## 4. The Mathematical Sublime

4.1 Aesthetic vs. Mathematical Estimation of Size

Now that we have a general sense of what Kant means by the sublime, it is time to discuss the more specific notion of the mathematical sublime. The mathematical sublime is a feeling of the sublime which we experience when we encounter something overwhelming in size. Thus, to understand what gives rise to the feeling of the mathematical sublime, we need to understand how Kant conceives of the mind to be capable of judging sizes.
Kant makes a distinction between two ways of estimating size: aesthetically and mathematically. An aesthetic estimation of size, says Kant, is something that occurs “in mere intuition (measured by eye).” (p. 134) When we run into a tall person on our way to the office, for example, we judge the person to be tall, not through the use of a ruler, but through what we consider to be the norm of a person’s size – i.e. through “the average magnitude of the people known to us.” (p. 133) An aesthetic estimation of size is thus something that we do every day, intuitively, without any attempt at calculation.

A mathematical estimation of size, on the other hand, is something that requires us to calculate (i.e. brings things into numerical relations with one another). When we want to judge whether a person’s height has broken the world record, for example, we do not simply eyeball the person’s height: we measure the person’s height quantitatively, “by means of numerical concepts.” (p. 134) Thus, in a mathematical estimation of size, instead of using the norms of intuition as our guide, we use the norms of reason (i.e. numerical relation) for our estimation. A mathematical estimation of size is thus an estimation of size based on quantitative reasoning, which requires us to compare the size of things based on their numerical relations to one another, rather than judging merely on the basis of our intuitive norms.

4.2 The Limits of Aesthetic Comprehension

Now, for Kant, the aesthetic estimation of size and the mathematical estimation of size are not equal in power. Kant points out that, in the mathematical estimation of size, there is no limit to how far we can go on measuring: no matter how large the object is, it is always possible to bring its size into numerical relation with other things. “For the mathematical estimation of magnitude,” says Kant, “there is… no greatest [i.e. maximum size] (for the power of numbers goes to infinity).” (p. 135) Thus, even if the size of the sun is much too large to be understood in terms of everyday norms, we can still give it precise numerical relations to the size of other objects: we can say, for example, that its volume is about 1,299,400 times that of the Earth.

In an aesthetic estimation of size, on the other hand, there is a limit to how far we can go on measuring. When I see a tall person, it is easy for me to intuitively assess his or her height in relation to my everyday norms; but when I encounter a building of truly monumental proportions, such as St. Peter’s Basilica in the Vatican, the task of bringing its size into relation with everyday norms becomes impossible – my mind simply cannot grasp the size of such things on an intuitive level. In such an encounter, I have reached the limits of aesthetic estimation and can say nothing precise about the size of the object before me, except that it is “absolutelygreat” [schlechthin groß] – i.e. its size has exceeded the maximum of aesthetic estimation, “an absolute measure, beyond which no greater is subjectively (for the judging subject) possible.” (p. 135)

Kant associates the mathematical estimation of size, in its pure form, with a mode of thought called “apprehension (apprehension).” The aesthetic estimation of size, on the other hand, he associates with “comprehension (comprehension aesthetica).” That the power of the former mode of thought exceeds that of the latter is made clear in the following passage:

There is no difficulty with apprehension, because it can go on to infinity; but comprehension becomes ever more difficult the further apprehension advances, and soon reaches its maximum, namely the aesthetically greatest basic measure for the estimation of magnitude. (p. 135)

Thus, for Kant, there is no limit to our mathematical apprehension of size (numerical concepts go up to infinity), but there is a limit to our aesthetic comprehension of size (intuitive norms reach a certain maximum). And, since Kant associates the former with the faculty of reason and the latter with the faculties of the imagination and the understanding, we can also say that, in our estimation of size, the power of our reason exceeds the limits of our imagination and understanding. This reformulation makes it clear how the estimation of size can be related to the experience of the sublime: since the act of estimating sizes show reason to be more powerful than the imagination and the understanding, we can see how the act of estimating sizes might lead us to experience that which characterizes the experience of the sublime – the feeling of respect for our faculty of reason.

