Counting, Computing, and the Representation of Numbers

2013

In ancient Egyptian mathematics, the algorithmic structure of the problem texts is characterized by the presence of two levels of calculation: the main algorithmic level, constituted by a series of operations executed step by step, and a second, “nested”, level of calculation, in which the individual operations of multiplication and division are executed in a scheme organized on two-columns. While for the main algorithmic level a methodology of mathematical rewriting that parallels the procedures of the ancient Egyptians is available, no completely effective methodology has been proposed for the “nested” level of calculation, which has been frequently read by means of anachronistic equations. The present article aims to fill this gap. The information flows develop along two directions: vertical and horizontal. Horizontal relations are in general implicit relations generated by operations of doubling, halving, etc., but, in some cases, both horizontal and vertical flows are involved in the procedures. This constitutes a sharp contrast with our modern, “one-way”, mentality.

Aestimatio: Critical Reviews in the History of Science

Atti del Convegno della Società Italiana di Storia della Scienza, 2023

Within medieval and early modern mathematics, arithmetic and algebra were closely related to each other in terms of rules, methods, and problems.As a result, they were characterized by several types of interactions-or contaminations-. In this paper, I focus on one of these contaminations, related to the study of numbers. I take into account several case studies from Mediterranean medieval mathematics, belonging to two different traditions of arithmetician-algebraists. The first is the Arabic tradition of the ḥussāb (calculators), represented here by the mathematician Abū Bakr al-Karajī (d. beg. 11th c.) and his successors al-Samaw'al (d. 12 th c.) and al-Zanjānī (d. mid 13 th c.). The second is the Italian tradition of the maestri d'abaco, within which I focus on two authors-Paolo Gherardi and an anonymous author-of the 14th century. Within my investigation of the notion of number, I concentrate on integers and fractions and-only in relation to the Arabic context-on expressible and surd numbers (i.e. rational and irrational numbers). First, I clarify the polysemic nature of the term "arithmetic" and mention some aspects related to the historical context that characterized my authors. Second, I analyze the primary sources and present some further observations that build upon earlier studies dedicated to these same sources. My main goal is to show that the development of algebra contributes to the expansion of the domain of numbers, and hence to the expansion of arithmetic.

2014

In this article I outline the analogies and the differences in number concept development in prehistoric Europe and in the Near East. Research on Near Eastern recording systems is far more advanced, and it provides us with a good theoretical approach. There are, however, more and more finds in Europe that deserve our attention when looking toward a theory of early number concepts, concepts of measure and mathematics. For archaeologists, there is an obvious requirement that such a theory has to be constructed on a material basis. Therefore in the second part of the text I describe some key finds from Europe that in my opinion allow us, on the one hand, to reference current theoretical approaches and, on the other, to connect theoretical considerations and their material basis.

Historia Mathematica, 2022

The discovery at the end of the 19th century of the mathematical cuneiform texts posed to historians the question of the nature of the numbers used in them, i.e. that of the sexagesimal place-value notation. This notation, although familiar to us today since it is the one we use to measure time, has, in the cuneiform texts, specificities which still raise challenges of interpretation. One of these specificities is the fact that the cuneiform writing does not indicate the order of magnitude of the numbers (for example, 1, 60, 1/60 or any other power of 60 are written in the same way). This article outlines the way in which historians of the late 19th and early 20th centuries interpreted this specificity. The focus here is on the interpretation proposed by the Assyriologist François Thureau-Dangin, who in 1930 considered numbers in sexagesimal place-value notation as “abstract numbers”, as opposed to “concrete numbers”.

Historia Mathematica, 2022

Numeracy and writing constitute different phenomena, whose paths of formation often appear intertwined. Here we reassess the theory that numeracy evolved universally from a concrete to an abstract concept of number, and that that shift is correlated with the invention of writing. First, we gather contemporary linguistic data and early Mesopotamian epigraphic evidence that indicates that the ‘concrete’ vs. ‘abstract’ dichotomy is not useful to understand the emergence of numbers. Then, we discuss evidence from other regions where writing was probably invented independently, in order to investigate the conceptualization and formation of early numerical notations.

Metascience, 2010

The essay surveys the history of the origin and spread of the Indian Numerals (1,2,3,...,9,0) through the ages , first to China and then, through the mediation of the Arabs, to other parts of Asia and Europe. The decimal system was adopted without any active promotion by any agency. It was because of the efficacy that it ultimately replaced all other number systems; no prejudice or narrow-mindedness could stand in its way.