The Golden Ratio: The Story of Phi, the World's Most Astonishing Number - Mario Livio (2003)
9. IS GOD A MATHEMATICIAN?
I should attempt to treat human vice and folly geometrically…the passions of hatred, anger, envy, and so on, considered in themselves, follow from the necessity and efficacy of nature. … I shall, therefore, treat the nature and strength of the emotion in exactly the same manner, as though I were concerned with lines, planes and solids.
—BARUCH SPINOZA (1632–1677)
Two and two the mathematician continues to make four, in spite of the whine of the amateur for three, or the cry of the critic for five.
—JAMES MCNEILL WHISTLER (1834–1903)
Euclid defined the Golden Ratio because he was interested in using this simple proportion for the construction of the pentagon and the pentagram. Had this remained the Golden Ratio's only application, the present book would have never been written. The delight we derive from this concept today is based primarily on the element of surprise. The Golden Ratio turned out to be, on one hand, the simplest of the continued fractions (but also the “most irrational” of all irrational numbers) and, on the other, the heart of an endless number of complex natural phenomena. Somehow the Golden Ratio always makes an unexpected appearance at the juxtaposition of the simple and the complex, at the intersection of Euclidean geometry and fractal geometry.
The sense of gratification provided by the Golden Ratio's surprising emergences probably comes as close as we could expect to the sensuous visual pleasure we obtain from a work of art. This fact raises the question of what type of aesthetic judgment can be applied to mathematics or, even more specifically, what did the famous British mathematician Godfrey Harold Hardy (1877–1947) actually mean when he said: “The mathematician's patterns, like the painter's or the poet's, must be beautiful.”
This is not an easy question. When I discussed the psychological experiments that tested the visual appeal of the Golden Rectangle, I deliberately avoided the term “beautiful.” I will adopt the same strategy here, because of the ambiguity associated with the definition of beauty. The extent to which beauty is in the eye of the beholder when referring to mathematics is exemplified magnificently by a story presented in the excellent 1981 book The Mathematical Experience by Philip J. Davis and Reuben Hersh.
In 1976, a delegation of distinguished mathematicians from the United States was invited to the People's Republic of China for a series of talks and informal meetings with Chinese mathematicians. The delegation subsequently issued a report entitled “Pure and Applied Mathematics in the People's Republic of China.” By “pure,” mathematicians usually refer to the type of mathematics that at least on the face of it has absolutely no direct relevance to the world outside the mind. At the same time, we should realize that Penrose tilings and random Fibonaccis, for example, provide two of the numerous examples of “pure” mathematics turning into “applied.” One of the dialogues in the delegation's report, between Princeton mathematician Joseph J. Kohn and one of his Chinese hosts, is particularly illuminating. The dialogue was on the topic of the “beauty of mathematics,” and it took place at the Shanghai Hua-Tung University.
Since, as this dialogue starkly indicates, there is hardly any formal, accepted description of aesthetic judgment in mathematics and how it should be applied, I prefer to discuss only one particular element in mathematics that invariably gives pleasure to nonexperts and experts alike—the element of surprise.
MATHEMATICS SHOULD SURPRISE
In a letter written on February 27, 1818, the English Romantic poet John Keats (1795–1821) wrote: “Poetry should surprise by a fine excess and not by Singularity—it should strike the Reader as a wording of his own highest thoughts, and appear almost a Remembrance.” Unlike poetry, however, mathematics more often tends to delight when it exhibits an unanticipated result rather than when conforming to the reader's own expectations. In addition, the pleasure derived from mathematics is related in many cases to the surprise felt upon perception of totally unexpected relationships and unities. A mathematical relation known as Benford's law provides a wonderful case study for how all of these elements combine to produce a great sense of satisfaction.
Take a look, for example, in the World Almanac, at the table of “U.S. Farm Marketings by State” for 1999. There is a column for “Crops” and one for “Livestock and Products.” The numbers are given in U.S. dollars. You would have thought that the numbers from 1 to 9 should occur with the same frequency among the first digits of all the listed marketings. Specifically, the numbers starting with 1 should constitute about one-ninth of all the listed numbers, as would numbers starting with 9. Yet, if you count them, you will find that the number 1 appears as the first digit in 32 percent of the numbers (instead of the expected 11 percent if all digits occurred equally often). The number 2 also appears more frequently than its fair share—appearing in 19 percent of the numbers. The number 9, on the other hand, appears only in 5 percent of the numbers—less than expected. You may think that finding this result in one table is surprising, but hardly shocking, until you examine a few more pages in the Almanac (the numbers above were taken from the 2001 edition). For example, if you look at the table of the death toll of “Some Major Earthquakes,” you will find that the numbers starting with 1 constitute about 38 percent of all the numbers, and those starting with 2 are 18 percent. If you choose a totally different table, such as the one for the population in Massachusetts in places of 5,000 or more, the numbers start with 1 about 36 percent of the time and with 2 about 16.5 percent of the time. At the other end, in all of these tables the number 9 appears first only in about 5 percent of the numbers, far less than the expected 11 percent. How is it possible that tables describing such diverse and apparently random data all have the property that the number 1 appears as the first digit 30-some percent of the time and the number 2 around 18 percent of the time? The situation becomes even more puzzling when you examine still larger databases. For example, accounting professor Mark Nigrini of the Cox School of Business at Southern Methodist University, Dallas, examined the populations of 3,141 counties in the 1990 U.S. Census. He found that the number 1 appeared as the first digit in about 32 percent of the numbers, 2 appeared in about 17 percent, 3 in 14 percent, and 9 in less than 5 percent. Analyst Eduardo Ley of Resources for the Future in Washington, D.C., found very similar numbers for the Dow Jones Industrial Average in the years 1990 to 1993. And if all of this is not dumfounding enough, here is another amazing fact. If you examine the list of, say, the first two thousand Fibonacci numbers, you will find that the number 1 appears as the first digit 30 percent of the time, the number 2 appears 17.65 percent, 3 appears 12.5 percent, and the values continue to decrease, with 9 appearing 4.6 percent of the time as first digit. In fact, Fibonacci numbers are more likely to start with 1, with the other numbers decreasing in popularity in precisely the same manner as the just-described random selections of numbers!
