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Exploratory Experimentation: Digitallyassisted Discovery and Proof
 Chapter in ICMI Study 19: On Proof and Proving in Mathematics Education. In press
, 2010
"... Abstract The mathematical community (appropriately defined) faces a great challenge to reevaluate 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 datamine on the internet. Added to those ..."
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Abstract The mathematical community (appropriately defined) faces a great challenge to reevaluate 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 datamine 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
The Life of Modern Homo Habilis Mathematicus: Experimental Computation and Visual Theorems. Chapter prepared for
, 2014
"... Long before current graphic, visualisation and geometric tools were available, John E. Littlewood (18851977) wrote in his delightful Miscellany: A heavy warning used to be given [by lecturers] that pictures are not rigorous; this has never had its bluff called and has permanently frightened its vic ..."
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Long before current graphic, visualisation and geometric tools were available, John E. Littlewood (18851977) wrote in his delightful Miscellany: A heavy warning used to be given [by lecturers] that pictures are not rigorous; this has never had its bluff called and has permanently frightened its victims into playing for safety. Some pictures, of course, are not rigorous, but I should say most are (and I use them whenever possible myself). [34, p. 53] Over the past five to ten years, the role of visual computing in my own research has expanded dramatically. In part this was made possible by the increasing speed and storage capabilities—and the growing ease of programming—of modern multicore computing environments. But, at least as much, it has been driven by my group’s paying more active attention to the possibilities for graphing, animating or simulating most mathematical research activities.
Exploratory experimentation in experimental mathematics: A glimpse at the PSLQ algorithm
"... From a philosophical viewpoint, mathematics has traditionally been distinguished from the natural sciences by its formal nature and emphasis on deductive reasoning. Experiments—one of the corner stones of most modern natural science—have had no role to play in mathematics. However, in the ..."
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From a philosophical viewpoint, mathematics has traditionally been distinguished from the natural sciences by its formal nature and emphasis on deductive reasoning. Experiments—one of the corner stones of most modern natural science—have had no role to play in mathematics. However, in the
DIGITALLYASSISTED DISCOVERY AND PROOF
"... I will argue that the mathematical community (appropriately defined) is facing a great challenge to reevaluate the role of proof in light of the power of current computer systems, of modern mathematical computing packages and of the growing capacity to datamine on the internet. With great challe ..."
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I will argue that the mathematical community (appropriately defined) is facing a great challenge to reevaluate the role of proof in light of the power of current computer systems, of modern mathematical computing packages and of the growing capacity to datamine 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. As the prospects for inductive mathematics blossom, the need to make sure that the role of proof is properly founded remains undiminished.