I am working now with the Pluhar translation of the Analytic of the Sublime from Kant's Critique of Judgment, found in Korsmeyer's textbook Aesthetics: The Big Questions. This is an extremely difficult bit of material, and it is abridged too! My focus is on the sentence "If the human mind is nonetheless to be able even to think the given infinite without contradiction, it must have within itself a power that is supersensible, whose idea of a noumenon cannot be intuited but can yet be regarded as the substrate underlying what is mere appearance, namely, our intuition of the world." So, to start off, I do not think that the mathematical sublime has much to do with mathematics and actually has a lot to do with the aesthetics of nature as well as the aesthetic experience of great works of art. It mainly is concerned with a particular kind of experience, one that is aptly illustrated by the example of the Egyptian pyramid. Kant makes a distinction between apprehension and comprehension, and by the first he simply means the progressive adding of units or numbers. Apprehension is associated with mathematics. It is the second concept, comprehension, that is interesting, and it is illustrated by the example of the pyramid. There is a sweet spot in our appreciation of an Egyptian pyramid: we cannot be too far away or we will not be able to appreciate the individual units of which it is made, but we cannot be too close as then we will not be able to include all of those units in one intuited whole I like to think of this in relation to appreciation of another great work of art known for another kind of complexity, The Night Watch by Rembrandt. The point is to try to hold the entire experience of such a work in one's mind, to contemplate it in such a way as to try to intuit it as a whole. This might be a matter of how one positions oneself spatially before it, although perhaps more important is the amount of time one spends contemplating it: again, not too much or too little. Although The Night Watch is not as enormous as an Egyptian pyramid, it is great in its complexity, and the number of its parts or appreciable aspects can be seen as, if not infinite, at least indeterminately large. Absolute magnitude, which, Kant believes, aesthetic experience of the sublime gives us, is just a matter of taking the whole thing with its indefinitely large number of appreciable parts in with one intuition. The problem with being too close to the pyramid is that, as one looks at the stones, the earlier parts of the experience are "extinguished." So, again, this is a temporal as well as a spatial matter.
Another factor in the experience of the mathematical sublime for Kant is that the judgment of the sublime only happens when the imagination is unable to exhibit the concept of the magnitude. This idea seems to conflict with the main theme of the preceding paragraph. Our intuition is supposed to grasp the object as a whole, in comprehension, and yet our imagination is not up to the task. I suspect that by "imagination" here, Kant is not referring to creative imagination but simply to apprehension (associative imagination), i.e. to the ability to put things together in an associative and relatively mechanical way. So, when the imagination in this mechanical sense gives out, then we have an intuition that hooks us up to something grander. I take it that this grander thing is much like Aristotle's notion of an organic whole. Aristotle brings in his concept of the organic whole in the Poetics when he discusses beauty. As with Kant, and perhaps Kant was thinking of Aristotle when discussing mathematical sublime, he sees magnitude as something valuable, although he associates it with beauty. He also associates it with the mind's ability to intuit something as a whole. A play is more beautiful if it has magnitude and is still comprehensible, that is, can be apprehended as an organic whole.
Kant goes on to argue that, unlike mathematics operating under the relatively mechanical concept of apprehension, when the mind listens to what he calls "reason" (Kant's notion of "reason" seems to bear little relation to what we mean by reason and does not for example have anything to do with logic or giving good reasons or arguing well) it demands "totality" even of magnitudes that we never can apprehend completely. It "demands comprehension in one intuition" and it requires that we experience the infinite in its entirety. But since the infinite is so large, everything else in relation to it is incredibly small, and to be able to think such an infinite one must go beyond imagination, which is based ultimately on sensation. There is, here, no determinate relation between things expressible in terms of numbers.
Again, this is not about mathematics but about the limits of mathematics, even of the mathematical concept of infinity. Although Kant never mentions the phrase, I find myself thinking of that old saying: "the whole is greater than the sum of the parts." The experience of the sublime is a matter of experiencing a whole that is greater than the sum of its parts. Kant speaks of the limitations of ordinary intuitions in terms of following standards of sense. So what is this mental power that goes beyond the standard of sense? It is the capacity to perceive totality. Kant explains this in one fascinating sentence quoted above, and quoted again here: "If the human mind is nonetheless to be able even to think the given infinite without contradiction, it must have within itself a power that is supersensible, whose idea of a noumenon cannot be intuited but can yet be regarded as the substrate underlying what is mere appearance, namely our intuition of the world." The substrate of our world of appearance is the idea of a noumenon, of a world of things in themselves. So the substrate that underlies appearance is the intuition of the world, i.e. the background thought that everything that happens happens in the world. This ability is something that goes beyond mere sensation or imagination or even imagination combined with the understanding to point at the noumenal realm, i.e. the realm of God, immortality and the soul, which is to say the realm referred to by religion.
Bear in mind that belief in God is not necessary for this all to work. Belief in an indeterminate whole that subsumes experience and that also makes wholes greater than the sums of their parts may be sufficient.