Astronomer and mathematician Simon Newcomb (1835–1909) first discovered this “first-digit phenomenon” in 1881. He noticed that books of logarithms in the library, which were used for calculations, were considerably dirtier at the beginning (where numbers starting with 1 and 2 were printed) and progressively cleaner throughout. While this might be expected with bad novels abandoned by bored readers, in the case of mathematical tables they simply indicated a more frequent appearance of numbers starting with 1 and 2. Newcomb, however, went much further than merely noting this fact; he came up with an actual formula that was supposed to give the probability that a random number begins with a particular digit. That formula (presented in Appendix 9) gives for 1 a probability of 30 percent; for 2, about 17.6 percent; for 3, about 12.5 percent; for 4, about 9.7 percent; for 5, about 8 percent; for 6, about 6.7 percent; for 7, about 5.8 percent; for 8, about 5 percent; and for 9, about 4.6 percent. Newcomb's 1881 article in the American Journal of Mathematics and the “law” he discovered went entirely unnoticed, until fifty-seven years later, when physicist Frank Benford of General Electric rediscovered the law (apparently independently) and tested it with extensive data on river basin areas, baseball statistics, and even numbers appearing in Reader's Digestarticles. All the data fit the postulated formula amazingly well, and hence this formula is now known as Benford's law.
Not all lists of numbers obey Benford's law. Numbers in telephone books, for example, tend to begin with the same few digits in any given region. Even tables of square roots of numbers do not obey the law. On the other hand, chances are that if you collect all the numbers appearing on the front pages of several of your local newspapers for a week, you will obtain a pretty good fit. But why should it be this way? What do the populations of towns in Massachusetts have to do with death tolls from earthquakes around the globe or with numbers appearing in the Reader's Digest? Why do the Fibonacci numbers also obey the same law?
Attempts to put Benford's law on a firm mathematical basis have proven to be much more difficult than expected. One of the key obstacles has been precisely the fact that not all lists of numbers obey the law (even the preceding examples from the Almanac do not obey the law precisely). In his Scientific American article describing the law in 1969, University of Rochester mathematician Ralph A. Raimi concluded that “the answer remains obscure.”
The explanation finally emerged in 1995–1996, in the work of Georgia Institute of Technology mathematician Ted Hill. Hill became first interested in Benford's law while preparing a talk on surprises in probability in the early 1990s. When describing to me his experience, Hill said: “I started working on this problem as a recreational experiment, but a few people warned me to be careful, because Benford's law can become addictive.” After a few years of work it finally dawned on him that rather than looking at numbers from one given source, the mixture of data was the key. Hill formulated the law statistically, in a new form: “If distributions are selected at random (in any unbiased way) and random samples are taken from each of these distributions, then the significant-digit frequencies of the combined sample will converge to Benford's distribution, even if some of the individual distributions selected do not follow the law.” In other words, suppose you assemble random collections of numbers from a hodgepodge of distributions, such as a table of square roots, a table of the death toll in notable aircraft disasters, the populations of counties, and a table of air distances between selected world cities. Some of these distributions do not obey Benford's law by themselves. What Hill proved, however, is that as you collect ever more of such numbers, the digits of these numbers will yield frequencies that conform ever closer to the law's predictions. Now, why do Fibonacci numbers also follow Benford's law? After all, they are fully determined by a recursive relation and are not random samples from random distributions.
Well, in this case it turns out that this conformity with Benford's law is not a unique property of the Fibonacci numbers. If you examine a large number of powers of 2 (21 = 2, 22 = 4, 23 = 8, etc.), you'll see that they also obey Benford's law. This should not be so surprising, given that the Fibonacci numbers themselves are obtained as powers of the Golden Ratio (recall that the nth Fibonacci number is close to ). In fact, we can prove that sequences defined by a large class of recursive relations follow Benford's law.
