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Оглавление2.Kant’s pre-critical notion of schema
Before dealing with the use of the notion of schema in the Critique of Pure Reason, it is important to realise that its use is not limited to Kant’s opus magnum, but it is present in pre-critical (as well as in later) works of Kant, as I will demonstrate in this second chapter.
The literature about the meaning and uses of the term ‘schema’ before the Critique of Pure Reason is very scarce. The Kant-Lexikon ignores the problem, while in the Historisches Wörterbuch der Philosophie from 1992 Stegmeier stresses the presence of the notion in two pre-critical works: the Nova Dilucidatio and the Dissertation from 1770, De mundi sensibilis atque intelligibilis forma et principiis. However, they do not go into the details of Kant’s uses and changes of the meaning of the term. Besides, an interesting paper written by Alba Jiménez Rodriguez from 2016 (Die Projektion des Schematismus in den vorkritischen Schriften Kants: Das Problem der mathematischen Konstruktion), focuses on the anticipation in the pre-critical works of a kind of schematism intended as a constructive process of the imagination similar to that of mathematical47 construction. More specifically, Rodriguez points out that Kant’s pre-critical use of ‘schema’ might be related to the Baconian concepts of schematism. As shown in the previous chapter, Bacon’s notion of schema refers to the structure of nature or the ways in which properties are related to each other and ordered in the different substances. According to Rodriguez, the Baconian meaning of ‘schema’ as ←41 | 42→transformation, a building process, has influenced Kant’s notion of schematism, rooted in his account of the mathematical process of construction in his pre-critical works. While Kant does not yet use the terminology of schematism, relevant aspects of the problem of applying pure concepts to experience are already present. In the Inquiry concerning the Distinctness of the Principles of Natural Theology and Morality (Untersuchung über die Deutlichkeit der Grundsätze der natürlichen Theologie und der Moral) from 1764, where the mathematical method is regarded as a way of developing and using rules of construction, in a way similar to the drawing of geometrical figures:
“There are two ways in which one can arrive at a general concept: either by the arbitrary combination of concepts, or by separating out that cognition which has been rendered distinct by means of analysis. Mathematics only ever draws up its definitions in the first way. For example, think arbitrarily of four straight lines bounding a plane surface so that the opposite sides are not parallel to each other. Let this figure be called a trapezium. The concept which I am defining is not given a priori to the definition itself; on the contrary, it only comes into existence as a result of that definition. Whatever the concept of a cone may ordinarily signify, in mathematics the concept is the product of the arbitrary representation of a right-angled triangle which is rotated on one of its sides. In this and in all other cases the definition obviously comes into being as a result of synthesis. The situation is entirely different in the case of philosophical definitions. In philosophy, the concept of a thing is always given, albeit confusedly or in an insufficiently determinate fashion. The concept has to be analysed; the characteristic marks which have been separated out and the concept which has been given have to be compared with each other in all kinds of contexts; and this abstract thought must be rendered complete and determinate.” (AA II, p. 276)48←42 | 43→
According to these remarks, the method used in mathematics is synthetic, insofar, as its concepts (for instance a ‘triangle’) are results of their definitions. In contrast, in philosophy concepts are already given but they are unclear and undetermined and therefore need to be analysed49. However, in philosophy there is a question that has to be answered using a method similar to a mathematical one: for instance, taking claims such as: “7+5=12”, philosophy has to show how they are related to experience. In order to achieve this task, Kant relies on a method through which objects are constructed following rules of the understanding, which is similar to the mathematical method (Jiménez Rodríguez 2016, p. 440). This is a point which Kant consistently sticks to, as shown in the Critique of Pure Reason and in a footnote contained in the late On a Discovery whereby any New Critique of Pure Reason is to be made Superfluous by an Older One (Über eine Entdeckung, nach der alle neue Kritik der reinen Vernunft durch eine ältere entbehrlich gemacht werden soll):
“Hence it is also requisite for one to make an abstract concept sensible, i.e. display the object that corresponds to it in intuition, since without this the concept would remain (as one says) without sense, i.e. without significance. Mathematics fulfils this requirement by means of the construction of the figure, which is an appearance present to the senses (even though brought about a priori). In the same science, the concept of magnitude seeks its standing and sense in number, but seeks this in turn in the fingers, in the beads of an abacus, or in strokes and points that are placed before the eyes. The concept is always generated a priori, together with the synthetic principles or formulas from such concepts; but their use and relation to supposed objects can in the end be sought nowhere but in experience, the possibility of which (as far as its form concerned) is contained in them a priori.”(KrV A240-B299)50←43 | 44→
