Philosophical Way to God’s Wisdom: Arithmetic and the Definitions of a Number in Early Medieval Texts (original) (raw)
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Numbers, in: Medieval Culture: A Compendium of Critical Topics (full text)
Medieval Culture: A Compendium of Critical Topics Fundamental Aspects and Conditions of the European Middle Ages, ed. Albrecht Classen (Berlin and New York: De Gruyter 2015), pp. 1205–1260, 2015
This article outlines a history of numerical knowledge. It focuses on the forms of use that were developed for dealing with numbers. I begin with the act of counting and the notation of numbers (B); move from the role of numerals in medieval writing practices (C.) to the practices of measuring (D.) and calculating (E.); and close with comments on medieval theory building (F.), on speculative interpretations of numbers (G.), and with a short conclusion (H.). In doing so, the article focuses on the phenomenality of number—i.e., number as a spoken counting word (B.); as a graphic figure carved in wood or written on parchment (B.); as a formal pattern that organizes books and memory (C.); as a verbal expression for the documentation of quantities (D.); as a manual or symbolic instrument of calculation (E.); as a formally structured multitude determined by arithmetic properties (F.), which as symbolic carriers organize interpretation processes as well as the production of texts and works of art (G.). Thus, the approach allows for an exploration of how these varying phenomenons of number incorporate different kinds of semantic saturation that derive from the various fields of its use. As a result, this focus also contributes to dissolving traditional disciplinary gaps, especially the opposition between the history of science, which traditionally deals with the history of numerical knowledge, and historical branches of the humanities, which handle semantic and narrative phenomena.
Number and early medieval thought
‘The Mysticism of Number in the Medieval Period before Eriugena’ in J.J. Cleary ed., The Perennial Tradition of Neoplatonism (Leuven University Press, Leuven 1997), pp. 397-416.
The Development of Mathematics in Medieval Europe: The Arabs, Euclid, Regiomontanus
Aestimatio: Critical Reviews in the History of Science, 2015
This is Menso Folkerts' second Variorum volume. The first was published in 2003 [see Høyrup 2007b for a review]; it contained papers dealing with the properly Latin tradition in European mathematics, that is, the kind of mathematics which developed (mainly on the basis of agrimensor mathematics and the surviving fragments of Boethius' translation of the Elements) before the 12th-century Arabo-Latin and Greco-Latin translations. This second volume deals with aspects of the development which took place after this decisive divide, from ca 1100 to ca 1500. Few scholars, if any, know more than Folkerts about medieval Latin mathematical manuscripts. It is, therefore, natural that the perspective on mathematics applied in the papers of this volume is on mathematics as a body of knowledge, in particular, as it is transmitted in and between manuscripts. To the extent that mathematics as an activity is an independent topic, it mostly remains peripheral, being dealt with through references to the existing literature-exceptions are the investigations of what Regiomontanus and Pacioli do with their Euclid [in articles VII and XI]-or it is undocumented, as when it is said that Jordanus de Nemore's De numeris datis was 'probably used as a university textbook for algebra' [VIII.413]. There should be no need to argue, however, that familiarity with the body of mathematical knowledge is fundamental for the study of mathematics from any perspective: whoever is interested in medieval Latin mathematics can therefore learn from this book.
2015
http://www.brepols.net/Pages/ShowProduct.aspx?prod\_id=IS-9780888441911-1 http://www.pims.ca/publications/new-and-recent-titles/publication/thierry-of-chartres-the-commentary-on-the-de-arithmetica-of-boethius Unlike the commentaries on Plato's Timaeus and on Boethius's Consolatio phlosophiae, medieval exegesis of Boethius's De arithmetica has seldom been subjected to comprehensive and systematic enquiry. Inhabiting the shifting boundary between philosophy and history of science, the De arithmetica itself has been neglected by most medievalists. Yet, from the Carolingian renaissance onward, when the scholarly curriculum came to be based on the seven liberal arts, Boethius's work soon became a canonical text for the study of arithmetic. Indeed, the growing interest in it during the twelfth century is attested by the large number of surviving commentaries in manuscript. The commentary on the De arithmetica preserved in Stuttgart, Württembergische Landesbibliothek, Cod. math. 4° 33 and edited here for the first time can be securely attributed to Thierry of Chartres. It belongs to a phase when the Chartrian master's interests were mainly directed toward the liberal arts. We can also discern in Thierry's commentary on the De arithmetica themes and problems developed in his Tractatus de sex dierum operibus and more elaborately in his commentaries on Boethius's Opuscula sacra. Indeed, the discovery of this commentary on the De arithmetica might legitimately be said to clarify not only the more intractable passages in the theological writings but also to illuminate Thierry's philosophical project as a whole. At the heart of that vision is a developing trend in twelfth-century philosophy that places number and proportion at the heart of the physical cosmos. In this profoundly 'mathematical Platonism,' all things are based on number and follow the rule of number; or, to quote Thierry himself, “creatio numerorum, rerum est creatio.”
2012
After Arabic into Latin in the Middle Ages: The Translators and Their Intellectual and Social Context and Magic and Divination in the Middle Ages: Two Texts and Techniques in the Islamic and Christian Worlds, this third Variorum volume from Charles Burnett's hand collects papers dealing with the period and process of adoption of the Hindu-Arabic numerals. The collection shows us the intricacies of this process, a process which was probably the 'most momentous development in the history of pre-modern mathematics' [IX.15]. Intricacies are certainly not unexpected in a process of this kind; but their precise portrayal can only be painted by someone as familiar as Burnett with the original documents, their languages, their style and context.
Particularis de Computis et Scripturis and Medieval Applied Mathematics
This paper seeks to elucidate the reason that the passage Particularis de Computis et Scripturis (PCS) appeared in the Summa, a mathematical work written by Luca Pacioli. Being PCS a work on commercial arithmetic, the antecedents of such works in Islamic and European mathematics are investigated, along with the extent of the practical use of mathematics by those societies. In addition, this work studies why Pacioli, who apparently had professional training in mathematics, became a well-known intellectual of his time.
Old Arithmetic Books: Mathematics in Spain in the First Half of the Sixteenth Century
International Electronic Journal of Mathematics Education
During the sixteenth century, a relevant number of books related to practical arithmetic were published in Spain. This paper presents a comparative study that aims to identify the object and target of five old mathematics books and the main contents of these books. In order to do so, all the contents and all the examples found in five arithmetic books written during the first half of the 16th century were identified and categorised. A historical-mathematical analysis using a content analysis technique was then applied. The results show the authors' concerns about mathematics at the time and the different contents that were included in some arithmetic books of this century.