Is God a Mathematician? (32 page)

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 existence.

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.

As I noted before, Lakoff and Núñez emphasize the role of metaphors in mathematics. Cognitive linguists also argue that all human languages use metaphors to express almost everything. Even more importantly perhaps, ever since 1957, the year in which the famous linguist Noam Chomsky published his revolutionary work
Syntactic Structures,
many linguistic endeavors have revolved around the concept of a
universal grammar
—principles that govern all languages. In other words, what appears to be a Tower of Babel of diversity may really hide a surprising structural similarity. In fact, if this were not the case, dictionaries that translate from one language into another might have never worked.

You may still wonder why mathematics is as uniform as it is, both in terms of subject matter and in terms of symbolic notation. The first part of this question is particularly intriguing. Most mathematicians agree that mathematics as we know it has evolved from the basic branches of geometry and arithmetic that were practiced by the ancient Babylonians, Egyptians, and Greeks. However, was it truly inevitable that mathematics would start with these particular disciplines? Computer scientist Stephen Wolfram argued in his massive book
A New Kind of Science
that this was not necessarily the case. In particular, Wolfram showed how starting from simple sets of rules that act as short computer programs (known as
cellular automata
), one could develop a very different type of mathematics. These cellular automata could be used (in principle, at least) as the basic tools for modeling natural phenomena, instead of the differential equations that have dominated science for three centuries. What was it, then, that drove the ancient civilizations toward discovering and inventing our special “brand” of mathematics? I don’t really know, but it may
have had much to do with the particulars of the human perceptual system. Humans detect and perceive edges, straight lines, and smooth curves very easily. Notice, for instance, with what precision you can determine (just by eye) whether a line is perfectly straight, or how effortlessly you are able to distinguish between a circle and a shape that is slightly elliptical. These perceptual abilities may have strongly shaped the human experience of the world, and may have therefore led to a mathematics that was based on discrete objects (arithmetic) and on geometrical figures (Euclidean geometry).

The uniformity in symbolic notation is probably a result of what one might call the “Microsoft Windows effect”: The entire world is using Microsoft’s operating system—not because this conformity was inevitable, but because once one operating system started to dominate the computer market, everybody had to adopt it to allow for ease in communication and for availability of products. Similarly, the Western symbolic notation imposed uniformity on the world of mathematics.

Intriguingly, astronomy and astrophysics may still contribute to the “invention and discovery” question in interesting ways. The most recent studies of extrasolar planets indicate that about 5 percent of all stars have at least one giant planet (like Jupiter in our own solar system) revolving around them, and that this fraction remains roughly constant, on the average, all across the Milky Way galaxy. While the precise fraction of
terrestrial
(Earth-like) planets is not yet known, chances are that the galaxy is teeming with billions of such planets. Even if only a small (but nonnegligible) fraction of these “Earths” are in the
habitable zone
(the range of orbits that allows for liquid water on a planet’s surface) around their host stars, the probability of life in general, and of intelligent life in particular, developing on the surface of these planets is not zero. If we were to discover another intelligent life form with which we could communicate, we could gain invaluable information about the formalisms developed by this civilization to explain the cosmos. Not only would we make unimaginable progress in the understanding of the origin and evolution of life, but we could even compare our logic to the logical system of potentially more advanced creatures.

On a much more speculative note, some scenarios in cosmology (e.g., one known as
eternal inflation
) predict the possible existence of multiple universes. Some of these universes may not only be characterized by different values of the
constants of nature
(e.g., the strengths of the different forces; the mass ratios of subatomic particles), but even by different laws of nature altogether. Astrophysicist Max Tegmark argues that there should even be a universe corresponding to (or that
is,
in his language) each possible mathematical structure. If this were true, this would be an extreme version of the “universe
is
mathematics” perspective—there isn’t just one world that can be identified with mathematics, but an entire ensemble of them. Unfortunately, not only is this speculation radical and currently untestable, it also appears (at least in its simplest form) to contradict what has become known as the
principle of mediocrity
. As I have described in chapter 5, if you pick a person at random on the street, you have a 95 percent chance that his or her height would be within two standard deviations from the mean height. A similar argument should apply to the properties of universes. But the number of possible mathematical structures increases dramatically with increasing complexity. This means that the most “mediocre” structure (close to the mean) should be incredibly intricate. This appears to be at odds with the relative simplicity of our mathematics and our theories of the universe, thus violating the natural expectation that our universe should be typical.

