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I know how to define addition and multiplication of natural numbers using set theory, but I think that before Cantor, mathematicians did not try to use set theory as a foundation for mathematics (I could be wrong). How did mathematicians define addition and multiplication of natural numbers before Cantor and Peano?

I chose mathematics before 1870 as an arbitrary cutoff, after Cauchy’s death and before Cantor and “modern” set theory, since rigorous mathematics started with Cauchy (I think). This era should have some somewhat rigorous definitions of addition and multiplication.

I also want to know how addition and multiplication were defined in Newton’s or ancient times (if there exist a definition).

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    $\begingroup$ You may want to clarify what you mean by "define" (and try your best to avoid imposing today's standards on past mathematicians). $\endgroup$ Jan 6 at 21:50
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    $\begingroup$ I don't know where to begin searching --- Google-book search words/phrases such as "arithmetic", "algebra", "calculus", "number system", etc. (continued) $\endgroup$ Jan 7 at 6:27
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    $\begingroup$ date restricted to 19th century (or otherwise; also French and/or German equivalents, if you can read those languages). Look at the prefaces (or introductions, note to the reader, etc.) and table of contents to see if a book might include comments about addition and multiplication of natural numbers. After some experience in doing this, you'll begin to learn about some of the most significant authors and more useful words/phrases to search with (just replace 'arithmetic', 'algebra', etc. in the URL's for those other searches, (continued) $\endgroup$ Jan 7 at 6:28
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    $\begingroup$ rather than entering into a google-search bar, unless you want to change the date range or something else). For two specific examples that might be useful, see the 3rd result (for me, at least) for the above "number systems" search, which is The Number Concept. Its Origin and Development by Levi Leonard Conant (1896), (continued) $\endgroup$ Jan 7 at 6:28
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    $\begingroup$ and see [5] in my answer here (where you might also find other books of possible relevance) -- The Spirit of Mathematical Analysis, and Its Relation to a Logical System which is an 1843 English translation of an 1842 German book by Martin [Marcin, Martinus] Ohm. $\endgroup$ Jan 7 at 6:28

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In the first appendix (Note I) to his 1821 book Analyse Algebrique, Cauchy discusses multiplication of signs in detail, and then gives the following definition of addition:

Ajouter au nombre A le nombre B, ou, en d'autres termes, faire subir au nombre A 1'accroissement + B ; c'est ce qu'on appelle faire une addition arithmetique. Le resultat de cette operation s'appelle somme.

A few pages later, he defines multiplication as follows:

Multiplier le nombre A par Ie nombre B, c'est operer sur Ie nombre A precisement comme on opere sur l'unite pour obtenir B. Le resultat de cette operation est ce qu'on appelle Ie produit de A par B.

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The addition and multiplication of numbers were seen as so intuitively obvious that no-one bothered to formalise them before the modern period with Peano's axioms and also the development of set theory. It's worth noting that set theory would be trivial if it was not for the axiom asserting the existence of an infinity.

This is not the case for geometry which was first formalised by Euclid though he was only completing a development that had preceded him and which also had a further period of development in the modern period by inclusion of the axioms of order which involves introducing a new primitive notion - betweeness. This was discovered by Pasch and developed further by Hilbert.

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