Re: Dimensional Analysis Via PI Versus Factor/Cluster/Scaling/PCA

From: Osher Doctorow (mdoctorow_at_comcast.net)
Date: 12/31/04 

Date: Fri, 31 Dec 2004 14:22:56 +0000 (UTC)

On 30 Dec 04 13:34:00 -0500 (EST), Osher Doctorow wrote:
>6) E*P = kI (k dimensionless constant)
>so that intention or decision I is a constant times emotion times
>perception. This seems to characterize decisions to withdraw from
>danger (or self-defense against danger) and decision to move toward
>positively perceived things like food. Although Knowledge is
>implicit in I, its explicit absence suggests that it may be useful
>to distinguish between intention (I) and Knowledge-based decision
>(K*).

There are several directions in which research can plausibly go
related to dimensional analysis, but there are also several errors
of past research that should be remembered. I'll give a brief
numbered list of both of them below.

1. Mathematical sociology and mathematical psychology should now
concentrate on dimensional analysis of their underlying concepts
including especially Knowledge/Learning, Emotion(-ality), Action/
Behavior. The interface between these two disciplines, like most
interfaces and interdisciplinary topics, is central according to
PI since boundaries (among other things) maximize PI under quite
general conditions. One of the best papers in this regard is Thomas
J. Fararo's (U. Pittsburgh Sociology Department) "Theoretical socio-
logy in the 20th Century," JOSS {Journal of Social Structure) Vol-
ume 2 April 2000, which is accessible on the internet under http://
www.cmu.edu/joss/content/articles/volume2/Fararo.html. Fararo is
a mathematical sociologist who wrote one of the early books on
mathematical sociology in the 1970s.

2. Quantities that we ordinarily take to be scalars, like prob-
abilities and angles and distances, should be studied for possible
(and in my opinion probable) dimensional rather than dimensionless
structure. Angles were already recognized as possibly having
dimensionality under certain conditions by Krantz, Luce, Tversky
and Suppes in one of their early volumes. But probabilities are even
more important, I think. Notice that probabilities are measures on
sets, and sets have dimensional characterizations of all kinds
outside of something like pure number theory, but the usual excuse
for not assigning a dimension to probability is the "relative freq-
uency" argument since if we regard a probability as a limiting
relative frequency of limit of a ratio of two frequencies (the
frequency of some event divided by the total frequencies of the
"universe" of events studied) it looks as though probability is just
a number divided by a number which has no dimension. What is
ignored by this argument is that probability is just as fundamental
as length (distance) which has a dimension (L), and that just as
the length of an object is a number and yet has a dimension, so a
limit of a relative frequency is a number and yet can have a dimen-
sion depending on how fundamental it is for example. Since the
set (A-->B) is even fundamental among sets, the probability P(A-->B)
should especially have a dimension.

3. Conformity in quantitative and scientific disciplines has missed
the boat in the two directions above and others as well. We may
well benefit from the old time type of Medical Doctor who used to
ignore publish-or-perish and visited sick people at home. In
sociological language, publishing makes a person "one of the boys"
not only in a positive sense but in a negative sense of conformity
and tendency to follow popular fads and popular trends in the field.
The danger of talking about things whose dimensions one does not
even dream of much less care about is very real. I see it in
politics when mathematicians/scientists jump into politics without
even interest in definitions near election times. In fact, it is
even in my opinion a general rule that if something is conformist,
then it is probably wrong (though a few things which most people
believe are probably correct, especially bodies of mathematical/
scientific Knowledge in the sense of "facts" or theorems or laws).

Osher Doctorow

Relevant Pages

  • Re: Spectrum, Markov, Large Deviations, and PI
    ... The connection of Probability-Statistics with volume and Lebesgue ... elementary or even first upper division probability or statistics ... Volume, area, and length in Euclidean ... capacity dimension on closed bounded (that is ...
    (sci.stat.math)
  • Re: Sci-Fi Noein - "Shrodingers Sand Box"
    ... the physics sound more complicated than it really was. ... we would need a complicated set of probability functions for our particles. ... dimension they want to reach. ...
    (rec.arts.anime.misc)
  • Re: Black Hole Bouncing vs Evaporation vs Control
    ... statistics, randomized decision rules are key in statistical decision ... words about Probability as a dimension after writing the new equation: ... It is typical to regard Probability as dimensionless because of the ...
    (sci.physics)
  • Re: how to compare the Gaussian probability of data with different dimensions?
    ... the covariance matrix could be singula and I have to deduce the ... dimension by projecting the data to a lower dimension to calculate ... how can I compare the probability ... want to add extra weight to the prior proobability of a singular model ...
    (sci.stat.math)
  • Re: Arbitrarily Many Nested Loops
    ... Sinan Unur ... Neither the code snippet above nor your verbal description makes any ... in another vector (one size per dimension). ... distribution where the success probability differs between trials. ...
    (comp.lang.perl.misc)