Synthese (2012) 186:231255DOI 10.1007/s11229-012-0066-2

Kant on geometry and spatial intuition

Michael Friedman

Received: 14 January 2010 / Accepted: 27 July 2010 / Published online: 7 March 2012 Springer Science+Business Media B.V. 2012

Abstract I use recent work on Kant and diagrammatic reasoning to develop a recon-sideration of central aspects of Kants philosophy of geometry and its relation to spatialintuition. In particular, I reconsider in this light the relations between geometrical con-cepts and their schemata, and the relationship between pure and empirical intuition. Iargue that diagrammatic interpretations of Kants theory of geometrical intuition can,at best, capture only part of what Kants conception involves and that, for example,they cannot explain why Kant takes geometrical constructions in the style of Euclidto provide us with an a priori framework for physical space. I attempt, along theway, to shed new light on the relationship between Kants theory of space and thedebate between Newton and Leibniz to which he was reacting, and also on the role ofgeometry and spatial intuition in the transcendental deduction of the categories.

Keywords Geometry Diagrammatic reasoning Space Intuition Schematism Transcendental deduction

An earlier version of this paper was presented at the second meeting of the Stanford-Paris workshop ondiagrams in mathematics in the Fall of 2008 from which the present special issue is drawn, and it wasoriginally inspired by a paper presented by Marco Panza on diagrammatic reasoning in Euclid at the firstmeeting of the Stanford-Paris workshop in the Fall of 2007. Panzas paper in the present issue is based, inturn, on his earlier presentation. Since Panzas paper, as it now appears, has since been substantiallyrevised, I have taken the opportunity substantially to revise my paper as well, and, in particular, I havechosen to take as my main target work of the Kant scholar Lisa Shabel that is very much in the spirit ofKenneth Manderss original discussion of the Euclidean diagram (note 1 below). I am also indebted, inthis connection, to comments on the earlier version of my paper from Jeremy Avigad. For helpfulcomments on the penultimate version of this paper I am further indebted to Daniel Sutherland and to ananonymous referee for Synthese.

M. Friedman (B)Department of Philosophy, Stanford University, Stanford, CA 94305, USAe-mail: mlfriedman@stanford.edu

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Kants philosophy of geometry can only be properly understood against the back-ground of two more general features of his philosophical position: his fundamentaldichotomy between the two basic cognitive faculties of the mind, sensibility andunderstanding, and his distinctive theory of space as the pure form of our outer sen-sible intuition. Kants conception of space and time as our pure forms of sensibleintuition (outer and inner) is central to his general philosophical position, which hecalls formal or transcendental idealism. And, although a fundamental dichotomybetween the two faculties of sense and intellect precedes Kant by many centuries, andis characteristic of all forms of traditional rationalism from Plato to Leibniz, Kantsparticular version of the dichotomy is entirely distinctive of him. For, in sharp contrastto all forms of traditional rationalism, Kant locates the primary seat of a priori math-ematical knowledge in sensibility rather than the intellect. In particular, our pure formof outer sensible intuitionspaceis the primary ground of our pure geometricalknowledge.

Kant characterizes the distinctive role of our pure intuition of space in geometryin terms of what he calls construction in pure intuition, and he illustrates this roleby examples of geometrical construction from Euclids Elements. It is natural, then,to turn to recent work on diagrammatic reasoning in Euclid originating with KennethManders to elucidate Kants conception.1 In particular, when Kant says that spatialintuition plays a necessary role in the science of geometry, we might take him to meanthat diagrammatic reasoning in the sense of Manders plays a necessary role. I shallargue that this kind of view of Euclidean geometry, as illuminating as it may be asan interpretation of the Elements, is not adequate as an interpretation of Kant, and,more generally, that recent work on diagrammatic reasoning can, at best, capture onlya part of what Kants conception of geometry involves. Most importantly, it cannotexplain why Kant took this conception crucially to involve a revolutionary new the-ory of spacethe very (three-dimensional) space in which we, and all other physicalobjects, live and move and have our being.

Kant, as I have said, diverges from traditional rationalism in locating the seat of puregeometry in sensibility rather than the understanding, and he thereby gives a centralrole in geometry to what he calls the pure productive imagination. Perhaps the mostimportant problem facing interpretations of Kants philosophy of geometry, then, is toexplain how, for Kant, sensibility and the imaginationfaculties traditionally asso-ciated with the immediate apprehension of sensible particularscan possibly yieldtruly universal and necessary knowledge. For example, in a well-known passage fromthe Discipline of Pure Reason in its Dogmatic Employment in the first Critique, Kantcontrasts philosophical cognition, as rational cognition from concepts, with math-ematical cognition, as rational cognition from the construction of conceptsand,Kant famously adds, to construct a concept is to present the intuition correspondingto it a priori (A713/B741).2 Kant concludes, [philosophy] confines itself merelyto universal concepts, [mathematics] can effect nothing by mere concepts, but hastens

1 Manderss classic paper, The Euclidean Diagram, has been widely circulating in manuscript form since1995. It has now finally appeared in print as (2008b), together with a new introduction to the subject (2008a).2 All translations from Kants writings are my own, and I cite them according to the standard conventions:all citations of the Critique of Pure Reason are to the pagination in the first, 1781 (A), and second, 1787(B),

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immediately to intuition, in which it considers the concept in concretonot, how-ever, empirically, but merely in an [intuition] that it presents a priori, that is, whichit has constructed, and in which that which follows from the universal conditionsof construction must also hold universally of the object of the constructed concept(A715716/B743744).

Exactly what, however, is a pure or non-empirical intuition corresponding to a gen-eral concepta singular instance of this concept that is nonetheless presented purelya priori ? Moreover, how can any singular instance of a general concept (no matterhow it is supposed to be produced) possibly be an additional source, over and abovepurely conceptual representation, of universal and necessary knowledge? Immediatelyafter the just quoted sentence defining the construction of a concept as the a prioripresentation of the corresponding intuition, Kant continues (A713/B741): For theconstruction of a concept we therefore require a non-empirical intuition, which con-sequently, as intuition, is a singular [einzelnes] object, but which nonetheless, as theconstruction of a concept (a universal representation), must express universal valid-ity, in the representation, for all possible intuitions that belong under this concept.But how, once again, can an essentially singular representation (no matter how it issupposed to be produced) possibly express such truly universal validity? Problems ofprecisely this kind underlie the contrary conviction, common to all traditional formsof rationalism, that mathematical knowledge must be conceptual or intellectual asopposed to sensible.

