# When did people start to state and justify properties of arithmetical operations?

I have question regarding the history of the idea of founding mathematics (specially arithmetic) on a logical basis.

What I'm interested in knowing is, at what point historically people started to make claims about the properties of the natural numbers let's say, like: commutativity of addition and multiplication (not necessarily general claims) ? And at what point people started to feel that those properties needed justification ?

• It was a very slow process which started long before Euclid. According to the tradition, Pythagoreans introduced such things as odd and even numbers, squares, triangular numbers etc., and studied their properties. Commented Apr 22, 2019 at 12:45
• dose statements like $2*3 = 3*2$ make sense to the ancient-greece for example ? and did they have an idea how to justify them ? Commented Apr 22, 2019 at 12:51
• Probably not. This was considered evident. It was much later that people started to state such properties explicitly. Commented Apr 22, 2019 at 12:58
• "dose statements like 2∗3=3∗2 make sense to the ancient-greece for example ?" --- This led me to an interesting idea, one that is probably well known but I don't dabble in very early mathematics history so it's new to me, namely that before these types of written expressions, it may be a category mistake to even ask about commutativity. For example, when counting how many spears were made by counting spears in a pile of spears, I doubt anyone considered the order that the spears were tossed into the pile. Commented Apr 24, 2019 at 16:57
• Maybe you are right in the case of addition, but the commutativity of multiplication is far from obvious (at least for me). ٍٍSpecially if we use the definition of $2*3 = 3+3$, and $3*2 = 2+2+2$ both equal 6. Why should this be true for all integers. I can not imagine how would they see that as obvious.We only see it obvious, because we learn it from an early stage.@Dave L Renfro Commented Apr 24, 2019 at 17:41

At the operational level, the use of properties of addition and multiplication can be traced to Euclid's Elements, and more algebraically, to Diophantus's Arithmetica. They are used by al-Khwarizmi, various medieval treatises on arithmetic, and, eventually symbolized by Vieta and Descartes.

But the conceptualizing of operations as such, and the study of their properties, only begins in the early 19-th century, when alternative "arithmetics" gain prominence. According to Jeff Miller's Earliest Known Uses of Some of the Words of Mathematics, Servois's memoir published in Annales de Gergonne (volume V, no. IV, October 1, 1814) mentions commutative and distributive laws. In England, "commutative law" appears in Gregory's 1841 textbook. Hamilton in "On a new Species of Imaginary Quantities connected with a theory of Quaternions" Royal Irish Academy, Proceedings, Nov. 13, 1843, vol. 2, 424-434 writes

"However, in virtue of the same definitions, it will be found that another important property of the old multiplication is preserved, or extended to the new, namely, that which may be called the associative character of the operation..."

In The Mathematical Analysis of Logic (1847) Boole formalized symbolic logic, with inspiration for properties of logical operations explicitly taken from those of arithmetic.

As for justification, 18th century algebraists were content with Leibniz's vague "generality of algebra" used to transplant idenities from known to new (say, complex) numbers, see Was 18th century algebra more symbolic/formal than the modern conception? Peacock in A Treatise on Algebra (1830) renamed it into the "principle of permanence", which he attempted to use, as we would say, axiomatically:

"Peacock attempted to put the theory of negative and complex numbers on a firm logical basis by dividing the field of algebra into arithmetical algebra and symbolic algebra. In the former the symbols represented positive integers; in the latter the domain of the symbols was extended by his principle of the permanence of equivalent forms. This principle asserts that rules in arithmetical algebra, which hold only when the values of the variables are restricted, remain valid when the restriction is removed."

His approach was continued by Hankel in Vorlesungen über die complexen Zahlen und ihre Functionen (1867), who also brought Grassman's 1844 work on vector algebras out of obscurity. With the proliferation of new algebraic systems in the 19th century (Boole's, Hamilton's, Grassman's etc.), and the new focus on rigor, the attention turned to rigorous construction of arithmetic, and arithmetization of analysis. Dedekind came up with the arithmetization program as early as 1858, and published Stetigkeit und Irrationale Zahlen in 1872. As far as reducing arithmetic to logic, that was a rather bold idea put forward and developed by Frege in Die Grundlagen der Arithmetik (1884). The modern axiomatization of arithmetic appears in Peano's Arithmetices Principia (1889), and is indebted to Grassman's ideas about deriving properties of arithmetic operations from induction on the successor operation, and to Peirce's (1881) and Dedekind's (1888) prior axiomatizations.