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From: zerr@cat50.CS.WISC.EDU (Troy Zerr)
Newsgroups: comp.graphics
Subject: Re: fractals as bad science
Summary: Let's frame the issue more precisely.
Keywords: Fractals, ugh!, mathematicians, pure, applied, non-
Message-ID: <3842@puff.cs.wisc.edu>
Date: 23 Nov 89 04:29:32 GMT
References: <19544@pasteur.Berkeley.EDU> <1619@crdos1.crd.ge.COM> <3775@celit.fps.com> <5383@orca.WV.TEK.COM>
Sender: news@puff.cs.wisc.edu
Reply-To: zerr@schaefer (Troy Zerr)
Organization: Mathematics Department, University of Wisconsin - Madison
Lines: 81

In article <5383@orca.WV.TEK.COM> brucec@demiurge.WV.TEK.COM (Bruce Cohen) writes:
>I think what you're seeing here is the classic
>antipathy of "pure" mathematicians for "applied" mathematics.  The quotes
>are because I don't think anyone can really tell where the line is drawn;
>maybe there aren't any real distinctions in the mathematics, and all you
>can say is that there are pure and applied mathematicians.  In any case, I
>believe that the Intelligencer is where I saw the quote: "Applied
>mathematics is bad mathematics." 
>brucec@orca.wv.tek.com
     
     You seem to be unclear about what "Applied mathematics" is. . .
The applied mathematician wishes to predict or explain some (usually)
physical situation by constructing a mathematical model and using the    
techniques of mathematical proof to derive certain properties of this
model -- which are then reinterpreted in a physical context.
(This definition is woefully incomplete, and excludes entire fields such
as numerical analysis -- but it suffices for the present discussion.)

     For example, suppose I wish to know the shape of a telephone wire
suspended from two telephone poles.  First I make some simplifying 
assumptions (e.g. the wire is one-dimensional and of uniform density)
and then reinterpret these assumptions as a differential equation.
I then solve the differential equation, and obtain an equation for  the
curve describing the shape that the (idealized) telephone wire will
assume at equilibrium.  Notice that once I have constructed (and justified
the applicability of) the appropriate differential equation, the
techniques used to derive a solution are as rigorous as those of one
performing "pure" mathematics.  (Theorem Proof Theorem Proof . . .)

     Constrast this with the activity of a prototypical "fractal
geometer", the sort whose books are often seen on bookstore shelves.
I program a computer to start with a triangle, choose
a random (well, not really random since my computer isn't very creative)
point in the triangle, move it above or below the plane of the triangle,
and add new edges to form a pyramid.  I then repeat the procedure with
each of the (Triangular) faces of the pyramid, ad infinitum.  (well, not
really ad infinitum since my computer will only do finitely many calculations
before I run out of coffee and doze off.)  With suitably chosen random numbers,
the resulting picture looks like an island.  (Well, on my crude display   
it looks like an island . . . on a more refined display it looks like
a bunch of spikes.)

     Am I to conclude that islands are constructed by some geological 
process of repeated subdivision?  Of course not!  All I have done is to
produce a picture which looks (to me) like an island.  


     Admittedly, I chose a particularly obnoxious example -- "fractals"
may indeed be good models of some physical phenomena.  I
agree that computer graphics are capable of generating interesting
mathematical QUESTIONS, and that Hausdorff measure and Hausdorff 
dimension are interesting on solely mathematical grounds.  But most of
what is popularized is more akin to "pretty pictures" with no real
underlying motivation, attaching numbers to physical phenomena with no
rigor or reason (What is the dimension of the coastline of Britain
WHO CARES!), and attempting to ANSWER mathematical questions with
computer graphics. (Question:  is the "Mandelbrot Set" connected?
Answer:  Well, whenever I look at a picture of it generated by my
computer, it looks connected, so it must be so.)

     Compare the methods of the Applied Mathematician and the Fractal
Geometer.  The applied mathematician (after providing adequate justification
for his model) gives proofs for his (or her) assertions just as any
other mathematician would.   Our fractal geometer (whose books reach
the populus and form their impression of mathematics) regards precise 
and well-founded definitions as an inconvenience
and rigouous proofs as esoteric and boring.   While our portrait of this
unfortunate soul is not applicable to every fractal geometer, that it is
applicable at all is unfortunate indeed. 

     The debate here is not between "pure mathematics" and "applied 
mathematics", nor is it between "good mathematics" and "bad mathematics."
It is between what is mathematics and what is not mathematics.  Mathematics,
simply defined, is what a mathematician does;  and what a mathematician does
and what our prototypical fractal geometer does differ dramatically.

-Troy Zerr
University of Wisconsin
Department of Mathematics
Madison, Wisconsin
zerr@math.wisc.edu