Readings
Is God a Mathematician?
Mercury, January/February 2003 Table of Contents
For some inexplicable reason, mathematics does an extraordinary job of explaining the universe.
By Mario Livio
Many outstanding physicists, most notably Albert Einstein, Eugene Wigner, and James Jeans, remarked that mathematics appears to be just too effective in explaining the universe. Wigner, in particular, wrote a remarkable paper in 1960 titled "The unreasonable Effectiveness of Mathematics in the Physical Sciences." He wrote, "The miracle of the appropriateness of the language of mathematics to the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve."
We may wonder, for example, why all the phenomena encompassed by electromagnetism, from the behavior of electrons to the nature of light, can be explained by a set of four differential equations known as Maxwell's equations. Equally puzzling is the fact that some geometrical curves like the ellipse, invented/discovered by the Greek mathematician Menaechmus around 350 BC, were found 2,000 years later to describe the orbits of planets around the Sun. Similarly, group theory proved to be essential in the understanding of both the organization of elementary (subatomic) particles, and the structure of solids. What is it that makes mathematics fit the observable universe like a glove?
The attempts to answer this question fall generally into two broad categories. According to one view, mathematics is in some sense the actual "language" of the universe. It exists independent of us humans, and we are merely discovering it in the workings of the cosmos. Proponents of this philosophy like to point out that even some of the more esoteric areas of mathematics, such as non-Euclidean geometries, were eventually found to provide cornerstones to cosmological models.
An even more impressive example, perhaps, is provided by Calabi-Yau manifolds.
These intricate-looking complex spaces were investigated by mathematicians Euginio Calabi at the University of Pennsylvania and Shing-Tung Yau at Harvard University years before the introduction of string theory as a potential framework for the unification of all the interactions in the universe (gravity, electromagnetism, and the strong nuclear and weak nuclear forces). In fact, already in 1957 Calabi hypothesized that a certain type of metric that appears in general relativity is admitted by these mathematical spaces. Yau provided a rigorous proof in 1977. Seven years later, four physicists demonstrated that the six, curled-up, extra spatial dimensions required by string theories (beyond those of our familiar three spatial dimensions and one time dimension) can be accommodated in a six0dimensional Calabi-Yau space.
The success of mathematics in explaining nature is no accident, if one accepts this "language of the universe" premise the cosmos has literally imposed mathematics upon humanity.
Many thinkers throughout history have espoused the above view. In II Saggiatore ( The Essayist), Galileo Galilei writes: "Philosophy is written in this grand book -- I mean the universe -- which stands continually open to our gaze, but it cannot be understood unless one first to learn the language and intepret the characters in which it is written. It is written in the language of mathematics, and its characters are triangles, circles, and other geometrical figures, without which it is humanly impossible to understand a single word of it, without these, one is wandering about in dark labyrinth." Given that these ideas put mathematics on a footing somewhat similar to that of religion -- both represent a relationship between humans and the universe -- it should come as no surprise that some religious natural philosophers regarded mathematics as a manifestation as divine thought. In the words of the great early 17th-century astronomer Johannes Kepler, "Geometry which before the origin of things was coeternal would not be God himself), supplied God with patterns for the creation of the world."
Even Hermann Wevl, one of the leading mathematicians of the 20th century,writes:
"...purely mathematical inquiry in itself, according to the conviction of many graet thinkers, by its special character, its certainty and stringency, li
fts the human mind into closer proximity with the divine than attainable through any other medium."
There exists, however, a very different view of mathematics, according to which mathematics is nothing but a human invention that has no real existence outside the human brain. In the words of the great German philosopher Immanuel Kant, "The ultimate truth of mathematics lies in the possibility that its concepts can be constructed by the human mind." "Theories" of the universe are, according to this view, only models, the utility of which is determined solely by their success in explaining natural phenomena.
The effectiveness of mathematics in the physical, biological and social sciences is in this case a direct consequence of evolution and natural selection of ideas. In other words, over the centuries mathematicians have produced a plethora of mathematical constructs, and models of the universe galore. Many of these models proved to be blind alleys (such as the Ptolemaic models of the solar system) and were eventually discarded. The successful ones have been continuously improved with superior data and new mathematical machinery becoming available. The road to our present theoretical thinking about the cosmos, according to this view (sometimes labeled "intuitionist"), has been very similar to the emergence of Homo sapiens via the tortuous evolution of species.
