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.

Communicating Kant’s Free Play

by Corry Shores
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“Taste is the faculty for judging an object or a kind of representation through a satisfaction or dissatisfaction without any interest. The object of such a satisfaction is called beautiful” (Critique of the Power of Judgment §5).

When an object or a representation gives us pleasure, our faculty of taste judges it as beautiful. This satisfaction derived from the beautiful cannot be one that bears personal interest in the object, because when one deems something beautiful, one presumes that the object is satisfying – and hence beautiful – for everyone (§6). Thus someone would not say, “‘This object. . .is beautiful for me.’ For he must not call it beautiful if it pleases merely him” (§7). We consider the beauty-judgment of an object to have universal validity, because we believe that no one else would think otherwise (§8).

But Kant wonders if first we feel the pleasure, then afterwards we judge the object as beautiful; or if instead, first we make the judgment before feeling the pleasure.

According to Kant, it could not be that the pleasure precedes the judgment of beauty. This judgment presupposes that it holds for all people, and thus we may communicate to others the shared satisfaction we would all have when apprehending the object. But if first we have just pleasure, then that is only valid for ourselves, and hence we would not initially have grounds to infer that the pleasure would be held by everyone else.

“Thus it is the universal capacity for the communication of the state of mind in the given representation which, as the subjective condition of the judgment of taste, must serve as its ground and have the pleasure in the object as a consequence” (§9).

Hence, although we might find the object agreeable because it pleases us subjectively, that is, aesthetically, we do not then follow-up this pleasure by judging that it holds for everyone else. Rather, an object we deem beautiful strikes us firstly as being universally pleasant, with ourselves naturally among those who would find it so. Yet,

“Nothing, however, can be universally communicated except cognition and representation so far as it belongs to cognition” (§9).

This is because only cognition is objective, while our feelings of pleasure are subjective.

So to say that something is beautiful is to say that the object would please anybody. But this pleasantness refers not to the object but to the subjective experience of cognizing the object. So when we consider such a judgment, our faculties are not at that moment encountering the beautiful pleasant object: there is no sensation, merely cognition. Thus:

“The powers of cognition that are set into play by this representation are hereby in a free play, since no determinate concept restricts them to a particular rule of cognition” (§9).

If we say, “roses are beautiful,” we are not thereby having some rose as a determinate and immediate representation. Instead, our powers of representation are free to play-around in cognizing the terms of the judgment. (“Thus the state of mind in this representation must be that of a feeling of the free play of the powers of representation in a given representation for a cognition in general” (§9).) Kant continues:

“Now there belongs to a representation by which an object is given, in order for there to be cognition of it in general, imagination for the composition of the manifold of intuition, and understanding for the unity of the concept that unifies the representations” (§9).

We apprehend things in parts, in a manifold. Imagination links the parts together into a unity that will match with a single concept in the understanding.

“This state of a free play of the faculties of cognition with a representation through which an object is given must be able to be universally communicated, because cognition, as a determination of the object with which given representations (in whatever subject it may be) should agree, is the only kind of representation that is valid for everyone.”

So this free play of the imagination to bring together intuitions for the sake of conceptualizing the judgment of the beautiful object must be communicable to everyone else, because for the judgment to be universally valid, the same cognitive process must be possible for anyone else.

This free play which brings about the representation of the object which is judged universally pleasing is the grounds to derive pleasure from that representation, because communicating our states of mind “carries a pleasure with it,” a fact that is demonstrated by “the natural tendency of human beings to sociability” (§9).

But, we may only subjectively relate the judgment of something’s beauty to the experience of its beauty, “thus that subjective unity of the relation can make itself known only through sensation” (§9). The faculties of imagination and cognition must be in “unison” with the whole cogitation that considers the beautiful object, and this harmony between the faculties of cognition and the whole process of cognition produces the sensation of pleasure. So when we know that our ability to consider a beautiful object, which is performed by faculties all human share universally, and the process of considering that object are together harmonious, then we know that this process would be harmonious for everyone else (because the object is not considered subjectively and the faculties are shared universally), and thus we know that anyone else who considers this object would likewise feel the pleasure of the “well-proportioned disposition” of facultative harmony (§9).

For Deleuze, communication happens not when an experience of an object is universally valid, because communication requires difference. When we see a Francis Bacon painting, our faculties are unable to synthesize it into a representation, hence there is no universality to either the object or to the experience of it. However, the creation of this artwork required an internal and external collision of order and chaos, and it produces in the viewer such a collision when sensing it, because the viewer both tries to organize her impressions while also being unable to. Communication can only happen when there is irreducible difference and heterogeneity mixed with our universally held tendencies to homogeneously organize something that suggests order.