Conceptual divergence—canons and taboos—and critique: reflections on explanatory categories

Abstract

Since the late 19th century it has been regularly discussed whether, e.g., the ancient Egyptian way to deal with fractions or the Greek exclusion of fractions and unity from the realm of numbers was a mere matter of imperfect notations or due to genuine “conceptual divergence,” that is, to a mathematical mode of thought that differed from ours. After a discussion of how the notion of a “mode of thought” can be made operational through the linking of concepts to mathematical operations and practices it is argued (1) that cases of conceptual divergence exist, but (2) that the discussion of notational imperfection versus conceptual divergence is none the less too simplistic, since differences may also be due to deliberate choices and exclusions on the part of the authors of the ancient texts—for instance because such a choice helps to fence off a profession, because it expresses appurtenance to a real or imagined tradition, or as a result of a critique in the Kantian sense, an elimination of expressions and forms of reasoning that are found theoretically incoherent. The argument is based throughout on historical examples.