TITLE: How did early mathematicians make it without Set theory?
QUESTION [51 upvotes]: It is said that Cauchy was a pioneer of rigour in calculus and a founder of complex analysis. Yet if baffles me as set theory was an invention of the 1870s, 20 years after the death of Cauchy. Currently the beginning of most concepts in mathematics begins with the concept of set. Furthermore the concept of groups whose foundations were laid by Galois and Abel were done so long before set theory.
I hope there is a genral way to answer these questions
1) We define functions with a domain and range both being sets. But when Cauchy used the symbol 'f(x)', what did it really mean to him? As Cauchy was notorious for his rigorous approach, it is hard to believe that he may have just used the word function ambiguously with intuitive satisfaction.
(If the following question makes the topic too broad I'd be more than happy to list it as a separate question.
2)To a certain extent I can even brush away the idea of functions before sets. But I simply cannot grasp how the concept of group was formulated without a set and I'm puzzled as to how Galois and Abel were independently able to frame methods to prove the unsolvability of the quintic (these days the proof makes generous use of set theory)without sets.
In these days where N, Z, Q and R all sets, how did the early masters do what they did? How on earth was calculus made rigorous without the sets of different numbers?

REPLY [4 votes]: Set theory is one of the most common ways in modern mathematics to justify the ontology of the entities mathematicians are dealing with, by providing a general framework where all (or most, to be safe) of these entities can be constructed from the empty set. On the other hand, the procedures of mathematics are able to stand, and rigorously at that, without the set-theoretic foundations. Cauchy was rigorous not in the sense that he had set theory (he didn't) or that he gave epsilon-delta definitions of continuity (he didn't), but because he adhered to the rigor of the geometry of Euclid as it was practiced for several centuries before him, and because he rejected the generality of algebra (roughly, cavalier summation of divergent series) of Euler and Lagrange. The distinction procedures versus ontology and how it sheds light on the history of mathematics is explored in the article "Toward a history of mathematics focused on procedures".