The Historical Origins of Algebraic Thinking

Abstract

Barely some fifteen years ago, a new and completely different interpretation of Mesopotamian mathematics arose due, on the one hand, to new philological analyses and, on the other, to the development of new methods in the historiography of mathematics; these have been increasingly incorporating anthropological and sociological categories in the understanding of the mathematics of the pasto The new results have been changing the traditional view of Ancient Near East mathematics and their influence on Greek mathematics and have led us to review old questions and to raise new ones. One of these questions dea'..; with the transition from arithmetic thinking to algebraic thinking. Another question is the role played by language and symbolism in algebraic thinking. ln this paper we will deal with these . 1 two questIOns . Concerning the first question, we shall attempt to show that (numeric) algebraic thinking emerged from proportional thinking as a short, direct and alternative way of solving 'non-practical' problems. ln order to do so, in the first section of the chapter, we briefly describe some of the features of Proportional Mesopotamian thinking. The second section is devoted to the elaboration of our thesis. The historical teehnical evidence and its argumentation is presented in following sections. Since a certain knowledge of Diophantus' algebraic methods is required, we give an overview of Diophantus' monumental treatise Arithmetica, and deal with Babylonian geometric algebra. The second question with which we deal in this paper that of the role of language and symbolism in algebraic thinking is submitted to the same methodological approach that we used in the previous sections deriving from our perception of what mathematics is: we see mathematics as a semiotic manifestation of the culture in which mathematics is practised. Consequently, we suggest that the role of symbolism in algebraic thinking needs to be studied through the social meaning of alge-