1

Kant defines the mathematical sublime as the ‘absolutely great’, i.e. a non-comparatively great magnitude (KU, §25, 5: 248). He distinguishes between ‘to be a magnitude (Größe) (quantitas)’ and ‘to be great (groß) (magnitudo)’. The Latin terms indicate the difference between ‘possessing a certain quantity’ and ‘being superior in terms of quantity’. For instance, both a mansion and a cottage are ‘magnitudes’ with measurable sizes, and the house is ‘greater’ in size.

For Kant, we cognize something to be a ‘magnitude’ (Größe) from cognizance of the thing itself, insofar as we regard a magnitude as a ‘unity’ constituted by a ‘multitude of homogeneous elements’ (KU, §25, 5: 248). Thus the magnitude in question is regarded as an extensive magnitude, in which ‘the representation of the parts makes possible the representation of the whole’ (A162/B203). In the Critique of Pure Reason, Kant describes a threefold synthesis that is essential for cognition of objects: first, the imagination ‘apprehends’ an object’s parts successively in the intuition; secondly, the imagination ‘reproduces’ the multitude of these parts altogether as one unity, which possesses an extensive magnitude; and thirdly, the understanding ‘recognizes’ the unity of the initially apprehended and then reproduced parts under a concept (A98–A110). The threefold synthesis grounds the ‘axioms of intuition’, for example, ‘all intuitions are extensive magnitudes’ (A161/B202).

Kant further distinguishes between two methods for estimating a magnitude ‘to be great’. By the first, we estimate the magnitude logically by comparing it with an objective measure, namely, its own part or another magnitude. For instance, we estimate a building as five times higher than each storey it contains, while the latter is two times higher than an average human being. But in this way, the determination is always relative and a greater magnitude is always possible; nothing in this leads us to a sense of anything sublime. And so, Kant introduces the second kind of estimation as follows:

This sentence is rather convoluted and bewildering. To begin with, let me examine a possible interpretation, which I shall soon dismiss. By ‘many others of the same kind’, Kant seems to refer to his notion of an ‘aesthetic normal idea’, that is, ‘an individual intuition (of the imagination) that represents the standard for judging it as a thing belonging to a particular species’ (KU, §17, 5: 233). Just as Kant characterizes the ‘merely subjective standard’ here as ‘assumed to be the same for everyone’, he declares an aesthetic normal idea to be ‘a universal standard’ (KU, §17, 5: 233). Moreover, while he exemplifies the former by ‘the average magnitude of the people known to us’ (KU, §25, 5: 249), he exemplifies the latter by ‘the average size’ of a thousand grown men (KU, §17, 5: 234). Accordingly, the judgement of the simply great is ‘aesthetic’ because it compares an object with a subjective standard or aesthetic normal idea, which does not refer to any individual object. And so, we estimate the object’s magnitude as superior but do not determine this superiority, insofar as we do not compare the magnitude with any determinate standard.

I believe this is what underlies Allison’s contention that, when characterizing something simply as great, we are implying that ‘its magnitude is greater than that of many other objects of the same kind, even though this superiority is not assigned a determinate numerical value’ (Allison Reference Allison2001: 312); in other words, we compare the magnitude of the object to that of its kindred ones but without mathematical precision. Crowther (Reference Crowther1989: 88), Park (Reference Park2009: 133) and Smith (Reference Smith2015: 102) offer similar readings.

But I find this approach untenable in two respects. First, by comparing a magnitude with an aesthetic normal idea, we can still determine its magnitude, with or without a determinate numerical value. The judgement ‘Jordan is much taller than an average American man’ is as determining as the judgement ‘Jordan is 23 cm taller than an average American male’ or ‘Jordan is 5 cm shorter than Williams’, for they all state an objective fact about Jordan’s magnitude. Hence, by an aesthetic normal idea, Kant refers to a non-conceptual intuition which is still useable for logical estimation.

