@article {heefferinterpretation, title = {On the interpretation of early Italian bartering problems}, year = {Submitted}, author = {Heeffer, Albrecht} } @incollection {HA_DuAlJ, title = {Dutch Algebra and Arithmetic in Japan before the Meiji Restoration}, booktitle = {Navigating across Mathematical Cultures and Times: Exploring the Diversity of Discoveries and Proofs}, year = {In Press}, publisher = {World Scientific Publishing Co.}, abstract = {

This paper gives an overview of the scarce occasions in which Japan came into contact with Western arithmetic and algebra before the Meiji restoration of 1868. After the refutation of persistent claims on the influence through Japanese students at Leiden during the seventeenth century, 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,2 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. Two Japanese textbooks on arithmetic from that period are viewed within the context of this naval training school.

}, author = {Heeffer, Albrecht}, editor = {Vandoulakis, I.M. and Dun, Liu} } @incollection {5673678, title = {Data-driven induction in scientific discovery: a critical assessment based on Kepler{\textquoteright}s discoveries}, booktitle = {Logic, reasoning, and rationality}, year = {2014}, pages = {59{\textendash}76}, publisher = {Springer}, abstract = {

Motivated by the renewed interest in knowledge discovery from data (KDD) by the artificial intelligence community, this paper provides a critical assessment of the model of data-driven induction for scientific discovery. The most influential research program using this model is developed by the BACON team. Two of the main claims by this research program, the descriptive and constructive power of data-driven induction, are evaluated by means of two historical cases studies: the discovery of the sine law of refraction in optics and Kepler{\textquoteright}s third law of planetary motion. I will provide evidence that the data used by the BACON program{\textendash}-despite the claims being made{\textendash}-does not correspond with the historical data available to Kepler and his contemporaries. Secondly, it is shown that for the two cases the method by which the general law was arrived at did not involve data-driven induction. Finally, the value of the data-driven induction as a general model for scientific discovery is being questioned.

}, isbn = {9789401790109}, doi = {http://dx.doi.org/10.1007/978-94-017-9011-6\_4}, author = {Heeffer, Albrecht}, editor = {Weber, Erik and Wouters, Dietlinde and Meheus, Joke} } @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} } @incollection {2116243, title = {The rule of quantity by {C}huquet and de la {R}oche and its influence on {G}erman {C}ossic algebra}, booktitle = {Pluralit{\'e} de l{\textquoteright}alg{\`e}bre {\`a} la Renaissance}, year = {2012}, pages = {127{\textendash}147}, publisher = {Honor{\'e} Champion}, 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{\textquoteright}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.

}, isbn = {9782745323989}, author = {Heeffer, Albrecht}, editor = {Rommevaux, Sabine and Spiesser, Maryvonne and Massa Esteve, Maria Rosa} } @incollection {1998274, title = {On the curious historical coincidence of algebra and double-entry bookkeeping}, booktitle = {Foundations of the formal sciences VII : bringing together philosophy and sociology of science}, series = {Studies in Logic}, volume = {32}, year = {2011}, pages = {109{\textendash}130}, publisher = {College Publications}, isbn = {9781848900493}, url = {http://www.collegepublications.co.uk/logic/?00021}, author = {Heeffer, Albrecht}, editor = {Fran{\c c}ois, Karen and L{\"o}we, Benedikt and M{\"u}ller, Thomas and Van Kerkhove, Bart} } @proceedings {1854115, title = {From problem solving to the teaching of algebra: the genesis of the algebra textbook}, journal = {11th international IHPST and 6th Greek history, philosophy and science teaching joint conference, Proceedings}, year = {2011}, pages = {344-347}, publisher = {Epikentro Publications}, isbn = {9789604583256}, author = {Heeffer, Albrecht}, editor = {Seroglou, Fanny and Koulountzos, Vassilis and Siatras, Anastasios} } @article {1891046, title = {Historical objections against the number line}, journal = {Science and Education}, volume = {20}, number = {9}, year = {2011}, pages = {863{\textendash}880}, abstract = {

