«Absolute, non comparative magnum»: The connotation of the mathematical in the Kantian Sublime
Paula Órdenes
Universität Heidelberg, Germany
ORCID: https://orcid.org/0000-0002-4180-9186
DOI: 10.24310/nyl.19.2025.21016
Abstract: In the Analytic of the Sublime in Kant's Critique of the Power of Judgment, the connection between pure aesthetic judgment and mathematics appears at first sight as both surprising and unprecedented within Kantian aesthetics. The sublime is mathematical (as well as dynamical), leaving the attentive reader to ponder how this relationship between the mathematical and a feeling that claims universal communicability is possible. This distinction between mathematical and dynamic forms of the sublime is entirely original and lacks historical precedent. Neither Pseudo-Longinus nor Joseph Addison nor Edmund Burke nor Henry Home nor John Baillie (Scheck 2009, pp. 34ff), nor even Kant himself in his pre-critical work, Observations on the Feelings of the Beautiful and the Sublime (1764),[1] made this distinction between two types of the sublime: mathematical and dynamical. Moreover, Kant emphasizes the autonomy of pure aesthetic judgments from both particular concepts and specific sensations, as neither of these representations constitutes the foundation of aesthetic judgment. Consequently, the origin of the association between the sublime and the mathematical, as well as the dynamical, remains opaque. This article will focus exclusively on the nexus between the mathematical and the sublime, invoking the synthesis of composition from Kant's First Critique as a possible explanation of this Kantian innovation.
Keywords: Kant, Sublime, Mathematical, Synthesis, Composition, Magnitude.
«Absolute non comparative magnum»: La connotación de lo matemático en lo Sublime kantiano
Resumen: En la Analítica de lo Sublime de la Crítica de la Facultad de Juzgar de Kant, la conexión entre el juicio estético puro y las matemáticas aparece, a primera vista, como algo tanto sorprendente como sin precedentes dentro de la estética kantiana. Lo sublime es matemático (además de dinámico), lo que deja al lector atento reflexionando sobre cómo es posible esta relación entre lo matemático y un sentimiento que pretende una comunicabilidad universal. Esta distinción entre las formas matemática y dinámica de lo sublime es completamente original y carece de precedentes históricos. Ni Pseudo-Longino ni Joseph Addison ni Edmund Burke ni Henry Home ni John Baillie (Scheck 2009, pp. 34 y siguientes) ni siquiera el propio Kant en su obra precrítica Observaciones sobre los sentimientos de lo bello y lo sublime (1764), hicieron esta distinción entre dos tipos de lo sublime: matemático y dinámico. Además, Kant subraya la autonomía de los juicios estéticos puros tanto de conceptos particulares como de sensaciones específicas, ya que ninguno de estas representaciones constituye la base del juicio estético. En consecuencia, el origen de la asociación entre lo sublime y lo matemático, así como lo dinámico, sigue siendo opaco. Este artículo se centrará exclusivamente en el nexo entre lo matemático y lo sublime, invocando la síntesis de la composición de la Primera Crítica de Kant como una posible explicación de esta innovación kantiana.
Palabras clave: Kant, Sublime, Matemático, Síntesis, Composición, Magnitud.
