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Conclusion

All the versions of experimental mathematics that we have dealt with so far have two characteristics: their main interest is in expanding our mathematical knowledge as rapidly as possible and none of them stray too far from the mainstream. In many cases this urgency leads to a temporary relaxation of rigor, a relaxation that is well documented and hopefully can be cleaned up afterwords. In other cases it may be intrinsic to the mathematics they wish to explore. When a field has been as wildly successful as mathematics has been in the past few centuries there is a reluctance to change. We have hoped to convince some of the readers that these changes are revolutionary only in the same sense that the earth revolves around the sun. We conclude with a definition of experimental mathematics. Experimental Mathematics is that branch of mathematics that concerns itself ultimately with the codification and transmission of insights within the mathematical community through the use of experimental (in either the Galilean, Baconian, Aristotelian or Kantian sense) exploration of conjectures and more informal beliefs and a careful analysis of the data acquired in this pursuit.

Results discovered experimentally will, in general, lack some of the rigor associated with mathematics but will provide general insights into mathematical problems to guide further exploration, either experimental or traditional. We have restricted our definition of experimental mathematics to methodological pursuits that in some way mimic Medawar's views of Gallilean experimentation. However, our emphasis on insight also calls for the judicious use of examples (Aristotelian experimentation).

If the mathematical community as a whole was less splintered, we would probably remove the word `codification' from the definition. That is to say that a great deal of time will need to be spent on developing a language for the expression of experimental results. Since there are real communications problems between fields and since the questions to be explored will be difficult, it seems imperative that experimental investigators make every effort to organize their insights and present their data in a manner that will be as widely accessible as possible.

With respect to reliability and rigor, the main tools here are already in place. We need to stress systematization of our exploration. As in our experimental project on normality, it is important to clearly define what has been looked at, how things have been examined, and what confidence the reader should have in the data. Although mathematicians may not like to admit it, ease of use will have to be a primary consideration if experimental results are to be of widescale use. As such, visualization and hypertextual presentations of material will become increasingly important in the future. We began by stealing some of the basic tools of scientific analysis and laying claim to them. As the needs of the community become more apparent one would expect these tools and others to evolve into a form better suited to the particular needs of the mathematical community. Someday, who knows, first year graduate students may be signing up for Experimental Methods in Mathematics I.

Contents Next: References Up: Experimental Mathematics: A Discussion Previous: Interpretation