*Misleading
fractals *

Professor Andrew Abbott's
usage of the word "fractal" is unconvincing ("Investigations,"
December/00). That word has been coined to have a specific meaning
beyond that in "subdivision" and "similarity."
Among its specific characteristics, a fractal stands for repeated
subdivisions with no end, a recursive similarity from one step
to the next in the subdivision, and a fractional dimension in
a well-defined mathematical sense. That sociologists, as in one
of his examples, can be divided into positivists and interpretivists,
each group again divisible in the same way for a certain number
of steps, is not enough to justify jazzing up the simple word
"divide" by using fractal instead. Not all categories
and sub-categories are fractals.

Matter, it was believed from
the time of the Greeks, is repeatedly divisible down to a final
scale of atoms. That does not make matter fractal. Closer to his
example would be to divide a group of people according to taller
or shorter than some prescribed height. Each sub-group could be
further divided into finer sub-scales, but again that does not
a fractal make.

The problem with inappropriate
usage is that words are metaphors, bringing along with them a
whole baggage of associated meanings, particularly in the case
of a word such as fractal which has been specifically coined as
a stand-in for some precise concepts. In a way, a fractal is boring
because it implies that no new aspects emerge at each stage of
the subdivision because of strict self-similarity. In today's
physics, we know atoms are further divisible, but each stage brings
in new features so that no physicist would use the word fractal
to describe this sub-structure. So too in the social sciences.
The over-selling ("Abbott pushes his argument beyond the
social sciences to academia in general and society at large"!)
of the word fractal may foreclose the possibility that, as sociologists
debate and make further distinctions between ideas, something
new appears on the stage at some (each?!) step. If not, what is
this game worth?

**Ravi
Prakash Rau, PhD'71**

Baton Rouge, Louisiana

I found the article by Sharla Stewart
about Professor Abbott's work very interesting, because I have
also made an effort to apply chaos theory to sociology.

Abbott's observation that sociologists
tend to split into positivists and humanists takes me back to
my conflicts in graduate school, but I'm not sure if it qualifies
as a "fractal," geometrically speaking. A fractal,
as I understand it, is a pattern generated by a nonlinear equation
that repeats itself not at *every* scale but at every *n*th
interval. For example, at every 5th interval of the Mandelbrot
set depicted in James Gleick's Chaos: *Making a New Science*
(Viking, 1987).

Instead of a fractal pattern, Abbott
seems to be describing a polarity that he says repeats itself
at every scale. This sounds like Sorokin's "law of polarization,"
except that he applied it to altruistic vs. egoistic responses
to crisis, not sociologists.

**Michel
Paul Richard, AB'51, AM'55**

Miami

*Professor Abbott replies*: It is not surprising that a
short summary of a complex argument should lead to misunderstandings.
Both Dr. Rau and Mr. Richard worry that I have used the concept
of fractal incorrectly. Dr. Rau thinks that my argument uses
the concept to refer to what is merely the use of nested dichotomies
to approximate either a linear scale (as in his height example)
or a simple inclusion/categorization system (as in his example
of the divisibility of matter). In discussing both these possibilities,
however, my book makes precisely Dr. Rau's argument that social
and cultural structures must be something more peculiar than
that to be called fractal. Mr. Richard worries that in emphasizing
the nearly continuous scalability of fractal distinctions, I
have lost sight of the discrete character of successive contraction
mappings. I haven't. The vast majority of the book's examples
do concern discrete mappings (mappings "at every *n*th
interval" in Mr. Richard's phrase). But the power of the
continuous scalability conception makes it worth trying out,
so I did so.

As
for using "fractal" as metaphor, I plead guilty. The
metaphor organized large masses of previously unaccountable
facts. That's good enough for me. Any reader is welcome to read
the book to see if he or she agrees.