Is God a Mathematician? (16 page)

The other point that was apparently still troubling Newton when he wrote the early draft of
De Motu
was the fact that he neglected the influence of the forces by which the planets attracted the Sun. In other words, in his original formulation, he reduced the Sun to a mere unmovable center of force of the type that “hardly exists,” in Newton’s words, in the real world. This scheme contradicted Newton’s own third law of motion, according to which “the actions of attracting and attracted bodies are always mutual and equal.” Each planet attracts the Sun precisely with the same force that the Sun attracts the planet. Consequently, he added, “if there are two bodies [such as the Earth and the Sun], neither the attracting nor the attracted body can be at rest.” This seemingly minor realization was actually an important stepping-stone toward the concept of a universal gravity. We can attempt to guess Newton’s line of thought: If the Sun pulls the Earth, then the Earth must also pull the Sun, with equal strength. That is, the Earth doesn’t simply orbit the Sun, but rather they both revolve around their mutual center of gravity. But this is not all. All the other planets also attract the Sun, and indeed each planet feels the attraction not just of the Sun, but also of all other planets. The same type of logic could be applied to Jupiter and its satellites, to the Earth and the Moon, and even to an apple and the Earth. The conclusion is astounding in it simplicity—
there is only one gravitational force, and it acts between any two masses, anywhere in the universe
. This was all that Newton needed. The
Principia
—510 dense Latin pages—was published in July of 1687.

Newton took observations and experiments that were accurate to only about 4 percent and established from those a mathematical law of gravity that turned out to be accurate to better than one part in a million. He united for the first time
explanations
of natural phenomena with the power of
prediction
of the results of observations. Physics and mathematics became forever intertwined, while the divorce of science from philosophy became inevitable.

The second edition of the
Principia,
edited extensively by Newton and in particular by the mathematician Roger Cotes (1682–1716),
appeared in 1713 (figure 30 shows the frontispiece). Newton, who was never known for his warmth, did not even bother to thank Cotes in the preface to the book for his fabulous work. Still, when Cotes died from violent fever at age thirty-three, Newton did show some appreciation: “If he had lived we would have known something.”

Curiously, some of Newton’s most memorable remarks about God appeared only as afterthoughts in the second edition. In a letter to Cotes on March 28, 1713, less than three months before the completion of
Principia
’s second edition, Newton included the sentence: “It surely does belong to natural philosophy to discourse of God from the phenomena [of nature].” Indeed, Newton expressed his ideas of an “eternal and infinite, omnipotent and omniscient” God in the “General Scholium”—the section he regarded as putting the final touch on the
Principia.

Figure 30

But did God’s role remain unchanged in this increasingly mathematical universe? Or was God perceived more and more as a mathematician? After all, until the formulation of the law of gravitation, the motions of the planets had been regarded as one of the unmistakable works of God. How did Newton and Descartes see this shift in emphasis toward scientific explanations of nature?

The Mathematician God of Newton and Descartes

As were most people of their time, both Newton and Descartes were religious men. The French writer known by the pen name of Voltaire (1694–1778), who wrote quite extensively about Newton, famously said that “if God did not exist, it would be necessary for us to invent Him.”

For Newton, the world’s very existence and the mathematical regularity of the observed cosmos were evidence for God’s presence. This type of causal reasoning was first used by the theologian Thomas Aquinas (ca. 1225–1274), and the arguments fall under the general philosophical labels of a
cosmological argument
and a
teleological argument.
Put simply, the cosmological argument claims that since the physical world had to come into existence somehow, there must be a First Cause, namely, a creator God. The teleological argument, or
argument from design,
attempts to furnish evidence for God’s existence from the apparent design of the world. Here are Newton’s thoughts, as expressed in
Principia:
“This most beautiful system of the sun, planets and comets, could only proceed from the counsel and dominion of an intelligent and powerful Being. And if the fixed stars are the centers of other like systems, these, being formed by the like wise counsel, must be all subject to the dominion of One.” The validity of the cosmological, teleological, and similar arguments as proof for God’s existence has been the subject of debate among philosophers for centuries. My personal impression has always been that theists don’t need these arguments to be convinced, and atheists are not persuaded by them.

Newton added yet another twist, based on the universality of his laws. He regarded the fact that the entire cosmos is governed by the
same laws and appears to be stable as further evidence for God’s guiding hand, “especially since the light of the fixed stars is of the
same nature
[emphasis added] with the light of the sun, and from every system light passes into all the other systems: and lest the systems of the fixed stars should, by their gravity, fall on each other mutually, he hath placed these systems at immense distances one from another.”

