Is God a Mathematician?

by Mario Livio

  The fact that Einstein had turned warped four-dimensional spacetime into the cornerstone of his new theory of the cosmos meant that he badly needed a mathematical theory of such geometrical entities. In desperation, he turned to his old classmate the mathematician Marcel Grossmann (1878–1936): “I have become imbued with great respect for mathematics, the more subtle parts of which I had previously regarded as sheer luxury.” Grossmann pointed out that Riemann’s non-Euclidean geometry (described in chapter 6) was precisely the tool that Einstein needed—a geometry of curved spaces in any number of dimensions. This was an incredible demonstration of what I dubbed the “passive” effectiveness of mathematics, which Einstein was quick to acknowledge: “We may in fact regard [geometry] as the most ancient branch of physics,” he declared. “Without it I would have been unable to formulate the theory of relativity.”

  General relativity has also been tested with impressive accuracy. These tests are not easy to come by, since the curvature in spacetime introduced by objects such as the Sun is measured only in parts per million. While the original tests were all associated with observations within the solar system (e.g., tiny changes to the orbit of the planet Mercury, as compared to the predictions of Newtonian gravity), more exotic tests have recently become feasible. One of the best verifications uses an astronomical object known as a double pulsar.

  A pulsar is an extraordinarily compact, radio-wave-emitting star, with a mass somewhat larger than the mass of the Sun but a radius of only about six miles. The density of such a star (known as a neutron star) is so high that one cubic inch of its matter has a mass of about a billion tons. Many of these neutron stars spin very fast, while emitting radio waves from their magnetic poles. When the magnetic axis is somewhat inclined to the rotation axis (as in figure 61), the radio beam from a given pole may cross our line of sight only once every rotation, like the flash of light from a lighthouse. In this case, the radio emission will appear to be pulsed—hence the name “pulsar.” In one case, two pulsars revolve around their mutual center of gravity in a close orbit, creating a double-pulsar system.

  There are two properties that make this double pulsar an excellent laboratory for testing general relativity: (1) Radio pulsars are superb clocks—their rotation rates are so stable that in fact they surpass atomic clocks in accuracy; and (2) Pulsars are so compact that their gravitational fields are very strong, producing significant relativistic effects. These features allow astronomers to measure very precisely changes in the light travel time from the pulsars to Earth caused by the orbital motion of the two pulsars in each other’s gravitational field.

  Figure 61

  The most recent test was the result of precision timing observations taken over a period of two and a half years on the double-pulsar system known as PSR J0737-3039A/B (the long “telephone number” reflects the coordinates of the system in the sky). The two pulsars in this system complete an orbital revolution in just two hours and twenty-seven minutes, and the system is about two thousand light-years away from Earth (a light-year is the distance light travels in one year in a vacuum; about six trillion miles). A team of astronomers led by Michael Kramer of the University of Manchester measured the relativistic corrections to the Newtonian motion. The results, published in October 2006, agreed with the values predicted by general relativity within an uncertainty of 0.05 percent!

  Incidentally, both special relativity and general relativity play an important role in the Global Positioning System (GPS) that helps us find our location on the surface of the Earth and our way from place to place, whether in a car, airplane, or on foot. The GPS determines the current position of the receiver by measuring the time it takes the signal from several satellites to reach it and by triangulating on the known positions of each satellite. Special relativity predicts that the atomic clocks on board the satellites should tick more slowly (falling behind by a few millionths of a second per day) than those on the ground because of their relative motion. At the same time, general relativity predicts that the satellite clocks should tick faster (by a few tens of millionths of a second per day) than those on the ground due to the fact that high above the Earth’s surface the curvature in spacetime resulting from the Earth’s mass is smaller. Without making the necessary corrections for these two effects, errors in global positions could accumulate at a rate of more than five miles in each day.

  The theory of gravity is only one of the many examples that illustrate the miraculous suitability and astonishing accuracy of the mathematical formulation of the laws of nature. In this case, as in numerous others, what we got out of the equations was much more than what was originally put in. The accuracy of both Newton’s and Einstein’s theories proved to far exceed the accuracy of the observations that the theories attempted to explain in the first place.

  Perhaps the best example of the astonishing accuracy that a mathematical theory can achieve is provided by quantum electrodynamics (QED), the theory that describes all phenomena involving electrically charged particles and light. In 2006 a group of physicists at Harvard University determined the magnetic moment of the electron (which measures how strongly the electron interacts with a magnetic field) to a precision of eight parts in a trillion. This is an incredible experimental feat in its own right. But when you add to that the fact that the most recent theoretical calculations based on QED reach a similar precision and that the two results agree, the accuracy becomes almost unbelievable. When he heard about the continuing success of QED, one of QED’s originators, the physicist Freeman Dyson, reacted: “I’m amazed at how precisely Nature dances to the tune we scribbled so carelessly fifty-seven years ago, and at how the experimenters and the theorists can measure and calculate her dance to a part in a trillion.”

