Actual infinity and God

Abstract

Greek thought gave birth to the two notions of infinity that dominated the history of philosophy and mathematics: the potential infinite and the actual infinite. Initially infinity was understood in the literal sense of one, another one, yet another one – and so on indefinitely, without an end, endlessly, infinitely. The discovery of the whole-parts relation by the school of Parmenides mediated the turn “inwards” – introducing infinite divisibility. This development is related to the contribution made by Zeno’s paradoxes: in particular, the bisecting paradox and the flying arrow. Aristotle developed a theory of continuity based upon two criteria that formally match the Weierstrass-Cantor-Dedekind understanding continuity –in spite of the fact that Aristotle rejected the actual infinite while Weierstrass-Cantor-Dedekind accepted it as the basis for their view of continuity. Mathematics faced it first crisis when the Pythagorean arithmetization was confronted with the discovery of irrational numbers. Aristotle’s objections to the actual infinite turned out to be hall-marks of infinity. The prevalent Greek view was that since the apeiron (the unbounded-unlimited) was formless, God (as thought thinking itself) cannot be infinite (formless). Gregor von Nyssa (fourth century AD) was the first thinker who positively asserted that God is infinite. Augustine provided a starting point for Cantor because he contrasted succession with simultaneity: God can oversee an infinity of numbers at once, whereas the human mind can only contemplate a multiplicity in succession. (Maimon formulated a similar view in 1792.) Nicholas of Cusa distinguishes between the absolute (actual) infinity of God and the endlessness of reality – in God all oppositions coincide (God is the coincidentia oppositorum). Compare the view of al-Ghaz?l? mentioned by Verhoef et.al. It turned out that Aristotle’s objections to the actual infinite are in fact characteristic features of infinity. (It is shown with reference to the smallest transfinite ordinal number, ?). According to Descartes the infinite is perfect and the finite is imperfect. Yet the potential infinite governed mathematics until the 19^th century. In his letter of July 12, 1831 to Schumacher Gauss stated that “in this manner I protest against the use of an infinite magnitude as something completed, which is never allowed in mathematics”. The limit concept gave rise to the second foundational crisis of mathematics – irrational numbers cannot be defined as the limits of converging sequences of rational numbers. Weierstrass, Cantor and Dedekind once more introduced the idea of the actual infinite. Cantor distinguishes between the potential infinite, the actual infinite and the absolute infinite (God). Hermann Weyl restricted infinity to the potential infinite, while leaving actual infinity open for God, for him God is the completed infinite. In line with the historical contours outlined, the idea of eternity also entered the theological domain in the form of two apparently opposing notions: eternity as an endless period of time, and eternity as timelessness. The line runs from Parmenides, Plotinus (Enneads III/7), Boethius, Kierkegaard (the nunc aeternum/the eternal now) and Schilder. Is infinity brought into mathematics on a Christian theological foundation? The important distinction between conceptual knowledge and concept-transcending knowledge (idea-knowledge) is introduced. At the same time Bernays rejected the attempt to reduce continuity to discreteness. Erasmus and Verhoef captures the actual infinite in familiar terms: an infinite totality, a completed whole with all members present all at once. The potential infinite and the actual infinite should rather be designated as the successive infinite and the at once infinite. In addition, a closer analysis of the notion of a “totality” and its connection to “all at once” and “(actual) infinity” was undertaken.