In his book
Opticks,
Newton made it clear that he did not believe that the laws of nature by themselves were sufficient to explain the universe’s existence—God was the creator and sustainer of all the atoms that make up the cosmic matter: “For it became him [God] who created them [the atoms] to set them in order. And if he did so, it’s unphilosophical to seek for any other Origin of the World, or to pretend that it might arise out of a Chaos by the mere Laws of Nature.” In other words, to Newton, God was a mathematician (among other things), not just as a figure of speech, but almost literally—the Creator God brought into existence a physical world that was governed by mathematical laws.

Being much more philosophically inclined than Newton, Descartes had been extremely preoccupied with proving God’s existence. To him, the road from the certainty in our own existence (“I am thinking, therefore I exist”) to our ability to construct a tapestry of objective science had to pass through an unassailable proof for the existence of a supremely perfect God. This God, he argued, was the ultimate source of all truth and the only guarantor of the reliability of human reasoning. This suspiciously circular argument (known as the
Cartesian circle
) was already criticized during Descartes’ time, especially by the French philosopher, theologian, and mathematician Antoine Arnauld (1612–94). Arnauld posed a question that was devastating in its simplicity: If we need to prove God’s existence in order to guarantee the validity of the human thought process, how can we trust the proof, which is in itself a product of the human mind? While Descartes did make some desperate attempts to escape from this vicious reasoning circle, many of the philosophers who followed him did not find his efforts particularly convincing. Descartes’ “supplemental proof” for the existence of God was equally questionable. It falls under the general philosophical label of an
ontological argument
.
The philosophical theologian St. Anselm of Canterbury (1033–1109) first formulated this type of reasoning in 1078, and it has since resurfaced in many incarnations. The logical construct goes something like this: God, by definition, is so perfect that he is the greatest conceivable being. But if God did not exist, then it would be possible to conceive of a greater being yet—one that in addition to being blessed with all of God’s perfections also exists. This would contradict God’s definition as the greatest conceivable being—therefore God has to exist. In Descartes’ words: “Existence can no more be separated from the essence of God than the fact that its angles equal two right angles can be separated from the essence of a triangle.”

This type of logical maneuvering does not convince many philosophers, and they argue that to establish the existence of anything that is consequential in the physical world, and in particular something as grand as God, logic alone does not suffice.

Oddly enough, Descartes was accused of fostering atheism, and his works were put on the Catholic Church’s Index of Forbidden Books in 1667. This was a bizarre charge in light of Descartes’ insistence on God as the ultimate guarantor of truth.

Leaving the purely philosophical questions aside, for our present purposes the most interesting point is Descartes’ view that God created all the “eternal truths.” In particular, he declared that “the mathematical truths which you call eternal have been laid down by God and depend on Him entirely no less than the rest of his creatures.” So the Cartesian God was more than a mathematician, in the sense of being the creator of both mathematics and a physical world that is entirely based on mathematics. According to this worldview, which was becoming prevalent at the end of the seventeenth century, humans clearly only
discover
mathematics and do not invent it.

More significantly, the works of Galileo, Descartes, and Newton have changed the relationship between mathematics and the sciences in a profound way. First, the explosive developments in science became strong motivators for mathematical investigations. Second, through Newton’s laws, even more abstract mathematical fields, such as calculus, became the
essence
of physical explanations. Finally, and perhaps most importantly, the boundary between mathematics and
the sciences was blurred beyond recognition, almost to the point of a complete fusion between mathematical insights and large swaths of exploration. All of these developments created a level of enthusiasm for mathematics perhaps not experienced since the time of the ancient Greeks. Mathematicians felt that the world was theirs to conquer, and that it offered unlimited potential for discovery.

The world doesn’t stand still. Most things around us are either in motion or continuously changing. Even the seemingly firm Earth underneath our feet is in fact spinning around its axis, revolving around the Sun, and traveling (together with the Sun) around the center of our Milky Way galaxy. The air we breathe is composed of trillions of molecules that move ceaselessly and randomly. At the same time, plants grow, radioactive materials decay, the atmospheric temperature rises and falls both daily and with the seasons, and the human life expectancy keeps increasing. This cosmic restlessness in itself, however, did not stump mathematics. The branch of mathematics called
calculus
was introduced by Newton and Leibniz precisely to permit a rigorous analysis and an accurate modeling of both motion and change. By now, this incredible tool has become so potent and all encompassing that it can be used to examine problems as diverse as the motion of the space shuttle or the spreading of an infectious disease. Just as a movie can capture motion by breaking it up into a frame-by-frame sequence, calculus can measure change on such a fine grid that it allows for the determination of quantities that have only a fleeting existence, such as instantaneous speed, acceleration, or rate of change.

