Revolutions in mathematics: an old debate revisited. Part 1

Abstract

The 1970s to 1990s constitute a crucial period in the history of the philosophy of mathemat­ics for it was the time when the philosophy of mathematical practice movement took shape. One of its most notable episodes arguably was the debate concerning the existence and meaning of revolutions in the history of mathematics which was triggered by the question whether T.S. Kuhn’s extremely influential theory of science applies to mathematics or not. The paper attempts at revisiting that debate in search of its outcome and possible signifi­cance for the philosophy of mathematics nowadays. The debate was initiated by a Crowe – Dauben controversy: while M. Crowe claimed that revolutions never occur in mathematics, J. Dauben objected that revolutions do occur within it. Tracking the course of the debate during the three decades in question, in this paper, I have concluded that only the nominal victory was Dauben’s while the real one was Crowe’s. The existence of “revolutions” in the history of mathematics was generally accepted, but for the most part not in the Kuhnian sense of the word for that acceptance was combined with the ubiquitous presence of the be­lief in the strictly progressive accumulation of the results throughout the history of mathe­matics. In contradistinction to the scholarly works that assert the fruitlessness of the debate on revolutions in mathematics, in my paper, some intellectual trends brought to the fore by it are recognised. These trends are still highly relevant to the philosophy of mathematical prac­tice; the positive ones call for further scrutiny while the negative ones for conscious opposi­tion.