In my last article “Origin of mass, gateway to the final physics” (http://prebabel.blogspot.com/2012/04/origin-of-mass-gateway-to-final-physics.html ), I have defined m (mass) axiomatically. Mass is the attribute which describes the internal structure (spacetime structure) of a wave-packet (with a wavelength L = lambda).
I also introduced the “Interlocking Thesis (IT)”: all Nature constant [Alpha, e (electric charge), c (light speed) and h-bar (Planck constant)] and Nature dimensions (space, time and mass) are interlocked (cross-checked). That is, all parameters of IT can be recursively defined. Thus, mass must also be defined with both space and time. And now, it is the time to define space axiomatically.
In fact, there is no way to define what the “space” is with the anchor-lock (the Alpha). With this anchor-lock, we can only define the “delta s”.
delta s = c * delta t, t is the time
Someone can object this definition quickly by pointing out that time is one-dimensional while the space has three dimensions. Thus, I will discuss these issues with three points.
One, the existential issue --- For any symbol (representing concrete, conceptual, abstract or all the whatnots), its existence consists of two parts.
1. It is physically there.
2. It carries meaning.
If a thing is physically there but is “never” interact with anything (including itself), it carries no meaning. At here, the “meaning” of a thing has nothing to do with consciousness. As long as it participates in an interaction, it provides a meaning to that interaction. Thus, a physical reality without meaning has no existential value (again, having nothing to do with consciousness). Black hole or dark matter has meaning as they participate in gravitational interaction.
On the other hand, a symbol without a corresponding reality does have existential value if it carries meaning. The concept of Heaven might not have a corresponding reality but has the existential value.
As the meaning of a thing arises from participation (not from consciousness), multi-level meanings can arise from multi-level participations. And, the trace or signal of the early participation can be erased by the later (higher level) interaction. Thus, although the trace of the early signal cannot be detected physically at the higher level after some history killing processes, its meaning must be still visible at that higher level. Thus, a seed-reality can always be defined with high level and visible entity. Space being a high level visible entity, its existential value is guaranteed.
Two, then, what is dimension? And, what are its attributes? What those attributes got to do with the universe?
We should look what are already known.
1. In 1870s, Georg Cantor proved that every n-dimensional space can always be brought into a one-to-one correspondence with the one dimensional line, that is, one dimensional line can give rise to n-dimensional space.
2. This dimensionality issue can also be understood with fractal geometry. In fractal geometry, there are many space-filling curves, an infinite number of them to be exact. With the Hilbert space-filling curve, it crosses every point on a plane without crossing itself, that is, a two dimensional plane is reduced to a one dimensional line.
From two facts above, all high-dimensions can always be reduced to one-dimension.
In physics, the dimension refers mostly as the spatial dimensions. In the String theory, it predicts some extra dimensions, and their existence can be tested by looking for the Kaluza-Klein (KK) partner particles. The LHC data thus far (April 21, 2012) has ruled out all KK particles. Yet, my definition of the term of “dimension” is significantly different from the one of String theory. It has a much bigger scope.
New definition: If “all” information of system B can be “wholly” described with n codes, then system B has n dimensions.
The above definition is a linguistics definition. Yet, it is similar to the base-dimension of a vector space, such as, “A location inside a two-dimensional space is specified by two pieces of information.”
With this new definition, the entire world must change.
1. Regardless of how many “physical dimensions” it has, the entire computable world has only two-dimensions, as it can be wholly described with two codes (such as, [0, 1]). This is guaranteed with the Two-code theorem of mathematics.
2. The outer surface of a ball has 4 dimensions, guaranteed with the four-color theorem.
3. The outer surface of a torus has 7 dimensions, calculated by the edge-equations or the Heawood Conjecture.
With this new definition, the consciousness and the intelligence could be dimensions, even the dimensions for the non-computable world.
Three, how can the spatial dimensions be defined with the above general definition? That is, how do the three spatial dimensions arise from Cantor’s theorem?
Cantor’s theorem is well-known, and its proof is very intense. However, I will show a shorter proof here in order to show the process of the rising of three spatial dimensions.
With Cantor’s theorem, there is one and only one dimension, the number line, and we can call it x-dimension. While this x-dimension has an infinite length, its thickness is zero.
For a plane, it is viewed to have two dimensions in physics, but it is, in fact, the x-dimension which acquires a width (y) for its dimension-line. And, this “width” is only a “trait” for this one-dimension of x. In essence, y is not a true dimension but is a “trait” of x. Perhaps, we can call it a “trait-dimension”.
For a physical three dimensions (x, y, z), it is again the result of the x-dimension which acquires two traits, the width (y) and the height (z).
With the above simple procedure, the Cantor’s theorem is proved. Every point in the three dimensional space is brought to a one-to-one correspondence to one-dimensional number line x.
Of course, there is no reason to re-prove Cantor’s theorem here. My point is trying to discuss the true meaning and the true essence on dimension. There is one and only one dimension, and any other dimensions are “trait-dimensions” and are extra-dimensions.
In physics, the time is the Cantor dimension which gives rise to three spatial dimensions. As these three spatial dimensions are the “trait” dimensions of time, there needs a matrix (N) to describe those traits. Thus, the space can be defined axiomatically as,
Delta s = (N) c * delta t, N is a trait matrix, c the light speed.
Thus, as soon as we know the N (trait matrix), the space is defined axiomatically. And, I will discuss the issue of N in the next posts.