Kant illustrates his meaning, in the continuation of our passage, by an example ofa Euclidean proof, Proposition I.32 of the Elements, where it is shown that the sum ofthe interior angles of a triangle is equal to the sum of two right angles:

A

B C D

E

Given a triangle ABC one extends the side BC (in a straight line) to D and draws theline CE parallel to AB. One then notes (by Proposition I.29) that the alternate anglesBAC and ACE are equal, and also that the angle ECD is equal to the internal andopposite angle ABC. But the remaining internal angle ACB added to the two anglesACE and ECD (whose sum is the external angle ACD) is equal to the sum of two rightangles (the straight line BCD), and the two angles ACE and ECD have just been shownto be equal, respectively, to the first two internal angles. Therefore, the three internalangles taken together also equal the sum of two right angles. This construction and

Footnote 2 continuededitions respectively; all of Kants other writings are cited by volume and page number in the Akademieedition of Kants collected writings, (1902-), abbreviated as Ak.

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proof obviously has universal validity for all triangles, because the required inferencesand auxiliary constructions (extending the line BC to D and drawing the parallel CEto AB) can always be carried out within Euclidean geometry, no matter what triangleABC we start with.

It appears, in fact, that the proof-procedure of Euclids Elements is paradigmatic ofconstruction in pure intuition throughout Kants discussion of mathematics in the firstCritiquewhich includes a fairly complete presentation of the elementary Euclideangeometry of the triangle. In the Transcendental Aesthetic, for example, Kant presentsthe corresponding side-sum property of trianglesthat two sides taken together arealways greater than the third (Proposition I.20)as an illustration of how geometri-cal propositions are never derived from universal concepts of line and triangle, butrather from intuition, and, in fact, [are thereby derived] a priori with apodictic cer-tainty (A25/B39). And the Euclidean proof of this proposition proceeds, just likeProposition I.32, by auxiliary constructions and inferences starting from an arbitrarytriangle ABC: we extend side BA (in a straight line) to D such that AD is equal to AC;we then draw CD and note (by Proposition I.5) that the two angles ACD and ADCare equal, so that BCD is greater than BDC; since (by Proposition I.19) the greaterangle is subtended by the greater side, it follows that BD is greater than BC; but BDis equal to the sum of BA and AD (= AC). Moreover, Kant refers to the Euclideanproof of Proposition I.5 itselfthat the angles at the base of an isosceles triangle areequalin a famous passage in the second (1787) edition Preface praising the charac-teristic method of mathematics introduced by the revolution in thought effected bythe Ancient Greeks; and this proof, too, proceeds by the expansion of an original (andarbitrary) triangle ABC into a more complicated figure by auxiliary constructions.3

Kants reliance on Euclid is thus very clear, and, once again, it is therefore natu-ral to turn to recent work on the diagrammatic reasoning found in the Elements forelucidating Kants view. With respect to the issue of how perception of an individualsensible particular (such as a concrete physical diagram) could possibly issue in uni-versally valid knowledge, for example, we can appeal to Manderss central distinctionbetween exact and co-exact properties of a Euclidean diagram. The former include themetrical relations of equality or inequality between lengths, angles, and areas, whereasthe latter include only the topological (or mereo-topological) relations of containmentbetween the regions defined by these magnitudes. We observe, for example, that thespecifically metrical features of the triangle used in the proof of Proposition I.32thelengths of its particular sides and the magnitudes of its particular anglesplay norole at all: it remains true for all continuous variations of these lengths and angles. Bycontrast, that the external angle ACD of the extended diagram (ABCDE) contains (as

3 The reference to Proposition I.5 is made explicit in a letter to Christian Schtz of June 25, 1787, whereKant corrects gleichseitiger in the printed text to gleichschenkligter (Ak. 10, 489). The passage, socorrected, reads as follows (Bxi-xii): A light dawned on the first man (whether he may have been Thalesor some other) who first demonstrated the isosceles triangle; for he found that what he had to do was not toinspect what he saw in the figure, or even in the mere concept of it, and, as it were, to read off its propertiestherefrom, but rather to bring forth what he had himself a priori injected in thought [hineindachte] and pre-sented (through construction), in accordance with concepts, and that, in order securely to know somethinga priori , he had to attribute nothing to the thing except that which followed necessarily from what he hadplaced in it himself in accordance with his concept.

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their sum) the two angles ACE and ECD is essential to the proof, and it, too, remainstrue for all continuous variations of the original sides and angles. Thus, by relyingonly on the co-exact properties of the extended diagram, we have indeed proved aproposition valid for all particular triangles whatsoever.

Aware of how his ideas had meanwhile proved attractive to Kant scholars,Manders briefly addresses the relationship between his conception of Euclidean proofand Kants conception of (pure) intuition in (2008a, p. 74): [My understanding ofEuclidean diagrams] respects Kants conception (cf. Shabel 2003, Goodwin (2003))that intuitions (diagrams) are particular, and connected to general claims via schema-tization (conceptualization via the diagram construction conditions). That diagram-based (co-exact) claims are stable under diagram distortion, hence independent of anyparticular empirical realization, might then motivate the necessity or aprioricity ofgeometrical intuition. Manders here refers to Lisa Shabels 1997 dissertation at theUniversity of Pennsylvaniapublished as Shabel (2003) in the Outstanding Disserta-tion Series: Studies in Philosophyalong with William Goodwins 2003 dissertationat the University of California-Berkeley.

Shabels basic idea is that a pure intuition is just an empirical intuition (an actuallydrawn particular figure) which functions in a certain way in geometrical demonstra-tionsin precisely such a way that it can then confer both apriority and universality onsuch demonstrations.4 She illustrates the characteristic function in question by distin-guishing between two different proofs of Euclid I.32: a mechanical demonstrationdue to Christian Wolff, based on making exact (metrical) comparisons between theangles in the extended figure (ABCDE) by transporting an open compass, and theoriginal Euclidean proof of I.32, which, as we have seen, Kant himself appeals to.The second mathematical demonstration succeeds in conferring both apriority anduniversality on its conclusion, for Shabel, precisely because it does not depend onexact metrical information. Thus, although she does not explicitly cite Manderssoriginal 1995 paper, Shabels analysis of the distinction between mechanical andmathematical demonstrations closely parallels his fundamental distinction betweenexact and co-exact properties of particular concrete diagrams.5 Accordingly, Sha-bel places the same kind of emphasis on the concrete individual diagram (the actually

4 See Shabel (2003, p. 94): I propose that Kant is here [A714/B742] showing how a pure intuition can beconstrued as actually drawn, and thus rendered empirically, without ceasing to function as a pure intuition.The three ways in which an empirical intuition can confer a priority are thus read as ways in which anindividual drawn figure can function purely. . . . [T]he pure intuitions which exhibit and construct mathe-matical concepts, and on which mathematical demonstrations are based, are intuitions of single, individual,sensible objects considered in conjunction with the procedure for the construction of those objects.5 See Shabel (2003, pp. 99100): By contrast [with the mechanical demonstration], the diagram con-structed for the mathematical demonstration yields no exact information, such as the comparative mea-sures of the interior and exterior angles of the triangle. The diagram [ABCDE] provides information aboutpart/whole (and consequently lesser/greater) relationships without determining strict equalities betweenparts. We might say that the diagram, considered mechanically, provides exact (though possibly imprecise)information regarding the measures of magnitudes; when considered mathematically the diagram providesinexact information regarding spatial containment of magnitudes. In the mechanical proof the claim thatthe angles ABC and BAC together equal the angle ACD is justified by measuring all three angles withinstruments and comparing the results, whereas in the mathematical proof the same claim is justified by thepreviously demonstrated relationships between angles contained by parallel lines and a transversal. Sha-bel concludes (p. 101): [T]hus, the mechanical demonstration is not distinguished fromthe mathematical

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drawn particular figure) as does Manders: we begin with the former and then con-nect it with general claims by (co-exact) diagram construction conditions.6 As aninterpretation of Kant, I believe that this emphasis is misplaced.