Benford's law provides yet another fascinating example of pure mathematics transformed into applied. One interesting application is in the detection of fraud or fabrication of data in accounting and tax evasion. In a broad range of financial documents, data conform very closely to Benford's law. Fabricated data, on the other hand, very rarely do. Hill demonstrates how such fraud detection works with another simple example, using probability theory. In the first day of class in his course on probability, he asks students to do an experiment. If their mother's maiden name begins with A through L, they are to flip a coin 200 times and record the results. The rest of the class is asked to fake a sequence of 200 heads and tails. Hill collects the results the following day, and within a short time he is able to separate the genuine from the fake with 95 percent accuracy. How does he do that? Any sequence of 200 genuine coin tosses contains a run of six consecutive heads or six consecutive tails with a very high probability. On the other hand, people trying to fake a sequence of coin tosses very rarely believe that they should record such a sequence.
A recent case in which Benford's law was used to uncover fraud involved an American leisure and travel company. The company's audit director discovered something that looked odd in claims made by the supervisor of the company's healthcare department. The first two digits of the healthcare payments showed a suspicious spike in numbers starting with 65 when checked for conformity to Benford's law. (A more detailed version of the law predicts also the frequency of the second and higher digits; see Appendix 9.) A careful audit revealed thirteen fraudulent checks for amounts between $6,500 and $6,599. The District Attorney's office in Brooklyn, New York, also used tests based on Ben-ford's law to detect accounting fraud in seven New York companies.
Benford's law contains precisely some of the ingredients of surprise that most mathematicians find attractive. It reflects a simple but astonishing fact—that the distribution of first digits is extremely peculiar. In addition, that fact turned out to be difficult to explain. Numbers, with the Golden Ratio as an outstanding example, sometimes provide a more instantaneous gratification. For example, many professional and amateur mathematicians are fascinated by primes. Why are primes so important? Because the “Fundamental Theorem of Arithmetic” states that every whole number larger than 1 can be expressed as a product of prime numbers. (Note that 1 is not considered a prime.) For example, 28 = 2 × 2 × 7; 66 = 2 × 3 × 11; and so on. Primes are so rooted in the human comprehension of mathematics that in his book Cosmos, when Carl Sagan (1934–1996) had to describe what type of signal an intelligent civilization would transmit into space he chose as an example the sequence of primes. Sagan wrote: “It is extremely unlikely that any natural physical process could transmit radio messages containing prime numbers only. If we received such a message we would deduce a civilization out there that was at least fond of prime numbers.” The great Euclid proved more than two thousand years ago that infinitely many primes exist. (The elegant proof is presented in Appendix 10.) Yet most people will agree that some primes are more attractive than others. Some mathematicians, such as the French François Le Lionnais and the American Chris Caldwell, maintain lists of “remarkable” or “titanic” numbers. Here are just a few intriguing examples from the great treasury of primes:
· The number 1,234,567,891, which cycles through all the digits, is a prime.
· The 230th largest prime, which has 6,400 digits, is composed of 6,399 9s and only one 8.
· The number composed of 317 iterations of the digit 1 is a prime.
· The 713th largest prime can be written as (10 1951)X(101975 + 1991991991991991991991991)+1, and it was discovered in—you guessed it—1991.
From the perspective of this book, the connection between primes and Fibonacci numbers is of special interest. With the exception of the number 3, every Fibonacci number that is a prime also has a prime subscript (its order in the sequence). For example, the Fibonacci number 233 is a prime, and it is the thirteenth (also a prime) number in the sequence. The converse, however, is not true: The fact that the subscript is a prime does not necessarily mean that the number is also a prime. For example, the nineteenth number (19 is a prime) is 4181, and 4181 is not a prime—it is equal to 113 × 37.
The number of known Fibonacci primes has increased steadily over the years. In 1979, the largest known Fibonacci prime was the 531st in the sequence. By the mid-1990s, the largest known was the 2,971st; and in 2001, the 81,839st number was shown to be a prime with 17,103 digits. So, is there an infinite number of Fibonacci primes (as there is an infinite number of primes, in general)? No one actually knows, and this is probably the greatest unsolved mathematical mystery about Fibonacci numbers.
THE UNREASONABLE POWER OF MATHEMATICS
The collection of dialogues Intentions contains the aesthetic philosophy of the famous playwright and poet Oscar Wilde (1854–1900). In that collection, the dialogue “The Decay of Lying” is a particularly provocative presentation of Wilde's ideas on “the new aesthetics.” In the conclusion of this dialogue, one of the characters (Vivian) summarizes:
Life imitates Art far more than Art imitates Life. This results not merely from Life's imitative instinct, but from the fact that the selfconscious aim of Life is to find expression, and that Art offers it certain beautiful forms through which it may realize that energy. It is a theory that has never been put forward before, but it is extremely fruitful, and throws an entirely new light upon the history of Art.
It follows, as a corollary from this, that external Nature also imitates Art. The only effects that she can show us are effects that we have already seen through poetry, or in paintings. This is the secret of Nature's charm, as well as the explanation of Nature's weakness.
We could almost substitute “Mathematics” for “Art” in this passage and obtain a statement that reflects the reality with which many outstanding minds have struggled. Mathematics appears at first glance to be just too effective. In Einstein's own words: “How is it possible that mathematics, a product of human thought that is independent of experience, fits so excellently the objects of physical reality?” Another outstanding physicist, Eugene Wigner (1902–1995), known for his many contributions to nuclear physics, delivered in 1960 a famous lecture entitled “The Unreasonable Effectiveness of Mathematics in the Physical Sciences.” We have to wonder, for example, how is it possible that planets in their orbits around the Sun were found to follow a curve (an ellipse) that had been explored by the Greek geometers long before Kepler's laws were discovered? Why does the explanation of the existence of quasi-crystals rely on the Golden Ratio, a concept conceived by Euclid for purely mathematical purposes? Is it not astounding that the structure of so many galaxies containing billions of stars follows closely Bernoulli's favorite curve—the magnificent logarithmic spiral? And the most astonishing of all: Why are the laws of physics themselves expressible as mathematical equations in the first place?