“In a general sense one may call construction all exhibition of a concept through the (spontaneous) production of a corresponding intuition. If it occurs through mere imagination in accordance with an a priori concept, it is called pure construction (such as must underlie all the demonstrations of the mathematician; hence he can demonstrate by means of a circle which he draws with his stick in the sand, no matter how irregular it may turn out to be, the properties of a circle in general, as perfectly as if it had been etched in copperplate by the greatest artist). If it is carried out on some kind of material, however, it could be called empirical construction. The first can also be called schematic, the second technical construction.” (AA VIII, p. 192)51
Besides Jiménez Rodriguez, Young Ahn Kang also remarks on the connection between mathematical construction and schematism:
“[…] the construction of a concept is an act of providing a concept with objective reality (cf. Entdeckung BA 10–11; Fortschritte A183). In other words, constructability is a semantic rule of mathematical cognition. It makes possible a meaningful use of mathematical concepts on the one hand, and it restricts the valid sphere of mathematical knowledge to the sensible world on the other (Prolegomena § 13 note). The presentation of a concept in intuition (mathematical schematism) provides the concept with ‘sense and meaning’ (Sinn und Bedeutung) (Prolegomena § 8). Thus, construction has the same function as the transcendental schema both in in its realizing and restricting of the pure concepts at the same time (A147/B187).” (Kang 1985, p. 51)
After these considerations, that stress that there are hints to the problem of schematism in works before the Critique, I now move on to the analysis of the passages ←44 | 45→in which Kant makes use of the term ‘schema’ in his pre-critical writings. In the first part of the chapter I will focus on the metaphysical meaning of ‘schema’ as presented in the New Elucidation, while in the second part I will consider the various meanings of the term in the Dissertation from 1770, namely: “adumbratio”, “outline”. I want to stress how the use of ‘schema’ changes from indicating a mere unclear outline to addressing the form through which the coordination of impressions is possible. Finally, since the notion of schema occurs in reference to space and time, I shall analyse how Kant positions himself in contemporary debates over realism and idealism about space and time.
2.1 The metaphysical notion of schema in the Nova Dilucidatio
The first appearance of Kant’s use of the term ‘schema’ is found in the Principiorum Primorum Cognitionis Metaphysicae Nova Dilucidato of the year 1755. The aim of this work, presented by Kant to receive permission to teach philosophy at the Faculty of Königsberg, is to clarify the first principles of knowledge. The New Elucidation (Nova Dilucidatio) deals with the value of the principles of non-contradiction and sufficient reason, from which Kant derives two principles of metaphysical knowledge: succession and coexistence. The former establishes that the possibility of change in a substance depends on its connection to other substances; the latter affirms that without a common principle of existence (the divine understanding), no relation among substances would be possible. It is first in the demonstration and second in the clarification of the latter that the noun ‘schema’ can be found.
According to the demonstration of the principle of succession each substance is separated and intelligible in itself and has no relation to the others, since they are not the cause of each other’s existence. Therefore, to explain the relation among substances, it is necessary to address their common cause, God, intended as a general principle of existence of all entities. However, this reference is not sufficient, because it might be the case that God caused the existence of separated entities, without them having relation to each other. For this reason a further clarification is needed, namely that God determines not only the existence but also the mutual relations of things and it is in this context where Kant speaks for the first time of a certain ‘schema’:
“But it does not follow from the fact that God simply established the existence of things that there is also a reciprocal relation between those things, unless the self-same schema of the divine understanding (intellectus divini schema), which gives existence, also ←45 | 46→established the relations of things to each other, by conceiving their existences as correlated with each other.”(AA I, p. 413)52
Later on, in the context of the clarification of the principle of coexistence, a second occurrence of the notion of a schema is found:
“The schema of the divine understanding, the origin of existences, is an enduring act (it is called preservation); and in that act, if any substances are conceived by God as existing in isolation and without any relational determinations, no connection between them and no reciprocal relation would come into being.” (AA I, p. 414)53
In this passage, Kant explains that God’s activity, which brings things into existence and mutual commerce, is not an instantaneous and punctual act, but rather an enduring one, called conservation, thus providing the reason why things endure and have relations persisting in time.
As demonstrated in lines mentioned above, the notion of schema in this work possesses a mere metaphysical sense: it refers to a divine project or organisation, and it can be regarded as a synonym for ‘divine understanding’. It is close to a general and common way of using the term as a synonym for order, structure and it is thus far from the epistemological and logical views of some of Kant’s predecessors.
