convincing himself that he really had an aptitude for mathematics. The translator turned out to be none other than the Dutch mathematician and scientist Isaac Beeckman (1588–1637), whose influence on Descartes’ physico-mathematical investigations continued for years. The next nine years saw Descartes alternating between the hurly-burly of Paris and service in the military corps of several armies. In a Europe in the throes of religious and political struggle and the commencement of the Thirty Years’ War, it was relatively easy for Descartes to find battles or marching battalions to join, be it in Prague, Germany, or Transylvania. Nevertheless, throughout this period he continued, as he put it, “over head and ears in the study of mathematics.”

  Figure 21

  On November 10, 1619, Descartes experienced three dreams that not only had a dramatic effect on the rest of his life, but which also marked perhaps the beginning of the modern world. When later describing the event, Descartes says in one of his notes: “I was filled with enthusiasm and discovered the foundations of a wonderful science.” What were these influential dreams about?

  Actually, two were nightmares. In the first dream, Descartes found himself caught in a turbulent whirlwind that revolved him violently on his left heel. He was also terrified by an endless sensation of falling down at each step. An old man appeared and attempted to present him with a melon from a foreign land. The second dream was yet another vision of horror. He was trapped in a room with ominous thunderclaps and sparks flying all around. In sharp contrast to the first two, the third dream was a picture of calm and meditation. As his eyes scanned the room, Descartes saw books appearing and disappearing on a table. They included an anthology of poems entitled Corpus Poetarum and an encyclopedia. He opened the anthology at random and caught a glimpse of the opening line of a poem by the fourth century Roman poet Ausonius. It read: “Quod vitae sectabor iter?” (“What road shall I pursue in life?”). A man miraculously appeared out of thin air and cited another verse: “Est et non” (“Yes and no” or “It is and it is not”). Descartes wanted to show him the Ausonius verse, but the entire vision disappeared into nothingness.

  As is usually the case with dreams, their significance lies not so much in their actual content, which is often perplexing and bizarre, but in the interpretation the dreamer chooses to give them. In Descartes’ case, the effect of these three enigmatic dreams was astounding. He took the encyclopedia to signify the collective scientific knowledge and the anthology of poetry to portray philosophy, revelation, and enthusiasm. The “Yes and no”—the famous opposites of Pythagoras—he understood as representing truth and falsehood. (Not surprisingly, some psychoanalytical interpretations suggested sexual connotations in relation to the melon.) Descartes was absolutely convinced that the dreams pointed him in the direction of the unification of the whole of human knowledge by the means of reason. He resigned from the army in 1621 but continued to travel and study mathematics for the next five years. All of those who met Descartes during that time, including the influential spiritual leader Cardinal Pierre de Bérulle (1575–1629), were deeply impressed with his sharpness and clarity of thought. Many encouraged him to publish his ideas. With any other young man, such fatherly words of wisdom might have had the same counterproductive effect that the one-word career advice “Plastics!” had on Dustin Hoffman’s character in the movie The Graduate, but Descartes was different. Since he had already committed to the goal of searching for the truth, he was easily persuaded. He moved to Holland, which at the time seemed to offer a more tranquil intellectual milieu, and for the next twenty years produced one tour de force after another.

  Descartes published his first masterpiece on the foundations of science, Discourse on the Method of Properly Guiding the Reason and Seeking for Truth in the Sciences, in 1637 (figure 22 shows the frontispiece of the first edition). Three outstanding appendices—on optics, meteorology, and geometry—accompanied this treatise. Next came his philosophical work, Meditations on First Philosophy, in 1641, and his work on physics, Principles of Philosophy, in 1644. Descartes was by then famous all over Europe, counting among his admirers and correspondents the exiled Princess Elisabeth of Bohemia (1618–80). In 1649 Descartes was invited to instruct the colorful Queen Christina of Sweden (1626–89) in philosophy. Having always had a soft spot for royalty, Descartes agreed. In fact, his letter to the queen was so full of expressions of courtly seventeenth century awe, that today it looks utterly ridiculous: “I dare to protest here to Your Majesty that she could command nothing to me so difficult that I would not always be ready to do everything possible to execute it, and that even if I had been born a Swede or a Finn, I could not be more zealous nor more perfectly [for you] than I am.” The iron-willed twenty-three-year-old queen insisted on Descartes giving her the lessons at the ungodly hour of five o’clock in the morning. In a land that was so cold that, as Descartes wrote to his friend, even thoughts froze there, this proved to be deadly. “I am out of my element here,” Descartes wrote, “and I desire only tranquility and repose, which are goods the most powerful kings on earth cannot give to those who cannot obtain them for themselves.” After only a few months of braving the brutal Swedish winter in those dark morning hours that he had managed to avoid throughout his entire life, Descartes contracted pneumonia. He died at age fifty-three on February 11, 1650, at four o’clock in the morning, as if trying to avoid another wake-up call. The man whose works announced the modern era fell victim to his own snobbish tendencies and the caprices of a young queen.

