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A.: — Of the Mathematically Sublime - Immanuel Kant, The Critique of Judgement 
Kant’s Critique of Judgement, translated with Introduction and Notes by J.H. Bernard (2nd ed. revised) (London: Macmillan, 1914).
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—Of the Mathematically Sublime
Explanation of the term “sublime”
We call that sublime which is absolutely great. But to be great, and to be a great something are quite different concepts (magnitudo and quantitas). In like manner to say simply (simpliciter) that anything is great is quite different from saying that it is absolutely great (absolute, non comparative magnum). The latter is what is great beyond all comparison.— What now is meant by the expression that anything is great or small or of medium size? It is not a pure concept of Understanding that is thus signified; still less is it an intuition of Sense, and just as little is it a concept of Reason, because it brings with it no principle of cognition. It must therefore be a concept of Judgement or derived from one; and a subjective purposiveness of the representation in reference to the Judgement must lie at its basis. That anything is a magnitude (quantum) may be cognised from the thing itself, without any comparison of it with other things; viz. if there is a multiplicity of the homogeneous constituting one thing. But to cognise how great it is always requires some other magnitude as a measure. But because the judging of magnitude depends not merely on multiplicity (number), but also on the magnitude of the unit (the measure), and since, to judge of the magnitude of this latter again requires another as measure with which it may be compared, we see that the determination of the magnitude of phenomena can supply no absolute concept whatever of magnitude, but only a comparative one.
If now I say simply that anything is great, it appears that I have no comparison in view, at least none with an objective measure; because it is thus not determined at all how great the object is. But although the standard of comparison is merely subjective, yet the judgement none the less claims universal assent; “this man is beautiful,” and “he is tall,” are judgements not limited merely to the judging subject, but, like theoretical judgements, demanding the assent of every one.
In a judgement by which anything is designated simply as great, it is not merely meant that the object has a magnitude, but that this magnitude is superior to that of many other objects of the same kind, without, however, any exact determination of this superiority. Thus there is always at the basis of our judgement a standard which we assume as the same for every one; this, however, is not available for any logical (mathematically definite) judging of magnitude, but only for aesthetical judging of the same, because it is a merely subjective standard lying at the basis of the reflective judgement upon magnitude. It may be empirical, as, e.g. the average size of the men known to us, of animals of a certain kind, trees, houses, mountains, etc. Or it may be a standard given a priori, which through the defects of the judging subject is limited by the subjective conditions of presentation in concreto; as, e.g. in the practical sphere, the greatness of a certain virtue, or of the public liberty and justice in a country; or, in the theoretical sphere, the greatness of the accuracy or the inaccuracy of an observation or measurement that has been made, etc.
Here it is remarkable that, although we have no interest whatever in an Object,—i.e. its existence is indifferent to us,—yet its mere size, even if it is considered as formless, may bring a satisfaction with it that is universally communicable, and that consequently involves the consciousness of a subjective purposiveness in the use of our cognitive faculty. This is not indeed a satisfaction in the Object (because it may be formless), as in the case of the Beautiful, in which the reflective Judgement finds itself purposively determined in reference to cognition in general; but [a satisfaction] in the extension of the Imagination by itself.
If (under the above limitation) we say simply of an object “it is great,” this is no mathematically definite judgement but a mere judgement of reflection upon the representation of it, which is subjectively purposive for a certain use of our cognitive powers in the estimation of magnitude; and we always then bind up with the representation a kind of respect, as also a kind of contempt for what we simply call “small.” Further, the judging of things as great or small extends to everything, even to all their characteristics; thus we describe beauty as great or small. The reason of this is to be sought in the fact that whatever we present in intuition according to the precept of the Judgement (and thus represent aesthetically) is always a phenomenon and thus a quantum.
But if we call anything not only great, but absolutely great in every point of view (great beyond all comparison), i.e. sublime, we soon see that it is not permissible to seek for an adequate standard of this outside itself, but merely in itself. It is a magnitude which is like itself alone. It follows hence that the sublime is not to be sought in the things of nature, but only in our Ideas; but in which of them it lies must be reserved for the Deduction.
