Heller M, Woodin W.H. (eds.) Infinity. New research frontiers (CUP, 2011)(ISBN 1107003873)(O)(327s)_MAml_

22 infinity as a transformative concept in science and theologyTable 1.1ActualityPotentialityFiniteness ———— nonexistent,finite lineself-contradictoryInfinity nonexistent, ————self-contradictory infinite lineway? That is, how can it be conceived in a way that transcends the religious feelingsabout the infinite in Anaximander’s account?Aristotle discussed infinity in the framework of his philosophical distinction betweenpotentiality and actuality. Within this context, Aristotle said that the infinite cannotexist as an actual infinity. He said that the infinite is only a potential infinity, because it isthe possibility of endless action. 11 Thus, the infinite exists in the mode of potentiality 12as an endless approximation. 13 Aristotle associates the latter mainly with mathematicalprocedures, such as addition or division. 14 It can be demonstrated in a Table 1.1. Actualinfinity and potential finiteness are nonexistent, because they are self-contradictory.Thus, Aristotle’s major claim is that infinity only exists in a potential manner.However, increasing a quantum in the process of change within potentiality andactuality also includes time and movement. Therefore, Aristotle argued at the end ofbook III of his Physics (about motion) that time and motion are infinite. 15 However, itis not the case with thinking. Although thinking relates also to the infinity of movementand time, thinking itself is not infinite. 16On the other hand, Aristotle denied the possibility of an infinite space. The reasonis that Aristotle did not have a notion of space that would be void of matter. Instead ofspace, Aristotle had the notion of place, which is always associated with matter. Thus,in Aristotle’s philosophy, we find a finite world with respect to space.This finiteness with regard to the process of thinking has an important consequence.Given that God can be understood as the thinking of thinking, the (Aristotle 1997, Metaphysics, XII, 9, 1074b33–35), God also must be understood asfinite.Nearly simultaneously and in close agreement with Aristotle’s philosophical treatmentof the issue, the Greek mathematician Eudoxos of Knidos (410 [408]–355 [347]BC), a contemporary of Aristotle, invented the method of exhaustion (Frank 1955;Merlan 1960; Lasserre 1966). In this method, an infinitely repeatable mathematicalprocess is used to measure surface area. This process corresponds strikingly with11 “The unlimited, then, is the open possibility of taking more, however much you have already taken” (Aristotle1993, Physics III, 6, 207a6; elsewhere in Aristotle 2003, Metaphysics IX, 6, 1048b10–18).12 “It results that the unlimited potentiality exists” [ ] (Aristotle 1993,Physics III, 6, 206a18).13 “The illimitable, then, exists only the way just described – as an endless potentiality of approximation byreduction of intervals” (Aristotle 1993, Physics III, 6, 206b15).14 “It never exists as a thing, as a determined quantum does. In this sense, then, there is also illimitable potentialityof addition” (Aristotle 1993, Physics III, 6, 206b16).15 “Time and movement are indeed unlimited, but only as processes, and we cannot even suppose their successivestretches to exist” [ ] (Aristotle 1993, Physics III, 8, 208a20).16 “ de , , ” (Aristotle 1993,Physics III, 8, 208a20).