Quantifying Aristotle. The Impact, Spread and Decline of the Calculatores Tradition
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Program

Wednesday, 23 May

TimeTopicChair
10:00 - 10:30 Welcome & Registration -
10:30 - 11:00 Introduction -
11:00 - 11:45 Elżbieta Jung: The New Interpretation of Aristotle. Richard Kilvington, Thomas Bradwardine and The New Rule of Motion Edith D. Sylla
11:45 - 12:30 Edit A. Lukács: The Mathematical Theory of Proportions in Thomas Bradwardine´s De causa Dei, Book I Edith D. Sylla
12:30 - 14:30 Lunch Break -
14:30 - 15:15 José Meirinhos: The Falsigraphus in Thomas Bradwardine’s De continuo Kärin Nickelsen
15:15 - 16:00 Magali Roques: Ockham and Bradwardine on propositions de incipit et desinit Kärin Nickelsen
16:00 - 16:30 Coffee Break -
16:30 - 17:15 Robert Podkoński: The Opuscula de motu ascribed to Richard Swineshead – the testimony of the ongoing development of the Oxford Calculators’s science of motion Sabine Rommevaux-Tani
17:15 - 18:00 Fabio Seller: The “rules” of motus difformis in Angelo da Fossambruno’s De tribus praedicamentis Hentisberi Sabine Rommevaux-Tani
18:15 - 19:30 Reception -

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Thursday, 24 May

TimeTopicChair
09:45 - 10:30 Stephen Read: The Calculators on the Insolubles: Bradwardine, Kilvington, Swyneshed, Heytesbury and Dumbleton Paolo Mancosu
10:30 - 11:00 Coffee Break -
11:00 - 11:45 Mark Thakkar: Wyclif and the Oxford Calculators Edit A. Lukács
11:45 - 12:30 Harald Berger: Zur Rezeption der englischen Logik an zentraleuropäischen Universitäten: Helmoldus de Zoltwedel (Prag, Leipzig) über die Lügner-Paradoxie Edit A. Lukács
12:30 - 14:30 Lunch Break -
14:30 - 15:15 Cecilia Trifogli: Thomas Wylton and Walter Burley on the Incipit and Desinit of an Instant of Time Peter Adamson
15:15 - 16:00 Mischa von Perger: Albert of Saxony and the idea of indivisible points of space and time Peter Adamson
16:00 - 16:30 Coffee Break -
16:30 - 17:15 Aníbal Szapiro: The Calculators' Tradition and the Geometrisation of Light in Oresme's De visione stellarum Elżbieta Jung
17:15 - 18:00 Daniel A. Di Liscia: The Geometrisation of Theology and the Metaphysics in the tradition de perfectione specierum Elżbieta Jung
19:30 Conference Dinner -

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Friday, 25 May

TimeTopicChair
09:00 - 09:45 Sabine Rommevaux-Tani: Les questions sur le mouvement local dans le traité anonyme De sex inconvenientibus Sabrina Ebbersmeyer
09:45 - 10:30 Joël Biard: Le calcul des variations de qualités et la physique aristotélicienne selon Blaise de Parme Sabrina Ebbersmeyer
10:30 - 11:00 Coffee Break -
11:00 - 11:45 Richard Oosterhoff: How Humanists Repurposed the Calculatores in Renaissance Paris Stephen Read
11:45 - 12:30 Henrique Leitão: Mathematics and Motion in Alvarus Thomas' Liber de triplici motu (1509) Stephen Read
12:30 - 14:30 Lunch Break -
14:30 - 15:15 Edith D. Sylla: Leibniz, the Liber Calculationum, and Mathematical Physics Daniel A. Di Liscia
15:15 - 16:00 Concluding discussion -

Abstracts

Harald Berger: Zur Rezeption der englischen Logik an zentraleuropäischen Universitäten: Helmoldus de Zoltwedel (Prag, Leipzig) über die Lügner-Paradoxie

Helmoldus Gledenstede de Zoltwedel, Magister artium, Doctor medicinae and Baccalaureus theologiae, was an important scholar of his times, he was active at the University of Prague from 1383 to 1409 and at the University of Leipzig from 1409 to his death in 1441.

