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Are dimensions real?
Yes, they are by definition tied to the real number within modern mathematical formalism. As such a one dimensional entity is bidirectional, and this poses a numerical error which is being passed down generation after generation and is in such a place that to challenge it is to challenge much of modern knowledge, for physics relies heavily upon the real number as fundamental; as does mathematics. Even simpler forms such as R+ are of course defined in terms of the real numbers. Mimicry rules us all, but variations (and especially variations which work) are worthy of investigation. This route outlined above is suggestive, and I don't generally take it, but what appears as rhetoric will be substantiated below so that the definition of dimension deserves to be challenged. Particularly the ill treatment of time by moderns, whose behavior is unidirectional; whose formal treatment is bidirectional; so much so that great modern physicists puzzle over the action of their equations under time reversal; this absurdity must end.
If spacetime is to be accurately unified then the mathematical formalism which develops spacetime as fundamental must be discovered. The generalization of sign produces such a route, so that a unidirectional number known as the one-signed numbers exists alongside the 'real' two-signed numbers, followed by three-signed numbers. None of these is any more fundamental than the other, nor their subsequent siblings. They exist as a family
P1, P2, P3, P4, ...
and lo and behold a behavioral arithmetic breakpoint does in fact exist after P3
P1, P2, P3 | P4, P5, P6, ...
and this boundary thus provides the arithmetic support for spacetime with unidirectional time even while the entirety is general dimensional and subdimensional.
A prone mind should be asking at this point what that breakpoint is exactly, but to develop this requires learning polysign numbers
but can be briefed here in short form. The usual behavior of the real numbers
- 1.0 + 1.0 = 0 or - x + x = 0
does allow for generalized sign so that for a three-signed system
- 1.0 + 1.0 * 1.0 = 0
and strangely enough for the one-signed numbers
- 1.0 = 0 .
This peculiarity on P1 might seem cause for rejection, but let us please recall the geometrical meaning of the real value so familiar and sensible between its numeric form and its geometric form. Two rays opposed form the real line such that minus one plus one equals zero. Which is more fundamental: the ray or the line? Because the line is formed of two rays we should choose the ray as fundamental, and this is a necessary choice in the development of polysign numbers and their geometrical representation. In order to provide graphical support for P3 we already have enough to go on just from the simple generalization posed above. Three rays arranged in balanced opposition form in a plane. We can draw these rays and label them '-', '+', and '*'. Travelling along the minus ray one unit, then along the + ray one unit, then along the star ray one unit brings one back to the origin. Thus we can argue the laws of balance posed above form the geometry or a rendering of the polysign numbers. It is apparent to the wary that the four-signed numbers will be three dimensional in nature, though this language of 'dimension' is being partially challenged at this point. With P1 we now have a zero dimensional unidirectional entity, whose correspondence to time still goes underappreciated; at least publically that is so. Is to publish to make public? Does an individual have such power? In a free modern society this ought to be true. Whether that is so or not is a valid concern as subauthoritatian power structures get exposed. Should I publish in a high priced journal that I have no access to? No, I should not. Certainly that is far less public than this method here.
The furtherance of polysign numbers shows that beyond the vector behaved simplex coordinate systems outlined above we have a sum and a product to cover, and until we study the product carefully the breakpoint at P4 cannot be exposed. The above construction is compact and a cause for further study, and if it is not readily understood then it should be gone over again. The simplicity thus far along with all of the consequences so far mentioned (except the breakpoint) develops multidimensional geometry from a singular equation
Sum over s ( s x ) = 0
where s is sign and x is magnitude. The geometry is inherent to this equation. No further constraints are required other than the instantiation as a family of number systems, but this comes naturally as one instantiates firstly the real (two-sign) number thencely the three-sign number, the four-sign number(whose rays emanate from the center of a tetrahedron outward to its vertices), and so forth. Oddly though there is enough already to puzzle over the little sister P1... or is she the elder?
Is multiplicity fundamental or ought it be derived via a generator? These are instances of paths to follow out which I have not exhausted though I certainly rely upon the multiplicity. Either way the real value, and so the language of dimension, no longer holds a special place, and this is the moral of the story.