Aristotle’s intermediates and Xenocrates’ mathematicals

This paper investigates the identity and function of τὰ μεταξύ in Aristotle and the Early Academy by focussing primarily on Aristotle’s criticisms of Xenocrates of Chalcedon, the third scholarch of Plato’s Academy and Aristotle’s direct competitor. It argues that a number of passages in Aristotle’s Metaphysics (at Β 2, Μ 1-2, and Κ 12) are chiefly directed at Xenocrates as a proponent of theories of mathematical intermediates, despite the fact that Aristotle does not mention Xenocrates there. Aristotle complains that the advocates for mathematical intermediates produce theories that are ontologically and epistemologically inefficient (related to, but not confined by, the “Uniqueness Problem”); that their so-called “intermediates” feature properties opposed to those of the forms on which those intermediates are thought to depend; and that what is μεταξύ must be between objects of a different genus. In all three cases, Aristotle’s criticisms are shown to be reactions to the metaphysics and cosmology of Xenocrates, especially his doctrine of the intermediary demonic isosceles triangle souls. Xenocrates emerges as a prime candidate for Aristotle’s critique of Platonist τὰ μεταξύ theories – equal to, and possibly even exceeding, Plato as target.