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

CHAPTER III. MATHEMATICS AND THE VIEW OF THE WORLD IN EARLY MODERN TIMES

21. PURPOSE: THE PRINCIPLE OF LEAST ACTION

In his laws, Newton ignored the purpose that Aristotle's approach required. He made do with formulating mathematical laws through which, using mathematical operations, properties of the element being studied could be derived and predicted. A comparison of the prediction with the actual outcome is what corroborates a mathematical law of nature. When Newton was asked why his laws were formulated as they were, he answered that there was no doubt that God created a world that followed clear and simple mathematical laws. Tradition, however, and the approach that predominated in science until then were deep-rooted enough to make other scientists try to describe natural occurrences via basic principles rather than mere equations.

One route followed in the search for a purpose derived from an empirical law. Scientists had long known that the path of a beam of light passing from one medium to another is refracted at the point of transition. Willebrord van Roijen Snell (1580–1626), a Dutch physicist, discovered empirically and formulated what is known today as Snell's law of refraction. The law states that the sines of the angles of refraction are in the same ratio to each other as are the speeds of light in the different media (see the diagram below, in which the speeds of light in the two media are notated v₁ and v₂). The great French mathematician Pierre de Fermat (1601–1665) showed that Snell's law was equivalent to saying that the light beam passes from one point to another in a space such that it minimizes the time to get there. In other words, if we choose two points, A and B, on the beam, the point of refraction will be that which reduces the time the light takes to get from A to B to its minimum.

One way of interpreting Fermat's principle is to say that the light beam in effect solves a mathematical problem: it chooses a route that will get it as quickly as possible from one point to another. Fermat proved that the solution to the mathematical problem was given by Snell's formula. Obviously neither Fermat nor his successors claimed that the light beam had the purpose of reaching its destination as quickly as possible. They saw this property as a basic principle beyond a mathematical description, and thus, in the Greek tradition, the principle of the least time served as the purpose underlying the law of nature.

It is interesting to note that the Greeks themselves used a similar rationale. As is generally known, the angle of reflection of a beam of light striking a mirror equals the angle at which it strikes the mirror (or the angle of incidence equals the angle of reflection). Hero (or Heron) of Alexandria (10–70 CE) proved that the identity of the two angles leads to the conclusion that a beam of light from point A toward a mirror is reflected by the mirror and reaches a point B such that it moves from A to B (via the mirror) in the shortest possible time. Hero's reasoning, however, was inverted by Fermat. Hero considered it axiomatic that the angle of incidence equals the angle of reflection and proved that the light took the quickest route. Fermat proposed as an axiom, as a purpose, the principle of the minimum time and concluded that, in the case of the beam reflected from a mirror, the angles had to be equal.

From the time Newton's laws were published, many scientists tried to generalize Fermat's least time principle so that it would also apply to new laws. The most famous of those scientists were Leibniz, Euler, Lagrange, and Hamilton. The modern formulation of this type of principle, that is, the least action principle, is attributed to the latter. In this, the purpose is achieved by minimizing the integral along the path the body takes, integral of the physical quantity called the momentum of the system, that is, the multiple of the mass and the speed. This principle too is corroborated by experiments. Moreover, Newton's equations of motion can be derived from this principle. Hence, at least as far as mechanics are concerned, complete equivalence has been achieved between the direct mathematical description of the equations of motion and the description of the system by means of the purpose inherent in the least action principle. The reason for nature revealing such “efficiency” is a question with no clear answer.

Moreover, sometimes we try to attribute to nature an element of efficiency that it apparently does not have. For example, parts of lava fields have hardened into hexagonal shapes, as can be seen in the photograph below, taken in Iceland. The explanation offered by the tour guide and by scientists I have asked about this is that covering an area with hexagons is a solution to the minimum energy problem, which is rather difficult to formulate. The hexagonal structure of honeycombs is explained in a similar fashion. The bees try to “solve” the problem of constructing a surface of cells of a given magnitude with minimum length of walls. The intention is to minimize the amount of wax required. The ancient Greeks already conjectured that hexagons offer a solution to this minimization problem, called the honeycomb conjecture. Many tried to resolve the issue, but a complete proof of this mathematical fact was published in only 2001 by Thomas C. Hales of the University of Michigan, more recently of the University of Pittsburgh.

In regard to lava hexagons, I offer an alternative explanation. The support offered by one hexagon to its neighbors makes the hexagonal structure the most stable, the most able to withstand external forces of displacement. At the time of the formation of the lava fields, various weird and wonderful shapes were formed, covering different areas, such as squares and triangles. Those small areas covered by hexagons best survived the shocks and earthquakes that occurred there, which is why that is the shape we can see today, millions of years after its formation. That is also why the area covered by hexagons is only a small part of the total lava fields, with only smaller areas covered by other shapes that survived too. With regard to honeycombs, evolution may have taught bees to solve the honeycomb problem, but it may also be the case that the stability aspect also played a role; in other words, evolution selected the bees that constructed stabler honeycombs.