Recibido: 17 de diciembre de 2024
Aceptado: 23 de diciembre de 2024
1. Preliminary remarks
Before we begin our search for the basis of the connection between the mathematical and the sublime, it is worth noting that the sublime is introduced in §24 of the Critique of the Power of Judgment as divided into the mathematical and the dynamical. The only textual explanation of this division relates it to the two dispositions (Stimmungen)[2] of the imagination (cf. KU, AA 05: 247). Although there is no further account elsewhere in Kantian philosophy of these dispositions, the mathematical-dynamical division appears at multiple significant moments of the critical philosophy, such as in the division of the categories of the understanding, of the principles of the understanding, of the antinomies etc. For this reason, I consider it apt to begin our search under the hypothesis that a systematic motive underlies such a division. The idea of tracing the division directly back to the division of categories and principles of the Critique of Pure Reason is held by various authors, but with different approaches. Although Park (2008, (pp. 124-135) refers to the possible origin of such a division in the First Critique, he does not explain how this duality applies to the judgment of the sublime. Drawing on the lessons of Kant, he ends up making a psycho-anthropological argument about the movement of the mind (Bewegung des Gemüts), appealing to the distinction between affection and emotion. This distinction is not, however, supported by evidence, as the sublime is neither an affect nor an emotion. Rather, it is an aesthetic judgment that can be instantiated in two distinct ways. Lyotard (1994, pp. 98, 106, 123), on the other hand, offers an explanation and application based on the synthetic differentiation in Kant's First Critique. However, he fails to consider that the sublime pertains to a type of judgment with two moments, and in doing so, he overlooks the fact that these are two distinct judgments that share a single object (the sublime in itself). This leads to a subreption, as they arise from two completely different types of phenomena: the greatness and the power of nature (neither of them represents the sublime in itself). Wenzel (2005, p. 108) hardly mentions the matter and does not pay much attention to it[3]. Elsewhere (Órdenes 2023, pp. 77-85) I have argued that this division is traceable to the type of synthesizing activity of the sensible manifold, since according to this activity it is possible to conceive two types of synthesis of the sensible manifold: homogeneous synthesis (compositio) that corresponds to the mathematical and heterogeneous or nonhomogeneous synthesis (nexus) that corresponds to the dynamical (cf. KrV, B 200-202). In this article I develop this idea further. The very difficulty lies in how to understand this synthetic difference in the experience of the sublime. In this paper I want to merely examine to what extent the criteria of the mathematical (homogeneous) synthesis are applicable to the mathematical sublime. Therefore I will only concentrate on the mathematical sublime.
2. In which sense is the sublime mathematical?
The nominal definition of the sublime is: «We call sublime that which is absolutely great» (KU, AA 05: 248). Something being absolutely great («absolute, non comparative magnum» (ibd.)) implies to be in possession of an absolute magnitude or measure to be able to consider it as such. But what is a magnitude for Kant? In the Analytics of the Sublime he specifies that being a magnitude (quantitas) and being great (magnitudo) are not the same just as it is not the same that something is the greatest in comparison to all other magnitudes: «That something is a magnitude (quantum) may be cognized from the thing itself, without any comparison with another; if, that is, a multitude of homogeneous elements together constitute a unity» (KU, AA 05: 248). To be able to call a «thing» a magnitude is due to the critical result of the application of the mathematical schemes (by means of the figurative synthesis and the synthesis of the pure apprehension of the manifold), which, in my opinion, isn’t conceivable without the homogeneous (mathematical) synthesis. The axioms of intuition and the anticipations of perception result from composition (Zusammensetzung). Mathematical schemes and principles establish the intuitive certainty of cognition via composition, because they concern the evidence and determination of all intuitions a priori (KrV, B 201). In other words, without them there is no formation of an intuition just as there is no effective (pure or empirical) affection. In the KrV Kant defines the first mathematical scheme of imagination on the basis of the pure images of space and time as follows: «The pure image of all magnitudes (quantorum) for outer sense is space; for all objects of the senses in general, it is time. The pure scheme of magnitude (quantitatis), however, as a concept of the understanding, is number» (KrV, A 142/ B 182). And the second mathematical scheme is defined thus: «the scheme of a reality, as the quantity of something insofar as it fills time, is just this continuous and uniform generation of that quantity in time» (KrV, A 143/ B 184). Every sensation has «a degree or magnitude, through which it can more or less fill the same time» (KrV, A 142/ B 182). This constitutes the reality of an object. «Reality is in the pure concept of understanding that to which sensation in general corresponds, that, therefore, the concept of which in itself indicates a being (in time)» (KrV, A 143/ B 182). Magnitude and degree together constitute all objects. Both are quantitas, the former extensive (in space), the latter intensive (in time). «All intuitions are extensive magnitudes» (KrV, B 202) is brought out as a principle of the axioms of intuition just as the principle of the anticipations of perception points to intensive magnitude: «In all appearances the real, which is an object of sensation, has intensive magnitude, i.e., a degree» (KrV, B 207). In mathematical synthesis a continuous and homogeneous manifold is first «apprehended» and then «composed», once directed to the external sense in «aggregation», once directed to the internal sense in «coalition». Mathematics find their transcendental basis in the aggregation and coalition of the homogeneous synthesis. The experience of the mathematical sublime seems to be one where the synthesis of aggregation in its external aesthetic articulation and the synthesis of coalition in its internal aesthetic articulation are brought to their maximum until they collapse. Both syntheses according to our hypothesis about the systematics of the classification of the sublime should occur in the mathematical sublime even if it proceeded empirically. For it is to be assumed that the conditions of the possibility of experience can be applied to actual experience. Kant explains the following in the Analytics of the Sublime:
The imagination, by itself, without anything hindering it, advances to infinity in the composition that is requisite for the representation of magnitude; the understanding, however, guides this by numerical concepts, for which the former must provide the schema; and in this procedure, belonging to the logical estimation of magnitude, there is certainly something objectively purposive in accordance with the concept of an end (such as all measuring is), but nothing that is purposive and pleasing for the aesthetic power of judgment. (KU, AA 05: 253)
The previously explained could lead us to assume that the faculty responsible for comparing magnitudes is the understanding. However, comparison is not a concept of understanding, but reflective judgment (cf. KrV, A 261/B 317).
