How Early Mathematicians Worked Without Set Theory

elementary-set-theorygroup-theorymath-historyreal-analysissoft-question

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?

Best Answer

Some hints...

You can start reading Cauchy's elucidation of function (1823) :

On nomme quantité variable celle que l'on considère comme devant recevoir successivement plusieurs valeurs différentes les unes des autres. On appelle au contraire quantité constante toute quantité qui reçoit une valeur fixe et déterminée.

[We name variable a quantity that receives successively many different values. We name constant a quantity that receives a fixed and determined value.]

Lorsque des quantités variables sont tellement liées entre elles, que, la valeur de l'une d'elles étant donnée, on puisse en conclure les valeurs de toutes les autres, on conçoit d'ordinaire ces diverses quantités exprimées au moyen de l'une d'entre elles, qui prend alors le nom de variable indépendante; et les autres quantités, exprimées au moyen de la variable indépendante, sont ce qu'on appelle des fonctions de cette variable .

[When some variable quantities are linked together in a way that, having fixed the value of one of them, all others quantities can be determined, on conceive these different quantities as expressed by way of one of them, named independent variable. The remaining quantities, expessed by way of the independent variable, are named functions of that variable.]

Thus, in a nutshell, the concept of "function" was a primitive one, like today for set. A function is a correspondence (a relation) between two "variable quantities".

It is worth noting that Cauchy's definition of "variable quantity" was already present into de L'Hôpital's textbook : Analyse des infiniment petits pour l'intelligence des lignes courbes (1st ed, 1696), the first calculus' textbook. See :

An early occurrence of "function" is in Leibniz, in De linea ex lineis numero infinitis ordinatim ductis inter se concurrentibus formata, easque omnes tangente, ac de novo in ea re Analysis infinitorum usu (1692), but a "reasonable" definition of function is available only with Johann Bernoulli, Remarques sur ce qu'on a donne jusqu'ici de solutions des problemes sur les isopdrimitres (1718) and Leonhard Euler, Introductio in analysin infinitorum (1748).

Regarding group we may see e.g. Arthur Cayley : he uses the name "set" in his definition of group (1854) :

A set of symbols : $1,α,β,\ldots$ all of them different, and such that the product of any two of them (no matter in what order), or the product of any one of them into itself, belongs to the set, is said to be a group.

Set here is not a mathematical object : no specific properties of sets are assumed.

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