ABSTRACT
Modern algebra organizes phenomena related to the structural properties of arbitrary sets of objects in which there are defined operations. The structural properties of modern algebra come from the objectification of means of organization of other phenomena through processes of abstraction, and they are the product of a long history of searching, using, and studying structural properties in arithmetic and algebraic problem solving.
This history is closely related to the history of the sign systems used in algebra from Babylonian protoalgebra to the symbolic sign system taught nowadays in Secondary schools. Different sign systems allowed for structures at different levels to be conceivable and operational, from relations between quantities in a problem to the ideas of “species of numbers”, canonical forms and complete sets of canonical forms in medieval Arabic algebra. The same is true of later periods until a fully symbolic sign system was in place in which algebraic expressions are icons of the relations between quantities, as Peirce pointed out.
