Mathematics and the Real World: The Remarkable Role of Evolution in the Making of Mathematics (2014)

CHAPTER IX. THE NATURE OF RESEARCH IN MATHEMATICS

65. THE BEAUTY, EFFICIENCY, AND UNIVERSALITY OF MATHEMATICS

It is generally accepted that beauty is a matter of taste, yet certain fundamentals of beauty and enjoyment are common to most human beings. Like other aspects of human conduct, the sources of enjoyment and the relation to beauty were formed by evolution. A prime example of the emotion inspired by beauty is that felt on recognizing patterns, symmetries, and so on. Recognizing a pattern or symmetry affords great pleasure, and, in surprising situations or places where one does not expect to find order, discovering them can give even greater delight. That is true in all areas of life, from natural scenery, to the plastic arts, to social and cultural environments, and so on. Patterns are also the basis of mathematical practice. Hence, every new theorem or new geometrical law that we encounter has the potential to give us pleasure. In most such cases, the patterns, and, hence, the enjoyment, are of the intellectual rather than visual sort. In many instances, before enjoying the mathematical pattern, we have to contend with terminology that is sometimes hard even for professionals to understand, learn special jargon, and also imbibe the mathematical content. These are some of the factors that keep much of the general public away from mathematics and its pleasures. I remember a cartoon showing three mathematicians in a jolly mood next to a blackboard covered with complex formulae, and one is saying to the others, “I knew you'd burst out laughing.” There are indeed professional jokes and professional pleasures in mathematics just as in other disciplines. I will nevertheless repeat a statement presented at the beginning of this book: just as one can derive pleasure from music without being able to read a score, so can one derive pleasure from mathematical patterns without being able to read the mathematical notes, and one certainly can get enjoyment from knowing how mathematics fulfills its role as a tool for explaining nature. I hope that “non-mathematical” readers who have reached this section will agree with that statement.

Nonetheless, the general public is sometimes presented with other examples of the enjoyment and beauty in mathematics. Not long ago I received an e-mail with the following tower-shaped table, alongside other similar tables, under the title “The Beauty of Mathematics.”

The table was presented in lovely colors, every digit in another color, and the colors accentuated the special pattern. Some of my acquaintances, of course including the person who had sent it to me, expressed their wonder at the beauty of this mathematics. I did not find anything interesting or special, in a mathematical sense, in this tower. It was pleasing to the eye, and its colors were attractive. The order that the arithmetic exercises reflected was also interesting and enjoyable, but it did not contain any marked element of surprise for me. I did not find any beauty in it from a mathematical aspect. I agree that the arithmetic reflected by the tower is aesthetic, but that is not the beauty of mathematics. If I were to delve deeper, or if someone were to show me that hidden within the structure is a special pattern, then I might find beauty in the equalities that constitute the tower. But I did not find it in the tower as it is.

Here is another example. One of Israel's daily newspapers published a diagram and puzzle under the heading “You Can Enjoy Geometry” (see my sketch of the puzzle below). The sketch shows four strips that could be considered bridges from one bank of an imaginary river to the other. The width of each band is the same and is constant along its whole length, but some are straight and some change direction, as in the diagram. The puzzle is: Which bridge would be the cheapest to paint, and which the most expensive?

The answer is that there is no difference in the costs. All the bridges have the same area. The solution can be seen “clearly” (the word used by the paper; one could argue with the use of that term) if the picture is turned onto its side. Every bridge consists of parallelograms with the same sized bases, and the widths of the bridges stays the same from one bank of the river to the other, and the sum of the heights of the parallelograms is the same for all the bridges. Hence (as anyone who learned and remembers basic geometry will know), the areas of the bridges are all equal. And I ask, “What is enjoyable in this?” I did not get any pleasure from it, firstly because I did not find the solution myself. Moreover, whoever set the puzzle thought that I ought to see the answer immediately, and that is another reason why I did not enjoy it. I was also left with the feeling that I had been misled. The question was which bridge would be the cheapest to paint, and which the most expensive. To my mind, if the question is worded like that, it implies that one of the bridges has the largest area and one has the smallest, and if that is not the case, then I have been misled. Even if I derived a little enjoyment from the trick that I learned, to place the picture on its side, it was mixed with some disappointment that it was after all just a trick, let alone the fact that you do not need to place a parallelogram on its side to calculate its area; but that is how it is taught in schools. I have no objections to those who do enjoy such puzzles. To anyone like me who does not enjoy them I would say, don't think that you cannot get pleasure from mathematics. It is possible to get enjoyment from tricks, or to be frustrated that you didn't manage to find the trick yourself, but tricks of this type are not an essential part of mathematics.

