When did the concept of model begin to be used in mathematics? This question appears at first somewhat surprising since “model” is such a standard term now in the discourse on mathematics and “modelling” such a standard activity that it seems to be well established since long. The paper shows that the term— in the intended epistemological meaning—emerged rather recently and tries to reveal in which mathematical contexts it became established. The paper discusses various layers of argumentations and reflections in (...) order to unravel and reach the pertinent kernel of the issue. The specific points of this paper are the difference in the epistemological concept to the usually discussed notions of model and the difference between conceptions implied in mathematical practices and their becoming conscious in proper reflections of mathematicians. (shrink)

This comment is analysing the last section of a paper by Piotr Blaszczyk, Mikhail G. Katz, and David Sherry on alleged misconceptions committed by historians of mathematics regarding the history of analysis, published in this journal in the first issue of 2013. Since this section abounds of wrong attributions and denouncing statements regarding my research and a key publication, the comment serves to rectify them and to recall some minimal methodological requirements for historical research.

ArgumentThis paper studies the evolution of mathematics teaching in France and Germany from 1900 to about 1980. These two countries were leading in the processes of international modernization. We investigate the similarities and differences during the various periods, which showed to constitute significant time units and this in a remarkably parallel manner for the two countries. We argue that the processes of reform concerning the teaching of this major school subject are not understandable from within mathematics education or even within (...) the school system. Rather, the evolution of the processes of reform prove to be intimately tied to changing conceptions of modernity according to respective social and cultural values and to changing epistemological conceptions of mathematics. It is particularly novel that we show the key impact of the changing social status of primary schooling for these modernization processes. (shrink)

By the end of the 19th century, mathematics teaching underwent a crisis of similar proportion in several European countries. Constrained by the values of classical Antiquity, which dominated in secondary education, mathematics was taught as a formal discipline enhancing elementary geometry and intended as some gymnastics of the mind. Moreover, a specific feature in Germany was that the country was split up into numerous independent states each having a separated educational system of its own. Many of these conferred an only (...) marginal status to mathematics instruction. This complex situation is analyzed for the traditional school system for boys, as well as for the emerging system for girls, and for primary schooling.The study describes the key initiatives by Felix Klein, who forged a dynamic movement of teachers for modernising secondary school teaching. Their culmination in the famous “Meran” reform program is depicted, a program which became the starting point for the first international reform movement in mathematics teaching. (shrink)

By the end of the 19th century, mathematics teaching underwent a crisis of similar proportion in several European countries. Constrained by the values of classical Antiquity, which dominated in secondary education, mathematics was taught as a formal discipline enhancing elementary geometry and intended as some gymnastics of the mind. Moreover, a specific feature in Germany was that the country was split up into numerous independent states each having a separated educational system of its own. Many of these conferred an only (...) marginal status to mathematics instruction. This complex situation is analyzed for the traditional school system for boys, as well as for the emerging system for girls, and for primary schooling.The study describes the key initiatives by Felix Klein, who forged a dynamic movement of teachers for modernising secondary school teaching. Their culmination in the famous “Meran” reform program is depicted, a program which became the starting point for the first international reform movement in mathematics teaching. (shrink)

The life and work of Hermann Günther Graßmann (1809–1877) attract not only ever again the attention of mathematicians, mathematical historians and those interested in the history of mathematics, they constitute also a challenge for the methodology of historiographical research. This challenge persists since Friedrich Engel’s biography of 1911; there, two sources were presented and interpreted in a not legitimate manner which even mislead since then various scholars. This paper faces the intricate task to unravel not only the methodological shortcomings of (...) Engel’s biography, but also to re-assess the misinterpretations – particularly of alleged influences on Graßmann’s approaches – induced by Engel’s misleading claims. Based upon broader historically contextualised analyses of Graßmann’s innovative elements in his theory of extension and upon new primary sources for his student times at the Gymnasium and Berlin University, the paper presents a novel assessment of Graßmann’s approaches. (shrink)

Enno Heeren Dirksen (1788–1850) was one of several mathematicians originating from Eastern Frisia at the end of the eighteenth century to acquire prominence in neohumanist reformed Prussia. While striving—like so many of his contemporaries—for more rigour in the foundations of calculus, his work is significant due to his unusual emphasis on the semiotic aspects of the basic concepts. As a clear-sighted propagator of Cauchy's conceptual innovations in Germany, his achievements in improving sign expressions of such concepts enabled him to clarify (...) hitherto not differentiated multiple limit processes—although less successfully in constructive concept formation itself. The contribution is based on the analysis of Dirksen's published texts and on the numerous texts which remained drafts for his projected voluminousOrganon der gesamten Analysis. (shrink)

The mathematical and epistemological origins of Hermann Graßmann’s innovative work have always attracted the interest of mathematicians and historians. Since Friedrich Engel’s biography, a favourite source for these interpretations has been two curriculum vitae, which were, however, only known from several excerpts. The complete texts are edited here for the first time. They are presented and commented on in their respective contexts, namely the examinations required for a career as Gymnasium teacher and as Protestant pastor. Graßmann’s relations to Schleiermacher’s theology (...) and to his philosophy, as well as Graßmann’s philosophical and mathematical motivations, are discussed in the light of these two documents. (shrink)