As already anticipated the literature on the use of ‘schema’ in the pre-critical writings is scarce. In the Kant-Lexikon Martin Schönefeld refers only indirectly to ‘schemata’, in order to explain the forms of the sensible world, without further inquiring if there is a distinction between the meaning of ‘schema’ in the Dissertation from 1770 and the Critique of Pure Reason. Moreover, in the Historisches Wörterbuch der Philosophie Stegmeier (Stegmeier 1992, pp. 1249–1252) only reports where the term ‘schema’ appears in the text, while Jiménez Rodríguez claims that the Dissertation from 1770 contains the first, ←46 | 47→clearest anticipation of the chapter of schematism of the Critique. More specifically, she points out that the work includes theories - such as: the definition of space and time as formal principles, the distinctions between empirical and pure intuitions, between receptivity and spontaneity as well as between form and content - that will be reechoed in the Critique (Jiménez Rodríguez 2016, p. 431).
In the following sections, I shall stress that after the Dilucidatio of 1755 ‘schema’ is taken up again only fifteen years later, namely in the dissertation De mundi intelligibilis atque sensibilis forma et principiis from 1770. As the term, here, refers to the notions of the forms of the worlds I will give a short overview of this topic.
2.2.1 Schema and the forms of the worlds
Thanks to his Dissertation Kant obtained the position of Professor of Logics and Metaphysics at Königsberg. One of the main themes of the De mundi intelligibilis atque sensibilis forma et principiis lies in the antinomical (Hinske 1980) contrast between the laws of the understanding and of the pure reason and those of the intuitive faculty, which implies a distinction between two kinds of knowledge (intellectual and empirical) and two kinds of entities: phenomena, objects “as they appear” in sensibility and things in themselves.
With this sharp distinction between intelligible and sensible levels, Kant situates himself in accordance with traditional views such as those of Alexander Gottlieb Baumgarten and Christian Wolff, while through his reference to the notion of form he distances himself from them, introducing a novelty regarding the theory of sensibility. According to Kant, the unity and organisation of empirical elements is provided not by the matter itself, but by formal principles, which, although not sensible, are implied in the constitution of the objects of experience. Although defined as forms, these principles are not conceived as the ancient ousia, as a static and immutable essence of a thing, rather they are dynamic relations, coordinative functions:
“Form, which consists in the co-ordination, not in the subordination, of substances. For co-ordinates are related to one another as complements to a whole, while subordinates are related to one another as caused and cause, or, generally, as principle and that which is governed by principle.’ The former relationship is reciprocal and homonymous, so that any correlate is related to the other as both determining it and being determined by it.” (AA II, p. 390)54←47 | 48→
It is precisely in reference to the explanation of the constitution of sensible objects, i.e. representations, that the first use of the term ‘schema’ in the work is found. As the author states, each sensible representation is given both by matter, which reveals the presence of something sensible, although it depends in its quality also on the nature of the subject, and by form:
“Moreover, just as the sensation which constitutes the matter of a sensible representation is, indeed, evidence for the presence of something sensible, though in respect of its quality is dependent upon the nature of the subject insofar as the latter is capable of modification by the object in question, so also the form of the same representation is undoubtedly evidence of a certain reference in what is sensed, though properly speaking it is not an outline or any kind of schema of the object, but only a certain law, which is inherent in the mind and by means of which it co-ordinates for itself that which is sensed from the presence of the object.” (AA II. p. 393)55
Here “schema” or “the outline” refers no longer to the divine understanding as it did in the Dilucidatio, but to an unclear image, a “shadow” (adumbratio in the Latin text), which is opposed to form, because the form is defined as an internal law of the mind, according to which the objects of experience can be structured and organised. But what is meant precisely by formal principles? What are their features? According to Kant, a principle is that which contains the reason of a relation. While the principle of the form of the intelligible world is an objective cause, the world of phenomena, i.e. of our experience, has a subjective principle only. The latter is regarded as a law of the mind (animo), according to which things appear as if they belong necessarily to a whole. This principle has no validity for objects that cannot be objects of our possible experience. More specifically, Kant states that there are two formal principles of the sensible world: space and time.←48 | 49→
“These formal principles of the phenomenal universe are absolutely primary and universal; they are, so to speak, the schemata and conditions of everything sensitive in human cognition. I shall now show that there are two such principles, namely, space and time.” (AA II, p. 398)56
In the above-mentioned passage, ‘schema’ assumes a new significance: it does not refer to something unclear, nor does it mention the divine understanding of the Dilucidatio. Instead, it refers to the conditions of sensibility and human knowledge, namely the formal principles of space and time, which are provided with characteristics, which will be reechoed in the Critique of Pure Reason.