His Opus Magnum is the Quaestiones parvorum logicalium, composed at Prague around 1390, which comprises nine parts and some 90 questions. The ninth and last part is devoted to the Insolubles, and the fifth question thereof to the so-called Liar. The complete work is extant in two manuscripts, qu. IX.5 alone is copied even in a third manuscript.

This large text is interesting and valuable, because Helmold presents ten contemporary positions and then his own as an eleventh one. Therefore, we have got here a source that is a bit earlier than the well-known list in Paul of Venice’s Logica magna from the end of the 1390ies (14 solutions plus Paul’s own).

In this text, Helmold quotes besides authors from Paris and Prague also some English logicians, such as Brinkley, Dumbleton, Heytesbury, Kilvington, what makes him a good example for the reception of English philosophers in Central Europe.

In my talk (in German), I shall present and discuss those eleven approaches to solve the Liar-paradox in Helmold’s text.top

Joël Biard: Le calcul des variations de qualités et la physique aristotélicienne selon Blaise de Parme

Blasius of Parma deals with the intension and the remission of forms, as well as with the concept of "latitude", which is related to it, in at least three texts: The Quaestiones de latitudinibus formarum, the Questio disputata de intensione et remission formarum and the question 10 on the book V of Physics. I will focus on the last two texts. Here, at the limit of physics and metaphysics, Blasius of Parma discusses several theses on the ontological status of the qualities and their relationship to the subject via the question of their intensification or remission. At the same time, he takes advantage of the calculation procedures that have been under discussion since the beginning of the 14th century, accentuating, in the wake of Jean Buridan, the purely operational character of the implemented concepts.
This talk will be given in french.
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Daniel A. Di Liscia: The Geometrisation of Theology and the Metaphysics in the tradition de perfectione specierum

The fact that the tradition of the calculators provided a quantified view of Aristotle is well known, though not in all its breadth or depth. While the vast majority of studies conducted so far has focused on the theory of motion, i.e. on the reinterpretation of the Physics, parts of De caelo, De generatione et corruptione and parallel passages in other works, in my talk I shall emphasise one of the new readings of the Metaphysics that the calculators tradition made possible. Starting from a theological background and first presenting the fundamental text on the subject under discussion, the Tractatus de perfectione specierum by Jacobus de Napoli, I shall offer a general view of some of the central issues regarding the widespread problem of “the great chain of being” in France, Germany and Italy. As we will see, one of the solutions to this problem consisted in the attempt of a thoroughgoing geometrisation of the set of all the entities in the universe according to a gradual system of perfection. I would like to show, however, that this attempt toward geometrisation is not based on pure geometry, but rather on a number of concepts and principles which were generated specifically within the tradition of calculators and which, as a matter of fact, emerged as a critical development within its own ranks. Finally, we will also examine the reaction of a ‘good Peripatetic’ of the Renaissance against these new developments toward a quantifying interpretation of Aristotle’s Metaphysics.top

Elżbieta Jung: The New Interpretation of Aristotle. Richard Kilvington, Thomas Bradwardine and The New Rule of Motion

The founders of the School of Oxford Calculators, i.e. Richard Kilvington and Thomas Bradwardine presented their new theory of motion respectively in commentary to Aristotle’s Physics and in Tractatus de proportionibus velocitatum in motibus. My long lasting study of these works shows that Bradwardine was a beneficiary from Kilvington’s works, and he successfully later developed his ideas. That is why, in this paper I would like first to focus on dispersed tradition of Kilvington’s Physics commentary; secondly I will describe mutual dependencies of their works, and finally I will present the new interpretation of Aristotelean “laws of motion”. These threefold aim will nicely show when, why and how the story begun.top

Henrique Leitão: Mathematics and Motion in Alvarus Thomas' Liber de triplici motu (1509)

