Background: Eberhard and the ‘Philosophisches Magazin’

It would be no exaggeration to claim that, by the end of the penultimate decade of the eighteenth century, Kant's Critical philosophy, and his Critique of Pure Reason (henceforth CPR)Footnote ¹ in particular, had brought about a revolution in German intellectual life. Inevitably, such a change was bound to be resisted, and the resistance from the dominant Wolffian school of metaphysics in the Leibnizian tradition was chiefly led by Johann August Eberhard (1739–1809). Eberhard took it upon himself, after the publication of the second edition of Kant's CPR in 1787, to organize a response to the spread of the new Critical philosophy. In 1788, he launched a new philosophical journal, the Philosophisches Magazin (PM), to which several Wolffians contributed.Footnote ² This journal was primarily designed to publish papers criticizing Kant's Critical philosophy from a Wolffian angle, and specifically aimed at opposing the views published in the Allgemeine Literatur-Zeitung (ALZ). The latter was a journal started in 1785 by C. G. Schütz, which was committed to the propagation and defence of the new Critical philosophy. Among its contributors were some of Kant's most prominent followers, especially Johann Friedrich Schultz (1739–1805)Footnote ³ and Karl Leonhard Reinhold (1757–1823),Footnote ⁴ who were keen to spare him the need to get involved in polemics, and thus allow him to concentrate fully upon the completion of the Critical system.Footnote ⁵ They therefore took up the gauntlet, and assumed the responsibility of coordinating a Kantian response to the criticisms that were aimed at his philosophical system from various directions.Footnote ⁶

Over a short period of time, a number of papers were published in ALZ that, even if not written by Kant himself, expressed Kantian rejoinders to the criticisms published by the faction around Eberhard. Only once did Kant himself put pen to paper to respond to articles by Eberhard that were published in the first volume of PM (1788–9). Kant deemed that the author was not only fundamentally mistaken about the meaning of his writings, but also had been particularly dishonest in dealing with them. Kant's response was published as a separate work titled On a Discovery According to which Any New Critique of Pure Reason Has Been Made Superfluous by an Earlier One (1790) (AA 8: 185–251). The main aim of this work was to counter Eberhard's attacks on two fronts, namely on the issue of the limits of knowledge, with particular emphasis on the problem of synthetic a priori judgements, and on the very distinction these limits rest upon, namely that between analytic and synthetic judgements.Footnote ⁷

For Kant's responses to other attacks from the Eberhard camp, we have to rely upon ALZ and Kant's correspondence with Schultz and Reinhold in particular, which indicates to what extent Kant's voice is speaking through them. One case in which the notes that Kant included as attachment in his reply to Schultz (2 August 1790; AA 11: 184) are, at Kant's suggestion, reproduced practically verbatim by Schultz in the ALZ less than two months later,Footnote ⁸ is in his discussion of three essays by the mathematician Abraham Gotthelf Kästner (1719–1800), professor of mathematics and physics in Göttingen, director of the old Göttingen observatory, and also a noted epigrammist (see Kästner Reference Kästner1797). These were published in the second volume of PM (1790) and bear the titles: ‘What is the Meaning of “Possible” in Euclid's Geometry?’ (Kästner Reference Kästner1790a), ‘On the Mathematical Concept of Space’ (Kästner Reference Kästner1790b) and ‘On Geometric Axioms’ (Kästner Reference Kästner1790c).

On Kästner

As Allison (Reference Allison1973: 13) notes, Kant treats Kästner with a great deal more respect than he does Eberhard. This is partly a strategic decision no doubt: Kästner was a very well-respected figure in mathematics, chiefly known for his textbooks which were widely used in German universities. And although Kästner's papers discuss issues in the philosophy of mathematics from a broadly Leibnizian perspective, it was shrewder to avoid an open attack and rather attempt to show that there were no grounds for dispute between them. Unlike Eberhard's very polemical writings, which questioned just about all the key features of the Critical project, Kästner's papers are aimed at a particular issue, and his style is dispassionate.

