"Commutative" and its derivatives -----------------------------
The concept of commutativity appeared in the 1660s in the work of Gottfried Wilhelm Leibniz (1646–1716), when he wrote in his project for a universal logical calculus: "Transpositio litterarum in eodem termino nihil mutat ut ab coincidet cum ba." This passage may be translated as: "Reversing letters within the same term changes nothing, so that, for example, ab has the same value as ba."
The word "commutative" combines the prefix cum with the root of the verb mutare. Both terms occur in Leibniz's sentence and mean "with" and "change," respectively. The word had already appeared in the work of Nicole Oresme (c. 1325–1382), though there it described a type of justice.
In mathematics, "commutative" reappeared around the middle of the 19th century. It is clearly stated in the American logician Benjamin Peirce's (1809–1880) Linear Associative Algebra, written in 1870 but published posthumously in 1882: "The principle that the value of a product is unchanged by the order of its factors is called 'the commutative principle' and is expressed by the equation AB = BA."
The term "associative" derives from the Latin word associare, itself formed from socius, meaning "companion." It was used in philosophy during the Renaissance, then disappeared before returning in mathematics—as the title of Benjamin Peirce's book shows—and later in other disciplines. Finally, it entered everyday French after the 1901 law on associations.