by Mario Livio
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:
A point of view that, from my experience, is not uncommon among creative mathematicians—namely that the mathematical structures that they arrive at are not artificial creations of the human mind but rather have a naturalness to them as if they were as real as the structures created by physicists to describe the so-called real world. Mathematicians, in other words, are not inventing new mathematics, they are discovering it. If this is the case then perhaps some of the mysteries that we have been exploring [the “unreasonable effectiveness”] are rendered slightly less mysterious. If mathematics is about structures that are a real part of the natural world, as real as the concepts of theoretical physics, then it is not so surprising that it is an effective tool in analyzing the real world.
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:
If one views the brain in its evolutionary context then the mysterious success of mathematics in the physical sciences is at least partially explained. The brain evolved in order to deal with the physical world, so it should not be too surprising that it has developed a language, mathematics, that is well suited for the purpose.
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.
Hamming’s second potential solution relies on the fact that humans select, and continuously improve the mathematics, to fit a given situation. In other words, Hamming proposes that we are witnessing what we might call an “evolution and natural selection” of mathematical ideas—humans invent a large number of mathematical concepts, and only those that fit are chosen. For years I also used to believe that this was the complete explanation. A similar interpretation was proposed by physics Nobel laureate Steven Weinberg in his book Dreams of a Final Theory. Can this be the explanation to Wigner’s enigma? There is no doubt that such selection and evolution indeed occur. After sifting through a variety of mathematical formalisms and tools, scientists retain those that work, and they do not hesitate to upgrade them or change them as better ones become available. But even if we accept this idea, why are there mathematical theories that can explain the universe at all?
Hamming’s third point is that our impression of the effectiveness of mathematics may, in fact, be an illusion, since there is much in the world around us that mathematics does not really explain. In support of this perspective I could note, for instance, that the mathematician Israïl Moseevich Gelfand was once quoted as having said: “There is only one thing which is more unreasonable than the unreasonable effectiveness of mathematics in physics, and this is the unreasonable ineffectiveness [emphasis added] of mathematics in biology.” I don’t think that this in itself can explain away Wigner’s problem. It is true that unlike in The Hitchhiker’s Guide to the Galaxy, we cannot say that the answer to life, the universe, and everything is forty-two. Nevertheless, there is a sufficiently large number of phenomena that mathematics does elucidate to warrant an explanation. Moreover, the range of facts and processes that can be interpreted by mathematics continually widens.
Hamming’s fourth explanation is very similar to the one suggested by Atiyah—that “Darwinian evolution would naturally select for survival those competing forms of life which had the best models of reality in their minds—‘best’ meaning best for surviving and propagating.”
Computer scientist Jef Raskin (1943–2005), who started the Macintosh project for Apple Computer, also held related views, with a particular emphasis on the role of logic. Raskin concluded that
human logic was forced on us by the physical world and is therefore consistent with it. Mathematics derives from logic. This is why mathematics is consistent with the physical world. There is no mystery here—though we should not lose our sense of wonder and amazement at the nature of things even as we come to understand them better.
Hamming was less convinced, even by the strength of his own argument. He pointed out that
if you pick 4,000 years for the age of science, generally, then you get an upper bound of 200 generations. Considering the effects of evolution we are looking for via selection of small chance variations, it does not seem to me that evolution can explain more than a small part of the unreasonable effectiveness of mathematics.
Raskin insisted that “the groundwork for mathematics had been laid down long before in our ancestors, probably over millions of generations.” I must say, however, that I do not find this argument particularly convincing. Even if logic had been deeply embedded in our ancestors’ brains, it is difficult to see how this ability could have led to abstract mathematical theories of the subatomic world, such as quantum mechanics, that display stupendous accuracy.
Remarkably, Hamming concluded his article with an admission that “all of the explanations I have given when added together simply are not enough to explain what I set out to account for” (namely, the unreasonable effectiveness of mathematics).
So, should we close by conceding that the effectiveness of mathematics remains as mysterious as it was when we started?
Before giving up, let us try to distill the essence of Wigner’s puzzle by examining what is known as the scientific method. Scientists first learn facts about nature through a series of experiments and observations. Those facts are initially used to develop some sort of qualitative models of the phenomena (e.g., the Earth attracts apples; colliding subatomic particles can produce other particles; the universe is expanding; and so on). In many branches of science even the emerging theories may remain nonmathematical. One of the best examples of a powerfully explanatory theory of this type is Darwin’s theory of evolution. Even though natural selection is not based on a mathematical formalism, its success in clarifying the origin of species has been remarkable. In fundamental physics, on the other hand, usually the next step involves attempts to construct mathematical, quantitative theories (e.g., general relativity; quantum electrodynamics; string theory; and so on). Finally, the researchers use those mathematical models to predict new phenomena, new particles, and results of never-before-performed experiments and observations. What puzzled Wigner and Einstein was the incredible success of the last two processes. How is it possible that time after time physicists are able to find mathematical tools that not only explain the existing experimental and observational results, but which also lead to entirely new discernments and new predictions?
I attempt to answer this version of the question by borrowing a beautiful example from mathematician Reuben Hersh. Hersh proposed that in the spirit of the analysis of many such problems in mathematics (and indeed in theoretical physics) one should examine the simplest possible case. Consider the seemingly trivial experiment of putting pebbles into an opaque vase. Suppose you first put in four white pebbles, and later you put in seven black pebbles. At some point in their history, humans learned that for some purposes they could represent a collection of pebbles of any color by an abstract concept that they had invented—a natural number. That is, the collection of white pebbles could be asso
ciated with the number 4 (or IIII or IV or whichever symbol was used at the time) and the black pebbles with the number 7. Via experimentation of the type I have described above, humans also discovered that another invented concept—arithmetic addition—represents correctly the physical act of aggregation. In other words, the result of the abstract process denoted symbolically by 4 7 can predict unambiguously the final number of pebbles in the vase. What does all of this mean? It means that humans have developed an incredible mathematical tool—one that could reliably predict the result of any experiment of this type! This tool is actually much less trivial than it might seem, because the same tool, for instance, does not work for drops of water. If you put four separate drops of water into the vase, followed by seven additional drops, you don’t get eleven separate drops of water in the vase. In fact, to make any kind of prediction for similar experiments with liquids (or gases), humans had to invent entirely different concepts (such as weight) and to realize that they have to weigh individually each drop or volume of gas.