From BENOIT MANDELBROT
It was a pleasure to see the feature devoted to ‘New-wave mathematics’
(3 August). But I was disappointed that William Bown and those he interviewed
described overly briefly the procedures followed in the actual practice
of experimental mathematics. As it is a new craft, it has few accepted rules
and can only be taught by example. Outsiders tend to think it is straightforward;
when they find that they do not necessarily see much on their own, they
tend to think it is ineffective, or even an imposture. To the contrary,
having practised it longer than anyone else alive, I know that it is truly
a craft: it requires a proper attitude and involves its own set of delicate
skills, but it is very effective.
Take the notion that there was a magical day when, to quote your feature,
‘suddenly, everyone could see the amazing things that Julia had conjured
in his head’. Such a day might have happened and perhaps should have happened,
but apparently it did not happen. I was not the first to use the computer
to draw Julia sets. The evidence is that Stanislaw Ulam was the first in
the 1950s. But Ulam thought that ‘mathematics is not really an observational
science and not even an experimental one’. Much additional evidence exists
to show that the computer was necessary to the new experimental mathematics,
but it was not sufficient. In my experience, to look at a picture very rarely
suffices to set out an ‘Eureka moment’ of any sort.
Let me also sharpen your feature on another point. Quite a number of
major fractal conjectures have already been vindicated by a full proof admired
by connoisseurs of mathematics, both ‘old-wave’ and ‘new-wave.’ The connectedness
of the M-set has long ceased to be an isolated example. Dr Tai Lai has proven
a refined form of one of my first observations, namely, that there is an
astonishing resemblance between a J-set and the corresponding piece of the
M-set. And just last month, Dr Shishikura proved an even more difficult
item on my list of observations-conjectures: that the boundary of the M-set
is of Hausdorff dimension 2. However, these proofs have done little to reduce
my personal collection of difficult mathematical conjectures involving fractals.
When one includes topics other than iteration, this list grows faster on
one end than it shrinks on the other. The best path to understanding continues
to be pluralistic. It uses observation and proof hand in hand, as described
in your feature.
Benoit Mandelbrot Yorktown Heights, New York
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