RésuméEn expliquant la relation entre hypothèses et images dans l'analogie de la ligne du livre Vl de laRépubliquede Platon, je m'attarde d'abordsur l'élucidation platonicienne de la nature des mathématiques telle que la conçoit le mathématicien lui-même. Je poursuis avec une critique des interprétations traditionnelles de cette relation, qui partent de l'assomption douteuse que les mathématiques s'occupent des Formes platoniciennes. Pour formuler mon point de vue sur cette relation, j'exploite la notion de «structure». Je montre comment les «hypothèses» comme principes de (...) preuve peuvent déterminer les structures des «images» par lesquelles les structures intelligibles correspondantes des objets mathématiques en viennent à être connues. (shrink)

My goal here is to explore the relationship between pure and applied mathematics and then, eventually, to draw a few morals for both. In particular, I hope to show that this relationship has not been static, that the historical rise of pure mathematics has coincided with a gradual shift in our understanding of how mathematics works in application to the world. In some circles today, it is held that historical developments of this sort simply represent changes in fashion, or in (...) social arrangements, governments, power structures, or some such thing, but I resist the full force of this way of thinking, clinging to the old school notion that we have gradually learned more about the world over time, that our opinions on these matters have improved, and that seeing how we reached the point we now occupy may help us avoid falling back into old philosophies that are now no longer viable. In that spirit, it seems to me that once we focus on the general question of how mathematics relates to science, one observation is immediate: the march of the centuries has produced an amusing reversal of philosophical fortunes. Let me begin there. In the beginning, that is, in Plato, mathematical knowledge was sharply distinguished from ordinary perceptual belief about the world. According to Plato’s metaphysics, mathematics is the study of eternal and unchanging abstract Forms,1 while science is uncertain and changeable opinion about the world of mere becoming. Indeed, in Plato’s lights, of the two, only mathematics deserves to be called ‘knowledge’ at all! Of course if sense perception cannot give us knowledge, if mathematics is not about perceivable things, then Plato owes us an account of how we ordinary humans achieve this wonderful insight into the properties of the abstract world of Forms. Plato’s answer is that we do not actually acquire mathematical knowledge at all; rather, we recollect it from a time before birth, when our souls, unencumbered by physical bodies, were free to commune with the Forms, and not just the mathematical ones, either – also Truth, Beauty, Justice, the Good, and so on.2 Whatever appeal this position may have held for the ancient Greeks, it will not begin to satisfy a contemporary, scientifically minded philosopher.. (shrink)

Die Frage, ob die Gültigkeit alternativer begrifflicher Strukturen empirisch entscheidbar oder eine Sache der willkürlichen Festsetzung ist, wird, eingeschränkt auf den Fall der physikalischen Geometrie, diskutiert. Die erkenntnistheoretischen Komponenten der empirischen Bestimmung von metrischen und topologischen Eigenschaften des physikalischen Raumes werden in der neueren Wissenschaftstheorie verfolgt. In Anschluß an die Auseinandersetzung zwischen A. Grünbaum und H. Putnam wird eine Interpretation des semantischen Status des Kongruenzprädikates vorgeschlagen, die Schwierigkeiten im Verhältnis von Intension und Extension dieses Terms beseitigen soll. Bei der Konfrontation (...) von Folgerungen aus der Konventionalitätsthese mit der dynamischen Raumtheorie, wie sie im Anschluß an Clifford von J. A. Wheeler entwickelt wurde, treten Unvereinbarkeiten auf, die in der Gegensätzlichkeit von absoluter und relationaler Raumauffassung gründen. (shrink)

What is the function of logic in al-Kindī's corpus? What kind of relation does it have with mathematics? This article tackles these questions by examining al-Kindī's theory of categories as it was presented in his epistle On the Number of Aristotle's Books, from which we can learn about his special attitude towards Aristotle theory of categories and his interpretation, as well. Al-Kindī treats the Categories as a logical book, but in a manner different from that of the classical Aristotelian tradition. (...) He ascribes a special status to the categories Quantity and Quality, whereas the rest of the categories are thought to be no more than different combinations of these two categories with the category Substance. The discussion will pay special attention to the function of the categories of Quantity and Quality as mediators between logic and mathematics. (shrink)

This paper presents a new interpretation of the objects of dianoia in Plato’s divided line, contending that they are mental images of the Forms hypothesized by the dianoetic reasoner. The paper is divided into two parts. A survey of the contemporary debate over the identity of the objects of dianoia yields three criteria a successful interpretation should meet. Then, it is argued that the mental images interpretation, in addition to proving consistent with key passages in the middle books of the (...) Republic, better meets those criteria than do any of the three main positions. (shrink)