The Collatz conjecture. A case study in mathematical problem solving

Jean Paul Van Bendegem



In previous papers (see Van Bendegem [1993], [1996], [1998], [2000], [2004], [2005], and jointly with Van Kerkhove [2005]) we have proposed the idea that, if we look at what mathematicians do in their daily work, one will find that conceiving and writing down proofs does not fully capture their activity. In other words, it is of course true that mathematicians spend lots of time proving theorems, but at the same time they also spend lots of time preparing the ground, if you like, to construct a proof.

Full Text:



Boolos, G.S., J.P. Burgess, and R.C. Jeffrey, Computability and Logic. Fourth edition. Cambridge University Press, Cambridge, 2002.

Conway, J.H., “Unpredictable Iterations”. In: Proceedings 1972 Number Theory Conference. University of Colorado, Boulder, CO, 1972, pp. 49–52.

Edwards, H.M., Riemann’s Zeta Function. Academic Press, New York, 1974 (Dover Publications, New York, 2001).

Hayes, B., “Computer Recreations. On the ups and downs of hailstone numbers.” Scientific American, vol. 250, nr. 1, 1984, 13–17.

Lagarias, J.C., “The 3x+1 problem and its generalizations”. American Mathematical Monthly, 92, 1985, 3–23.

Lagarias, J.C., “The 3x+1 problem and its generalizations”. 1996. Web-address:

Lagarias, J.C., “The 3x+1 problem: an annotated bibliography”. 2004. Webaddress:

Lakoff, G., and R.E. Nunez, Where Mathematics Comes From. How the Embodied Mind Brings Mathematics into Being. Basic Books, New York, 2000.

Lenat, Douglas B., “AM: Discovery in Mathematics as Heuristic Search”. In: R. Davis and D.B. Lenat (eds.), Knowledge-Based Systems in Artificial Intelligence. New York: McGraw-Hill, 1980, pp. 3–228.

Ribenboim, P., 13 Lectures on Fermat’s Last Theorem. Springer, New York, 1979.

Van Bendegem, J.P., “Real-Life Mathematics versus Ideal Mathematics: The Ugly Truth”. In: E.C.W. Krabbe, R.J. Dalitz and P.A. Smit (eds.), Empirical Logic and Public Debate. Essays in Honour of Else M. Barth, Rodopi, Amsterdam, 1993, pp. 263–272.

Van Bendegem, J.P., “Mathematical Experiments and Mathematical Pictures”. In: I. Douven and L. Horsten (eds.), Realism in the Sciences. Proceedings of the Ernan McMullin Symposium Leuven 1995. Louvain Philosophical Studies 10. Leuven University Press, Louvain, 1996, pp. 203–216.

Van Bendegem, J.P., “What, if anything, is an experiment in mathematics?” In: D. Anapolitanos, A. Baltas and S. Tsinorema (eds.), Philosophy and the Many Faces of Science, (CPS Publications in the Philosophy of Science), Rowman and Littlefield, London, 1998, pp. 172–182.

Van Bendegem, J.P., “Analogy and Metaphor as Essentials Tools for the Working Mathematician”. In: F. Hallyn (ed.), Metaphor and Analogy in the Sciences, (Origins: Studies in the Sources of Scientific Creativity), Kluwer Academic, Dodrecht, 2000, pp. 105–123.

Van Bendegem, J.P., “The Creative Growth of Mathematics”. In: D. Gabbay, S. Rahman, J. Symons and J.P. Van Bendegem (eds.), Logic, Epistemology and the Unity of Science (LEUS), Volume 1, Dordrecht: Kluwer Academic, 2004, pp. 229–255.

Van Bendegem, J.P., “Proofs and Arguments: The Special Case of Mathematics”. In: Logics of Scientific Cognition. Essays in Debate With Theo Kuipers, Poznan Studies, Rodopi, Amsterdam, 2005, to appear.

Van Kerkhove, B., and J.P. Van Bendegem: “The Unreasonable Richness of Mathematics”. Journal of Cognition and Culture, 2005, to appear.

ISSN: 1425-3305 (print version)

ISSN: 2300-9802 (electronic version)

Partnerzy platformy czasopism