Mathematics and the Real World (29 page)

Image courtesy of Hanuš Seiner
.

This effect, that is, nature
approximating as closely as possible
to a mathematical solution that is impossible from a physical aspect, has since then been discovered in many situations and has provided a mathematical interpretation for previously observed effects. The approach also succeeded in predicting new effects that were then confirmed in the laboratory. Such a mathematical achievement is illustrated in the picture above, from the laboratory of Hanuš Seiner, of Prague. The lower part of the picture shows alternate microscopic layers of two states of metal, layers constituting a microscopic approximation to the required mathematical solution in which both layers appear simultaneously (the length of the picture represents an actual length of 2 millimeters). The upper left part of the picture shows the classic state of the metal. The possibility that there would be anastomosis between the mathematical approximation and the classic state of the metal (i.e., that they would have a common edge) is not self-evident. The transition from one state to another was predicted by mathematics, by John Ball and his colleagues, and was identified in Seiner's laboratory.

34. THE SCIENTIFIC METHOD: IS THERE AN ALTERNATIVE?

As we have seen, the scientific method that developed over thousands of years to describe the physical world is based
on mathematics. This dependence on mathematics is common to all parts of the scientific world. Professor Herbert Jäckle of the Max Planck Institute in Germany, in a critique of the quality of research in a particular department of biology, wrote in 2010, “The aim is to go to the ultimate step, just as physics has done before: to describe development in mathematical terms and to model the system. Otherwise we have not understood the process and it remained a phenomenon, a miracle.” That is a pictorial way, a variation of a claim by Aristo, of paraphrasing the view that without mathematics it is impossible to understand the effects seen in nature.

The question arises nevertheless: Are there other ways to describe nature, ways that will not depend solely on a mathematical model? And there is a similar question: In order to understand those effects that current mathematics does not describe well enough, or to which it is even not applicable, do existing mathematical methods have to be developed further, or must a more appropriate mathematics be found? Or, in order to understand those effects, should we perhaps adopt nonmathematical methods?

It is difficult to give definite answers to these questions. It is clear that mathematical methods have not been exhausted, and it is also clear that there is room for new mathematical methods beyond the discovery of new equations or new criteria such as the least action principle.

A novel approach being tried by various researchers is to change the equations that describe the state of nature for algorithms that bring about that state of nature. Darwin's theory of evolution is an example of an algorithmic law of nature. To predict what the state of nature will be at a particular time, it will then be necessary to perform a simulation, no doubt with the aid of computers. This approach is still in its infancy.

Another recent approach is to employ the so-called fractals. These geometrical objects have the property that a portion, even a small portion, of the body looks like a reduced-scale version of the entire body. Such bodies are called self-similar. Of course, a segment of a line is self-similar, but already in the beginning of the twentieth century it was discovered that such geometric figures may have very complicated structures, and their mathematical properties were investigated. It was Benoit Mandelbrot (1924–2010), a Polish-born who got his education in Paris and in the
United States and worked most of his scientific life for IBM, who discovered the resemblance of these figures to objects in the real world and coined the term
fractals
. Mandelbrot pointed out that, for instance, a cloud, a forest, a coastline, all have the property that a part of the object looks like the entire object. Of course, a self-similar body in nature lacks the property that any small part of it is similar to the entire body, as the mathematical analysis requires. A too-small portion of a cloud will contain only few molecules and will not resemble a cloud. Yet, the mathematical theory of fractals helps our understanding of the real-life objects. We may consider it as another example of Applied Platonism: nature is trying to mimic the fractal structure to the extent possible. Beyond a mere description of an object in nature, the approach has some concrete successes. For instance, it is difficult to even define the length of a coastline, since the self-similarity of the coastline of a rocky structure implies that the naive definition of a length would result in the coastline being infinitely long. The theory suggests an alternative measure, the fractal dimension of the line (that is too technical to review here). In the future the theory of fractals may provide more useful tools, helping to examine complicated structures in nature.

With regard to the possibility of complementing or even substituting mathematics with alternative approaches, we would observe first that the status that mathematics achieved in describing nature was not achieved easily or without competition. We could even claim that mathematics reached its current state almost as part of the evolutionary struggle. The competition in ancient times was against the various oracles, for example. Later, the competitor was astrology. The practice of using astrology displays some signs of a scientific activity. There are rules according to which predictions are reached. There are predictions even backed by claims of success in statistical tests regarding their reliability. The “missing” part is a mathematical model, a model that would explain how the planetary system affects us. Despite that lack, some people today still believe in and use astrology to explain and predict various effects, including effects in nature. As an argument supporting the validity of astrology, its proponents point out that the best scientists in the early and even later modern era believed in astrology. That is correct. But the argument
ignores the fact that astrology was abandoned by science because of its lack of success.

