Exploratory Experimentation and Computation 1. Exploratory Experimentation

I believe our research community is facing a great challenge to re-evaluate the role of proof in light of the growing power of current computer systems, of modern mathematical computing packages, and of the growing capacity to data-mine on the Internet. Add to that the enormous complexity of many modern capstone results such as the Poincaré conjecture, Fermat's last theorem, and the Classification of finite simple groups. As the need and prospects for inductive mathematics blossom, the requirement to ensure the role of proof is properly founded remains undiminished. I share with George Polya (1887-1985) the view that while learned, " intuition comes to us much earlier and with much less outside influence than formal arguments " [12, 2 p. 128]. Polya went on to reaffirm, nonetheless, that proof should certainly be taught in school. I turn to observations, many of which have been fleshed out in my recently coauthored books The Computer as Crucible [7], Mathematics by Experiment [5], and Experimental Mathematics in Action [3]. I want to argue for the changing nature of mathematical knowledge and in consequence to ask questions such as " How do we come to believe and trust pieces of mathematics? " , " Why do we wish to prove things? " and " How do we teach what and why to students? " I like various notions of embodied cognition. Smail [14, p. 113] writes: " the large human brain evolved over the past 1.7 million years to allow individuals to negotiate the growing complexities posed by human social living. " We find various modes of argument more palatable than others, and are more prone to make certain kinds of errors than others. Likewise, Steve Pinker's observation about language [11, p. 83] as founded on ". .. the ethe-real notions of space, time, causation, possession, and goals that appear to make up a language of thought. " This remains so within mathematics. The computer offers scaffolding both to enhance mathematical reasoning, as with the computation with E 8 , and to restrain mathematical error.