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

25. AND THEN CAME MAXWELL

James Clerk Maxwell was born in 1832 in Edinburgh, Scotland, to a well-established yet not particularly wealthy family. His father, John Clerk, was a fairly successful lawyer who inherited from childless relatives a country estate in the area of Glenlair, not far from the capital, Edinburgh, on the condition that he adopted their family name, Maxwell. This he did, and he moved to the estate with his wife, Francis, and his firstborn son, James. James's mother died when he was eight years old, and his first years of schooling were at home with a young private tutor. It was only when James reached the age of ten that relatives persuaded his father to send him to a normal school in Edinburgh. In school he was found to be an outstanding student, and he continued his studies at Edinburgh University. There too he excelled, and in the course of his studies he moved to Cambridge University in England, where he completed his studies cum laude, second among all the graduates. Heading the list was his friend Edward Routh (1831–1907), who also became a distinguished mathematician and actually researched areas in which Maxwell had laid the foundations. While at Cambridge, Maxwell wrote several important papers that earned him a fellowship at Trinity College in Cambridge. His father's death brought Maxwell back to Scotland at the age of twenty-five, and he held the position of chair of natural philosophy at Marischal College, Aberdeen. There he wrote an important paper on the stability of Saturn's rings, a paper that earned him the prestigious Adams Prize. There he also married his sweetheart, Katherine Mary Dewar, the daughter of the principal of Marischal College. Neither his family connection nor his scientific fame helped him when the college merged two departments and a professor from the other department was chosen to stay on. Maxwell had to leave, and he returned to England, to a post at King's College, London.

His achievements in London were astounding. There he completed his work on colors, a subject that had absorbed and fascinated him for many years (we will expand on this below), and, his crowning glory, he published his study on electricity and magnetism. His success and fame, among other things his appointment to the Royal Society, did not change his modest, family-oriented personality. After suffering from an infection resulting from a cut incurred when he fell from a horse, in 1865 Maxwell chose to return to Scotland and lived on his estate in Glenlair. He continued with his academic work and published two seminal papers, one on the structure of stabilizers and the other on what is today called statistical mechanics; on these too we will expand below. Maxwell did not succeed in obtaining a position at St. Andrews, and in 1871 he decided to move back to England to head the Cavendish Laboratory of Experimental Physics in Cambridge. He continued his theoretical work and published several important papers and books. He died in Cambridge in 1879.

Maxwell's greatest achievement was his contribution to electricity and magnetism, but it should be noted that other research he carried out led to major changes in other areas of science as well. The composition of the colors of the spectrum was a subject that occupied Maxwell for many years. Newton showed how the elements of white light can be revealed, that is, all the colors of the spectrum, by passing the light through a prism. After Newton, many attempted to understand the links between the different colors and the rules according to which, by mixing various colors, we create new ones. It was Maxwell who discovered and proposed the structure known by the initials he used, RGB (i.e., the three primary colors, red, green, and blue), which when mixed in the correct proportions can produce all the colors the human eye can perceive. Even today this system is used in our daily lives, including television broadcasts and printing of pictures. Maxwell was also the first, in 1861, while at King's College, London, to produce a color picture. Another area of his work laid the foundation of the area of physics called statistical mechanics, the mathematical laws according to which gases move. Maxwell's ideas were later developed further by the Austrian mathematician Ludwig Boltzmann (1844–1906) at the University of Vienna. Today the equations defining the movement of gases are called the Maxwell-Boltzmann equations, or Boltzmann equations. Yet another subject in which Maxwell set the foundations was the theory of stability in control systems; he wrote this in an article in 1868 as a solution to an engineering problem that engineers of steamships had encountered, and asked for his help (we will review this fascinating topic in section 64, in the context of a discussion of applied mathematics).

