Books by Albrecht Heeffer
The novel use of symbolism in early modern mathematics poses both philosophical and historical qu... more The novel use of symbolism in early modern mathematics poses both philosophical and historical questions. How can we trace its development and transmission through manuscript sources? Is it intrinsically related to the emergence of symbolic algebra? How does symbolism relate to the use of diagrams? What are the consequences of symbolic reasoning on our understanding of nature? Can a symbolic language enable new forms of reasoning? Does a universal symbolic language exist which enable us to express all knowledge?
This book brings together a collection of papers that address all these and related questions which were initially posed at a conference held in Ghent (Belgium) in August 2009. Scholars working on philosophy of science, history of philosophy and history of mathematics provide an insight into the role and function of symbolic representations in the development of early modern mathematics. The papers cover the period from early abbaco arithmetic and algebra (14h century) up to Leibniz (early 18th century).
Papers by Albrecht Heeffer
The Chinese rings puzzle is one of those recreational mathematical problems known for several cen... more The Chinese rings puzzle is one of those recreational mathematical problems known for several centuries in the West as well as in Asia. Its origin is difficult to ascertain but is most likely not Chinese. In this paper we provide an English translation, based on a mathematical analysis of the puzzle, of two sixteenth-century witness accounts. The first is by Luca Pacioli and was previously unpublished. The second is by Girolamo Cardano for which we provide an interpretation considerably different from existing translations. Finally, both treatments of the puzzle are compared, pointing out the presence of an implicit idea of non-numerical recursive algorithms.
Data-Driven Induction in Scientific Discovery: A Critical Assessment Based on Kepler’s Discoveries
Logic, Argumentation & Reasoning, 2014
The Henri Bosmans Bibliography
How algebra spoiled recreational problems: A case study in the cross-cultural dissemination of mathematics
Historia Mathematica, 2014
The Genesis of the Algebra Textbook: From Pacioli to Euler
Almagest, 2012
The Mathematical Intelligencer, 2015
is the prototype of a successful textbook on elementary algebra. The selection of problems by Eul... more is the prototype of a successful textbook on elementary algebra. The selection of problems by Euler displays a great familiarity with the typical recreational and practical problems of Renaissance and sixteenth-century algebra books. A detailed study into the sources of Euler revealed that he copied most of his problems from Christoff Rudolff"s Coss which was first published in 1525 and reissued in 1553 by Michael Stifel. Why would Euler found his popular textbook on algebra on a book published 250 years before? Part of the motivation could be sentimental. Euler was taught mathematics by his father using Stifel"s edition of the Coss, and the young Euler spent several years studying the problems from the book. However, we propose an explanation based on the evolving rhetorical function of problems in algebra textbooks since the first printed book on algebra by Pacioli (1494). We discern six stages in the evolution from abbacus problem solving to algebraic theory. The first theory emerged through the extraction of general principles from the practice of problem solving. The algebra textbooks of the eighteenth century close a circle of continuous rhetorical development by using problems for practicing general principles and applying the algebraic language. Euler"s Algebra is a prime example of the new rhetoric of problems still prominent in today"s textbooks.
Récréations Mathématiques au Moyen Âge by Jacques Sesiano
The Mathematical Intelligencer, 2015
In Medieval and Renaissance arithmetic we find several instances of references to body parts or a... more 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. 12
Validating Concepts from Automated Acquisition Systems
Ijcai, 1985
Henri Bosmans S.J. (1852–1928) – grondlegger van de geschiedenis van de wiskunde in België
Studium, 2013
Boekbesprekingen/Comptes Rendus
Studium, 2014
Mathematics education benefits from an integration of the history of mathematics within the mathe... more Mathematics education benefits from an integration of the history of mathematics within the mathematics curriculum. We provide three basic arguments for such integration. The first is epistemological and addresses a contextual view on mathematical knowledge. The second concerns the phylogenic aspects of the development of mathematics. Conceptual difficulties with teaching children mathematics often correspond with historical periods of conceptual crisis in mathematics. A third, historical argument, draws on the vast repository of experience in mathematics education. We provide examples for each of these arguments from the history of algebra. In this paper we argue for the integration of the history of mathematics in mathematics education. Our motive for the study of the emergence of symbolic algebra is mainly epistemological. How are concepts formed in mathematics? Which factors influence or change the meaning of concepts? Is there an internal logic and order in the development of m...
From Problem Solving to Argumentation: Pacioli's Appropriation of Abbacus Algebra
... E quella 13 sira la summa de ditte quantita ne virra la 2a parte. ... Pacioli's Summa an... more ... E quella 13 sira la summa de ditte quantita ne virra la 2a parte. ... Pacioli's Summa and Cardano's Practica Arithmeticae had a decisive influence and the two works together ... Klein, Jacob (1968) Greek mathematical thought and the origin of algebra, MIT Press, Cambridge, 1968. ...
this paper we cannot go into the history of optics, but will give a detailed account of Kepler... more this paper we cannot go into the history of optics, but will give a detailed account of Kepler's study of refraction based on his Ad Vitellionern paralipomena of 1604. Thanks to his diligent reporting style Kepler's writings are grateful objects of study and allow us to reconstruct his path of reasoning in detail. But we will first critically review a previous model of the discovery of the sine law of refraction
The use of symbolism in mathematics is probably the mostly quoted reason people use for explainin... more 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...
On Remembering Cardano Anew
The Mathematical Intelligencer, 2014
Mathematics education benefits from an integration of the history of mathematics within the mathe... more Mathematics education benefits from an integration of the history of mathematics within the mathematics curriculum. We provide three basic arguments for such integration. The first is epistemological and addresses a contextual view on mathematical knowledge. The second concerns the phylogenic aspects of the development of mathematics. Conceptual difficulties with teaching children mathematics often correspond with historical periods of conceptual crisis in mathematics. A third, historical argument, draws on the vast repository of experience in mathematics education. We provide examples for each of these arguments from the history of algebra.
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Books by Albrecht Heeffer
This book brings together a collection of papers that address all these and related questions which were initially posed at a conference held in Ghent (Belgium) in August 2009. Scholars working on philosophy of science, history of philosophy and history of mathematics provide an insight into the role and function of symbolic representations in the development of early modern mathematics. The papers cover the period from early abbaco arithmetic and algebra (14h century) up to Leibniz (early 18th century).
Papers by Albrecht Heeffer
This book brings together a collection of papers that address all these and related questions which were initially posed at a conference held in Ghent (Belgium) in August 2009. Scholars working on philosophy of science, history of philosophy and history of mathematics provide an insight into the role and function of symbolic representations in the development of early modern mathematics. The papers cover the period from early abbaco arithmetic and algebra (14h century) up to Leibniz (early 18th century).