Counter-revolutions in mathematics

The paper deals with two polar-opposite approaches to the study of the history of mathematics, that of the mathematician, tackling the history of his discipline, and that of the historian. It argues for, something denied by the typical mathematician, the existence of revolutions in the development of mathematics, accompanied by upheavals in the history and, historically grounded, philosophy of mathematics. The main historical examples it analyzes are drawn from the history of Hellenistic mathematics, showing their fundamental disparity with their modern analogues. It rejects the conception of notation as a mere, neutral, contentually-indifferent, abbreviatory device in the development and elucidation of mathematical cultures.