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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 were 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?

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?