Jonathan Borwein, FRSC Canada Research Chair in Collaborative Technology Laureate Professor Newcastle, NSW ``intuition comes to us much earlier and with much less outside influence than formal arguments which we cannot really understand unless we have reached a relatively high level of logical experience and sophistication. George Polya ( ) Revised 30/08/2008 Two Lectures on Experimental Mathematics (ANU, November 13-14, 2008) Therefore, I think that in teaching high school age youngsters we should emphasize intuitive insight more than, and long before, deductive reasoning.” Digitally-assisted Discovery and Proof Revised 16/10/08

ABSTRACT Jonathan M. Borwein Dalhousie and Newcastle “The object of mathematical rigor is to sanction and legitimize the conquests of intuition, and there was never any other object for it.” – Jacques Hadamard ( ) I will argue that the mathematical community (appropriately defined) is facing a great challenge to re-evaluate the role of proof in light of the power of current computer systems, of modern mathematical computing packages and of the growing capacity to data-mine on the internet. With great challenges come great opportunities. I intend to illustrate the current challenges and opportunities for the learning and doing of mathematics.

II. Computer-assisted Discovery and Proof ( esp. of Generating Functions for Riemann’s Zeta ) Jonathan M. Borwein Dalhousie and Newcastle David H Bailey Lawrence Berkeley Labs “All truths are easy to understand once they are discovered; the point is to discover them.” – Galileo Galilei Details are in Excursions in Experimental Mathematics, Bailey, Borwein et al, A.K. Peters, (MAA Short Course) ABSTRACT I shall describe further some of the methods that are now available for computer- assisted discovery and proof of highly non- trivial mathematical formulas and identities.