Aristotle: Metaphysics Mu

In the first argument of Metaphysics Μ.2 against the Platonist introduction of separate mathematical objects, Aristotle purports to show that positing separate geometrical objects to explain geometrical facts generates an ‘absurd accumulation’ of geometrical objects. Interpretations of the argument have varied widely. I distinguish between two types of interpretation, corrective and non-corrective interpretations. Here I defend a new, and more systematic, non-corrective interpretation that takes the argument as a serious and very interesting challenge to the Platonist.

The opening argument in the Metaphysics M.2 series targeting separate mathematical objects has been dismissed as flawed and half-hearted. Yet it makes a strong case for a point that is central to Aristotle’s broader critique of Platonist views: if we posit distinct substances to explain the properties of sensible objects, we become committed to an embarrassingly prodigious ontology. There is also something to be learned from the argument about Aristotle’s own criteria for a theory of mathematical objects. I hope to (...) persuade readers of Metaphysics M.2 that Aristotle is a more thoughtful critic than he is often taken to be. (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)

Do Ideas exist and can we prove it ? Do proofs of their existence have all the same value or not ? Aristotle addresses these issues in two famous documents of the controversy that pitted supporters of the theory of Forms against its opponents within Plato’s Academy : his lost work, quoted by Alexander of Aphrodisias by the title of Peri Ideon, and the lengthy thrust against Ideas that can be read, with some minor variations, in books A, chapter 9, (...) and M, chapter 4, of his Metaphysics. As we only have fragments of the first, the second being laconic and little more than a summary, there has been much speculation about the exact number and nature of the arguments for and against the Forms. Since the pioneering works of Léon Robin, Paul Wilpert et Harold Cherniss, one problem in particular has attracted the attention of specialists : what arguments does Aristotle accuse of either producing Ideas on relatives for one or of dragging in the « Third Man » for another ? Why does he consider these arguments to be more rigorous than the others ? If we are not dealing with the same arguments, how can these be more or better argued than those mentioned by Aristotle in the same breath, namely the arguments Plato’s followers took from the sciences, the one over many and the thought about things that have perished ? Through detailed analysis of texts from the Corpus Aristotelicum and Alexander’s commentary on Aristotle’s Metaphysics, Rationes ex machina develops a new interpretation of the controversial file of the akribesteroi tôn logôn as well as a micrological solution to the puzzle that has come to be a sort of compulsory figure of the exegesis of the Aristotelian criticism of « Plato’s Ideas ». (shrink)