Infinity and givenness: Kant on the intuitive origin of spatial representation

Abstract

I advance a novel interpretation of Kant's argument that our original representation of space must be intuitive, according to which the intuitive status of spatial representation is secured by its infinitary structure. I defend a conception of intuitive representation as what must be given to the mind in order to be thought at all. Discursive representation, as modelled on the specific division of a highest genus into species, cannot account for infinite complexity. Because we represent space as infinitely complex, the spatial manifold cannot be generated discursively and must therefore be given to the mind, i.e. represented in intuition.

Keywords::

• Kant
• intuition
• sensibility
• space
• infinity
• continuum
• singular representation

Acknowledgements

The debts of gratitude I have gladly incurred in writing this paper are too many to list. For valuable comments, questions and encouragement I owe particular thanks to Ian Blecher, Matt Boyle, Jim Conant, Robert Pippin, Anat Schechtman, Daniel Sutherland, Clinton Tolley and an anonymous reviewer.

Notes

^1. All translations are my own, though I have consulted the standard editions. Kant's emphases are in bold, my own are in italics and noted parenthetically. References to the Critique of Pure Reason refer to the 1781 (A) and 1787 (B) edition pagination. References to Kant's other writings cite the volume and page number of the Akademieausgabe of Kants Gesammelte Schriften and are preceded by an abbreviated title of the cited work. I abbreviate the titles of Kant's works as follows: Physical Monadology ( = The Employment in Natural Philosophy of Metaphysics Combined with Geometry, of which Sample 1 Contains the Physical Monadology); Only Argument ( = The Only Possible Argument in Support of a Demonstration of the Existence of God); Inquiry ( = Inquiry Concerning the Distinctness of the Principles of Natural Theology and Morality); De Mundi ( = On the Form and Principles of the Sensible and the Intelligible World); Critique ( = Critique of Pure Reason); Prolegomena ( = Prolegomena to Any Future Metaphysics that Will Be Able to Come Forward as Science); Groundwork ( = Groundwork of the Metaphysics of Morals); Discovery ( = On a Discovery According to which Any New Critique of Pure Reason is to be made Superfluous by an Older One); Prominent Tone ( = On a Recently Prominent Tone of Superiority in Philosophy); Logic ( = Jäsche Logic); DW-Logic ( = Dohna-Wundlacken Logic); Kästner ( = Über Kästners Abhandlungen). Reflections are preceded by an ‘R’ and include Adickes's estimate of their date, where applicable.

^2. This impression is perhaps encouraged by the fact that the sensibility/understanding distinction is first presented ‘by way of introduction’ as a ‘preliminary reminder [Vorerinnerung]’ (A15/B29). The opening claims of the Transcendental Logic are liable to reinforce this impression, because it is hardly any easier to find an explicit, non-question-begging argument for the details of Kant's concept/intuition distinction in the vicinity of his much celebrated declaration that ‘thoughts without content are empty, intuitions without concepts are blind’ (A51/B75; cf. A271/B327).

^3. I will refer to the numbered sections of the Metaphysical Exposition of the Concept of Space in the B edition simply as the ‘MEs’. There are significant similarities between these and the corresponding discussions of the concept of time, so much of what I say about the one will apply mutatis mutandis to the other. But the two are sufficiently different to warrant separate treatments.

^4. My guiding assumption throughout is that Kant's claims pertain to our representation of space – i.e. to space as we represent it to be. Without this implicit qualification, the question whether ‘space is […] a discursive […] concept’ already presupposes that space itself is a mere representation (and, hence, ideal), which is rather supposed to be one of the ‘Conclusions from the Above Concepts’ (A26/B42). The MEs are expressly concerned with our concept of space – i.e. with space as we conceive of it. Their task is to provide a ‘clear (if not complete) representation of what belongs to [this] concept’ (B38; see Section 2).

^5. The chart provides only the roughest glosses on these notions, and it would be hasty to attribute them, as stated, either to Kant or to any particular commentator's account of his views. They are meant only to indicate some directions in which precise criteria for intuitive and conceptual representation might be sought. My discussion aims to accommodate differences of opinion about how these criteria are to be spelled out and about how they interrelate. With one exception – namely the purportedly atomic containment structure of conceptual representation, discussed below – I do not dispute that the listed features are criterial for intuitive and conceptual representation. What I dispute is that these criteria may be legitimately invoked to generate a cogent reconstruction of the argument in question.