  But accuracy is not the only claim to fame of mathematical theories—predictive power is another. Let me give just two simple examples, one from the nineteenth century and one from the twentieth century, that demonstrate this potency. The former theory predicted a new phenomenon and the latter the existence of new fundamental particles.

  James Clerk Maxwell, who formulated the classical theory of electromagnetism, showed in 1864 that the theory predicted that varying electric or magnetic fields should generate propagating waves. These waves—the familiar electromagnetic waves (e.g., radio)—were first detected by the German physicist Heinrich Hertz (1857–94) in a series of experiments conducted in the late 1880s.

  In the late 1960s, physicists Steven Weinberg, Sheldon Glashow, and Abdus Salam developed a theory that treats the electromagnetic force and weak nuclear force in a unified manner. This theory, now known as the electroweak theory, predicted the existence of three particles (called the W, W–, and Z bosons) that had never before been observed. The particles were unambiguously detected in 1983 in accelerator experiments (which smash one subatomic particle into another at very high energies) led by physicists Carlo Rubbia and Simon van der Meer.

  The physicist Eugene Wigner, who coined the phrase “the unreasonable effectiveness of mathematics,” proposed to call all of these unexpected achievements of mathematical theories the “empirical law of epistemology” (epistemology is the discipline that investigates the origin and limits of knowledge). If this “law” were not correct, he argued, scientists would have lacked the encouragement and reassurance that are absolutely necessary for a thorough exploration of the laws of nature. Wigner, however, did not offer any explanation for the empirical law of epistemology. Rather, he regarded it as a “wonderful gift” for which we should be grateful even though we do not understand its origin. Indeed, to Wigner, this “gift” captured the essence of the question about the unreasonable effectiveness of mathematics.

  At this point, I believe that we have gathered enough clues that we should at least be able to try answering the questions we started with: Why is mathematics so effective and productive in explaining the world around us that it even yields new knowledge? And, is mathematics ultimately invented or discovered?

  CHAPTE
R 9

  ON THE HUMAN MIND, MATHEMATICS, AND THE UNIVERSE

  The two questions: (1) Does mathematics have an existence independent of the human mind? and (2) Why do mathematical concepts have applicability far beyond the context in which they have originally been developed? are related in complex ways. Still, to simplify the discussion, I will attempt to address them sequentially.

  First, you may wonder where modern-day mathematicians stand on the question of mathematics as a discovery or an invention. Here is how mathematicians Philip Davis and Reuben Hersh described the situation in their wonderful book The Mathematical Experience:

  Most writers on the subject seem to agree that the typical working mathematician is a Platonist [views mathematics as discovery] on weekdays and a formalist [views mathematics as invention] on Sundays. That is, when he is doing mathematics he is convinced that he is dealing with an objective reality whose properties he is attempting to determine. But then, when challenged to give a philosophical account of this reality, he finds it easiest to pretend that he does not believe in it after all.

  Other than being tempted to substitute “he or she” for “he” everywhere, to reflect the changing mathematical demographics, I have the impression that this characterization continues to be true for many present-day mathematicians and theoretical physicists. Nevertheless, some twentieth century mathematicians did take a strong position on one side or the other. Here, representing the Platonic point of view, is G. H. Hardy in A Mathematician’s Apology:

  For me, and I suppose for most mathematicians, there is another reality, which I will call “mathematical reality”; and there is no sort of agreement about the nature of mathematical reality among either mathematicians or philosophers. Some hold that it is “mental” and that in some sense we construct it, others that it is outside and independent of us. A man who could give a convincing account of mathematical reality would have solved very many of the most difficult problems of metaphysics. If he could include physical reality in his account, he would have solved them all.

  I should not wish to argue any of these questions here even if I were competent to do so, but I will state my own position dogmatically in order to avoid minor misapprehensions. I believe that mathematical reality lies outside us, that our function is to discover or observe it, and that the theorems which we prove, and which we describe grandiloquently as our “creations,” are simply our notes of our observations. This view has been held, in one form or another, by many philosophers of high reputation from Plato onwards, and I shall use the language which is natural to a man who holds it.

  Mathematicians Edward Kasner (1878–1955) and James Newman (1907–66) expressed precisely the opposite perspective in Mathematics and the Imagination:

  That mathematics enjoys a prestige unequaled by any other flight of purposive thinking is not surprising. It has made possible so many advances in the sciences, it is at once so indispensable in practical affairs and so easily the masterpiece of pure abstraction that the recognition of its pre-eminence among man’s intellectual achievements is no more than its due.

  In spite of this pre-eminence, the first significant appraisal of mathematics was occasioned only recently by the advent of non-Euclidean and four-dimensional geometry. That is not to say that the advances made by the calculus, the theory of probability, the arithmetic of the infinite, topology, and the other subjects we have discussed, are to be minimized. Each one has widened mathematics and deepened its meaning as well as our comprehension of the physical universe. Yet none has contributed to mathematical introspection, to the knowledge of the relation of the parts of mathematics to one another and to the whole as much as the non-Euclidean heresies.