Alvarus Thomas’ Liber de triplici motu (Paris, 1509) is one of the finest examples of the later phase of the calculatores tradition. An ambitious but somewhat daunting work, the book is still comparatively unknown by historians, despite being frequently mentioned. In recent years, however, new studies have provided elements that shed new light on the book and confirm its influence. The objective of this presentation is to look into the Liber de triplici motu in some detail in order to better characterize the shape of the calculatores tradition at the beginning of the sixteenth century, and, more specifically, within the context of the university of Paris. I will describe the structure and main goals of the book, the contents and form of presentation, and will analyse selected portions of the text, related to the theory of motion. I will also comment on its sources and its influence.top

Edit A. Lukács: The Mathematical Theory of Proportions in Thomas Bradwardine´s De causa Dei, Book I

In De causa Dei I, 1, corollary 40, Thomas Bradwardine aligns different, mathematically sophisticated arguments in favor of the temporal beginning and end of the world. One of these arguments (per viam motus localis) employs the mathematical theory of proportions. In this talk, I will explore the theological use of the new science of motion in focusing on Bradwardine’s report of the Aristotelian account for motion, and compare it with the Tractatus de proportionibus.top

José Meirinhos: The Falsigraphus in Thomas Bradwardine’s De continuo

Thomas Bradwardine’s De continuo (c. 1328-35) follows the model of Euclid’s works, starting from 24 definitiones and 9 suppositiones, from which, in 151 conclusiones, he removes any chance of allowing that the continuum is made up of indivisibles. Besides relying on geometry and physics to assert the impossibility of indivisibles in both geometry and mathematics (conclusiones 1-34), Bradwardine also ranges through the various sciences so as to ascertain the impossibility of the continuum being made up of a finite number of indivisibles (conclusiones 57-114). Moreover, he was intent on showing that it would be impossible for the continuum to be made up of indivisibilia, even if these were immediately connected (conclusiones 35-56) or in infinite number (conclusiones 115-151). The arguments do not follow the usual form of questions or disputations. Preferring the procedures of geometry and mathematics, Bradwardine avails himself of authorities like Aristotle and Euclid, but also Archimedes and Boethius, or even opposing the opinions of some contemporaries like Walter Chatton or Henri of Harclay. In addition to innovation in the literary form, Bradwardine innovates also in argument, in order to prove the falsehood of the idea that the continuum is made up of indivisibles. In this talk we examine the use, in at least 15 cases, of the figure of the falsigraphus, to whom are attributed arguments painstakingly refuted by Bradwardine, pointing out their self-contradictions or absurd consequences. Contrived arguer or disguise for an unnamed author, the falsigraphus has a long tradition in geometry works. In Bradwardine, the falsigraphus seems to be a fiction, bringing back into the core of argument procedures of refutation and falsification we find in scholastic disputations.top

Richard Oosterhoff: How Humanists Repurposed the Calculatores in Renaissance Paris

Jacques Lefèvre d’Étaples (c. 1455-1536), normally a model of urbane virtue, reserved his harshest words for the calculatores and those who followed them. This fits his reputation today as a founding figure of French humanism, well connected with eminences such as Giovanni Pico della Mirandola, Marsilio Ficino, and Erasmus. But Lefèvre was a creature of the university, and worked with a generation of students to reimagine the entire university curriculum—with a particular focus on mathematics. (As the editor of the Opera of Cusanus published in 1514, Lefèvre praised mathematics for its philosophical and theological value.) At the beginning of that project Lefèvre wrote a set of dialogues on physics that deploy precisely those mathematical techniques associated with the calculatores. I shall argue that this was by no means inconsistent. Indeed, the genre of dialogue aimed a critique at what were thought to be the calculatores’ vicious habits, while repurposing what was useful. Furthermore, I shall suggest that this approach helped found a tradition in Paris of using mathematics to make physical claims, among Lefèvre’s followers such as Charles de Bovelles and Pedro Ciruelo.top

Mischa von Perger: Albert of Saxony and the idea of indivisible points of space and time