The apparently miraculous applicability (to models of the universe) of some mathematical tools originally conceived with no application in mind, reflects, in this case, a mere overproduction of ideas, of which physics has selected only the appropriate ones. The latter point of view has become increasingly popular, especially with psychologists and researchers in the field of embodied cognition. For example, Berkeley linguist George Lakoff and Frieburg University psychologist Rafael Nunez write (in Where Mathematics Comes From): "Sometimes human physicists are successful in fitting human mathematics as they conceptualize it to their human conceptualization of the regularities they observe in the physical world." Similarly, cognitive neuropsychologist Stanislas Dehaene concludes in The Number Sense.
"There is one instrument on which scientists rely so regularly that they sometimes forget its very existence their own brain ... Is the universe really written in mathematical language, as Galileo contended? I am inclined to think instead that this is the inly language with which we [the emphasis is mine] can try to read it."
So, was mathematics discovered or invented? In his recent thought-provoking book A New Kind of Science, computer scientist Stephen Wolfram strongly argues for mathematics being a human invention. Wolfram shows that computer programs and cellular automata can embody more general rules than afforded by mathematical equations, and that the systems of axioms representing conventional mathematics cover only a tiny fraction of th huge range of all possible abstract systems. Any attempt to describe mathematics as a pure discovery would thus leave us entirely in the dark with respect to two major questions: (1) Can the human mind get access to that mythical space in which mathematics presumably exists? (2) How can we call the choice of a small number of sets of rules (out of an immense range of possibilities) a "discovery"?
However, if we were to conclude that mathematics is entirely a human invention, this would raise two different questions: (1) How can we explain all the unanticipated theorems that emerged from the systems of axioms that were chose (including ones that we still don't know how to prove)? (2) Why did the Italian mathematician Gluseppe Peano, after all, choose a particular set of carefully crafted axioms for the theory of numbers, and not any other set (and similar questions about the axioms of geometry, etc.)?
There is no doubt that even extremely simple systems of rules or axioms can generate highly complex behavior or unexpected "theorems". In this sense, mathematician does take on a life of its own once the basic rules are specified, and humans do have to discover its endless list of properties. Mathematics is therefore a human invention that intrinsically contains discoveries. Why did humans come up with this particular version of mathematics, and why is it so effective in explaining the universe? The answer to these two questions may actually be intimately related. Human may have developed branches of mathematics (such as arithmetic and geometry) that are largely based on the human perception of the universe. Arithmetic may reflect the human ability to discern discrete objects and geometry may represent the human brain's sponse to edges and lines. If this is true, then the effectiveness of mathematics may
indeed be a consequence of the fact that the universe has imposed, in some sense, a particular brand of mathematics on humans.
Jef Raskin, who helped create the Macintosh computer, goes even farther. He thinks that human logic, from which mathematics has presumably emerged, was shaped and essentially forced on us by the workings of the universe, via Darwinian natural selection. Raskin's argument goes something like this: human can assert the truth or falsehood of most propositions encountered in everyday life. If some creatures were to develop, with a logic allowing them to assert "true and not true" for certain propositions, such creature might jump off the edge of a cliff thinking that nothing would happen to it, even if it had seen others jumping to their inevitable deaths before it.As strange as this may sound, astronomy could, in principle, provide a more definitive answer to the question about the effectiveness of mathematics. Imagine that we discover many extraterrestrial intelligent civilizations, all of which evolved independently of each other. Imagine further that we find all of these civilizations recognize, for example, the value of π, and the prime numbers. One could then argue that mathematics as we know it, is, in this sense, "universal". These particular astronomical discoveries may prove, however, to be entirely infeasible. Not only is the discovery of extraterrestrial civilization extremely difficult; even the definition of an "intelligent civilization" may prove to be an insurmountable task, of other civilizations are very different from us. In the meantime, therefore, we are forced to continue to use our mathematics, while the question of the cause of its effectiveness remains somewhat unresolved.
Back to my work page