Contributed Papers

Greek Science and Its Perception

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Presenter 1

A Hamiltonian Interpretation of the Antikythera Mechanism

Alessandro Amabile

Università degli Studi di Napoli Federico II

Abstract

In March 1964 Richard Feynman held a lecture at California Institute of Technology about the motion of planets around the Sun. In his lecture Feynman uses a purely geometrical approach whose origin goes back to the work of Sir William Hamilton. Inspired by Feynman and Hamilton, we propose a new interpretation of Antikythera Mechanism (AM), the only surviving example of the ancient Greek art of building reduced models of the cosmos (sphairopoiia). If our interpretation is correct, the theory embedded in the AM was phenomenologically equivalent to newtonian mechanics. Moreover, new hints about the working of the lost planetary mechanisms are discussed, along with a conjecture about the Hellenistic origin of Ptolemy's astronomical models as they're presented in the Almagest.

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Presenter 2

Aristotle’s Wheel: A Peripatetic Paradox of Motion

Arthur Harris

University of Cambridge

Abstract

The twenty-fourth problem of the pseudo-Aristotelian Mechanical Problems, known as 'Aristotle's Wheel', challenged the minds of mathematicians and natural philosophers from the Renaissance to the eighteenth century and to this day has no universally accepted solution. The problem's intractability was once so notorious that it became the subject of a scholastic proverb: Rota Aristotelis magis torquet quo magis torquetur ('The more Aristotle's Wheel turns, the more it torments'). Nevertheless, no modern historical studies of the problem have been published since Israel Drabkin's pioneering work in the late 1930s (published in 1950 Osiris), which focussed on early modern solution attempts.
I offer a novel assessment of the problem in the context of ancient Greek empirical, mathematical and philosophical inquiry. I argue the problem presents a fiendishly sophisticated paradox that derives an absurd contradiction from plausible premises. Specifically, the paradox challenges the basic principles of mechanics as practiced by the early Peripatetics. This helps to account for its appearance near the end of a collection of Peripatetic mechanical explanations: the paradox dramatically threatens to upend the preceding project. The long and detailed solution offered in the Mechanical Problems involves an appeal to principles of natural science to dissolve what is presented as an essentially mathematical problem. It follows that scholars should be cautious in assessing the relation of the Mechanical Problems to Aristotle's methodological strictures in the Posterior Analytics, according to which mathematics can explain natural phenomena, but not vice versa. A reassessment of the status of mechanical inquiry, from an Aristotelian point of view, may be in order.

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Presenter 3

Conceptions of Science in the Later Islamic Intellectual History (1250-1550)

Kenan Tekin

Harvard University/Yalova University

Abstract

Later Islamic intellectual history (ca. 1250-1550), which witnessed the commentary and gloss tradition, has been neglected as being devoid of any interesting intellectual developments. In this paper, I will challenge this bias against the commentary and gloss literature of the period by looking at conceptions of science. It is well-known that the Arabic-Islamic philosophical thought owes much to the Greek heritage. The same can be said with regard to the notion of science. However, I hope to show that there were some interesting developments, and much fine grained discussions concerning the nature of science in the commentary and gloss tradition of the later periods. I will look at the prolegomena of a chain of commentaries and glosses on a base text in order to follow debates on the nature of science. Close reading of such texts shows that they contain some noteworthy developments and departures from the peripatetic notion of science. For one thing, the commentators and glossators not only acknowledged the Aristotelian notion that the absolute unknown cannot be sought, they also provided further nuances of what exactly is involved in seeking a thing, be it knowledge or otherwise. Jurjani (d. 1413), for instance, pointed out two necessary, and three customary things that are involved in seeking a science. Another noteworthy development in the literature I look at is the challenge to peripatetic notion that sciences essentially consist of three elements, that is subject matter, principles, and problems. Some of them, such as Qutb al-Din al-Razi (d. 1365), state that sciences essentially consist of problems, and the other two elements are added for convenience. This is an obvious criticism of the Aristotelian-Avicennian assertion on the issue.

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Presenter 4

Meanings of the Right Triangle in Ancient Geometry

John Z. Wee

University of Chicago

Abstract

Already in southern Iraq during the Old Babylonian Period (c. 20th to 16th centuries BCE), mathematicians employed the so-called 'Pythagorean Rule' to solve geometrical problems involving right-(angled) triangles. Famous manuscripts such as 'YBC 7289' and especially 'Plimpton 322' relied with precision on sets of 'Pythagorean triples'-a group of three numbers representing the width (w), the length (l), and the diagonal (d) of a given triangle. Under Greek influence in Babylonia during the Seleucid Period (c. 3rd to 1st centuries BCE), the historian Jens Høyrup (2002) suggested possible "innovations" in cuneiform mathematics, and observed the "preferential use of the identity {w-squared = (d + l) times (d − l)}" for right triangles.
The popular expression of the Rule today, using the formula {w-squared + l-squared = d-squared}, was likely also one of the better-known versions in Greco-Roman antiquity. Its proof was demonstrated by Euclid-who treated the triangle's sides not merely as numerical values raised to their second power, but as the borders of actual square areas-though later authors like Plutarch, Diogenes Laertius, and Athenaeus ascribed the Rule's invention to Pythagoras instead. Interestingly, Proclus' commentary on Euclid mentions alternative methods for deriving Pythagorean triples, attributed to Pythagoras and Plato, which appear to be anchored in models and imagery original to Babylonian mathematics.
Using algebraic operations, one can easily show that the different formulas for the Rule are fully reconcilable with each other. By paying close attention to phraseological choices in mathematical language and their underlying conceptual models, however, I explore the changing meanings of the right triangle over time, and argue that geometrical ideas-axiomatic and universal at first glance-can in fact be features of particular mathematical cultures.

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