Secondly, Kant states that the aesthetic or ‘merely subjective standard’ in the simply-called-great can be ‘given a priori’. One such example is the magnitude of a ‘certain virtue’ (KU, §25, 5: 249), which is, paradoxically, conceptual or objective in relation to the moral law. In what sense does Kant call such an objective standard ‘subjective’ in the aesthetic judgement? Apparently in terms of its usage for subjective (i.e. aesthetic) estimation of magnitude. Put differently, it is the standard’s very involvement in the subjective estimation that enables Kant to call the standard ‘subjective’ – not the other way around. Therefore, the estimation must be aesthetic on account of some sensible element other than the ‘subjective standard’ involved, in reference to ‘many others of the same kind’.

What is this sensible element? Kant gives a hint: when something is simply judged as great, a magnitude is ‘attributed to it to a superior extent’. Why does he not straightforwardly state that ‘we attribute a superior magnitude to the object’ or ‘we judge the object’s magnitude to be superior’? To show the subtlety in Kant’s phrasing, I propose that we break his underlying reasoning into three steps.

To estimate something simply as great: first, we represent the object as having an extensive magnitude (i.e. as a multitude of units) and perceive some non-conceptual consciousness therein, which is presumably a kind of sensation. Insofar as the representation does not determine the object, it cannot be a sensation of the object, such as the sweetness of sugar, which would constitute a kind of knowledge of sugar; rather, it must be an inner sensation, which derives from a representational act in our inner sense.

Secondly, we compare the degree of this sensation in representing this object with something else as its measure, namely, the degrees of sensations in representing many other objects of the same kind, and we judge the former to be superior.

Thirdly, we represent the first object superiorly, that is, when we represent it as having a magnitude, we represent it by way of a superior sensation. And what we estimate and compare are in fact really not extensive magnitudes of objects but only intensive magnitudes or degrees of sensations. Put differently, what is superior is not the object’s magnitude itself but the degree of sensation in representing this magnitude, such that the judgement is not determining but only reflecting with regard to the object. It would be a subreption to mistake the superiority in the subject’s sensation as a characteristic of the object, even though the former is related to the latter, much in the same way that one’s satisfaction in sugar is related to its sweetness.

To illustrate: when we represent the average magnitude of most buildings under normal circumstances, we perceive some inner sensation; then, when we represent the magnitude of the Eiffel Tower from an aircraft at high altitude, we also perceive a sensation which might be inferior to the former in degree. Now, by comparing these two degrees of sensations, we describe the tower simply (i.e. aesthetically rather than conceptually) as small. In other words, we attribute our representation of the tower’s magnitude to the tower inferiorly, insofar as the representation is accompanied with an inferior sensation. Yet even a child can estimate vaguely, without precise numerical value, that the tower is objectively much higher than most buildings. To describe something simply as great, we represent its magnitude with a superior sensation; we do not determine the magnitude itself insofar as we do not directly compare it with another magnitude. The subjective superiority in aesthetic estimation should be strictly distinguished from the objective superiority in logical estimation.

There must then be a certain kind of inner sensation that facilitates aesthetic estimation of magnitude in general. But what would be this sensation? Moreover, by comparing degrees of sensations, we seem to lack an ‘absolute concept of magnitude’ as much as we do in logical estimation. What would be the absolutely great in aesthetic estimation? Sections 2 and 3 will investigate these two problems in turn.

2

This section explains the mental operation in aesthetic estimation. As we shall see, the operation involves the sensation of a temporal tension in the imagination.

Logical estimation of magnitude presupposes an objective measure, but estimation of this measure requires still another measure, and so on and so forth. Thus Kant claims that, ultimately, the basic measure must be obtained in an aesthetic representation (KU, §26, 5: 251). He then distinguishes between two actions in an aesthetic representation, namely, the imagination’s ‘apprehension (apprehensio)’ and its ‘comprehension (comprehensio aesthetica)’ (KU, §26, 5: 25).

