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Implications of Experimental Mathematics for the Philosophy of Mathematics,” chapter to appear
 Current Issues in the Philosophy of Mathematics From the Viewpoint of Mathematicians and Teachers of Mathematics, 2006. [Ddrive Preprint 280
"... Christopher Koch [34] accurately captures a great scientific distaste for philosophizing: “Whether we scientists are inspired, bored, or infuriated by philosophy, all our theorizing and experimentation depends on particular philosophical background assumptions. This hidden influence is an acute emba ..."
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Christopher Koch [34] accurately captures a great scientific distaste for philosophizing: “Whether we scientists are inspired, bored, or infuriated by philosophy, all our theorizing and experimentation depends on particular philosophical background assumptions. This hidden influence is an acute embarrassment to many researchers, and it is therefore not often acknowledged. ” (Christopher Koch, 2004) That acknowledged, I am of the opinion that mathematical philosophy matters more now than it has in nearly a century. The power of modern computers matched with that of modern mathematical software and the sophistication of current mathematics is changing the way we do mathematics. In my view it is now both necessary and possible to admit quasiempirical inductive methods fully into mathematical argument. In doing so carefully we will enrich mathematics and yet preserve the mathematical literature’s deserved reputation for reliability—even as the methods and criteria change. What do I mean by reliability? Well, research mathematicians still consult Euler or Riemann to be informed, anatomists only consult Harvey 3 for historical reasons. Mathematicians happily quote old papers as core steps of arguments, physical scientists expect to have to confirm results with another experiment. 1 Mathematical Knowledge as I View It Somewhat unusually, I can exactly place the day at registration that I became a mathematician and I recall the reason why. I was about to deposit my punch cards in the ‘honours history bin’. I remember thinking “If I do study history, in ten years I shall have forgotten how to use the calculus properly. If I take mathematics, I shall still be able to read competently about the War of 1812 or the Papal schism. ” (Jonathan Borwein, 1968) The inescapable reality of objective mathematical knowledge is still with me. Nonetheless, my view then of the edifice I was entering is not that close to my view of the one I inhabit forty years later. 1 The companion web site is at www.experimentalmath.info
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"... A C.I.P. record for this book is available from the Library of Congress. Paperback ISBN 9077874747 Published by: Sense Publishers, ..."
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A C.I.P. record for this book is available from the Library of Congress. Paperback ISBN 9077874747 Published by: Sense Publishers,
Revised
"... ``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. Therefore, I think that in teaching high school age youngsters we should emphasiz ..."
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``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. Therefore, I think that in teaching high school age youngsters we should emphasize intuitive insight more than, and long before, deductive reasoning.”
Honours Seminar based on MAA Summer Seminar Experimental Math in Action
"... ``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. Therefore, I think that in teaching high school age youngsters we should emphasiz ..."
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``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. Therefore, I think that in teaching high school age youngsters we should emphasize intuitive insight more than, and long before, deductive reasoning.”
CreaComp: Experimental Formal . . .
, 2007
"... CreaComp provides an electronic environment for learning and teaching mathematics that aims at inspiring the creative potential of students. During their learning process, students are encouraged to engage themselves in various kinds of interactive experiments, both of visual and purely formal mathe ..."
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CreaComp provides an electronic environment for learning and teaching mathematics that aims at inspiring the creative potential of students. During their learning process, students are encouraged to engage themselves in various kinds of interactive experiments, both of visual and purely formal mathematical nature. The computeralgebra system Mathematica powers the visualization of mathematical concepts and the tools provided by the theorem proving system Theorema are used for the formal counterparts. We present a case study on the concept of equivalence relations and set partitions, in which we demonstrate the entire bandwidth of computersupport that we envision for modern learning environments for mathematics.
Mathematics by Experiment, I & II: Plausible Reasoning in the 21st Century
, 2003
"... Abstract. In the first of these two lectures I shall talk generally about experimental mathematics. In Part II, I shall present some more detailed and sophisticated examples. The emergence of powerful mathematical computing environments, the growing availability of correspondingly powerful (multipr ..."
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Abstract. In the first of these two lectures I shall talk generally about experimental mathematics. In Part II, I shall present some more detailed and sophisticated examples. The emergence of powerful mathematical computing environments, the growing availability of correspondingly powerful (multiprocessor) computers and the pervasive presence of the internet allow for research mathematicians, students and teachers, to proceed heuristically and `quasiinductively'. We may increasingly use symbolic and numeric computation visualization tools, simulation and data mining. Many of the benefits of computation are accessible through lowend `electronic blackboard ' versions of experimental mathematics [1, 8]. This also permits livelier classes, more realistic examples, and more collaborative learning. Moreover, the distinction between computing (HPC) and communicating (HPN) is increasingly moot. 2 The unique features of our discipline make this both more problematic and more challenging. For example, there is still no truly satisfactory way of displaying mathematical notation on the web; and we care more about the reliability of our literature than does any other science. The traditional role of proof in mathematics is arguably under siege. Limned by examples, I intend to pose questions ([9]) such as: ffl What constitutes secure mathematical knowledge? ffl When is computation convincing? Are humans less fallible? ffl What tools are available? What methodologies? 3 ffl What about the `law of the small numbers '? ffl How is mathematics actually done? How should it be? ffl Who cares for certainty? What is the role of proof? And I shall offer some personal conclusions.