Playing with Personae. Men of Mathematics in Commentary.
Greek Mathematical writers, above all, Euclid, tend to present their theorems/proofs as decontextualized and, thus, as timeless, which has become the standard modus of textual presentation in theoretical mathematics. Mathematical commentators, however, provide their readers with personal names and historical information in order to elucidate problems, provide contexts of discovery, or construct doxographies, among other objectives. Such information has been used by modern readers for the construction of histories of science. When we look at such passages, however, we see that personal names and information about mathematicians past can serve to quite a range of different goals, such as the strategic self-positioning of the commentator vis-à-vis the past or the present of mathematics, the education of the reader, mathematical or moral, the construction of the history of the field, etc. Not only does the commentator present a persona of himself to the reader, he can also turn colleagues and predecessors into personae. The talk will offer a preliminary survey of such plays of and with personae.
Abstracts
Forms and Functions of Mathematical Commentaries in Islamicate Societies
Commentaries in the mathematical sciences appear in two basic forms: as independent texts or as entries in the margins, between the lines, or on flyleaves of the text commented on. Commentaries as independent texts follow different formats. One representative of a commentary that carries with it the entire text of a work commented on is al-Nayrizi’s Commentary on the Elements. I will analyze al-Nayrizi’s writing practice. Other commentaries, in particular in the form of glosses and super glosses, only connect with the commented on work through words or brief phrases of reference. Further differences consist in the issues of content that are discussed - the content of the entire work commented on or selected problems only. One of the main reason for these variations is, in my view, the two primary functions of this type of literature: discussion among senior scholars and professionals and training of students. I will present at least one example of commentaries in a school context. Beside these formal and functional differences, however, there is also a remarkable further difference which concerns the mathematical subdisciplines or themes on which commentaries were written. I will try to offer some impressionistic views on the distribution of commentaries across mathematical fields as far as I am familiar with the respective literature.
Following the Traces of the Early History of Algebra: Evidence of Practicing Algebra in the Greek Speaking World (4th-10th Century CE)
In addition to the debates concerning the proper understanding of the algebraic character of Diophantus' mathematical practice there is also another debate concerning the path or paths through which this practice was transmitted and was received in early medieval times. So far the transmission of Diophantine ideas has being studied only through the transmission of the text of the Arithmetica (translations, copies and commentaries to the text). However, in premodern mathematics the transmission of texts was not the only channel through which ideas and practices circulated. Oral tradition and teaching-and- learning activities played a very important role, especially when concerning series of problems and methods for problem solving. In this sense, the influence of Diophantus extends further than the transmission of the text of Diophantus itself. Thus collecting evidence about algebraic practices throughout the late antique and early medieval times will help represent a clearer image of the transmission of Diophantus’ mathesis and will permit us to complete a chapter in the early history of algebra. The aspect of the transmission of the ideas of Diophantus has not been so far studied adequately. Therefore there is room for research on this topic. Our contribution will rely on (a larger research program that relies on) the following axes: (a) collecting testimonia about Diophantus in Greek scholarship in late antique and medieval times, (b) recording the preserved examples from this period of problem solving via algebra, especially including the scholia to the arithmetical epigrams of the Anthology. We will be elaborating on some of the above issues presenting a case study, within our larger ongoing research project.
Book II and Renaissance Justifications of Algebraic Rules
I would like to discuss how Renaissance mathematicians used Euclid's Elements to justify algebra. The need to justify algebraic solutions of equations can be traced back to Al Kwarizmi, but the motivations to do it and the ways in which it was done are quite different and deserve to be described in detail. For, they reveal mathematical perceptions and practices of mathematics quite idiosyncratic for readers of our century. They also differ for purpose, from geometrical intuition to logical foundation, from mathematical soundness to intelligibility. As a consequence, the Euclidean passages mentioned as a justification, theorems and problems were also different. This debate interested sixteenth-century mathematicians because it could give a classical status to a discipline which had been presented in vernacular and commercial texts. Furthermore, it contributed to a larger discussion about the logic of mathematics, the reducibility of Euclidean proofs to syllogisms and the status of algebra with respect to a general art of thinking. We shall try to look at these various aspects in connection with present days debates.