The notion ‘comprehension’ here, as an action of the imagination, might seem problematic; for Kant also defines comprehension as ‘the synthetic unity of the consciousness of this manifold [of intuition] in the concept of an object (apperceptio comprehensiva)’, which requires not only the imagination but also the ‘understanding’ (EEKU, §7, 20: 220). But we should notice that Kant specifies the comprehension in aesthetic estimation as ‘comprehensio aesthetica’, while he claims that mathematical or logical estimation of magnitude involves ‘comprehensio logica’ (KU, §26, 5: 254). As I see it, by apprehension (Auffassung) the mind ‘seizes on’ a multitude of impressions or elements of intuition, and by comprehension (Zusammenfassung) it further ‘takes’ them ‘altogether’. Therefore, I identify comprehension in general with a higher stage of synthesis than apprehension: it is either ‘aesthetica’ and corresponds to the imagination’s reproduction of apprehended elements,Footnote ⁴ or ‘logica’ and corresponds to the understanding’s recognition of the reproduced elements under a concept.

According to Kant, while the imagination’s apprehension may advance towards infinity, its aesthetic comprehension becomes more and more difficult (KU, §26, 5: 251–2). He elaborates the whole mental operation in a crucial but rather dense text:

I break Kant’s reasoning into four steps as follows.

First, apprehension is successive. For Kant, to apprehend a manifold of intuition, we must ‘distinguish the time in the succession of impressions on one another’ (A99).Footnote ⁶ The distinction of time is necessary not because the existence of the impressions are objectively successive, but because, to regard them as individual elements, we must apprehend them one by one in different moments. To illustrate: in observing a house, I may first take notice of the door, then the window, and lastly the roof. Even though I may eventually recognize these elements as objectively coexistent, I must apprehend them successively in the first place; otherwise I would only obtain one impression of the whole house rather than a multitude of impressions of its parts. The imagination’s apprehension always relies on this temporal condition, even though the lapses between successive moments could be minimal (provided that the moments are still distinguishable).

Secondly, comprehension is regressive and successive. Since the ‘comprehension’ in question concerns not ‘thought’ but only ‘intuition’, I take it as the imagination’s aesthetic comprehension (comprehensio aesthetica) or its reproduction, which is a ‘regression’. According to the first Critique, a ‘regressive’ synthesis proceeds from the conditioned towards more and more remote conditions, while a ‘progressive’ synthesis proceeds in the opposite direction (A411/B438). To apprehend the individual parts of an object is also to apprehend the spaces they occupy and to measure a space, which is ‘a progression’. For example, in measuring the space occupied by the house, our imagination apprehends the door, the window and then the roof progressively in three successive moments. On this basis, we reproduce and retain the apprehended impressions and their corresponding spatial parts in a reverse order, as we always start from the impression we are now apprehending, move to the one just apprehended, and then to another one apprehended even earlier, and so on and so forth. In this way, the imagination reproduces the roof, the window and lastly the door regressively in three successive moments, and this regression happens through retaining earlier times rather than merely tracing them. Thus successiveness applies to both stages of synthesis: the longer the progressive apprehension takes, the longer the regressive reproduction or comprehension.Footnote ⁷

Thirdly, aesthetic comprehension, qua regressive and successive reproduction, is nevertheless simultaneous. The imagination aims to comprehend the apprehended elements simultaneously as one unity. This simultaneity does not conflict with the successive apprehension, for comprehension is a higher stage of synthesis than apprehension. But the simultaneity indeed conflicts with or ‘cancels’ the successive time-condition underlying both the progression and the regression. Since all our imaginative representations, qua ‘modifications of the mind’, belong to inner sense (A98), the form of which is time (A33/B49), the aesthetic comprehension does ‘violence’ (Gewalt) to the condition of inner sense.Footnote ⁸ Put differently, we are to reproduce individual elements regressively one after another while comprehending them altogether in one intuition. The latter movement violates the former, which means a tension between the two time-conditions.

