Kant's Mathematical Sublime (§26)

by Corry Shores
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When we estimate a magnitude numerically, this is mathematical; when we estimate by mere intuition, ‘measured by the eye,’ this is aesthetic (§26 134d).

But even mathematical estimation relies on an aesthetic estimate of a common measure which may be computed variously (135b). (This aesthetic estimate is obtained by means of averaging).

In mathematical estimation, there can be no greatest magnitude, because numbers succeed to infinity. But on account of the limitations of our faculties, there can be a greatest estimated aesthetic magnitude: if there is a point to which anything greater is beyond our comprehension, it does not matter how much greater, because all will seem absolutely great. This caries with it the idea of the sublime and produces an emotion “which no mathematical estimation of magnitudes by means of numbers can produce” (135bc).

In order to use a quantum in the imagination to estimate a magnitude, there must be two facultative activities: apprehension and comprehension. We apprehend an object when it is given part-by-part in series, and we then comprehend them through synthetic unification. But we have only so much capacity to retain apprehensions, so when a certain quantity has been obtained, we begin to forget those furthest back in our retention. So we cannot comprehend something whose magnitude is beyond our retentional capacities (135c-d).

These aesthetic judgments must be pure, and thus not involve teleology. Hence the sublime is not shown in products of art, which were made with a human end in mind that determines its form and magnitude; nor is the sublime found in natural things whose “concept already brings with it a determinate end,” as for example animals with known natural determinations. Rather, the mathematical sublime can only be found in raw nature merely insofar as it contains magnitudes, and not whether it by itself alone brings charm or the emotion felt in times of danger (136bc).

“An object is monstrous if by its magnitude it annihilates the end which its concept constitutes.” An object is colossal when the mere presentation of its concept is almost too great for all presentation. “A pure judgment on the sublime, however, must have no end of the object as its determining ground if it is to be aesthetic and not mixed-up with any judgment of the understanding or of reason” (136-137).

When magnitudes are estimated conceptually, there is no force that pushes the imagination to comprehend the infinite, because the imagination can schematize in a way that presents the entire magnitude in such a manner that it can be grasped as one abstract quantity (137c-d).

But in aesthetic apprehensions, the “voice of reason” requires that all given magnitudes be comprehended into intuition, and demands that even the infinite be comprehended (138ab). However, the infinite is absolutely great, and cannot be compared to a standard measure, and hence cannot be comprehended into one homogenous quantity. “But what is most important is that even being able to think of it as a whole indicates a faculty of the mind which surpasses every standard of sense;” and “even to be able to think the given infinite without contradiction requires a faculty in the human mind that is itself supersensible” (138b).

“Nature is thus sublime in those of its appearances the intuition of which brings with them the idea of its infinity” (138d).

Normally in aesthetic judgment, the free play of the imagination connects comprehensions to concepts in the understanding: the synthesized representation of a rose is matched with its concept. But we have no concept of the absolute whole, because we have never comprehended it; however, the faculty of reason contains the a priori idea of the absolute whole, and so reason steps-in to restore harmony among the faculties by matching the representation of the faculties’ limitations with this idea of the absolute totality.

Kant, Immanuel. Critique of the Power of Judgment. Transls. & Eds. Paul Guyer & Eric Matthews. Cambridge: Cambridge University Press, 2000.