A similar message is derived from this well-known story about John von Neumann. A friend asked him the following question: Two trains start moving toward each other at the same time from positions a hundred kilometers apart at a speed of fifty kilometers an hour. At the same time a bee starts flying at three hundred kilometers an hour from one train toward the other until it reaches it, and then starts flying back to the first train, until it reaches it, and then turns around again. This continues until the trains meet. What total distance does the bee fly? Before we give von Neumann's answer, we observe that there are two ways of solving the problem. The simple way is to note that the trains meet after exactly one hour, and in that time the bee will have flown three hundred kilometers. The second, more complicated, method is to calculate the distance that the bee will have flown until it first meets the second train (this requires solving a not-particularly difficult equation), then to calculate the distance it flies until it is back at the first train, and continue in this manner. This gives an infinite series that can be added relatively simply (for anyone who remembers the formula), and the result, if no mistakes are made, is three hundred kilometers. Now back to von Neumann. He thought briefly (he was known to be a very quick thinker) and then gave the right answer, three hundred kilometers. The friend confirmed that von Neumann's answer was indeed correct and then complimented him, saying, “One can see that you are an excellent mathematician. Some fools,” he continued, “calculate the sum of the infinite series in order to get to the answer.” Von Neumann looked at him in surprise and asked, “Is there another way?”

The message I am trying to convey is that mathematics is not a collection of tricks. On the contrary; the essence of mathematics is ordered and well-organized analysis of patterns already revealed or that still need to be found. We saw an example of the difference between a trick and a theory in the instance of the brachistochrone in the previous section. It is acceptable and possible to benefit from tricks, but the real beauty of mathematics derives from the patterns and rules found in mathematics, and the pleasure from, and sometimes amazement at, the link to nature and to applications that the patterns and rules open up for us. I believe that nearly all the material discussed in the book up until now reflects the beauty of that sort in mathematics.

Another aspect related to the beauty and pleasure of mathematics is the sometimes-wonderful efficiency of mathematics as an instrument for describing and explaining natural phenomena. The question why is it just mathematics that is so appropriate to the description of nature arose already in ancient times, and it reappeared even more pointedly following the revolution led by Maxwell, which led to the theory of relativity and quantum theory.

Until that revolution, which symbolizes the start of the modern era in science, the question was not why does mathematics describe nature so well, but why do natural phenomena occur according to any system of rules? Why does nature itself have to follow laws that we call laws of nature? Why are there such clear patterns in nature? Why is the law of gravity on Earth the same as the law of gravity on the Moon, and apparently the same as the law of gravity in other galaxies? Why is a small, finite number of particles the component that constitutes all known materials in nature? And why can we find ordered patterns, to the extent even of a structure of mathematical groups, that determine how those particles arrange themselves? We could extend this list of questions. There are no standard scientific answers to these questions, and interest and involvement in them is usually placed in what philosophers call the transcendental sphere. For example, the saying that nature is based on mathematics is attributed to Galileo Galilei, and even the Christian Church at the time of the Renaissance agreed that God in his great wisdom used mathematics to create such a wonderful world. That of course is not a satisfactory answer to the question of why mathematics is the appropriate instrument to be used in describing and analyzing nature. We should bear in mind that the very understanding that nature expresses clear sets of laws is not natural or self-explanatory. The ancient civilizations discovered that such laws existed but did not delve into them or try to fathom them. Also, the Greeks, who declared that the laws of nature could be described successfully through logic, needed hundreds of years to really absorb that understanding into their scientific view of the world. The fact that the earthly laws of motion were the same as the heavenly laws of motion was discovered, and then accepted, only a little more than three hundred years ago.

