Uncertainty, rule-bending, and the near-miss three lessons from the dialectics of mathematical practice

Abstract

This paper develops a framework for thinking about mathematical creativity that takes uncertainty as a structural condition of mathematical work, rule-bending as a methodological move that is more pervasive than usually acknowledged, and near-misses, in the sense of Marvin Minsky, as the cognitive unit through which mathematical learning advances. Two case studies - the origami construction of √23, and the zeta-regularised assignment 1 + 2 + 3 + ⋯ = -1/12 - are used to illustrate the rule-bending move at very different scales of the mathematical enterprise. The framework is then applied to the pedagogy of mathematics, with a particular focus on the Pathology Principle: that the objects historically labelled "pathological" by mathematicians have very often been near-misses in the precise Minsky sense, and that their pedagogical value lies in exactly this fact. The paper concludes with concrete pedagogical implications for the teaching of mathematics at the secondary and early-tertiary level.