On Mathematical Warrants: Proof Does Not Always Warrant, and a Warrant May Be Other Than a Proof

Abstract

The nature of the justification for a learner's belief in mathematical propositions is central to the question of whether the learner could be said to have mathematical knowledge. In this paper, the philosophical concept of "warrant" is interpreted for the mathematics education context and then applied to 2 central questions: (a) In what sense does mathematical proof warrant? and (b) Can there be warrants for mathematical knowledge other than deductive proofs? Applying Johnson's (1987) concept of "embodiment" to ideas from Wang (1986) and Pólya (1945/1990), the 1st question is answered, briefly, by the adage "When basic proof becomes an embodied process." To address the 2nd question, Giaquinto's (1992) concept of "visualization" is presented, and its claim to be able to serve as an insightful, a-linguistic mathematical warrant is discussed with reference to dynamic geometry.