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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 André 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. [ABSTRACT FROM AUTHOR]

Felix Klein and Abraham Fraenkel each formulated a criterion for a theory of infinitesimals to be successful, in terms of the feasibility of implementation of the Mean Value Theorem. We explore the evolution of the idea over the past century, and the role of Abraham Robinson's framework therein. [ABSTRACT FROM AUTHOR]

In teaching infinitesimal calculus we sought to present basic concepts like continuity and convergence by comparing and contrasting various definitions, rather than presenting \the definition" to the students as a monolithic absolute. We hope that our experiences could be useful to other instructors wishing to follow this method of instruction. A poll run at the conclusion of the course indicates that students tend to favor infinitesimal definitions over ∈-δ ones. [ABSTRACT FROM AUTHOR]