  Figure 22

  Descartes was buried in Sweden, but his remains, or at least part of them, were transferred to France in 1667. There, the remains were displaced multiple times, until they were eventually buried on February 26, 1819, in one of the chapels of the Saint-Germain-des-Prés cathedral. Figure 23 shows me next to the simple black plaque celebrating Descartes. A skull claimed to be that of Descartes was passed from hand to hand in Sweden until it was bought by a chemist named Berzelius, who transported it to France. That skull is currently at the Natural Science Museum, which is part of the Musée de l’Homme (the Museum of Man) in Paris. The skull is often on display opposite the skull of a Neanderthal man.

  Figure 23

  A Modern

  The label “modern,” when attached to a person, usually refers to those individuals who can converse comfortably with their twentieth (or by now, twenty-first) century professional peers. What makes Descartes a true modern is the fact that he dared to question all the philosophical and scientific assertions that were made before his time. He once noted that his education served only to advance his perplexity and to make him aware of his own ignorance. In his celebrated Discourse he wrote: “I observed with regard to philosophy, that despite being cultivated for many centuries by the best minds, it contained no point which was not disputed and hence doubtful.” While the fate of many of Descartes’ own philosophical ideas was not going to be much different in that significant shortcomings in his propositions have been pointed out by later philosophers, his fresh skepticism of even the most basic concepts certainly makes him modern to the core. More important from the perspective of the present book, Descartes recognized that the methods and reasoning process of mathematics produced precisely the kind of certainty that the scholastic philosophy before his time lacked. He pronounced clearly:

  Those
long chains, composed of very simple and easy reasonings, which geometers customarily use to arrive at their most difficult demonstrations, gave me occasion to suppose that all the things which fall within the scope of human knowledge are interconnected in the same way [the emphasis is mine]. And I thought that, provided we refrain from accepting anything as true which is not, and always keep to the order required for deducing one thing from another, there can be nothing too remote to be reached in the end or too well hidden to be discovered.

  This bold statement goes, in some sense, even beyond Galileo’s views. It is not only the physical universe that is written in the language of mathematics; all of human knowledge follows the logic of mathematics. In Descartes’ words: “It [the method of mathematics] is a more powerful instrument of knowledge than any other that has been bequeathed to us by human agency, as being the source of all others.” One of Descartes’ goals became, therefore, to demonstrate that the world of physics, which to him was a mathematically describable reality, could be depicted without having to rely on any of our often-misleading sensory perceptions. He advocated that the mind should filter what the eye sees and turn the perceptions into ideas. After all, Descartes argued, “there are no certain signs to distinguish between being awake and being asleep.” But, Descartes wondered, if everything we perceive as reality could in fact be only a dream, how are we to know that even the Earth and the sky are not some “delusions of dreams” installed in our senses by some “malicious demon of infinite power”? Or, as Woody Allen once put it: “What if everything is an illusion and nothing exists? In that case, I definitely overpaid for my carpet.”

  For Descartes, this deluge of troubling doubts eventually produced what has become his most memorable argument: Cogito ergo sum (I am thinking, therefore I exist). In other words, behind the thoughts there must be a conscious mind. Paradoxically perhaps, the act of doubting cannot itself be doubted! Descartes attempted to use this seemingly slight beginning to construct a complete enterprise of reliable knowledge. Whether it was in philosophy, optics, mechanics, medicine, embryology, or meteorology, Descartes tried his hand at it all and achieved accomplishments of some significance in every one of these disciplines. Still, in spite of his insistence on the human capacity to reason, Descartes did not believe that logic alone could uncover fundamental truths. Reaching essentially the same conclusion as Galileo, he noted: “As for logic, its syllogisms and the majority of its other percepts are of avail rather in the communication of what we already know…than in the investigation of the unknown.” Instead, throughout his heroic endeavor to reinvent, or establish, the foundations of entire disciplines, Descartes attempted to use the principles that he had distilled from the mathematical method to ensure that he was proceeding on solid ground. He described these rigorous guidelines in his Rules for the Direction of the Mind. He would start with truths about which he had no doubt (similar to the axioms in Euclid’s geometry); he would attempt to break up difficult problems into more manageable ones; he would proceed from the rudimentary to the intricate; and he would double-check his entire procedure to satisfy himself that no potential solution has been ignored. Needless to say, even this carefully constructed, arduous process could not make Descartes’ conclusions immune to error. In fact, even though Descartes is best known for his monumental breakthroughs in philosophy, his most enduring contributions have been in mathematics. I shall now concentrate in particular on that one brilliantly simple idea that John Stuart Mill referred to as the “greatest single step ever made in the progress of the exact sciences.”