Kästner did not make a name for himself through original research in mathematics (cf. Sinaceur Reference Sinaceur1974). His main contribution was in the writing of textbooks (he published Anfangsgründe der Mathematik in 4 volumes, 1758–61) as well as about the applications and the history of mathematics. His interest in geometry is, however, noteworthy, and of particular interest here. Specifically, Kästner proposed an alternative axiomatization of Euclid's geometry. In so doing, he attempted to find grounds for the selection of the parallel postulate (the fifth postulate) in Euclidean geometry. However, he did not follow those (including Kant's contemporary J. H. Lambert, 1728–77Footnote ⁹) who unsuccessfully tried to derive it from the other postulates. His interest in this postulate is historically relevant insofar as among Kästner's pupils we find Bolyai's father and Lobachevsky's teacher. We can thus see a direct line of influence connecting Kästner to these two great originators of the first non-Euclidean geometry to be published, namely hyperbolic geometry.

Although Kästner's philosophical sympathies were Leibnizian, Kant would have had no grounds for considering him an enemy in the same vein as Eberhard. Aside from their having mutual friends, such as the physicist and aphorist G. C. Lichtenberg (1742–99), who was Kästner's Ph.D. student, and his long-standing admiration for Kästner,Footnote ¹⁰ Kant also corresponded with Kästner, having asked him to arbitrate in his dispute with Eberhard (see AA 11: 186). Although Kästner declined to take on such a role, he advised Kant to aim for more clarity in exposing his ideas (AA 11: 214), advice which Kant apparently took heed of in writing his Religion within the Limits of Reason alone (AA 11: 427). As Förster and Rosen report in their edition of Kant's Opus Postumum (Kant Reference Kant1993: 267–8), Kästner can hardly be said to have converted to the Critical philosophy, but at least he viewed Kant, the author of Towards Eternal Peace, as a ‘wise man’ (Kästner Reference Kästner1797: 100).Footnote ¹¹

Kästner's Articles

The first two topics discussed in Kästner's PM articles (Kästner Reference Kästner1790a, Reference Kästnerb and Reference Kästnerc) were of particular interest to Kant: the notion of possibility in geometry, as defined through construction, and the characterization of space as infinite. In the first article (1790a), Kästner differentiates between those propositions which Euclid takes as postulates, and those whose possibility is inferred by deriving them from the first, and voices his scepticism that any further grounding of the possibility of the postulates can be achieved (Kästner Reference Kästner1790a: 392–5). The notions introduced with these postulates are however sufficiently clear so that no further grounding is needed since, for Kästner, the absence of any contradiction is sufficient to establish possibility. Kästner explains that what is conceivable to ‘make’ (machen) according to well-defined and non-contradictory geometric rules, as are found in Euclid's geometry, is thereby established as possible (Kästner Reference Kästner1790a: 391–2), and even ‘necessarily possible’ (Kästner Reference Kästner1790a: 400). And the operation of the understanding by which I grasp the conceivability of ‘making’ the geometric entity, is just what ‘making’ in one's understanding amounts to. Together with the Leibnizian claim that what is in the understanding is real, this entails that what is possible in geometry is real (Kästner Reference Kästner1790a: 400). In these claims, Kästner wants to deny any essential role to any auxiliary figures which may be used (Kästner Reference Kästner1790a: 398), a point which Eberhard will have taken as grist to the mill of his particularly vehement attack on Kant's assigning a key role to a priori intuition in mathematical knowledge. Indeed, in the preceding issue of the same volume of PM, Eberhard published an article in which he expressly attacks the very notion of a priori intuition (Eberhard Reference Eberhard1789: 84–90). Kästner's claims are: (i) that it is impossible to have an image of the infinite as all images are determinate; and, for the same reason, (ii) that no general image of space is possible. Kästner's paper stresses the fact that certainty does not reside in any image which may correspond to a concept, but in the understanding (Kästner Reference Kästner1790b: 419).

But of course, Eberhard's and Kästner's claims can only be viewed as critiques of Kant if the latter's notion of intuition and image are identifiable, a point that Kant unequivocally denies in his response to Kästner (AA 20: 413). In line with his general attitude towards Kästner, Kant further seeks to defuse any other appearance of dissent from his Critical philosophy of mathematics by cleverly translating Kästner's notion of ‘making’ into one of ‘construction’ carried out by the imagination a priori (AA 20: 411), thus altering Kästner's operation of the understanding to one that is carried out by the imagination under a rule prescribed by the understanding.

