@article {3144548, title = {Epistemic justification and operational symbolism}, journal = {Foundations of science}, volume = {19}, number = {1}, year = {2014}, pages = {89{\textendash}113}, abstract = {

By the end of the twelfth century in the south of Europe, new methods of calculating with Hindu-Arabic numerals developed. This tradition of sub-scientific mathematical practices is known as the abbaco period and flourished during 1280{\textendash}1500. This paper investigates the methods of justification for the new calculating procedures and algorithms. It addresses in particular graphical schemes for the justification of operations on fractions and the multiplication of binomial structures. It is argued that these schemes provided the validation of mathematical practices necessary for the development towards symbolic reasoning. It is shown how justification schemes compensated for the lack of symbolism in abbaco treatises and at the same time facilitated a process of abstraction.

}, issn = {1233-1821}, doi = {http://dx.doi.org/10.1007/s10699-012-9311-x}, author = {Heeffer, Albrecht} } @conference {heeffer2010estienne, title = {Estienne de la {R}oche’s appropriation of {C}huquet (1484)}, booktitle = {3rd International Conference of the European Society for the History of Science}, year = {2010}, abstract = {

The importance of Larismethique of de La Roche, published in 1520, has been seriously underestimated. One reason for the neglect is related to the inscrutable way he is referred to. Buteo and Wallis called him Stephanus {\`a} Rupe de Lyon. Other obscure references, such as Gosselin calling him Villafrancus Gallus have been overlooked by many commentators. His influence can be determined in several works that do not credit him but use problems or definitions from the Larismethique. However, most damaging for its historical assessment was Aristide Marre?s misrepresentation of the Larismethique as a grave case of plagiarism. Marre discovered that the printed work of 1520 by Estienne de la Roche contained large fragments that were literally copied from Chuquet?s manuscript of the Triparty. Especially on the Appendice, which contains the solution to a large number of problems, Marre writes repeatedly that it is a literal copy of Chuquet. However, he fails to mention that the structure of the text of de la Roche, his solution methods and symbolism differs significantly from Chuquet. De la Roche introduces several improvements, especially with regards to the use of the second unknown. We provide an in-depth comparison of some problems solved by the so-called regle de la quantite by Chuquet with those of de la Roche. We further report on the surprising finding that Christoff Rudolff?s solution to linear problems by means of the second unknown in his Behend vnnd Hubsch Rechnung of 1525 depends on Chuquet and de la Roche. As it is generally considered that algebra was introduced in Germany through Italy this provides a new light on the transmission of algebraic knowledge from France to the rest of Europe.

}, author = {Heeffer, Albrecht}, editor = {Hunger, Hermann} } @article {427773, title = {The Emergence of Symbolic Algebra as a Shift in Predominant Models}, journal = {Foundations of Science}, volume = {13}, number = {2}, year = {2008}, pages = {149{\textendash}161}, abstract = {

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.

}, issn = {1233-1821}, url = {http://dx.doi.org/10.1007/s10699-008-9124-0}, author = {Heeffer, Albrecht} }