The Abbaco Tradition (1300-1500): its role in the development of European algebra

Abbaco algebra is a coherent tradition of algebraic problem solving mostly based in the merchant cities of fourteenth and fifteenth-century Italy. This period is roughly situated between two important works dealing with algebra: the Liber Abbaci by Fibonacci (1202) and the Summa di Arithmetica et Geometria by Lucca Pacioli (1492). Such continuous tradition of mathematical practice was hardly known before the first transcriptions of extant manuscripts by Gino Arrighi from the 1960’s and the ground-breaking work by Warren van Egmond (1980). After some decades of manuscript study and the recent assessment of Jens Høyrup (2007) we now have a better understanding of this tradition. In this paper we provide an overview of the basic characteristics of the abbaco tradition and discuss the role it played towards the new symbolic algebra as it emerged in sixteenth-century Europe. We argue that its influence on the sixteenth century has largely been ignored and that the new ars analytica from the French algebraists should be understood as establishing new foundations for the general practice of abbaco problem solving

The Emergence of Symbolic Algebra as a Shift in Predominant Models

Historians of science find it difficult to pinpoint to an exact period in which symbolic algebra came into existence. This can be explained partly because the historical process leading to this breakthrough in mathematics has been a complex and diffuse one. On the other hand, it might also be the case that in the early twentieth century, historians of mathematics over emphasized the achievements in algebraic procedures and underestimated the conceptual changes leading to symbolic algebra. This paper attempts to provide a more precise setting for the historical context in which this decisive step to symbolic reasoning took place. For that purpose we will consider algebraic problem solving as model-based reasoning and symbolic representation as a model. This allows us to characterize the emergence of symbolic algebra as a shift from a geometrical to a symbolic mode of representation. The use of the symbolic as a model will be situated in the context of mercantilism where merchant activity of exchange has led to reciprocal relations between money and wealth

The body in Renaissance arithmetic: from mnemonics to embodied cognition

In Medieval and Renaissance arithmetic we find several instances of references to body parts or actions involving body parts. In this paper we will address the question on the historical functions of body parts in mathematics and discuss its relation to the currently prevailing practice of symbolic mathematics.1

Algebraic symbolism as a conceptual barrier in learning mathematics

The use of symbolism in mathematics is probably the mostly quoted reason people use for explaining their lack of understanding and difficulties in learning mathematics. We will consider symbolism as a conceptual barrier drawing on some recent findings in historical epistemology and cognitive psychology. Instead of relying on the narrow psychological interpretation of epistemic obstacles we use the barrier for situating symbolism in the ‘ontogeny recapitulates phylogeny’-debate. Drawing on a recent study within historical epistemology we show how early symbolism functioned in a way similar to concrete operational schemes. Furthermore we will discuss several studies from cognitive psychology which come to the conclusion that symbolism is not as abstract and arbitrary as one considers but often relies on perceptually organized grouping and concrete spatial relations. We will use operations on fractions to show that the reliance on concrete spatial operations also provides opportunities for teaching. We will conclude arguing that a better conceptual understanding of symbolism by teachers will prepare them for possible difficulties that students will be confronted with in the classroom

Dutch arithmetic, samurai and warships: teaching of Western mathematics in pre-Meiji Japan

This paper discusses the scarce occasions in which Japan came into contact with Western arithmetic and algebra before the Meiji restoration of 1868. It concentrates on the reception of Dutch works during the last decades of the Tokugawa shogunate and the motivations to study and translate these books. While some studies based on Japanese sources have already been published on this period, this paper draws from Dutch sources and in particular on witness accounts from Dutch officers at the Nagasaki naval school, responsible for the instruction of mathematics to selected samurai and rangakusha