Kästner's understanding of the infinite follows from his understanding of geometric space as abstracted from our representations of outer sense (Kästner Reference Kästner1790b: 407): the infinite is just that which is unlimited, in the sense that there are no limits as to how far geometric figures extend (Kästner Reference Kästner1790b: 407–8). In this sense, he opposes Schultz's theory of the infinite, expounded in Versuch einer genauen Theorie des Unendlichen (1788), which, in distinguishing different actual infinite magnitudes, prefigures Cantor's groundbreaking work on the infinite. Although Kant is aware of Kästner's views on Schultz's theory (cf. AA 11: 184), he stresses his agreement with Kästner on the issue of the geometric notion of space, which is indeed a notion of potential infinite, and differentiates it from the actual infinity of metaphysical space. In so doing, he draws unexpected support from Joseph Raphson, a seventeenth-century English mathematician, whom Kästner refers to, claiming that Raphson understands the actual infinity of space to be in the mind. However, as Michel Fichant (Reference Fichant1997b: 42, 44–5) points out, Kant appears to make a rare interpretative mistake here, as it is most unlikely that the Newtonian Raphson would hold such a view of space.Footnote ¹²

Kant's strategy is therefore two-fold in responding to Kästner: he accommodates Kästner's views on infinity through his distinction between metaphysical and geometric space, suggesting it is implicitly assumed by Kästner (AA 20: 411–12, 416); and he appeals to the difference between intuition and image to neutralize Kästner's statements about the irrelevance of images to mathematical truth (AA 20: 422), thereby cleverly emphasizing that the error is Eberhard's. But there remains a clear gulf between Kästner's claims and the Critical philosophy's outlook on mathematical truth (see Fichant Reference Fichant1997a: 8–9). First, Kästner sticks to the Leibnizian-Wolffian line according to which mathematical truth relies entirely upon the criterion of non-contradiction. For Kant, the non-contradictory nature of a mathematical concept does not guarantee its truth. The question of its being constructible in a priori intuition, rather, is the key to mathematical truth. And second, Kästner espouses the rationalist doctrine that mathematics, and geometry in particular, defines a world of objects for the understanding (Kästner Reference Kästner1790a: 400). Establishing a mathematical truth is a matter of exploring this world. For Kant, there are no geometric objects, only constructions in a priori intuition according to geometrical concepts.Footnote ¹³

Contemporary Relevance of Kant's text

Kant's text on Kästner, which for the first time appears in an integral English translation below, is important for our understanding of Kant's conception of space. The key point is Kant's insistence on a clear distinction between metaphysical and geometric space. The first is a given infinite, while the second is a potential infinite. Fichant's translation into French of this text in the 1990s (Kant Reference Kant1997; Fichant Reference Fichant1997a) came at a time when Béatrice Longuenesse's magnum opus on CPR (Longuenesse Reference Longuenesse1998a) had just established itself as a serious alternative approach to the dominant interpretative strands in the analytic tradition.Footnote ¹⁴ A key feature of Longuenesse's work is its claim that space is, in effect, a product of the faculty of productive imagination, and therefore, given that according to Kant the productive imagination is an effect of the understanding on sensibility (B151–2), a product of the faculty of understanding (Longuenesse Reference Longuenesse1998a: 219, 222–3). The stress Kant puts in the article on Kästner upon the independence of a notion of space from any conceptualization such as occurs in geometry certainly provided ammunition for an ‘anti-conceptualist’ backlash. In French Kant scholarship, this was led by Fichant's translation of Kant's reply to Kästner, as well as a paper underlining its importance for an understanding of Kant on space (Fichant Reference Fichant1997b). Replies followed from Longuenesse (Reference Longuenesse1998b, Reference Longuenesse2005), and from philosophers upholding the stronger conceptualist stance of Marburg neo-Kantianism (Dufour Reference Dufour2003).