Historical studies on the development of mathematical concepts will help mathematics teachers to relate their students{\textquoteright} difficulties in understanding to conceptual problems in the history of mathematics. We argue that one popular tool for teaching about numbers, the number line, may not be fit for early teaching of operations involving negative numbers. Our arguments are drawn from the many discussions on negative numbers during the seventeenth and eighteenth centuries from philosophers and mathematicians such as Arnauld, Leibniz, Wallis, Euler and d{\textquoteright}Alembert. Not only does division by negative numbers pose problems for the number line, but even the very idea of quantities smaller than nothing has been challenged. Drawing lessons from the history of mathematics, we argue for the introduction of negative numbers in education within the context of symbolic operations.

}, issn = {0926-7220}, doi = {http://dx.doi.org/10.1007/s11191-011-9349-0}, author = {Heeffer, Albrecht} } @article {1931707, title = {Hypocrisie binnen de wiskunde?}, journal = {Wiskunde \& onderwijs}, volume = {148}, year = {2011}, pages = {306{\textendash}315}, issn = {2032-0485}, author = {Heeffer, Albrecht} } @article {Heeffer2011, title = {The Rhetoric of Problems in Algebra Textbooks from Pacioli to Euler}, year = {2011}, abstract = {

The selection of problems by Euler in his Vollst{\"a}ndige Anleitung zur Algebra 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 reveals 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? We propose an explanation based on the evolving rhetorical function of problems in algebra textbooks since the Renaissance. We discern six stages in the evolution from abacus 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.

}, author = {Heeffer, Albrecht} } @article {1019205, title = {Algebraic partitioning problems from {L}uca {P}acioli{\textquoteright}s {P}erugia manuscript (Vat. Lat. 3129)}, journal = {Sources and Commentaries in Exact Sciences}, volume = {11}, year = {2010}, pages = {3{\textendash}51}, issn = {1345-4617}, author = {Heeffer, Albrecht} } @conference {924838, title = {The body in Renaissance arithmetic: from mnemonics to embodied cognition}, booktitle = {Proceedings of the Thirty Sixth Annual Convention of the Society for the Study of Artificial Intelligence and Simulation}, year = {2010}, publisher = {The Society for the Study of Artificial Intelligence and the Simulation of Behaviour}, organization = {The Society for the Study of Artificial Intelligence and the Simulation of Behaviour}, abstract = {

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

}, isbn = {1902956931}, author = {Heeffer, Albrecht}, editor = {Pease, Alison and Guhe, Markus and Smaill, Alan} } @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} } @incollection {1043033, title = {From the second unknown to the symbolic equation}, booktitle = {Philosophical Aspects of Symbolic Reasoning in Early Modern Mathematics}, volume = {26}, year = {2010}, pages = {57{\textendash}102}, publisher = {College Publications}, abstract = {

The symbolic equation slowly emerged during the course of the sixteenth century as a new mathematical concept as well as a mathematical object on which new operations were made possible. Where historians have of- ten pointed at Francois Vi{\`e}te as the father of symbolic algebra, we would like to emphasize the foundations on which Vi{\`e}te could base his logistica speciosa. The period between Cardano’s Practica Arithmeticae of 1539 and Gosselin’s De arte magna of 1577 has been crucial in providing the necessary build- ing blocks for the transformation of algebra from rules for problem solving to the study of equations. In this paper we argue that the so-called “second unknown” or the Regula quantitates steered the development of an adequate symbolism to deal with multiple unknowns and aggregates of equations. Dur- ing this process the very concept of a symbolic equation emerged separate from previous notions of what we call “co-equal polynomials”.

