In Aristotelian philosophy, quantity – together with substance, quality, and relation – is one of the four principal categories. Its relation to corporeal substance was a matter of dispute. Descartes’ assertion, which agrees with the Nominalists’ position, that quantity and the thing quantified differ only in reason (AT VIIIA 45, CSM I 226), was often referred to in theological condemnations of Cartesianism, on the grounds that it rendered transubstantiation unintelligible (Armogathe 1977).
Scholastic questions on quantity standardly distinguish intensive from extensive quantity. Intensive quantity is the quantity associated with qualities admitting of degree, such as heat. Descartes has little explicitly to say about it. Extensive quantity is the quantity associated with things that occupy or are located in space (temporal duration is also an extensive quantity). Continuous extensive quantity, or extension, is said to be in essence spatially distributed, divisible, and impenetrable (Suárez 1856, 26:99). An extended thing is said, following Ockham, to have partes extra partes (parts within parts), a phrase that occurs in the Fifth Objections (AT VII 337, CSM II 234) and in the correspondence with More (see, e.g., AT VII 270, CSMK 362). According to Suárez, quantity need not be actually extended, and so it need not actually occupy space; its essence is to be potentially extended (Des Chene 1996, 104). Descartes rejects that view (AT VIIIA 42, CSM I 224; AT X 447, CSM I 62).
Descartes introduces two innovations in his treatment of quantity. In the Rules, he enlarges the notion of dimension so as to include any manner in which a thing may be considered measurable (AT X 447, CSM I 62). In the Geometry, he redefines the multiplication of quantities so that the product of two lines, traditionally represented by area, is a line; similarly the product of three lines is again a line, not a volume (AT VI 370). That innovation allowed quartic and higher-degree powers to be treated on an equal footing with squares and cubes. In natural philosophy, on the other hand, Descartes’ conception of quantity remains geometrical: the product of two quantities is represented by a rectangle, not a line. Relations of quantities are construed in terms of geometrical proportions, not symbolically (Des Chene 1996, 302).
See also Body, Divisibility, Extension, Geometry