Wigner’s Enigma

“Is mathematics created or discovered?” is the wrong question to ask because it implies that the answer has to be one or the other and that the two possibilities are mutually exclusive. Instead, I suggest that mathematics is partly created and partly discovered. Humans commonly invent mathematical concepts and discover the relations among those concepts. Some empirical discoveries surely preceded the formulation of concepts, but the concepts themselves undoubtedly provided an incentive for more theorems to be discovered. I should also note that some philosophers of mathematics, such as the American Hilary Putnam, adopt an intermediate position known as
realism
—they believe in the objectivity of mathematical discourse (that is, sentences are true or false, and what makes them true or false is external to humans) without committing themselves, like the Platonists, to the existence of “mathematical objects.” Do any of these insights also lead to a satisfactory explanation for Wigner’s “unreasonable effectiveness” puzzle?

Let me first briefly review some of the potential solutions proposed by contemporary thinkers. Physics Nobel laureate David Gross writes:

In other words, Gross relies here on a version of the “mathematics as a discovery” perspective that is somewhere between the Platonic world and the “universe
is
mathematics” world, but closer to a Platonic viewpoint. As we have seen, however, it is difficult to philosophically support the “mathematics as a discovery” claim. Furthermore, Platonism cannot truly solve the problem of the phenomenal accuracy that I have described in chapter 8, a point acknowledged by Gross.

Sir Michael Atiyah, whose views on the nature of mathematics I have largely adopted, argues as follows:

This line of reasoning is very similar to the solutions proposed by the cognitive scientists. Atiyah also recognizes, however, that this explanation hardly addresses the thornier parts of the problem—how does mathematics explain the more esoteric aspects of the physical world. In particular, this explanation leaves the question of what I called the “passive” effectiveness (mathematical concepts finding applications long after their invention) entirely open. Atiyah notes: “The skeptic can point out that the struggle for survival only requires us to cope with physical phenomena at the human scale, yet mathematical theory appears to deal successfully with all scales from the atomic to the galactic.” To which his only suggestion is: “Perhaps the explanation lies in the abstract hierarchical nature of mathematics which enables us to move up and down the world scale with comparative ease.”

The American mathematician and computer scientist Richard Hamming (1915–98) provided a very extensive and interesting discussion of Wigner’s enigma in 1980. First, on the question of the nature of mathematics, he concluded that “mathematics has been made by man and therefore is apt to be altered rather continuously by him.” Then, he proposed four potential explanations for the unreasonable effectiveness: (1) selection effects; (2) evolution of the mathematical tools; (3) the limited explanatory power of mathematics; and (4) evolution of humans.

Recall that selection effects are distortions introduced in the results of experiments either by the apparatus being used or by the way in which the data are collected. For instance, if in a test of the efficiency of a dieting program the researcher would reject everyone who drops out of the trial, this would bias the result, since most likely the ones who drop out are those for whom the program wasn’t working. In other words, Hamming suggests that at least in some cases, “the original phenomenon arises from the mathematical tools we use and not from the real world…a lot of what we see comes from the glasses
we put on.” As an example, he correctly points out that one can show that any force symmetrically emanating from a point (and conserving energy) in three-dimensional space should behave according to an inverse-square law, and therefore that the applicability of Newton’s law of gravity should not be surprising. Hamming’s point is well taken, but selection effects can hardly explain the fantastic accuracy of some theories.