The foregoing explanation can be thus expressed: the sublime is that in comparison with which everything else is small. Here we easily see that nothing can be given in nature, however great it is judged by us to be, which could not if considered in another relation be reduced to the infinitely small; and conversely there is nothing so small, which does not admit of extension by our Imagination to the greatness of a world, if compared with still smaller standards. Telescopes have furnished us with abundant material for making the first remark, microscopes for the second. Nothing, therefore, which can be an object of the senses, is, considered on this basis, to be called sublime. But because there is in our Imagination a striving towards infinite progress, and in our Reason a claim for absolute totality, regarded as a real Idea, therefore this very inadequateness for that Idea in our faculty for estimating the magnitude of things of sense, excites in us the feeling of a supersensible faculty. And it is not the object of sense, but the use which the Judgement naturally makes of certain objects on behalf of this latter feeling, that is absolutely great; and in comparison every other use is small. Consequently it is the state of mind produced by a certain representation with which the reflective Judgement is occupied, and not the Object, that is to be called sublime.
We may therefore append to the preceding formulas explaining the sublime this other: the sublime is that, the mere ability to think which, shows a faculty of the mind surpassing every standard of Sense.
Of that estimation of the magnitude of natural things which is requisite for the Idea of the Sublime
The estimation of magnitude by means of concepts of number (or their signs in Algebra) is mathematical; but that in mere intuition (by the measurement of the eye) is aesthetical. Now we can come by definite concepts of how great a thing is, [only]1 by numbers, of which the unit is the measure (at all events by series of numbers progressing to infinity); and so far all logical estimation of magnitude is mathematical. But since the magnitude of the measure must then be assumed known, and this again is only to be estimated mathematically by means of numbers,—the unit of which must be another [smaller] measure,—we can never have a first or fundamental measure, and therefore can never have a definite concept of a given magnitude. So the estimation of the magnitude of the fundamental measure must consist in this, that we can immediately apprehend it in intuition and use it by the Imagination for the presentation of concepts of number. That is, all estimation of the magnitude of the objects of nature is in the end aesthetical (i.e. subjectively and not objectively determined).
Now for the mathematical estimation of magnitude there is, indeed, no maximum (for the power of numbers extends to infinity); but for its aesthetical estimation there is always a maximum, and of this I say that if it is judged as the absolute measure than which no greater is possible subjectively (for the judging subject), it brings with it the Idea of the sublime and produces that emotion which no mathematical estimation of its magnitude by means of numbers can bring about (except so far as the aesthetical fundamental measure remains vividly in the Imagination). For the former only presents relative magnitude by means of comparison with others of the same kind; but the latter presents magnitude absolutely, so far as the mind can grasp it in an intuition.
In receiving a quantum into the Imagination by intuition, in order to be able to use it for a measure or as a unit for the estimation of magnitude by means of numbers, there are two operations of the Imagination involved: apprehension (apprehensio) and comprehension (comprehensio aesthetica). As to apprehension there is no difficulty, for it can go on ad infinitum; but comprehension becomes harder the further apprehension advances, and soon attains to its maximum, viz. the aesthetically greatest fundamental measure for the estimation of magnitude. For when apprehension has gone so far that the partial representations of sensuous intuition at first apprehended begin to vanish in the Imagination, whilst this ever proceeds to the apprehension of others, then it loses as much on the one side as it gains on the other; and in comprehension there is a maximum beyond which it cannot go.
Hence can be explained what Savary1 remarks in his account of Egypt, viz. that we must keep from going very near the Pyramids just as much as we keep from going too far from them, in order to get the full emotional effect from their size. For if we are too far away, the parts to be apprehended (the stones lying one over the other) are only obscurely represented, and the representation of them produces no effect upon the aesthetical judgement of the subject. But if we are very near, the eye requires some time to complete the apprehension of the tiers from the bottom up to the apex; and then the first tiers are always partly forgotten before the Imagination has taken in the last, and so the comprehension of them is never complete.— The same thing may sufficiently explain the bewilderment or, as it were, perplexity which, it is said, seizes the spectator on his first entrance into St. Peter’s at Rome. For there is here a feeling of the inadequacy of his Imagination for presenting the Ideas of a whole, wherein the Imagination reaches its maximum, and, in striving to surpass it, sinks back into itself, by which, however, a kind of emotional satisfaction is produced.
I do not wish to speak as yet of the ground of this satisfaction, which is bound up with a representation from which we should least of all expect it, viz. a representation which lets us remark its inadequacy and consequently its subjective want of purposiveness for the Judgement in the estimation of magnitude. I only remark that if the aesthetical judgement is pure (i.e. mingled with no teleological judgement or judgement of Reason) and is to be given as a completely suitable example of the Critique of the aesthetical Judgement, we must not exhibit the sublime in products of art (e.g. buildings, pillars, etc.) where human purpose determines the form as well as the size; nor yet in things of nature the concepts of which bring with them a definite purpose (e.g. animals with a known natural destination); but in rude nature (and in this only in so far as it does not bring with it any charm or emotion produced by actual danger) merely as containing magnitude. For in this kind of representation nature contains nothing monstrous (either magnificent or horrible); the magnitude that is apprehended may be increased as much as you wish provided it can be comprehended in a whole by the Imagination. An object is monstrous if by its size it destroys the purpose which constitutes the concept of it. But the mere presentation of a concept is called colossal, which is almost too great for any presentation (bordering on the relatively monstrous); because the purpose of the presentation of a concept is made harder [to realise] by the intuition of the object being almost too great for our faculty of apprehension.— A pure judgement upon the sublime must, however, have no purpose of the Object as its determining ground, if it is to be aesthetical and not mixed up with any judgement of Understanding or Reason.
