Descartes’s most extensive discussion of the law of refraction and the shape of the anaclastic lens is contained in Rule 8 of "Rules for the Direction of the Mind". Few reconstructions of Descartes’s discovery of the law of refraction take Rule 8 as their basis. In Rule 8, Descartes denies that the law of refraction can be discovered by purely mathematical means, and he requires that the law of refraction be deduced from physical principles about natural power or force, the (...) nature of the action of light, and the behavior of light rays in a variety of transparent media. For over a century, however, there has been broad agreement that Descartes discovered the law of refraction by purely mathematical means, and that he only later provided the relevant physical rationale (via comparisons or analogies) in Dioptrics II. I execute each step in Descartes’s proposed deduction of the law of refraction and the shape of the anaclastic lens in Rule 8 and concretely show how Descartes could have discovered the law of refraction and the shape of the anaclastic by its means. Rule 8, I argue, reflects Descartes’s actual path to the discovery of the law of refraction and the shape of the anaclastic lens. (shrink)

Descartes’ Rules for the direction of the mind presents us with a theory of knowledge in which imagination, considered as an “aid” for the intellect, plays a key role. This function of schematization, which strongly resembles key features of Proclus’ philosophy of mathematics, is in full accordance with Descartes’ mathematical practice in later works such as La Géométrie from 1637. Although due to its reliance on a form of geometric intuition, it may sound obsolete, I would like to show that (...) this has strong echoes in contemporary philosophy of mathematics, in particular in the trend of the so called “philosophy of mathematical practice”. Indeed Ken Manders’ study on the Euclidean practice, along with Reviel Netz’s historical studies on ancient Greek Geometry, indicate that mathematical imagination can play a central role in mathematical knowledge as bearing specific forms of inference. Moreover, this role can be formalized into sound logical systems. One question of general epistemology is thus to understand this mysterious role of the imagination in reasoning and to assess its relevance for other mathematical practices. Drawing from Edwin Hutchins’ study of “material anchors” in human reasoning, I would like to show that Descartes’ epistemology of mathematics may prove to be a helpful resource in the analysis of mathematical knowledge. (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)

This paper opens by reviewing Aristotle’s conception of the natural slave and then familiar treatments of the internal conflict between the ruling and subject parts of the soul in Aristotle and Plato; I highlight especially the figurative uses of slavery and servitude when discussing such problems pertaining to incontinence and vice—viz., being a ‘slave’ to the passions. Turning to Campanella, features of the City of the Sun pertaining to slavery are examined: in sketching his ideal city, Campanella both rejects Aristotle’s (...) natural slave and is critical of the European institutions of slavery. The fact that slavery has no place in the City of the Sun takes on added significance when complemented with Campanella’s remarks on dominion and servitude in his Quaestiones de politiciis: I show that Campanella’s conception of slavery is intimately bound to sin, and consider whether his employment of the trope of slavery within the context of vice diverges from familiar usage. (shrink)

O caráter protocolar desempenhado pelas matemáticas na formulação do conceito cartesiano de ciência é amplamente difundido e frequentemente reinvocado na literatura especializada quando se trata de abordar a exigência apodítica inerente a este conceito. No entanto, pouco se explora o que a diversidade das disciplinas matemáticas bem como a relação entretida por elas permite trazer de elucidação à noção cartesiana de ciência. Nosso propósito consiste, aqui, em tomar posição quanto a um debate acerca do estatuto da álgebra e da geometria (...) como disciplinas que fixam os lineamentos da teoria cartesiana do conhecimento ao exibirem o que Descartes reputaria como a aplicação ideal do intelecto. Trata-se de insistir, por vias que assinalam uma decisão teórica em favor de uma concepção mais intelectualista e menos imaginativa da ciência, que não há hesitação quanto ao papel primitivo e fundacional da álgebra se comparada à geometria. (shrink)

Ever since its foundation in 1540, the Society of Jesus had had one mission—to restore order where Luther, Calvin and the other instigators of the Reformation had brought chaos. To stop the hemorrhage of believers, the Jesuits needed to form a united front. No signs of internal disagreement could to be shown to the outside world, lest the congregation lose its credibility. But in 1570s two prominent Jesuits, Cristophorus Clavius and Benito Perera, had engaged in a bitter controversy. The issue (...) at stake had apparently nothing to do with the values on which Ignazio of Loyola had built the Society of Jesus. And yet the dispute between Clavius and Perera was matter of concern for the entire Jesuit community. They were... (shrink)