In the mid-first century BC Geminus of Rhodes, a scientist and philosopher close to Posidonius, composed a comprehensive Theory of Mathematical Sciences, in the surviving fragments of which the numerous characters are referred to plainly by name, with some of them being namesakes of other, more well-known mathematicians and philosophers. This paper tries to set apart the namesakes of Geminus, of which there are four in his fragments: Theodorus, Hippias, Oenopides, and Menaechmus.

In this paper, we examine Sextus Empiricus' treatise Against the geometers . We first set this treatise in the overall context of the sceptic's polemics against the liberal arts. After a discussion of Sextus' attitude to the quadrivium , we discuss the structure, the sources and the target of the Against the geometers . It appears that Euclid is not Sextus' source, and neither he, nor the professional geometers, seem to be Sextus' main targets. Of course, Sextus never really makes (...) clear his precise target, but his attacks are rather directed against geometry as a means of modelling the physical world, thus ruining the support geometry was intended to bring to the physical part of dogmatic philosophy. (shrink)

L’article met en lumière la continuité intellectuelle de l’Académie à propos d’une question précise, les rapports entre philosophie et géométrie. On soutient d’abord que, dans les livres VI-VII de la République, Platon ne cherche pas à réformer les pratiques des géomètres mais identifie les contraintes incontournables de leurs raisonnements (constructions, hypothèses), qui constituent et limitent leur objectivité. On montre ensuite que cette analyse constitue le cadre des réflexions académiciennes ultérieures sur la géométrie. Speusippe reprend et développe l’analyse platonicienne des constructions (...) géométriques, qui est appliquée (de Speusippe à Crantor) à la cosmologie et même à la doctrine des principes, afin d’expliquer comment elles saisissent leurs objets éternels depuis le devenir, de manière indirecte. Si la Nouvelle Académie d’Arcésilas et de Carnéade critique les mathématiques, on fait l’hypothèse qu’il s’agissait pour elle – dans le contexte des débats philosophiques hellénistiques sur les usages des mathématiques –, de rappeler les limites de leur objectivité contre toute idéalisation dogmatique de la méthode géométrique. (shrink)

Dans cet article, nous analysons le passage dit « mathématique » de l’Épinomis. Dans le programme de formation proposé pour les futurs membres du conseil vespéral de vigilance (990c5-991b4), certains interprètes modernes ont cru voir un témoignage capital pour l’histoire des mathématiques grecques anciennes portant sur la question de l’irrationalité. L’analyse du lexique et du mode de composition du texte – un collage maladroit d’expressions reprises aux loci mathematici platoniciens –, la confrontation avec la littérature mathématique conservée et ce que (...) l’on sait de l’histoire de cette discipline avant Euclide montrent que cette interprétation, pourtant largement répandue, est sans fon­dement. En revanche, il est possible de reconstituer une intention philosophique compréhensible et mieux à même de rendre compte de certaines particularités du passage en consultant les commentateurs anciens (Nicomaque de Gérase, Théon de Smyrne, Jamblique de Chalcis, Proclus de Lycie). En privilégiant le passage conclusif sur le « lien » des mathématiques (991d8-992a1), ils identi­fiaient une tout autre question : celle de l’unité des mathématiques, à partir de laquelle le passage mathématique prenait à leurs yeux tout son sens. (shrink)

ArgumentThis article is the sequel to an article published in the previous issue ofScience in Contextthat dealt with homeomeric lines (Acerbi 2010). The present article deals with foundational issues in Greek mathematics. It considers two key characters in the study of mathematical homeomery, namely, Apollonius and Geminus, and analyzes in detail their approaches to foundational themes as they are attested in ancient sources. The main historiographical result of this paper is to show thatthere wasa well-establishedmathematicalfield of discourse in “foundations of (...) mathematics,” a fact that is by no means obvious. The paper argues that the authors involved in this field of discourse set up a variety of philosophical, scholarly, and mathematical tools that they used in developing their investigations. (shrink)

: The Hellenistic reception of Babylonian horoscopic astrology gave rise to the question of what the planets really do and whether astrology is a science. This question in turn became one of defining the Greco-Latin science of astronomy, a project that took Aristotle's views as a starting-point. Thus, I concentrate on one aspect of the various definitions of astronomy proposed in Hellenistic times, their demarcation of astronomy and physical theory. I explicate the account offered by Geminus and its subordination of (...) astronomy to arguments made in physical theory about what really is the case. I then show how Ptolemy treats the same topic but maintains that this science is sufficient on its own to determine the realia it studies. In this way, I identify two moments in an obvious process of intellectual change that had profound consequences for the history of astronomy and cosmology over the next 1500 years. My hope is that this will advance our understanding of the reception of horoscopic astrology in Hellenistic times and also serve to locate Ptolemy more fully in his intellectual context. (shrink)