Although Aristotle is usually thought to deny the existence of physical objects with perfect geometric properties, J. Lear has argued that for Aristotle geometric properties can be perfectly instantiated in the physical world. In support of this thesis Lear has pointed mainly to
de An.
403a10- 6, where Aristotle seems to admit the existence of physical objects with so perfect geometric properties that the edge of one touches the spherical surface of the other at a point. In this paper I argue that
de An.
403a10- 6 does not commit Aristotle to the perfect instantiation of geometric properties in the physical world because the two objects assumed to touch each other at a point in this passage are not physical, as Lear takes it, but geometric: consequently,
de An.
403a10-6 cannot be taken as evidence that geometric properties are perfectly instantiated in physical objects, from which geometric objects are abstracted. In
Cael.
287b14-21, however, Aristotle notes that unlike the heaven a sphere made by a craftsman cannot be perfectly spherical and, in general, that no human artifact of whatever shape can be as geometrically perfect as the spherical heaven. This passage leaves no doubt that Aristotle denies the perfect instantiation of geometric properties in the sublunary region of his universe: some geometric properties are perfectly instantiated only in the superlunary region where the
aether
, the material of the heaven as well as of the celestial spheres that produce the apparent motions of each planet (the sun and the moon included), forms geometrically perfect spheres.
Geometric integrators and the Hamiltonian Monte Carlo method
