In the *"….Of Great Mathematicians...."* blog I discussed Einstein and current researchers' difficulties in creating an appropriate Mathematical Framework. I'd like to add to that.

First, I will quote from Maxwell (1831-1879) “A Dynamical Theory of the Electromagnetic Field”, completed in 1864, the purpose is “to explain the [electromagnetic] action between distant bodies without assuming the existence of forces capable of acting directly at suitable distances. The Theory I propose may therefore be called a theory of the Electromagnetic Field...” .

So began the Field Theory and a search for its unification, because after the main body of the text, Maxwell abruptly changes tact and turns to gravitation, writing: “After tracing the action of the surrounding medium both the magnetic and the electric attractions and repulsions, and finding them to depend on the inverse square of the distance, we are mutually let to inquire whether the attraction of gravitation, which follows the same law of distance, is not also traceable to the action of a surrounding medium”.

“But how can one explain,” Maxwell asks, “that gravitational force is attractive whereas the force between electric charges of the same sign is repulsive?” Maxwell notes that this requires and ad hoc change of sign when going from the electromagnetic to the gravitational ponderomotive force. Therefore the gravitational energy also needs an additional minus sign. This leads to paradoxes: “the presence of dense bodies influences the medium so as to diminish the energy [of the medium] wherever there is a resultant attraction. As I am unable to understand in what way a medium can possess such properties, I cannot go any further in this direction in searching for the cause of gravitation.” With such wise words began the search, which to this day is as yet unresolved due primarily, in my opinion, to what Eugine Wigner called “the unreasonable effectiveness of mathematics”.

Enter Einstein. Among Einstein's minor blunders, in my opinion, were that he did not study the works of his contemporary Felix Klein and his disagreement with Walter Ritz, who tragically died in 1909. Ritz did not accept special relativity (Poincare [1854-1912] had similar misgivings), but rather believed in the need to give up the concept of a field described by partial differential equations. The issue they disagreed upon was whether advanced and retarded solutions of the electromagnetic field equations are both admissible types of solutions.

Their joint 1909 paper, in which they both agreed to disagree states: ”Ritz considers the restrictions to the … retarded potentials as one of the roots of the second law [of thermodynamics], whereas Einstein believes that the irreversibility rests exclusively on probability grounds”. Further note that it was Ritz who devised the combination principal for Line Spectra. Tragically, Einstein also died before Group Theory, one of the only outstanding mathematical developments of the 20th century, fully began to emerge.

On Maxwell, Einstein wrote that: “I incline to the belief that physics will not be permanently satisfied with...an indirect description of reality, even if the [quantum] theory cam be fitted successfully to General Relativity postulates. They would then be brought back to the attempt to realise that program which may suitably be called Maxwell's: the description of Physical Reality by fields which satisfy without singularity a set of partial differential equations”. Maxwell himself knew his work was incomplete and indeed it took him nine years to come up with just a handful of equations! But on these, incomplete, set of equations the whole of Western Technological and Full Spectrum Dominance rests.

In his final letter to his lifelong friend and confident Besso, Einstein wrote in 1954, “I consider it quite possible that physics cannot be based on the field concept, i.e. on continuous structures. In that case, nothing remains of my entire castles in the air, gravitation theory included, [and of] the rest of modern physics.”. Einstein died on the 18th of April 1955. For Einsteins “continuous structures” here, read Sinusoids and Superposition. Both are Mathematical Illusions. Einstein always referred to the Wave Equation as the Psi – function. A wise man indeed. The other poignant tragedy was that Klein died one year before Wave Mechanics came to prominence.

Previously I have raised the issue of the electron being split but what if a photon can be split and indeed what is light? No one really knows. Therefore I have always wondered what Einstein, Klein, Maxwell or Poincare would have made of the concept of a Phased Locked Loop and the modern concept of Direct Digital Frequency Synthesis. Or Phase Noise and the relationship with the Five Power Laws.

The mathematician, Tom Apostol, outlined the human requirements for developing (not discovering) new frontiers in mathematics: the guide, the explorer, and the diggers. Lots of diggers are needed! In this respect Einstein was a true guide but in the realm of Matrix Mechanics, the guide was Max Born, Heisenberg was the explorer and Jordan the first digger. Of Schrodinger, the less said the better. QFT and QED aside, at the heart of the matter lies the mathematicians continuous-discrete divide. Both approaches are as yet to be reconciled and have been since the time of the Greeks.

Recall that Einsteins most famous solution written out reads: Energy equals a positive OR negative mass multiplied by the theoretical constant of the speed of light raised to the power of two.

The historian of mathematics E.T. Bell, wrote (in 1937) that the “major task of mathematics to-day is to harmonise the continuous and the discrete, to include them in one comprehensive mathematics, and to eliminate obscurity from both”. Mathematicians have failed in this task. Bell also wrote that “An impartial account of western mathematics, including the award to each man [and woman] and to each nation of its just share in the intricate development, could only be written by a Chinese historian. He alone would have the patience and the detached necessary for disentangling the curiously perverted pattern to discover whatever truth may be concealed in our variegated occidental boasting”

So, if you were interested in leveling the playing field, and sorting out the mess western mathematicians have left us in, where would you advise a Chinese mathematician to start?

Well, western mathematicians and philosophers talk about a “limit case” - the point where a system breaks down. Where we've taken an idea as far as it can go. Perhaps a Chinese mathematician should start there.

Barry McKeown

15 November 2018