Distributivity-like Results in Euclid's Elements and in the Euclidean Medieval Traditions: Between Geometry and Arithmetic
My talk will explore such results in Euclid's Elements that, retrospectively seen, embody treatments of "distributivity-like" properties of multiplication over addition and the basic conceptions associated with them. It will also discuss some of the significant changes underwent by these basic conceptions in some influential medieval mathematical treatises, such as Liber Mahameleth, and those of Fibonacci, Jordanus Nemorarius, Campanus de Novara, and Gersonides. This discussion affords interesting insights concerning both Greek and medieval attitudes towards the questions of what are numbers, what are their basic defining properties, and what is the right way to provide clear foundations for arithmetic as a solid mathematical field of knowledge.
The Development of Axiomatics in the Commentaries to Euclid’s Elements.
I discuss the medieval and early modern tradition of commentaries to Euclid’s Elements and in particular the different systems of principles that were devised to ground elementary mathematics. The modifications to Euclid’s list of postulates and common notions, in fact, represented one of the most important textual interventions on the Elements in the modern age, and engendered important changes in both mathematics and epistemology.
Ziyadat in Arabic Euclidean texts: Explorations in an Underappreciated Form of Commentary
Ziyādāt (additions) in medieval mathematical commentaries seem to have at least two purposes: to fill logical lacunae in the original treatise and to demonstrate the author’s mathematical prowess. I will focus on the additions to the early Euclidean commentary of Abū al-Qāsim `Alī b. Aḥmed al-Anṭākī (died 367 / 987), and especially a collection of added propositions placed at the end of book VI. These ziyādāt extend Euclid’s discussion of proportionality in several directions. Until recently, al-Anṭākī’s commentary has been known only in an incomplete manuscript (Oxford, Bodleian Library, Huntington 70) containing books V -- XIII of his commentary. An anonymous commentary (Hyderabad, Oriental Manuscripts Library and Research Institute, riyāḍī 2 and Hyderabad, Osmania University Library, ms. QA 510 A Sh-R, accession number 375, which is probably a direct copy from it) includes frequent quotations from the commentary of al-Anṭākī. These quotations allow us to reconstruct the content of some of the now-lost first section of al-Anṭākī’s commentary. The relationship between these quotations and the commentary is complex – the verbal features of the block of propositions added to book VI sometimes differ significantly between the two testimonia and diagram construction too is sometimes distinctly different.
Strategies of Discussion in Proclus’ Commentary on Euclid’s Elements
At the end of the second introduction to his commentary on the first book of Euclid’s Elements, Proclus briefly compares his commentary to his predecessors’ commentaries, saying that unlike them he focuses on theoretical matters that contribute to philosophy as a whole. This statement notwithstanding, Proclus’ discussions of Euclid’s principles and proofs are not confined to philosophical matters but show a diversity of interest and aims. In my talk, I try to offer a typology of Proclus’ various discussions and examine the question whether his commentary is shaped by a unified exegetical approach.
Basic Quantitative Analysis of the Corpus of Greek Mathematical Commentaries Using Visualization tools
The Corpus of Greek Mathematical texts has usually been studied from a qualitative point of view and from a content perspective. The formal and quantitative aspects have only been recently discussed. In this talk I will introduce two tools to visualize quantitative data of the Corpus of Greek Mathematical texts. The first one is a timeline displaying the distribution of either works or words of the corpus in a selected timespan. I will explain the meaning of this chart and its use. The second is a zoomable sunburst or multi-level pie chart that allows viewing the dataset of works in a hierarchical form, organized from the content, genre and style point of view. With these tools we will show some basic quantitative information of the Corpus of Greek Mathematical Commentaries, as well as a comparison with other stylistic corpus.