Since the last occurrence of the term ‘schema’ is found in reference to the elucidation of the forms of space and time, I will briefly introduce the characteristics of these forms. Kant first clarifies the notion of time, because it is more general than space: each experience, is at least temporal (“internal” such as emotions), while some are also spatial (“external” for instance: representations of objects and events). Because time is the most general condition of experience, it does not derive from the senses, but it is a presupposition of them and for this reason it is non-discursive, in opposition to thoughts, which are abstract and derivative.
Moreover, time must be one, singular, identical, and homogeneous (quantum continuum) in order to explain the experiences of succession and simultaneity of the material elements that are related. Time is the possibility of this relation in itself and for this reason, time cannot be regarded as belonging to the same level of sensible features. In conclusion, time is defined as a pure subjective intuition (and not a discursive concept), which does not belong to nor derives from matter. The reference to time as “subjective” might be misleading: it seems, that time identifies a sort of natural human capacity as if Kant is providing a naturalistic or anthropological explanation of the process of experience. However, the allusion to time as a pure intuition suggests that such an interpretation has to be put aside. Time, then, is the condition of all sensible experience, or the universal form of all phenomena, through which they are perceived as existent and can be coordinated. It is this feature, which clearly and deeply differentiates Kant’s approach from, say, the empirical explanation of the process of knowledge by Johann Christian Lossius. In his Physischen Ursachen des Wahren of 1775 Lossius states that the principles of logic can be understood only through the reference to the organs that are implied in the production of ideas. This point of view is similar to that of Tetens, from which Kant explicitly distances himself:←49 | 50→
“Tetens investigates the concepts of pure reason merely subjectively (human nature), I objectively. The former analysis is empirical, the latter transcendental” (AA XVIII, p. 23)57
Similarly to time, space also cannot be regarded as a concept, as something induced from experience but rather as a law, a function, presupposed in each perception as a condition of its organisation. As anticipated, here Kant refers to space as ‘schema’:
“Space is not something objective and real nor is it a substance, nor an accident, nor a relation; it is, rather, subjective and ideal; it issues from the nature of the mind in accordance with a stable law as a schema, so to speak, for co-ordinating everything which is sensed externally.” (AA II, p. 403)58
Kant here expresses himself no further on ‘outline’ (the English translation of the Latin ‘schema’), which will be later investigated in more detail in the Critique of Pure Reason. However, from the content of the Dissertation, it is possible to argue that it might be regarded as a condition of the order of perception, a ‘schema’ in the sense of a pattern that provides unity and coordination and that space and time are linked to sensibility but do not derive from it.
Besides the above-mentioned occurrences of ‘schema’, there is another passage in the Dissertation which is of great interest for my inquiry:
“But pure (human) intuition is not a universal or logical concept under which, but a singular concept in which, all sensible things whatever are thought, and thus it contains the concepts of space and time. These concepts, since they determine nothing as to the quality of sensible things, are not objects of science, except in respect of quantity. These concepts, since they determine nothing as to the quality of sensible things, are not objects of science, except in respect of quantity. Hence, PURE MATHEMATICS deals with space in GEOMETRY, and time in pure MECHANICS. In addition to these concepts, there is a certain concept which in itself, indeed, belongs to the understanding but of which the actualisation’ in the concrete requires the auxiliary notions of time and space (by successively adding a number of things and setting them simultaneously side by side). This is the concept of number, which is the concept treated in ARITHMETIC.” (AA II, 397)59←50 | 51→
In these lines, Kant does not refer to schemata but to a number, which will be one of the particular, transcendental schemata exposed in the Critique of Pure Reason. This reference is important because it alludes to the way in which schematism will be developed: namely, the concepts of the understanding (such as number), can be actualised through time (and space).
After considering these references to the notion of schema, it is possible to synthesise its significance as the following: schema is no more a metaphysical concept (as it was presented in the Dilucidatio), but rather it obtains a more epistemic significance. It refers to the forms of sensibility, which are necessary conditions for providing the material elements of experience with organisation, thus explaining the possibility of experience and knowledge. This characterisation of space and time as schemata (or “quasi” schemata, as if Kant is using the noun schema without a proper definition) as conditions shares similarities, but also differences, with the doctrines of the Critique of Pure Reason: on the one hand, space and time are defined in the Transcendental Aesthetic as conditions of the possibility of the intuition, on the other, they become defined as pure intuitions and not as schemata (that will have a different function as illustrated in the Transcendental Logic). Nevertheless, there are passages in the Critique of Pure Reason in which they are still regarded as conditions (KrV A140/B179), although in a different sense as the forms of intuitions of space and time.
The value of the Dissertation might lie precisely in the doctrine of space and time and in their definitions as forms, schemata, which provide a solution to an important debate of the time, namely the conflict between empiricism and rationalism. However, Kant’s theory has its limits, which the author himself soon becomes aware of and which lead him to write the Critique of Pure Reason.