If (under the above-mentioned restriction) we say of an object absolutely that it is great, this is not a mathematically determining judgment but a mere judgment of reflection about its representation, which is subjectively purposive for a certain use of our cognitive powers in the estimation of magnitude, and in that case we always combine a kind of respect with the representation, just as we combine contempt with that which we call absolutely small. (KU, AA 05: 249)
This means, first, that the power of judgment is responsible for comparisons[4]. Then it is said that it is about an aesthetic subjective-purposeful and not a logical mathematical-determining measurement. Thus, the power of judgment acts without the help of numbers (the magnitude of multiplicity), i.e., not by comprehensio logica but with the help of another measure (the magnitude of unity), i.e., by comprehensio aesthetica, a standard «that one presupposes can be assumed to be the same for everyone, but which is not usable for any logical (mathematically determinate) judging of magnitude but only for an aesthetic one.» (KU, AA 05: 249). Following this an appreciation is stated, namely, that which anyone can call great is associated with «a kind of respect», while that which is called small is associated with a representation of «contempt.»
In apprehension (Auffassung) the logical estimation of magnitude can proceed to infinity «in accordance with an assumed principle of progression» but «not comprehensively» (KU, AA 05: 254). No matter if this principle of progression represents a scale that «can be grasped in a glance, e.g. a foot or a rod, or whether it chooses a German mile or even a diameter of the earth» (KU, AA 05: 254), the imagination cannot aesthetically comprehend the infinite as a totality. The understanding is «equally well served and satisfied» (KU, AA 05: 254), if the imagination chooses a scale for the unity of a magnitude.
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. (KU, AA 05: 252)
The imagination can proceed to infinity in the apprehension (apprehensio) as well as in the logical comprehension («comprehensio logica into a number concept») but not in the aesthetic comprehension (comprehensio aesthetica) (KU, AA 05: 254):
For when apprehension has gone so far that the partial representations of the intuition of the senses that were apprehended first already begin to fade in the imagination as the latter proceeds on to the apprehension of further ones, then it loses on one side as much as it gains on the other, and there is in the comprehension a greatest point beyond which it cannot go. (KU, AA 05: 252)
What is an aesthetic measure? First, it turns out that the «measure by eye» (Augenmaß) is the aesthetic measure - but with a twist[5]. If the measure is the sight, even this needs another measure to orient itself. What is its measure? This can be nothing other than the body, the sensorial substrate of the subject. But if people are of different bodily constitution and have different powers of sight, how can we still lay claim to an aesthetic universal measure of magnitude in that regard? If the sublime is to surpass all ordinary objects in magnitude, we should before making the aesthetic judgment have had an experience of all the objects in the world, so that we can judge their aesthetic magnitude and determine that this X is sublime. In nature all estimates of size are relative, even when we consider merely the scale of ordinary objects. How then can something be called absolutely great?
If, however, we call something not only great, but simply, absolutely great, great in every respect (beyond all comparison), i.e., sublime, then one immediately sees that we do not allow a suitable standard for it to be sought outside of it, but merely within it. It is a magnitude that is equal only to itself. That the sublime is therefore not to be sought in the things of nature but only in our ideas follows from this. (KU, AA 05: 250)
From this it follows clearly that the sublime as absolute greatness has its own measure and that it is not to be looked for in nature, but only in the faculty of the ideas. Which is its own measure? Reason.
That is, it is a law (of reason) for us and part of our vocation to estimate everything great that nature contains as an object of the senses for us as small in comparison with ideas of reason; and whatever arouses the feeling of this supersensible vocation in us is in agreement with that law. (KU, AA 05: 257)
The ultimate surpassing measure is then not the measure by eye, but by reason by means of its ideas. Now we have two measures: nature and reason, with reason being the superior one. But if the sublime is to be found only in our ideas, what role does the contemplation of a colossal object play at all?
