Now what the description of the below proof is in mathematics, is called a "closed algebra set". The rationals in mathematics are a closed algebra set in that if we take any two rationals and add them, we do not arrive at a "new number" not already a rational. So that in the proof that the Maxwell Equations cannot have a 4th dimension because a 4th perpendicular cannot yield a new solution for the already preexisting solutions gained from 3 perpendiculars of 3rd dimension.
Now the Natural Numbers: 0, 1, 2, 3, 4, 5, . . are not closed to division because we can take 1 and divide it by 5 and gain a new number not already existing. So that if the Maxwell Equations could take a 4th new perpendicular and arrive at a new solution not already existing with the 3 perpendiculars of 3rd dimension, then the 4th dimension exists, but it does not exist for the reasons stated above.
I wrote in sci.physics, a few minutes ago:
Maxwell Equations proof that 4th dimension is phony baloney #1461 New Physics #1811 ATOM TOTALITY 5th ed
So how does the Maxwell Equations prevent the existence of 4th dimension or higher?
It does so in the fact that the Maxwell Equations are linear and so the thought of a **new perpendicular** giving rise to a solution not already existing by the 3 perpendiculars of 3rd dimension is only wishful thinking. Every new perpendicular besides the existing 3 perpendiculars is a preexisting solution already contained in the Maxwell Equations.
So that for example if we have a alleged 4th new perpendicular of a moving bar magnet of Faraday's law coming into a Ampere/Maxwell law set-up, that this new alleged 4th perpendicular is already covered by the existing Ampere/Maxwell law.
In short, the Maxwell Equations cannot accommodate a new 4th perpendicular and that all solutions to the Maxwell Equations are handled by the existing 3 perpendiculars.
What Ptolemy proved that only 3 dimensions can exist in mathematics could not be a general proof because Ptolemy could not bring in Physics and the human mind unable to perceive of a 4th dimension. But when we anchor the proof in the Maxwell Equations we get rid of that last objection, because the Maxwell Equations cannot fit a new 4th perpendicular.