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The genesis of this article was a simple question:
``How can one use the computer in dealing with computationally
approachable but otherwise
intractable problems in
mathematics?''
We began our current exploration of
experimental mathematics
by examining a number of very long--standing conjectures and strongly held beliefs
regarding decimal and continued fraction expansions of certain elementary
constants. These questions are uniformly considered to be hopelessly
intractable given present mathematical technology. Unified field theory or
cancer's ``magic bullet'' seem accessible by comparison. But like many of
the most tantalizing problems in
mathematics their statements are beguilingly simple. Since our experimental
approach
was unlikely to result in any new discoveries, we focused on two aspects of experimentation:
systematization and communication.
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For our attempted systematization of experimental mathematics we were
concerned with producing data that were `completely' reliable and
insights that could be quantified and effectively communicated.
We initially took as our model experimental physics. We were particularly
interested in how physicists verified their results and the efforts they
took to guarantee the reliability of their data.
The question of reliability is undoubtedly
central to mathematicians and here we believe we can draw a useful
distinction between experimental physics and mathematics. While it is
clearly impossible to extract perfect experimental data from nature such is not
the case with mathematics. Indeed, reliability of raw mathematical data
is far from the most vexing issue.
Let us turn to our second and primary concern: insight.
All experimental sciences turn on
the intuitions and insights uncovered through modeling and the use of probabilistic,
statistical and visual analysis. There is really no other way to proceed,
but this process even when applied to mathematics
inevitably leads to some considerable loss of exactness.
The communication of insight, whether derived from mathematical
experiment or not, is a complex issue.
Unlike most experimentalized
fields, Mathematics does not have a `vocabulary' tailored to the
transmission of condensed data and insight.
As in most physics experiments the amount of raw data obtained from
mathematical experiments is, in general, too large for anyone to grasp.
The collected data needs to be compressed and compartmentalized.
To make up for this lack of unifying vocabulary we have borrowed
heavily from
statistics and data analysis to interpret our results.
For now we have used restraint in the
presentation of our results in what we hope is an intuitive, friendly and
convincing manner. Eventually what will probably be required is
a multi-leveled hyper-textual presentation of mathematics, allowing
mathematicians from diverse fields to quickly examine and interpret the results
of others -- without demanding the present level of specialist knowledge.
[Not only do mathematicians have trouble communicating with lay audiences, but
they have significant difficulty talking to each other.
There are hundreds of distinct mathematical languages.
The myth of a universal language of mathematics is just that.
Many subdisciplines
simply can not comprehend each other.]
Contents
Next: Unifying Themes
Up: Introduction
Previous: Introduction