From the Practice of Explanation to the Ideology of Demonstration: An Informal Essay

The following discusses the practice of mathematical argument or demonstration— at first based on what I shall speak of as “the locally obvious”, that is, presuppositions which the interlocutor—or, in case of writing, the imagined or “model” reader—will accept as obvious; next in its interaction with critique, investigation of the conditions for the validity of the seemingly obvious as well as the limits of this validity. This is done, in part through analysis of material produced within late medieval Italian abbacus culture, in part from a perspective offered by the Old Babylonian mathematical corpus—both sufficiently distant from what we are familiar with to make phenomena visible which in our daily life go as unnoticed as the air we breathe; that is, they allow Verfremdung. These tools are then applied to the development from argued procedure toward axiomatics in ancient Greece, from the midfifth to the mid-third century bce. Finally is discussed the further development of ancient demonstrative mathematics, when axiomatization, at first a practice, then a norm, in the end became an ideology. The whole is rounded off by a few polemical remarks about present-day beliefs concerning the character of mathematics.