In English-speaking Kant scholarship, a related discussion has been taking place about the sense to be given to Kant's claim that constructions in spatial intuition are required to derive geometric truths. A key text in this debate is Friedman (Reference Friedman1992). Friedman claims that Kant's requirement is, in effect, a consequence of the limitations of the monadic logic he used. Friedman (Reference Friedman1992: 63–4) points out that, while Kant thought he needed to carry out iterative constructions to construct new points as part of a geometric proof, existential quantifiers in the setting of a polyadic logic could have been used instead, had such a logic been available to Kant. For Friedman, this shows that Kant's view of geometry as synthetic was conditioned by the limits of the logic at his disposal. Friedman's reading of the Metaphysical Exposition (1992: 68–70) has it that it is only because of its role in grounding geometry that the representation of space must be an a priori intuition. Carson (Reference Carson1997: 495–7) takes up this point and shows that the infinity of space described by Kant in the Metaphysical Exposition is not that of infinite iteration, but of an infinite given space, and that it therefore is irreducible to logic. In so doing, she is upholding an emphasis upon the phenomenological character of space first made by Parsons (Reference Parsons1992: 72), which parallels the anti-conceptualist thrust of Fichant's position. Partly in response to Carson, Friedman revised his position (Friedman Reference Friedman2000, Reference Friedman2003, Reference Friedman2012) leading to some convergence with Carson's position, eventually abandoning in effect the fully conceptualist take upon space which he still held in 2000, as we shall see below, while other authors reinforced the need to take the phenomenological notion of space seriously (e.g. Kjosavik Reference Kjosavik2009).Footnote ¹⁵ It is noteworthy that Kant's response to Kästner is discussed by Friedman in the last of those three papers, and indeed its key distinction between metaphysical space that is a given infinite and space as the topic of geometry is what is at the heart of this particular debate around the importance of the phenomenology of space.

For simplicity, we shall describe all positions which uphold a key role for the faculty of the understanding in defining the unity of space as ‘conceptualist’, while we are aware that is, strictly speaking, a misnomer in some cases.Footnote ¹⁶

The Response to Kästner Confronted with B160–1n in the Critique of Pure Reason

A key text used by the conceptualist reply is footnote B160–1n in the B-version of the Transcendental Deduction of the Categories. And this is hardly surprising given its emphasis upon the understanding in defining the unity of the formal intuition of space: B160–1n is thus often read as a reinterpretation of the Transcendental Aesthetic in the light of the results of the Analytic (e.g. Longuenesse Reference Longuenesse1998a: 214ff.). On the conceptualist reading, the unity of space discussed in the Aesthetic is now to be assigned to the unity of the understanding. It is certainly the case that some statements in B160–1n lend themselves to this interpretation. For example, Kant writes: ‘In the Aesthetic, I ascribed this unity merely to sensibility . . . though to be sure it presupposes a synthesis, which does not belong to the senses but through which all concepts of space and time first become possible.’ The conceptualist reading according to which the unity of space requires the understanding is given further support towards the end of B160–1n when Kant, ambiguously, talks of space and time being ‘given as intuitions’, ‘as the understanding determines sensibility’.

These statements appear to be flatly contradicted by Kant's assertions in On Kästner's Treatises. There, Kant insists upon a notion of ‘metaphysically, i.e. originally, nonetheless merely subjectively given space, which (because there is no plurality thereof) cannot be brought under any concept which would be constructible, but to be sure contains the ground of the construction of all possible geometrical concepts’ (AA 20: 420, our emphasis). This space is described as existing in ‘the pure form of the sensible mode of representation of the subject’ (AA 20: 421). As such, it is defined independently of any synthesis. In the response to Kästner, this is contrasted with space as the topic of geometry, which is discussed in B160–1n, where Kant considers ‘[s]pace, represented as object (as is really required in geometry)’.