^6. A smattering of examples should suffice to confirm this. Vaihinger treats the concept/intuition distinction as exclusive and invokes singularity as a necessary and sufficient criterion of intuition (1892, 211f., 223; citing De Mundi and the Nachlass), generality as a sufficient condition of concepts (211f.; citing the Nachlass) and atomic containment structure as a sufficient condition of concepts (219; citing the Logic). Kemp Smith (1923, 105, 107) treats the distinction as exclusive and exhaustive and invokes singularity and immediacy as severally necessary and sufficient conditions of intuition, and then generality, mediacy and atomic containment structure as severally sufficient conditions of concepts. Paton (1936, I:115) presupposes the exclusivity of the distinction and invokes generality as a sufficient condition of concepts (citing the Logic) and declares singularity a sufficient condition of intuitions. Ewing (1950, 37) invokes singularity as a sufficient condition of intuition and atomic containment structure as a necessary condition of concepts. Strawson (1966, 64) claims that the distinction is exclusive and exhaustive and invokes singularity as a necessary and sufficient criterion of intuitions and generality as a necessary condition of concepts. Parsons (1992, 63, 69f., 1998, 46) invokes singularity and immediacy as severally sufficient conditions of intuition (citing the Stufenleiter at A320/B376f., the Logic and the Aesthetic). Pippin (1982, 64ff.) invokes singularity and immediacy as severally necessary and sufficient conditions of intuition (citing the Stufenleiter and B136n.). Allison (1983, 90, 2004, 109) presents the argument as consisting of ‘two steps’, the first of which presupposes the exhaustiveness of the concept/intuition distinction and invokes singularity as a sufficient condition of intuition. The second step Allison identifies (1983, 91, 2004, 110) presupposes the exclusiveness of the concept/intuition distinction and invokes atomic containment as a sufficient criterion of conceptual representation. Guyer (1987, 346, 348) invokes singularity as a necessary and sufficient condition of intuition. Falkenstein (1995, 218) invokes singularity as part of the ‘definition’ of intuition and thus, presumably, as at least a necessary condition for intuition (citing De Mundi, Logic and Discovery). Falkenstein (1995, 230, 234f.) also describes atomic containment structure as a necessary condition of concepts (oddly citing Kant's conception of substance (not any theory of concepts) in the Physical Monadology and De Mundi). Carson (1997, 494, 496, 498) implicitly invokes singularity as a sufficient criterion of intuition and atomic containment structure as a necessary condition of conceptual representation, thereby following Allison in presenting two distinct argumentative routes to the desired conclusion. Gardner (1999, 78) invokes singularity and immediacy as severally sufficient conditions of intuition. Rosenberg (2005, 66) invokes only the mediacy or discursivity of concepts. Buroker (2006, 52) invokes generality as a necessary condition of conceptual representation. Shabel (2010, 100, 102) invokes only singularity as a sufficient condition of intuition, implicitly treating the concept/intuition distinction as exclusive.

^7. Kant is often interpreted as asserting that there is (necessarily) exactly one space – a claim some challenge by arguing that the idea of wholly dissociated spaces is coherent (cf. Quentin 1962; Hollis 1967). The basic worry is that Kant commits a fallacious quantifier inversion. The fact that every space is part of some greater space does not entail that there is an all-encompassing space of which every space is part. But the weaker point may suffice for Kant's purposes because he is ultimately concerned with space as the framework of our outer experience. The unity of space, then, is a function of the unity of experience (cf. B136n., B138, B144n, B160, B161n.). The point is that it is part of our concept of experiential space that it cannot be fragmentary and that anything we might possibly experience as outer must be located within it. Falkenstein (1995, 219) accordingly interprets Kant as claiming only that space (and anything in it) is a particular, but not that it is unique. In any case, a possible plurality of spaces only seems to threaten Kant's position if one takes his argument to turn on the singularity of intuition, and it is my present task to expose the flaws of this interpretation. Section 2 gives an account of intuition that does not presuppose singularity. Section 4 then indicates how the singularity of intuition can be viewed as a consequence of Kant's argument.

^8. Guyer and Wood translate ‘einig’ as ‘single’. Now ‘einig’ can indeed mean single or solitary, as in Luther's translation of Genesis 22.2: ‘nim Isaac deinen einigen son’ [‘take Isaac, your only son’]. But it can also signify unity or undividedness, as in Kant's claim that the necessary being ‘is unified [einig] in its essence, simple [einfach] in its substance’ (Only Argument 2:89). Indeed, in its primary colloquial sense, ‘einig’ is used (and was used at the time) to signify agreement or solidarity (i.e. unity) among various parties, and this is the sense that most frequently appears in Kant's writings. Kemp Smith preserves the ambiguity between these senses of ‘einig’ by translating it as ‘one’ or eliding it altogether. Though I take ‘einig’ here to mean unified, I have opted for ‘unitary’ in order to preserve the ambiguity of the original and to avoid begging any questions. If Kant had used ‘einzig’ rather than ‘einig’, ‘single’ would be apt. Since he did not, a neutral translation seems preferable. See the entry for ‘einig’ in the Grimm Wörterbuch, from which the above examples are drawn.

^9. See, for example, Kemp Smith (1923, 107), Allison (1983, 90–91, 2004, 109–110), Parsons (1992, 70) and Falkenstein (1995, 67–69). Vaihinger mentions, but does not evaluate this objection, attributing it to Riehl, while also citing Trendelenburg (1892, 213). Commentators typically cite singular ideas of reason as potential counterexamples – e.g. < the world>, < nature> or < God>. But the full force of this objection only becomes clear when one considers the wide range of ostensibly singular concepts Kant's system allows, as I do below.

^10. I follow the common practice of using chevrons around words to mention the concepts they signify: thus, < Pferd> and < horse> denote the same concept, viz. the concept of the natural species equus ferus.

^11. For a compelling account of why this should be so, see Anderson (2004).

^12. E.g. B15 (twice), B16 and Prolegomena 4:268.

