Does Mathematical Beauty Pose Problems for Naturalism?1

Does Mathematical Beauty Pose Problems for Naturalism?1 Russell W. Howell Professor of Mathematics Westmont College, Santa Barbara, CA [email protected] Numerous events occurred in 1960 whose eects could hardly have been predicted at the time: several African Americans staged a sit-in at a Greensboro lunch counter, the Soviet Union shot down Gary Powers while he was ying a U2 spy plane, the US FDA approved the use of the rst oral contraceptive, AT&T led with the Federal Communications Commission for permission to launch an experimental communications satellite, and four Presidential debates between John Kennedy and Richard Nixon aired on national television. Less well known was the publication of a paper by the physicist Eugene Wigner. Ap- pearing in Communications in Pure and Applied Mathematics, a journal certainly not widely read by the general public, it bore the mysterious title The Unreasonable Eectiveness of Mathematics in the Natural Sciences. (Wigner, 1960) Like our cultural examples of the 1960's, it has had eects beyond what most people would have imagined. Our purpose here is to tease out some strains of an important question that has emerged from Wigner's work. Wigner begins with a story about two friends who were discussing their jobs. One of them, a statistician, was working on population trends. He mentioned a paper he had produced, which contained the Gaussian distribution equation near the beginning. The statistician attempted to explain the meaning it, as well as other mathematical symbols. His friend was a bit perplexed, and was not quite sure whether the statistician was pulling his leg. How can you know that? he repeatedly asked. And what is this symbol here? Oh, said the statistician, this is pi. What is that? The ratio of the circumference of a circle to its diameter. Well, now you are pushing your joke too far. Surely the population has nothing to do with the circumference of the circle. Wigner uses this story to introduce two issues: (1) the surprising phenomenon that we have used mathematics so often to build successful theories; (2) the nagging Kuhnian-like question, How do we know that, if we made a theory which focuses its attention on phenomena we disregard and disregards some of the phenomena now commanding our attention, that we could not build another theory which has little in common with the present one but which, nevertheless, explains just as many phenomena as the present theory? (Wigner, 1960) Regarding Wigner's rst point, he concedes that much of mathematics, such as Eu- clidean Geometry, was developed because its axioms were created on the basis of what appeared to be true of reality. From this viewpoint the applicability of mathematics to the physical world is hardly surprising. But how much of mathematics actually progresses in this manner? A strong argument can be made that other notions guide the formation of a large body of higher mathematical theories. Take the eld of complex analysis as just one example. In the 1500's the notion of √ −1 seemed odd to mathematicians. At that time, negative numbers by themselves were still being treated with some suspicion, so taking square roots of them was all the more problematic. But mathematicians kept using their imagination and pressed forward. Serious investigation of complex numbers dates to the mid-fourteenth century, when Scipione del Ferro of Bologna solved the depressed cubic equation, which is a cubic equation without an x2 term, such as x3 − 15x − 4 = 0. The solution was independently discovered some thirty years later by Niccolo Fontana. Girolamo Cardano subsequently extended the Ferro- Fontana formula to obtain a solution of the general cubic equation, which he published in Ars Magna in 1545. Then, in 1572, Rafael Bombelli used Cardano's published work to interpret the form of solutions to some depressed cubic equations. Prior to Bombelli, these forms had been impossible to decipher. Bombelli derived his solutions by using complex numbers. For example, Bombelli's techniques, when applied to the depressed cubic equation x3 −15x−4 = 0, yielded a solution of the form x = (2 + i) + (2 − i). Simple arithmetic then gave x = (2 + i) + (2 − i) = 4. That x = 4 was a correct solution to the original equation was indisputable, as it could be checked easily. However, it was only arrived at via a detour through the uncharted territory of complex numbers. The story that details the entire development of complex numbers is quite intricate, and it wasn't until the end of the nineteenth century that they became rmly entrenched in the corpus of mathematical literature. For our purposes, it is important to note that complex numbers were studied because they were useful for mathematical and not for physical purposes. To be sure, solving equations had the potential to be of great practical value, but no physical phenomenon guided the investigation of complex numbers. Success came mainly from