Over the years several attempts were made to formulate other methods to complement the method of mathematical models. One example is that of Immanuel Velikovsky (1895–1979). Velikovsky was born in Russia, studied physics in the Hebrew University of Jerusalem, and studied psychiatry under the guidance of Sigmund Freud. He did not deny the scientific method but proposed adding to it; the essence of the addition was reliance on ancient literary and religious sources, including the Bible, and archeological finds. The most famous of his far-reaching hypotheses was his claim that the planet Venus was different than the other planets in the solar system as it was not created together with the rest of the system but was previously a wandering body in the galaxy that was caught by the gravity of the Sun. The biblical narrative of the “Sun standing still upon Gibeon and the Moon in the valley of Ayalon” was understood by Velikovsky as a physical occurrence that supported his claims. In addition, Velikovsky made several physical predictions, some of which were substantiated by measurements. For example, of all the hypotheses about the temperature on Venus, before that was measured by the spacecraft sent there, Velikovsky's was the most accurate. Nonetheless, his approach to scientific research was not accepted (to use an understatement; in fact, it was rejected vigorously by the scientific establishment). His opponents said that there was nothing wrong with using ancient sources or any other means to obtain inspiration, but those texts cannot be allowed to serve as a substantiation of a theory.

An area in which mathematical models have not yet proved themselves is that of life sciences. Much effort has been invested in attempts to apply the mathematical approach to biology and its scientific subgroups, some of which have met with much, but still only very partial, success. Experimental findings show that biology is far more complex than physics. Is this a temporary situation that will prevail until the correct mathematics will be discovered to describe biology? We should remember that until Kepler and Newton, the physics of the celestial bodies seemed like an enormous
collection of findings independent of a simple mathematical framework. It may be, however, that the complexity in biology cannot be solved using the simple mathematics that we know, and new principles in biology need to be deciphered before mathematics can be successfully applied.

Mathematics also shows only partial success in the social sciences and humanities (we will present some approaches in the next chapters of this book). Here too there are competing theories, some respected, like the diagnosis of a person's mental state using psychological analysis, and some less respected, such as graphology to determine someone's character and palm reading for foretelling the future. Some of these also have some “scientific” aspects and apparent statistical support. Such support has no basis in science because until a mechanism, preferably mathematical, has been suggested as a basis for the method, it has no scientific validity. The existence of such methods, in particular the respected ones, raises the following question: Is the mathematical approach the right one for analyzing the social sciences and humanities?

Can birds calculate probabilities? • How can you calculate the chances of winning an unfinished game? • Is it worthwhile to believe in God? • Why did the municipality of Amsterdam almost go bankrupt? • Who murdered Mrs. Simpson? • Why didn't dinosaurs develop gills against dust? • What are the chances of the Ayalon Highway being flooded? • Is there a “hot hand” in basketball?

35. EVOLUTION AND RANDOMNESS IN THE ANIMAL WORLD

The title of this chapter does not refer to the random part of the process of evolution but to the question that has been with us throughout this book, and that is, to what extent did evolution prepare us to analyze intuitively and understand situations in which randomness plays a part? The question is reasonable. Indeed, uncertainty and randomness appear frequently in nature and were part of the environment of species in the evolutionary struggle. We may assume, therefore, that the evolutionary competition gave rise to the development of intuition with respect to uncertainty.

First, however, we will examine more closely the difference between randomness and uncertainty. A situation of uncertainty is one in which we do not know what the result of an occurrence will be and we do not know the circumstances prevailing at the time of the occurrence. Randomness occurs when we do not know for sure what will occur, but we do know
that what happens is controlled by a process with given probabilities. For example, when we throw a die with six faces, we do not know in advance which face will be uppermost, but we do know that each of the six faces has the same probability of being the one. The same applies to tossing a coin, with each of the sides, “heads” or “tails,” having an equal chance of falling uppermost. The probabilities do not have to be equal, but the process will still be random. Thus, if four sides of a cube are blue and the other two sides are red, in a random throw of the cube the chances of a blue side appearing uppermost will be two-thirds, and of a red side, one-third. The probabilities are not the same, but the process is a random one. In contrast, in a situation of general uncertainty the result may not be determined by a probability process. For example, a committee is about to make a decision, and we have no idea of the procedure through which it reaches its decisions. That is a situation of uncertainty where, usually, we cannot assign probabilities to the outcome of the decision-making process.

One way of examining whether the relation to uncertainty in general and randomness in particular are rooted in evolution is to study reactions to such situations in the animal world. We used the same approach with the relation to arithmetic in section 2. The result with regard to randomness is very clear: animals succeed in discerning random states. Moreover, they sometimes use randomness to improve their situation. They also manage to recognize situations of uncertainty unrelated to randomness. Furthermore, they are sometimes aware of the fact that there is something they do not know, and then their behavior is similar to that of humans in similar situations. The conclusion to be drawn is that the relation to randomness and uncertainty derives from evolution. We will describe briefly a number of experiments from among the many that confirm this conclusion.