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

from potential infinity to actual infinity 371 2 3 4 5 6 7 8 9 10 11 121 4 9 16 25 36 49 64 81 100 121 144Figure 1.1oppositorum” to purely mathematical problems. 100 It can be seen in Nicholas of Cusa’swork De Mathematica Perfectione (Nikolaus von Kues 1979, pp. 160–77). In thiswork he used the concept of “coincidentia oppositorum” as a tool for creating the newmathematical procedure of infinite approximation.1.4.2.3.3 Approximate Mathematical Illustration of Infinity, Second New EpistemologicalApproach. In his mathematical work De Mathematica Perfectione,Nicholas of Cusa gave a more elaborate account of his metaphysical principle concerninginfinity. In fact, he was quite ambitious in wanting to bring mathematics to itsperfection and completion by the application of the “coincidentia oppositorum.” 101 Heaccomplished it by an approximate process of straightening a segment of a circle intoa straight line.By means of this new methodological approach of infinite approximation, he triedto calculate the circumference of a circle (Cantor 1892, pp. 176ff). In this way, infinityhad become for him a methodological tool in mathematics. This elaboration of approximateprocesses in mathematics corresponded completely with his understanding ofepistemology as an approximate process toward truth. Nicholas of Cusa substitutedan infinite process for St. Thomas Aquinas’s famous definition of truth as “veritas estadaequatio rei et intellectus.” The epistemological consequences can hardly be overestimated,because, in the final analysis, he liberated rationality in theology from thetrap of apophatic theology. Apophatic theology claimed that the infinite God is notaccessible rationally. Therefore, rationality was not esteemed very highly in apophatictheology. Nicholas of Cusa, however, preserved the place of rationality in theology bytransforming the ontological infinity of God into an infinity of epistemological processes,in mathematics, physical sciences, as well as philosophy. Thus, he created theepistemological prerequisites of modern natural science.1.4.2.3.4 Infinity and the Real Numbers. In dealing with the question of the applicabilityof the category of quantity to infinity, Nicholas of Cusa argued – as we haveseen – that infinity defies being measured in terms of “bigger” or “smaller,” or, to putit differently, he argued for the uniqueness of infinity (Knobloch 2002, p. 231). In thisrespect, he preceded the insight of Galileo Galilei – who might have been familiarwith Nicholas of Cusa’s writings – that the integers are as many as the numbers of thepower two, although the numbers of the power two are obviously less than the integers(Rucker 1989, p. 19; Knobloch 2002, pp. 231ff).Galileo Galilei could not solve this conceptual difficulty of comparing two infinitesets by using the quantitative comparison of “bigger” and “smaller” (>,