^13. One might think the singularity of such mathematical concepts can be explained by invoking construction in intuition and the singularity of intuition. There is surely something right about this, but it does not advance our understanding of A25/B39. A first difficulty with the proposal is that some mathematical concepts admit of multiple references: e.g. < √4> refers both to 2 and to − 2. Kant implies that square roots have multiple solutions in Negative Magnitudes (2:173) in arguing that combining negative magnitudes is a case of addition rather than subtraction (i.e. that − 2 and − 2 yield − 4). Because the product of − 2 and 2 is just the sum of − 2 and − 2 (i.e. − 4), the product of − 2 and − 2 should have the opposite sign, i.e. = 4 (cf. Euler 1911, Vollständige Anleitung zur Algebra, §33). Thus, 2 and − 2 are referents of < √4>. Kästner (1758) also argues that the product of two negative numbers is positive in §105 of his Anfangsgründe der Arithmetik, which Kant mentions approvingly in Negative Magnitudes (2:170). Moreover, the solution of physical problems involving scalar quantities will require that we allow the products of negative magnitudes to be positive. Multiply referential mathematical concepts show that construction in intuition does not ensure singularity in the sense of reference to exactly one object (unless we construe the referent of < √4> as a set or an unordered pair, which seems unjustifiably anachronistic). Moreover, this plurality of reference seems to result from the relevant mathematical operation, not from its construction in intuition, which supports my contention that singular mathematical concepts owe their singularity to the content of the mathematical functions they invoke (and not to their having been put to a singular use). A second difficulty with the proposal is that the theory of mathematical construction in intuition (along with the singularity of intuition itself, I would argue) is something the Aesthetic is supposed to help establish, not something it can take for granted as a premise. So, one cannot appeal to this theory in order to defuse counterexamples to a proposed reconstruction of Kant's argument in the MEs. It is more natural to suppose that the order of argumentation in the Critique is precisely the reverse: geometrical concepts, for example, must be constructed in intuition precisely because space is the form of outer sense and geometry is the pure science of space. Finally, even if Kant were entitled to invoke this theory of mathematical construction in establishing the originally intuitive status of spatial representation, the role of construction in mathematics no more undermines the genuinely discursive nature of mathematical concepts than the originally intuitive status of spatial representation undermines the discursive character of the concept < space>. So we would still be lacking an account of singularity on which only intuitions are singular and/or an account of generality such that all concepts are general, which is what the mooted reconstruction of the argument calls for.

^14. Recall that Kant explicitly argues that the concept of the most real being is singular in Only Argument (2:83f.).

^15. It should be obvious that the singular purport of < entis realissimi> cannot be borrowed from intuition, because it is ‘a concept which we can never exhibit in concreto in accordance with its totality’ (A573/B601).

^16. The idea that < totality> can itself constitute a mark (Merkmal) of more specific concepts not only affords us a procedure for generating essentially singular discursive representations consistent with Kant's theory of concepts, but it also suggests a way of reconciling these peculiar representations with Kant's claims that concepts are intrinsically general (e.g. Logic, 9:91). Although many concepts which contain < totality> as a mark are, for that reason, essentially singular, < totality> is not itself a singular representation. (There are, after all, innumerably many totalities.) This ought to remind us that individual marks are essentially general, for they are precisely ‘that which is common to many objects’ (Logic, 9:58, my italics, cf. A320/B377). Marks very often impart their generality to the discursive representations mediated through them. But as we have observed, one can generate a singular discursive representation either by including a mark like < totality> in its intension, or by combining other marks in such a way that it is (logically, metaphysically, or mathematically) impossible for more than one object to exhibit those marks. Because all concepts (as discursive) are mediated through marks, one can see the sense in insisting that even < ens realissimum> is an ‘intrinsically [an sich] general concept’ despite its being ‘the representation of an individual’ (A576/B604). Moreover, this explains why the Stufenleiter does not actually characterize concepts as ‘general’ but instead says that they are ‘mediated by means of a mark that can be common to many things’ (A320/B377). It also sheds light on Kant's remark that ‘[t]he generality or general validity of the concept does not rest on the fact that it is a partial-concept [Theilbegriff], but on the fact that it is a ground of cognition’ (Logic, 9:95). If it is impossible for a certain concept to serve as the ground of cognition of more than a single thing, that concept is essentially (though not intrinsically) singular, notwithstanding its discursive mediation through general Merkmale. What a certain concept can enable us to cognize is precisely the sort of consideration from which we abstract in pure, general logic (cf. A58/B83; Logic 9:13). Within the context of such an investigation, then, it makes sense to treat all discursive representations as (tautologically) general. Once we are concerned with the conditions of the possibility of knowledge of objects (e.g. in the Critique), such abstractions are illegitimate and we must ask of every discursive representation whether and of how many objects it can serve as a ground of cognition. Sometimes, we can answer both questions (e.g. in the case of < sum of 7 and 5>, or < largest integer>). Other times, we can answer one but not the other. We may demonstrate that, if anything can be cognized through the concept in question (e.g. < complete series of causes>), only one thing can be cognized through it, even though the objective validity of the concept as a ground of cognition must remain forever problematic for human reason. The crucial point is that, in the context of a critique of our ability to know objects, the discursivity of a representation is not a matter of its generality, but a matter of its mediation through marks and its consequent inability to give us the object(s) it represents. It is, I shall suggest, precisely this feature that fundamentally distinguishes concepts from intuitions.

^17. The link between the forms (and functions) of judgement and the categories that is supposed to be established in the Metaphysical Deduction clearly involves some reference to intuition in general: ‘[The categories] are concepts of an object in general, through which the intuition of that object is regarded as determined with respect to one of the logical functions to judge’ (B128). This might tempt one to yet again (see note 13) attempt to explain away the singular purport of such totality-concepts by invoking their connection to intuition (and the singularity of intuition). There is surely something right about this, but the strategy faces a number of complications. First, the category < totality> remains a discursive representation that originates in the understanding alone (cf. A137/B176, B377; Prolegomena 4:330; Logic 9:92), so totality-concepts still seem to provide examples of essentially singular discursive content, even if that content bears some intrinsic link to intuition. Second, the strategy does not neatly apply to a number of totality-concepts (such as < ens realissimum>), which cannot, in principle, be exhibited in intuition and which, therefore, cannot derive their singular purport from intuition in any straightforward manner. Finally, it is interpretively suspect to draw on the results of the Metaphysical Deduction (and, one suspects, the notion of figurative synthesis from the Transcendental Deduction) in reconstructing Kant's argument in the MEs. Even if it is possible to explain away the ostensible singularity of totality-concepts (and their kin) by exploiting argumentative resources found elsewhere in Kant's corpus, it strains credulity to claim that these argumentative resources are available at the outset of the Aesthetic. So while it may be possible to resolve the apparent tension between such singular discursive representations and Kant's claims that concepts are intrinsically general (perhaps along the lines I suggest in note 16), it is illegitimate to employ such an account in reconstructing Kant's argument at A25/B39.

