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

CHAPTER IV. MATHEMATICS AND THE MODERN VIEW OF THE WORLD

33. ANOTHER LOOK AT PLATONISM

We return now to the discussion of the connection between mathematics and nature and first remind the reader of the main difference between Plato's and Aristotle's (and their successors’) approaches to the essence of mathematics as a description of nature. Plato claimed that mathematics has an independent existence in the world of ideas in which mathematical truth is absolute. Humans can reveal this truth via logic. The starting point of research that will reveal the truth is axioms, which must be derived from nature, but it must be borne in mind that nature itself only mimics the ideal mathematical truth. Aristotle, however, claimed that mathematics itself has no independent significance or even existence. If axioms are found that reflect the truth of nature, the conclusions that can be drawn by means of mathematical formalism will apply to the description of nature. Specifically, the closer the axioms are to the truth of nature, the better the mathematical conclusions to describe nature will be. Plato and Aristotle agreed that there would be differences between mathematical conclusions and actual observations of nature. For Plato these differences were disturbances. For Aristotle the differences derived from inaccurate axioms, or in more modern terms, the inaccurate construction of the mathematical model. Neither of them discusses the essence of the difference between mathematics and nature.

The Aristotelian approach to the applications of mathematics can be summarized by the statement that mathematics is a very good approximation of nature. Furthermore, when a mathematical model does not describe nature as it really is, the model must be corrected. As stated, this is Aristotle's approach according to which mathematics does not have an independent existence. To reach the correct description of nature, we start with an approximate model and correct it and make it consistent with reality by comparing the results derived from the model with empirical data.

New research into mathematics and its applications suggest another way of looking at the relation between mathematics and nature, one that is closer to Platonism and may be called Applied Platonism. This approach can be summarized by the statement that nature is a very good approximation of mathematics. Moreover, in some cases Platonic mathematics in the world of ideas inherently contains contradictions to basic laws of nature. Nevertheless, nature tries to copy it and has no choice but to approximate mathematics. The following is an example of this.

We have referred to the least action principle as the purpose underlying motion in nature. In the same way the minimization-of-energy principle also serves a purpose. Objects in nature strive to reach a situation of minimum energy, at least a local minimum; in other words, they will stay in the local minimum state unless they are subject to an external force. John Ball of the University of Oxford, England, and Richard James of the University of Minnesota in the United States examined the structure of an elastic object under stress. Their approach was a mathematical one. They wrote the expression for the energy of a body under stress and looked for the structure that would minimize the energy. They succeeded in solving the mathematical problem, and the result was that the mathematical solution is not applicable in nature. Mathematics required that the molecules in the elastic body simultaneously arrange themselves in two different ways. Clearly that is impossible in nature. Does it mean that the minimization-of-energy principle is not correct in this case? Laboratory experiments provided the surprising answer: The structure of the object in nature is an approximation of the mathematical result. The volume that the object occupies divides into microscopic parts so that in each of the tiny parts the molecules arrange themselves in one of the forms that constitutes the mathematical solution. Specifically, in each relatively large microscopic volume, both arrangements that together constitute the minimum energy appear and in the right proportions so that the average over the macroscopic surface is very close to the mathematical minimum. Nature tries to converge to the ideal solution that mathematics found, but it is not achievable.

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.