Because everything which is to give disinterested pleasure to the merely reflective Judgement must bring with the representation of it, subjective and, as subjective, universally valid purposiveness—although no purposiveness of the form of the object lies (as in the case of the Beautiful) at the ground of the judgement—the question arises “what is this subjective purposiveness?” And how does it come to be prescribed as the norm by which a ground for universally valid satisfaction is supplied in the mere estimation of magnitude, even in that which is forced up to the point where our faculty of Imagination is inadequate for the presentation of the concept of magnitude?
In the process of combination requisite for the estimation of magnitude, the Imagination proceeds of itself to infinity without anything hindering it; but the Understanding guides it by means of concepts of number, for which the Imagination must furnish the schema. And in this procedure, as belonging to the logical estimation of magnitude, there is indeed something objectively purposive,—in accordance with the concept of a purpose (as all measurement is),—but nothing purposive and pleasing for the aesthetical Judgement. There is also in this designed purposiveness nothing which would force us to push the magnitude of the measure, and consequently the comprehension of the manifold in an intuition, to the bounds of the faculty of Imagination, or as far as ever this can reach in its presentations. For in the estimation of magnitude by the Understanding (Arithmetic) we only go to a certain point whether we push the comprehension of the units up to the number 10 (as in the decimal scale) or only up to 4 (as in the quaternary scale); the further production of magnitude proceeds by combination or, if the quantum is given in intuition, by apprehension, but merely by way of progression (not of comprehension) in accordance with an assumed principle of progression. In this mathematical estimation of magnitude the Understanding is equally served and contented whether the Imagination chooses for unit a magnitude that we can take in in a glance, e.g. a foot or rod, or a German mile or even the earth’s diameter,—of which the apprehension is indeed possible, but not the comprehension in an intuition of the Imagination (not possible by comprehensio aesthetica, although quite possible by comprehensio logica in a concept of number). In both cases the logical estimation of magnitude goes on without hindrance to infinity.
But now the mind listens to the voice of Reason which, for every given magnitude,—even for those that can never be entirely apprehended, although (in sensible representation) they are judged as entirely given,— requires totality. Reason consequently desires comprehension in one intuition, and so the presentation of all these members of a progressively increasing series. It does not even exempt the infinite (space and past time) from this requirement; it rather renders it unavoidable to think the infinite (in the judgement of common Reason) as entirely given (according to its totality).
But the infinite is absolutely (not merely comparatively) great. Compared with it everything else (of the same kind of magnitudes) is small. And what is most important is that to be able only to think it as a whole indicates a faculty of mind which surpasses every standard of Sense. For [to represent it sensibly] would require a comprehension having for unit a standard bearing a definite relation, expressible in numbers, to the infinite; which is impossible. Nevertheless, the bare capability of thinking this infinite without contradiction requires in the human mind a faculty itself supersensible. For it is only by means of this faculty and its Idea of a noumenon,— which admits of no intuition, but which yet serves as the substrate for the intuition of the world, as a mere phenomenon,—that the infinite of the world of sense, in the pure intellectual estimation of magnitude, can be completely comprehended under a concept, although in the mathematical estimation of magnitude by means of concepts of number it can never be completely thought. The faculty of being able to think the infinite of supersensible intuition as given (in its intelligible substrate), surpasses every standard of sensibility, and is great beyond all comparison even with the faculty of mathematical estimation; not of course in a theoretical point of view and on behalf of the cognitive faculty, but as an extension of the mind which feels itself able in another (practical) point of view to go beyond the limit of sensibility.