^18. It is worth noting that Kant nowhere characterizes intuitions as singular in the text leading up to the MEs. He says that intuitions alone give us objects (A16/B29, A19/B33), he says they relate immediately to the objects they give us (A19/B33), and he says that they contain a manifold and have a hylomorphic structure (A20/B34). But singularity does not figure in the characterizations of intuitive representation that open the Critique. Allison (1983, 2004) points to Kant's parallel discussion of time to justify invoking the singularity of intuition in reconstructing Kant's argument. There, Kant says that ‘the representation which can only be given through a single object [einen einzigen Gegenstand] is intuition’ (A32/B47). But this does not state that intuitions alone are singular representations. What it literally says is that intuitions are the only representations that can solely arise through isolated affections by objects. Whether or not concepts can arise through affection by a single object (and it is doubtful that they can), it is clear that they also (and paradigmatically) arise when we reflect on what many objects (which we have compared) have in common, and when we abstract from certain of their features to common ones (perhaps while recombining these with others; cf. Logic, 9:95). Thus, it is not true that concepts can only arise through cognitive contact with a single object. This is at once what enables concepts to represent objects and their properties in the absence of those objects and what disables them from guaranteeing (on their own) the objective validity of what they represent. It is because of this discursivity or mediacy that concepts cannot give us objects. (Cf. note 16 above.) Given the exhaustiveness of the concept/intuition distinction, this means that intuitions are the only representations that can only arise through affection, which is just what A32/B47 says. It thus recapitulates the opening sentences of the Aesthetic, which connect the idea that intuitions give us objects with the idea that intuitions arise through affection: ‘The latter [sc. intuition] only takes place [findet nur statt], insofar as the object is given to us; and this, in turn, is only possible, for us humans at least, if it [sc. the object] affects the mind in a certain way’. (A19/B33)

In Section 4, I will argue that the singularity of intuition is not mentioned before the MEs precisely because it is a consequence of them: since all intuition is spatiotemporal and since spacetime is a unique, unitary structure, all intuition is singular, for it represents unique portions of that structure, as such.

^19. For example, in the first edition version of the final MEs concerning time, Kant writes: ‘But where the parts themselves and every magnitude of an object can only be determinately represented through limitations, there the entire representation cannot be given through concepts (for there [sc. in concepts] the partial representations are prior)’ (A32, my italics). However, Kant changes this parenthetical phrase in the second edition to read ‘(for they [sc. concepts] contain only partial representations)’. This change reflects Kant's more substantial revision of the corresponding argument about space in light of his realization that the concept/intuition distinction does not turn on the priority of part versus whole, but on the kind of relation that obtains between contained and container. See note 50 for my reading of the final MEs.

^20. Readers may also have misgivings about (ii), inasmuch as the text of A25/B39 seems to speak not about our representation of space and its parts, but about the parts of space itself. But see note 4 and Section 2.

^21. The sense in which concepts may be ‘given’ differs from the sense in which objects may be given (in intuition). With respect to objects, the contrast is between being given and being (merely) thought. Here, the contrast is between conceptus dati, which are given, and conceptus factitii, which are made or fabricated (cf. A730/B758; Logic 9:93; Vienna Logic 24:913–918). Fabricated concepts are ‘willkürlich’ or ‘arbitrary’ in that their marks are determined by our elective choice (arbitrium). They accordingly have an atomic containment structure: their parts (marks) are prior to the whole. But not all concepts are fabricated and concept fabrication presupposes our possession of ‘given’ concepts, which, I argue above, are not atomically, but holistically articulated. Paradigmatic fabricated concepts are those of mathematics and the technical concepts of natural science. The former can be defined, in Kant's ambitious sense, for they are generated through the construction of their objects in pure intuition, and we can therefore secure their objective validity a priori (cf. A730-2/B758-60; Logic 9:63f.). A non-mathematical, fabricated concept cannot be defined because merely combining given marks does not indicate ‘whether [the fabricated concept] has an object’ (A729/B757). It will have the logical form of a concept but may, for all that, ‘[have] no meaning [Sinn] and [be] completely devoid of content’ (A239/B298). A would-be definition of such a concept is thus a ‘declaration (of my project)’ (A729/B757) of demonstrating its application to objects of experience – namely by ‘compel[ing] nature to answer [my] questions’ (B xiii; cf. Logic 9:36f.). This is the project of natural science, which extends our knowledge of the phenomenal world even as it secures the meaningfulness of our fabricated concepts.

^22. Cf. A727/B755; Logic 9:140–145.

^23. For example, by noting the contradiction involved in thinking of a thoroughly permeable material body, we can establish that < impenetrable> is a mark of the concept < body>. For if something is completely permeable, it cannot properly be said to occupy a given space, because its ‘presence’ in no way prevents anything else from occupying that space. But every material body must occupy a space. Thus, by reflecting on competent, non-trivial uses of concepts in judgements and by drawing on our comparatively primitive ability to recognize contradictions, valid inferences, and so forth, we can establish that certain marks are necessary criteria of a given concept's application. Yet we cannot, by this means, establish what marks are sufficient criteria for a given concept's application. And that is why given concepts cannot be defined.