In the Axioms of Intuition (the principles of pure understanding corresponding tothe categories of quantityunity, plurality, and totality), Kant considers the Euclideanconstruction of a triangle in general from any three lines such that two taken togetherare greater than the third (Proposition I.22: the restriction is obviously necessarybecause of what has just been proved in Proposition I.20). This makes it clear, in myview, that the construction in pure intuition of the concept of a triangle in general, forKant, just is the Euclidean construction demonstrated in Proposition I.22where, inKants words, I have here the mere function of the productive imagination, which candraw the lines greater or smaller, and thereby allow them to meet at any and all arbi-trary angles (A164-5/B205). Moreover, in the chapter on the Schematism of the PureConcepts of the Understanding, Kant carefully distinguishes the general schema of apure sensible concept (i.e., a mathematical concept) from any particular image fall-ing under this concept that may be produced by the general schema (A140/B179-180):I call [the] representation of a general procedure of the imagination [Einbildungsk-raft] for providing a concept with its image [Bild] the schema of this concept. Kantthen illustrates this idea, once again, with the example of a triangle:

In fact, schemata rather than images of objects are what lie at the basis of ourpure sensible concepts. No image at all would ever be adequate to the conceptof a triangle in general. For it would never attain the universality of the concept,which makes it hold for all triangles, whether right-angled, acute-angled, andso on, but would always be limited to only a part of this sphere. The schema ofthe triangle can never exist anywhere but in thought, and it signifies a rule ofsynthesis of the imagination with respect to pure figures in space. (A140-1/B180)

This rule of synthesis, therefore, appears to be nothing more nor less than the Euclid-ean construction of an arbitrary triangle considered in the Axioms of Intuition as amere [universal] function of the productive imagination.

Footnote 5 continueddemonstration by virtue of a distinction between an actually constructed figure and an imagined figure, butrather by the way in which we operate on and draw inferences from that actually constructed figure.6 For Shabel, this priority of the concrete individual diagram is expressed in her view that a pure intuitionis just an empirical intuition functioning purely. Compare Shabel (2003), p. 102: Despite the fact thatthe figures constructed in the mechanical and mathematical demonstrations of proposition I.32 are iden-tical, the former figure is, in Kantian terms, a case of empirical intuition, and the latter of pure intuition.Since they are not distinguished by the way they appear, nor by the medium in which or tools with whichthey are constructed, they must be distinguished by their function in the demonstration. The endnote adds(p. 160): [T]he pure intuition might be empirical insofar as it is (or can be) of an actually drawn figure,and not a merely imagined one. But it is an empirical intuition that functions purely. This coheres with herearlier idea (note 4 above) that pure intuitions are intuitions of single, individual, sensible objects consid-ered in conjunction with the procedure for the construction of those objects (emphasis added). Shabel laterexplains that the relevant procedures for construction are what Kant means by schemata, and so a schema,on her interpretation, is a general condition by which a concrete individual diagram is seen as expressinguniversality. I shall return to Shabels interpretation of the schematism below.

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More generally, then, we can take the Euclidean constructions corresponding tothe fundamental geometrical concepts (line, circle, triangle, and so on) as what Kantmeans by the schemata of such concepts.7 We can understand the schema of the con-cept of triangle as a function or constructive operation which takes three arbitrarylines (such that two together are greater than the third) as input and yields the triangleconstructed out of these three lines as output (in accordance with Proposition I.22);we can understand the schema of the concept of circle as a function which takes anarbitrary point and line segment having this point as one of its endpoints as input andyields the circle with the given point as center and the given line segment as radiusas output (in accordance with Postulate 3); and so on.8 Such constructive operationshave all the generality or universality of the corresponding concepts: they yield, withappropriate inputs, any and all instances of these concepts. Unlike general conceptsthemselves, however, the outputs of a schema are indeed singular or individual rep-resentationsparticular instances, or what Kant calls images, which fall under theconcept in question. The outputs of a schema, therefore, are not conceptual or logicalentities like propositions or truth-values.

This last point is crucial for understanding why Kant takes pure mathematicsessentially to involve non-discursive or non-conceptual cognitive resources, which,nonetheless, possess all the universality and necessity of purely conceptual thought.Characteristic of conceptual thinking, for Kant, is the logical procedure of subsump-tion, whether of an individual under a general concept or of a less general concept(species) under a more general concept (genus). Characteristic of mathematical rea-soning, by contrast, is the procedure of substitutionby which, as we would now put it,an object is inserted into the argument place of a function, yielding another object thatcan be inserted into the argument places of further functions, and so on. Reasoning bysubstitution is therefore essentially iterative, and it is precisely such iterative thinking,

7 I articulate this interpretation of geometrical schemata in Friedman (1992, pp. 9091, n. 59) and, morefully, (1992, pp. 122129). Shabel develops a closely analogous reading, based on many of the same pas-sages, in (2003, pp. 109114). The main difference, as already suggested, is that Shabel views such a schemaas a general condition for seeing a particular image as expressing universality (compare note 6 above). Asshe herself puts it (2003, p. 114): [T]he pure intuition that is the basis for a mathematical demonstration ofproposition I.32 is a universalizable image since it is intuited with, and only with, the specified procedure forits construction in imagination . . . Because mathematical cognition considers the universal in the particular. . . (which is to say that the schematized mathematical concept provides the rule for constructing a pureand universalizable intuition), the individual pure intuition so constructed can be understood as general.On my reading, by contrast, the notion of a universalizable image is an oxymoron, since an image (asopposed to a schema) is precisely that which is not universal and thus can never be adequate to the conceptof a triangle in general. Compare A140/B179 (emphasis added): The schema in itself is always only aproduct of the imagination; yet, in so far as the synthesis of the imagination aims at no individual intuition,but rather at unity in the determination of sensibility alone, the schema is to be distinguished from theimage.8 See A234/B287: Now a postulate in mathematics is the practical proposition that contains nothing butthe synthesis by which we first give to ourselves an object and generate its concepte.g., to describe acircle with a given line from a given point on a planeand such a proposition cannot be proved, becausethe procedure it requires is precisely that by which we generate the concept of such a figure.

