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

CHAPTER I. EVOLUTION, MATHEMATICS, AND THE EVOLUTION OF MATHEMATICS

8. WHAT MOTIVATED THE GREEKS?

Why did the Greeks ask questions whose purposes were not clear and try to answer them via nonintuitive methods?

One reason suggested in the literature is a technical one. The Greeks found mistakes, inconsistencies, and contradictions in the calculations of the Babylonians and the Egyptians, and in order to resolve these they developed a more exact form of mathematics. I am not convinced that this was the reason. If you are uncertain about which of two calculations is more correct or you doubt the accuracy of a calculation, it is reasonable to assume that you would try to perform the calculation more accurately yourself and thus arrive at the right answer. In addition, the Greeks knew that in many areas the calculations of the Babylonians and the Egyptians were more accurate than their own.

A more plausible explanation relates to the political and economic situation in ancient Greece. Although it was a period when many wars were fought between cities and between small kingdoms, in general, a democratic aura prevailed, and political and social philosophies were highly developed. In an environment in which the study of philosophy is important, when there is no single ruler or government that requires instant achievements from its subjects, when there are no government-appointed committees that determine priorities for research, in an atmosphere when one can question and doubt everything and curiosity-driven study is a highly valued pursuit, in such an atmosphere enormous achievements can be made, even if it takes a very long time to derive benefit from them. To these considerations may be added the fact that the main contributors to developments in research came from established families, and they could study without being concerned with their livelihood and subsistence, a fact that clearly helped them to advance along unorthodox channels. These considerations explain how basic research developed, but they do not explain why it developed in nonintuitive directions and contrary to what evolution would have determined.

A documented explanation for the path followed by the Greeks derives from what are known as illusions. We will expand on this point because it will be relevant in the following chapters. The Greeks were familiar with geometrical or optical illusions, and they therefore tried to prove mathematical propositions without relying on appearances, in other words, relying only on axioms and logical deduction. We will describe two famous illusions from a later period.

The first is known as the Müller-Lyer illusion, named after the scientist who published it in 1889. The upper line in the diagram seems to be shorter than the lower one, despite the fact that they are of the same length (see the diagram). The usual explanation is that generally, in nature, we see a shape similar to the upper line when looking from the outside, for instance, looking closely at an edge of a three-dimensional cube, whereas we see the lower line as the more distant edge, looking into the cube. The brain uncontrollably corrects the signals that the eye receives, shortens the upper line and lengthens the lower one, in order to obtain the “right” length.

The answer lies in evolution. A correct interpretation of the signals gave an evolutionary advantage and therefore is embedded in the genes. Therefore the way the brain analyzes the information cannot be changed. By directing the eye to see only the horizontal lines we may be able to get the brain to see that the lines shown in a particular way, such as in the diagram, are equal, but we will not be able to do that with lines in situations where the brain interprets spontaneously. In any case, it would be inadvisable to change the way the brain interprets what the eye sees, because if we were to do so we would cause errors in the many situations in which the upper line is actually shorter than the lower one.

The second example is what is known as the Poggendorff illusion, published in 1860. To the untrained observer, the diagonal lines in the following figure do not seem to be sections of one broken straight line, but it can easily be shown that they are. In this instance too there is an explanation why the brain “misleads” us. The brain developed in such a way that it compares angles, not lines. The angles between the diagonals and the vertical lines make the brain create an illusion. Here the illusion does not derive from a correction the brain makes to the data it receives, in other words, a “software” correction; it derives from the “hardware” that the brain employs. The means by which the brain views geometry lead to such errors. In this case too the brain can be trained to avoid the error in specific cases, but it is not possible to carry out a repair that will prevent all such errors.

Illusions such as these occupied artists and engineers throughout the generations, and they used their knowledge to impress anyone interested in visual effects. My friend and mathematician Arrigo Cellina from Milan drew my attention to the apse in the San Satiro church, which is on Via Torino, close to the Piazza del Duomo (Cathedral Square), the main city square of Milan. The church was built in the fifteenth century. From the entrance one sees the nave, the pulpit, and, behind it, a large deep apse with a ceiling decorated with interesting paintings. On approaching the pulpit, however, one can see that the apse, its depth, and its dome are a very interesting optical illusion. Highly recommended.

The possibility of visual errors and illusions so captivated the Greeks that it drove them to extremes. In order not to rely on appearance or what one sees, for example in mathematical proofs about triangles, they would draw the triangles with sides that were not straight lines but curves. The purpose was to rely only on axioms, and as far as possible to avoid errors based on appearances or eyesight. Nevertheless, even the Greeks could not avoid having to relate to some extent to sketches and drawings. On the face of things, drawings are not relevant in cases of deductive proofs based only on axioms, but it seems that the brain cannot cope with abstract axioms without the help of metaphors or without reference to a model or previous experience. This characteristic of the brain, or perhaps this limitation, arises every time abstract mathematics is used to describe geometry or a natural phenomenon. In mathematical depictions of nature, as we will see in the next chapters, even though mathematics can stand alone and does not need a visual model, the brain does need a model or a metaphor to enable it to analyze and absorb the mathematics.