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.”
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Notes
- 1.
All definitions below are taken from the Collin’s Dictionary of Mathematics which I co-authored. It is available as software—with a version of Student Maple embedded in it—at http://www.mathresources.com/products/mathresource/index.html.
- 2.
With the growing realisation of the importance of gesture in mathematics “as the very texture of thinking,” (Sfard 2009, p. 92) it is time to seriously explore tactile devices.
- 3.
A cross-section of such resources is available through http://ddrive.cs.dal.ca/∼isc/portal/.
- 4.
In Achenblog http://blog.washingtonpost.com/achenblog/ of July 1 2008.
- 5.
http://www.snre.umich.edu/eplab/demos/st0/stroopdesc.html has a fine overview.
- 6.
- 7.
In Dewey’s introduction to his book (1910). Dewey, a leading pragmatist (or instrumentalist) philosopher and educational thinker of his period, is also largely responsible for the Trotsky archives being at Harvard, through his activities on the Dewey Commission.
- 8.
This was said in an interview in Regis (1986), not only in Kuhn’s 1962 The Structure of Scientific Revolutions, which Brown notes is “the single most influential work in the history of science in the twentieth century.” In Brown’s accounting (2008) Kuhn bears more responsibility for the slide into PIS than either Dewey or Popper. An unpremeditated example of digitally assisted research is that—as I type—I am listening to The Structure of Scientific Revolutions, having last read it 35 years ago.
- 9.
Quoted by R. C. Leowontin, in Science p. 1264, Feb 16, 2001 (the Human Genome Issue).
- 10.
Michael Moncur’s (Cynical) Quotations #255 http://www.quotationspage.com/collections.html
- 11.
Richard Hamming’s philosophy of scientific computing appears as preface to his influential 1962 book (1962).
- 12.
Science, August 3, 2007, p. 579: “documenting equipment losses of more than $94 million over the past 10 years by the agency.”
- 13.
In The flight From Science and Reason. See Science, Oct. 11, 1996, p. 183.
- 14.
- 15.
The online Inverse Symbolic Calculator http://isc.carma.newcastle.edu.au/ was newly web-accessible in the same year, 1995.
- 16.
In his 23 Mathematische Probleme lecture to the Paris International Congress, 1900 (Yandell 2002).
- 17.
Eureka was an undergraduate Cambridge University journal.
- 18.
Leshner, the publisher of Science, was speaking at the Canadian Federal Science & Technology Forum, October 2, 2002.
- 19.
Available at http://www.cinderella.de.
- 20.
From p. 53 of the 1953 edition of Littlewood’s Miscellany and so said long before the current fine graphic, geometric, and other visualisation tools were available; also quoted in Inglis and Mejia-Ramos (2009).
- 21.
In his famous Mathematician’s Apology of 1940. I can not resist noting that modern digital assistance often makes more careful referencing unnecessary and sometimes even unhelpful!
- 22.
As quoted in Barad (2007, p. 54) with a footnote citing The Philosophical Writings of Niels Bohr (1998). (1885–1962).
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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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Borwein, J.M. (2012). Exploratory Experimentation: Digitally-Assisted Discovery and Proof. In: Hanna, G., de Villiers, M. (eds) Proof and Proving in Mathematics Education. New ICMI Study Series, vol 15. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-2129-6_4
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