On Smith’s reading, when the mind fails to take up the intuition ‘simultaneously’, our imagination as ‘temporally progressive’ finds itself to be ‘opened’, such that it will ‘advance towards infinity’ (Smith Reference Smith2015: 114). I consider this interpretation untenable in two respects. First, since Kant explicitly states that there is ‘no difficulty with apprehension, because it can go on to infinity’ (KU, §26, 5: 251–2), the imagination’s progressive apprehension does not need to be ‘opened’ at all. Second, since apprehension and reproduction are two distinct stages in the threefold synthesis, the imagination’s successive progression can be neither cancelled nor ‘opened’ by its simultaneous (and yet successive) regression.

My reading of the ‘temporal tension’ might appear counterintuitive, as it seems quite natural for us to comprehend several elements simultaneously without perceiving any succession. For example, once we apprehend three colours in a flag, we seem to comprehend them all separately and instantly without any noticeable tension. This leads to the final point.

Fourthly and lastly, the tension intensifies only gradually when we comprehend more and more units in one intuition. The tension is ‘all the more marked’ when the quantum is aesthetically ‘greater’. In the flag example, in fact, the imagination must recollect the three colours in three different moments, which means a succession of events in a succession of moments. And yet, we may take them as one moment insofar as the succession is almost undiscernible. After all, the distinction between these three moments, which enables the distinction between the colours qua three individual elements, is not so much phenomenal as logical. Just as we may neglect this tension when it is minimal, we are able to perceive but tolerate it to some extent, which makes cognition possible in the first place; for otherwise we would not be able to comprehend even two elements. Nevertheless, when the imagination takes a significant time to apprehend progressively a significant number of elements (say, ten colours), it must equally take some significant time to reproduce them regressively, which conflicts with its task of simultaneous comprehension.

The sensation of this temporal tension in inner sense, then, may be reasonably supposed to be the sensible element or the subjective aspect of consciousness that grounds aesthetic estimation in general. The remaining question is how to acquire the ‘absolute concept of magnitude’ in aesthetic estimation, to which my next section will turn.

3

On aesthetic comprehension, Kant writes:

As discussed, the successive time-condition in the imagination’s apprehension also applies to its simultaneous comprehension. Therefore, the more representations or elements the progressive apprehension obtains, the greater tension the regressive comprehension undergoes.Footnote ⁹ Suppose the imagination already yields its maximal capacity and becomes incompetent to regress any further or to reproduce any more ‘representations of the intuition’; the representations ‘apprehended first’ must then remain un-reproduced and begin to ‘fade’. In the first Critique, Kant also writes that ‘if I were always to lose the preceding representations … from thoughts and not reproduce them when I proceed to the following ones, then no whole representation … could ever arise’ (A102). When the imagination reaches a ‘greatest point’, it comprehends and ‘gains’ a newly apprehended impression on one side but fails to reproduce and thus ‘loses’ a previously apprehended impression on the other. In this case, the temporal tension and the sensation thereof must be absolutely great, which provides us with an ‘absolute concept of magnitude’ in aesthetic estimation, namely, the mathematical sublime.

To illustrate, in the aesthetic comprehension of an Egyptian pyramid, suppose the imagination is only capable of reproducing nine stone tiers in one intuition, then, once the mind apprehends the tenth tier in the pyramid, it is only able to reproduce regressively from the tenth to the second tier, while the tier apprehended first begins to fade. Otherwise, the mind would have to reproduce ten impressions successively and also simultaneously in one intuition, and the tension would exceed the imagination’s limit. Consequently, we fail to represent the complete form of the pyramid. Indeed, the mathematical sublime is to be found in the formlessness of things (KU, §23, 5: 244).