In his Sophismata Albert of Saxony sometimes employs mathematical imagination in order to determine the truth value of propositions in which the words “infinite”, “begins to …” or “stops doing something” occur. In these cases imagination works counterfactually and thereby makes long chains of arguments unnecessary. So imagination is given a dialectical function. Albert refers directly to the relevant sophisms of Richard Kilvington, who ranges among the Oxford calculators. Therefore we have to consider how Albert might have been influenced by Kilvington’s way of imagining points of space and time.top

Robert Podkoński: The Opuscula de motu ascribed to Richard Swineshead – the testimony of the ongoing development of the Oxford Calculators’s science of motion

The presentation concerns two short treatises De motu preserved in the codices: MS. Cambridge, Peterhouse 499/268 and MS. Seville, Biblioteca Colombana 7-7-29, that, thanks to their explicits are ascribed to Richard Swineshead – the Calculator. Both these texts seem to be succesive draft versions of the same work that ultimately become treatise XIV of his Liber calculationum: De motu locali. At the same time, they exhibit close analogues in William Heytesbury’s De tribus predicamentis. The presentation, however, focuses on the most intriguing, and even surprising features of these treatises that suggest not only the development of the theory of local motion in Swineshead’s own views, as well as in the Oxford Calculators’s school.top

Stephen Read: The Calculators on the Insolubles: Bradwardine, Kilvington, Swyneshed, Heytesbury and Dumbleton

The most exciting and innovative period in the discussion of the logical paradoxes before the twentieth century occurred in the second quarter of the fourteenth in Oxford, and at its heart were many of the Calculators. It was prompted by Thomas Bradwardine's iconoclastic ideas about the insolubles in the early 1320s. Framed largely within the context of the theory of obligations, it was continued by Richard Kilvington, Roger Swyneshed, William Heytesbury and John Dumbleton, each responding to Bradwardine's analysis, particularly his idea that propositions had additional hidden and implicit meanings, in different ways. Kilvington identified an equivocation in what was said; Swyneshed preferred to modify the account of truth rather than signification; Heytesbury exploited the respondent's role in obligational dialogues to avoid Bradwardine's tendentious closure postulate on signification; and Dumbleton relied on other constraints on signification to give new life to an old account of insolubles that Bradwardine had summarily dismissed. The talk will focus on the central thesis of each thinker's response to the insolubles and their interaction.top

Sabine Rommevaux-Tani: L'étude du mouvement local dans le De sex inconvenientibus: un exemple d'héritage des Calculatores

The anonymous treatise De sex inconvenientibus, written in the 1330s, is structured around four main questions, which ask how to determine the speed of generation, alteration, increase and local movement. Each of these main questions has three auxiliary questions. Concerning the local movement, the author seeks in the first auxiliary question to determine the cause of the acceleration of the fall of a heavy body, in the second he wonders if the movement of a sphere depends on a point or on a space, and finally, in the third he proposes several demonstrations of the mean speed theorem. We will analyze each of these questions, placing the arguments of the anonymous author in the context of Aristotelian physics as reworked by the Oxford Calculators.
This talk will be given in french.top

Magali Roques: Ockham and Bradwardine on propositions de incipit et desinit

In my paper, I intend to discuss Lauge Nielsen’s interpretation of Ockham’s and Bradwardine’s positions on incipit- and desinit- propositions in his 1982 paper “Thome Bradwardini Angli de incipit et desinit,” ed. and introd. L. A. Nielsen, Cahiers de l’Institut du Moyen Âge Grec et Latin 42, p. 1-83. I will suggest that Ockham does not reply to Bradwardine’s criticism in the Sum of Logic, II, 19 and that in this chapter he does not offer a solution that is incompatible with the one he proposes in the Sum of Logic, I, 75. At best, we may think of Ockham's account in SL II, 19 as a revision and a deepening of his former account. As a consequence, I will argue that Nielsen has no good argument to support his claim that the treatise he attributes to Bradwardine should be dated to 1323. I will conclude that the evidence available is too tenuous to attribute the treatise to Bradwardine with certainty. Since two of the four manuscripts attribute it to Thomas Maulfelt, the possibility that the treatise was written by Thomas Maulfelt should not be excluded.top

Fabio Seller: The “rules” of motus difformis in Angelo da Fossambruno’s De tribus praedicamentis Hentisberi