^24. Kant's emphasis on the standing possibility of human ignorance and error in identifying concepts’ marks might prompt the objection that use (in judgements) concerns only the ratio cognoscendi, not the ratio essendi of concepts’ marks – i.e. how we come to know a concept's marks but not what it is to be a mark of a concept. Yet such fallibility does not suggest that there is some standard for the content of concepts apart from their knowledgeable use. Granted, we can be no more certain of the marks of our concepts than we can be certain that our apparently virtuous motives are not surreptitiously corrupted by self-love (cf. Groundwork 4:407). Yet, in neither case does pervasive uncertainty vitiate the fundamentally self-conscious nature of the representation or the internality of the standard against which it is measured. Just as a good will is an expression of reason's self-knowledge, so too is the full content of a concept an expression of the understanding's self-understanding – i.e. its knowledge of its own activity in applying concepts in potentially knowledgeable judgements. Despite the fallibility of our efforts at self-knowledge (e.g. through analysis of our concepts), our use of concepts in knowledgeable judgements is constitutive of their meaning precisely because their meaning consists in the contribution they are capable of making to our knowledge. Since knowledge is essentially self-conscious (epistemic hiccups notwithstanding), for concepts, esse is concipi.

^25. This comports well with Kant's emphasis that a mark is not only a ‘Partialvorstellung’ but a ‘partial-representation insofar as it is considered [a] ground of cognition of the whole representation’ (Logic 9:58, my italics). The cognitive contribution of the (whole) concept is the standard against which a mark's membership in its intension is determined. Cf. also Logic, 9:35f., 58f., 64, 95, 96, 145; Critique B39f., A69/B94, B133f.n., A728/B756; Discovery 8:199; Prominent Tone 8:399.

^26. That the marks of a concept are posterior to and dependent on the whole of that concept is thus the complement of Kant's view that concepts are essentially ‘predicates of possible judgments’ (A68/B93).

^27. Longuenesse (1998, 24n.13) seems to understand the immediacy criterion in this sense. Parsons (1992) advances an alternative conception of immediacy as ‘phenomenological presence to the mind’ (66). I prefer to reserve the term ‘immediacy’ for non-mediation through discursive marks, but I do think Parsons's account gets closer to the nervus probandi of Kant's reasoning here. For, as I will argue, Kant's conclusion does turn on a sort of cognitive presence to the mind – albeit one that should not be construed phenomenologically.

^28. I argue for this in detail in Chapter 3 of my dissertation (Smyth, n.d.), which is indebted to Engstrom 2006. I go beyond his account in locating a sound argument for Kant's conception of our cognitive finitude in the opening passages of the Critique, as befits Kant's ‘synthetic’ method (cf. Prolegomena 4:274 and note 35).

^29. The A edition Introduction opens with the same thought, though it is already entangled with a particular picture of the dependence of human knowledge on experience, which Kant excises from the B edition: ‘Experience is, without doubt, the first product which our understanding produces, in working up the raw material [Stoff] of sense impressions’ (A1).

^30. For an intuitive intellect, however, the faculty of thought is also a faculty of intuition. There is an intuitive moment in all theoretical knowledge (finite or infinite), because all knowledge, for Kant, relates to objects. Rosenkoetter (unpublished manuscript) fruitfully suggests that Kant takes knowledge to be of objects (rather than states of affairs, say) in order to explain the meaningfulness of false propositions.

^31. See A19/B33 and notes 16 and 18, above.

^32. This suggests an analogous functional characterization of the understanding as that faculty which secures the aspects of our knowledge which could not possibly be given to us. The Transcendental Analytic initiates a (synthetic) transition from a ‘merely negative’ conception of the understanding as a ‘non-sensible cognitive faculty’ (A67/B92) to a richer, positive conception of the understanding as a spontaneous capacity. This synthetic enrichment of our conception of the understanding begins by noting features of our knowledge that our passive sensibility cannot account for, e.g. combination in general (cf. B129).

^33. For a refreshingly detailed account of Kant's conception of exposition as analysis, see Messina (forthcoming).

^34. Falkenstein (1995) rightly emphasizes that the MEs should be read in light of Kant's ‘blindness thesis’. This corrects a widespread misconception that the Aesthetic presupposes cognitive access to unconceptualized intuitions, which the Analytic denies is possible. Yet while he insists that unconceptualized intuitions are ‘for us as good as nothing’ (A111), Falkenstein still thinks Kant implies the existence of such intuitions. Kant does hold that we can ‘isolate’ (A22/B36) sensibility's distinctive contribution to cognition by abstracting from cognitive features due to discursive thought, once ‘extended practice has made us attentive to [them] and skilled in separating [them] out’ (B1f.). But such notional separability need not imply that these aspects can exist or be conceived on their own. Nor do I see what explanatory role such nugatory intuitions could play in Kant's transcendental epistemology and hence why we should read him as committed to their existence.

^35. Carson (1997, 495) objects that these judgements cannot include the claims of geometry, for that would ‘[go] against Kant's explicit assertion in the Prolegomena that in the Critique, he is pursuing the “synthetic method” which is “based on no data except reason itself”’ [4:274]. I agree that Kant cannot legitimately appeal here to geometrical principles as objectively valid cognitions we can endorse as true, but I do think he can appeal to them as merely thinkable contents. Such thinkable content provides a sufficient basis for the ‘clear (if incomplete)’ analyses that make up the Metaphysical and Transcendental Expositions (B52). These analyses nevertheless contribute to a ‘synthetic’ argument, i.e. one that progresses from relatively simple principles to relatively complex ones and their consequences (Logic 9:149; DW-Logic 24:779; R3831 (from 1769) 17:353; cf. also R3343 (c. 1772-5) 16:789, and §422 of Meier's 1752, Auszug aus der Vernunftlehre). For example, as I suggest below, the marks identified in the MEs enable us to conclude that intuitions are not only immediate and object-giving, but also singular representations. Or we can infer any of the notorious claims Kant makes in the section ‘Conclusions from the Above Concepts’, e.g. that space does not represent any property of things in themselves (A26/B42). Each of these inferences constitutes a synthetic step in that it enriches the principles we began with (e.g. our concept of intuition, or our concept of space). Yet insofar as they depend solely on analyses of concepts (i.e. contents treated as merely thinkable), they do not ‘rest upon any fact’ (4:274). This is how I interpret Kant's claim that, in philosophy, ‘we can conclude various things from a few marks drawn from an incomplete analysis [i.e. an exposition] before we arrive at the complete exposition, i.e. the definition’ (A730/B758; cf. Logic 9:145). Even while progressing synthetically with the ‘assured gait of science’, there are moments of analysis and exposition when one foot is planted, enabling the next step forward.