Nature is therefore sublime in those of its phenomena, whose intuition brings with it the Idea of their infinity. This last can only come by the inadequacy of the greatest effort of our Imagination to estimate the magnitude of an object. But now in mathematical estimation of magnitude the Imagination is equal to providing a sufficient measure for every object; because the numerical concepts of the Understanding, by means of progression, can make any measure adequate to any given magnitude. Therefore it must be the aesthetical estimation of magnitude in which it is felt that the effort towards comprehension surpasses the power of the Imagination to grasp in a whole of intuition the progressive apprehension; and at the same time is perceived the inadequacy of this faculty, unbounded in its progress, for grasping and using, for the estimation of magnitude, a fundamental measure which could be made available by the Understanding with little trouble. Now the proper unchangeable fundamental measure of nature is its absolute whole; which, regarding nature as a phenomenon, would be infinity comprehended. But since this fundamental measure is a self-contradictory concept (on account of the impossibility of the absolute totality of an endless progress), that magnitude of a natural Object, on which the Imagination fruitlessly spends its whole faculty of comprehension, must carry our concept of nature to a supersensible substrate (which lies at its basis and also at the basis of our faculty of thought). As this, however, is great beyond all standards of sense, it makes us judge as sublime, not so much the object, as our own state of mind in the estimation of it.
Therefore, just as the aesthetical Judgement in judging the Beautiful refers the Imagination in its free play to the Understanding, in order to harmonise it with the concepts of the latter in general (without any determination of them); so does the same faculty when judging a thing as Sublime refer itself to the Reason in order that it may subjectively be in accordance with its Ideas (no matter what they are):—i.e. that it may produce a state of mind conformable to them and compatible with that brought about by the influence of definite (practical) Ideas upon feeling.
We hence see also that true sublimity must be sought only in the mind of the [subject] judging, not in the natural Object, the judgement upon which occasions this state. Who would call sublime, e.g. shapeless mountain masses piled in wild disorder upon each other with their pyramids of ice, or the gloomy raging sea? But the mind feels itself elevated in its own judgement if, while contemplating them without any reference to their form, and abandoning itself to the Imagination and to the Reason—which although placed in combination with the Imagination without any definite purpose, merely extends it—it yet finds the whole power of the Imagination inadequate to its Ideas.
Examples of the mathematically Sublime of nature in mere intuition are all the cases in which we are given, not so much a larger numerical concept as a large unit for the measure of the Imagination (for shortening the numerical series). A tree, [the height of] which we estimate with reference to the height of a man, at all events gives a standard for a mountain; and if this were a mile high, it would serve as unit for the number expressive of the earth’s diameter, so that the latter might be made intuitible. The earth’s diameter [would supply a unit] for the known planetary system; this again for the Milky Way; and the immeasurable number of milky way systems called nebulae,—which presumably constitute a system of the same kind among themselves—lets us expect no bounds here. Now the Sublime in the aesthetical judging of an immeasurable whole like this lies not so much in the greatness of the number [of units], as in the fact that in our progress we ever arrive at yet greater units. To this the systematic division of the universe contributes, which represents every magnitude in nature as small in its turn; and represents our Imagination with its entire freedom from bounds, and with it Nature, as a mere nothing in comparison with the Ideas of Reason, if it is sought to furnish a presentation which shall be adequate to them.
Of the quality of the satisfaction in our judgements upon the Sublime
The feeling of our incapacity to attain to an Idea, which is a law for us, is respect. Now the Idea of the comprehension of every phenomenon that can be given us in the intuition of a whole, is an Idea prescribed to us by a law of Reason, which recognises no other measure, definite, valid for every one, and invariable, than the absolute whole. But our Imagination, even in its greatest efforts, in respect of that comprehension, which we expect from it, of a given object in a whole of intuition (and thus with reference to the presentation of the Idea of Reason), exhibits its own limits and inadequacy; although at the same time it shows that its destination is to make itself adequate to this Idea regarded as a law. Therefore the feeling of the Sublime in nature is respect for our own destination, which by a certain subreption we attribute to an Object of nature (conversion of respect for the Idea of humanity in our own subject into respect for the Object). This makes intuitively evident the superiority of the rational determination of our cognitive faculties to the greatest faculty of our Sensibility.
The feeling of the Sublime is therefore a feeling of pain, arising from the want of accordance between the aesthetical estimation of magnitude formed by the Imagination and the estimation of the same formed by Reason. There is at the same time a pleasure thus excited, arising from the correspondence with rational Ideas of this very judgement of the inadequacy of our greatest faculty of Sense; in so far as it is a law for us to strive after these Ideas. In fact it is for us a law (of Reason), and belongs to our destination, to estimate as small, in comparison with Ideas of Reason, everything which nature, regarded as an object of Sense, contains that is great for us; and that which arouses in us the feeling of this supersensible destination agrees with that law. Now the greatest effort of the Imagination in the presentation of the unit for the estimation of magnitude indicates a reference to something absolutely great; and consequently a reference to the law of Reason, which bids us take this alone as the supreme measure of magnitude. Therefore the inner perception of the inadequacy of all sensible standards for rational estimation of magnitude indicates a correspondence with rational laws; it involves a pain, which arouses in us the feeling of our supersensible destination, according to which it is purposive and therefore pleasurable to find every standard of Sensibility inadequate to the Ideas of Understanding.
