A New Framework Based on the Methodology of Scientific Research Programs for Describing the Quality of Mathematical Activities

This paper proposes a new framework for describing the quality of mathematical activities under radical constructivism. It is based on Lakatos' philosophy of science, instead of his philosophy of mathematics. We focus on a structural similarity between mathematical problem-solving activities and scientific research programs. While Lakatos 'philosophy of mathematics is only a model of a progressive activity, the new framework can distinguish between progressive and degenerative activities. To show its usefulness, we provide a sample analysis. Based on the analysis, we hypothesize that the zig-zag process of solving a mathematical problem is driven by a hard core: A set of one's unrevised assumptions that one would like to continue to maintain. The necessity of further research with the proposed framework is suggested.