Continuing in Newton’s and Leibniz’s giant footsteps, mathematicians of the Age of Reason (the late seventeenth and eighteenth centuries) extended calculus to the even more powerful and widely applicable branch of
differential equations
. Armed with this new weapon, scientists were now able to present detailed mathematical theories of phenomena ranging from the music produced by a violin string to the transport of heat, from the motion of a spinning top to the flow of liquids and gases. For a while, differential equations became the tool of choice for making progress in physics.

A few of the first explorers of the new vistas opened by differential equations were members of the legendary Bernoulli family. Between the mid-seventeenth century and the mid-nineteenth century, this family produced no fewer than eight prominent mathematicians. These gifted individuals were almost equally known for their bitter intrafamily feuds as they were for their outstanding mathematics. While the Bernoulli quarrels were always concerned with competition for mathematical supremacy, some of the problems they argued about may not seem today to be of the highest significance. Still, the solution of these intricate puzzles often paved the way for more impressive mathematical breakthroughs. Overall, there is no question that the Bernoullis played an important role in establishing mathematics as the language of a variety of physical processes.

One story can help exemplify the complexity of the minds of two of the brightest Bernoullis—the brothers Jakob (1654–1705) and Johann (1667–1748). Jakob Bernoulli was one of the pioneers of
probability theory,
and we shall return to him later in the chapter. In 1690, however, Jakob was busy resurrecting a problem first examined by the quintessential Renaissance man, Leonardo da Vinci, two centuries earlier: What is the shape taken by an elastic but inextensible chain suspended from two fixed points (as in figure 31)? Leonardo sketched a few such chains in his notebooks. The problem was also presented to Descartes by his friend Isaac Beeckman, but there is no evidence of Descartes’ trying to solve it. Eventually the problem became known as the problem of the
catenary
(from the Latin word
catena,
meaning “a chain”). Galileo thought that the shape would be parabolic but was proven wrong by the French Jesuit Ignatius Pardies (1636–73). Pardies
was not up to the task, however, of actually solving mathematically for the correct shape.

Figure 31

Just one year after Jakob Bernoulli posed the problem, his younger brother Johann solved it (by means of a differential equation). Leibniz and the Dutch mathematical physicist Christiaan Huygens (1629–95) also solved it, but Huygens’s solution employed a more obscure geometrical method. The fact that Johann managed to solve a problem that had stymied his brother and teacher continued to be an immense source of satisfaction to the younger Bernoulli, even as late as thirteen years after Jakob’s death. In a letter Johann wrote on September 29, 1718, to the French mathematician Pierre Rémond de Montmort (1678–1719), he could not hide his delight:

After shamelessly taking ownership of even the suggestion of the problem, Johann continued with unconcealed glee:

In case you ever needed proof that mathematicians are humans after all, this story amply provides it. The familial rivalry, however, does not take anything away from the accomplishments of the Bernoullis. During the years that followed the catenary episode, Jakob, Johann, and Daniel Bernoulli (1700–1782) went on not only to solve other similar problems of hanging cords, but also to advance the theory of differential equations in general and to solve for the motion of projectiles through a resisting medium.

The tale of the catenary serves to demonstrate another facet of the power of mathematics—even seemingly trivial physical problems have mathematical solutions. The shape of the catenary itself, by the way, continues to delight millions of visitors to the famous Gateway Arch in St. Louis, Missouri. The Finnish-American architect Eero Saarinen (1910–61) and the German-American structural engineer
Hannskarl Bandel (1925–93) designed this iconic structure in a shape that is similar to that of an inverted catenary.