But this is not all. Mathematician John Forbes Nash (now world famous as the subject of the book and film biography A Beautiful Mind), for example, shared the 1994 Nobel Prize in economics because his mathematical dissertation (written at age twenty-one!) outlining his “Nash Equilibrium” for strategic noncooperative games inaugurated a revolution in fields as diverse as economics, evolutionary biology, and political science. What is it that makes mathematics work so well?
The recognition of the extraordinary “effectiveness” of mathematics even made it into a hysterically funny passage in Samuel Beckett's novel Molloy, about which I have a personal story. In 1980, two colleagues from the University of Florida and I wrote a paper about neutron stars, which are extremely compact and dense astronomical objects that result from the gravitational collapse of the cores of massive stars. The paper was more mathematical than the garden variety of astronomical papers, and consequently we decided to add an appropriate motto on the first page. The motto read:
Extraordinary how mathematics help you…
—SAMUEL BECKETT, Molloy
—SAMUEL BECKETT, Molloy
The line was cited as being taken from the first of the trilogy of novels Molloy, Malone Dies, and The Unnamable by the famous writer and playwright Samuel Beckett (1906–1989). All three novels, incidentally, represent a search for self—a hunt for identity by writers through writing. We are led to observe the characters in states of decay while they pursue a meaning for their existence.
Papers in astrophysics very rarely have mottoes. Consequently, we received a letter from the editor of The Astrophysical Journal informing us that while he liked Beckett, too, he did not quite see the necessity of including the motto. We replied that we would leave the decision of whether to publish the motto or not entirely to him, and the paper eventually was published with the motto in the December 15 issue. Here, however, is the full passage from Molloy:
And in winter, under my greatcoat, I wrapped myself in swathes of newspaper, and did not shed them until the earth awoke, for good, in April. The Times Literary Supplement was admirably adapted to this purpose, of a neverfailing toughness and impermeability. Even farts made no impression on it. I can't help it, gas escapes from my fundament on the least pretext, it's hard not to mention it now and then, however great my distaste. One day I counted them. Three hundred and fifteen farts in nineteen hours, or an average of over sixteen farts an hour. After all it's not excessive. Four farts every fifteen minutes. It's nothing. Not even one fart every four minutes. It's unbelievable. Damn it, I hardly fart at all, I should never have mentioned it. Extraordinary how mathematics help you to know yourself.
The history of mathematics has produced at least two attempts, philosophically very different, to answer the question of the incredible power of mathematics. The answers are also related to the fundamental issue of the actual nature of mathematics. A comprehensive discussion of these topics can fill entire volumes and is certainly beyond the scope of this book. I will therefore give only a brief description of some of the main lines of thought and present my personal opinion.
One view on the nature of mathematics, traditionally dubbed the “Platonic view,” is that mathematics is universal and timeless, and its existence is an objective fact, independent of us humans. According to this Platonic view, mathematics has always been out there in some abstract world, for humans to simply discover, just as Michelangelo thought that his sculptures existed inside the marble and he merely uncovered them. The Golden Ratio, Fibonacci numbers, Euclidean geometry, and Einstein's equations are all a part of this Platonic reality that transcends the human mind. Supporters of this Platonic view regard the famous Austrian logician Kurt Gödel (1906–1978) also as a wholehearted Platonist. They point out that not only did he say about mathematical concepts that “they, too, may represent an aspect of objective reality” but that his “incompleteness theorems” by themselves could be taken as arguments in favor of the Platonic view. These theorems, probably the most celebrated results in the whole of logic, show that for any formal axiomatic system (e.g., number theory) there exist statements formulable in its language that it cannot either prove or disprove. In other words, number theory, for example, is “incomplete” in the sense that there are true statements of number theory that the theory's methods of proof are incapable of demonstrating. To prove them we must jump to a higher and richer system, in which again other true statements can be made that cannot be proved, and so on ad infinitum. Computer scientist and author Douglas R. Hofstadter phrased this succinctly in his fantastic book Gödel, Escher, Bach: An Eternal Golden Braid: “Provability is a weaker notion than truth.” In this sense, there will never be a formal method of determining for every mathematical proposition whether it is absolutely true, any more than there is a way to determine whether a theory in physics is absolutely true. Oxford's mathematical physicist Roger Penrose is among those who believe that Gödel's theorems argue powerfully for the very existence of a Platonic mathematical world. In his wonderfully thought-provoking book Shadows of the Mind Penrose says: “Mathematical truth is not determined arbitrarily by the rules of some 'man-made' formal system, but has an absolute nature, and lies beyond any such system of specifiable rules.” To which he adds that: “Support for the Platonic viewpoint… was an important part of Gödel's initial motivations.” Twentieth-century British mathematician G. H. Hardy also believed that the human function is to “discover or observe” mathematics rather than to invent it. In other words, the abstract landscape of mathematics was there, waiting for mathematical explorers to reveal it.
