Mathematics and the Real World (26 page)

Albert Einstein was by all accounts the most famous scientist of the modern era and, hence, the most written about. Here we will present just a few of the central facts about his life and work relevant to our account. Einstein was born in 1879 in the town of Ulm, then in the Kingdom of Württemberg, part of the German Empire, and he moved with his parents to Munich when he was one year old. As a child and youth, he did not stand out as a student, but nor did he lag behind (as rumored later on). When he was fifteen years old, his family moved to Italy for economic reasons. Albert joined them but did not acclimatize well to the new environment, and he was sent to complete his secondary schooling in Aarau, in northern Switzerland. In 1896 he was accepted into the Swiss Federal Polytechnic (known today as the Swiss Federal Institute of Technology) in Zurich, and he graduated in 1900. In his university studies he did not shine either, mainly because he concentrated on subjects that interested him, physics, mathematics, and philosophy, and in these too he did not persevere with the studies themselves but invested his time and energy in independent reading. On completing his studies he tried for some years to obtain a teaching post, unsuccessfully, and eventually, in 1903, was given the position of examiner in the Swiss patent office in Bern. There he had to evaluate many patent applications for electromagnetic devices, whose uses in engineering were constantly increasing.

Einstein had come across Maxwell's theory in his studies, and he continued to be interested in it and to involve himself in the scientific side of the theory, as he did in other scientific and philosophical subjects, but
not in any formal academic framework. At the same time, he studied and carried out research at the University of Zurich, being awarded his doctorate in 1905. In that same year, while still working as a patents examiner, he published four groundbreaking papers that have left a deep imprint on science. The first paper gave an explanation of the photoelectric effect, and we will return to this in the next section. Two of the papers dealt with what is now referred to as the special theory of relativity: the first dealt with the laws of mechanics that were derived from the new geometry that we described above, and the second with the equivalence of energy and matter. The fourth paper in the series, which resulted in the year 1905 becoming known as Einstein's annus mirabilis (miracle year), described the mathematical basis of the motion of particles similar to Brownian motion. This is the random motion of microscopic particles reported in various situation, named after the eighteenth-century Scottish botanist Robert Brown. This paper of Einstein's served as the springboard for the mathematical subject called random motion, a ramified area of mathematics still of active interest today.

These outstanding contributions brought Einstein broad academic acclaim, which led to his being offered an associate professorship at the University of Zurich. His reputation in the academic world did not filter through to the general public fast enough. When Einstein resigned from the patent office in 1909, four years after his annus mirabilis, explaining that it was because of the offer of a teaching post in the University of Zurich, his superior in the office reacted by saying, “Einstein, stop fooling around. Tell me the real reason for your resignation.” At that time he had already started to work on the theory of gravity, to which he devoted about ten years that ended with the publication in 1916 of his paper presenting the general theory of relativity. Meanwhile he served for short periods as a professor at the University of Prague and at the Swiss Polytechnic in Zurich.

In 1913 he received a personal invitation from two of the best-known scientists in the world at that time, the physicist Max Planck and the chemist Walther Nernst, who came to Zurich to persuade Einstein to accept the position of head of the Kaiser Wilhelm Institute of Physics in Berlin, and Einstein moved there in 1914. As stated in the previous section,
in 1919 the general theory of relativity was confirmed, an event that spread Einstein's fame worldwide.

He was awarded the 1921 Nobel Prize in Physics for his contribution to the understanding of the photoelectric effect, not for the theory of relativity. The Royal Swedish Academy of Sciences of course does not publish reasons why it does
not
award the prize for a particular achievement, but unofficially it was explained that according to the will of Alfred Nobel, the prize is supposed to be awarded for achievements of practical value to the welfare of humanity, and the theory of relativity was not considered to have such value. Obviously even such a narrow guideline was incorrect with regard to the theory of relativity. According to other rumors, some of the members of the prize committee of the Royal Swedish Academy of Sciences, although recognizing the greatness of the achievement, were still not convinced that the theory of relativity was correct.

Einstein spent his time in Berlin mainly in research in attempts to find a single theory that would explain both the mechanics of gravity and the quantum theory that had been developed in the meantime. He continued with these attempts, without much success, up to his final years in the United States, where he had moved due to the Nazis’ rise to power in 1933. Luckily he was not in Germany (he was on a visit to the United States) at the time of the regime change; under the Nazi government his property was confiscated, he lost his German citizenship, and his theory was declared an incorrect Jewish theory. He spent some time at the California Institute of Technology (also known as Caltech) in Pasadena, not far from Los Angeles, and later joined the Institute for Advanced Study in Princeton, New Jersey. He became an American citizen in 1940.

Experimental confirmation of mass-energy equivalence came only in the 1930s. The outbreak of World War II led to the accelerated development of the technique of converting mass to energy, a development whose peak was the manufacture of the atom bomb. Dropping the atom bombs on Hiroshima and Nagasaki led to the end of the war.

In general Einstein distanced himself from politics, but he did not hesitate to express his liberal pacifist views. Nevertheless, in the Second World War he signed a letter in favor of the development of a nuclear bomb in
order to achieve nuclear power before the Germans. Einstein, who was a secular Jew all his life, identified with his Jewishness and the Jewish people, supported the creation of the State of Israel, and was even invited to become its president after the death of its first president, Chaim Weizmann. He turned the offer down politely, as he did not consider himself suitable to the position, stating that he lacked “both the natural aptitude and the experience to deal properly with people and to exercise official functions.” He died in Princeton in 1955.

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.