Newton's Principia begins with eight formal definitions and a scholium, the so-called scholium on space and time. Despite a history of misinterpretation, scholars now largely agree that the purpose of the scholium is to establish and defend the de fi nitions of key concepts. There is no consensus, however, on how those definitions differ in kind from the Principia's formal definitions and why they are set-off in a scholium. The purpose of the present essay is to shed light on the (...) scholium by focusing on Newton's notion and use of de fi nition. The resulting view is developmental. I argue that when Newton first wrote the Principia, he viewed the scholium's definitions as items of “natural philosophy.” By the time of the third edition, however, he came to view their methodological status differently; he viewed them as belonging to the more qualified “manner of geometers.” I explicate the two methods of natural inquiry and draw out their implications for Newton's account of space. (shrink)

According to the standard and largely traditional interpretation, Aristotle’s conception of nous, at least as it occurs in the Posterior Analytics, is geared against a certain set of skeptical worries about the possibility of scientific knowledge, and ultimately of the knowledge of Aristotelian first principles. On this view, Aristotle introduces nous as an intuitive faculty that grasps the first principles once and for all as true in such a way that it does not leave any room for the skeptic to (...) press his skeptical point any further. Thus the tradi- tional interpretation views Aristotelian nous as having an internalist justificatory role in Aristotelian epistemology. In contrast, a minority (empiricist) view that has emerged recently holds the same internalist justificatory view of nous but rejects its internally certifiable infallibility by stressing the connection between nous and Aristotelian induction. I argue that both approaches are flawed in that Aristotle’s project in the Posterior Analytics is not to answer the skeptic on internalist justificatory grounds, but rather lay out a largely externalist explication of scientific knowledge, i.e. what scientific knowledge consists in, without worrying as to whether we can ever show the skeptic to his satisfaction that we do ever possess knowledge so defined. (shrink)

This article focusses on the generality of the entities involved in a geometric proof of the kind found in ancient Greek treatises: it shows that the standard modern translation of Greek mathematical propositions falsifies crucial syntactical elements, and employs an incorrect conception of the denotative letters in a Greek geometric proof; epigraphic evidence is adduced to show that these denotative letters are ‘letter-labels’. On this basis, the article explores the consequences of seeing that a Greek mathematical proposition is fully general, (...) and the ontological commitments underlying the stylistic practice. (shrink)

Book 1 of Euclid's Elements opens with a set of unproved assumptions: definitions, postulates, and ‘common notions’. The common notions are general rules validating deductions that involve the relations of equality and congruence. The attested postulates are five in number, even if a part of the manuscript tradition adds a sixth, almost surely spurious, that in some manuscripts features as the ninth, and last, common notion. The postulates are called αἰτήματα both in the manuscripts of the Elements and in the (...) ancient exegetic tradition. It is not said, however, that this denomination is original, as it coincides with the nomen rei actae associated with the verbal form introducing the postulates themselves. Since antiquity it has been recognized that the postulates naturally split into two groups of quite different character: the first three are rules licensing basic constructions, the fourth and the fifth are assertions stating properties of particular geometric objects. I transcribe postulates 1–3 in the text established by Heiberg : ᾐτήσθω ἀπὸ παντὸς σημείου ἐπὶ πᾶν σημεῖον εὐθεῖαν γραμμὴν ἀγαγεῖνκαὶ πεπερασμένην εὐθεῖαν κατὰ τὸ συνεχὲς ἐπ᾽ εὐθείας ἐκβαλεῖνκαὶ παντὶ κέντρῳ καὶ διαστήματι κύκλον γράϕεσθαι. (shrink)

Dans un célèbre ouvrage surContemplation et Vie contemplative selon Platon, A.J. Festugière donnait de la dialectique platonicienne une interprétation selon laquelle celle-ci constituait une véritable expérience mystique possédant presque tous les traits de la contemplation chrétienne. La dialectique platonicienne y était présentée, surtout dans son acte final, comme une sorte d'extase, une union d'ordre mystique, un contact de l'âme perdue dans son objet, contact qui suscite en elle un sentiment qui dépasse tout l'ordre normal de la connaissance. Le Bien ou (...) l'Un en tant qu'objet dernier de l'ascension dialectique y était conçu comme quelque chose d'ineffable et d'indéfinissable. (shrink)

I argue that these inconsistencies in wording and practice reflect the existence of two distinct Aristotelian views of inquiry, one peculiar to the Posterior Analytics and the other put forward in the Physics and practiced in the Physics and in other treatises. Although the two views overlap to some degree (e.g. both regard a rudimentary understanding of the subject as an essential first stage), the view of the syllogism as the workhorse of scientific investigation and the related view of inquiry (...) as a search for the ‘missing middle terms’ turn out to be ideas peculiar to the Analytics. Conversely, the techniques of analysis and differentiation highlighted in the Physics account receive only cursory attention in the Analytics. However, when we consider the character of Aristotle’s own inquiries, on both scientific and philosophical topics, it becomes clear that it is the Physics rather than the Posterior Analytics that gives us Aristotle’s considered view the path to knowledge. (shrink)

