Exploratory expermentation : digitally-assisted discovery and proof

摘要

Our community (appropriately defined) 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 datamine on the internet. Add to that the enormous complexity of many modern mathematical results such as the Poincare conjecture, Fermat's last theorem, and the classification of finite simple groups. As the need and prospects for inductive mathematics blossom, the need to ensure that the role of proof is properly founded remains undiminished. I share with Polya the view that [I]ntuition comes to us much earlier and with much less outside influence than formal arguments ... Therefore, I think that in teaching high school age youngsters we should emphasize intuitive insight more than, and long before, deductive reasoning.