}, isbn = {978-1-84890-017-2}, author = {Heeffer, Albrecht}, editor = {Heeffer, Albrecht and Van Dyck, Maarten} } @book {1042976, title = {Philosophical Aspects of Symbolic Reasoning in Early Modern Mathematics}, series = {Studies in Logic}, volume = {26}, year = {2010}, pages = {302}, publisher = {College Publications}, organization = {College Publications}, abstract = {

This book presents a selection of peer-reviewed papers which were presented on a conference organized in Ghent, Belgium, from 27 till 29 August, 2009. The conference was given the title Philosophical Aspects of Symbolic Reasoning in Early modern Science and Mathematics (PASR). For this book we selected papers which deal with the consequences for mathematics in particular, hence the omission of {\textquoteleft}science{\textquoteright} in the title of this book. Another selection, dealing with the understanding of nature and a broader range of topics, will appear in the journal Foundations of Science. The conference was sponsored by the Research Foundation Flanders (FWO) and Ghent University, which indirectly made this book possible. We also have to thank the other members of the programme committee Marco Panza, Chikara Sasaki, and Erik Weber and our keynote speakers Jens H{\o}yrup, Doug Jesseph, Eberhard Knobloch, Marco Panza, Mathias Schemmel and Michel Serfati. Five of their papers are included in this volume. Most of the papers benefited from valuable and sometimes substantive comments by our referees which must remain anonymous. Special thanks to Michael Barany who assisted in the editorial process.

}, isbn = {978-1-84890-017-2}, editor = {Heeffer, Albrecht and Van Dyck, Maarten} } @incollection {1029910, title = {The Reception of Ancient Indian Mathematics by Western Historians}, booktitle = {Ancient Indian Leaps into Mathematics}, year = {2010}, pages = {135{\textendash}152}, publisher = {Birkh{\"a}user Verlag}, abstract = {

While there was an awareness of ancient Indian mathematics in the West since the sixteenth century, historians discuss 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 the 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.

}, isbn = {978-0-8176-4694-3}, author = {Heeffer, Albrecht}, editor = {Yadav, B.S. and Mohan, Man} } @incollection {1217131, title = {The symbolic model for algebra: functions and mechanisms}, booktitle = {Model-Based Reasoning in Science and Technology}, volume = {314}, year = {2010}, pages = {519{\textendash}532}, publisher = {Springer}, abstract = {

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.

}, isbn = {9783642152221}, url = {http://dx.doi.org/10.1007/978-3-642-15223-8\_29}, author = {Heeffer, Albrecht}, editor = {Magnani, Lorenzo and Carnielli, Walter A. and Pizzi, Claudio} } @article {666246, title = {The Abbaco Tradition (1300-1500): its role in the development of {E}uropean algebra}, journal = {Suuri Kaiseki Kenkyuujo koukyuuroku}, volume = {1625}, year = {2009}, pages = {23{\textendash}33}, abstract = {

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{\textquoteright}s and the ground-breaking work by Warren van Egmond (1980). After some decades of manuscript study and the recent assessment of Jens H{\o}yrup (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.

}, issn = {1880-2818}, author = {Heeffer, Albrecht} } @article {heeffer2009abbacus, title = {The Abbacus Tradition: the Missing Link between Arabic and Early Symbolic Algebra?}, year = {2009}, abstract = {

Abbaco algebra is a coherent tradition of arithmetical and algebraic problem solving, mostly based in the merchant cities of fourteenth and fifteenth-century Italy. This period is roughly situated between two impor- tant works dealing with algebra: the Liber Abbaci by Fibonacci (1202) [5] and the Summa di Arithmetica et Geometria by Luca Pacioli (1492) [20]. Such continuous tradition of mathematical practice was hardly known be- fore the first transcriptions of extant manuscripts by Gino Arrighi from the 1960{\textquoteright}s (most notably [1], [2], [3] and [4]) and the ground-breaking work by Warren van Egmond [28]. After some decades of manuscript study and the recent assessment of Jens H{\o}yrup [10] we now have a bet- ter understanding of this tradition. Here 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.