^36. Hopefully, this mention of singularity no longer tempts us to short-circuit the argument at this point. Recall that intuitions have not yet been characterized as singular and that many discursive representations are essentially singular. See notes 7 and 18.

^37. The priority here is obviously not supposed to be temporal, for all parts of space are simultaneous (B40). Nor is it likely that the priority in question is ‘ontological dependence’, for Kant emphatically rejects the Newtonians’ reification of space: space is not a ‘thing’, not an ‘object’ or a ‘substance’ (see, e.g. A39/B56; A291/B374). Despite his own idealism about space, Leibniz makes the same mistake as the Newtonians in applying the principles of a substance ontology to space – e.g. in claiming that the parts of a continuum depend on the whole of it as the modes of a substance depend on that substance (cf. Leibniz 1923, A6.3:502, 520, and 553). Kant rather seems to hold that the part-whole dependence relation in continua pertains to the sortal-identity of the parts – to what they are, not merely that they are. It is in virtue of being (represented as) situated within the whole of space that its parts are (represented as) spatial in the first place. I think this is why Kant expresses himself by saying that the parts of space ‘can only be thought as in it’ (A25/B39, my italics), rather than by saying that they can only exist as in it. Nor is his point simply that the numerical identity or difference of distinct spaces is secured by their co-membership in one unitary space (per Melnick 1973, 9). Questions of numerical identity can only arise once the elements in question have been identified as homogeneous – e.g. once the ‘parts’ of space are identified as spaces (as spatial). It is a subsidiary point that spatial disjointness (at a given time) suffices for the numerical distinctness of spaces and spatial objects, as such: the question of what they are is prior to the question of which they are or whether they are.

^38. Cf. A290/B346, A659/B687; Logic 9:59, 146. See also Kant's notes on Logic 24:755 (cited by Carson 1997) and the Vienna Logic (24:912). For a helpful discussion of Kant's conception of specific differentiation, see Anderson (2004, 507–14).

^39. This line of thought bears some structural similarity to Descartes's so-called ‘causal’ argument for the existence of an infinite being (namely God) in the third Meditation (see Descartes 1964–1974, AT 7:40ff.). Descartes maintains that (i) we have an idea of an infinite being (i.e. God), (ii) we could not have derived this idea from anything but an infinite being and (iii) because we know ourselves to be finite beings, we cannot have derived the idea from ourselves, therefore, (iv) there must be an infinite being (God), which is distinct from us, from which we have derived this idea. Accordingly, Descartes and Kant face structurally similar objections. Yet while it is plausible for Descartes's critics to argue (against (i)) that we do not in fact have a genuine or adequate idea of an infinite being, and (against (ii)) that an idea of an infinite being can be derived from our ideas of finite beings, these challenges lose much of their force against Kant. For it is not similarly plausible to maintain (against a Kantian analogue of (i)) that we do not represent space as infinite in geometry, physics or everyday reasoning. Moreover, if I am correct in arguing that the continuity of space secures its holistic structure (see below and notes 37, 44 and 51), then it is demonstrably false to claim (against a Kantian analogue of (ii)) that the idea of space's infinite complexity can be derived from an idea of finitely complex space(s). Thus, Kant's responses to these objections are different from (and arguably stronger than) any responses available to Descartes. If Kant's argument is to be overturned, it must be on the grounds that human spontaneity can indeed account for the infinitary structure of space (time, etc.). This, I take it, is the strategy both Hegel and, in an entirely different manner, Michael Friedman pursue. For a compelling account of Descartes's argument, see Schechtman, 2014.

^40. Despite her well-placed and novel emphasis on the twofold infinity of space, Carson's overall reading still falters because it presupposes the essential singularity of intuition. See note 6.

^41. Kant operates with a number of different conceptions of infinity. He sometimes says that ‘the collection [Menge] which is not a part’ is infinite [R4764 (from 1770s) 17:721]. Space would meet this criterion, though it clearly does not entail what we would call ‘metrical infinity’. More frequently, Kant characterizes infinity as ‘great beyond all measure [über alle Maße groß]’ or ‘greater than any number’ (Critique A32/B48, A432n./B460n.; Kästner 20:419ff.; R4673 (from 1774) 17:637, R5338 (from 1770s) 18:155). This would imply metrical infinity. To establish that space is infinite in this sense, however, it is not enough to show that every space is surrounded by more; one must show that there is no upper limit on the magnitude of spaces that are delimitations of the whole of space. Kant clearly believes this, but his reasons are not entirely apparent.

^42.De Mundi 2:403n.; Critique A169f./B211f.; R4183 (early 1770s) 17:448; R4424 (c. 1771) 17:541 and R5831 (c.1783/4) 18:365; see also his corollary view that spatial points, which are simple and thus indivisible, are mere limits (cf. A169/B211, B419, A438/B466, A439/B467). It is probable that Kant (like his contemporaries) did not clearly distinguish between infinite divisibility (i.e. denseness) and continuity. Nevertheless, Kant is clearly committed to both the denseness and continuity of space, for he held that the possibility of continuous motion entailed the continuity of the spatiotemporal manifold. See note 49.

