Process and the Emergence of Limit: Revisiting the Foundations of Infinitesimal Calculus

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Keywords: Philosophy of mathematics and mathematical sciences, calculus, Newton, Leibniz, Cusanus, infinitesimal, actual and potential infinity, conce...

Дата загрузки:2021-05-02T23:10:10+0000

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Keywords: Philosophy of mathematics and mathematical sciences, calculus, Newton, Leibniz, Cusanus, infinitesimal, actual and potential infinity, concept of limit, temporal differentiation

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.

In my presentation, I will discuss how the philosophically observed dichotomy in the concept of infinitesimals—namely the dichotomy between the contradicting notions of actual and potential infinitesimal—historically necessitated the development of the concept of limit in the Early Modern Age. In the Aristotelian distinction, the problem of infinity and infinitesimal emerged aporetically, since the existence of both actual and potential infinitesimal eludes human understanding and imagination. In other words, these contrasting modes of the infinitesimal form neither ontologically final entities nor epistemologically final justifications of infinity. The ontological aporia of the infinitesimal presupposes the problem of its recognisability; that is, how the being of infinitesimal could be conceived both in a continuous process-mode of potential infinity as well as in a finite discretion of actual infinity.

These contradicting notions of infinity, historically bequeathed, were synthesised 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 (found in his major works De Docta Ignorantia and De coniecturis). 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).

Would it be possible to rehabilitate these historically and paradigmatically suppressed factors in the original method of differentiation, so that it could be developed into a form of temporal differentiation? The predetermination of the limit seems to be a problematic assumption in the original and historically established mode of differentiation. 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 the method of 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.
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