Exploratory Experimentation: Digitally-Assisted Discovery and Proof

  • Jonathan Michael BorweinEmail author
Part of the New ICMI Study Series book series (NISS, volume 15)


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.”


Continue Fraction Computer Algebra System Minimal Polynomial Deductive Reasoning Scientific Revolution 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



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.


  1. Anderson, C. (2008). The end of theory. Wired,
  2. Avigad, J. (2008). Computers in mathematical inquiry. In P. Mancuso (Ed.), The philosophy of mathematical practice (pp. 302–316). Oxford: Oxford University Press.CrossRefGoogle Scholar
  3. Bailey D. H., & Borwein, J. M. (2011, November). Exploratory experimentation and computation. Notices of the AMS. November 1410–1419.Google Scholar
  4. Bailey, D., Borwein, J., Calkin, N, Girgensohn, R., Luke, R., & Moll, V. (2007). Experimental mathematics in action. Wellesley: A K Peters.Google Scholar
  5. Bailey, D. H., Borwein, J. M., Broadhurst, L. M., & Glasser, L. (2008). Elliptic integral representation of Bessel moments. Journal of Physics A: Mathematics & Theory, 41, 5203–5231.Google Scholar
  6. Bailey, D. H., Borwein, J. M., & Crandall, R. E. (2010, March). Advances in the theory of box integrals. Mathematics of Computation , 79, 1839–1866.Google Scholar
  7. Baillie, R., Borwein, D., & Borwein, J. (2008). Some sinc sums and integrals. American Mathematical Monthly, 115(10), 888–901.Google Scholar
  8. Barad, K. M. (2007). Meeting the universe halfway. Durham: Duke University Press.Google Scholar
  9. Borwein, J. M. (2005). The SIAM 100 digits challenge. Extended review in the Mathematical Intelligencer, 27, 40–48. [D-drive preprint 285].Google Scholar
  10. Borwein, J. M., & Bailey, D. H. (2008). Mathematics by experiment: Plausible reasoning in the 21st century (2nd ed.). Wellesley: A K Peters.Google Scholar
  11. Borwein, J. M., & Devlin, K. (2009). The computer as crucible. Wellesley: A K Peters.Google Scholar
  12. Borwein, J. M., Bailey D. H., & Girgensohn, R. (2004). Experimentation in mathematics: Computational pahts to discovery. Wellesley: A K Peters.Google Scholar
  13. Brown, R. D. (2008). Are science and mathematics socially constructed? A mathematician encounters postmodern interpretations of science. Singapore: World Scientific.Google Scholar
  14. de Villiers, M. (2004). The role and function of quasi-empirical methods in Mathematics. Canadian Journal of Science, Mathematics and Technology Education, 4, 397–418.CrossRefGoogle Scholar
  15. Dewey, J. (1910). The influence of darwin on philosophy and other essays. New York: Henry Holt.Google Scholar
  16. Ewing, J. H., & Gehring, F. W. (1991). Paul Halmos. Celebrating 50 years of mathematics. New York: Springer.CrossRefGoogle Scholar
  17. Franklin, L. R. (2005). Exploratory experiments. Philosophy of Science, 72, 888–899.CrossRefGoogle Scholar
  18. Giaquinto, M. (2007). Visual thinking in mathematics. An epistemological study. Oxford: Oxford University Press.CrossRefGoogle Scholar
  19. Gregory, J., & Miller, S. (1998). Science in public, communication, culture and credibility. Cambridge: Basic Books.Google Scholar
  20. Hamming, R. W. (1962). Numerical methods for scientists and engineers. New York: McGraw-Hill.Google Scholar
  21. Havel, J. (2003). Gamma: Exploring Euler’s constant. Princeton: Princeton University Press.Google Scholar
  22. Hersh, R. (1997). What is mathematics, really? New York: Oxford University Press.Google Scholar
  23. Ifrah, G. (2000). The universal history of numbers. New York: Wiley.Google Scholar
  24. Inglis, M., & Mejia-Ramos, J. P. (2009). The effect of authority on the persuasiveness of mathematical arguments. preprint.Google Scholar
  25. Jaffe, A., & Quinn, F. (1991). “Theoretical mathematics”: Toward a cultural synthesis of mathematics and theoretical physics. Bulletin of the American Mathematical Society, 29(1), 1–13.CrossRefGoogle Scholar
  26. Livio, M. (2009). Is God a mathematician?. New York:Simon and Schuster.Google Scholar
  27. Lucas, S. K. (2009). Approximations to π derived from integrals with nonnegative integrands. American Mathematical Monthly, 116(10), 166–172.CrossRefGoogle Scholar
  28. Pinker, S. (2007). The stuff of thought: Language as a window into human nature. London: Allen Lane.Google Scholar
  29. Pólya, G. (1981). Mathematical discovery: On understanding, learning, and teaching problem solving (Combined ed.). New York: Wiley.Google Scholar
  30. Regis, E. (1986). Who got Einstein’s office? Reading: Addison-Wesley.Google Scholar
  31. Rice A. (1999). What makes a great mathematics teacher? American Mathematical Monthly, 106(6), 534–552.CrossRefGoogle Scholar
  32. Sfard, A. (2009). What’s all the fuss about gestures: A commmentary. Special issue on gestures and multimodality in the construction of mathematical meaning. Educational Studies in Mathematics, 70, 191–200.Google Scholar
  33. Smail, D. L. (2008). On deep history and the brain. Berkeley: Caravan Books/University of California Press.Google Scholar
  34. Sørensen, H. K. (2010). Exploratory experimentation in experimental mathematics: A glimpse at the PSLQ algorithm. In B. Löwe & T. Müller (Eds.), PhiMSAMP. Philosophy of mathematics: Sociological aspects and mathematical practice. Texts in Philosophy (Vol. 11, pp. 341–360). London: College Publications.Google Scholar
  35. Yandell, B. (2002) The honors class. Natick: A K Peters.Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.Centre for Computer-Assisted Research Mathematics and its Applications, CARMAUniversity of NewcastleCallaghanAustralia

Personalised recommendations