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We feel that many of these problems can be addressed through
the development
of a rigorous notion of experimental mathematics.
In keeping with the positivist tradition,
mathematics is viewed as the most exact of sciences and
mathematicians have long taken pride in this. But as
mathematics has expanded, many mathematicians have begun to feel constrained
by the bonds placed upon us by our collective notion of proof.
Mathematics has grown explosively during our century
with many of the seminal developments in
highly abstract seemingly non-computational
areas. This was partly from taste and the power of abstraction
but, we would argue, equally much from the lack of an alternative.
Many intrinsically more concrete areas were, by 1900,
explored to the limits of pre-computer mathematics.
Highly computational, even ``brute--force'' methods were of
necessity limited but the computer has changed all that. A re-concretization
is now underway. The
computer--assisted proofs of the four color theorem are a prime example of
computer--dependent methodology
and have been highly controversial despite the fact
that such proofs are much more likely to be error
free than, say, even the revised proof of Fermat's Last Theorem.
Still, these computerized proofs need offer no insight. The Wilf/Zeilberger
algorithms for `hypergeometric' summation and integration, if properly
implemented, can rigorously prove very large classes of identities.
In effect, the algorithms encapsulate parts of mathematics.
The question raised is: `` How can one make full use of these very powerful
ideas?'' Doron Zeilberger has expressed his ideas on experimental
mathematics in a paper dealing with what he called `semi-rigorous'
mathematics. While his ideas as presented are somewhat controversial,
many of his ideas have a great deal of merit.
The last problem is perhaps the most surprising.
As mathematics has continued to grow there has been a recognition
that the age of the mathematical generalist is long over.
What has not been so readily acknowledged is just how specialized
mathematics has become. As we have already observed,
sub-fields of mathematics have become more and
more isolated from each other. At some level, this isolation is
inherent but it is imperative that communications
between fields should be left as wide open as possible.
As fields mature, speciation occurs.
The communication of sophisticated proofs will never transcend all
boundaries since many boundaries mark true conceptual
difficulties. But experimental mathematics, centering on the use of
computers in mathematics, would seem to provide a common ground for the
transmission of many insights.
And this requires a common meta--language
.
While such a language may develop largely
independent of any conscious direction on the part of the mathematical
community, some focused effort on the problems of
today will result in fewer growing pains tomorrow.
Contents
Next: Experimental Mathematics
Up: Introduction
Previous: Our Goals