# Exploratory Experimentation: Digitally-Assisted Discovery and Proof

## Abstract

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.”

## Keywords

Continue Fraction Computer Algebra System Minimal Polynomial Deductive Reasoning Scientific Revolution## Notes

### Acknowledgements

I owe many people thanks for helping refine my thoughts on this subject over many years. Four I must mention by name: my long-standing collaborators Brailey Sims, Richard Crandall and David Bailey, and my business partner Ron Fitzgerald from *MathResources*, who has taught me a lot about balancing pragmatism and idealism in educational technology—among other things. I also thank Henrik Sørensen whose thought-provoking analysis gave birth to the title and the thrust of the paper, and my student Chris Maitland who built most of the *Cinderella* applets.

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