Thus nothing that can be an object of the senses is, considered on this footing, to be called sublime. But just because there is in our imagination a striving to advance to the infinite, while in our reason there lies a claim to absolute totality, as to a real idea, the very inadequacy of our faculty for estimating the magnitude of the things of the sensible world awakens the feeling of a supersensible faculty in us; and the use that the power of judgment naturally makes in behalf of the latter (feeling), though not the object of the senses, is absolutely great, while in contrast to it any other use is small. Hence it is the disposition of the mind resulting from a certain representation occupying the reflective judgment, but not the object, which is to be called sublime. (KU, AA 05: 250)
So, can we (intentionally) produce aesthetic purposiveness for ourselves simply by thinking of the ideas of reason? No - the sublime needs the object as an instance of displeasure for the evocation of the ideas in the mind. To think of something and thereby feel a pure mixed pleasure is not one and the same; even if I could feel pleasure in the evocation of a discursive idea this pleasure would necessarily (according to the declination of pleasures) not be a pure aesthetic one but only one associated with interest.
Now the greatest effort of the imagination in the presentation of the unity for the estimation of magnitude is a relation to something absolutely great, and consequently also a relation to the law of reason to adopt this alone as the supreme measure of magnitude. Thus the inner perception of the inadequacy of any sensible standard for the estimation of magnitude by reason corresponds with reason’s laws, and is a displeasure that arouses the feeling of our supersensible vocation in us, in accordance with which it is purposive and thus a pleasure to find every standard of sensibility inadequate for the ideas of the [reason][6]. (KU, AA 05: 258)
The collapse of the empirical synthesis of the comprehension in so far as an appearance is so great can be considered at the same time as a kind of collapse of the synthesis of the composition (Zusammensetzung). Mathematically-schematically, this «sublime» appearance of nature brings the imagination to provide an image of all space and flowing time as a scheme, on the one hand, for the extensive magnitude, the progressive infinity in aggregation, which remains unitary-intangible, and on the other hand, for the intensive magnitude (degree), the affirmation (reality) and negation in coalition, which remains momentary-intangible. Aggregation and coalition fail, because the phenomenon cannot be perceived in an instant as a unity by the imagination. Despite all efforts in apprehension, it fails in an objective view of this whole in the aesthetic comprehension. This inadequacy to achieve one intuition evokes another inadequacy to achieve another representation. Hence, the mind moves, and it is no longer clear whether the lack of the aesthetic comprehension of a phenomenon or the lack of a presentation (Darstellung) of an idea causes the inadequacy of the mind (displeasure):
But now the mind hears in itself the voice of reason, which requires totality for all given magnitudes, even for those that can never be entirely apprehended although they are (in the sensible representation) judged as entirely given, hence comprehension in one intuition, and it demands a presentation for all members of a progressively increasing numerical series, and does not exempt from this requirement even the infinite (space and past time), but rather makes it unavoidable for us to think of it (in the judgment of common reason) as given entirely (in its totality). (KU, AA 05:254)
The mathematical synthesis fails, no image results from this, and the mind has to appeal to something else: «But even to be able to think the given infinite without contradiction requires a faculty in the human mind that is itself is supersensible» (KU, AA 05: 254). Through this faculty «the infinite of the sense world is completely comprehended in the pure intellectual estimation of magnitude under a concept, even though it can never be completely thought in the mathematical estimation of magnitude through numerical concepts» (KU, AA 05: 255). Since this faculty can think «the infinite of supersensible as given, (in its intelligible substratum) surpasses any standards of sensibility, and is great beyond all comparison even with the faculty of mathematical estimation» (KU, AA 05: 255). But not within the theoretical use of the cognitive faculty, because in that case reason must be restrained regarding its cognitive demands, «but still as an enlargement of the mind which feels itself to empowered to overstep the limits of sensibility from another (practical) point of view» (KU, AA 05: 255). The mind is thus moved (bewegt) in its totality, the totality of the inner sense is awakened due to the unattainability of a whole in the comprehension, the being is evoked by the non-being of the subject, whereby the subject feels itself. But it recognizes itself not in the progression of the manifold, but in the claim to totality of its capacity; that claim turns out not to belong to nature, thus the subject remembers its own intelligible, not sensible, nature. Without the metaphysical disposition of the subject, the reversal of the first moment from collapse to breakthrough moment is not possible.
