Mathematics and the Real World (63 page)

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.

Is it helpful to join a class in mathematical thinking? • How can you learn mathematics to a mother-tongue level? • How did prehistoric man discover mathematics? • What is the essence of a triangle? • Is there a link between a mathematical tree and a botanical one? • How does a centipede walk? • Do students have difficulty in understanding the parallel-lines axiom? • What are the chances that in a family with three children the third will be a boy?

66. WHY LEARN MATHEMATICS?

There is no doubt that mathematics is considered one of the hardest subjects in the education system, from elementary school to university, and achievements in the study of mathematics generally fall short of expectations. However, before we turn to the question of how to improve mathematics teaching, we should clarify what we expect from it. Once we agree on its aims, we may be able to examine the extent to which teaching practice achieves those objectives and improve the system to enhance its performance. The targets that we will formulate relate to students in elementary and secondary schools. We will also relate to higher education, from the aspect of teacher training.

The first objective is to provide the student with the
basic mathematical tools
needed to function in the modern world. Functioning clearly includes the ability to operate in a world driven by commerce, money, loans, investments,
and the like. It is also advisable to be able to understand a world in which we are assailed by a mass of statistics, some of them important, and some unreliable. It is also important to possess the ability to estimate and calculate areas and volumes, that is, the ability to use basic geometric tools relevant to our environment. Lastly, in order to be able to function properly in the modern world it is worthwhile gaining at least an elementary mathematical understanding of the developing technological world.

The second objective is to become familiar with a
system of logic that rigorously checks claims
. This applies despite the fact that we have argued several times in this book that our natural tendency, resulting from evolution, is to believe what we are told. Yet when there is doubt whether the claims put to you have a firm basis, it is worthwhile being able to check them. No system can match mathematics in being able to clarify the difference between an assumption and a conclusion, between a conclusion drawn by deduction and a hypothesis reached via induction, and so on. Mathematics develops the ability to perform an ordered analysis using the tools of logic, to examine assumptions with minimal use of metaphors, and to query the quality of the information by using very precise language. It is true that people's day-to-day activity is based on intuition, and that is something that is impossible and unnecessary to change. Yet it is important that a graduate of the education system should have the ability to follow a logical analysis of claims, and even to perform such a logical analysis in cases where it is important.