There is little agreement about Aristotle’s philosophy of geometry, partly due to the textual evidence and partly part to disagreement over what constitutes a plausible view. I keep separate the questions ‘What is Aristotle’s philosophy of geometry?’ and ‘Is Aristotle right?’, and consider the textual evidence in the context of Greek geometrical practice, and show that, for Aristotle, plane geometry is about properties of certain sensible objects—specifically, dimensional continuity—and certain properties possessed by actual and potential compass-and-straightedge drawings qua quantitative and (...) continuous. For their part, the objects of stereometry are potential sensible three-dimensional figures qua quantitative and continuous. (shrink)

In Book K of the Metaphysics Aristotle raises a problem about a very persistent concern of Greek philosophy, that of the relation between the one and the many , but in a rather peculiar context. He asks: ‘What on earth is it in virtùe of which mathematical magnitudes are one? It is reasonable that things around us [i.e. sensible things] be one in virtue of [their] ψνχ or part of their ψνχ, or something else; otherwise there is not one but (...) many, the thing is divided up. But [mathematical] objects are divisible and quantitative. What is it that makes them one and holds them together?’. (shrink)

This paper aims to reassess the philosophical puzzle of the “applicability of mathematics to physical sciences” as a misunderstood disciplinary interplay. If the border isolating mathematics from the empirical world is based on appropriate criteria, how does one explain the fruitfulness of its systematic crossings in recent centuries? An analysis of the evolution of the criteria used to separate mathematics from experimental sciences will shed some light on this question. In this respect, we will highlight the historical influence of three (...) major disciplinary paradigms. According to the Aristotelian classification of the sciences, the separation of mathematics from physics is based on their respective objects of study. The Baconian system distinguishes these sciences by the type of knowledge involved in each field. Finally, the Whewellian disciplinary layout categorises these disciplines by their respective methods. In this paper, we argue that the cascading effect of such disciplinary categorisations—based successively on ontological, epistemological, and heuristic criteria—resulted in a profound redefinition of the border between mathematics and physics, with the consequence of obscuring the foundations of the interplay between these fields. Our approach intends to put forward such a mechanism as a constitutive piece of the puzzle of the applicability of mathematics. (shrink)

I argue that intelligible matter, for Aristotle, is what makes mathematical objects quantities and divisible in their characteristic way. On this view, the intelligible matter of a magnitude is a sensible object just insofar as it has dimensional continuity, while that of a number is a plurality just insofar as it consists of indivisibles that measure it. This interpretation takes seriously Aristotle's claim that intelligible matter is the matter of mathematicals generally – not just of geometricals. I also show that (...) intelligible matter has the same meaning in all three places where it is explicitly invoked: Z.10, Z.11, and H.6. Since the H.6 passage involves a mathematical definition, this requires determining what the mathematician defines and how she defines it. I show that, as with natural scientific definitions, there must be a matterlike element in mathematical definitions. This element is not identical with, but rather refers to, intelligible matter. (shrink)

Aristotle explicitly speaks of intelligible matter in three passages only, all from theMetaphysics, in the context of the analysis of definition as the formula that expresses the essence:Metaph.Z10, 1036 a8-11;Metaph.Z11, 1037 a5;Metaph.H6, 1045 a34-36 and 45 b1. In the case of the occurrences of Z10 and Z11, there is almost unanimous consensus that Aristotle uses the expression in a technical way, to indicate the matter of that particular type of objects that are intelligible compounds, of which mathematical objects are paradigmatic (...) instances. By contrast, there is no agreement on how to understand the expression in the case of H6. Here, ‘intelligible matter’ would denote, as stated by some interpreters, the generic element of the definition. The aim of this paper is to show that ὕλη νοητή has the same technical meaning in all the three explicit occurrences in theMetaphisics. Contrary to the interpretation that there is a shift in meaning from Z10-11 to H6, my goal is to illustrate that in both contexts, Aristotle uses the expression ‘intelligible matter’ to designate the matter of mathematical objects insofar as they are paradigmatic intelligible compounds. (shrink)

