Is God a Mathematician?

by Mario Livio

  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.

  According to Lakoff and Núñez, a major tool for advancement beyond these innate abilities is the construction of conceptual metaphors—thought processes that translate abstract concepts into more concrete ones. For example, the conception of arithmetic is grounded in the very basic metaphor of object collection. On the other hand, Boole’s more abstract algebra of classes metaphorically linked classes to numbers. The elaborate scenario developed by Lakoff and Núñez offers interesting insights into why humans find some mathematical concepts much more difficult than others. Other researchers, such as cognitive neuroscientist Rosemary Varley of the University of Sheffield, suggest that at least some mathematical structures are parasitic on the language faculty—mathematical insights develop by borrowing mind tools used for building language.

  The cognitive scientists make a fairly strong case for an association of our mathematics with the human mind, and against Platonism. Interestingly, though, what I regard as possibly the strongest argument against Platonism comes not from neurobiologists, but rather from Sir Michael Atiyah, one of the greatest mathematicians of the twentieth century. I did, in fact, mention his line of reasoning briefly in chapter 1, but I would now like to present it in more detail.

  If you had to choose one concept of our mathematics that has the highest probability of having an existence independent of the human mind, which one would you select? Most people would probably conclude that this has to be the natural numbers. What can be more “natural” than 1, 2, 3,…? Even the German mathematician of intuitionist inclinations Leopold Kronecker (1823–91) famously declared: “God created the natural numbers, all else is the work of man.” So if one could show that even the natural numbers, as a concept, have their origin in the human mind, this would be a powerful argument in favor of the “invention” paradigm. Here, again, is how Atiyah argues the case: “Let us imagine that intelligence had resided, not in mankind, but in some vast solitary and isolated jelly-fish, buried deep in the depths of the Pacific Ocean. It would have no experience of individual objects, only with the surrounding water. Motion, temperature and pressure would provide its basic sensory data. In such a pure continuum the discrete would not arise and there would be nothing to count.” In other words, Atiyah is convinced that even a concept as basic as that of the natural numbers was created by humans, by abstracting (the cognitive scientists would say, “through grounding metaphors”) elements of the physical world. Put differently, the number 12, for instance, represents an abstraction of a property that is common to all things that come in dozens, in the same way that the word “thoughts” represents a variety of processes occurring in our brains.

  The reader might object to the use of the hypothetical universe of the jellyfish to prove this point. He or she may argue that there is only one, inevitable universe, and that every supposition should be examined in the context of this universe. However, this would be tantamount to conceding that the concept of the natural numbers is in fact somehow dependent on the universe of human experiences! Note that this is precisely what Lakoff and Núñez mean when they refer to mathematics as being “embodied.”

  I have just argued that the concepts of our mathematics originate in the human mind. You may wonder then why I had insisted earlier
that much of mathematics is in fact discovered, a position that appears to be closer to that of the Platonists.

  Invention and Discovery

  In our everyday language the distinction between discovery and invention is sometimes crystal clear, sometimes a bit fuzzier. No one would say that Shakespeare discovered Hamlet, or that Madame Curie invented radium. At the same time, new drugs for certain types of diseases are normally announced as discoveries, even though they often involve the meticulous synthesis of new chemical compounds. I would like to describe in some detail a very specific example in mathematics, which I believe will not only help clarify the distinction between invention and discovery but also yield valuable insights into the process by which mathematics evolves and progresses.

  In book VI of The Elements, Euclid’s monumental work on geometry, we find a definition of a certain division of a line into two unequal parts (an earlier definition, in terms of areas, appears in book II). According to Euclid, if a line AB is divided by a point C (figure 62) in such a way that the ratio of the lengths of the two segments (AC/CB) is equal to the whole line divided by the longer segment (AB/AC), then the line is said to have been divided in “extreme and mean ratio.” In other words, if AC/CB AB/AC, then each one of these ratios is called the “extreme and mean ratio.” Since the nineteenth century, this ratio is popularly known as the golden ratio. Some easy algebra can show that the golden ratio is equal to