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for Kant, that underlies both pure geometry (in the guise of Euclidean proof) and themore general calculative manipulation of magnitudes in algebra and arithmetic.9

Kants conception of the essentially non-conceptual character of geometrical rea-soning is thus especially sensitive to the circumstance that, in Euclids formulationof geometry, the iterative application of initial constructive operations represents theexistential assumptions we would express by explicit quantificational statements inmodern formulations following Hilbert. Thus, for example, whereas Hilbert repre-sents the infinite divisibility of a line by an explicit quantificational axiom stating thatbetween any two points there exists a third, Euclid represents the same idea by showinghow to construct a bisection function for any given line segment (Proposition I.10): ourability to iterate this construction indefinitely then represents the infinite divisibility ofthe same segment. More generally, Euclid constructs all the points in his plane by theiterative application of three initial constructive operations to any given (arbitrary) pairof points: connecting any two points by a straight line segment (Postulate 1), extendingany given line segment in a straight line (Postulate 2), constructing a circle with anypoint as center and any given line segment having this point as one of its endpoints asradius (Postulate 3). This procedure yields all points constructible by straight-edge andcompass, which, of course, comprise only a small (denumerable) subset of the full two-dimensional continuum whose existence is explicitly postulated by Hilbert.10 In thissense, the existential assumptions needed for Euclids particular proof-procedurethevery assumptions needed to justify all the auxiliary constructions needed along thewayare given by Skolem functions for the existential quantifiers we would use informulating a Hilbert-style axiomatization in modern quantificational logic, where (inEuclid) all such Skolem functions can be explicitly constructed by finite iterations ofthe three initial constructive operations laid down in the first three postulates.

Following Leibniz, Kant takes the discursive structure of the understanding orintellect to be delimited by the logical forms of traditional subject-predicate logic. Inexplicit opposition to Leibniz, however, Kant takes these logical forms to be strictlylimited to essentially finitary representations: there are, for Kant, no Leibnizean com-plete concepts comprising within themselves (that is, within their defining sets ofmarks [Merkmale] or partial concepts [Teilbegriffe]) an infinite manifold of furtherconceptual representations. But mathematical representations (including the mathe-matical representation of space) can and do contain an infinite manifold of further(mathematical) representations within themselves (as in the representation of infinitedivisibility). So such representations, for Kant, are not and cannot be conceptual.11 Of

9 For further discussion of algebra and arithmetic from this point of view see Friedman (1992, pp. 8389,104122). For a contrasting view see Shabel (1998). Compare also Sutherland (2006).10 More precisely, we can represent all the points constructible by straight-edge and compass constructionin the Euclidean plane by the Cartesian product of a square-root extension field of the rationals (aptly calleda Euclidean field) with itself, whereas the full set of points generated by a true (second-order) continuityaxiom is of course represented by R2, where R is the real numbers. An important intermediate case, studiedin Tarski (1959), uses a (first-order) continuity schema and is represented by a Cartesian product over anyreal closed field.11 This is the burden of the fourth argument in the Metaphysical Exposition of Space in the second editionTranscendental Aesthetic (B3940): Space is represented as an infinite given quantity. Now one mustcertainly think every concept as a representation which is contained in an infinite aggregate of different

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course, we now have an entirely different conception of logic from Kants, one that ismuch more powerful than anything either he, or even Leibniz, ever envisioned. Nev-ertheless, we can still understand Kants fundamental insight, from our own point ofview, if we observe that no infinite mathematical structure (such as either the space ofEuclidean geometry or the number series) can possibly be represented within monadicquantificational logic. Such infinite structures, in modern logic, are represented by theuse of nested sequences of universal and existential quantifiers using polyadic logic.These same representations, from Kants point of view, are instead made possibleby the iterative application of constructive functions in the productive imagination,where, as we have seen, Skolem functions for the existential quantifiers we would usein our formulations are rather explicitly constructed.

We now see, from Kants point of view, why mathematical thinking essentiallyinvolves what he calls the pure productive imagination, and why, accordingly, thistype of thinking essentially exceeds the bounds of purely conceptual, purely intellec-tual thought. My first problem with using the diagrammatic interpretations of Euclid inthe style of Manders to interpret Kants notion of construction in pure intuition, there-fore, is that they do not square with Kants understanding of the relationship betweenconceptual thought and sensible intuition. They do not square, more specifically, withhis developed view of the relationship between general (geometrical) concepts, theircorresponding general schemata, and the particular sensible images (particular geo-metrical figures) which then result by applying these schemata. In particular, whereassuch diagrammatic accounts of the generality of geometrical propositions, as we haveseen, begin with particular concrete diagrams and then endeavor to explain how we canabstract from their irrelevant particular features (specific lengths of sides and angles,say) by relying only on their co-exact features, Kant begins with general concepts asconceived within the Leibnizean (logical) tradition and then shows how to schema-tize them sensibly by means of an intellectual act or function of the pure productiveimagination. Both the general concepts in question and their corresponding generalschemata are pure rather than empirical representations; and a particular concrete fig-ure occurs, as it were, only incidentally for Kant, at the end of a process of intellectualdetermination of pure (rather than empirical) sensibility.

The more general point underlying these considerations is that pure intuition, forKant, is the form of (empirical) intuition: it lies in wait prior to the reception of all sen-sationsthe corresponding matter of (empirical) intuitionas an a priori conditionof the possibility of all sensory perceptions and their objects.12 Actually perceived

Footnote 11 continuedpossible representations (as their common mark), and it therefore contains these under itself. But no con-cept, as such, can be so thought as if it were to contain an infinite aggregate of representations within itself.However space is thought in precisely this way (for all parts of space in infinitum exist simultaneously).Therefore, the original representation of space is an a priori intuition, and not a concept. For further dis-cussion see Friedman (1992, pp. 6671). As I shall explain below, however, I now think that the relationshipbetween the mathematical (i.e., geometrical) representation of space and the original representation ofspace described in the Metaphysical Exposition is a bit more subtle: the latter grounds the former but is notsimply identical with it.12 Kant explains this at the beginning of the Transcendental Aesthetic (A20/B34): I call that in the appear-ance that corresponds to sensation the matter of appearance, but that which brings it about that themanifold

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concrete diagrams therefore presuppose the prior structure of pure intuition just asmuch as all other sensibly perceived objects, and so it is very misleading, at best,to interpret a Kantian pure intuition as a certain kind of empirical intuition. On thecontrary, we need to connect Kants conception of geometrical reasoning, in the firstinstance, with the pure intuitions of space and timenot with particular spatial figuresdrawn on paper or a blackboard but with space and time themselves, as pure ratherthan empirical intuitions.13 And it is precisely here, as I have intimated, that Kant alsoengages Newtons conception of space (and time) as it figures in his controversy withLeibniz. Space, for Newton, is a great ontological receptacle, as it were, for both allpossible geometrical figures and all possible material objects, and Kants theory ofspace as a pure form of intuition is supposed to be an alternativeas we shall seetoprecisely this Newtonian conception.

