DISCREPANCY BETWEEN MAXWELL’S THEORY AND NEWTON’S THEORY - MATHEMATICS AND THE MODERN VIEW OF THE WORLD - The Remarkable Role of Evolution in the Making of Mathematics - Mathematics and the Real World

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

26. DISCREPANCY BETWEEN MAXWELL'S THEORY AND NEWTON'S THEORY

Maxwell's equations were amazing in their predictions of the discoveries that came later and fitted the observations and measurements performed in relation to those discoveries. Likewise, Newton's original equations and other equations that were derived using the mathematical instruments that he developed fitted physical reality perfectly. Yet there was a discrepancy between their equations, a discrepancy that threw doubt on whether both described the same physical world. We will describe that discrepancy.

Newton's second law, F = ma, states that the acceleration a of an object is proportional to the force F exerted upon it. The law does not relate to the speed of the object while the force is being exerted on it, but to its acceleration, that is, the change in its speed. Neither does the law of gravity relate to the speed of the object. The gravitational force acting on a body does not vary if it is moving or stationary. This non-dependence on the speed of the object is a blessing to mathematicians and physicists, and of course to engineers, because when measuring a change in speed as a result of the application of a force, there is no difference whether it is measured in relation to the Earth or in relation to another system of coordinates, such as a moving train. Newton presumed that the world had an absolute system of coordinates that can be used to measure the speed of every moving object. Although we cannot accurately identify this system, we are fortunate that it is not of any importance, since the laws of motion do not change if the speeds are measured with reference to some speed that is constant relative to the absolute system. Such systems are called inertial systems. The laws of motion do not change if one moves from one inertial system to another.

This property is lacking in Maxwell's equations. A transition from one system of coordinates to another, even if the second is moving at a constant speed relative to the first, alters the equation. We did not state Maxwell's equation explicitly, but it is not necessary to know its details to understand why a move from one system to another changes it. Maxwell's equation describing electromagnetic waves makes use of the speed of the wave. When the speed appears explicitly in the equation, a change in the coordinates according to which the speed is measured changes the equation. In other words, Maxwell's equations are inconsistent with the invariance with respect to the inertial system in which Newton's equations are formulated.

The state of physics at that time can be summarized as follows: Newton's equations, which relate to a geometry that is consistent with our senses and whose relation has been accepted for hundreds and even thousands of years, accurately described many physical aspects of motion, including the propagation of waves in an earthly medium such as air or water. The movement described by the equations fits our intuition regarding the movement of objects, intuition that was formed in the course of the development of the human species. Maxwell's equations, on the other hand, predicted the existence of waves without indicating the medium in which those waves move, and their formulation is inconsistent with the known and familiar geometry. Those equations, however, were also astonishingly effective in predicting the physical effects relevant to them.

How do we proceed from here? One possibility is not to try to reconcile the two approaches. No discrepancy has been found between the two sets of equations. They describe different physical effects, and nobody can guarantee that there is one mathematical theory that covers the different effects in physics. A second possibility is to try to change one of the systems of equations for another system whose structure is consistent with that of the second. Many attempts were indeed made to replace Maxwell's relatively new equations with others in order to remove the discrepancy; none succeeded. Then Einstein came along and proposed a surprising third solution: he changed the geometric depiction of the world, that is to say he suggested a description of the geometry of the world that differs from what we feel it is but that reconciles the two systems of equations. Einstein's contribution was a huge breakthrough. It too came about following findings in physics, which we will discuss in section 28, and research in mathematics over many years into the geometry of the world. The salient aspects of that mathematical research are described in the next section.