Exploratory experimentation: digitally-assisted discovery and proof

The mathematical community (appropriately defined) faces a great challenge to re-evaluate the role of proof in light of the power of current computer systems, the sophistication of modern mathematical computing packages, and the growing capacity to data-mine on the internet. Added to those are the enormous complexity of many modern mathematical results such as the Poincaré conjecture, Fermat’s last theorem, and the classification of finite simple groups. With great challenges come great opportunities. Here, I survey the current challenges and opportunities for the learning and doing of mathematics. As the prospects for inductive mathematics blossom, the need to ensure that the role of proof is properly founded remains undiminished. Much of this material was presented as a plenary talk in May 2009 at the National Taiwan Normal University Workshop for ICMI Study 19 “On Proof and Proving in Mathematics Education.”