To underline both the novelties and the limits of Kant’s doctrine of space and time in the Dissertation and to understand why he will develop his theory of form and schema in the Critique of Pure Reason, it is helpful to refer to the main theories that he encountered concerning space and time, namely those of Isaac Newton and Gottfried Wilhelm Leibniz.←51 | 52→
In his famous treatise on light, Opticks (Newton 1730 (in part. Book III, question 31, pp. 350–382) Newton states that physics must abandon the study of qualities, which characterised the old Aristotelian view, focusing instead only on principles which can be empirically demonstrated. Principles must be distinct from obscure metaphysical causes (Newton 1730, p. 377): while the former are either immediately evident or proved by induction, the latter are often obscure or impenetrable, their relation to the events they are supposed to produce is unknown and lies beyond our possibility of knowledge. However, Newton seems to abandon the rigid separation between science and metaphysics when he refers to the existence of space and time as absolute in his Philosophiae naturalis principia matematica from 1687. This work led at first to the atheistic interpretation of Newton’s doctrine. Due to being accused of atheism, Newton was obliged to add a Scholium Generale to later publications of the work in order to defend himself.
The Scholium to the Definition of the Principles opens with the distinction between absolute and relative quantities, namely, of time and space. Time can be regarded as absolute, independent from the existence of things of experience but also as relative, as a measurement or limitation (hours, days, years) of the infinite duration of absolute time. In his attempt to provide an explanation of the existence of absolute space, Newton relies on the first law of motion: since the possibility of a rectilinear, uniform motion lies in the absence of acceleration, a reference to absolute time, which has no limitations and so can explain the possibility of such infinite movement, is needed. Together with absolute time, absolute space is presupposed, intended as the field in which bodies are situated: it is not a relation between objects, but rather a primary location, unique and with no relation to anything, but containing all relations within itself.
As presented, Newton alternates between the need to free science from metaphysical assumptions and the reference to principles whose nature cannot be scientifically justified. How can this ambiguity be explained? As Ernst Cassirer remarks, Kant will avoid the risk of mixing the sensible and intelligible realms (for instance, by referring predicates such as ‘where’ and ‘when’ to objects of the pure world, like God, and by grounding relative space and time on metaphysical principles): “The ‘infection’ the contagium, of the intelligible by the sensible, which emerges so clearly in Newton’s theory concerning God, is avoided;” ←52 | 53→(Cassirer 1922, p. 121, transl. L.S.)60. Maybe the clearest passage in which this problem can be seen is the following:
“He is not eternity or infinity, but eternal and infinite; he is not duration or space, but he endures and is present. He endures forever, and is everywhere present; and by existing always and everywhere, he constitutes duration and space. Since every particle of space is always, and every indivisible moment of duration is everywhere, certainly the Maker and Lord of all things cannot be never and nowhere.” (Newton 1687, transl. A. Motte, p. 441)61
In Newton’s work, on the one hand it seems that one of his main attempts consists in providing objective grounds to science through the reference to demonstrated claims, on the other hand these claims seem not to be sufficient, and need the reference to principles which belong to other fields of knowledge. Another and maybe easier solution is to stress the influence exerted on Newton by metaphysicians and theologians of the time, such as Henry More, and, in general, by his attempt to find a conciliation between science and religion so as to defend himself against the accusation of atheism.
Confronting the same question concerning the nature of space and time, Leibniz situated himself in direct opposition to Newton. His Epistolary with Samuel Clarke (between 1715 and 1716) can be regarded as emblematic of the contemporary focus on the relation between metaphysics and sciences and the nature of the principles of knowledge. The epistolary originates from a letter sent by Leibniz to Caroline of Wales, in which he distanced himself from the Newtonian theory of absolute space and time. Then she put the philosopher in contact with Clarke, a theologian of Westminster and defender of Newton’s perspective. Influenced by the recent publication of the paradoxes of Zenon in Pierre Bayle’s Dictionnaire historique et critique (Bayle 1702) Leibniz affirms that space cannot be absolute, otherwise there would be something that cannot be explained by a cause, as required by the principle of sufficient reason: if space were uniform, absolute, there would be no difference between one point and another and consequently insufficient reason to explain why God situated bodies in these points and not in others. Similarly, if time were independent from things, there ←53 | 54→would be no reason why things happen in one moment rather than in another. But, then, what are time and space? Leibniz proposes not considering them as absolute positions, but as relations: space is an order of coexistences, while time is one of successions. As he affirms they are no-things nor attributes, but rather idealitas, in the sense that they consist of relations, orders, abstracted from material objects and then provided with universality and necessity in opposition to the overflowing matter of senses. In this sense, they are defined as idealitas and not objects, res. The problem is: if space and time derive from sensibility, then the sciences based on them (physics and mathematics) depend on sensible objects, which are contingent, and cannot be universal or necessary. Is the claim of mathematics and physics to be considered as universal and necessary only an illusion, or can it be justified?