Throughout the generations mathematics was in competition with other approaches that proposed different understandings of how nature worked, including idol worship, astrology, and other strange theories, some of which are long forgotten. Today we view the mathematical structure of nature as obvious, but that was not the case in the not so distant past. What made it easier to instill the understanding that mathematics was the right instrument for describing the world was the fact that up to Maxwell's revolution, mathematics described mainly effects that people experienced and could measure. The Greeks saw that planets moved in the sky and enlisted geometry to describe the manifestation. Newton constructed infinitesimal calculus to describe the laws of motion, motion that we can observe. From the moment that mathematics was found to be an efficient instrument for describing events that we perceive with our senses, it is no surprise that it was developed, and is still developing, to elicit more and more from it.

What is more surprising is that the same instrument that serves to describe phenomena that we measure and experience also manages to foresee new ones, including phenomena of whose existence we are unaware and which we feel no need to try to understand. How did Maxwell's equations, created “only” to give uniform expression to the link between electricity and magnetism, foretell the existence of electromagnetic waves? How did the change in the variables proposed by Hendrik Lorentz to provide a formal explanation for the results of the Michelson-Morley experiment lead to the realization that everything is relative and that the geometry of the world is not what we see and experience? And how is the behavior of particles, which we also cannot perceive directly but whose effects we can measure, best described using solutions relating to waves, despite the fact that we cannot identify any medium in which these virtual waves move? Again, currently there are no answers to these questions that are not transcendental, that is, are not “beyond” the realm of scientific discussion.

We may be able to obtain partial understanding of these questions by recognizing that all the new phenomena that we discover are consistent with what evolution prepared us for. Even when mathematics, followed by experiments, leads us to the discovery of completely new phenomena, in order to understand them we translate them into a language and metaphors that we are familiar with. We speak of electromagnetic waves because we are familiar with waves in the sea and because we know what sound waves are. We describe the laws of relativity by means of geometry because that is something we know. Although we cannot perceive that the geometry of the world is not Euclidean, we have encountered non-Euclidean geometry in other experiences, such as the geometry of the surface of a ball. Thus, it may well be that nature seems to us to be subject to such elementary laws because that is what we are searching for, the same search and law-based system that are the products of evolution. Einstein said that the laws of nature are characterized by their simplicity, and a simple law must be preferable to a complex one. Could it be, however, that Einstein's statement expresses wishful thinking rather than a description of reality? Even when we discover manifestations that suggest a lack of clear laws, such as the phenomenon of mathematical chaos (in chaos theory), we nonetheless focus our efforts on finding order in the chaos. We will expand a little on this.

In 1961 Edward Lorenz (1917–2008), an MIT mathematician and a meteorologist, performed computerized experimental simulations of equations relating to weather forecasts. He found that although the equations were relatively simple, the results of the simulations were unpredictable. The reason was that minor changes in the data caused great changes in the results. This is critical in the context of computer simulations because in such calculations, perfect accuracy is never achieved. It soon became clear, however, that the deviations were beyond the range of mathematical calculations. Lorenz's discovery was related to the mathematical results that Poincaré had indicated previously, results that showed that the heavenly orbits, such as the Sun and its planets, were subject to significant irregularities. Poincaré's results also found expression in more-general equations of motion, for example, those by Jacques Hadamard and others relating to the movement of billiard balls. Another development related to these was made by Steve Smale, who showed that certain equations that satisfy relatively simple conditions (embodied in the function known as the Smale horseshoe) lead to the result that the dynamics represented by the equation is extremely intricate. An important step in that direction was taken by James Yorke and his student Tien-Yien Li of the University of Maryland. They found the simplest condition for the equation that results in the most complex dynamics, in which minute changes in the data result in huge changes in the dynamics. Li and Yorke also coined the term chaos for their result, and that word appeared in the title of their paper. An anecdote that also points to the nature of research and the nature of man relates that, not long after the publication of Li and Yorke's findings, it came to light that their mathematical result was a particular case of a far more advanced result published some years earlier by the Ukrainian Oleksandr Sharkovsky. The title of his paper, however, was not sufficiently eye-catching to draw the attention that the article deserved. After some time, Yorke said jokingly that his own contribution to chaos theory consisted merely in providing the name for the theory. In truth, however, the important contribution was in drawing attention to the connection between the mathematical expressions and intuition about the complexity of the dynamics. The appearance of Li and Yorke's paper sparked off very extensive research on the chaos effect, research that spread into the spheres of philosophy and social science.