}, author = {Heeffer, Albrecht} } @article {weber2009laudation, title = {Laudatio Jens H{\o}yrup}, journal = {Sartoniana}, volume = {22}, year = {2009}, pages = {5{\textendash}7}, author = {Weber, Erik and Heeffer, Albrecht} } @conference {678587, title = {On the Nature and Origin of Algebraic Symbolism}, booktitle = {New Perspectives on Mathematical Practices. Essays in Philosophy and History of Mathematics.}, year = {2009}, publisher = {World Scientific Publishing}, organization = {World Scientific Publishing}, isbn = {978-981-281-222-3}, url = {http://www.worldscibooks.com/etextbook/6810/6810\_chap01.pdf}, author = {Heeffer, Albrecht}, editor = {Van Kerkhove, Bart} } @incollection {417858, title = {A conceptual analysis of early arabic algebra}, booktitle = {The unity of science in the arabic tradition : science, logic, epistemology and their interactions}, year = {2008}, pages = {89{\textendash}128}, publisher = {Kluwer Academic Publishers}, address = {Dordrecht}, abstract = {

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.

}, isbn = {978-1-4020-8404-1}, url = {http://dx.doi.org/1854/11276}, author = {Heeffer, Albrecht}, editor = {Rahman, Shahid and Street, Tony and Tahiri, Hassan} } @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} } @article {420317, title = {An introduction to wasan, native {J}apanese mathematics}, journal = {History and Pedagogy of Mathematics}, volume = {68}, year = {2008}, pages = {20{\textendash}24}, issn = {N/A}, url = {http://dx.doi.org/1854/11413}, author = {Heeffer, Albrecht} } @conference {428001, title = {Negative numbers as an epistemic difficult concept. Some lessons from history.}, booktitle = {History and Pedagogy of Mathematics. Satellite Meeting of International Congress on Mathematical Education 11, 14 - 18 July 2008.}, year = {2008}, publisher = {Centro Cultural del M{\'e}xico Contemporan{\'e}o}, organization = {Centro Cultural del M{\'e}xico Contemporan{\'e}o}, abstract = {

Historical studies on the development of mathematical concepts will serve mathematics teachers to relate their students’ difficulties in understanding to conceptual problems in the history of mathematics. We argue that one popular tool for teaching about numbers, the number line, may not be fit for early teaching of operations involving negative numbers. Our arguments are drawn from the many discussions on negative numbers during the seventeenth and eighteenth centuries from philosophers and mathematicians as Arnauld, Leibniz, Wallis, Euler and d’Alembert. Not only the division by negative numbers poses problems for the number line, but also the very idea of quantities smaller than nothing has been challenged. Drawing lessons from the history of mathematics we argue for the introduction of negative numbers in education within the context of symbolic operations.

}, author = {Heeffer, Albrecht}, editor = {Tzanakis, C.} } @article {1092888, title = {Regiomontanus and Chinese mathematics}, journal = {PHILOSOPHICA}, volume = {82}, number = {2}, year = {2008}, pages = {87{\textendash}114}, abstract = {

This paper critically assesses the claim by Gavin Menzies that Regiomontanus knew about the Chinese Remainder Theorem (CRT) through the Sh{\`u} sh\={u} Ji\v{u} zh\={a}ng (SSJZ) written in 1247. Menzies uses this among many others arguments for his controversial theory that a large fleet of Chinese vessels visited Italy in the first half of the 15th century. We first refute that Regiomontanus used the method from the SSJZ. CRT problems appear in earlier European arithmetic and can be solved by the method of the Sun Zi, as did Fibonacci. Secondly, we pro-vide evidence that remainder problems were treated within the European abbaco tradition independently of the CRT method. Finally, we discuss the role of recre-ational mathematics for the oral dissemination of sub-scientific knowledge.