Kant’s 1770 conception of space and time as forms might be interpreted as a first, successful attempt to provide an original solution to this question. If some sciences (such as geometry and arithmetic) are based on space and time and if Kant’s forms of sensibility do not derive from senses, then it is possible to justify their universal and objective value. However, Kant’s doctrine still has some limitations.
Although Kant’s conception of space and time distinguishes him from previous traditions, he is still very influenced by the Wolffian division between inferior and superior faculties, receptivity and spontaneity, thus generating incoherences, or at least ambiguities in his doctrine - for instance, regarding his account of sensibility and the distinction between phaenomena and noumena -. More specifically, Kant defines sensibility as receptivitas, the faculty through which the subject can be affected by the object, while the understanding is conceived as the faculty of representing what is not present in the senses. Whilst the former deals with things as they appear (uti apparent), the latter focuses on things as they are (sicuti sunt). This division reechoes the traditional separation between primary qualities, which belong to the things in themselves and are grasped through the understanding62, and secondary qualities that depend on the subject and its sensibility. But, as already stressed, sensibility cannot be defined as passive, since it is ←54 | 55→characterised by the pure forms of intuitions, through which impressions do not affect the subject, as it were, automatically, but according to an order.
Then, if on the one hand the doctrine of space and time exposed in the Dissertation opens the way to a new approach to the problem of knowledge, on the other hand it contains limitations and incoherences, as underlined by Kant himself. As he states in the famous letter to Herz from 21st February 1772, the most important philosophical question focuses on the link between representations and objects. What causes this link? While it might be easy to explain it in reference to sensibility (as it could be possible to affirm that representations reflect objects as they are produced by their affection) on the other hand the relation between the intellectual action and objects is harder to justify. In the letter to Herz, Kant provides only a negative definition of the understanding’s activity: it is the faculty of representing things we are not affected by, it is not an abstraction from the senses, and neither is it a production such as the efficient causality of an intuitive understanding. Unfortunately, Kant does not delve into detail with this kind of action, nor does he explain whether, and if so how, such activity possesses an objective significance: how can it not simply be a product of the imagination? How is its reference to actual things (sicuti sunt) justified? Are there forms and schemata similar to the ones of sensibility also for the understanding? If the reference to space and time as schemata allows Kant to explain the possibility of the objective value of sciences and of our sensible experience as well, why should not a similar solution be valid also for the activity of the understanding? As Kant himself puts it in the letter to Marcus Herz from 21st February 1772:
“In my Dissertation I was content to explain the nature of intellectual representations in a merely negative way, namely, to state that they were not modifications of the soul brought about by the object. However, I silently passed over the further question of how a representation that refers to an object without being in any way affected by it can be possible. I had said The sensuous representations present them as they are. But by what means are these things given to us, if not by the way in which they affect us? And if such intellectual representations depend on our inner activity, whence comes the agreement that they are supposed to have with objects […]?” (AA X, p. 130–31)63←55 | 56→
Moreover, another limitation of the Dissertation can be seen in the fact that, although it states that the difference between understanding and sensibility is not a qualitative one, it is difficult to avoid the thought that the knowledge concerning things uti apparent is somehow inferior to that of the things sicuti sunt (Cassirer 1918). It seems that there are two worlds and that the intelligible one is absolutely separated and opposed to the sensible one, kind of a Platonic ideal world, which can be grasped only when the understanding is freed from the bounds of senses, thus revealing itself as a traditional, metaphysical, perfect realm. As Cassirer states (Cassirer 1918, p. 137) in the Dissertation, Kant assumes a distinction between a purely creative understanding and a purely receptive one. But since our understanding falls not under these two kinds, a new concept of understanding needs to be elaborated by Kant. The overcoming of the separation between these two worlds and accounts of the understanding will be one of the findings of the Critique of Pure Reason.←56 | 57→
47 I do not intend to delve into the question concerning the validity of Kant’s account of mathematics, but just to touch upon the criticism made by Koriako. According to Koriako, Kant presents mathematics and its constructive method “with philosophical eyes” (Koriako 1999, p. 3), thus presenting a misleading view of mathematics. Consequently, if Koriako is right, the particularity of the mathematical construction should no longer be a valid solution to the problem concerning the method to apply pure concepts to experience. Even if this is the case, as it seems, that Kant’s account of mathematics has philosophical grounds, my historical claim, that links mathematical construction and schematism, should not suffer. Within Kant’s perspective, if mathematics deals with the application of pure concepts, then, mathematical construction should be regarded as a procedure comparable to that of schematism. The question concerning the actual mathematical validity of such a constructive method is not my concern.
