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.