Je suis tellement pour l’infini actuel, qu’au lieu d’admettre, que la nature l’abhorre, comme l’on dit vulgairement, je tiens qu’elle l’affecte par-tout, pour mieux marquer les perfections de son Auteur.
— Leibniz, Opera omnia studio Ludov. Dutens., Tom. II, part x, p. 243
Certainly most of the paradoxical assertions which we meet with in the area of mathematics, though not all of them as Kästner suggests, are propositions that either contain the concept of infinity directly, or depend on it in some way for their attempted proof. It is even more indisputable that precisely those mathem- atical paradoxes which deserve our greatest attention are of this kind. This is because decisions on very important questions in many another subject, such as metaphysics and physics, depend on a satisfactory resolution of their apparent contradictions.
This is the reason why in the present work I am dealing exclusively with the consideration of the paradoxes of the infinite. But it is self-evident that it would not be possible to recognize the appearance of contradiction which is attached to these mathematical paradoxes for what it is, a mere appearance, if we did not make abundantly clear what concept we actually associate with the infinite. Therefore we do this first.
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