Read Is God a Mathematician? Online
Authors: Mario Livio
Is God a Mathematician? (30 page)
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?
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 logician 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.
With the enormous progress in the cognitive sciences in recent years, it was only natural to expect that neurobiologists and psychologists would turn their attention to mathematics, in particular to the search for the foundations of mathematics in human cognition. A cursory glance at the conclusions of most cognitive scientists may initially leave you with the impression that you are witnessing an embodiment of Mark Twain’s phrase “To a man with a hammer, everything looks like a nail.” With small variations in emphasis, essentially all of the neuropsychologists and biologists determine that mathematics is a human invention. Upon closer examination, however, you find that while the interpretation of the cognitive data is far from being unambiguous, there is no question that the cognitive efforts represent a new and innovative phase in the search for the foundations of mathematics. Here is a small but representative sample of the comments made by the cognitive scientists.
The French neuroscientist Stanislas Dehaene, whose primary interest is in numerical cognition, concluded in his 1997 book
The Number Sense
that “intuition about numbers is thus anchored deep in our brain.” This position is in fact close to that of the intuitionists, who wanted to ground all of mathematics in the pure form of intuition of the natural numbers. Dehaene argues that discoveries about the psychology of arithmetic confirm that “number belongs to the ‘natural objects of thought,’ the innate categories according to which we apprehend the world.” Following a separate study conducted with the Mundurukú—an isolated Amazonian indigenous group—Dehaene and his collaborators added in 2006 a similar judgment about geometry: “The spontaneous understanding of geometrical concepts and maps by this remote human community provides evidence that core geometrical knowledge, like basic arithmetic, is a universal constituent of the human mind.” Not all cognitive scientists agree with the latter conclusions. Some point out, for instance, that the success of the Mundurukú in the recent geometrical study, in which they had to identify a curve among straight lines, a rectangle among squares, an ellipse among circles, and so on, may have more to do with their visual ability to spot the odd one out, rather than with an innate geometrical knowledge.
The French neurobiologist Jean-Pierre Changeux, who engaged in a fascinating dialogue on the nature of mathematics with the mathematician (of Platonic “persuasion”) Alain Connes in
Conservations on Mind, Matter, and Mathematics,
provided the following observation:
The reason mathematical objects have nothing to do with the sensible world has to do…with their generative character, their capacity to give birth to other objects. The point that needs emphasizing here is that there exists in the brain what may be called a “conscious compartment,” a sort of physical space for simulation and creation of new objects…In certain respects these new mathematical objects are like living beings: like living beings they’re physical objects susceptible to very rapid evolution; unlike living beings, with the particular exception of viruses, they evolve in our brain.
Finally, the most categorical statement in the context of invention versus discovery was made by cognitive linguist George Lakoff and psychologist Rafael Núñez in their somewhat controversial book
Where Mathematics Comes From.
As I have noted already in chapter 1, they pronounced:
Mathematics is a natural part of being human. It arises from our bodies, our brains, and our everyday experiences in the world. [Lakoff and Núñez therefore speak of mathematics as arising from an “embodied mind”]…Mathematics is a system of human concepts that makes extraordinary use of the ordinary tools of human cognition…Human beings have been responsible for the creation of mathematics, and we remain responsible for maintaining and extending it. The portrait of mathematics has a human face.
The cognitive scientists base their conclusions on what they regard as a compelling body of evidence from the results of numerous experiments. Some of these tests involved functional imaging studies of the brain during the performance of mathematical tasks. Others examined
the math competence of infants, of hunter-gatherer groups such as the Mundurukú, who were never exposed to schooling, and of people with various degrees of brain damage. Most of the researchers agree that certain mathematical capacities appear to be innate. For instance, all humans are able to tell at a glance whether they are looking at one, two, or three objects (an ability called
subitizing
). A very limited version of arithmetic, in the form of grouping, pairing, and very simple addition and subtraction, may also be innate, as is perhaps some very basic understanding of geometrical concepts (although this assertion is more controversial). Neuroscientists have also identified regions in the brain, such as the angular gyrus in the left hemisphere, that appear to be crucial for juggling numbers and mathematical computations, but which are not essential for language or the working memory.