Albrecht Heeffer

A Conceptual Analysis of Early Arabic Algebra

Arabic algebra derives its epistemic value not from proofs but from correctly performing calculations using coequal polynomials. This idea of ‘mathematics as calculation’ had an important influence on the epistemological status of European mathematics until the seventeenth century. We analyze the basic concepts of early Arabic algebra such as the unknown and the equation and their subsequent changes within the Italian abacus tradition. We demonstrate that the use of these concepts has been problematic in several aspects. Early Arabic algebra reveals anomalies which can be attributed to the diversity of influences in which the al-jabr practice flourished. We argue that the concept of a symbolic equation as it emerges in algebra textbooks around 1550 is fundamentally different from the ‘equation’ as known in Arabic algebra.

The Symbolic Model for Algebra: Functions and Mechanisms

The symbolic mode of reasoning in algebra, as it emerged during the sixteenth century, can be considered as a form of model-based reasoning. In this paper we will discuss the functions and mechanisms of this model and show how the model relates to its arithmetical basis. We will argue that the symbolic model was made possible by the epistemic justification of the basic operations of algebra as practiced within the abbaco tradition. We will also show that this form of model-based reasoning facilitated the expansion of the number concept from Renaissance interpretations of number to the full notion of algebraic numbers.

The Reception of Ancient Indian Mathematics by Western Historians

While there has been an awareness of ancient Indian mathematics in the West since the sixteenth century, historians discussed the Indian mathematical tradition only after the publication of the first translations by Colebrooke in 1817. Its reception cannot be comprehended without accounting for the way that new European mathematics was shaped by Renaissance humanist writings. We sketch this background and show with one case study on algebraic solutions to a linear problem, how the understanding and appreciation of Indian mathematics was deeply influenced by the humanist prejudice that all higher intellectual culture, in particular all science, had risen from Greek soil.

Learning Concepts Through the History of Mathematics

The adolescent’s notion of rationality often encompasses the epistemological view of mathematics as knowledge which offers absolute certainty. Several findings such as Godel’s theorem and the construction of a strict finite arithmetic, however, provide strong arguments against that view. The static and unalterable mode of presentation of concepts in the mathematics curriculum, rather than lack of knowledge of metatheory, contributes to this misconception. I will argue that the conceptual history of mathematics provides excellent opportunities to convey the basic epistemological and ontological questions of the philosophy of mathematics in mathematics education. In particular, the emergence of the concept of an equation will be presented in a historical context. Such examples will alert students of the relativity of mathematical methods, truth, and knowledge, and will put mathematics back in the perspective of time, culture, and context.

ON THE NATURE AND ORIGIN OF ALGEBRAIC SYMBOLISM

Ever since Nesselmann’s study on “Greek algebra” (1842), historical accounts on algebra draw a distinction in rhetorical, syncopated and symbolic algebra. This tripartite distinction has become such a common-place depiction of the history of algebraic symbolism that modern-day authors even fail to mention their source (e.g., Boyer 1968, 201; Flegg and Hay 1985; Struik 1987). The repeated use of Nesselmann’s distinction in three Entwickelungstufen on the stairs to perfection is odd because it should be considered a highly normative view which cannot be sustained within our current assessment of the history of algebra. Its use in present-day text books can only be explained by an embarrassing absence of any alternative models. There are several problems with Nesselmann’s approach.

Identifying Adequate Models in Physico-Mathematics: Descartes’ Analysis of the Rainbow

The physico-mathematics that emerged at the beginning of the seventeenth century entailed the quantitative analysis of the physical nature with optics, meteorology and hydrostatics as its main subjects. Rather than considering physico-mathematics as the mathematization of natural philosophy, it can be characterized it as the physicalization of mathematics, in particular the subordinate mixed mathematics. Such transformation of mixed mathematics was a process in which physico-mathematics became liberated from Aristotelian constraints. This new approach to natural philosophy was strongly influenced by Jesuit writings and experimental practices. In this paper we will look at the strategies in which models were selected from the mixed sciences, engineering and technology adequate for an analysis of the specific phenomena under investigation. We will discuss Descartes’ analysis of the rainbow in the eight discourse of his Meteorology as an example of carefully selected models for physico-mathematical reasoning. We will further demonstrate that these models were readily available from Jesuit education and literature.