abstraction, and the manipulation of mathematical symbols in accordance with specied rules of algebra. Complex numbers, however, now play a pivotal role in helping physicists understand the quantum world. For Wigner, it is not just the utility of complex numbers in quantum mechanics that is surprising; it is that, time and time again, leaps of theory seem to be successfully guided by mathematical formalisms rather than experimentation. According to Wigner, 2 Quantum mechanics originated when Max Born noticed that some rules of com- putation, given by Heisenberg, were formally identical with the rules of compu- tation with matrices. Born, Jordan, and Heisenberg then proposed to replace by matrices the position and momentum variables of the equations of classical mechanics. The results were quite satisfactory. However, there was . no rational evidence that their matrix mechanics would prove correct under more realistic conditions. As a matter of fact, the rst application of their mechanics to a realistic problem, that of the hydrogen atom, was given several months later, by Pauli. This application gave results in agreement with experience. This was . understandable because Heisenberg's rules of calculation were abstracted from problems which included the old theory of the hydrogen atom. The miracle occurred only when matrix mechanics . was applied to problems for which Heisenberg's calculating rules were meaningless. Heisenberg's rules presupposed that the classical equations of motion had solutions with certain periodicity properties; and the equations of motion of the two electrons of the helium atom, or of the even greater number of electrons of heavier atoms, simply do not have these properties, so that Heisenberg's rules cannot be applied to these cases. Never- theless, the calculation of the lowest energy level of helium . [agreed] with the experimental data within the accuracy of the observations, which is one part in ten million. (Wigner, 1960) Surely, Wigner concludes, in this case we `got something out' of the equations that we did not put in. Wigner cites other examples: Newton's law of motion, formulated in terms that appear simple to mathematicians, but which proved to be accurate beyond all reasonable expectations; quantum electrodynamics; and the pure mathematical theory of the Lamb shift. He nally concludes his paper with the following observation. The enormous usefulness of mathematics in the natural sciences is something bor- dering on the mysterious, and . there is no rational explanation for it. The miracle of the appropriateness of the language of mathematics for the formula- tion of the laws of physics is a wonderful gift which we neither understand nor deserve. We should be grateful for it and hope that it will remain valid in future research and that it will extend, for better or for worse, to our pleasure, even though perhaps also to our baement, to wide branches of learning. (Wigner, 1960) In 1980 Richard Wesley Hamming took up the issues raised by Wigner, and oered four partial explanations that could account for the applicability of mathematics. (Ham- ming, 1980) Following is a brief survey of Hamming's ideas. First, we see what we look for. Mathematicians craft postulates so that they will pro- duce theories that conform to their prior observations. The Pythagorean theorem, Hamming 3 claims, drove the formation of the geometric postulates, and not vice versa. Furthermore, if we insist on looking at the world through a mathematical lens, it is not entirely surprising that we wind up describing the world in mathematical terms. Second, we select the kind of mathematics to use. Hamming simply means here that we adopt the mathematical theories that seem, ahead of time, to be a good t for a physical phenomenon being investigated. The same type of mathematical theory does not work everywhere; dierent theories are selected in accordance with the phenomenon they seem to describe. Because we habitually force mathematics onto particular situations, it is only natural that we subsequently nd mathematics in general to be so widely applicable. These rst two explanations are quite similar, although part of the rst one is more in line with Wigner's Kuhnian-like question, which we will discuss later on. Regarding the selection of mathematics to t our perception of reality, we have already acknowledged that some of mathematics develops this way. To the extent that not all applicable mathematical theories are generated out of concern for applicability, further work must be done, and Hamming indeed gives some additional food for thought. Hamming's third response is that science in fact answers comparatively few problems. To the extent that this assertion is true, the less of a miracle the success of mathematics would appear to be. Wigner, as a physicist, certainly lived with mathematics as an indispensable tool, but other sciences do not share this same reliance on mathematics.

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