Here is another story relevant to the sometimes-strange interpretations given to mathematical results. To illustrate the dependence of a significant event on a very small change, the phrase “the butterfly effect” has been applied. According to this metaphor, the tiny movement of the wings of a butterfly in Southeast Asia could cause a hurricane in the Atlantic Ocean. I heard a television commentator say that butterflies in Asia cause hurricanes in the Gulf of Mexico, as if that were something well known and obvious. The commentator did not appreciate the difference between “could cause” and “causes.” The commentator can be reassured: no hurricane in the Atlantic Ocean was ever caused by the fluttering of a butterfly's wings in Southeast Asia. The subject of chaos has become a wide-ranging and productive branch of mathematics, with many applications in physics and other sciences. However, most of that research itself focuses on finding the order within the appearances of chaos, either in the way chaos is created, or in the characteristics of the statistical rules governing its occurrences, that is to say, the same type of patterns that we generally seek. To repeat what has already been said, the essence of mathematical research is indeed the search for patterns, and we usually find them among those we already know.

We can even stretch the point and go further and ask, might there be laws of nature that we have not found because the human brain is limited to identifying patterns and rules of a certain type that are consistent with the way evolution molded our brains? In my opinion, the answer is yes, our brain is limited in that way. Indeed, Poincaré and Einstein both expressed the opinion that the laws of nature that we identify through mathematics are limited to the metaphors that our brains can create, and that in nature itself there exist phenomena that are beyond our capacity to understand. Those views are in line with the modern understanding of the way our brain, the product of evolution, perceives the world around us. The problem is that as long as research is carried out under the guidance of a human brain and is examined by a human brain, it is not clear how we can deal with that restriction.

And here we reach the last question in our current quest, the universal question of mathematics. Is mathematics that would have been developed (or that was developed) by various societies, either isolated human societies detached from our civilization or societies in another galaxy or another world, necessarily the same as our mathematics? The prevailing opinion is that the answer is yes. Mathematics developed independently and under different conditions may have different emphases, and certainly different symbols and a different language. However, the logical basis and the basic technical elements, such as the natural numbers and the operation of their addition, would be the same in all versions of mathematics. That is why when a spacecraft was sent into space in the hope that it would be taken by a foreign space civilization, it contained a board with signs for the numbers 1, 2, 3,…as if to announce, “We can count.”

Nevertheless, I will allow myself to raise another possibility that, by its very nature, is no more than speculation, and which I can see no way to prove or disprove. Maybe one can question the belief that mathematics is uniform. The natural numbers are the result of the fact that our world is made up of items that we can count, and counting lends itself to the operation of addition. In a perfectly continuous world in which entities are not defined separately or as distinct units, there is no reason for the natural numbers to have any meaning (this insight is accredited to Sir Michael Atiyah, one of the most prominent mathematicians of our time). There is no reason that mathematicians in such worlds should be able to understand, say, the Peano axioms, let alone such concrete operations as 2 + 2 = 4. In other worlds that developed differently than we did, there may be other rules of logic. Even if our brains cannot imagine the concept of another logic, there is no guarantee that our logic will be relevant to societies in other worlds. Even if we do not wish to go so far as to speculate about other worlds, we should bear in mind that the logic we use, including the elementary rules of inference, is the product of our brains. And that product is the outcome of experience accumulated over the course of evolution that formed our brains.

It is clear, nevertheless, that the fact that another logic might exist in another world with another mathematics in another society does not lead to the conclusion that the logic we use is faulty or that we must look for and try other methods, new and old. The mathematics we know and are continuing to develop has proved that it is correct, fine, and efficient.