On this interpretation of the experience of the mathematical sublime the mathematical schemata and principles seem to be partially applicable to experience and this is how the mathematical aspect of the sublime should be understood. Nevertheless, it should be emphasized that the parallelism with the schematism and principles chapters of the Transcendental Doctrine of the Power of Judgment of KrV can only be restrictively applied to the aesthetic judgment. In my opinion, the restriction is based on two central aspects of the articulation of the aesthetic judgment: 1. The aesthetic judgment is not a logical one, i.e., it does not determine the object (no subsumption of the aesthetic representation under a concept). 2. The aesthetic judgment is disinterested. It refers to a purposiveness without purpose (neither the existence nor the concept of the object play a role in its pure judgment). Both aspects should be preserved, otherwise the sublime loses its pure status. Kant clearly states in the Deduction of the Judgment of Taste that «the imagination schematizes without concept» (KU, AA 05: 287). Of course no objects can be given to the imagination without the synthetic performance of our spontaneity, but in the aesthetic determination «no concept of the object is here the ground of the judgment» (KU, AA 05: 287), otherwise the ground of pleasure is an indirect representation of the object, and not the formal immediacy of the latter. The mathematical sublime might then be schematized – but only on the assumption that, given the absence of its own deduction, some criteria of the Deduction of Taste can be applied to the judgment of the sublime – but without concepts[7].
3. Conclusion
In summary, the judging of the mathematical sublime takes place as follows: although the imagination can continue in the apprehension (progressus) into the infinite (as continuum), it has an aesthetic maximum in the comprehension (regressus) of the apprehended manifold, which is quickly reached. It fails because it cannot accomplish the synthesis of the reproduction of the apprehended manifold without losing the progressus of the same. It is simply too much for the imagination to apprehend and comprehend all this in a «moment». The manifold of an empirical whole cannot be brought to unity in the view, this produces displeasure in the mind:
we feel ourselves in our mind as aesthetically confined within borders; but with respect to the necessary enlargement of the imagination to the point of adequacy to that which is unlimited in our faculty of reason, namely the idea of the absolute whole, the displeasure and thus the contrapurposiveness of the faculty of imagination is yet represented as purposive for the ideas of reason and their awakening. (KU, AA 05: 259-260)
The object appears as contrapurposive for our aesthetic power of judgment, because its presence points to a contrapurposiveness of the sensible faculty of representation. But this arises in two ways: from the object to the imagination (in the form of displeasure) and from reason to the imagination. Two sensible contrapurposivenesses constitute the subjective purposiveness (pleasure) because both demand the enlargement of the imagination: «So do imagination and reason produce subjective purposiveness [of the powers of the mind] through their conflict»; their subjective play is represented «as harmonious even in their contrast» (KU, AA 05: 258). The expansion of the imagination to the idea of the supersensible frees it from sensibility, where its cognitive limits lie and opens the way for it to a realm where the supersensible acquires meaning not only as a limiting concept, but where it is postulated and a system of virtuos action is built upon it - morality:
This idea of the supersensible, however, which of course we cannot further determine, so that we cannot cognize nature as a presentation of it but can only think it, is awakened in us by means of an object the aesthetic judging of which stretches imagination to its limit, whether that of enlargement (mathematically) or of its power over the mind (dynamically), in that it is grounded in the feeling of a vocation of the mind that entirely oversteps the domain of the former (the moral feeling), in regard to which the representation of the object is judged as subjectively purposive. (KU, AA 05: 268)
This thought alone, to imagine nature as a whole at the presentation of a huge appearance of it[8], expands the «mathematical disposition» of the imagination. Its «dynamical disposition» is also expanded by occasion of a natural phenomenon. This will be dealt with on another occasion. Here our only intention was to reinterpret the meaning of the mathematical in the determination of the sublime in Kant´s Philosophy. To finish with our view of the mathematical sublime: the imagination tries to comprehend the parts of an object given in the intuition, the magnitude of which it cannot summarize in a single sensible representation. To accomplish this task the aesthetic power of judgment will need an idea of reason capable of summing up all possible sensible magnitude. Thus, synthesis concludes in the mathematical judgment of the sublime. The synthesis is homogeneous (therefore, mathematical) because it composes two sensuous non-presentations (Nichtdarstellbarkeiten), one does not appear completely in the mind and the other does not appear in the world.
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Paula Órdenes
paula.ordenes@uni-heidelberg.de