At Aristotle,MetaphysicsE.1, 1026a14, Schwegler’s conjectural emendation of the manuscript reading ἀχώριστα to χωριστά has been widely adopted. The objects of physical science are therefore here ‘separate’, or ‘independently existent’. By contrast, the manuscripts make them ‘not separate’, construed by earlier commentators as dependent on matter. In this paper, I offer a new defense of the manuscript reading. I review past defenses based on the internal consistency of the chapter, explore where they have left supporters of the emendation unpersuaded, and attempt (...) to strengthen their appeal. I challenge Schwegler’s central case, developed by Ross and others, that the construction μὲν ἀλλ’ οὐκ demands an implausible ‘logical antithesis’ between inseparability and mobility. This is arguably the fundamental obstacle to the manuscript reading, and counterexamples have not to date convinced emenders. I offer a new, systematic review of Aristotle’s use of the phrase, including relevant cases from theRhetoric, to show how its usual meaning in Aristotle supports the transmitted text; I also reply to possible objections. Finally, I explore the implications of this defense for the classification of the sciences in E.1. (shrink)

아리스토텔레스는 『형이상학』 13권 2장에서 기하학적 대상은 실체적으로 존재할 수 없음을 증명한다. 플라톤주의자들은 기하학적 대상이 실체적으로 감각대상 안에 있거나, 감각대상과 떨어져서 존재한다고 주장하는 자들이다. 아리스토텔레스는 13권 3장에서 기하학적 대상은 질료적으로 감각대상 안에 존재한다고 주장한다. 필자는 ‘질료적으로’의 의미를 ‘부수적으로’와 ‘잠재적으로’로 이해한다. 기하학적 대상은 감각대상 안에 있지만, 실체적으로가 아니라 부수적으로 존재하는 것들이다. 기하학적 대상은 그 자체로 변화를 겪을 수 없다. 변화를 겪는 직접적인 주체는 감각대상이다. 이 감각대상이 분할되거나, 또 다른 감각대상과 결합할 때 기하학적 대상은 간접적으로 변화를 겪는다. 기하학적 대상의 잠재성은 지성에 의해 추상과정을 (...) 거쳐 기하학의 영역에서 현실화된다. 물리적인 영역에서 이것들은 현실적으로 존재하는 것들이 아니다. 기하학적 대상은 분할 불가능한 형상적 측면과 분할 가능한 질료적 측면을 모두 갖는다. 기하학적 대상이 갖는 질료적 측면은 기하학적 대상이 갖는 양적인 측면으로, 형상적 측면은 경계적 측면으로 드러난다. 기하학적 대상이 n차원이라면, 기하학적 대상의 경계는 n-1차원을 갖는다. 입체의 경계는 면이며, 면의 경계는 선, 선의 경계는 점이다. 아리스토텔레스에게는 3차원의 입체가 가장 완전하므로 입체 자체는 경계가 되지 않는다. 또한 면과 선은 질료적 측면과 경계적 측면을 모두 가지는 반면, 점 자체는 경계적 측면만 갖는다. 경계적 측면에 주목하면, 아리스토텔레스가 자연철학에서, 무한, 장소, 시간, 연속과 관련하여 제기한 중요 난점들을 해결하는데 실마리를 얻을 수 있다. (shrink)

Over time, various thematic classifications have been put forward to organize science into a coherent system of specialized areas of research. From an analysis of the historical evolution of the criteria used to distinguish the sciences from one another, I propose in this paper a quadripartite typology for the different thematic classification systems propounded by scholars throughout the centuries. Basically, I argue that the criteria used to differentiate the sciences have been alternately drawn from their respective subject matters, kinds of (...) knowledge, methods and aims. Then, I show that several reclassifications occurred in the thematic structure of science. Finally, I argue that such changes in the structure of learning displaced the modalities of contact between the objects, knowledge, methods and aims of the various branches of science, with the result of outlining reshaped intellectual territories conducive to the emergence of new areas of research. (shrink)

The aim of this paper is to focus on the reception of Plotinus’ concept of matter in the Renaissance philosopher Giordano Bruno and his early Italian dialogue De la causa, principio et uno. I argue that Bruno’s concept of materia in De la causa, principio et uno reflects Plotinus’ theory of intelligible matter in Ennead ii 4 [12] 2–5 as well as Plotinus’ positive view of the perceptible world in Enneads ii 9 [33] and iv 8 [6]. It is suggested (...) that Bruno interprets Plotinus as an ancient philosopher who supported the unity and homogeneity of matter. Giordano Bruno’s reception of Plotinus in De la Causa, principio et uno could enlighten contemporary discussions in Neoplatonic studies concerning the question of matter in the Enneads. (shrink)