PERSONAL AND PUBLIC ASPECTS OF FORMAL PROOF: A THEORY AND A SINGLE-CASE STUDY

within a framework of students’ personal cognitive development and the view of proof in the mathematical community. The cognitive framework is based on a theory of three worlds of mathematics in which worlds of embodiment and symbolism in elementary mathematics are reconstructed as a world of formal definition and deduction. The view of proof in the mathematical community is framed by a distinction between logical formal proof, the logically-sound transformation of formal sentences, and mathematical formal proof, that is what mathematicians actually do to formulate and communicate their ideas. The theory is illustrated by a descriptive case study of an individual mathematics student, which is part of a wider multiple-case study.