^43. There is some evidence that Kant might have granted that these claims were unsupported by argument – not because they were dogmatic, but because they were indemonstrable. In Inquiry, Kant calls it ‘the most important business of higher philosophy’ to adumbrate ‘indemonstrable fundamental truths’. The examples of such truths he provides recur throughout the critical corpus: the externality of spaces to one another, the non-substantiality of the spatial manifold, and the three-dimensionality of space, ‘etc.’: ‘Such propositions can very well be elucidated [erläutern] if one examines them in concreto so as to cognize them intuitively; but they can never be proved’ (Inquiry 2:281). The problem Carson's Kant faces, of course, is not just that his assertions are indemonstrable, but that they may be false.

^44. In the New Essays, with which Kant was familiar, Leibniz offers a précis of his solution to the ‘labyrinth of the continuum’: ‘The true infinite […] precedes all composition and is not formed by the addition of parts’ (1981, 2.17.1, A 6.6:157). This non-compositional priority of the whole over its parts is precisely what leads Leibniz to regard continua (and, a fortiori, space and time) as ideal (not real): ‘It follows from the very fact that a [continuous] mathematical body cannot be resolved into primary constituents that it is also not real but something mental and designates nothing but the possibility of parts, not anything actual. […] [T]he parts are only possible and completely indefinite. […] But in real things, that is, bodies, the parts are not indefinite – as they are in space, which is a mental thing – but actually specified in a fixed way’. (Letter to de Volder, June 30, 1704; Leibniz 1978, G 2:268f.; cf. G 2:276f., 2:281f., 3:611f., 4:394, 4:491f., 5:17, 6:394, 7:563; for a helpful discussion, see Levey 1998, 58–68). Leibniz also denies that the infinite can be a ‘genuine whole’, but this should not be taken to suggest he denies the essential unity of space. Leibniz follows Locke (1959, Essay 2.17.7-8, vol. 1, 281) in distinguishing the ‘infinity of space’ from ‘space infinite’ – i.e. the idea that, since any determinate space may be extended further, space itself is greater than any assignable quantity, from the idea that there is an all-encompassing space whose measure is actually infinite. To view space as a complete whole is, for Locke and Leibniz, to view it as consisting of a determinate number of parts (given magnitudes).Yet to view it as an infinite whole is to view it as having more parts than can be expressed in a determinate number. Thus, ‘it would be a mistake to try to suppose an absolute space which is an infinite whole made up of parts. There is no such thing: it is a notion which implies a contradiction’ (1981, 2.17.5, A 6.6:158). Leibniz's denial that space is a whole does not imply that space is not a unity (in Kant's sense), but only that the status of its ‘parts’ (as indeterminate, potential and abstract) ensures that it cannot have the (atomic) compositional structure of a ‘real thing’. There is evidence that Kant also took the infinitary structure of space and time to entail their ideality: ‘The mathematical properties of matter, e.g. infinite divisibility, proves that space and time belong not to the properties of things but to the representations of things in sensible intuition’ (R5876 (c. 1783/4) 18:374). I explore the Leibnizian roots of Kant's idealism in Chapter 1 of my dissertation (Smyth, n.d.).

^45. These temporal metaphors are only meant to capture the asymmetric dependence of a composite on the parts that compose it. They neither imply that composition is a temporal process (psychological or otherwise) nor that Kant or Leibniz treat it as one (though Locke may well conceive things this way; cf. Locke 1959, Essay 2.17.7, vol. 1, 281).

^46. The priority relation in this regress can be interpreted in various ways. If it is a relation of sortal-identity dependence, as I take it to be, then the regress is clearly vicious, since, if it does not terminate, one literally does not know what one is talking about at any given stage in the regress. The regress would arguably also be vicious if such priority expressed ontological dependence, but see note 37.

^47. E.g. Physical Monadology 1:479; Critique A169f./B211, A439/B467.

^48. The variables’ values could be assigned on analogy with a popularized version of Cantor's diagonal proof:

(x, y, z) 1…2…3…The point is that there is a recursive procedure for picking

	11/11/21/3out spatial points with triplets of rational Cartesian coordinates.
	22/12/22/3We can thus ‘construct’ an unbounded and infinitely divisible
	33/13/23/3(though not continuous) manifold out of points. Friedman(1992, 66ff.) offers a similar model. Grünbaum 1952 gives a consistent conception of the linear continuum as an aggregate of unextended points, but a discussion of that account and its relation to Kant is out of place here.

^49. Both the continuity and infinite extendability of a line segment are secured by Euclid's second postulate, from which the continuity and infinite expanse of space itself follow easily (Euclid 1908, I:154). Kant's favoured geometrical proof of the infinite divisibility of space is drawn from either Jacques Rohault or John Keill (cf. Physical Monadology 1:478; Kant 1992, 422n.6) and that proof depends, in turn, on the infinite extendability of line segments (Kant himself emphasizes this point in R5901 (c.1783/4) 18:379). The physicist's use of infinitesimal calculus in modelling physical motions similarly presupposes the continuity of space (cf. Metaphysical Foundations, 4:501, 503, 505, 508, 521f., 551, 557), which Kant explicitly recognizes in noting that the generation of a line (in time) through a fluxion entails the continuity of the spatial manifold (see his notes (R13 and R14 14:53–59) for his 1790 letter to Rehberg (11:207–210), cited by Friedman (1992, 76n.29) and originally highlighted by Parsons 2012). Kant registers the connection between continuity and motion in a figurative but telling manner in the Anticipations of Perception: ‘[Continuous] magnitudes can also be called flowing [fließende] because the synthesis (of the productive imagination) in their generation is a process in time, the continuity of which is paradigmatically designated by the expression ‘flowing’ (‘elapsing’)’ (A170/B211f.).