Ptolemy’s Theory of Harmonia
In the Harmonics, after he has completed his exposition of music theory, Ptolemy steps back and examines what harmonia is, how it functions in the cosmos, and how it affects not only the relations among musical pitches but also the order in the human soul and the movements and configurations of the stars. In this talk, I will analyze Ptolemy’s definition of harmonia and demonstrate how it impacts his ethical theory, especially as described in the Almagest. Moreover, I will show how Ptolemy’s harmonic ethics respond to and build upon the relation of harmonics, astronomy, and ethics presented in Plato’s Timaeus.
Aristotle and the Meno’s Paradox
Meno (in the homonymous dialogue of Plato) states a sophistic paradox according to which any research is impossible. Plato tries to solve this paradox via the introduction of the theory of recollection. Aristotle refers twice to the Meno’s paradox in the Posterior Analytics I, 1 and in the Prior Analytics II, 21. Aristotle gives a solution to this paradox within the framework of his logic. However, it is not certain what Aristotle has in mind when he refers to the Meno’s paradox. In this paper, I am going to examine and formulate logically Aristotle’s answer to this paradox and to compare it with what Plato says in the Meno.
Diophantus, al-Karajī, and Quadratic Equations
In the beginning of his Arithmetica Diophantus promises to show how to solve three-term quadratic equations. The rules are not in the extant part of his work, but al-Karajī (early 11th c.) provides clues as to how Diophantus may have presented them. In his algebra book al-Fakhrī al-Karajī gives different rules for solving three-term equations, including the standard Arabic rule, the variation practiced by Diophantus, and a rule borrowed from Abū Kāmil for finding the square of the unknown directly. After proofs of these rules he gives a derivation that he calls ``the method of Diophantus''. By examining proofs and problems in several of al-Karajī's works we propose that Diophantus derived his rules for solving three term equations by this ``method of Diophantus'', and that al-Karajī learned these rules and their derivations from the Arithmetica.
Pushing and Tugging et the Mechanical Tradition: Aristotle’s Physics VII
Aristotle’s Physics H, on caused motion, is not a pristine Greek mathematical treatise, but that doesn’t mean that it isn’t some sort of mathematical treatise. Furthermore, ch. 2, a theorem by cases, fits squarely into a mechanical tradition (just as does ch. 5, the laws for caused motion), albeit one that we have merely hints of.
Diagrams of the Arithmetical Books of Euclid’s Elements
When I published the diagrams of all the geometrical books of the Elements, I left out the diagrams of books VII-IX (as well as those of books V and X). These diagrams are, nonetheless, as interesting as those in the geometrical books and, moreover, one may find considerably different diagrams from one ms. to another. Thus, their investigation, combined with an analysis of the mathematical content of the corresponding propositions, may provide suggestions about the history and tradition of the arithmetic of the Elements.
Premodern Algebra as a Historiographical Category
The aim of my talk is to revisit Al-Nayrizi’s commentary on Euclid’s Elements—and, more specifically, the so-called ‘semi-algebraic’ demonstrations of II.1-10 attributed to Hero of Alexandria—in the light of the newly contextualized historiographical category of ‘premodern’ algebra. The discussion will, hopefully, also shed some light on one of the most controversial debates in the field of Greek mathematics, namely the debate on ‘geometrical algebra’.
Commentating on an Early Arabic Version of Euclid's Data
In this talk, I will discus various question that arise in making a commentary to the version of Euclid's Data produced by Thabit ibn Qurra. Already by the late ancient period, the Data was being read in two different mathematical contexts: (1) on the one hand, it was a foundational text in geometric analysis, allowing mathematicians to make inferences - through "chains of givens" known as resolutions - from an assumed solution to a problem to a configuration that was known to be constructable, and (2) on the other hand it was a preliminary text to reading Ptolemy's Almagest, providing justification for Ptolemy's practice of establishing computability by constructing "chains of givens" known as metrical analyses. A commentary to the medieval Data must, at the least, elucidate these two different practices.