48 “Man kann zu einem jeden allgemeinen Begriffe auf zweierlei Wege kommen, entweder durch die willkürliche Verbindung der Begriffe, oder durch Absonderung von demjenigen Erkenntnisse, welches durch Zergliederung ist deutlich gemacht worden. Die Mathematik faßt niemals anders Definitionen ab als auf die erstere Art. Man gedenkt sich z. E. willkürlich vier gerade Linien, die eine Ebene einschließen, so daß die entgegenstehende Seiten nicht parallel sind, und nennt diese Figur ein Trapezium. Der Begriff, den ich erkläre, ist nicht vor der Definition gegeben, sondern er entspringt allererst durch dieselbe. Ein Kegel mag sonst bedeuten, was er wolle; in der Mathematik entsteht er aus der willkürlichen Vorstellung eines rechtwinkligen Triangles, der sich um eine Seite dreht. Die Erklärung entspringt hier und in allen andern Fällen offenbar durch die Synthese. Mit den Definitionen der Weltweisheit ist es ganz anders bewandt. Es ist hier der Begriff von einem Dinge schon gegeben, aber verworren oder nicht genügsam bestimmt. Ich muß ihn zergliedern, die abgesonderte Merkmale zusammen mit dem gegebenen Begriffe in allerlei Fällen vergleichen und diesen abstrakten Gedanken ausführlich und bestimmt machen.”
49 On the one hand, it is true that both philosophy and mathematics deal with the problem of construction or exhibition of pure concepts in intuition, but on the other hand philosophy, differently from mathematics, does not produce its objects: “Mathematical objects have a sort of definite presence or ideal existence that no philosophical or empirical concepts can attain. But this existence differs from the ordinary notion of existence that can be given only in experience. Neither a dog not a transcendental concept can be arbitrarily constructed and thus exhibited completely in intuition.” (Ferrarin 1995, p. 151). And later in the text: “[…] in mathematics what we produce is the form and content of the object. Experience in this case shows that they belong to an appearance, that they are the form of an appearance, In philosophy we do not produce any intuition, but merely show the possibility of our pure concepts referring to objects under the condition of sensibility.” (Ferrarin 1995, p. 158)
50 “Daher erfordert man auch, einen abgesonderten Begriff sinnlich zu machen, d. i. das ihm korrespondierende Objekt in der Anschauung darzulegen, weil ohne dieses der Begriff (wie man sagt) ohne Sinn, d. i. ohne Bedeutung, bleiben würde. Die Mathematik erfüllt diese Forderung durch die Construction der Gestalt, welche eine den Sinnen gegenwärtige (obzwar a priori zu Stande gebrachte) Erscheinung ist. Der Begriff der Größe sucht in eben der Wissenschaft seine Haltung und Sinn in der Zahl, diese aber an den Fingern, den Corallen des Rechenbretts, oder den Strichen und Punkten, die vor Augen gestellt werden. Der Begriff bleibt immer a priori erzeugt sammt den synthetischen Grundsätzen oder Formeln aus solchen Begriffen; aber der Gebrauch derselben und Beziehung auf angebliche Gegenstände kann am Ende doch nirgend, als in der Erfahrung gesucht werden, deren Möglichkeit (der Form nach) jene a priori enthalten.”
51 “In allgemeiner Bedeutung kann alle Darstellung eines Begriffs durch die (selbstthätige) Hervorbringung einer ihm korrespondierenden Anschauung Construction heißen. Geschieht sie durch die bloße Einbildungskraft einem Begriffe a priori gemäß, so heißt sie die reine (dergleichen der Mathematiker allen seinen Demonstrationen zum Grunde legen muß; daher er an einem Cirkel, den er mit seinem Stabe im Sande beschreibt, so unregelmäßig er auch ausfalle, die Eigenschaften eines Cirkels überhaupt so vollkommen beweisen kann, als ob ihn der beste Künstler im Kupferstiche gezeichnet hätte). Wird sie aber an irgendeiner Materie ausgeübt, so würde sie die empirische Construction heißen können. Die erstere kann auch die schematische, die zweite die technische genannt werden.”
52 “Quoniam vero inde sempliciter ipsarum staibiliverit existentiam, mutuus inter easdem respectus etiam non consequiturm nisi idem, quod existentiam dat, intellectus divini schema, quatenus existentias ipsarum correlatas concepit, eorum respectus firmaverit universal rerum omnium commercium huius divinae ideae conceptui soli acceptum ferri, liquidissime apparet”.