It is centrally important to Kants philosophy of geometry that all possible objects ofhuman sense-perception, all objects of what Kant calls empirical intuition, must nec-essarily conform to the a priori principles of mathematics established in pure intuition(A165166/B206): The synthesis of spaces and times, as the essential form of all intu-ition, is that which, at the same time, makes possible the apprehension of appearance,and thus every outer experience, [and] therefore all cognition of the objects thereof;and what mathematics in its pure employment demonstrates of the former necessarilyholds also of the latter.14 In order to appreciate the role that pure geometry plays inour perception of empirical objects, then, we need explicitly to connect the functions

Footnote 12 continuedof appearance can be ordered in certain relations I call the form of appearance. Since that wherein thesensations are alone ordered, and can be placed in a certain form, cannot itself be sensation in turn, it isonly the matter of all appearance that can be given to us a posteriori; but the form of all appearance mustlie ready for them [the sensations] in the mind a priori , and it can therefore be considered separately fromall sensations.13 Manders (2008a, pp. 7071) is explicit that Euclidean diagrams, on his view, are individual physicalobjectswhich suggests that Kantian pure intuitions, understood in terms of Manderss conception ofdiagrammatic reasoning, are also individual physical objects (compare the passage to which note 4 aboveis appended). Shabel comes very close to this view in insisting that Kantian pure intuitions, in geometry,are intuitions of single, individual, sensible objects (note 4 above, emphasis added). In the Preface addedto the published version of her dissertation, Shabel explains that her interpretation of Kant has since beenfurther clarified and elaborated (2003, p. xi): My current project includes an attempt to understand the roleof mathematical construction in the context of a full investigation of Kants theory of sensibility, includinghis theory of pure intuition as articulated in the Transcendental Aesthetic. I did not pursue this more generalstrategy in the dissertation, which resulted in an incomplete and, at times, unclear account of both theschematism and the distinction between pure and empirical intuition as modes of sensible representation.I invite the interested reader to consult Shabels later writings on the subject and to compare (and contrast)them with the account presented here. See, for example, Shabel (2006), together with the works cited there.14 Compare the important passage at A223224/B272: It seems, to be sure, as if the possibility of a tri-angle could be cognized from its concept in itself (it is certainly independent of experience); for we can infact give it an object completely a priori , i.e., construct it. However, because this is only the form of anobject, it would remain forever only a product of the imagination, and the possibility of its object would stillremain doubtfulas that for which something more is still required, namely, that such a figure be thoughtunder pure conditions on which all objects of experience rest. Now, that space is a formal a priori conditionof outer experiences; that precisely the same image-forming [bildene] synthesis by which we construct atriangle in the imagination is completely identical with that which we exercise in the apprehension of anappearance, in order to make for ourselves an empirical concept of itit is this alone that connects thisconcept [of a triangle] with the possibility of such a thing. Thus, the formal conditions of all sensible or

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of the pure productive imagination expressed in the construction of geometrical con-cepts with the Kantian forms of pure intuition (space and time), as they are describedin the metaphysical expositions of space and time in the Transcendental Aesthetic.15

In the course of his controversy with the Leibnizean philosopher Johann AugustEberhard in 1790, Kant develops a contrast between the (successively constructed)space of the geometer and the subjectively given space of our pure form of outersensible intuition. Kant begins by asserting that to say that a line can be continued toinfinity means that the space in which I describe the line is greater than any line that Imay describe in it, so that the geometer grounds the possibility of his problemtoincrease a space (of which there are many) to infinityon the original representationof a single, infinite, subjectively given space. [G]eometrical and objective space,Kant continues, is always finite, for the latter is only given in so far as it is generated[gemacht]. And this geometrical space is then explicitly contrasted with what Kantcalls metaphysical space:

To say, however, that the metaphysical, i.e., original, but merely subjectivelygiven spacewhich (because there are not many of them) can be brought underno concept, which would be capable of a construction, but which still containsthe ground of the construction of all possible geometrical conceptsis infinite,means only that it consists in the pure form of the mode of sensible represen-tation of the subject, as a priori intuition; hence in this form of intuition, assingular [einzelnen] representation, the possibility of all spaces, which proceedsto infinity, is given. (Ak. 20, 420421)

Thus, metaphysical space is the space considered in the Metaphysical Expositionof Space in the Transcendental Aesthetic, whereas geometrical space consists of the

Footnote 14 continuedempirical intuitions include not only pure space and time themselves, as it were, but also the pure synthesesof the productive imagination expressed in the a priori constructions (schemata) of geometrical concepts. Itis only by presupposing that the latter are already available that the former (sensible or empirical intuitions)first become possible.15 It follows from this analysis (especially note 14 above) that the pure productive imagination is priorto all empirical intuitions, and thuscontrary to Shabel (compare notes 4 and 5 above)that the differ-ence between an actually drawn figure and a merely (purely productively) imagined one is indeed centralto Kants distinction between pure and empirical intuition. Shabel is perfectly correct, of course, that aconcrete empirical figure (even if badly drawn) can function as a Kantian pure intuition in the context ofexecuting an actual geometrical proof (compare note 6 above). But it can do so, on my reading, only becauseall empirical intuitions (including this one) take place in accordance with, and against the background of,the pure syntheses of the productive imagination. Immediately following the passage at A713/B741 withwhich we began our consideration of construction in pure intuition (see the passage to which note 2 aboveis appended, together with its continuation in the following paragraph), Kant continues (ibid., emphasisadded): Thus I construct a triangle, in so far as I present this concept with a corresponding object, eitherthrough mere imagination in pure intuition, or, in accordance with this [pure intuition], also on paper inempirical intuitionin both cases, however, completely a priori , without having derived its model fromany experience. The crucial point, once again, is that the activities of the productive imagination in pureintuition are prior to actually drawing a figure on paper in empirical intuition. (I shall return below to whatexactly this priority consists in.) NB: The generally excellent Guyer-Wood translation, which Shabel quotesto introduce her discussion (2003, pp. 9192), omits the in accordance with this phrasebut Shable (2003,p. 105) suggests an alternative reading of what cognizing an empirical intuition in accordance with theconditions of pure intuition might mean nonetheless.

