Mathematical Considerations

One of the fallacious assumptions of classical physics is the belief that physical events and phenomena occur within a simple and precise mathematical framework. This idea is an extension of the principle of Occam’s Razor, which states that the simplest idea that can account for all the facts is preferred over an unnecessarily complex one. While this may be true for conceptual ideals, it cannot be extended to mathematics.

When James Clerk Maxwell first tried to understand and explain the electromagnetic interaction, he developed a system in which the magnetic forces were enacted by hollow tubes extending from the magnetic bodies. The interaction occurred when the hollow tubes from one body came in contact with hollow tubes from another body. However, when he tried to apply mathematical analysis to this idea, he was hindered from proceeding by a wall of mathematical complexity beyond his abilities. Finally, in desperation, he abandoned the whole idea in favor of the concept of the “field.” When Maxwell applied mathematics to the idea of the electromagnetic field, he finally attained a set of beautiful differential equations that described the results of his electromagnetic experiments. This discovery was viewed by many to be the first great revelation toward the ultimate field explanation of everything.

The great beauty of the field in theoretical physics is that the mathematics involved to describe an event are simple and concise when compared with the mathematics of same event in non-field terminology, such as Maxwell attempted to do with his “hollow tube” explanation of magnetic force. The requirement of theoretical physics, that all fundamental theories be presented within a concise mathematical framework, virtually prevented the serious theoretician from ever considering a non-field theory because of its mathematical complexities. Non-field theories are denied serious consideration because they cannot be given a precise mathematical description. Even if someone was able to develop a precise set of non-field equations, they would likely be so complex and mystifying that only the most gifted mathematician would be able to understand them.

Prevailing logic maintains that as long as we can explain the all-pervading electromagnetic force as a field with such beautiful mathematics, then the truth of the field’s existence cannot be denied. Therefore, if the true underlying nature of electromagnetism is a field, then all other forces must also be field events at their basic level.

Albert Einstein readily embraced this line of reasoning when he developed the theory of General Relativity, which depicted the force of gravity being transferred between objects through a “gravitational field” that each body of mass-generated.

Johannes Kepler spent a good many years studying Tycho Brahe’s detailed observations of planetary motion in an effort to resolve their seemingly complex movements into a simple system which could be applied to all of them. Finally, armed with the Copernican assumption that the earth and the planets revolved around the sun, he was able to develop his four laws of planetary motion that were able to explain these motions in a simple and concise mathematical way.

Later, Isaac Newton was able to take Kepler’s laws and combine them with his own three laws of motion, his invention of calculus, and his theory of gravitation, to develop a simple and very precise system that was able not only to describe the motions of the planets but the motions of falling apples and the trajectories of cannon balls as well.

For many years Kepler’s laws and Newton’s equations were offered as proof of the idea that the underlying truths of complex natural phenomena could ultimately be expressed in simple and concise mathematical form. Then along came Einstein with his theory of General Relativity, which offered a far more complex system of mathematics that was more accurate than Newton’s system. Newton’s equations were thus shown to be a mere generalization of gravitational phenomena that had nothing whatsoever to do with any kind of underlying truth.

From its earliest beginnings, particle physics has developed and been built upon two basic assumptions: the “point particle” and the “field.” These two ideas have persisted to the present day not so much because they have been fruitful in revealing underlying truth but because they can be so easily manipulated mathematically.

Recently it has been proposed by some that the idea of the point particle be discarded and replaced with the idea of the “string particle.” String theories have shown much promise, but the extremely complex mathematics needed to manipulate them has severely limited their progress. The closer string theories approach the truth of matter, the more complex the mathematical treatment of matter’s actual dynamics become.

The idea that ultimate truth can be represented with simple mathematical equations is probably totally false. A simple example of this is the familiar series of circular waves that move away from the point where a pebble is dropped into a quiet pool of water. While these waves can be described in a general way with a simple set of mathematical equations, any true and precise mathematical description of this event would have to include the individual motion of each molecule within this body of water. Such an equation would require more than the world’s supply of paper to print and its complexity would make it virtually meaningless.

The idea of the circlon is easy to describe and illustrate. However, any kind of mathematical description of its complex internal dynamics is presently beyond my abilities. This deficiency does not mean that circlon theory cannot compete with the mathematically simplistic point-particle and field theories of matter. It simply means that perhaps ultimate truth is not as easily accessible to a mathematical format as was once hoped.

In the last portion of this book, nuclear structure is described in what I feel is a very precise mathematical presentation. While these nuclear equations are not what might be called “real” mathematical equations, they describe very accurately the entire body of experimental knowledge surrounding the 2000 or so known isotopes of the chemical elements.