53 “Schema intellectus divini, existentiarum origo, est actus perdurabilis (conservationem appellitant), in quo si substantiae quaevis solitario et abaque determinationum relatione a Deo conceptae sunt, nullus inter eas nexus nullusque respectus mutuus orietur.”
54 “FORMA, quae consistit in substantiarum coordinatione, non subordinatione. Coordinata enim se invicem respiciunt ut complementa ad totum, subordinata ut causatum et causa, s. generatim ut principium et principiatum. Prior relatio est reciproca et homonyma, ita, ut quodlibet correlatum alterum respiciat ut determinans, simulque ut determinatum, posterior est heteronyma, nempe ab una parte nonnisi dependentiae, ab altera causalitatis.”
55 “Porro, quemadmodum sensatio, quae sensualis repraesentationis materiam constituit, praesentiam quidem sensibilis alicuius arguit, sed quoad qualitatem pendet a natura subiecti, quatenus ab isto obiecto est modificabilis; ita eiusdem repraesentationis forma testator utique quondam sensorium respectum aut relationem, verum proprie non est adumbration aut schema quoddam obecti, sed non nisi lex quaedam memnti insita, sense ab obiecti praesentia orta sibimet coordinandi.”
56 “Haec principia formalia Universi phaenomeni ansolute prima, catholica et cuiuslibet praeterea in cognition humana sensitive quasi schemata et condiciones, bina esse, Tempus et Spatium, iam demonstrabo.”
57 “Tetens untersucht die Begriffe der reinen Vernunft bloss subjektiv (menschliche Natur), ich objektiv. Jene Analysis ist empirisch, diese transzendental.”
58 “Spatium non est aliquid obiectivi et realis nec substantia. Nec accidens, nec relatio; sed subiectivum et ideale er e natura mentis stabili lege proficiscens veluti schema, omnia omnino externe sense sibi coordinandi.”
59 “Intuitus autem purus (humanus) non est conceptus universalis s. logicus, sub quo, sed singularis, in quo sensibilia quaelibet cogitantur, ideoque continet conceptus spatii et temporis; qui, cum quoad qualitatem nihil de sensibilibus determinent, non sunt obiecta scientiae, nisi quoad quantitatem. Hinc MATHESIS PURA spatium considerat in GEOMETRIA, tempus in MECHANICA pura. Accedit hisce conceptus quidam, in se quidem intellectualis, sed cuius tamen actuatio in concreto exigit opitulantes notiones temporis et spatii (successive addendo plura et iuxta se simul ponendo), qui est conceptus numeri, quem tractat ARITHMETICA.”
60 “Die - Ansteckung - das contagium des Intelligiblen durch das Sinnliche, wie sie so deutlich in Newtons Gotteslehre hervortrat, ist beseitigt;”
61 “Non est aeternitas & infinitas, sed aeternus & infinitus; non est duratio & spatium, sed durat & adest. Durat semper, & adest ubique, & existendo semper & ubique durationem & spatium constituit. Cum unaquaeque spatii particula sit semper, & unumquodque durationis indivisibile momentum ubique, certe rerum omnium fabricator ac dominus non erit nunquam, nusquam”.
62 More specifically, Kant defines the understanding as the superior faculty of the soul characterised by two uses: usus realis, through which concepts of things and their relation are given, and usus logicus, proper to sciences, through which it is possible to subordinate inferior concepts to superior ones, related according to the logical principle of non-contradictoriness. Through the usus logicus appearance turns to experience: the understanding, confronts and coordinates empirical contents towards universality. But this does not imply that the sensible content can develop into an intellectual one; the intelligible, the pure ideas, are grasped (and not abstracted) from the understanding as symbols in its usus realis.
63 “Ich hatte mich in der Dissertation damit begnügt die Natur der intellectual Vorstellungen bloß negativ auszudrücken: daß sie nämlich nicht modifikationen der Seele durch den Gegenstand wären. Wie aber denn sonst eine Vorstellung die sich auf einen Gegenstand bezieht ohne von ihm auf einige Weise affiziert zu seyn möglich überging ich mit Stillschweigen. Ich hatte gesagt: die sinnliche Vorstellungen stellen die Dinge vor, wie sie erscheinen, die intellectuale wie sie sind. Wodurch aber werden uns denn diese Dinge gegeben, wenn sie es nicht durch die Art werden, womit sie uns affizieren und wenn solche intellectuale Vorstellungen auf unsrer innern Thätigkeit beruhen, woher kommt die Übereinstimmung die sie mit Gegenständen haben sollen”.