  (1 + √5) / 2 = 1.6180339887…

  The first question you may ask is why did Euclid even bother to define this particular line division and to give the ratio a name? After all, there are infinitely many ways in which a line could be divided. The answer to this question can be found in the cultural, mystical heritage of the Pythagoreans and Plato. Recall that the Pythagoreans were obsessed with numbers. They thought of the odd numbers as being masculine and good, and, rather prejudicially, of the even numbers as being feminine and bad. They had a particular affinity for the number 5, the union of 2 and 3, the first even (female) and first odd (masculine) numbers. (The number 1 was not considered to be a number, but rather the generator of all numbers.) To the Pythagoreans, therefore, the number 5 portrayed love and marriage, and they used the pentagram—the five-pointed star (figure 63)—as the symbol of their brotherhood. Here is where the golden ratio makes its first appearance. If you take a regular pentagram, the ratio of the side of any one of the triangles to its implied base (a/b in figure 63) is precisely equal to the golden ratio. Similarly, the ratio of any diagonal of a regular pentagon to its side (c/d in figure 64) is also equal to the golden ratio. In fact, to construct a pentagon using a straight edge and a compass (the common geometrical construction process of the ancient Greeks) requires dividing a line into the golden ratio.

  Figure 62

  Plato added another dimension to the mythical meaning of the golden ratio. The ancient Greeks believed that everything in the universe is composed of four elements: earth, fire, air, and water. In Timaeus, Plato attempted to explain the structure of matter using the five regular solids that now bear his name—the Platonic solids (figure 65). These convex solids, which include the tetrahedron, the cube, the octahedron, the dodecahedron, and the icosahedron, are the only ones in which all the faces (of each individual solid) are the same, and are regular polygons, and where all the vertices of each solid lie on a sphere. Plato associated each of four of the Platonic solids with one of the four basic cosmic elements. For instance, the Earth was associated with the stable cube, the penetrating fire with the pointy tetrahedron, air with the octahedron, and water with the icosahedron. Concerning the dodecahedron (Figure 65d), Plato wrote in Timaeus: “As there still remained one compound figure, the fifth, God used it for the whole, broidering it with designs.” So the dodecahedron represented the universe as a whole. Note, however, that the dodecahedron, with its twelve pentagonal surfaces, has the golden ratio written all over it. Both its volume and its surface area can be expressed as simple functions of the golden ratio (the same is true for the icosahedron).

  Figure 63

  Figure 64

  History therefore shows that by numerous trials and errors, the Pythagoreans and their followers discovered ways to construct certain geometrical figures that to them represented important concepts, such as love and the entire cosmos. No wonder, then, that they, and Euclid (who documented this tradition), invented the concept of the golden ratio that was involved in these constructions, and gave it a name. Unlike any other arbitrary ratio, the number 1.618…now became the focus of an intense and rich history of investigation, and it continues to pop up even today in the most unexpected places. For instance, two millennia after Euclid, the German astronomer Johannes Kepler discovered that this number appears, miraculously as it were, in relation to a series of numbers known as the Fibonacci sequence. The Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233,…is characterized by the fact that, starting with the third, each number is the sum of the previous two (e.g., 2 = 1 + 1;3 = 1 + 2;5 = 2 + 3; and so on). If you divide each number in the sequence by the one immediately preceding it (e.g., 144 ÷ 89; 233 ÷ 144;…), you find that the ratios oscillate about, but come closer and closer to the golden ratio the farther you go in the sequence. For example, one obtains the following results, rounding the numbers to the sixth decimal place): 144 ÷ 89 1.617978; 233 ÷ 144 1.618056; 377 ÷ 233 1.618026, and so on.

  Figure 65

  In more modern times, the Fibonacci sequence, and concomitantly the golden ratio, were found to figure in the leaf arrangements of some plants (the phenomenon known as phyllotaxis) and in the structure of the crystals of certain aluminum alloys.

  Why do I consider Euclid’s definition of the concept of the golden ratio an invention? Because Euclid’s inventive act singled out this ratio and attracted the attention of mathematicians to it. In China, on the other hand, where the concept of the golden ratio was not invented, the mathematical literature contains essentially no reference to it. In India, where again the concept was not invented, there are only a few insignificant theorems in trigonometry that peripherally involve this ratio.

