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From: Osher Doctorow on 8 Aug 2010 14:22
From Osher Doctorow

Superstring Theory regards 3 of its 10 Dimensions as being 3 Spatial
Dimensions, which in Dimensional Analysis would be symbolized for
example as Lx, Ly, Lz, where the subscripts x, y, z are ideally in
orthogonal directions usually. That leaves only 5 Dimensions to be
added to Dimensional Analysis' current 5 typical Fundamental
Dimensions of M (mass), L (length), T (time), Q ((Electric) Charge),
theta (temperature, if we use Superstring Theory's Lx, Ly, Lz - but
then we no longer have 5 Fundamental Dimensions of Dimensional
Analysis, since L changes to Lx, Ly, Lz. Something is wrong with
this picture.

In Dimensional Analysis applications, unless 2 or more directions like
x and y or x and y and z are all relevant to the particular problem
and have DIFFERENT TYPES OF INFLUENCE on the problem, L is used as the
Fundamental Dimension. Lx, Ly, Lz are SUBTYPES OF L, not
replacements for L. Otherwise, we could claim that different types
of Mass corresponding to different Fundamental or other important
particles are different Fundamental dimensions, yielding dimensions of
Mp (mass of proton), Me (mass of electron), and on and on.

Theoretical Physics uses both viewpoints, and quite often problems are
simplified by choosing their setting in one spatial direction like x
and then generalizing if necessdary.

So how many Fundamental Dimensions do we need in Dimensional Analysis
to correspond to the 10 dimensions of Superstring Theory or the 11 of
Supersymmetry? Collapsing the 3 Superstring dimensions of space
into one Fundamental Dimension L yields only two spacetime observable
dimensions correspond to L, T (time), instead of 4 (x, y, z, t), so we
only need 3 more Fundamental Dimensions in addition to M, L, T, Q,
theta to reach 8 Fundamental Dimensions with L, T added. The 3
subtypes Lx, Ly, Lz of L if they were to replace L among these 8
(which is analogously done in Superstring Theory but without declaring
them Fundamental Dimensions) would yield 2 additional dimensions to
yield 10 Dimensional Analysis "Fundamental Dimensions".

The 3 Fundamental Dimensions that we need in addition to M, L, T, Q,
theta, can still be built from the Repulsive vs Attractive or
Expansive vs Contractive or the Probabilistic dimensions discussed in
the last few sections and subsections. One might prefer
distinguishing between attractive "length" L1 and repulsive length L2,
attractive mass M1 and repulsive mass M2, and attractive charge Q1 and
repulsive charge Q2. Or one might prefer the Probabilistic types
such as P(A-->B), P(B|A), P(A) for example.

Osher Doctorow
From: Osher Doctorow on 8 Aug 2010 14:37
From Osher Doctorow

If we do use the Probability route mentioned, then P(A-->B), P ' (A--
>B), and either P(A)P(B) or P(A<-->B) would seem to be better than P(B|
A) and P(A). In fact, it is easy to show that P(A ' --> A) = P(A),
P(universe --> A) = P(A), etc.

In physical language, we would be introducing the dimensions of
Probable Causation/Influence (P(A-->B)), Internality-Externality (P(A--
>B) vs P ' (A-->B) above), and Dependence-Independence (P(A)P(B) vs
the others), or if we use the last quantity P(A<-->B) we would have
Probable Correlation. Notice that Internality-Externality has the
nice topological quality of being important in that field and in
geometry as well. Causation, Influence and Probability are certainly
important in a very large variety of physical problems, and similarly
for Independence vs Dependence.

Osher Doctorow
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