  Aristotle occasionally refers to mathematical objects as things by, in, from, or through removal (in different works Aristotle uses different expressions: ta aphairesei, ta en aphairesei, ta ex aphaireseôs, ta di’ aphaireseôs). It is also clear that this usage relates to logical discussions in the Topics of definitions where one can speak of adding a term or deleting a term from an expression and seeing what one gets as a result. Our principal task is to explain what this logical/psychological removal is and how it solves the four puzzles. Aristotle starts with the class of perceptible or physical magnitudes. The examination of these is a part of physics (cf. Physics iii.4). The ontological status of these does not concern him, but we may suppose that they consitute the category of quantities: the bodies, surfaces, edges, corners, places, and times, sounds, etc. of physical substances (Categories 6).