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

29. THE DISCOVERY OF THE QUANTUM STATE OF NATURE

Aristotle believed that matter was continuous. Other Greek scientists, with Leucippus and Democritus at their head, claimed that matter consists of atoms that cannot be split. The approach of the Greeks who supported the atomic structure was based on philosophical considerations with no experimental substantiation. The approach of the opponents of the atomic theory was consistent with what our senses teach us, and therefore their view held sway until the sixteenth century. At the end of that century and at the beginning of the seventeenth, experimental results led to the recognition that matter was not continuous. The best-known contributors to this revelation were the British chemist and philosopher Robert Boyle (1627–1691), who identified the atoms and introduced the concept of the molecule that consists of atoms, and the chemist and physicist John Dalton (1766–1844), who also was British and who developed the theory that all matter is composed of atoms, and the type of atom determines its properties. Dalton also introduced the concept of molecular weight based on the relative weight of the atoms that make up the molecules, which enables us to identify, and sometimes refine, different materials.

Another significant breakthrough was made by the Russian chemist Dmitri Mendeleev (1834–1907), who constructed the first version of the periodic table of the elements. In Mendeleev's time, sixty elements were known, and based on their properties he created a partial table and managed to predict the existence of other chemical elements, which were discovered soon after. Mendeleev's story is indeed wonderful, but from our point of view it is important to state that his discovery was based on the assumption of aestheticism, symmetry, and simplicity. He did not suggest a physical explanation for this periodicity. The electrons, whose paths currently explain the periodic table, were as yet unknown, and it was believed that atoms could not be divided.

The picture changed with the discovery of particles with negative electric charge, that is, electrons. It was then understood that an electric current consists of the movement of electrons whose source is in the atoms, so the atom has different parts. Moreover, different atoms have different numbers of electrons, but in general their charge is balanced by an equal number of particles with positive electric charge, called protons. The number of protons did not explain the ratios of the molecular weights of different atoms. The British physicist Ernest Rutherford (1871–1927), who was awarded the Nobel Prize in Chemistry in 1908, proposed in 1910 both the existence of particles without an electric charge (neutrons) and a model of the atom, a model still used today: a nucleus with protons and neutrons in it and electrons moving around it (the existence of neutrons was not confirmed by experiments until 1930).

The number of protons and electrons determines the electrical properties of the atoms, while the number of neutrons explains the difference between the atomic weight and the number of protons. Rutherford presented the appropriate mathematical calculations together with his model, but it was not a mathematical model that could serve to explain the situation, but rather a metaphorical model based on intuition derived from the solar system. (It is interesting to contemplate what Rutherford would have suggested if Ptolemy's model had still prevailed.) The need for such a model is clear. The human brain needs to arrange relevant information in patterns with rules, and the framework for those patterns is generally taken from patterns known previously.

Rutherford's model, however, suffered from important shortcomings. The main one was that if the electron is a particle with a normal electric charge, the act of its constant revolving around the nucleus would create radiation and loss of energy until eventually it would collapse into the nucleus, something that was not seen to occur in reality. A far-reaching hypothesis was proposed by the German physicist Max Planck (1858–1947), who in his research into electromagnetic radiation came across a surprising fact. He found that the energy is transmitted only in quantities that are multiples of one basic quantity, still today called the Planck constant. His discovery of energy quanta earned him the Nobel Prize in 1918.

The idea was so innovative that it was hard to adopt it, until Einstein used that hypothesis to explain the photoelectric effect. The effect was that when a beam of light illuminates a metal board, electrons are ejected from the board not continuously but according to jumps in the energy of the light. Einstein's explanation, for which he was awarded the 1921 Nobel Prize as mentioned previously, was that the electrons around the nucleus of the atom can exist only at pre-given levels of energy, which are multiples of the Planck constant, and the light itself consists of discrete photons with the same energy level.

Einstein's explanation, together with the results of other experiments, led the Danish physicist Niels Bohr (1885–1962) to put forward an improved model of the atom, for which he was awarded the Nobel Prize in 1922. In Bohr's model the electrons can be found around the nucleus of the atom only at certain energy levels or on certain paths, and on those paths they do not lose energy. The transition from one level to another depends on the loss or gain of energy from outside at a quantity that is a multiple of Planck's constant. Bohr went on to calculate the number of electrons at every level and the different levels themselves. The calculations matched the data obtained until then and were also used to predict the results of experiments, which naturally increased faith in the model. Bohr's model related to photons and electrons as particles and ignored the wave motion of light. The French scientist Louis-Victor de Broglie (1892–1987) tried to bridge this gap, claiming that all types of matter have properties both of waves and of particles, and the properties of matter as particles predominate the greater the size of the matter. He even gave a formula for the amplitude of the wave as dependent on the size of the element and showed that indeed for large bodies the amplitude of the wave is so small that it is impossible to perceive it. This was an important finding that explains the fact that although we are all to some extent waves, we do not feel that we are. De Broglie was awarded the Nobel Prize in Physics in 1929.

All these findings and insights gave a detailed description of the known facts about the structure of the atom and the particles, including numerical calculations that matched the observations. Yet the model did not provide a mathematical explanation, and without mathematics, as we keep stressing, there is no understanding.