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indefinitely extendible (but always finite) manifold of geometrical constructions whichmay (at any finite stage) be actually carried out starting from some (arbitrary) initialpair of points.16

This important passage, unlike the Metaphysical Exposition, articulates a clear andexplicit connection between space as the pure form of outer intuition and geometri-cal construction. So let us now turn to the first two arguments of the MetaphysicalExposition itself, where, I believe, the nature of this connection is nonetheless implic-itly suggested.17 These arguments are intended to show, in particular, that space is anecessary a priori representation that precedes all empirical perceptionsnot a repre-sentation that can in any way be abstracted from our empirical perceptions of (outer)spatial objects.

The first argument attempts to show that space is an a priori rather than empiri-cal representation by arguing that all perception of outer (empirical) objects in spacepresupposes the representation of space:

Space is no empirical concept that has been derived from outer experiences.For, in order that certain sensations are related to something outside me (thatis, to something in another place in space than the one in which I find myself),and, similarly, in order that I be able to represent them as outside of and nextto one anotherand thus not merely as different but as in different placestherepresentation of space must already lie at the basis. Therefore, the representa-tion of space cannot be obtained from the relations of outer appearance throughexperience; rather, this outer experience is itself only possible in the first placeby means of the representation in question. (A23/B38)

This argument emphasizes that space as the form of outer sense enables us to representobjects as outer precisely by representing them as spatially external to the perceivingsubject, so that the space in question contains the point of view from which the objectsof outer sense are perceived and around which the objects of outer sense are arranged.Empirical spatial intuition or perception occurs when an object spatially external tothe point of view of the subject affects this subjectalong a spatial line of sight, asit wereso as to produce a corresponding sensation; and it is in this sense, therefore,that the pure form of (spatial) sensible intuition expresses the manner in which we areaffected by (outer) spatial objects.18 Let us call this structure perspectival space.

The second argument goes on to claim that space is a necessary a priori represen-tation, which functions as a condition of the possibility of all outer experience:

16 The controversy in question is discussedand many relevant texts are translatedin Allison (1973). Inparticular, the entire passage (from Ak. 20, 419421) is translated in Allison (1973, pp. 175176).17 I develop this analysis, in response to Parsons (1992) and Carson (1997), in Friedman (2000)where, inparticular, I attempt to reconcile what I call the logical interpretation of Kants philosophy of geometry (asdeveloped by Evert Beth, Jaakko Hintikka, and my earlier self) with the phenomenological interpretationarticulated by Parsons and Carson. The basic idea of my attempted reconciliation is to embed the purelylogical understanding of geometrical constructions (as Skolem functions) within space as the pure form ofour outer sensible intuition (as described in the Transcendental Aesthetic).18 See again the beginning of the Transcendental Aesthetic (A1920/B3334): In whatever manner andby whatever means a cognition may relate to objects, that by which it is related to them immediately, andtowards which all thinking as a means is directed, is intuition. But this takes place only in so far as theobject

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Space is a necessary a priori representation, which lies at the basis of all outerintuition. One can never make a representation [of the supposed fact] that there isno space, although one can very well think that no objects are to be found therein.It must therefore be viewed as the condition of the possibility of appearances,not as a determination depending on them, and is an a priori representation,which necessarily lies at the basis of outer appearances. (A24/B38-9)

The crux of this argument is that one cannot represent outer objects without space,whereas one can think this very same space as entirely empty of such objects. And,since the first conjunct may appear tautological, the burden of the argument falls on thesecond conjunct. What exactly does it mean, therefore, to represent space as empty ofouter objects, and in what precise context, moreover, do we succeed in doing this? Avery natural suggestion is that we think space as empty of outer (empirical) objects justwhen we are doing pure geometry.19 This would accord very well, in particular, withthe concluding claim that space thereby functions as a necessary a priori condition ofthe possibility of outer appearances, for they would then all be subject to the a priorinecessary science of pure geometry.20

What is the precise relationship between the a priori structure attributed to space inthe first argument (perspectival space) and that attributed to space in the second (thestructure of pure geometry)? It is natural, in the first place, to view the former struc-ture as itself a priori, since it does not depend at all on the particular (empirical) outerobjects actually perceived from any particular point of view. On the contrary, this per-spectival structure is invariant under all changes in both the objects perceived and thepoint of view from which they are perceived, and, in this sense, it thereby expresses theform rather than the matter or content of outer intuition. Moreover, and in the secondplace, these possible changes in perspective themselves constitute what we now taketo be a mathematical structure: namely, a group of (Euclidean) motions or transfor-mations, comprising all possible translations of our initial point of view through spaceand all possible rotations of the perspective associated with this point of view around

Footnote 18 continuedis given to usand this, in turn, at least for us humans, is only possible in so far as the mind is affectedin a certain way. The capacity (receptivity) to obtain representations through the manner in which we areaffected by objects, is sensibility . . .. The effect of an object on the faculty of representation, in so far as weare affected by them, is sensation. That intuition which relates to the object through sensation is empirical.The undetermined object of an empirical intuition is appearance.19 Parsons (1992, p. 69) offers this as an obvious idea, although he does not embrace it unreservedly.20 This point also allows us to answer a well-known objection to the first argument first raised by J. G. Maa(a colleague of Eberhards), according to which it does not follow from the fact that one representation pre-supposes another that the latter representation is a priori: in order to recognize red objects, for example,one must first have the concept of red (and, more generally, color), but it of course does not follow from thisthat red (or color) is an a priori rather than empirical concept. See, e.g., the discussion of this objection inAllison (1983, pp. 8286. The crucial difference, I believe, is that we do have a necessary a priori scienceof space (geometry), whereas we do not have such an a priori science in other cases (like color). I am hereindebted to discussions with Graciela De Pierris concerning the first two arguments of the MetaphysicalExposition; for her own discussion see De Pierris (2001).

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the given point.21 In particular, any perceptible spatial object, located anywhere inspace, can thereby be made accessible by an appropriate sequence of such translationsand rotations starting from any initial point of view and associated perspective.