One of the proposed solutions to the mystery of the effectiveness of mathematics in explaining nature relies on an intriguing modification of the Platonic ideas. This “modified Platonic view” argues that the laws of physics are expressed as mathematical equations, the structure of the universe is a fractal, galaxies arrange themselves in logarithmic spirals, and so on, because mathematics is the universe's language. Specifically, mathematical objects are still assumed to exist objectively, quite independent of our knowledge of them, but instead of placing mathematics entirely in some mythical abstract plane, at least some parts of it would be placed in the real cosmos. If we want to communicate with intelligent civilizations 10,000 light-years away, all we have to do is transmit the number 1.6180339887… and be sure that they will understand what we mean, because the universe has undoubtedly imposed the same mathematics on them. God is indeed a mathematician.
This modified Platonic view was precisely the belief expressed by Kepler (colored by his religious inclinations), when he wrote that geometry “supplied God with patterns for the creation of the world, and passed over to Man along with the image of God; and was not in fact taken in through the eyes.” Galileo Galilei had similar thoughts:
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 learns to comprehend the language and interpret 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 a dark labyrinth.
The mystic poet and artist William Blake had a rather different opinion of this mathematician God. Blake utterly despised scientific explanations of nature. To him, Newton and the scientists who followed him merely conspired to unweave the rainbow, to conquer all mysteries of human life by rules. Accordingly, in Blake's powerful etching “The Ancient of Days” (Figure 128; currently at the Pierpont Morgan Library, New York), he depicts an evil God who wields a compass not to establish universal order but rather to clip the wings of imagination.
Kepler and Galileo, however, were definitely not the last mathematicians to adopt this “modified” version of the Platonic view, nor were such views limited to those who, like Newton, took for granted the existence of a Divine Mind. The great French mathematician, astronomer, and physicist Pierre-Simon de Laplace (1749–1827) wrote in his Théorie Analitique des Probability (Analytic theory of probabilities; 1812):
Given for one instant an intelligence which comprehends all the
forces by which nature is animated and the respective positions of
the beings which compose it, if moreover this intelligence were vast enough to submit these data to analysis, it would embrace in the same formula both the movements of the largest bodies in the universe and those of the lightest atom.
This was the same Laplace who replied to Napoleon Bonaparte: “Sire, I have no need for that hypothesis,” when the emperor remarked that there is no mention of the creator in Laplace's large book on celestial mechanics.
Very recently, IBM mathematician and author Clifford A. Pickover wrote in his lively book The Loom of God: “I do not know if God is a mathematician, but mathematics is the loom upon which God weaves the fabric of the universe.… The fact that reality can be described or approximated by simple mathematical expressions suggests to me that nature has mathematics at its core.”
Supporters of the “modified Platonic view” of mathematics like to point out that, over the centuries, mathematicians have produced (or “discovered”) numerous objects of pure mathematics with absolutely no application in mind. Decades later, these mathematical constructs and models were found to provide solutions to problems in physics. Penrose tilings and non-Euclidean geometries are beautiful testimonies to this process of mathematics unexpectedly feeding into physics, but there are many more.
There were also many cases of feedback between physics and mathematics, where a physical phenomenon inspired a mathematical model that later proved to be the explanation of an entirely different physical phenomenon. An excellent example is provided by the phenomenon known as Brownian motion. In 1827, British botanist Robert Brown (1773–1858) observed that when pollen particles are suspended in water, they get into a state of agitated motion. This effect was explained by Einstein in 1905 as resulting from the collisions that the colloidal particles experience with the molecules of the surrounding fluid. Each single collision has a negligible effect, because the pollen grains are millions of times more massive than the water molecules, but the persistent bombardment has a cumulative effect. Amazingly, the same model was found to apply to the motions of stars in star clusters. There the Brownian motion is produced by the cumulative effect of many stars passing by any given star, with each passage altering the motion (through gravitational interaction) by a tiny amount.
There exists, however, an entirely different view (from that of the modified Platonic view) on the nature of mathematics and the reason for its effectiveness. According to this view (which is intricately related to dogmas labeled “formalism” and “constructivism” in the philosophy of mathematics), mathematics has no existence outside the human brain. Mathematics as we know it is nothing but a human invention, and an intelligent civilization elsewhere in the universe might have developed a radically different construct. Mathematical objects have no objective reality—they are imaginary. 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.” In other words, Kant emphasizes the freedom aspect of mathematics, the freedom to postulate and to invent patterns and structures.