With the systematic aim of clarifying the phenomenon sometimes described as “the intellectual apprehension of first principles,” Descartes’ first principle par excellence is interpreted before the historical backcloth of Aristotle’s Posterior Analytics. To begin with, three “faces” of the cogito are distinguished: (1) the proto-cogito (“I think”), (2) the cogito proper (“I think, therefore I am”), and (3) the cogito principle (“Whatever thinks, is”). There follows a detailed (though inevitably somewhat conjectural) reconstruction of the transition of the mind from (1) (...) via (3) to (2) and back again to (3). What emerges is, surprisingly, a non-circular, non-logical, and ultimately non-mysterious process by which first principles implicitly contained in a complex intuition are gradually rendered explicit (and, if abstract, grasped in their abstract universality). This process bears a striking family resemblance to that intuitive induction (“grasping the universal in the particular”) which Aristotle scholars have distinguished from empirical forms of induction. (shrink)

Although there are many questions to be asked about philosophy of mathematics, the fundamental questions to be asked will be questions about what the mathematical object is in view of being and what the mathematical reasoning is in view of knowledge. It is clear that other problems will develop in parallel within the framework of the answers to these questions. For this reason, when we approach Aristotle's philosophy of mathematics over these two basic problems, we come up with the concept (...) of abstraction. In our work, I will try to explain the mathematical abstraction that Aristotle has developed to understand mathematical philosophy. (shrink)

The article comments on the epistemological foundations of medieval Arabic science and philosophy, as presented in five earlier communications, and attempts to draw some guidelines for the study of its social history. At the very beginning the notion of "Islam" is discounted as a meaningful explanatory category for historical investigation. A first part then looks at the applied sciences and notes three major characteristics of their epistemological approach: they were functionalist and based on experience and observation. The second part looks (...) at the theoretical sciences and notes that their epistemology was based on geometrical as well as (b) syllogistic modes of proof, and that the particular approach of scientists and philosophers was determined by (c) whether the motivations of each were scientific or non-scientific (e.g., religious, dogmatic, eristic, etc.). A third part concludes that for medieval Arabic-writing scholars with a scientific motivation the epistemological bases of their work were common to scientists to all cultures, and that the causes of advance or decline of Arabic science and philosophy have to be sought in social factors. In this regard three factors are presented as instrumental: (a) the level of patronage and wealth in any given society within the medieval Islamic world, (b) the cultural attitude of ideological laissezfaire in Sunni Islam which neither imposed nor adopted authoritative solutions in scientific and philosophical issues, and (c) the temporal and local specificity of scientific activity that makes the social contextualization of each science necessary in historical work. (shrink)

Three distinctly different interpretations of Aristotle’s notion of a sullogismos in Prior Analytics can be traced: (1) a valid or invalid premise-conclusion argument (2) a single, logically true conditional proposition and (3) a cogent argumentation or deduction. Remarkably the three interpretations hold similar notions about the logical relationships among the sullogismoi. This is most apparent in their conflating three processes that Aristotle especially distinguishes: completion (A4-6)reduction(A7) and analysis (A45). Interpretive problems result from not sufficiently recognizing Aristotle’s remarkable degree of metalogical (...) sophistication to distinguish logical syntax from semantics and, thus, also from not grasping him to refine the deduction system of his underlying logic. While it is obvious that Aristotle most often uses ‘sullogimos’ to denote a valid argument of a certain kind, we show that at Prior Analytics A4-6, 7, 45 Aristotle specifically treats a sullogismos as an elemental argument pattern having only valid instances and that such a pattern then serves as a rule of deduction in his syllogistic logic. By extracting Aristotle’s understanding of three proof-theoretic processes, this paper provides new insight into what Aristotle thinks reasoning syllogistically is and, moreover, it resolves three problems in the most recent interpretation that takes a sullogismos to be a deduction. (shrink)

Although there are many questions to be asked about philosophy of mathematics, the fundamental questions to be asked will be questions about what the mathematical object is in view of being and what the mathematical reasoning is in view of knowledge. It is clear that other problems will develop in parallel within the framework of the answers to these questions. For this rea­ son, when we approach Aristotle's philosophy of mathematics over these two basic problems, we come up with the (...) concept of abstraction. In our work, I will try to explain the mathematical abstraction that Aristotle has developed to understand mathematical philosophy. (shrink)