4.3 The Supersensible Faculty

When we enter St. Peter’s Basilica and stand before its majestic proportions, two things happen to us. First, we try to aesthetically comprehend the massive size of our surroundings – i.e. the size of the walls, the arches, and the domes – in our intuition. When we do this, our imagination is at work: we look at the massive size of our surroundings and try to imagine its relation to our everyday norms. But this process soon collapses into a feeling of “bewilderment” and “embarrassment,” says Kant, for the mind quickly reaches the limits of aesthetic comprehension and come to realize the “inadequacy” of the imagination in presenting the interior of St. Peter’s as a unified entity. (Kant, p.136) St. Peter’s, in other words, is too large for us to grasp its size intuitively: we simply cannot comprehend its immensity without the help of mathematical reasoning.

At this point in our attempt to assess the size of St. Peter’s Basilica, another thing happens to us: we become aware of an idea of totality, which we demand of ourselves, even when we fail to find it in the immensity of our surroundings. In our failure to aesthetically comprehend the proportions of St. Peter’s, we hear an inner-voice in our mind, which tells us that we ought to seek a sense of totality in the intuition of our surroundings, even if we fail to cognize it in our intuition. Consequently, we become aware of a “supersensible” faculty within us – a faculty that is not determined by our senses and yet is able to make demands upon them. “The very inadequacy of our faculty for estimating the magnitude of the things of the sensible world,” says Kant, “awakens the feeling of a supersensible faculty in us.” (Kant, p. 134) The limits of our aesthetic comprehension, in other words, make us aware of our ability to reason independently of the senses.

The supersensible faculty of the mind that Kant is talking about is of course the faculty of reason. It is reason which tells us to seek a sense of totality in our experience of the St. Peter’s Basilica, even when we fail to grasp this totality in our intuition. “The mind hears in itself the voice of reason, which requires totality for all given magnitudes,” says Kant. (Kant, p. 138)

Now, for Kant, there is an important relationship between the faculty of the imagination and the faculty of reason in our judgment of the sublime. Just as our judgment of the beautiful is the result of a particular relationship between the imagination and the understanding, so is our judgment of the sublime the result of a particular relationship between the imagination and reason:

[J]ust as the aesthetic power of judgment in judging the beautiful relates the imagination in its free play to the understanding, in order to agree with its concepts in general (without determination of them), so in judging a thing to be sublime the same faculty is related to reason, in order to correspond subjectively with its ideas (Kant, p. 139)

Unlike the relationship between the imagination and the understanding in our judgment of the beautiful, however, the relationship between the imagination and reason in our judgment of the sublime is not harmonious. After all, the former has failed to achieve the unified totality that the latter demands. Thus in our judgment of the sublime, there is no free play of the mind: there is only a feeling of displeasure in failing to cognize what reason demands of us, followed by a burst of pleasure in becoming aware of the activity of the supersensible faculty of reason. This movement from displeasure to pleasure in the judgment of the sublime is what Kant calls “a momentary inhibition of the vital powers and the immediately following and all the more powerful outpouring of them.” (p. 128)

4.4 Elevation and Affirmation of Reason

The failure of the imagination to match reason’s idea of totality is what brings about the feeling of elevation that characterizes our experience of the sublime. As Kant puts it: “The mind feels itself raised [gehoben]… abandoning itself to the imagination and to the reason… which finds the whole power of imagination inadequate to its ideas.” When we enter St. Peter’s Basilica and feel overwhelmed by its sheer immensity, we feel small and insignificant in our incapacity to aesthetically comprehend the object of our intuition. But this experience of humility is precisely what makes us appreciate the power of our reason, which demands us to seek totality and thus unifies our experience under its idea even when there is no unity to found in the object of intuition. We feel elevated by the fact that we are not completely dependent on our senses. Even when our imagination fails to give unity to the object of our intuition, we are able to unify our experience through ideas that exceed the bounds of sensibility.

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