When Aristotle argues at the Metaphysics Z.17, 1041b11–33 that a whole, which is not a heap, contains ‘something else’, i.e. the form, besides the elements, it is not clear whether or not the form is a proper part of the whole. I defend the claim that the form is not a proper part within the context of the relevant passage, since the whole is divided into elements, not into elements and the form. Different divisions determine different senses of ‘part’, and (...) thus the form is not a part in the same sense as the elements are parts. I object to Koslicki’s (2006) interpretation, according to which the form is a proper part along the elements in a single sense of ‘part’, although she insists that the form and the elements belong to different categories. I argue that Koslicki’s reading involves a category mistake, i.e. the conjunction of items that do not belong to the same category (Goldwater 2018). Since for Aristotle parthood presupposes some kind of similarity of parts, the conjunction of form and elements requires treating these items as somehow belonging to the same category, e.g. ‘being’, but no such category exists. (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)

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)