  The Mathematics of a New York City Map

  Take a look at the partial map of Manhattan in figure 24. If you are standing at the corner of Thirty-fourth Street and Eighth Avenue and you have to meet someone at the corner of Fifty-ninth Street and Fifth Avenue, you will have no trouble finding your way, right? This was the essence of Descartes’ idea for a new geometry. He outlined it in a 106-page appendix entitled La Géométrie (Geometry) to his Discourse on the Method. Hard to believe, but this remarkably simple concept revolutionized mathematics. Descartes started with the almost trivial fact that, just as the map of Manhattan shows, a pair of numbers on the plane can determine the position of a point unambiguously (e.g., point A in figure 25a). He then used this fact to develop a powerful theory of curves—analytical geometry. In Descartes’ honor, the pair of intersecting straight lines that give us the reference system is known as a Cartesian coordinate system. Traditionally, the horizontal line is labeled the “x axis,” the vertical line the “y axis,” and the point of intersection is known as the “origin.” The point marked “A” in figure 25a, for instance, has an x coordinate of 3 and a y coordinate of 5, which is symbolically denoted by the ordered pair of numbers (3,5). (Note that the origin is designated (0,0).) Suppose now that we want to somehow characterize all the points in the plane that are at a distance of precisely 5 units from the origin. This is, of course, precisely the geometrical definition of a circle around the origin, with a radius of five units (figure 25b). If you take the point (3,4) on this circle, you find that its coordinates satisfy 32 + 42 = 52. In fact, it is easy to show (using the Pythagorean theorem) that the coordinates (x, y) of any point on this circle satisfy x2 + y2 = 52. Furthermore, the points on the circle are the only points in the plane for whose coordinates this equation (x2 + y2 52) holds true. This means that the algebraic equation x2 + y2 = 52 precisely and uniquely characterizes this circle. In other words, Descartes discovered a way to represent a geometrical curve by an algebraic equation or numerically and vice versa. This may not sound exciting for a simple circle, but every graph you have ever seen, be it of the weekly ups and downs of the stock market, the temperature at the North Pole over the past century, or the rate of expansion of the universe, is based on this ingenious idea of Descartes’. Suddenly, geometry and algebra were no longer two separate branches of mathematics, but rather two representations of the same truths. The equation describing a curve contains implicitly every imaginable property of the curve, including, for instance, all the theorems of Euclidean geometry. And this was not all. Descartes pointed out that different curves could be drawn on the same coordinate system, and that their points of intersection could be found simply by finding the solutions that are common to their respective algebraic equations. In this way, Descartes managed to exploit the strengths of algebra to correct for what he regarded as the disturbing shortcomings of classical geometry. For instance, Euclid defined a point as an entity that has no parts and no magnitude. This rather obscure definition became forever obsolete once Descartes defined a point in the plane simply as the ordered pair of numbers (x,y). But even these new insights were just the tip of the iceberg. If two quantities x and y can be related in such a way that for every value of x there corresponds a unique value of y, then they constitute what is known as a function, and functions are truly ubiquitous. Whether you are monitoring your daily weight while on a diet, the height of your child on consecutive birthdays, or the dependence of your car’s gas mileage on the speed at which you drive, the data can all be represented by functions.

  Figure 24

  Figure 25

  Functions are truly the bread and butter of modern scientists, statisticians, and economists. Once many repeated scientific experiments or observations produce the same functional interrelationships, those may acquire the elevated status of laws of nature—mathematical descriptions of a behavior all natural phenomena are found to obey. For instance, Newton’s law of gravitation, to which we shall return later in this chapter, states that when the distance between two point masses is doubled, the gravitational attraction between them always decreases by a factor of four. Descartes’ ideas therefore opened the door for a systematic mathematization of nearly everything—the very essence of the notion that God is a mathematician. On the purely mathematical side, by establishing the equivalence of two perspectives of mathematics (algebraic and geometric) previously considered disjoint, Descartes expanded the horizons of
mathematics and paved the way to the modern arena of analysis, which allows mathematicians to comfortably cross from one mathematical subdiscipline into another. Consequently, not only did a variety of phenomena become describable by mathematics, but mathematics itself became broader, richer, and more unified. As the great mathematician Joseph-Louis Lagrange (1736–1813) put it: “As long as algebra and geometry traveled separate paths their advance was slow and their applications limited. But when these two sciences joined company, they drew from each other fresh vitality and thenceforward marched on at a rapid pace towards perfection.”

  As important as Descartes’ achievements in mathematics were, he himself did not limit his scientific interests to mathematics. Science, he said, was like a tree, with metaphysics being the roots, physics the trunk, and the three main branches representing mechanics, medicine, and morals. The choice of the branches may appear somewhat surprising at first, but in fact the branches symbolized beautifully the three major areas to which Descartes wanted to apply his new ideas: the universe, the human body, and the conduct of life. Descartes spent the first four years of his stay in Holland—1629 to 1633—writing his treatise on cosmology and physics, Le Monde (The World). Just as the book was ready to go to press, however, Descartes was shocked by some troubling news. In a letter to his friend and critic, the natural philosopher Marin Mersenne (1588–1648), he lamented: