The Social Construction of Mathematics as Characterized by Mathematical Structure, with Special Reference to Natural Numbers and Positional Notation

Abstract

　　The purpose of this paper is to elaborate on social perspectives provided by recent studies in mathematics education, and to consider what makes the social construction of mathematics necessarily valid.

　　First, we review previous research on social construction in mathematics education and point out that they have not sufficiently dealt with the mathematical aspect of this social construction.　The application of Lakatos’s logic of mathematical discovery (LMD) to mathematics education might not adequately capture the inherent characteristics of mathematics, because an ideal LMD does not always occur in mathematics classrooms.

　　Second, we consider the terms“ necessary truth” and“ structure.”In philosophy, a necessary truth is distinguished from a contingent truth, in that the former is a truth that could not have been false.　As for structure, in mathematical logic, a structure is a collection of interpretations of mathematical statements.　For example, a positional notation provides a structure for natural numbers.　With special reference to this example, we consider that (i) if a universal set is finite, the truth of a mathematical statement may be established by checking all elements, but also that (ii) justification focusing on a mathematical structure make mathematical statements not only true, but necessarily true.

　　On the basis of the above points, we argue as follows: (i) justification focusing on mathematical structure makes a mathematical statement necessarily true; (ii) the valid social construction of mathematics emerges from such necessary truths; and (iii) in mathematics education, we must give students a chance to experience social construction paying attention to mathematical structures.