The search for the laws of physics underlying electricity and magnetism occupied Maxwell and many of his colleagues for several years. After working intensively, Maxwell constructed a system of differential equations whose solutions described all the observable effects that could be measured at that time. The mathematics he used to formulate the equations was based on the calculus proposed by Newton and the further developments of the system, especially partial differential equations, such as the wave equation discussed in section 22. The equations developed by Maxwell related to the same magnetic and electrical fields that Faraday had suggested and proposed relations between them, from which the effects and the different forces observed in the laboratory could be derived. The equations, however, like Newton's law of gravity, did not include an explanation of how the electric and magnetic forces pass through space from the source of the force to the object on which it is exerted. Moreover, unlike the case with the law of gravity, the ether could not provide an explanation for the transition of the electrical and magnetic forces. The reason is “technical” (and can be skipped). The direction of the magnetic field created by the current is perpendicular to the direction of the current, while the force exerted on the current by the magnetic field is perpendicular to the magnetic field and varies according to whether the field gets stronger or weaker (readers who have studied electricity and magnetism will certainly remember the left-hand and right-hand rules to describe the directions of these forces). Such a combination is impossible if the force passes through mechanical stress in any type of material.

At this stage Maxwell proposed a mechanical model through which the electrical and magnetic force would pass. And indeed, after working on this for more than a year, Maxwell succeeded in constructing a theoretical system of gears and bearings of different sizes and directions (see the diagram) whose revolutions, if the space was covered by them, would explain the directions of the transfer of the electric and magnetic force as they were then known. Examining this model of bearings, Maxwell noticed that the same bearings and gears would probably provide a new type of solution to his equations, namely, waves in the magnetic fields themselves. He called these electromagnetic waves.

Maxwell published his gear model as one according to which the magnetic and electrical forces could operate, but he did not claim that a material with the same properties as the gears in fact existed. Yet, he published his hypothesis that the new solutions he found to the equations, namely, electromagnetic waves, exist in reality and added another bold hypothesis that light itself is an electromagnetic wave.

The fact that light has the properties of a wave had been discovered long before by Christiaan Hyugens (whom we will meet again further on). Since the days of Newton there had been a debate as to whether light consisted of particles or had wavelike properties. Newton was of the opinion that light was made up of particles, but many others championed the wave model. The wave model was based mainly on the results of work by the French physicist Augustin-Jean Fresnel (1788–1827), who gave a precise explanation, with the aid of the wave model, of light diffraction through the creation of a mosaic of light and bands of shadow when light is passed through narrow slits. Fresnel and others showed that diffraction can be clearly explained as the passage of light like a wave through that same intangible substance, the ether. Although it was acceptable to consider the ether as a medium for dispersing light, and although he himself showed that the ether cannot explain his electromagnetic theory, Maxwell ventured to claim that light is an electromagnetic wave.

Maxwell's conjectures were received coolly. Nobody had felt or seen the electromagnetic waves that Maxwell claimed existed, nor had anyone seen their activity. The similarity between light and the electromagnetic wave was perceived simply as a coincidence. The dispersion of light could be explained with the help of the ether, but the use of the ether contradicted the assumption that light satisfied Maxwell's equations. He himself proposed his gears as a metaphor and did not claim that some kind of gears actually pave space. Even his supporters among his professional colleagues tried to encourage him to find alternative explanations for the way in which magnetic and electrical forces are interwoven. In the approach that had prevailed since the Greeks, through Newton, and on until the nineteenth century, acceptable explanations for the application of a force had to indicate the means through which it was transmitted.

Maxwell then offered an improved formulation of his equations describing the connection between magnetic and electrical forces that was more complete and symmetrical than his earlier one. The development of the equations depended on only basic rules of dynamics, such as the conservation of energy, and totally ignored the mechanism through which the force was transmitted or the energy conserved. The only justification for the existence of the waves was the fact that they provided a solution to the equations, solutions that have wavelike characteristics. Unlike waves in the sea or sound waves, however, they are not necessarily the result of motion in any medium. The equation defines them as waves. Maxwell persisted in his view that light was an electromagnetic phenomenon. (The formulation in use today is a further simplification and improvement developed later by Oliver Heaviside, 1850–1925.)