  There are many other examples that demonstrate that the question “Is mathematics a discovery or an invention?” is ill posed. Our mathematics is a combination of inventions and discoveries. The axioms of Euclidean geometry as a concept were an invention, just as the rules of chess were an invention. The axioms were also supplemented by a variety of invented concepts, such as triangles, parallelograms, ellipses, the golden ratio, and so on. The theorems of Euclidean geometry, on the other hand, were by and large discoveries; they were the paths linking the different concepts. In some cases, the proofs generated the theorems—mathematicians examined what they could prove and from that they deduced the theorems. In others, as described by Archimedes in The Method, they first found the answer to a particular question they were interested in, and then they worked out the proof.

  Typically, the concepts were inventions. Prime numbers as a concept were an invention, but all the theorems about prime numbers were discoveries. The mathematicians of ancient Babylon, Egypt, and China never invented the concept of prime numbers, in spite of their advanced mathematics. Could we say instead that they just did not “discover” prime numbers? Not any more than we could say that the United Kingdom did not “discover” a single, codified, documentary constitution. Just as a country can survive without a constitution, elaborate mathematics could develop without the concept of prime numbers. And it did!

  Do we know why the Greeks invented such concepts as the axioms and prime numbers? We cannot be sure, but we could guess that this was part of their relentless efforts to investigate the most fundamental constituents of the universe. Prime numbers were the basic building blocks of numbers, just as atoms were the building blocks of matter. Similarly, the axioms were the fountain from which all geometrical truths were supposed to flow. The dodecahedron represented the entire cosmos and the golden ratio was the concept that brought that symbol into exis
tence.

  This discussion highlights another interesting aspect of mathematics—it is a part of the human culture. Once the Greeks invented the axiomatic method, all the subsequent European mathematics followed suit and adopted the same philosophy and practices. Anthropologist Leslie A. White (1900–1975) tried once to summarize this cultural facet by noting: “Had Newton been reared in Hottentot [South African tribal] culture he would have calculated like a Hottentot.” This cultural complexion of mathematics is most probably responsible for the fact that many mathematical discoveries (e.g., of knot invariants) and even some major inventions (e.g., of calculus) were made simultaneously by several people working independently.

  Do You Speak Mathematics?

  In a previous section I compared the import of the abstract concept of a number to that of the meaning of a word. Is mathematics then some kind of language? Insights from mathematical logic, on one hand, and from linguistics, on the other, show that to some extent it is. The works of Boole, Frege, Peano, Russell, Whitehead, Gödel, and their modern-day followers (in particular in areas such as philosophical syntax and semantics, and in parallel in linguistics), have demonstrated that grammar and reasoning are intimately related to an algebra of symbolic logic. But why then are there more than 6,500 languages while there is only one mathematics? Actually, all the different languages have many design features in common. For instance, the American linguist Charles F. Hockett (1916–2000) drew attention in the 1960s to the fact that all the languages have built-in devices for acquiring new words and phrases (think of “home page”; “laptop”; “indie flick”; and so on). Similarly, all the human languages allow for abstraction (e.g., “surrealism”; “absence”; “greatness”), for negation (e.g., “not”; “hasn’t”), and for hypothetical phrases (“If grandma had wheels she might have been a bus”). Perhaps two of the most important characteristics of all languages are their open-endedness and their stimulus-freedom. The former property represents the ability to create never-before-heard utterances, and to understand them. For instance, I can easily generate a sentence such as: “You cannot repair the Hoover Dam with chewing gum,” and even though you have probably never encountered this sentence before, you have no trouble understanding it. Stimulus-freedom is the power to choose how, or even if, one should respond to a received stimulus. For instance, the answer to the question posed by singer/songwriter Carole King in her song “Will You Still Love Me Tomorrow?” could be any of the following: “I don’t know if I’ll still be alive tomorrow”; “Absolutely”; “I don’t even love you today”; “Not as much as I love my dog”; “This is definitely your best song”; or even “I wonder who will win the Australian Open this year.” You will recognize that many of these features (e.g., abstraction; negation; open-endedness; and the ability to evolve) are also characteristic of mathematics.