Now mathematics is better equipped to deal with and understand the concept of closure of an axiomatic system. What I mean is, that mathematicians routinely deal with the boundary and limits of a set of axioms. For example, the Rational numbers are closed under the operations of add, subtract, multiply, divide, for if we are given any two rationals and perform such operations we end up with a Rational number as a final answer.
The question here is whether the Maxwell Equations are a closed set of axioms? Well the answer is that without magnetic monopoles, the Maxwell Equations are open ended into an infinite regress. With magnetic monopoles, the Maxwell Equations are a closed set of axioms because the Magnetic Monopoles disrupt the regression. What I need to show is that the Maxwell Equations force the Magnetic Monopoles to become "wire entities".
A few minutes ago in sci.physics, I wrote:
electrons spiraling into the electron; Maxwell Equations axioms over physics means chemistry dominates over nuclear interactions #1750 Atom Totality 5th ed
Alright, stand back for a minute or two and look at the entire Universe and ask yourself what interactions between atoms dominates the Cosmos. Is it interactions of chemistry such as H bonding to H to make H2 or is it nuclear interactions such as H --> He?
What I am asking is how much Chemistry is going on in the world at large compared to how much nuclear transformation is going on?
The answer is quite obvious that Chemistry dominates atomic interactions and not nuclear transformations.
The physics of the 20th century was preoccupied with electrons spiraling into the nucleus of atoms and in their (sic) hatred and haste of this spiraling, they tossed out the Maxwell Equations and invented a brand new physics called quantum mechanics. When we do things in haste and desperation we often get it all wrong.
If the dominate interaction of atoms is not nuclear transformations but is rather, that of electron transformations which is the science of Chemistry itself, then the worry of the 20th century should not have been a spiraling into the nucleus, but a spiraling of one electron of an atom into a second electron of another atom-- that is, chemistry.
I am near the end of my analysis of how the Maxwell Equations are the axioms over all of Physics. I have one last hurdle that needs to be overcome, (I am sure more hurdles in the future will come, but one major hurdle at this moment).
The hurdle I need to overcome is to incorporate Magnetic Monopoles into the Faraday law and Ampere/Maxwell law so as to form "wires" and those wires create the Hund's rule.
It is this last hurdle and obstacle to overcome and I am pretty well along that solution.
When we take the Maxwell Equations as axioms over all of Physics, they form a natural barrier of those equations. What I mean is that the Faraday law starts with magnetism and produces a electric current while Ampere/Maxwell law starts with electric current and produces a magnetic field with a displacement current.
Now there is a barrier because if we take those two together we would expect that the Faraday law produces a electric current which by the Ampere/Maxwell law would produce another yet tinier magnetic field with displacement current which in turn produces another electric current ad infinitum.
So there is something blocking the Maxwell Equations from devolving into an infinity of production of current to magnetism and over and over again.
What is blocking them from being an infinite regress, and the answer is that Magnetic Monopoles put a end to the regression.
The Magnetic Monopoles form a wire loop, a closed wire loop in the Ampere/Maxwell law and that is what the displacement current is confined to whereas the conduction current is confined to the copper closed loop of wire.