As a result, we have a contrast between, on the one hand, a metaphysical space that is subjectively given and, on the other, geometric spaces defined by conceptual determinations of regions of this space. This reading of the response to Kästner provides direct support to an anti-conceptualist understanding of the representation of space, such as Fichant's for instance (see Fichant Reference Fichant1997b: 36–7, 42–3; cf. 1997b: 24), a position that could be taken to be in tension with the claims made in B160–1n.Footnote ¹⁷

Indeed, what is striking in Kant's response to Kästner, as Fichant quite rightly emphasizes, is the importance accorded to a notion of metaphysical space as a subjective form of sensibility. While footnote B160–1n in the Transcendental Deduction appears to play down the role of the form of intuition as it ‘merely gives the manifold’, On Kästner's Treatises repeatedly emphasizes the key notion of metaphysical space as ‘space . . . considered in the way it is given, before all determination of it’.Footnote ¹⁸ To be sure, this reflects the different purposes of these two texts, but it certainly represents a worry for conceptualist readings. On the latter, the representation of space is a product of the understanding on its own (Dufour Reference Dufour2003), or on Longuenesse's reading, of the understanding together with some unknown ground of spatiality providing the innate seed for the process of epigenesis of the form of intuition of space. Additionally, an unfortunate consequence of such interpretative stances is that they end up deleting any meaningful distinction between the notions of ‘form of intuition’ and ‘formal intuition’, as Fichant (Reference Fichant1997b: 36) points out, whereas the point of the footnote B160–1n is precisely to draw a clear distinction here.

But, most importantly, as far as the response to Kästner is concerned, the issue is not simply one of a difference in emphasis: the biggest stumbling block to a conceptualist reading of Kant's response must be the topic which is the focus of Kästner's article, namely the infinity of space (see Fichant Reference Fichant1997b: 25–7). Eberhard saw Kästner's article as providing support from the mathematical establishment for his Leibnizian take upon the issue of the infinity of space. Leibniz's space, which is relational, is not infinite, and indeed Leibniz argues against the notion of an infinite space in his correspondence with Samuel Clarke. The motives for the avoidance of such a notion were partly theological: there can only be one infinite, God. The Leibnizian view of mathematical truth, moreover, does not require any appeal to infinite space. For Leibniz – and Eberhard later takes this up to attack Kant – the mathematician proceeds merely on the basis of the principle of non-contradiction (cf. AA 20: 413–14) or ‘mediately through inferences’ (AA 20: 411), hence purely conceptually, without beforehand ascertaining, through construction in intuition, the reality of the object of a geometric concept. But in On Kästner's Treatises Kant responds that the geometer already presupposes the pure forms of sensibility by means of which an object is in fact first constructed, and that this guarantees its reality, which no amount of conceptual analysis can bring about. Kant is therefore keen to suggest that this is something Kästner himself, as mathematician, effectively already presupposes (AA 20: 411–12, 416). He thereby identifies the actual infinity of space as a requirement for the possibility for the mathematician of constructing, to infinity, any geometric figure, so that the mathematician need only ever concern himself with the potential infinity of his constructions (AA 20: 419–20). For instance, considering Euclid's second postulate which states that any straight line can be extended indefinitely, an infinite space must first be given for such a construction to be possible. It is hard to see how any conceptualist understanding of space could do justice to this notion of the given infinity of space.Footnote ¹⁹

Footnote B160–1n also crucially introduces the notions of ‘form of intuition’ and ‘formal intuition’. The proposed reading of the response to Kästner might suggest an identification of metaphysical space with the form of intuition, and of the representation of the space of geometry with the formal intuition of space. But we must be careful here, because Kant talks of geometrical spaces in the plural (AA 20: 419), so that equating these spaces with space as object that is the topic of B160–1n, as such an identification would imply, is not straightforward. In his response to Kästner, Kant does not suggest that these geometric spaces are somehow dependent upon one geometric space that would be space considered as object; rather, he clearly states of metaphysical space that ‘hence in this, as singular representation, the possibility of all spaces, which goes to infinity, is given’ (AA 20: 421), and ‘there are (many) spaces, of which the geometer however, in accord with the metaphysician, must admit as a consequence of the foundational representation of space, that they can only be thought as parts of the single original space’ (AA 20: 419). The conceptually determinate geometric spaces are therefore parts of metaphysical space, not of one geometric space. Conceptually determinate geometric spaces are ‘a proper subset of the determinations of the single metaphysical space’ (Patton Reference Patton2011: 282–3). Such statements as in Kant's response to Kästner suggest that metaphysical space could then also be identified with what B160–1n describes as ‘[s]pace, represented as object (as is really required in geometry)’ and which is identified as the formal intuition of space. In view of these issues, it would seem that what the response to Kästner describes as metaphysical space is represented, in B160–1n, in terms of both notions of ‘form of intuition’ and ‘formal intuition’. In other words, Kant's distinction in the Kästner piece between metaphysical and geometric space does not separate out ‘form of intuition’ and ‘formal intuition’: both refer to metaphysical space and the distinction between the senses of these expressions is not directly relevant to the Kästner text.

