A ristotle on Intelligible Matter

In Metaphysics K, Aristotle talks of mathematical objects having matter: 'ows oG & &op7JueLe -Ls av rrOLctS ETLV E'nLaT?LRS To &avopTatX spi TIjS TWV aOLvrrxZv 'Xils (l059b14-16). There can be only one kind of matter that he has in mind here. Sensible matter is ruled out because this is the proper object of physics whereas, as he points out (1059b20-21), the matter of mathematical objects is the proper subject of first philosophy. Nor can prime matter be at issue here, since this is a purely limiting notion introduced to account for changes between contraries in sensible matter, such as generation and corruption. This leaves us with what Aristotle calls iSX'q voyrr noetic or 'intelligible' matter and in Metaphysics Z we are told explicitly that intelligible matter is present in the objects of mathematics: vih 8E' &' [v auOLL'r EaTLV i' b vorip, O1NTiq pV OtOV XtXXos xai ivXov xvi ani XI q V, VOi 8' ' 'V TOtLS OdToZS i'VrTpXovaa R I xLcI&1a, OLOV T'a ,LCtO7hLatXTLXa (1036a9-12). Insofar as this claim relates to geometry it has occasionally caused puzzlement, but there is a relatively straightforward solution to the puzzle. Insofar as it relates to arithmetic it is acutely problematic, but Aristotle nowhere even suggests that it does not cover the whole of mathematics. My main concern in this paper is to make sense of the idea of numbers having intelligible matter but, for reasons that I shall mention below, this requires that we first make clear what is involved in the doctrine that geometrical figures have intelligible matter.