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)

Frege argues that number is so unlike the things we accept as properties of external objects that it cannot be such a property. In particular, (1) number is arbitrary in a way that qualities are not, and (2) number is not predicated of its subjects in the way that qualities are. Most Aristotle scholars suppose either that Frege has refuted Aristotle's number theory or that Aristotle avoids Frege's objections by not making numbers properties of external objects. This has led some (...) to conclude that Aristotle's accounts of arithmetical and geometrical objects differ substantially. I close this supposed gap by showing that Aristotle's arithmetical objects, like geometrical objects, are just certain sensible things qua certain properties they in fact possess. Specifically, numbers are pluralities qua quantitative or relational properties like ten units or ten. I show that this view is resistant to the Fregean concerns about arbitrariness and numerical predication. (shrink)

아리스토텔레스의 수학적 소박실재론에 따르면 수학적 대상은 감각적 대상의 속성으로서 존재한다. 이와 같은 아리스토텔레스의 수학적 소박실재론의 문제점 중 하나는 수학적 대상들은 다른 학문의 대상들과 달리 감각적 개별자들에 의해 완벽하게 예화 되어 있지 않다는 것이다. 감각적 대상들은 수학적 대상의 정의를 만족시키지 않기 때문이다. 이러한 문제점에도 불구하고 아리스토텔레스가 그의 수학적 소박실재론을 유지할 수 있었던 이유는 유독수스의 새로운 천문학 이론 덕택이었다. 유독수스는 각각 따로 공전하는 네 개의 천구로 이루어진 천체를 가정함으로써, 불규칙하며 불완전하게 보이는 행성들의 운동이 사실은 규칙적이며 완전한 기하학적 원을 그린다는 것을 수학적으로 증명하였다. (...) 그러나 그의 이론을 받아들이더라도 여전히 수학적 소박실재론이 정당화되지는 못한다. 천체와 행성들의 운동이 모든 종류와 크기의 기하학적 대상을 예화 하지는 않기 때문이다. (shrink)