This view of mathematics as a human invention has become popular in particular with modern psychologists. For example, French researcher and author Stanislas Dehaene concludes in his interesting 1997 book The Number Sense that “intuitionism [which to him is synonymous with mathematics as a human invention] seems to me to provide the best account of the relations between arithmetic and the human brain.” Similarly, the last sentence in the book Where Mathematics Comes From (2000) by the University of California, Berkeley, linguist George Lakoff and psychologist Rafael E. Núñez reads: “The portrait of mathematics has a human face.” These conclusions are based primarily on the results of psychological experiments and on neurological studies of the functionality of the brain. Experiments show that babies have innate mechanisms for recognizing numbers in small sets and that children acquire simple arithmetical capabilities spontaneously, even without much formal instruction. Additionally, the inferior parietal cortex has been identified as the area of the brain that hosts the neural circuitry involved in symbolic numerical capabilities. This area of both cerebral hemispheres is located anatomically at the junction of neural connections from touch, vision, and audition. In patients suffering from a rare form of seizure while performing arithmetic manipulations (known as epilepsia arithmetices), brain wave measurements (electroencephalograms) show abnormalities in the inferior parietal cortex. Similarly, lesions in this region affect mathematical ability, writing, and spatial coordination.
Even if based on physiology and psychology, the view of mathematics as a human invention of no intrinsic reality still needs to answer the two intriguing questions: Why is mathematics so powerful in explaining the universe, and how is it possible that even some of the purest products of mathematics are found eventually to fit physical phenomena like a glove?
The “human inventionist” reply to both of these questions is also based on a biological model: evolution and natural selection. The idea here is that progress in understanding the universe and the formulation of mathematical laws that describe phenomena within it have been achieved via an extended and tortuous evolutionary process. Our current model of the universe is the result of a long evolution that involved many false starts and blind alleys. Natural selection has weeded out mathematical models that did not fit the observations and experiments and left only the successful ones. According to this view, all “theories” of the universe are in fact nothing but “models” whose attributes are determined solely by their success in fitting the observational and experimental data. Kepler's crazy model of the solar system in Mysterium Cosmographicum was acceptable, as long as it could explain and predict the behavior of the planets.
The success of pure mathematics turned into applied mathematics, in this picture, merely reflects an overproduction of concepts, from which physics has selected the most adequate for its needs—a true survival of the fittest. After all, “inventionists” would point out, Godfrey H. Hardy was always proud of having “never done anything 'useful.' This opinion of mathematics is apparently espoused also by Marilyn vos Savant, the “world record holder” in IQ—an incredible 228. She is quoted as having said “I'm beginning to think simply that mathematics can be invented to describe anything, and matter is no exception.”
In my humble opinion, neither the modified Platonic view nor the natural selection view provides a fully satisfactory answer (at least in the way both are traditionally formulated) to the mystery of the effectiveness of mathematics.
To claim that mathematics is purely a human invention and is successful in explaining nature only because of evolution and natural selection ignores some important facts in the nature of mathematics and in the history of theoretical models of the universe. First, while the mathematical rules (e.g., the axioms of geometry or of set theory) are indeed creations of the human mind, once those rules are specified, we lose our freedom. The definition of the Golden Ratio emerged originally from the axioms of Euclidean geometry; the definition of the Fibonacci sequence from the axioms of the theory of numbers. Yet the fact that the ratio of successive Fibonacci numbers converges to the Golden Ratio was imposed on us—humans had no choice in the matter. Therefore, mathematical objects, albeit imaginary, do have real properties. Second, the explanation of the unreasonable power of mathematics cannot be based entirely on evolution in the restricted sense. For example, when Newton proposed his theory of gravitation, the data that he was trying to explain were at best accurate to three significant figures. Yet his mathematical model for the force between any two masses in the universe achieved the incredible precision of better than one part in a million. Hence, that particular model was not forced on Newton by existing measurements of the motions of planets, nor did Newton force a natural phenomenon into a preexisting mathematical pattern. Furthermore, natural selection in the common interpretation of that concept does not quite apply either, because it was not the case that five competing theories were proposed, of which one eventually won. Rather, Newton's was the only game in town!
The modified Platonic view, on the other hand, faces different types of challenges.
First, there is the important conceptual issue that the modified Platonic view does not really offer any explanation to the power of mathematics. The question is simply transformed into a belief in the mathematical underpinning of the physical world. Mathematics is simply assumed to be the symbolic counterpart of the universe. Roger Penrose, who as I noted before is himself a powerful supporter of the Platonic world of mathematical forms, agrees that the “puzzling precise underlying role that the Platonic mathematical world has in the physical world” remains a mystery. Oxford University physicist David Deutsch turns the question somewhat around. In his insightful 1997 book The Fabric of Reality, he wonders: “in a reality composed of physics and understood by the methods of science, where does mathematical certainty come from?” Penrose adds to the effectiveness of mathematics two more mysteries. In his book Shadows of the Mind, he wonders: “How it is that perceiving beings can arise from out of the physical world,” and “how it is that mentality is able seemingly to 'create' mathematical concepts out of some kind of mental model.” These intriguing questions, which are entirely outside the scope of the present book, deal with the origin of consciousness and the perplexing ability of our rather primitive mental tools to gain access into the Platonic world (which to Penrose is an objective reality).
The second problem encountered by the modified Platonic view is related to the question of universality. To what extent are we certain that the laws that the universe must obey have to be presented by mathematical equations of the type we have formulated? Until very recently, probably most physicists on the face of the Earth would have argued that history has shown that equations are the only way in which the laws of physics can be expressed. This situation may change, however, with the impending publication of the book A New Kind of Science by Stephen Wolfram. Wolfram, one of the most innovative thinkers in scientific computing and in the theory of complex systems, has been best known for the development of Mathematica, a computer program/system that allows a range of calculations not accessible before. After ten years of virtual silence, Wolfram is about to emerge with a provocative book that makes the bold claim that he can replace the basic infrastructure of science. In a world used to more than three hundred years of science being dominated by mathematical equations as the basic building blocks of models for nature, Wolfram proposes simple computer programs instead. He suggests that nature's main secret is the use of simple programs to generate complexity.