This was a revolutionary approach to understanding nature and to the link between mathematics and nature. The Greeks employed the mathematics already known to them to describe natural occurrences accurately and logically, mainly to avoid errors and optical illusions. Newton and his successors, including D’Alembert and Euler whom we came across in connection with the wave equation, used new mathematics to describe quantities and effects that humans can perceive or feel and about which they can even develop intuition, for example the oscillations of a string. Maxwell's improved equations describe electromagnetic waves without reference to the physical nature of the wave. Maxwell's was a twofold innovation. First, the equations ignored how the factors they described operated, that is, the physics of the phenomenon. Second, no one had seen, heard, or felt the result predicted by the equations, the electromagnetic waves, and in particular there was no intuition regarding how they operated.

The same problem of lack of intuition explaining a remote effect apparently exists with regard to gravity and Newton's laws. The important word here is apparently. The existence of the ether, that unseen, intangible medium that can transfer a force, explained the working of gravity. It was only in the wake of Maxwell's revolution that it became accepted that gravity also acts without a material medium. The predictions provided by Newton's equations, for example about the existence of the planet Neptune, were predictions about bodies of a known type with a firmly established intuition about their movement in nature. Maxwell's equations described physical phenomena using only mathematics. Maxwell in effect changed the way in which the world is described by mathematics. He forwent the physical explanation based on known physical quantities. He published his equations, as if declaring this is physics, the physics is inherent in mathematics. Although we cannot see or perceive the elements appearing in the mathematical equations in any other way, they are there, in nature. We can measure the effects of these elements on other bodies, measure the electricity they generate, utilize the force they exert, and so on, without being able to relate directly to their physical entity.

This conceptual revolution was not welcomed with open arms. Michael Faraday, then one of Britain's most eminent physicists, wrote to Maxwell in relatively blunt language (in this author's words, not verbatim): Would the mathematicians kindly translate the mathematical hieroglyphics in which their conclusions are written into comprehensible intuitive language so that the physicists might understand? As far as we know, Maxwell did not respond. In any case, the answer would have been negative. Physics is inherent in the mathematics that describes it.

Beyond the revolution in the approach, the difficulty in adopting Maxwell's theory lay in the fact that at that time all the elements that could be measured and that substantiated the correctness of the equations could also be explained using the physical mathematical relations that were known and accepted. Although there was not one comprehensive equation incorporating all the phenomena, the overall equation that Maxwell proposed had no physical basis in terms of those days, and it predicted effects that no one had seen previously. It took twenty-five years from the publication of Maxwell's work, and eight years after his death, until Heinrich Hertz (1857–1894), a German physicist from the University of Bonn, succeeded in 1887 in producing an electromagnetic wave in the laboratory. There is no need to detail the importance of this physical discovery. Soon afterward, Maxwell's claim that light is an electromagnetic wave with a frequency we can perceive, that is, see, was confirmed. Its uses include radio transmissions, television, cellular telephones, microwave ovens, x-rays, and many others, all based on electromagnetic waves.

We should note that although Maxwell presented his equations without reference to the ether, he did not claim that the ether did not exist. The existence of the ether was still “required,” for example as a medium for Newton's gravitational forces. Maxwell claimed only that one could relate to how electromagnetic waves propagated by means of the mathematical equation they satisfied, without explaining the essence of the waves.

Maxwell's contribution was comparable to that of the Greeks in adopting mathematics to describe nature, and to that of Newton, who had the courage to develop a new mathematics to do so. Maxwell effectively changed the paradigm of the mathematical description of nature. No longer would a quantitative formulation of entities that the senses perceive directly or via a medium be the only way to proceed. Rather, the mathematical handling of abstract quantities, whose existence is justified simply by the fact that they explain effects that we can measure, became fully acceptable. Maxwell's contribution also changed the definition of a physicist. Today, if in the course of a conversation or a lecture I refer to Maxwell as a mathematician, some of my colleagues correct me, claiming that he was a physicist. Yet Faraday, one of the greatest physicists living in the time of Maxwell, refers to him as a mathematician who ought to translate his claims into the language of physics. Instead of doing that, Maxwell altered the definition of physics. Heinrich Hertz said, “Maxwell's theory is Maxwell's equation,” a statement that gives the essence of modern physics. Physics is inherent in the mathematics that describes it.