The work of Angelo da Fossambruno, as a magister logicae, at the Universities of Bologna and Padua between the end of the XIV and the beginning of the XV century, represents one of the main pathways through which English logic spreads into Italian academic learning. His commentary on Heytesbury’s De tribus praedicamentis (written when he was teaching in Bologna) consists in a carefully reasoned analysis of different types of motions, investigated in the framework provided by the canonical standards of English logic. This commentary clearly shows the influence of Bradwardine and Swineshead although these authors are never explicitly quoted by name. My essay aims at offering an account of the discussion about motus difformis, where the author, through the exemplification of several case studies, applies the rules established by Bradwardine. Apparently, Angelo can be placed among the authors who were inclined to adopt a “terministic” solution to the problem of variation of intensities, which aroused the reaction of Aristotelian schoolmen, committed to a more traditional metaphysical approach.top

Edith D. Sylla: Leibniz, the Liber Calculationum, and Mathematical Physics

Unlike many of the reformers of seventeenth-century science, Gottfried Wilhelm Leibniz adopted a conciliatory attitude toward scholastic Aristotelians. Early in his life Leibniz somehow learned about Richard Swineshead’s Liber Calculationum and proposed to republish the book in a series of reprints of treasures (cimelia) of early printing. On a trip to Italy in 1689 he managed to see, briefly, a copy of the book at the monastery of San Marco in Florence. Ultimately, after 1700, he persuaded Pinsson des Rolles to make a hand copy of the Venice, 1520, printed edition, which copy remains today in the Niedersächische Landesbibliothek. According to Leibniz, Swineshead had begun to do mathematics within a scholastic framework. How does the work of the Oxford Calculators appear if it is seen from the perspective of Leibniz rather than Galileo? The Calculatores introduced mathematics into natural philosophy, as Leibniz pointed out, and it did not presuppose Platonism.top

Aníbal Szapiro: The Calculators' Tradition and the Geometrisation of Light in Oresme's De visione stellarum

In De Visione Stellarum, a treatise that remained anonymous until the 1960s, Nicole Oresme analyzes whether the stars are seen where they are. To answer this, he jointly considers astronomical and optical issues, with special attention to atmospheric refraction. In doing this he deals with the problem of ignoring if there is a surface separating air and ether or if, instead of that, there is change without discontinuity. Facing it he establishes a relation between a uniformly difform density and a surface separating two media with given densities that bears resemblances to the Merton Rule. In this work, I present the features of this likely application of the Merton Rule to atmospheric refraction and I argue that its study contributes to the understanding of the role played by geometry and by the Calculators in medieval studies of light.top

Mark Thakkar: Wyclif and the Oxford Calculators

It is well known that the influence of the Oxford calculators lasted well beyond their heyday in the second quarter of the 14th century. Research on the subsequent history of the calculatorial tradition has, however, tended to focus on developments in Italy in the 15th and 16th centuries. My talk will instead explore the works of John Wyclif, and especially his Three Treatises on Proofs of Propositions, which I am currently re-editing. In the third of these treatises, Wyclif – who was educated at Oxford in the 1350s – complains that he “once sweated hard for no good reason” over certain calculatorial problems involving continuous condensation and rarefaction that are impossible according to the atomism he now favours. His recurrent preoccupation with such matters is not confined to his logical works, though; perhaps surprisingly, it can be felt as far afield as his Treatise on the Incarnation of the Word.top

Cecilia Trifogli: Thomas Wylton and Walter Burley on the Incipit and Desinit of an Instant of Time

Thomas Wylton (died ca. 1327) and Walter Burley (ca. 1275-after 1344) both dealt extensively with the so-called incipit and desinit problem, that is, the problem of assigning first and last instants of being to various kind of entities. In my talk I will present how Wylton and Burley apply their general accounts of the incipit and desinit to solve a puzzle raised by Aristotle about the ceasing to be of the present instant of time. I shall show that, although the general accounts of the incipit and desinit problem provided by the two philosophers are very similar, they adopt different strategies to solve the Aristotelian puzzle about the instant of time.