Mathematical estimation of magnitude, using numerical concepts only, would not allow us to think infinity in its entirety. So transcending mere mathematical estimation is actually the solution to the problem that Kant originally posed at the beginning of the Critique of Judgment, how to find a bridge between the realm of experience and the noumenal realm. His answer is that the mind can expand itself beyond the barrier of sensible experience as long as it has a practical, i.e. a moral, aim, as opposed to one that is focused on cognition. So nature is sublime insofar as the idea of infinity is combined with these its intuition. The introduction of the idea of morality into this discussion seems gratuitous and must have to do with other projects and worries of Kant.
What we are describing is a view of the aesthetics of nature quite contrary to that of those contemporary aestheticians who believe that the proper appreciation of nature is cognitive in a science-like way. Such cognition is bound by its inability to, as Kant would put it, comprehend the infinite. John Muir, as I have argued in earlier posts, manages to go beyond this relatively narrow perspective. If Kant and Muir are right than the cyclical view of nature appreciation I have recently advocated may require an addendum, that although cycles are important, there are also high points in the appreciation of nature, or of anything great, even in art, and these occur when the intuition has within it the idea of totality or something infinite, and the attendant pleasure.
So, the relatively mechanical imagination can't truly judge the magnitude of an object, since a true judgment would be one that would give us the experience of the sublime. So, when the imagination tries to comprehend in a way that surpasses its own ability to encompass all of this in an holistic intuition then we have the experience of the sublime, since at this point reason takes over, as it were, and with it the idea of something that underlies all of our experience, i.e. the intuition of ourselves as experiencing things within an entire world, i.e. the whole of nature. Kant writes that the "basic measure of nature is the absolute whole of nature, which in the case of nature as appearance, is infinity comprehended."
But then it turns out that this basic measure is contradictory, even on Kant's account. There cannot be absolute totality of a progression that is endless. Does Kant realize that he is deconstructing or at least undercutting himself here? He seems to be implying that the experience of the sublime is based on an illusion or an illusory experience. I have no problem with this Nietzschean idea, although it is surprising to find it in Kant. So, when the imagination tries fruitlessly to comprehend the magnitude of some natural thing, this contradiction leads us to think of a supersensible substrate underlying both nature and our ability to think, one that we must judge as sublime. (Why need we think this?) What is sublime in this case, Kant argues anthropocentrically, is not nature itself but the attunement within our minds when we judge such an object, the imagination then being referred to reason in such a way as to harmonize with its ideas, i.e. the indeterminate ideas of the noumenal realm. Kant naively thinks this will be attuned with moral judgments and the influence of moral demands on feeling. I can't follow him here.
So he finds true sublimity in the mind of the person who has this mental attunement of imagination and reason, and not in nature at all! This leads him to the rather counter-intuitive claim that "we would not want to call sublime such things as shapeless mountain masses piled on one another in wild disarray." Rather, in this case, what is happening is that the mind "feels elevated in judgment of itself" because it can contemplate these things without thinking about their form. Instead it can revel in its own imagination and its harmonious connection with reason in its attempt to arrive at noumenal things, although the imagination is inadequate to such ideas. Some harmony! Why would he even think that the experience of shapeless mountains and the sublime pleasure we get from this could be connected with the harmony of the defeated imagination in the face of reason's demand for unity even of the infinite?
Now one important aspect of reason, on Kant's account, is that it demands unity. So, as Kant puts it, "the idea of comprehending every appearance that may be given us in the intuition of a whole is an idea enjoined on us by a law of reason," which depends on the absolute whole as its measure, a measure that is valid for everyone. So the imagination, in trying to obey this law, fails. The feeling for the sublime is respect for our vocation in the sense that what humans can do best is to have the feeling of the sublime that comes from this failure. In having sublime experience we are respecting the idea of humanity in ourselves and not really the magnitude of the volcano or waterfall that we find sublime. This is just what Kant calls a "subreption." We realize then that the "rational vocation of our cognitive powers" is superior to the greatest power of sensibility, i.e. of our imagination even in conjunction with the understanding. So we feel displeasure because our imagination fails in its estimate of magnitude, i.e. of infinity or absolute totality, which reason poses to it, and yet feel pleasure insofar as we realize that such inadequate power is at least harmonious with reason in trying to do this impossible thing. (I see no reason why pleasure should win out over pain in this instance.) So we estimate large objects in nature, for example the galaxy seen through the Milky Way, as small in relation to absolute totality, and more important, in relation to the ideas of reason, for example the idea of God. What arouses in us a feeling of our human vocation is in harmony with this law.
What is valuable in all of this? I am not sure. I feel attracted to certain ideas, the idea that there is a high point in the aesthetics of nature, that this cannot be tied in any simple way with mathematical or scientific forms of cognition, that this also ties to our appreciation of works of art like not only the pyramid but more interestingly The Night Watch, that that background idea of the world as totality is somehow combined with or tied to our experience of things as wholes, and finally that there is something tragic and painful as well as pleasurable about the failure of our mechanical imagination to capture the sublime, which work needs the supplement of the creative imagination or what Kant calls "reason" or the principle that the whole is greater than the sum of the parts.