 This conception stands in opposition to that of ‘actual infinity’, which would result if some infinite processes could be completed, carried out ‘all at once’, as it were. If actual infinity were real, then one could have infinitely long bodies, infinitely long or infinitely small segments, the totality of natural numbers, an infinite number, infinitely many instants of time, etc. Aristotle rejected the notion of the infinite as a primordial substance, as we have encountered in Anaximander, and most of his discussion of the infinite takes place within a physical context, namely one relating to spatio-temporal features of reality. As a consequence, Aristotle’s discussion of the infinite fell squarely in what we have characterized as the ‘mathematical’ notion of infinity, where infinity applies first of all to magnitudes (continuous or discrete) and what is quantifiable (time, extension, numbers etc.). His Physics discusses the infinitely large, excluded because the world is finite; and the infinitely small, excluded because the division of matter can only be potentially infinite and thus finite at each stage, never reaching an infinitesimal quantity—one that is less than any finite quantity, while being something. The exclusion of the infinitely large also has as a consequence that Aristotle cannot allow a potential infinity by addition in an unqualified manner (for otherwise any finite extension could be added to itself sufficiently many times to become larger than the size of the world). Infinity by addition, then, is to be conceptualized as a sort of inverse operation to infinity by division which gives us the primary evidence for the existence of the potential infinite. This is the implicit force of the contrastive “but” in the following quote. Aristotle writes (our emphasis):