^50. This may seem to collapse the final two arguments of the MEs, but it does not. The penultimate argument, which we have been considering, turns on the difference in cardinality between the elements we represent space as containing, on the one hand, and the elements conceptual representations are capable of containing, on the other. By contrast, the final argument hinges not on the cardinality of elements contained in each type of representation, but on the different kinds of relation that obtain between those elements. Parts of space stand in compositional (viz. mereological) relations: two perfectly homogeneous spaces can compose to form a third space, distinct from the other two. By iterating a line segment, one can produce a new segment twice the size of the original. The sort of relation that obtains between the discursive marks of a concept is entirely different; it is, one might say, ‘information-theoretical’. I may ‘repeat’ a discursive mark as many times as I care to within the intension of a concept, without at all altering the content of the latter: the concept < rational animal> is identical to the concept < rational animal with reason>. Although Kant invokes the infinitary case in order to highlight this difference in the internal articulation of intuitive versus discursive representations, the point he is making does not strictly require that he do so. Mereological relations can also hold among the elements of a discrete and finite manifold (e.g. building blocks). So the final two MEs make complementary, but distinct points. The introduction of compositionality (mereology) further enriches the sense in which an intuition represents an individual: it not only refers to a single manifold, as such, but it also represents all the parts of that manifold as themselves individuals. This is what enables Kant to say that ‘space and time and all their parts are intuitions’ (B137n.). For more on this contrast, see Wilson (1975), Sutherland (2004) and Anderson (2004, forthcoming).

^51. It is tempting to think we can derive an idea of infinite complexity by reflecting on finitely complex representations – namely by observing that their complexity can be increased without (apparent) limit. If, as Locke argues, ‘it be so, that our Idea of Infinity be got from the Power, we observe in ourselves, of repeating without end our own Ideas’, then why should it be impossible for us to arrive at the idea of infinite space by extrapolating, through our own mental activity, on our ideas of finite spaces? (Locke 1959, Essay 2.17.6, vol. 1, 279; cf. Leibniz 1981, New Essays 2.17.3, A6.6:158). Yet this neglects the holistic mereological structure of space and, in particular, the sortal-identity-dependence of the parts of space on the whole (cf. note 37). This strategy for recursively generating (a representation of) infinite space is a non-starter, for one cannot help oneself to the base of the recursion – namely a (representation of any) finite space – without already presupposing the result the recursion is supposed to generate – viz. the (representation of the) whole of space, of which the part is a delimitation. One cannot even think of a finite space except as in the whole of space (A25/B39).

^52. Cf. also A429n./B457n., A431/B459; Discovery 8:222; Progress 20:267; R4673 (from 1774), 17:638f.

^53. A412f./B439f., A432/B306, A499f./B527f., A518-20/B546-8, A524/B552, A527/B555.

^54. Hence, Kant is prepared to argue against Eberhard that ‘if something is an object of the senses and of sensation, all its simple parts must be as well, even if clarity in their representation is lacking’ (Discovery 8:205, cf. also Discovery 8:209f., 212, 217; A522/B550; Logic 9:35). Of course, Kant denies that sensible intuition contains simple parts, but the principle still holds: every part of an object of the senses is represented in sensibility, even if we cannot be conscious of it, since it exceeds our powers of phenomenological discrimination. Thus, the Critique holds that ‘space and time and all their parts are intuitions’ (B137n., my italics). See also Kant's letter to Reinhold (19 May, 1789) 11:45f. (cited by Domski 2008). To make phenomenological features (such as ‘vivacity’ or ‘phenomenological presence’) or logical characteristics (such as ‘confusion’ or ‘obscurity’) criterial for the sensible status of a representation is to lapse into pre-critical conceptions of sensibility.

^55. The opening of the Aesthetic also introduces the idea that intuition depends on affection, which effectively marks out human intuition as a distinctively sensible species of cognitive receptivity: ‘thought must ultimately relate […] in our case, to sensibility’ (A19/B33, my italics). Sensible receptivity might be contrasted with a non-sensible receptivity for representations through divine implantation or some other ‘hyperphysical influx’ (cf. B167f.; letter to Herz, 21 February, 1772 (10:131); R5421 and 5424 (c.1776-8) 18:178)). The latter might characterize Aquinas's angels, whose representations derive from those of the divine mind, though they are not acquired through causal affection or sensation, since angels have no bodies (cf. Summa Q.55, Art.2). This suggests that the rationale behind Kant's claim may have something to do with the fact that we are embodied, or at least with the fact that our receptive representations are associated with a finite spatiotemporal perspective. At any rate, one significant consequence of the claim that our intuitions depend on causal affection is that it enriches the thought-independence of intuition into a more general form of mental-act-independence. The Introduction already demonstrated that knowable objects are independent of our acts of thinking them. The causal notion of affection implies a further distinctness – not just from our acts of thought, but also from our act of intuiting the object in question. The objects that affect us (i.e. the objects we are given to know), therefore, exist independently of the mind's acts of representing them (in thought or intuition). (At this point, acts of reason, acts of productive imagination, etc. are of a piece with acts of the understanding in that they are all classified as spontaneous acts of thought, as opposed to actualizations of receptivity.) This picture is complicated by the fact that the mind (Gemüt) can affect itself in inner sense. Yet even self-affection satisfies the relevant type of mental-act-independence, for states represented in inner sense still exist independently of their being thus represented (or subsequently reflected on in second-order thought).

^56. Cf. A23-4/B38-9, A30-2/B46-8, A163/B203f., A526f./B554f.

^57. Cf. A25/B39, A31f./B47, B137n.