Cauchy's real line and math philosophy till XIX

Question

I have to write an essay concerning philosophy of mathematics until the end of $XIX$ century. I've heard that the reason why the Cauchy's theorem (if continuous functions $f_n \rightarrow f$ then $f$ is continuous too) is wrong nowadays is the fact that his real numbers line was different then the modern one. It is indeed a very interesting topic but I couldn't find anything specific about it. Could you direct me to some sources or suggest a different topic?

Answer 1

Some scholars have argued that Cauchy's conception of the number line incorporated infinitesimals and his notion of convergence therefore imposes additional conditions (for example, convergence at infinitesimal points).

Thus, historian and mathematician Detlef Laugwitz argued that Cauchy's theorem was correct as stated.

Actually Cauchy himself published an article in 1853 where he gives additional details that make it clear that at least in this article his notion of convergence is stronger.

There is a review of these issues in this recent article.