Is God a Mathematician? (31 page)

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.