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

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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 nth 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

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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 nth 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.

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