The astounding success of the physical sciences in discovering mathematical laws that govern the behavior of the cosmos at large raised the inevitable question of whether or not similar principles might also underlie biological, social, or economical processes. Is mathematics only the language of nature, mathematicians wondered, or is it also the language of human nature? Even if truly universal principles do not exist, could mathematical tools, at the very least, be used to model and subsequently explain social behavior? At first, many mathematicians were quite convinced that “laws” based on some version of calculus would be able to accurately predict all future events, large or small. This was the opinion, for instance, of the great mathematical physicist Pierre-Simon de Laplace (1749–1827). Laplace’s five volumes of
Mécanique céleste
(
Celestial Mechanics
) gave the first virtually complete (if approximate) solution to the motions in the solar system. In addition, Laplace was the man who answered a question that puzzled even the giant Newton: Why is the solar system as stable as it is? Newton thought that due to their mutual attractions planets had to fall into the Sun or to fly away into free space, and he invoked God’s hand in keeping the solar system intact. Laplace had rather different views. Instead of relying on God’s handiwork, he simply proved mathematically that the solar system is stable over periods of time that are much longer than those anticipated by Newton. To solve this complex problem, Laplace introduced yet another mathematical formalism known as
perturbation theory,
which enabled him to calculate the cumulative effect of many small perturbations to each planet’s orbit. Finally, to top it all, Laplace proposed one of the first models for the very origin of the solar system—in his influential
nebular hypothesis
, the solar system formed from a contracting gaseous nebula.

Given all of these impressive feats, it is perhaps not surprising that in his
Philosophical Essay on Probabilities
Laplace boldly pronounced:

Just in case you wonder, when Laplace talked about this hypothetical supreme “intelligence,” he did not mean God. Unlike Newton and Descartes, Laplace was not a religious person. When he gave a copy of his
Celestial Mechanics
to Napoleon Bonaparte, the latter, who had heard that there was no reference to God in the work, remarked: “M. Laplace, they tell me you have written this huge book on the system of the universe and have never even mentioned its creator.” Laplace immediately replied: “I did not need to make that hypothesis.” The amused Napoleon told the mathematician Joseph-Louis Lagrange about this reply, and the latter exclaimed: “Ah! That is a beautiful hypothesis; it explains many things.” But the story doesn’t end there. When he heard about Lagrange’s reaction, Laplace commented dryly: “This hypothesis, Sir, explains in fact everything, but does not permit to predict anything. As a scholar, I must provide you with works permitting predictions.”

The twentieth century development of quantum mechanics—the theory of the subatomic world—has proven the expectation for a fully deterministic universe to be too optimistic. Modern physics has in fact demonstrated that it is impossible to predict the outcome of every experiment, even in principle. Rather, the theory can only predict the probabilities for different results. The situation in the social sciences
is clearly even more complex because of a multiplicity of interrelated elements, many of which are highly uncertain at best. The researchers of the seventeenth century realized soon enough that a search for precise universal social principles of the type of Newton’s law of gravitation was doomed from the start. For a while, it seemed that when the intricacies of human nature are brought into the equation, secure predictions become virtually impossible. The situation appeared to be even more hopeless when the minds of an entire population were involved. Rather than despairing, however, a few ingenious thinkers developed a fresh arsenal of innovative mathematical tools—
statistics
and
probability theory.

The Odds Beyond Death and Taxes

The English novelist Daniel Defoe (1660–1731), best known for his adventure story
Robinson Crusoe,
also authored a work on the supernatural entitled
The Political History of the Devil.
In it, Defoe, who saw evidence for the devil’s actions everywhere, wrote: “Things as certain as death and taxes, can be more firmly believed.” Benjamin Franklin (1706–90) seems to have subscribed to the same perspective with respect to certainty. In a letter he wrote at age eighty-three to the French physicist Jean-Baptiste Leroy, he said: “Our Constitution is in actual operation. Everything appears to promise that it will last; but in this world nothing can be said to be certain but death and taxes.” Indeed, the courses of our lives appear to be unpredictable, prone to natural disasters, susceptible to human errors, and affected by pure happenstance. Phrases such as “[- - - -] happens” have been invented precisely to express our vulnerability to the unexpected and our inability to control chance. In spite of these obstacles, and maybe even because of these challenges, mathematicians, social scientists, and biologists have embarked since the sixteenth century on serious attempts to tackle uncertainties methodically. Following the establishment of the field of statistical mechanics, and faced with the realization that the very foundations of physics—in the form of quantum mechanics—are based on uncertainty, physicists of the twentieth and twenty-first centuries have enthusiastically joined the battle. The weapon researchers
use to combat the lack of precise determinism is the ability to calculate the odds of a particular outcome. Short of being capable of actually predicting a result, computing the likelihood of different consequences is the next best thing. The tools that have been fashioned to improve on mere guesses and speculations—statistics and probability theory—provide the underpinning of not just much of modern science, but also a wide range of social activities, from economics to sports.