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

from potential infinity to actual infinity 23Aristotle’s understanding of the working of a potential infinity in a process. Eudoxos’smethod was taken up by Euclid (365–300 BC) into his elements (books V andXII). It was preserved as mathematical heritage, and it was refined by Archimedes(287–212 BC), who studied in Alexandria under the supervision of the successors ofEuclid. 17In sum, Aristotle seems to be a thinker who stressed finiteness. It is not clear if herejected an all-encompassing being, as Anaximander did with the , which hequotes in his Physics. It may well be that Aristotle’s account of potential infinity, asquoted in his Physics, and Anaximander’s account of the cannot be reconciled.Aristotle rejected the ability of the human mind, the , to think the actual infinite,and he contended that God is not infinite. However, he held that the human mind canpotentially think in infinite processes, and that time and motion are infinite, even thoughspace is finite. What makes Aristotle’s account important is his distinction betweenpotential infinity and actual infinity. This distinction proved to be of great influencein the subsequent intellectual struggle about infinity. However, Aristotle also createda major obstacle for scientific progress in thinking about infinity. This obstacle washis metaphysical framework. It stressed finiteness, at least in his thinking about God’sfiniteness. Christian theology broke through this limitation of Aristotle’s metaphysicsand paved the way to a new understanding of infinity. This view will be outlined inthe following section. Aristotle’s account of infinity did not enter into the theologicalconsiderations of Christian theology until Aristotle’s work was received for theologyby St. Thomas Aquinas in the thirteenth century.1.4 From Potential Infinity to Actual Infinity1.4.1 The Ecstatic Move toward Infinity – PlotinusIn late antiquity a slow process of transformation was about to emerge for understandingGod as infinite. It was partly based on biblical writings, and partly based on syncretism.Thus, the Hellenistic Jew, Philo of Alexandria, referred to God along the lines of theJewish tradition as absolutely transcendent. That meant that God is strictly to becontrasted with all mundane properties (Siegfried 1970, p. 199sq). 18 Thus, God mustbe thought of as not determined, which opposes the Greek philosophical tradition. Thelatter identified rational access and determinateness (). In this sense, one canargue that Philo is the founder of apophatic theology, which necessarily entails infinityas a property of God. 19 However, this argument is not compelling, as Philo neverapplies to God in his writings (Mühlenberg 1966, pp. 58–64). Furthermore,the idea that God is not measurable with respect to space and time, and that God isineffable, occurred during this late period of antiquity in the Mandean religion (Brandt1889), as well as in the philosophy of Apuleius (130 BC) (Zeller 1963, bd. III, 23, 210),17 The method of exhaustion to calculate the surface area of a circle is described in detail in Wilson (1995,pp. 293–97).18 Philo assigns properties to God that are all in contrast to mundane qualities like: uncreated, name, space, butencompasses all space, cognizable, and irrational affections (Siegfried 1970, pp. 199sq, 203, 207).19 This argument was exposed by Guyot (1906, pp. 31–32).