But there is a clear connection between this modern, group-theoretical structureand geometry in Kants sense; for, as Kant himself explicitly emphasizes in his con-troversy with Eberhard, the two fundamental Euclidean constructions of drawing astraight line and constructing a circle are generated precisely by translations and rota-tionsas we generate a line segment by the motion (translation) of a point and thenrotate this segment (in a given plane) around one of its endpoints.22 On the presentinterpretation, therefore, it is precisely this relationship between perspectival spaceand geometrical space which links Kants theory of space as the form of outer intui-tion or perception with his conception of pure mathematical geometry in terms of thesuccessive execution of Euclidean constructions in the pure productive imagination.23

The same relationship between perspectival and geometrical space appears to playa central role in the second edition version of the Transcendental Deduction of theCategories.24 In a central step of the argument, entitled On the Application of theCategories to Objects of the Senses as such ( 24), Kant introduces what he callsthe figurative synthesis (synthesis speciosa) or transcendental synthesis of the imag-

21 In a modern mathematical setting, the concept of a group of Euclidean rigid motions (translations androtations) need not involve the notions of perspective and point of view. The latter notions were introduced,in this context, by Hermann von Helmholtz and Henri Poincar as part of a program for explaining how our(mathematical) concepts of space and geometry can be grounded in and arise from our actual perceptualexperience. I here aim to apply these ideas to the interpretation of Kants conception of space and geometry:see note 23 below.22 See Ak. 20, 410-411 (not translated by Allison): [I]t is very correctly said that Euclid assumes the pos-sibility of drawing a straight line and describing a circle without proving itwhich means without provingthis possibility through inferences. For description, which takes place a priori through the imagination inaccordance with a rule and is called construction, is itself the proof of the possibility of the object . . ..However, that the possibility of a straight line and a circle can be proved, not mediately through inferences,but only immediately through the construction of these concepts (which is in no way empirical), is dueto the circumstance that among all constructions (presentations determined in accordance with a rule in apriori intuition) some must still be the firstnamely, the drawing or describing (in thought) of a straightline and the rotating of such a line around a fixed pointwhere the latter cannot be derived from the former,nor can it be derived from any other construction of the concept of a magnitude. (NB: In accordance withthe passage quoted in note 14 above, mathematical construction can only demonstrate the real possibility ofthe corresponding mathematical concept against the background of the Transcendental Deductiona pointto which I shall return below.) Straight lines and circles thereby appear as what we call the orbits (confinedto any two dimensional plane) of the Euclidean group of rigid motions in space. (For the construction ofa circle compare the passage from A234/B287 quoted in note 8 above. For the construction of a line, andmore generally, compare also A162163/B203204: I can represent no line to myself, no matter howsmall, without drawing it in thought, that is, gradually generating all its parts from a point . . .. On thissuccessive synthesis of the productive imagination in the generation of figures is based the mathematics ofextension (geometry), together with its axioms, which express the conditions of a priori sensible intuitionunder which alone the schema of a pure concept of outer appearance can arise.)23 As explained in Friedman (2000), an advantage of this reading is that it then allows us to connect Kantstheory of pure geometrical intuition with the later discussions of Helmholtz and Poincar (who were self-consciously influenced by Kant)although there can of course be no question of attributing to Kant himselfan explicit understanding of the group-theoretical approach to geometry.24 I begin to develop this connection in Friedman (2000) and (2003). I shall indicate below where I nowgo beyond and correct these earlier accounts.

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ination. This synthesis establishes the first connection between the understanding orthe transcendental unity of apperception and sensibility, and, Kant explains, it is thusan action of the understanding on sensibility, and its first application (at the sametime the ground of all others) to objects of the intuition that is possible for us (B152).Kant continues:

As figurative, it is distinguished from the intellectual synthesis merely throughthe understanding, without any [use of the] imagination. In so far as the imag-ination is spontaneity, I sometimes also call it the productive imagination, andthereby distinguish it from the reproductive imagination, whose synthesis is sub-ject solely to empirical laws, namely, those of associationand which thereforecontributes nothing to the explanation of the possibility of a priori cognition, andfor this reason belongs not to transcendental philosophy but rather to psychology.(B152)

Thus, the synthesis of the pure productive imagination is not only non-empirical and apriori, but it is also what Kant calls transcendental: viz., constitutive of the explanationof possibility of a priori cognition.25

What is especially striking, however, is how Kant then goes on to illustrate thistranscendental synthesis:

We also always observe this in ourselves. We can think no line without drawing itin thought, no circle without describing it. We can in no way represent the threedimensions of space without setting three lines at right angles to one anotherfrom the same point. And we cannot represent time itself without attending, inthe drawing of a straight line (which is to be the outer figurative representationof time), merely to the action of synthesis of the manifold, through which wesuccessively determine inner sense, and thereby attend to the succession of thisdetermination in it. Motion, as action of the subject (not as determination of anobject*), and thus the synthesis of the manifold in spaceif we abstract fromthe latter and attend merely to the action by which we determine inner sensein accordance with its form[such motion] even first produces the concept ofsuccession. (B154155)

And in the footnote Kant explicitly links motion in the relevant sense with the imagi-native description of space underlying the axioms of geometry (in the construction oflines and circles):

*Motion of an object in space does not belong in a pure science and thus notin geometry. For, that something is movable cannot be cognized a priori but

25 See the beginning of the Transcendental Logic at A56/B8081: I here make a remark whose influenceextends over all the following considerations, and which one must keep well in mind, namely, that notevery a priori cognition should be called transcendental, but only that by which we cognize that and howcertain representations (intuitions or concepts) are applied, or are possible, wholly a priori (i.e., the a prioripossibility or use of cognitions). Thus, neither space nor any a priori geometrical determination of spaceis a transcendental representation; rather, only the cognition that these representations are in no way ofempirical origin, and the possibility that they can nevertheless relate a priori to objects of experience, canbe called transcendental.

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only through experience. But motion, as the describing of a space, is a pureact of successive synthesis of the manifold in outer intuition in general throughthe productive imagination, and it belongs not only to geometry, but even totranscendental philosophy.

Thus motion in the relevant sensethe pure act of successive synthesis in space as tran-scendental activity of the subjectgrounds or underlies geometry by also belongingto the metaphysical consideration of space characteristic of transcendental philoso-phy.26

But what is the precise connection between the transcendental synthesis of theimagination, as an action of the understanding on sensibility, and the metaphysi-cal consideration of space in the Transcendental Aesthetic? The concluding argumentof the second edition Deduction, entitled Transcendental Deduction of the Univer-sally Possible Employment in Experience of the Pure Concepts of the Understanding( 26), crucially depends on this connection:

We have a priori forms of outer and inner sensible intuition in the representationsof space and time, and the synthesis of apprehension of the manifold of appear-ances [through which perception becomes possible] must always accord withthem, for it can only take place in accordance with this form. But space andtime are represented a priori, not merely as forms of sensible intuition, but asintuitions themselves (which contain a manifold) and thus [represented a pri-ori] with the determination of the unity of this manifold (see the TranscendentalAesthetic*). Therefore, unity of the synthesis of the manifold, outside us or inus, and thus a combination with which everything that is to be represented inspace or time as determined must accord, is itself already given simultaneously,with (not in) these intuitions. But this synthetic unity can be no other than thatof the combination of the manifold of a given intuition in general in an originalconsciousness, in accordance with the categories, only applied to our sensibleintuition. (B160161)