Wolfram's book was not out yet at the time of this writing, but from a long conversation I had with him and from an interview he gave to science writer Marcus Chown, I can safely conclude that his work has many far-reaching implications. From the restricted point of view of its reflection on Platonism, however, Wolfram's work points out that at the very least, the particular mathematical world that many thought exists out there, and which was believed to underlie physical reality, may not be unique. In other words, there definitely can exist descriptions of nature that are very different from the one we have. Mathematics as we know it captures only a tiny part of the vast space of all possible simple sets of rules that might describe the workings of the cosmos.
If both the modified Platonic view and the natural selection interpretation have difficulties in attempting to explain the striking effectiveness of mathematics, is there an exposition that works?
I believe that the explanation has to rely on concepts borrowed from both points of view rather than on adopting one or the other. The situation here is very similar to the historical attempts in physics to explain the nature of light. The lesson from this piece of history of science is so profound that I will describe it now briefly.
Newton's first paper was on optics, and he continued to work on this subject for most of his life. In 1704 he published the first edition of his book Opticks, which he later revised three times. Newton proposed a “particle theory of light,” in which light was assumed to be made up of tiny, hard particles, that obey the same laws of motion as do billiard balls. In Newton's words: “Even the rays of light seem to be hard bodies.” Two famous experiments at the beginning of the twentieth century discovered the photoelectric effect and the Compton effect, and provided strong support for the idea of particles of light. The photoelectric effect is a process in which electrons in a piece of metal absorb sufficient energy from light to allow them to escape. Einstein's explanation of this effect in 1905 (which won him the 1922 Nobel Prize for Physics) showed that light delivers the energy to the electrons in a grainy fashion, in indivisible units of energy. Thus, the photon—the particle of light—was introduced. Physicist Arthur Holly Compton (1892–1962) analyzed in 1918 to 1925 the scattering of X rays from electrons both experimentally and theoretically. His work (which won him the 1927 Nobel Prize for Physics) further confirmed the existence of the photon.
But there was another theory of light—a wave theory—in which light was assumed to behave like waves of water in a pond. This theory was most strongly advocated by the Dutch physicist Christiann Huygens (1629–1695). The wave theory did not have much going for it until the physicist and physician Thomas Young (1773–1829) discovered interference in 1801. The phenomenon itself is quite simple. Suppose you dip the index fingers of both hands periodically into the water in a pond. Each finger will create a sequence of concentric ripples; crest and trough will follow each other in the form of expanding rings. At points where a crest emanating from one finger intersects a crest from the other, you get the two waves to enhance each other (“constructive interference”). At points where a crest overlaps with a trough, they annihilate each other (“destructive interference”). A detailed analysis of the fixed pattern that emerges shows that along the central line (between the two fingers), there is constructive interference. To either side, lines of destructive interference alternate with lines of constructive interference.
In the case of light, destructive interference simply means dark lines. Young, a child prodigy who spoke eleven languages by age sixteen, performed an experiment in which he passed light through two slits and demonstrated that the light on the viewing surface was “divided by dark stripes.”
Young's results, which were followed by impressive theoretical work by French engineer Augustin Fresnel in 1815 to 1820, initiated a conversion of physicists to the wave theory. Later experiments conducted by the French physicist Léon Foucault in 1850 and by American physicist Albert Michelson in 1883 showed unambiguously that the refraction of light as it passes from air to water also behaves precisely as predicted by the wave theory. More important, the Scottish physicist James Clerk Maxwell (1831–1879) published in 1864 a comprehensive theory of electromagnetism that predicted the existence of propagating electromagnetic waves moving at the speed of light. He went on to propose that light itself is an electromagnetic wave. Finally, between 1886 and 1888, the German physicist Heinrich Herz proved experimentally that light was indeed the electromagnetic wave predicted by Maxwell.
So, what is light? Is it a pure bombardment by particles (photons) or a pure wave? Really, it is neither. Light is a more complicated physical phenomenon than any single one of these concepts, which are based on classical physical models, can describe. To describe the propagation of light and to understand phenomena like interference, we can and have to use the electromagnetic wave theory. When we want to discuss the interaction of light with elementary particles, however, we have to use the photon description. This picture, in which the particle and wave descriptions complement each other, has become known as the wave-particle duality. The modern quantum theory of light has unified the classical notions of waves and particles in the concept of probabilities. The electromagnetic field is represented by a wave function, which gives the probabilities of finding the field in certain modes. The photon is the energy associated with these modes.
Returning now to the question of the nature of mathematics and the reason for its effectiveness, I believe that the same type of complementarity should be applied. Mathematics was invented, in the sense that the “rules of the game” (the sets of axioms) are man-made. But once invented, it took on a life of its own, and humans had (and still have) to discover all of its properties, in the spirit of the Platonic view. The endless list of unexpected appearances of the Golden Ratio, the numberless mathematical relations obeyed by the Fibonacci numbers, and the fact that we still do not know if there are infinitely many Fibonacci primes provide ample evidence for this discovery quest.