The mind feels itself moved in the representation of the Sublime in nature; whilst in aesthetical judgements about the Beautiful it is in restful contemplation. This movement may (especially in its beginnings) be compared to a vibration, i.e. to a quickly alternating attraction towards, and repulsion from, the same Object. The transcendent (towards which the Imagination is impelled in its apprehension of intuition) is for the Imagination like an abyss in which it fears to lose itself; but for the rational Idea of the supersensible it is not transcendent but in conformity with law to bring about such an effort of the Imagination, and consequently here there is the same amount of attraction as there was of repulsion for the mere Sensibility. But the judgement itself always remains in this case only aesthetical, because—without having any determinate concept of the Object at its basis—it merely represents the subjective play of the mental powers (Imagination and Reason) as harmonious through their very contrast. For just as Imagination and Understanding, in judging of the Beautiful, generate a subjective purposiveness of the mental powers by means of their harmony, so [here1 ] Imagination and Reason do so by means of their conflict. That is, they bring about a feeling that we possess pure self-subsistent Reason, or a faculty for the estimation of magnitude, whose pre-eminence can be made intuitively evident only by the inadequacy of that faculty [Imagination] which is itself unbounded in the presentation of magnitudes (of sensible objects).
The measurement of a space (regarded as apprehension) is at the same time a description of it, and thus an objective movement in the act of Imagination and a progress. On the other hand, the comprehension of the manifold in the unity,—not of thought but of intuition,—and consequently the comprehension of the successively apprehended [elements] in one glance, is a regress, which annihilates the condition of time in this progress of the Imagination and makes coexistence intuitible.2 It is therefore (since the time-series is a condition of the internal sense and of an intuition) a subjective movement of the Imagination, by which it does violence to the internal sense; this must be the more noticeable, the greater the quantum is which the Imagination comprehends in one intuition. The effort, therefore, to receive in one single intuition a measure for magnitudes that requires an appreciable time to apprehend, is a kind of representation, which, subjectively considered, is contrary to purpose: but objectively, as requisite for the estimation of magnitude, it is purposive. Thus that very violence which is done to the subject through the Imagination is judged as purposive in reference to the whole determination of the mind.
The quality of the feeling of the Sublime is that it is a feeling of pain in reference to the faculty by which we judge aesthetically of an object, which pain, however, is represented at the same time as purposive. This is possible through the fact that the very incapacity in question discovers the consciousness of an unlimited faculty of the same subject, and that the mind can only judge of the latter aesthetically by means of the former.
In the logical estimation of magnitude the impossibility of ever arriving at absolute totality, by means of the progress of the measurement of things of the sensible world in time and space, was cognised as objective, i.e. as an impossibility of thinking the infinite as entirely given; and not as merely subjective or that there was only an incapacity to grasp it. For there we have not to do with the degree of comprehension in an intuition, regarded as a measure, but everything depends on a concept of number. But in aesthetical estimation of magnitude the concept of number must disappear or be changed, and the comprehension of the Imagination in reference to the unit of measure (thus avoiding the concepts of a law of the successive production of concepts of magnitude) is alone purposive for it.—If now a magnitude almost reaches the limit of our faculty of comprehension in an intuition, and yet the Imagination is invited by means of numerical magnitudes (in respect of which we are conscious that our faculty is unbounded) to aesthetical comprehension in a greater unit, then we mentally feel ourselves confined aesthetically within bounds. But nevertheless the pain in regard to the necessary extension of the Imagination for accordance with that which is unbounded in our faculty of Reason, viz. the Idea of the absolute whole, and consequently the very unpurposiveness of the faculty of Imagination for rational Ideas and the arousing of them, are represented as purposive. Thus it is that the aesthetical judgement itself is subjectively purposive for the Reason as the source of Ideas, i.e. as the source of an intellectual comprehension for which all aesthetical comprehension is small; and there accompanies the reception of an object as sublime a pleasure, which is only possible through the medium of a pain.
[1 ][Second Edition.]
[1 ][Lettres sur l’Égypte, par M. Savary, Amsterdam, 1787.]
[1 ][Second Edition.]
[2 ][With this should be compared the similar discussion in the Critique of Pure Reason, Dialectic, bk. ii. c. ii. § 1 , On the System of Cosmological Ideas.]