On Cauchy's rigorization of complex analysis

In this paper, I look at Cauchy’s early (1814–1825) rigorization of complex analysis. I argue that his work should not be understood as a step in improving the deductive methods of mathematics but as a clear, innovative and systematic stance about the semantics of mathematical languages. His approach is contrasted with Laplace’s “no- tational inductions,” influenced by Condillac’s ideas about the language of algebra. Cauchy’s opposition is then not to be seen as stemming from a comeback of geometric and synthetic methods, but as a rejection of the key Condillacian doctrines that algebra is about abstract quantities and that its rules provide means of discovering new mathematical truths. He thereby paved the way for the arithmetization of calculus and fruitfully extended his approach to complex analysis like no one before him. I finish by discussing lessons we can draw about how mathematical rigour differs from rigour in other sciences.