That this is a plausible interpretation of the relations between the two texts should, moreover, come as no surprise. The topic in the response to Kästner is very much that of the Transcendental Aesthetic: metaphysical space is essentially space as it is described in the Metaphysical Exposition (see also Friedman Reference Friedman2012: 241–3), whereas the Transcendental Exposition identifies the role space plays in grounding geometry.Footnote ²⁰ The discussion of the Transcendental Deduction, in which the issue is that of how objects in space and time stand under the categories, is a different topic which Kant would certainly have been wise not to bring up in the discussion of Kästner's views about geometric space if he wanted to achieve the kind of clarity that Kästner himself advised Kant to aim for in his writings, as we saw above. As Kant explains in the footnote B160–1n, he had ascribed the unity of space ‘merely to sensibility’ in the Transcendental Aesthetic because of the nature of the investigation in that particular chapter, namely establishing the distinctly non-conceptual nature of space, i.e. ‘in order to note that it precedes all concepts’. The account in the Transcendental Deduction demands however that the role of the understanding in the grasp of space as a unity be investigated further, and this gives rise to footnote B160–1n, in which two distinct notions of the representation of space are differentiated. As this would have been of no use in responding to Kästner, it therefore seems reasonable to view the way the Transcendental Aesthetic refers to space as indifferent between its characterization as form of intuition or as formal intuition, and this perhaps explains the simple claim that the representation of space is an ‘a priori intuition’ (AA 20: 421), exactly as in the Transcendental Aesthetic. What the response to Kästner and the Transcendental Aesthetic do not address, therefore, is the full story of how it is that the subjectively given metaphysical space can indeed provide a grounding for the objective science of geometry: such a full story must explain the role of the understanding (and imagination), and this is where the footnote B160–1n comes into play.

Outlining an Interpretation of B160n in the Light of the Response to Kästner

Even though the topic of the present article is not footnote B160–1n, to which we have devoted a comprehensive study (Onof and Schulting, Reference Onof and Schultingforthcoming), our reading of Kant's response to Kästner must, nevertheless, be able to give some account of how the notions of ‘form of intuition’ and ‘formal intuition’ are distinguished, insofar as this completes the account of how subjectively given metaphysical space functions as ground of the objectivity of geometry by focusing upon the role of faculties other than sensibility in this grounding. This interpretation will serve the purpose of shedding further light upon our proposed reading of On Kästner's Treatises as not introducing any distinction corresponding to that between ‘form of’ and ‘formal’ intuition in B160–1n. The space of our pure form of outer sensible intuition has a unity which Kant describes in the Metaphysical Exposition of space. It is not a conceptual unity because it is not a unity of a multiplicity of representations that are contained under a higher one (A69/B94; A78/B104),Footnote ²¹ and there are no grounds for viewing the understanding as involved in defining this unity. Rather, it is the internal unity of the space in which manifolds in intuition are first represented.Footnote ²² The pure receptivity of our faculty of sensibility defines a unity which Kant describes at A25/B39 as the ‘single all-encompassing space’ whose parts are ‘only thought in it’.Footnote ²³ This characterizes the metaphysical space of On Kästner's Treatises in its original givenness. Pace Friedman (Reference Friedman2012), it is not the set of all possible spatial perspectives (what he calls ‘perspectival space’),Footnote ²⁴ but rather a precondition for the very possibility of considering such a multiplicity of perspectives as belonging to one space. The unity of this unitary space is the unity that Kant refers to when he claims that a single representation is always a unity (A99): i.e. a single representation is always an ‘absolute unity’, which is different from the unity that the understanding brings to the manifold.

