Our Goals

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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.

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.]

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