}, issn = {0379-8402}, author = {Heeffer, Albrecht}, editor = {Weber, Erik} } @article {heeffer2008text, title = {Text production, reproduction and appropriation within the abbaco tradition: a case study}, journal = {SCIAMVS}, volume = {9}, year = {2008}, pages = {101-145}, author = {Heeffer, Albrecht} } @incollection {381663, title = {Abduction as a strategy for concept formation in mathematics: {C}ardano postulating a negative.}, booktitle = {Abduction and the process of scientific discovery}, year = {2007}, pages = {179{\textendash}194}, publisher = {Centro de Filosofia das Ci{\^e}ncias da Universidade de Lisboa}, abstract = {

When dealing with abductive reasoning in scientific discovery, historical case studies are focused mostly on the physical sciences, as with the discoveries of Kepler, Galilei and Newton. We will present a case study of abductive reasoning in early algebra. Two new concepts introduced by Cardano in his Ars Magna, imaginary numbers and a negative solution to a linear problem, can be explained as a result of a process of abduction. We will show that the first appearance of these new concepts fits very well Peirce’s original description of abductive reasoning. Abduction may be regarded as one important strategy for the formation of new concepts in mathematics.

}, author = {Heeffer, Albrecht}, editor = {Pombo, Olga and Gerner, Alexander} } @article {370073, title = {Father {H}enri {B}osmans (S.J.) A {B}elgian pioneer in the history of mathematics}, journal = {History and Pedagogy of Mathematics Newsletter}, volume = {65}, year = {2007}, pages = {12{\textendash}15}, issn = {N/A}, author = {Heeffer, Albrecht} } @incollection {370857, title = {Learning concepts through the history of mathematics: the case of symbolic algebra.}, booktitle = {Philosophical dimensions in mathematics education}, year = {2007}, pages = {83{\textendash}103}, publisher = {Springer}, isbn = {978-0-387-71571-1}, author = {Heeffer, Albrecht}, editor = {Fran{\c c}ois, Karen and Van Bendegem, Jean Paul} } @article {381861, title = {Mathematics Education Through Lesson Study in the {A}sia-{P}acific Region}, journal = {History and Pedagogy of Mathematics Newsletter}, volume = {66}, year = {2007}, pages = {4{\textendash}6}, issn = {N/A}, author = {Heeffer, Albrecht} } @booklet {heeffer2007near, title = {Near-optimal strategies for the game of Logik}, year = {2007}, abstract = {

Logik is an interesting variation of the game of Mastermind. For the latter several strategies have been proposed. We present some computational results for existing and new strategies applied to Logik. Our results give some indication on the scalability and applicability of these strategies to similar games.

}, author = {Heeffer, Albrecht and Heeffer, Harold} } @article {382923, title = {The tacit appropriation of {H}indu algebra in renaissance practical arithmetic}, journal = {Ganita Bharati}, volume = {29}, number = {1-2}, year = {2007}, pages = {1{\textendash}60}, issn = {0970-0307}, author = {Heeffer, Albrecht} } @article {heeffer2006logic, title = {The logic of disguise: {D}escartes{\textquoteright} discovery of the Sine Law}, journal = {Historia scientiarum. International Journal of the History of Science Society of Japan}, volume = {16}, number = {2}, year = {2006}, pages = {144{\textendash}165}, abstract = {

The discovery of the sine law of refraction by Decartes has been a source of controversy since its first publication in 1637. Scholars describe Descartes’ demonstration of the law as inadequate and implausible or see in his proof an attempt to cover up plagiarism. Most of this critique stems from a lack of understanding the Cartesian theory of explanation. Descartes aims in the Discourse to find undoubtable knowledge that serves as a basis for his natural philosophy. In his Principles of Philosophy he deduces from his metaphysics several laws of nature and additional rules. However, for the explanation on physical phenomena, Descartes adopts a method different from the one proposed in his Discourse. This paper discusses the logical structure of the demonstration and provides an analysis of the premises and conclusions used in the derivation of the sine law. It is argued that Descartes’s explanation of refraction is a continuation of the perspectivists tradition and, ironically, Fermat’s account of 1662 is better application of the Cartesian method.

