Is God a Mathematician? (13 page)

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 fig
ure 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 3
²
+ 4
²
= 5
²
. In fact, it is easy to show (using the Pythagorean theorem) that the coordinates (
x, y
) of any point on this circle satisfy
x
²
+
y
²
= 5
²
. Furthermore, the points on the circle are the only points in the plane for whose coordinates this equation (
x
²
+
y
²
5
²
) holds true. This means that the algebraic equation
x
²
+
y
²
= 5
²
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: