Experimental Mathematics: Examples, Methods and Implications

Recent years have seen the flowering of “experimental” mathematics, namely the utilization of modern computer technology as an active tool in mathematical research. This development is not limited to a handful of researchers, nor to a handful of universities, nor is it limited to one particular field of mathematics. Instead, it involves hundreds of individuals, at many different institutions, who have turned to the remarkable new computational tools now available to assist in their research, whether it be in number theory, algebra, analysis, geometry or even topology. These tools are being used to work out specific examples, generate plots, perform various algebraic and calculus manipulations, test conjectures, and explore routes to formal proof. Using computer tools to test conjectures is by itself a major time saver for mathematicians, as it permits them to quickly rule out false notions. Clearly one of the major factors here is the development of robust symbolic mathematics software. Leading the way are the Maple and Mathematica products, which in the latest ∗Lawrence Berkeley National Laboratory, Berkeley, CA 94720. [email protected]. This work was supported by the Director, Office of Computational and Technology Research, Division of Mathematical, Information, and Computational Sciences of the U.S. Department of Energy, under contract number DE-AC03-76SF00098. †Faculty of Computer Science, Dalhousie University, Halifax, NS, B3H 2W5, Canada. [email protected]. This work supported in part by NSERC and the Canada Research Chair Programme. Quoted at length in E. Borel, Lecons sur la theorie des fonctions, 1928. Kurt Gödel, Collected Works, Vol. III, 1951.