}, author = {Heeffer, Albrecht} } @conference {353364, title = {The Methodological Relevance of the History of Mathematics for Mathematics Education}, booktitle = {Proceedings of the International Conference on 21st Century Information Technology in Mathematics Education}, year = {2006}, abstract = {

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.

}, author = {Heeffer, Albrecht}, editor = {Dhompongsa, G and Bhatti , F and Kitson , Q} } @article {361180, title = {The origin of the problems in {E}uler{\textquoteright}s {A}lgebra}, journal = {Bulletin of the Belgian Mathematical Society - Simon Stevin}, volume = {13}, number = {5}, year = {2006}, pages = {949{\textendash}952}, issn = {1370-1444}, author = {Heeffer, Albrecht} } @article {353368, title = {R{\'e}cr{\'e}ations math{\'e}matiques (1624): a study on its authorship, sources and influence}, journal = {Gibeci{\`e}re}, volume = {1}, number = {2}, year = {2006}, pages = {77{\textendash}167}, abstract = {

In 1624 a small octavo was published in the French university town Pont-{\`a}-Mousson. It was the first time a reference was made to ‘recreational mathematics’ in the title of a book. This work is pivotal in the history of science and mathematics. It brings together two sixteenth-century traditions, mercantile arithmetic and natural magic, and creates two new ones: recreational mathematics and popular science. The booklet did not abound in new ideas. Several of the recreational problems treated, can be traced back to Babylonian, Greek and Hindu sources and the infatuation with mechanical contrivances dates from Hero of Alexandria. But the fact that the book stands on the crossroad of traditions, its popularity with the natural philosophers of the seventeenth century and its complex history makes it a grateful subject for study. Some of the complexities about the numerous editions and confusing claims about its authorship will be clarified. The author’s direct sources will be exposed. The arithmetical and combinatorial problems were copied from Bachet, problems on practical geometry from Jean Errard. Salomon de Caus was the source of inspiration for problems on perspective, mechanical devices and fountains. Several problems were recipies by Alexis of Piemont from the classic book by Ruscelli. The book was influencial on early seventeenth-century natural philosophers such as Descartes, Mersenne and Leibniz.

}, issn = {1558-8149}, author = {Heeffer, Albrecht} } @phdthesis {PhD_Albrecht, title = {Van Precepten tot Vergelijkingen. De Conceptuele Ontwikkeling van Symbolische Algebra in de Zestiende Eeuw}, year = {2006}, month = {February 22}, publisher = {Ghent University}, type = {phd}, author = {Heeffer, Albrecht} } @incollection {217593, title = {Kepler{\textquoteright}s near discovery of the sine law: a qualitative computational model.}, booktitle = {Computer modeling of scientific reasoning}, year = {2003}, pages = {93{\textendash}102}, publisher = {Universidad Nacional Del Sur. EDIUNS}, abstract = {

Computational models offer an excellent tool for the study and analysis of scientific discovery processes. The study of failures provides an insight into the history and philosophy of science as valuable as the study of successful discoveries. Using a computational model I analyzed Kepler’s approach in formulating a quantitative law for refraction. Although Kepler ultimately failed in discovering the sine law, the model shows that his basic hypothesis as well as his approach by geometrical reasoning was a correct one. This went largely unnoticed by commentators on the history of optics. Based on this analysis I provide new evidence that Descartes and Snell found in Kepler’s main hypothesis everything needed to deduce the sine law by pure geometrical reasoning. Our computational model is based on geometrical knowledge as contrasted with previous quantitative approaches. It has been implemented as a Prolog program.

}, isbn = {N/A}, author = {Heeffer, Albrecht}, editor = {Delrieux, Claudio and Legris, Javier} }