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Nicholas of Cusa and the Infinite

nicholas of cusa

Nicholas of Cusa

“Nicholas introduces a mathematical analogy to explain his metaphysical ideas. Just as the definite polygon cannot measure the continuous circle, our finite minds cannot know the Infinite. All we can know of the Infinite is that we cannot know the Infinite. To the extent that we can understand the Infinite at all, Nicholas argued, we must understand it through the coincidence of opposites. For example, Nicholas taught that, in the Infinite, the circle coincides with the line. He illustrated this paradoxical statement by considering a sequence of circles of larger and larger diameters…

Nicholas of Cusa had proposed ideas that appear to anticipate both Copernicus and Kepler. Like Copernicus, he proposed that the earth was not at the center of the cosmos and was in motion. Like Kepler, he proposed that the motions of the planets are not uniform or circular. Nicholas, however, did not base his arguments on empirical data, nor did he develop his ideas into mathematical models that could be tested against experience. Nicholas did not, like Copernicus, show that his ideas could account for astronomical observations with a respectable (although not perfect) amount of accuracy. Nor did Nicholas come even remotely close to Kepler’s discovery that the planets precisely follow elliptical orbits and sweep out equal areas in equal times. Although Nicholas does not deserve credit for these amazing discoveries themselves, his thought dramatically expanded the intellectual horizon of his time, and opened up possibilities of thought that allowed Copernicus and Kepler to make their breakthroughs….

Nicholas of Cusa’s thought also opened up possibilities in mathematics that paved the way for calculus and a mathematics of the continuum. Ever since the Pythagoreans discovered that certain geometrical magnitudes could not be expressed in terms of arithmetic ratios, mathematics had been divided into two incommensurate branches: arithmetic and geometry. This division expressed a fundamental division between the infinite (the geometrical continuum) and the finite (the arithmetic of whole numbers). The Arabs, however, did not let a lack of rigorous theoretical foundations prevent them from freely assuming the existence of irrational numbers and using them in calculations…”

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