This paper argues that Aristotle’s Metaphysics Z.3 deploys a reductio against the claim that ‘substances underlie by being the subjects of predication’, in order to demonstrate the need for a new explanation of how substances underlie. Z.13 and H.1 corroborate this reading: both allude to an argument originally contained in Z.3, but now lost from our text, that form, matter and compound ‘underlie’ in different ways. This helps explain some of Z’s peculiarities—and it avoids committing Aristotle to self-contradiction about whether (...) matter is substance, a claim denied in the reductio but endorsed elsewhere. (shrink)

Both literalism, the view that mathematical objects simply exist in the empirical world, and fictionalism, the view that mathematical objects do not exist but are rather harmless fictions, have been both ascribed to Aristotle. The ascription of literalism to Aristotle, however, commits Aristotle to the unattractive view that mathematics studies but a small fragment of the physical world; and there is evidence that Aristotle would deny the literalist position that mathematical objects are perceivable. The ascription of fictionalism also faces a (...) difficult challenge: there is evidence that Aristotle would deny the fictionalist position that mathematics is false. I argue that, in Aristotle's view, the fiction of mathematics is not to treat what does not exist as if existing but to treat mathematical objects with an ontological status they lack. This form of fictionalism is consistent with holding that mathematics is true. (shrink)