Clewis proposes that Kant may be using ‘a concept of form that is frequently found in eighteenth-century aesthetics’, which designates ‘symmetry, harmony, proportion, or unity’, such that a sublime object ‘would have an external shape or form detectable by the senses, but would be formless insofar as it lacks symmetry, harmony, and so on’ (Clewis Reference Clewis2009: 71).Footnote ¹⁰ For Guyer (Reference Guyer1996: 264), it is also the sublime object that is formless and unbounded by, say, a picture frame. To the contrary, I ascribe the formlessness to a cognitive inadequacy in the judging subject. For Kant, the sublime may ‘appear in view of the form’ (der Form nach Footnote ¹¹) to be ‘unsuitable for our faculty of presentation’ (KU, §23, 5: 245), and the judgement of the sublime makes ‘use of certain sensible intuitions in view of their form (ihrer Form nach)’ (EEKU, §12, 20: 250).Footnote ¹² Accordingly, we judge an object to be sublime not ‘for its unsuitable form’ (i.e. wegen seiner Form) but in view of its formlessness, that is, when we fail to grasp its form in one intuition.

In line with Allison (Reference Allison2001: 312), I thus consider the judgement of the mathematical sublime to be a special case of aesthetic estimation in general. To the contrary, Park (Reference Park2009: 134) argues that the simply great cannot be a prototype of the mathematical sublime, because in judging an object simply as great ‘the imagination can apprehend its form, especially its extended shape’. But in my view, even when the imagination aesthetically comprehends an object’s entire form, we still perceive a temporal tension or ‘violence to the inner sense’. When the tension becomes so great that the imagination cannot grasp the object’s form, we estimate its magnitude simply or aesthetically as sublime. As Kant puts it, in simply judging an object to be great, we may feel satisfaction ‘even if (selbst wenn) it is considered as formless’ (KU, §25, 5: 249; my emphasis), which means we do not necessarily consider it as formless.

This analysis enables us to understand Savary’s report on Egyptian Pyramids, as quoted by Kant: ‘in order to get the full emotional effect of the magnitude of the pyramids one must neither come too close to them nor to be too far away’ (KU, §26, 5: 252). Kant’s subsequent comments are intriguing but controversial. I consider the two scenarios as follows.

First, Kant claims that, when we stand too far away, ‘the parts that are apprehended (the stones piled on top of one another) are represented only obscurely, and their representation has no effect on the aesthetic judgment of the subject’ (KU, §26, 5: 252). When the stone tiers appear puny in vision and undistinguishable from each other, we cannot apprehend them one by one and thus do not reproduce them qua individual elements in different moments; instead, we apprehend the entire pyramid as a multitude of merely a few impressions or simply as one blur, the regressive reproduction of which takes merely a few moments or just one instant and does not challenge the time-condition of simultaneity in aesthetic comprehension.

Second, when we stand too close, says Kant, ‘the eye requires some time to complete its apprehension from the base level to the apex, but during this time the former always partly fades before the imagination has taken in the latter, and the comprehension is never complete’ (KU, §26, 5: 252). This assertion bewilders commentators. Crowther (Reference Crowther1989: 103) contends that the text is ‘puzzling’ and effectively ‘disproving’ Savary’s claim by showing how ‘our capacity for comprehension is soon overwhelmed’ when we come too close. Similarly, Pillow (Reference Pillow2000: 74) states that the fact that ‘the comprehension is never complete’ signifies exactly the maximum of the imagination’s capacity. Accordingly, we should encounter the same inadequacy in judging a proximate pyramid just as in judging a magnitude which we would call sublime. Should this be true, Kant would be undermining his own theory – in order to get the full emotional effect, we should approach the pyramid as close as possible.

