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ART MATRIX  PO 880  Ithaca, NY  14851-0880  USA
(607) 277-0959, Fax (607) 277-8913

'The Paths of Lovers Cross in the Line of Duty.'

IN DEFENSE OF THE MASTER

Copyright (C) 1991 Homer Wilson Smith

'In the preface to The Science of Fractal Images, Mandelbrot
suggests that fractal geometers also use computer graphics to develop
hypotheses and conjectures.  But the difference is that the hypotheses
and conjectures are (like the objects which they study) self-
referential.  One generates the pictures to learn more about the
pictures, not to attain deeper understanding.  That the pictures have
occasionally inspired fine mathematicians to prove good theorems seems
serendipitous at best.' Steven G Krantz, The Mathematical
Intelligencer, Vol 11, No 4.

I have probably burned more CPU cycles than most in the search
for pretty pictures, thus I can sympathize with Krantz's deploring
such use of computer time, however I have also worked long and closely
with Dr.  Hubbard during many of the years that he was first
interested in fractal mathematics and there is something that needs to
be said here.

In the first place Dr.  Hubbard is the first to tell his graduate
students that pretty pictures are a waste of time unless they can
PROVE something about them.  Proof, it would seem, is the coin of the
realm.  I know this for a fact because I have had to listen to the
endless woes of poor graduates students who can produce the most
amazing pictures, but who have a very hard time proving anything.
'Hubbard wants me to PROVE something', they complain.

Further my own experience working with Dr.  Hubbard has given me
a direct and personal insight into the relationship between pretty
pictures and deep mathematical cognition or proof.

In the first place, no picture can ever prove a conjecture, just
as one example can never prove a hypothesis.  However one measly
picture CAN DISPROVE a conjecture in no time flat.  Dr.  Hubbard is a
fine one for coming up with endless conjectures about iteration
theory, and he uses the images that I make for him to scan anxiously
for the one that will prove the conjecture wrong in an absence of
immediate analytical ability to prove or disprove it formally.

It is an enormous waste of time trying to prove something right
that is indeed wrong, and if one can bypass that effort by making a
few pictures to see if the disproof is easily forthcoming, then making
such pretty pictures is well worth the time.

Secondly, we were recently working on a problem of the
intersection of two quadratics and the behavior of Newton's Method in
this space to find the points of intersection.  Dr.  Hubbard directed
me to make a whole slew of movies, each movie had a 1000 frames, that
scanned the parameters from low to high.

During the viewing of these movies it became apparent that there
was a line in the space that Dr.  Hubbard could not immediately prove
should be there.  Thinking about it some he suddenly came up with the
conjecture that this line was invariant under Newton's method which
meant that any iteration starting on the line would forward iterate to
another point still on that line.

He got very animated suddenly, dragged me over to a table and
said 'This is amazing, I can't believe how stupid I am to not have
thought of this before, I wonder if I can PROVE it is true!'

He then proceeded to drag me through the entire procedure of
working through the proof, possibly for the first time in human
history.

Now THIS is the stuff of real mathematics, all of Steven Krantz to
the contrary.  And I was there.

Homer
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