  As a result of the valiantly critical spirit which engendered the heresies, we have overcome the notion that mathematical truths have an existence independent and apart from our own minds. It is even strange to us that such a notion could ever have existed. Yet this is what Pythagoras would have thought—and Descartes, along with hundreds of other great mathematicians before the nineteenth century. Today mathematics is unbound; it has cast off its chains. Whatever its essence, we recognize it to be as free as the mind, as prehensile as the imagination. Non-Euclidean geometry is proof that mathematics, unlike the music of the spheres, is man’s own handiwork, subject only to the limitations imposed by the laws of thought.

  So, contrary to the precision and certitude that are the hallmark of statements in mathematics, here we have a divergence of opinions that is more typical of debates in philosophy or politics. Should we be surprised? Not really. The question of whether mathematics is invented or discovered is actually not a question of mathematics at all.

  The notion of “discovery” implies preexistence in some universe, either real or metaphysical. The concept of “invention” implicates the human mind, either individually or collectively. The question therefore belongs to a combination of disciplines that may involve physics, philosophy, mathematics, cognitive science, even anthropology, but it is certainly not exclusive to mathematics (at least not directly). Consequently, mathematicians may not even be the best equipped to answer this question. After all, poets, who can perform magic with language, are not necessarily the best linguists, and the greatest philosophers are generally not experts in the functions of the brain. The answer to the “invented or discovered” question can therefore be gleaned only (if at all) from a careful examination of many clues, deriving from a wide variety of domains.

  Metaphysics, Physics, and Cognition

  Those who believe that mathematics exists in a universe that is independent of humans still fall into two different camps when it comes to identifying the nature of this universe. First, there are the “true” Platonists, for whom mathematics dwells in the abstract, eternal world of mathematical forms. Then there are those who suggest that mathematical structures are in fact a real part of the natural world. Since I have already discussed pure Platonism and some of its philosophical shortcomings quite extensively, let me elaborate a bit on the latter perspective.

  The person who presents what may be the most extreme and most speculative version of the “mathematics as a part of the physical world” scenario is an astrophysicist colleague, Max Tegmark of MIT.

  Tegmark argues that “our universe is not just described by mathematics—it is mathematics” [emphasis added]. His argument starts with the rather uncontroversial assumption that an external physical reality exists that is independent of human beings. He then proceeds to examine what might be the nature of the ultimate theory of such a reality (what physicists refer to as the “theory of everything”). Since this physical world is entirely independent of humans, Tegmark maintains, its description must be free of any human “baggage” (e.g., human language, in particular). In other words, the final theory cannot include any concepts such as “subatomic particles,” “vibrating strings,” “warped spacetime,” or other humanly conceived constructs. From this presumed insight, Tegmark concludes that the only possible description of the cosmos is one that involves only abstract concepts and the relations among them, which he takes to be the working definition of mathematics.

  Tegmark’s argument for a mathematical reality is certainly intriguing, and if it were true, it might have gone a long way toward solving the problem of the “unreasonable effectiveness” of mathematics. In a universe that is identified as mathematics, the fact that mathematics fits nature like a glove would hardly be a surprise. Unfortunately, I do not find Tegmark’s line of reasoning to be extremely compelling. The leap from the existence of an external reality (independent of humans) to the conclusion that, in Tegmark’s words, “You must believe in what I call the mathematical universe hypothesis: that our physical reality is a mathematical structure,” involves, in my opinion, a sleight of hand. When Tegmark attempts to characterize what mathematics really is, he says: “To a modern logician, a mathematical structure is precisely this: a set of abstract entities with relations between them.” But this modern logicia
n is human! In other words, Tegmark never really proves that our mathematics is not invented by humans; he simply assumes it. Furthermore, as the French neurobiologist Jean-Pierre Changeux has pointed out in response to a similar assertion: “To claim physical reality for mathematical objects, on a level of the natural phenomena we study in biology, poses a worrisome epistemological problem it seems to me. How can a physical state, internal to our brain, represent another physical state external to it?”

  Most other attempts to place mathematical objects squarely in the external physical reality simply rely on the effectiveness of mathematics in explaining nature as proof. This however assumes that no other explanation for the effectiveness of mathematics is possible, which, as I will show later, is not true.

  If mathematics resides neither in the spaceless and timeless Platonic world nor in the physical world, does this mean that mathematics is entirely invented by humans? Absolutely not. In fact, I shall argue in the next section that most of mathematics does consist of discoveries. Before going any further, however, it would be helpful to first examine some of the opinions of contemporary cognitive scientists. The reason is simple—even if mathematics were entirely discovered, these discoveries would still have been made by human mathematicians using their brains.