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

38 infinity as a transformative concept in science and theologyarguing that infinity is a paradoxical concept, because both sets are obviously equal,but at the same time the integers are more than the numbers of the power two. 102 Thisview opened up a completely new conceptual space, into which such different figuresas Galileo Galilei, Bernard Bolzano, Georg Cantor, Kurt Gödel, and Gerhard Gentzen(transfinite induction) entered.1.4.2.3.5 “Coincidentia Oppositorum” and the Relativity of Motion. Nicholas ofCusa’s concept of the “coincidentia oppositorum” in infinity had an impact not onlyon mathematics but also on the physical sciences. This impact of his “coincidentiaoppositorum” in infinity on the physical universe is due to a major shift in his teaching.He no longer held that only God is infinite, but because the world is the mirror of God, 103therefore the world must also be conceived as infinite, especially with respect to spaceand motion. Nicholas of Cusa argued that the world cannot have a center, because inan infinite world the center coincides with the circumference. 104 A consequence is thatthe world must be regarded as infinite with respect to space. It also means, of course,that the earth can no longer be the center of the world – as taught by Aristotle – andit must have some kind of motion. 105 Thus, in the final analysis, Nicholas of Cusa’sconcept of infinity led to the concept of the relativity of motion, because a centralpoint of reference in the world is denied. The principle of the relativity of motion wasinvented by purely philosophical considerations of infinity about 800 years before itwas formulated by Albert Einstein, who saw it as a principle of the physical sciences.To sum up, it can be argued that Nicholas of Cusa overcame the intellectual deadend of apophatic theology, by trying to illustrate infinity by means of mathematicalsymbols, and by applying the new thought of “coincidentia oppositorum” to God’sinfinity. However, this statement is only half of the truth. By also applying this newconcept of “coincidenita oppositorum” to scientific problems, he paved the way to akind of secularization of infinity. In different ways it became a scientific concept. Theexamples are (1) infinity of space, (2) the relativity of motion, (3) the approximateprocesses in mathematics and epistemology, and (4) the conceptual difficulties ofdealing with infinity in terms of quantity. Thus, he paved the way to the modernsciences by thinking about infinity in many different ways.1.4.2.4 The Mathematics of Actual Infinity: Georg CantorThe chapter will conclude with some remarks about Georg Cantor and his achievementsin giving a rational, mathematical account of infinity. Until Georg Cantor in the102 Later Cantor would argue that, in infinite sets, other arithmetic rules must be observed.103 “Consensere omnes sapientissime nostri et divinissimi doctores visibilia veraciter invisibilium imagines esseatque creatorem ita cognoscibiliter a creaturis videri posse quasi in speculo et in aenigmate” (Nikolausvon Kues 1964, book I, chap. 11, p. 228; Cassirer 1991, p. 24); “Quis melius sensum Pauli quam Paulusexprimeret? Invisibilia alibi ait aeterna esse. Temporalia imagines sunt aeternorum. Ideo si ea, quae facta sunt,intelliguntur invisibilia Dei conspiciuntur uti sunt sempiternitas virtus eius et divinitas. Ita a creatura mundifit Dei manifestatio” (Nikolaus von Kues 1966, p. 270; Cohn 1960,p.88).104 “Centrum igitur mundi coincidit cum circumferentia” (Nikolaus von Kues 1964, book II, chap. 11, p. 390).105 “Terra igitur, quae centrum esse nequit, motu omni career non potest” (Nikolaus von Kues 1964, book II,chap. 11, p. 390).