Mathematical conquerors, Unguru polarity, and the task of history

I compare several approaches to the history of mathematics recently proposed by Blåsjö, Fraser–Schroter, Fried, and others. I argue that tools from both mathematics and history are essential for a meaningful history of the discipline.

In an extension of the Unguru–Weil controversy over the concept of geometric algebra, Michael Fried presents a case against both Andr ́e Weil the “privileged observer” and Pierre de Fermat the “mathematical conqueror.” Here I analyze Fried’s version of Unguru’s alleged polarity between a historian’s and a mathematician’s history. I identify some axioms of Friedian historiographic ideology, and propose a thought experiment to gauge its pertinence.

Unguru and his disciples Corry, Fried, and Rowe have described Freudenthal, van der Waerden, and Weil as Platonists but provided no evidence; here I provide evidence to the contrary I also analyze how the various historiographic approaches play themselves out in the study of the pioneers of mathematical analysis including Fermat, Leibniz, Euler, and Cauchy.