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.
Paul Natorp (2004) writes that the first argument ‘deserves to be paraded in all its foolishness as one of the crassest pieces of evidence for the naivety with which Aristotle tried to make clear to himself the being-in-itself of the ideas’ (384) and Julia Annas (1976) objects that the argument makes ‘anti-platonist assumptions’ (141) and that Aristotle characterizes his opponents’ view somewhat crudely. Ross attributes an asymmetry in the argument to Aristotle’s ‘carelessness’ (1924 413). Hippocrates G. Apostle 1952 offers a detailed reconstruction of the argument but his version produces many more sets of ideal mathematical objects than Aristotle claims to have produced in his final tally at 1076b29-33. (Apostle’s version generates seven sets of ideal lines and fifteen sets of ideal points while Aristotle only claims to have generated four sets of ideal lines and five sets of ideal points.) Pettigrew 2009 discusses the argument in passing (242) but does not explain Aristotle’s difficult claims about the priority relations between the various ideal objects.
Like Annas Ross and others Crubellier199492takes ἀσύνθετος to mean ‘incomposite’ and so he takes Aristotle to be comparing one ideal geometrical object of n dimensions to prior geometrical objects of < n dimensions (e.g. a solid with prior planes). But these are not the objects Aristotle says he is comparing here with respect to their priority. He is comparing two kinds of ideal geometrical object both of n dimensions to one another.
See for example ProclusIn primum Euclidis elementorum librum commentarii246.10-12 Friedlein: πολλὰ γὰρ ἀσύνθετα µὲν ὄντα οὐκ ἀναλύεται συντεθέντα δὲ µόνως εὑοδίαν παρέχει πρὸς τὴν ἐπὶ τὰς ἀρχὰς ἀνάλυσιν (‘For often things left uncompounded do not lend themselves to analysis and only when put together provide an easy way of getting back to first principles’; trans. Morrow 1970 my emphasis). Here ἀσύνθετος2 is being paired and contrasted with σύγκειµαι2.