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

from to potential infinity: aristotle 21as far as , and it is the central notion for understanding the as being finiteand ordered. The is irrational, so it is not accessible and should be avoided.This view becomes clear in the philosophy of Parmenides (540–480 BC), in whichhis teaching of being, as the primary object of philosophy, is clearly identified withthe . 3 This view is also still the case in Plato’s philosophy. The is neitherdesirable nor rationally accessible. However, Plato encourages us to extend therealm of rational accessibility, as he argues in his Philebos. 4 There he writes that the is opposed to the and should be ignored only when all rational possibilitiesare realized. 5 This process is to be followed because all rationally accessible being isbeing in between the and the , which are not rationally accessible, and it isa mixture of and . 6It would be interesting to find out why and how the transformative process enteredthe Greek mind. This approach made it possible to think about the existentiallyin a positive way, and it led toward the possibility of making it rationally accessible inscience and mathematics.1.3 From to Potential Infinity: AristotleAristotle was the first philosopher of ancient Greece to give a rational account ofinfinity. 7 In book III of his Physics, 8 which is about motion, he discussed whetheror not the exists at all. More importantly, he transformed it into a scientificconcept, as opposed to the more or less mythological and religious notion of the in Anaximander, to whom Aristotle alluded in his Physics, referring to it as somethingdivine. 9 Because Aristotle could only think in his own philosophical categories, suchas substance and accident, he claimed that the infinite could not exist in the same wayas an infinite body does. 10 How then must the infinite be conceived of in a rational3 “Denn die machtvolle Notwendigkeit () hält es (= das Seiende) in den Banden der Grenze (),die es rings umzirkt, weil das Seiende nicht ohne Abschluss ( ) sein darf.” ParmenidesB 8, 30–32; see also B 8, 42–49.4 Philebos 16d–e; Philebos 23–25 (Plato 1972).5 “Des Unendlichen Begriff ( ) soll man an die Menge nicht eher anlegen, bis einer dieZahl derselben ganz übersehen hat, die zwischen dem Unenlichen und dem Einen liegt ( ), und dann erst jede Einheit von allem in die Unendlichkeit freilassen und verabschieden”(Philebos, 16d–e; Plato 1972). For an overview about the Greek , see Turmakin (1943).6 Philebos 23b–26 (Plato 1972).7 This corresponds strikingly with the definition of time in Aristotle’s philosophy. He avoids any allusion to theentanglement of time and guilt, as was the case with Anaximander. For Aristotle, time, as well as the ,can be subjected to rational measurement. His definition of time demonstrates this point: “ , ” (Aristotle 1993, Physics IV, 11, 219b2–3).8 “The study of Nature is concerned with extension, motion, and time; and since each one of these three mustbe either limited or unlimited ( ) ( . . . ), it follows that the student of Nature mustconsider the question of the unlimited (), with a view to determining whether it exists at all, and, ifso, what is its nature” (Aristotle 1993, Physics III, 4, 202b30–36).9 “ ’ (=) , ʼ ” (Aristotle 1993, Physics III, 203b14–15).10 “It is further manifest that infinity () cannot exist as an actualized entity () and as substance() or principle ()” (Aristotle 1993, Physics III, 5, 204a21); “From all these considerations it isevident that an unlimited body ( ) cannot exist in accomplished fact” (Aristotle 1993, PhysicsIII, 5, 206a8–9; whole argument in 204a–206a9).