Die zeitliche Differenzierung

Abstract

In its historical development, the notion of infinitesimal proved to be an ambiguous entity both in philosophy as well as in mathematics. The Aristotelian distinction between actual and potential infinity seemed to be predicated upon mechanical considerations (owing to its original relation to Zeno´s paradox). The discourse or debate on the mode of existence of the infinitesimal developed within the context of theoretical philosophy, mathematics, and mathematical sciences, as represented in the invention of the method of calculus by Newton and Leibniz. These contradicting notions of infinity, historically bequeathed, were synthesized into the early-modern concept of limits. The discretion of the limit did not exclude the potential infinitesimal but, on the contrary, presupposed it in the infinite continuity of a process of diminution. The historical origins of such a synthesis can be observed in the late Middle Ages, namely in Nicolaus Cusanus´notion of infinite processes and their discrete bounds, as represented in his geometrical demonstrations of the doctrine coincidentia oppositorum. However, the introduction of the principle of limits in the original geometric-mathematical method of differentiation led to a problem in the philosophy of mathematics. The infinitesimal was originally created in an infinitely continuous process of differentiation, but this (once admitted) element of potential infinitesimal was eliminated in a method of limits in favour of a discrete entity of differential, i.e. in favour of actual infinitesimal. This strategic elimination would also mean the elimination of two basically indispensable factors, or elements, from the limit process in differential calculus, namely the movement and the time (immanent in the movement). Moreover, the predetermination of the limit seems to be a problematic assumption in the original and historically established mode of differentiation. In reality, however, the actual limit itself evolves from a movement of an infinite retardation. When the limit is not predetermined in discretion but generates itself in a process-mode, it rehabilitates the (originally eliminated) potential infinitesimal in a method of temporal differentiation. An adequate substantiation of the self-generating principle of limit, that integrates the factors of space, time, and movement, becomes possible through the synthesis of actual and potential infinity.