Wolfram holds very similar views. I asked him specifically whether he thought mathematics was “invented” or “discovered.” He replied: “If there wasn't much choice in selecting this particular set of rules then it would have made sense to say that it was discovered, but since there was much choice, and our mathematics is merely historically based, I have to say that it was invented.” The phrase “historically based” in this context is crucial since it implies that the system of axioms on which our mathematics is based is the one that happened to emerge because of the arithmetic and geometry of the ancient Babylonians. This raises two immediate questions: (1) Why did the Babylonians develop these particular disciplines and not other sets of rules? And a rephrasing of the question on the effectiveness of mathematics: (2) Why were these disciplines and their offspring found to be useful at all for physics?
Interestingly, the answers to both of these questions may be related. Mathematics itself could have originated from a subjective human perception of how nature works. Geometry may simply reflect the human ability to easily recognize lines, edges, and curves. Arithmetic may represent the human aptitude to resolve discrete objects. In this picture, the mathematics that we have is a feature of the biological details of humans and of how they perceive the cosmos. Mathematics thus is, in some sense, the language of the universe—of the universe discerned by humans. Other intelligent civilizations out there might have developed totally different sets of rules, if their mechanisms of perception are very different from ours. For example, when one drop of water is added to another drop or one molecular cloud in the galaxy coalesces with another, they make only one drop or one cloud, not two. Therefore, if a civilization that is somehow fluid based exists, for it, one plus one does not necessarily equal two. Such a civilization may recognize neither the prime numbers nor the Golden Ratio. To give another example, there is hardly any doubt that had even just the gravity of Earth been much stronger than it actually is, the Babylonians and Euclid might have proposed a different geometry than the Euclidean. Einstein's theory of general relativity has taught us that in a much stronger gravitational field, space around us would be curved, not flat—light rays would travel along curved paths rather than along straight lines. Euclid's geometry emerged from his simple observations in Earth's weak gravity. (Other geometries, on curved surfaces, were formulated in the nineteenth century.)
Evolution and natural selection definitely played a cardinal role in our theories of the universe. This is precisely why we don't continue to adhere today to the physics of Aristotle. This is not to say, however, that the evolution was always continuous and smooth. The biological evolution of life on Earth was neither. Life's pathway was occasionally shaped by chance occurrences like mass extinctions. Impacts of astronomical bodies (comets or asteroids) several miles in diameter caused the dinosaurs to perish and paved the way for the dominance of the mammals. The evolution of theories of the universe was also sporadically punctuated by quantum leaps in understanding. Newton's theory of gravitation and Einstein's General Relativity (“I still can't see how he thought of it,” said the late physicist Richard Feynman) are two perfect examples of such spectacular advances. How can we explain these miraculous achievements? The truth is that we can't. That is, no more than we can explain how, in a world of chess that was used to victories by margins of half a point or so, in 1971 Bobby Fischer suddenly demolished both chess grandmasters Mark Taimanov and Bent Larsen by scores of six points to nothing on his way to the world championship. We may find it equally difficult to comprehend how naturalists Charles Darwin (1809–1882) and Alfred Russel Wallace (1823–1913) independently had the inspiration to introduce the concept of evolution itself—the idea of a descent of all life from a common ancestral origin. We must simply recognize the fact that certain individuals are head and shoulders above the rest in terms of insight. Can, however, dramatic breakthroughs like Newton's and Einstein's be accommodated at all in a scenario of evolution and natural selection? They can, but in a somewhat less common interpretation of natural selection. While it is true that Newton's theory of universal gravitation had no contending theories to compete with at the time, it would not have survived to the present day had it not been the “fittest.” Kepler, by contrast, proposed a very short-lived model for the Sun-planet interaction, in which the Sun spins on its axis flinging rays of magnetic power. These rays were supposed to grab on the planets and push them in a circle.
When these generalized definitions of evolution (allowing for quantum jumps) and natural selection (operating over extended periods of time) are adopted, I believe that the “unreasonable” effectiveness of mathematics finds an explanation. Our mathematics is the symbolic counterpart of the universe we perceive, and its power has been continuously enhanced by human exploration.
Jef Raskin, the creator of the Macintosh computer at Apple, emphasizes a different aspect—the evolution of human logic. In a 1998 essay on the effectiveness of mathematics, he concludes that “Human logic [emphasis added] 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.”
In the play Tamburlaine the Great, a tale about a Machiavellian hero-villain who is at the same time sensitive and a vicious murderer, the great English playwright Christopher Marlowe (1564–1593) recognizes this human aspiration for understanding the cosmos:
Nature that framed us of four elements,
Warring within our breasts for regiment,
Doth teach us all to have aspiring minds:
Our souls, whose faculties can comprehend
The wondrous Architecture of the world:
And measure every wandering planet's course,
Still climbing after knowledge infinite,
And always moving as the restless spheres…
The Golden Ratio is a product of humanly invented geometry. Humans had no idea, however, into what magical fairyland this product was going to lead them. If geometry had not been invented at all, then we might have never known about the Golden Ratio. But then, who knows? It might have emerged as the output of a short computer program.