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

From: Osher Doctorow (mdoctorow_at_comcast.net)
Date: 12/30/04 

Date: Thu, 30 Dec 2004 13:42:40 +0000 (UTC)

On 29 Dec 04 19:03:22 -0500 (EST), Osher Doctorow wrote:
>Let's turn to a more familiar situation, a college classroom where
>learning Knowledge is supposed to take place. One variable is
>Knowledge or Learning Knowledge, which I'll abbreviate as Knowledge.
>It is also certain required that the student attend the class, so
>the Action of attending is a second variable - rather obviously
>identical with kinetic energy of a type. A third requirement for
>the lecture/seminar/discussion to function is inhibition of inter-
>ference with other students' and one's own Learning, which includes
>not interfering with the teacher. So we have:

We can make a very slight change with quantities having mostly the
same dimensions and obtain a much more useful equation. We let L
continue to be Learning Knowledge, but instead of having the action
variable involve interference or inhibition of interference with
Learning, let's have I be the amount of attendance in college (or
a particular course) per unit Learning Knowledge. A is attendance
again. Now:

1) [K(A/K)]/A = k (dimensionless constant) or [KI]/A = k

says the same thing in variable terms as the dimensional equation:

2) [KEK^(-1)]E^(-1) = [1]

and multiplying both sides of (1) by A yields:

3) A = (1/k)[KI]

To see that this is not trivial, notice that 1/I is the amount of
Learning Knowledge per unit attendance, which reflects study habits,
background, instructor clarity and ability, etc. Abbreviating 1/I
by S for study habits-faculty habits, and solving (3) for K with
S substituted, we get:

4) K = kAS

so that attendance (A) and study habits-faculty habits account for
Learning Knowledge.

Simple as this may seem, almost nobody if not nobody has actually
studied this to my knowledge! Yet it is entirely consistent with
George C. Homans' views. See Homans' Presidential Address at the
American Sociological Association in Montreal, September 2, 1964,
"Bringing men back in," American Sociological Review 29(5), December,
1964, 809-920. Homans was a prominent Harvard sociologist who, as
might be guessed, was a pro-psychological nonconformist. His equa-
tions involving the variables of activity, interaction, and senti-
ment from his volume The Human Group of the early 1960s (I'll have
to look up the date and publisher) were made into differential equa-
tions by the mathematical sociologist Coleman (James Coleman, if
I recall correctly - I'll try to find references, or you can look him
up on the internet). I was at this time in both anthropology and
sociology (1963-1964), having published "Group structure and author-
ity" in the April 1963 American Anthropologist and doing a book
review on Mathematical Sociology in Sociology and Social Research
(a journal). My paper in the American Anthropologist dealt with
communication (related to interaction) and authority (related to both
activity and sentiment) which came from my M.A. Thesis at the
University of London in 1961. The book review dealt with game theory
and kinship equations.

Osher Doctorow