I find Pillow’s argument problematic. Indeed, if it takes too long for us to look upwards, the previous impressions of the pyramid’s base begin to ‘fade’ when the apprehension progresses to its apex. But we should distinguish between two uses of the term ‘fade’ in Kant’s text: first, there is ‘fading’ due to the imagination’s limit, as in Kant’s writing of how the elements apprehended earlier ‘begin to fade in the imagination as the latter proceeds on to the apprehension of further ones’ (KU, §26, 5: 252, my emphasis). Second, there is ‘fading’ as in Kant’s comments on Savary’s report: when we stand too close, the element apprehended first ‘fades before the imagination has taken in’ a new element (my emphasis). On my reading, the fading in the second case is not due to the imagination’s limit; rather, it is our memory that is inadequate to maintain the impressions in mind during such a long period. In other words, when we try to comprehend all the stone tiers, the earlier impressions of the bottom tiers are already too faint or simply forgotten and for this reason cannot be reproduced. As a result, ‘the comprehension is never complete’, as it never obtains a complete multitude. The partial multitude is not too great but rather too small for the imagination to yield its full capacity. Even from an ideal stance for judging the sublime – suppose we apprehend the tiers very slowly, say, one per minute – the early impressions would also fade in memory when the eyes advance to the apex.Footnote ¹³

In contrast to aesthetic comprehension, Kant declares the tension to be relieved in a logical comprehension, where the imagination provides schemata for the understanding’s numerical concepts (KU, §26, 5: 253). Kant defines a schema as the ‘representation of a general procedure of the imagination for providing a concept with its image’ (A140/B179). In accordance with a concept, a schema describes the method or rule for presenting images. The schema of magnitude is number, namely, ‘a representation that summarizes the successive addition of one (homogeneous) unit to another’ (A142/B182). The schema of the number ten does not refer to any particular image, such as ten dots or ten people; it only describes the method of tenfold successive additions of homogeneous elements. The understanding’s concept of ten guides the imagination to generate this schema, regardless of which particular impressions should realize the ten elements in an image.

Therefore, to comprehend the pyramid logically, the imagination still apprehends the tiers successively but ascribes them to a numerical concept rather than intuitions. In other words, when the imagination counts the tenth tier, it comprehends this tier along with the schema of the number nine (which corresponds to the concept of nine) and thus brings only two elements (i.e. the tenth tier and the schema) into a unity, which is then the schema of the number ten and referred to the concept of ten. The reproduction of merely two elements is hardly challenging. Relying on schemata and concepts, the imagination is barely enlarged, however great a number it counts. Thus Kant proceeds to claim that the logical comprehension can proceed ‘unhindered to infinity’ (KU, §26, 5: 254). By contrast, the aesthetic comprehension is ‘not of thought but of intuition’, in which case the imagination reproduces the tenth tier and the intuitions of the previous nine through ten moments, yetalso in one moment.

In the Transcendental Deduction in the first Critique, Kant already indicates the two time-conditions in cognition, as he writes that we add the units or parts of an object to each other ‘successively’ so that they hover before our senses ‘now’, that is, simultaneously (A103). But the temporal tension is relieved when we comprehend or reproduce the units according to a concept, such that we do not perceive any tension in a conceptual comprehension of an object. Nevertheless, insofar as logical estimation is always relative and its primary measure or basic unit must be acquired in aesthetic estimation, ‘in the end all estimation of magnitude of objects of nature is aesthetic’ (KU, §26, 5: 251), that is, involving a felt tension.

4

I have shown how the sensation of the imagination’s temporal tension grounds aesthetic estimation in general and how, when it comes to involve a sense of attempting to exceed the imagination’s limit, the sensation reaches its maximal degree and provides us with the ‘absolute concept of magnitude’. In this section, I will further investigate how this absolutely great sensation evokes the ‘negative pleasure’ in the judgement of the mathematical sublime.

That the feeling is ‘negative’ is easy to explain. When the imagination is unable to comprehend an object’s form, it fails to achieve a cognitive aim. For Kant, ‘The attainment of every aim is combined with the feeling of pleasure’ (KU, 5: 187). The failure of such an attainment, then, constitutes the displeasure in the sublime feeling. On the other hand, when the imagination successfully comprehends a form and fulfils its task, the temporal tension only causes violence to inner sense but not any displeasure.