So far, we have identified a purely subjective receptivity to manifolds of representations of outer sense, of which the necessary, subjective form is space with its sui generis unity. If cognition is to be possible, as Kant explains in A99–100, this manifold must be apprehended, i.e. I must take the manifold as manifold for my cognition. And he explicitly indicates that this brings a unity to my intuition ‘as, say, in the representation of space’ (A99). The manifold is now regarded as objective (although the object is not as yetFootnote ²⁵ thereby determined), i.e. as independent of any particular subjective perspective. The space in question here is of course still the same metaphysical space that was subjectively given, but now grasped as a conceptually differentiated (rather than merely absolute, undifferentiated) unity, through a synthetic act of the understanding. Kant adds that there is a ‘pure synthesis of apprehension’ (A100) involved in grasping space as a unity, but later indicates that it is in fact the imagination ‘whose action exercised immediately upon perceptions I call apprehension’ (A120), in line with his assertion that ‘[t]he synthesis of apprehension is . . . inseparably combined with the synthesis of reproduction’, i.e. ‘the transcendental faculty of the imagination’ (A102). Focusing upon the B-Deduction, Friedman (Reference Friedman2012: 248) correctly indicates that it is the transcendental synthesis of the imagination which is at stake here, also in line with Longuenesse's (Reference Longuenesse1998a) assigning a central role to synthesis speciosa. But importantly, in contrast to Longuenesse's reading, the unity brought about by the imagination is not equivalent to the unity and singularity of space. Rather, it is a grasp of space, with its already independently characterized singularity and unity, as a unity for my cognition: in other words, we have moved from a mere subjective reception of a manifold in space (with its internal sui generis unity) to the consideration of this manifold as objective, which involves unifying all possible perspectives upon it.

What is thus obtained is the formal intuition of space. It does indeed precede all concepts insofar as the apprehension is logically prior to bringing the manifold under a concept (in the terms of the A-Deduction: synthesis of apprehension is logically prior to, or a necessary condition of, the synthesis of recognition). However, pace Longuenesse, there are no grounds for considering the latter synthesis as involving a separate act of the understanding. That is, the act of apprehension is just one aspect of the synthesis which will bring the manifold under a certain concept, and thus it also requires the synthesis of reproduction and finally the synthesis of recognition; all of the different aspects of synthesis are equiprimordially and simultaneously involved. This can be seen by considering that apprehension is merely taking the manifold as manifold for my cognition. But a manifold is never taken as manifold for my cognition without also being determined in some way, i.e. taken as manifold of representations of some object. To suppose otherwise would be equivalent to claiming that an ‘I think’ could accompany my representations, without it being an ‘I think that . . .’, i.e. an ‘I think’ which is necessarily followed by a clause defining a fully fledged epistemic claim (see A102–4).

The role of the categories in the synthesis that brings about the formal intuition of space is to ensure that space is taken as a unity across representations, as explained above. While the manifold in space will thereby be determined under some concept (by means of the synthesis of recognition), this does not amount to a conceptualization of space itself as indeed there is only one space, which, given Kant's denial of the possibility of a singular concept, cannot as a unique individual be brought under a concept. Rather, what the synthesis achieves is to bring space under the categories, thereby making conceptual determinations of any manifold in space possible.

Kant does not shed much light on how a synthesis gives rise to the unity of space, but we can look at the case of time in the chapter on Schematism: ‘The schema of a pure concept of the understanding . . . is a transcendental product of the imagination, which concerns the determination of the inner sense in general, in accordance with conditions of its form (time) in regard to all representations, insofar as these are to be connected together a priori in one concept in accord with the unity of apperception’ (B181/A142). Here, as explained above, we note how the determination of inner sense must accord with the conditions of its form, time, while at the same time this determination is in accord with the unity of apperception. The two sets of conditions of accordance are only possible insofar as the unity of apperception is in the driving seat as it were: this requires that the form of time be brought under the categories. These determinations of time are essentially ways of enabling what is in time to be determined under the categories. The same applies mutatis mutandis to the determinations of space (as in geometric constructions): they have to accord with both the form of space whose unity has been examined in the Metaphysical Exposition of the Transcendental Aesthetic, and the unity of apperception. And this is achieved through the determination of the unity of the form of space under the categories. Kant does not provide us with further insights into this, but one can infer what this might amount to from the analysis of the form of space in the Aesthetic; this issue is discussed in further detail in Onof and Schulting (Reference Onof and Schultingforthcoming).

