Stuck in the Middle: Cauchy’s Intermediate Value Theorem and the History of Analytic Rigor

Intermediate Values With the restoration of King Louis XVIII of France in 1814, one revolution had come to an end, but another was just beginning. Historians often describe the French Revolution of 1789, along with its reactions and repercussions, as the start of the modern era. For many historians of mathematics, however, the modern era began with the Bourbon Restoration and the mathematics of Augustin-Louis Cauchy. In a 1972 talk at a Mathematical Association of America sectional meeting, Judith Grabiner offered an important interpretation of this critical juncture in the history of mathematics [5]. Asking whether mathematical truth was time dependent, Grabiner argued that while truths themselves may not depend on time, our knowledge certainly does. She stressed the revolutionary character of Cauchy and his contemporaries’ efforts to set mathematical analysis on a rigorous footing, which required applying a fundamentally new point of view to the problems and methods of their eighteenth-century forebears. One of Grabiner’s leading illustrations of this point, both in her talk and in her 1981 book on Cauchy’s calculus [6], is Cauchy’s proof of