What is more interesting is Kant’s account of pleasure ‘from the correspondence of this very judgment of the inadequacy of the greatest sensible faculty in comparison with ideas of reason’ (KU, §27, 5: 257). Two questions arise: (1) how is the judging of something sensible, such as a pyramid, referred to ideas of reason? (2) why should this reference bring about pleasure? The answers are indicated in two key texts respectively. The first:

For Kant, reason’s ideas give the understanding’s concepts ‘that unity which they can have in their greatest possible extension, i.e. in relation to the totality of series’ (A643/B671). The totality of all appearances would have a magnitude that comprises an infinite multitude of units.Footnote ¹⁴ Since this multitude cannot be entirely given in our intuition, it is the object of an idea. Kant ascribes to this idea a ‘necessary regulative use’ in directing our understanding to a cognitive ‘goal’ (A644/B672). The mind hears this ‘voice of reason’ and attempts to present the idea, that is, to apprehend and then comprehend all units of this series aesthetically or ‘in one intuition’.Footnote ¹⁵ But what is crucial, beyond this, for our present purposes is that when our imagination fails to grasp the form of a certain finite magnitude in aesthetic comprehension, we thereby also come to ‘judge’ or ‘think of’ the infinite as ‘given entirely’.

Thus the situation is the following. On the one hand, we must judge that, if the infinite were given in our sensibility, its aesthetic comprehension would occasion an absolutely great temporal tension in the imagination. On the other hand, in representing the finite magnitude, our imagination encounters an inadequacy or ‘greatest point’ where it cannot proceed any further, such that the tension it undergoes is also absolutely great. Hence, when we compare the temporal tension in comprehending the finite magnitude with the supposed tension in comprehending the infinite, we consider them equivalent in degree. And so, in aesthetic estimation, we describe the finite magnitude to the same ‘superior extent’ (KU, §25, 5: 249) as we would describe the infinite; put differently, we judge the former aesthetically, in terms of the absolutely great sensation, to be tantamount to the latter.

While we ‘think of’ the infinite as entirely given, what is actually given is only the absolutely great tension (in comprehending something finite) rather than the absolutely great object (i.e. the infinite). Although the infinite ‘can never be entirely apprehended’, it is nonetheless in a certain sense ‘in the sensible representation’, namely in the aesthetic comprehension of the finite, ‘judged as entirely given’. Strictly speaking, therefore, what we judge as mathematically sublime is only the maximal tension and the maximal sensation thereof. It is not the infinite, which is absolutely great but never given to us, letalone the finite magnitude, which is given but never absolutely great in itself.Footnote ¹⁶

In the second key text, Kant then explains how this sort of reference to infinity generates pleasure:

In judging the mathematical sublime, we think of the infinite as ‘given in sensible representation’ and regard the imagination’s failure as an unsuccessful attempt to comprehend the infinite. Since ‘striving’ for ideas of reason is ‘a law for us’, the imagination’s inadequacy for presenting the idea of infinity (and, by extension, ideas in general) is a mental disposition that ‘corresponds with reason’s laws’. As such, it reveals the imagination’s ‘supersensible vocation’, namely, its determination by reason for ‘adequately realizing’ ideas (KU, §27, 5: 257). Given that we also strive to realize practical ideas in the sensible world, Kant describes this disposition as akin to or compatible with ‘that which the influence of determinate (practical) ideas on feeling would produce’ (KU, §26, 5: 256). As Allison points out, the feeling of the superiority of theoretical reason to sensibility ‘serves as a reminder’ of a similar superiority of practical reason and thus of our moral autonomy (Allison Reference Allison2001: 326; cf. Matherne Reference Matherne, Heide and Tiffanyforthcoming). Put differently, the judgement of the mathematical sublime is an indirect representation of the practical purposiveness in ourselves.

