Kant, Calculus, Consciousness, and the Mathematical Infinite in Us

Abstract

Kant, Calculus, Consciousness, and the Mathematical Infinite in Us John H. Smith Der Begriff des Unendlichkleinen, darauf die Mathematik so öfters hinauskommt, wird mit einer angemaßten Dreistigkeit so geradezu als erdichtet verworfen, anstatt daß man eher vermuten sollte, daß man noch nicht genug davon verstände, um ein Urteil darüber zu fällen. Die Natur selbst scheint gleichwohl nicht undeutliche Beweistümer an die Hand zu geben, daß dieser Begriff sehr wahr sei. Denn wenn es Kräfte gibt, welche eine Zeit hindurch kontinuierlich wirken, um Bewegungen hervorzubringen, wie allem Ansehen nach die Schwere ist, so muß die Kraft, die sie im Anfangsaugenblicke oder in Ruhe ausübt, gegen die, welche sie in einer Zeit mitteilt, unendlich klein sein. Es ist schwer, ich gestehe es, in die Natur dieser Begriffe hineinzudringen; aber diese Schwierigkeit kann allenfalls nur die Behutsamkeit unsicherer Vermutungen, aber nicht entscheidende Aussprüche der Unmöglichkeit rechtfertigen.1 [The concept of the infinitely small, which comes up so often in mathematics, is rejected straight out with presumptuous audacity as a fiction, instead of assuming that it is not well enough understood to form a judgment about it. Nature itself seems, however, to provide clear proofs that there is truth to this concept. For if there are forces that work continuously through time in order to produce movements, as it would appear is the case with gravity, then the force that effects this movement in an initial instant (or rest) must be infinitely small as opposed to the one that imparts movement in time. It is difficult, I confess, to penetrate into the nature of these concepts; but this difficulty can at most justify cautiously avoiding unfounded suppositions and not claiming decisively that it is impossible.] The question that I address in this essay is simple: what is the mathematical infinite doing at crucial moments in Kant’s philosophy? By doing I mean the way that specific notions of the infinitely small—the infinitesimal, the differential, infinite approximation, continuity—serve as metaphors in a strong Aristotelian sense; that is, they attempt to “bring before the eyes” something abstract and thereby contribute a kind of intuition to the unintuitable.2 And yet, there is something doubly ironic going on in this effort. On the one hand, although mathematics and the mathematization of nature have been criticized by Husserl for abstracting modern thought from the concrete life-world of experience, I will look to places where there is a turn to the mathematical infinite in order to be concrete.3 On the other hand, the clear [End Page 95] intuitions associated with the mathematical infinite can easily get caught up in contradictions that undermine the initial clarity.4 By exploring these moments of metaphoric and paradoxical representation, we can see how major thinkers grappled with a notion of the “immanence of the infinite” that characterizes the epochal turn of modernity.5 First, let me give a general sense of what is at stake in the notion of the mathematical infinite and the issues of representation associated with infinitesimal calculus, which was up through the nineteenth century arguably the most powerful tool for understanding the physical world. Although Kant rarely addresses calculus as such, this tool has particular relevance for him given the fundamental role that both its inventors, Leibniz and Newton, play in the development of his critical project. Consider the problem of a line tangent to a curve—that is, a straight line that touches any curved line at precisely one point. As I hope and expect has just occurred in the mind of any reader at this point, a somewhat-clear image has taken shape. This image can be rendered mathematically precise thanks to the technique of calculus; indeed, calculus was developed in large part to address this problem. The reason is that such a tangent can be considered the rate of change of the curve at any instant, which can be calculated given the curve’s function. Thus, say, a soaring baseball defines a parabolic arc at a changing rate of speed (fast at first, slowing at the peak, then ever faster as it returns to the ground). The tangents at any point indicate...