With this interpretation, we can therefore explain how metaphysical space is presented by Kant both as subjectively given, and as conforming to a concept of object to the extent that space is categorially determinable: it is the same space that is considered, first as mere form of receptivity, i.e. as a receptacle for manifolds of outer sense, and second, in the grasp of its unity by means of synthesis (through the imagination, and hence involving an application of the categories) through which any such manifolds are taken as manifolds for my cognition. Through this grasp, that which is a sui generis unity inherent to the structure of spaceFootnote ²⁶ is grasped as a unity for the understanding.

This might still leave it unclear why the intuition of space is said in B160–1n to require a synthesis, i.e. insofar as Kant says that it is ‘as the understanding determines sensibility’ that ‘space and time are first given as intuitions’ (Kant's emphasis). Of course, it is the notion of formal intuition which Kant means here, but it now seems that this is somewhat different from the notion of ‘intuition’ as it was originally presented in the Transcendental Aesthetic. This apparent ambiguity around the way the term ‘intuition’ is used is not restricted to this passage, as Kant says, for instance in the Stufenleiter at A320/B376–7, that there are two types of ‘objective perception’, intuitions and concepts, which is prima facie puzzling for, on Kant's theory of cognition (cf. A51–2/B75–6), if an intuition is objective, then it must be united with concepts and therefore involve some role for synthesis by means of the transcendental unity of apperception. However, there is in fact no ambiguity in such terminology. Rather, there is a clear distinction between the mere intuition as the a priori form in which the manifold of representations is received (‘form of intuition’), and the intuition of an object, namely of an appearance (B34/A20), which is determined by means of a priori synthesis (‘formal intuition’). When considering our mere receptivity for manifolds, i.e. when Kant considers manifolds in intuition (e.g. A97), the intuition involves only sensibility, i.e. ‘form of intuition’. When the intuition is that of an object, this assumes that, although the intuition provides us with the immediate relation to the object (A68/B93), it has been synthesized by means of the transcendental unity of apperception, in order for such an intuition to amount to knowledge of an object (i.e. intuition as ‘formal intuition’). As indicated in the discussion above, in the Transcendental Aesthetic such distinctions are not yet relevant as the role of the understanding has not yet been examined.

Final Remarks

In the case of space, just as in the case of time, the conceptualist must explain how the determination of the form of intuition, which is at the heart of the chapter on Schematism, should warrant such attention if the form of intuition were nothing but a ‘projection’ of the understanding. Why then, in the case of time, would this determination be described by Kant as ‘a hidden art in the depths of the human soul’ (B180–1/A141)? On the contrary, this issue becomes central to the success of the Critical project if space (and time), as metaphysically given infinite (cf. AA 20: 420–1), has its own essential structure that is to be grasped by the faculty of understanding. It is part of the goal, in fact, of the so-called second step of the B-Deduction to explain, by means of a deduction of the ‘concept of space’ (B120–1/A88), how we can conceive of the unity of space without the latter being thereby reducible to a conceptual or purely formal analytical unity, since space (and time), as Kant had explicitly asserted in the Aesthetic (A24–5/B39), is not itself a concept. As we explain in Onof and Schulting (Reference Onof and Schultingforthcoming), this grasp of the form of space by the understanding is possible because, as Kant shows in the Metaphysical Exposition's analysis of the form of space, we have an intimate phenomenological acquaintance with space as a given infinite magnitude, with what in the response to Kästner is called metaphysical space. It is for this reason that, in the Metaphysical Exposition, it is possible to describe (necessarily in conceptual terms) the essential properties of what is not conceptual, namely space as a priori form of intuition, and therefore that it is possible to bring the unity of this form under the unity of apperception.Footnote ²⁷