Insofar as practical reason aims to determine (bestimmen) our acts in the sensible world, its successive determination or vocation (Bestimmung) with regard to sensibility fulfils an aim, and consciousness of this fulfilment is pleasure. Even consciousness of the vocation of theoretical reason, as in the case of the mathematical sublime, is an indirect representation of the practical fulfilment and thus pleasure. On the other hand, Kant considers the judgement of the dynamical sublime to be the direct, though still reflecting, representation of practical purposiveness, as it relates the mental movement not to an idea of theoretical reason (i.e. the infinite) but immediately to the faculty of desire (KU, §24, 5: 247).

Thus far I have limited myself to an interpretation of Kant’s theory, but now I would like to point out a problem therein. Kant claims that judgements of the mathematical sublime are ‘necessary’ and ‘universally valid’ (KU, §24, 5: 247)Footnote ¹⁷, because we find in them

A judgement of the mathematical sublime represents the purposive relation between the imagination and reason, insofar as the former strives to present the latter’s idea of infinity but fails. Reason is superior to sensibility. This relation grounds the will a priori, because the will as such is to determine our power of choice a priori, even though the latter can be affected by sensibility. ‘Hence’, the relation is purposive a priori.

So far so good, until Kant states that this purposive relation ‘immediately contains’ the deduction of the judgement of the sublime. His reasoning appears to be that, insofar as the judgement represents a kind of a priori purposiveness, it must be based a priori and thus universally necessary. However, it is one thing that the purposiveness is universal and necessary, but quite another that the representation of this purposiveness is also universal and necessary. A judgement of the mathematical sublime represents the a priori purposive relation not in thoughts but through a sensible element: the temporal tension in an aesthetic comprehension reaches the limit of one’s imagination and brings about a certain sort of feeling of maximal violence. Accordingly, universality and necessity of the judgement would presuppose universality and necessity of one’s imagination’s maximum. But this is problematic.

Cassirer argues that the judgement’s deduction consists in its exposition because the latter ‘has shown that the human mind, as possessed of imagination and Reason, is capable of relating them to each other, of becoming aware of its supersensible capacity on the presentation of a sensible object’ (Reference Cassirer1938: 249–50).Footnote ¹⁸ But the problem is exactly the ‘becoming aware’, which presupposes the sensation of the imagination’s inadequacy in judging certain finite magnitudes. This inadequacy, however, seems to be private and contingent.

Kant writes that ‘[aesthetic] comprehension becomes ever more difficult the further apprehension advances, and soon reaches its maximum’ (KU, §26, 5: 252), but he provides no deduction for this maximum. For Kant, ‘Every necessity has a transcendental condition as its ground’ (A106). I have shown that aesthetic comprehension involves an inevitable tension that consists in two a priori temporal conditions of reproductive synthesis, implied in the Transcendental Deduction in the first Critique and then expounded in the third Critique. The tension is indeed universal and necessary but can be tolerated in cognition to some extent, for otherwise no comprehension would be possible. The imagination reaches a maximum if it is incompetent to overcome a tremendous tension, but why must the maximum be the same to all judging subjects? Moreover, in representation of sensible objects, why must there be a maximum at all? Why can the imagination not reproduce in one intuition more and more elements with greater and greater hindrance but still advance towards infinity?

Certainly, we would universally and necessarily fail in an attempt at aesthetic comprehension of the infinite. However, we encounter the mathematical sublime in representing certain finite magnitudes such as a pyramid. While the imagination’s inferiority to reason and its inability for presenting ideas are indeed a priori, its inadequacy for the aesthetic comprehension of certain sensible objects is only a posteriori and, therefore, neither universal nor necessary.Footnote ¹⁹ Paradoxically, it is exactly by means of a private and contingent experience that one reveals the necessary and universal supersensible vocation in humanity.