Kant here wants to show that all possible objects of our spatio-temporal intuitionare necessarily subject to the transcendental unification of all representations in oneconsciousness in accordance with the categories, and he does this by appealing tothe unity of space and time themselves as already established in the Aesthetic. Sowhat is the precise connection, we now need to ask, between the unity of space and

26 According to the main passage in the text at B154155, the transcendental synthesis of the imaginationnot only grounds the science of geometry (in terms of the drawing of a straight line and the describing ofa circle), it also grounds the concept of succession and what Kant calls the general doctrine of motion(B4849). Compare B291292: How it may . . . be possible that an opposed state follows from a given stateof the same thing is not only inconceivable to any reason without example, but it is not even understandablewithout intuitionand this intuition is the motion of a point in space, whose existence in different places(as a sequence of opposed determination) alone makes alteration intuitive to us in the first place. For, inorder that we may afterwards make even inner alterations intuitive, we must make time, as the form of innersense, intelligible figuratively as a line, and inner alteration by the drawing of this line (motion), and thusthe successive existence of our self in different states by outer intuition. In this way, the transcendentalsynthesis of the imagination also explains the possibility mathematical physics: for further discussion seeFriedman (2003).

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time themselves and the synthesizing activities (via the transcendental synthesis of theimagination) of the understanding?

Kant explains in the footnote to this passage, which is unusually difficult even byKantian standards:

*Space represented as object (as is actually required in geometry) contains morethan the mere form of intuitionnamely, [it contains] the [act of] putting together[Zusammenfassung] the manifold, given in accordance with the form of sensi-bility, in an intuitive representation, so that the form of intuition gives merely amanifold, but the formal intuition [also] gives unity of representation. In the Aes-thetic I counted this unity [as belonging] to sensibility, only in order to remarkthat it precedes all concepts, although it in fact presupposes a synthesis that doesnot belong to the senses but through which all concepts of space and time firstbecome possible. For, since through it (in that the understanding determines sen-sibility) space or time are first given, the unity of this a priori intuition belongsto space and time, and not to the concept of the understanding (24).

Two points are especially mysterious here. On the one hand, it is the burden of thethird argument of the Metaphysical Exposition of Space (in the second edition) to showthat the characteristic unity of space cannot be a conceptual unity.27 It would seem,therefore, that this unity must be intuitive rather than intellectualand so how cansuch a distinctively intuitive unity possibly illustrate the synthesizing activities of theunderstanding? On the other hand, if the synthesis responsible for the unity of space(and time) does belongs to the understanding, why does it precede all concepts?And why, in particular, does the unity in question belong to space and time, and notto the concept of the understanding?

The above interpretation of the distinction between metaphysical (that is, perspec-tival) and geometrical space, as articulated in the controversy with Eberhard, helpsus to answer these questions. Metaphysical spacethe space of our pure form ofouter sensible intuitionconsists in the totality of possible perspectives from whichthe subject can be affected by outer objects. What unites this totality into a singleall-encompassing space, therefore, is the transcendental unity of apperception, whichentails that any possible outer object is in principle perceivable by the same subjectby an appropriate sequence of translations and rotations starting from any particular

27 See A24/B39: Space is not a discursive, or, as one says, general concept of relations of things in general,but a pure intuition. For, first, one can only represent to oneself a single [einigen] space, and if one speaks ofmany spaces, one understands by this only parts of one and the same unique [alleinigen] space. These partscannot precede the single all-encompassing [einigen allbefassenden] space, as it were as its constituents(out of which its composition would be possible); rather, they can only be thought in it. It is essentiallysingle [einig]; the manifold in it, and the general concept of spaces as such, rests solely on limitations. Fromthis it follows that an a priori intuition (that is not empirical) underlies all concepts of space. Thus, spaceis not a conceptual representation because, first, there is necessarily only one particular individual fallingunder it and, second, the parts of spaceunlike the parts (marks) of a conceptare not constituents (outof which its composition would be possible). A related asymmetry between the whole-part structure ofconcepts and that of intuitions underlies the immediately following fourth argument: see note 11 above,together with the paragraph to which it is appended.

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initial perspective.28 This singular, all-encompassing, and infinite space then groundsthe possibility of geometrical constructions, which are based, as we have seen, onour ability, in pure intuition, to draw a line by the translation of a point and to rotatesuch a line (in a plane) around one of its endpoints.29 The exercise of this ability, inturn, is an expression of the transcendental synthesis of the imagination, which is anaction of the understanding on sensibility, and its first application (at the same time theground of all others) to objects of the intuition that is possible for us (B152). Thus,the synthesis responsible for the characteristic unity and singularity of space (as thepure form of outer sensible intuition) does indeed belong to the understanding. It doesnot follow, however, that the unity in question is a conceptual unity.

For, in the first place, this action of the understanding on sensibility precedes allparticular geometrical constructions, and thus all particular spaces (spatial regions)since these are constructed within the singular, all-encompassing, and infinite space ofpure intuition by an indefinitely extendible (but always finite) sequence of particularacts of the pure productive imagination. Therefore, in the second place, the originaltranscendental synthesis of the imagination responsible for the characteristic unity andsingularity of space also precedes all geometrical concepts (of triangle, circle, and soon), since these concepts are generated by particular geometrical constructions inaccordance with their schemata.30 Finally, and in the third place, the same originalsynthesis precedes all (schematized) categories or pure concepts of the understand-ing, and therefore precedes all (schematized) concepts whatsoever, since each of theformer has its own particular schema in pure intuition (as a particular transcendentaldetermination of time)none of which are identical with the action of the under-standing on sensibility that first gives both space and time their characteristic unity

28 This is how I interpret the third argument of the Metaphysical Exposition (note 27 above). And it isin explaining the characteristic unity and singularity of space in terms of what I call perspectival space,and thus in terms (ultimately) of the transcendental unity of apperception, that I differ from the accountsof this characteristic unity offered by Parsons and Carson (see note 17 above). In particular, I do not takethis all-encompassing unity as a given quasi-perceptual fact, but base it on the prior condition that allpossible outer objects be perceivable, in principle, by the same perceiving subject. For further discussionsee Friedman (2000).29 See notes 21, 22, and 23 above, together with the paragraphs to which they are appended. That thelimitations mentioned in the penultimate sentence of the passage at A24/B39 quoted above (note 27)involve geometrical construction is suggested by its immediate continuation (ibid.): So, too, all geometri-cal principles, e.g., that in a triangle two sides together are always greater than the third, are never derivedfrom universal concepts of line and triangle, but rather from intuition, and, in fact, [are thereby derived]a priori with apodictic certainty. As we have seen (in the paragraph to which note 3 above is appended),Kant is here referring to Euclid I.20.30 I am here indebted to an illuminating conversation with Vincenzo De Risi. That geometrical construc-tions generate the concept constructed is explicitly stated at A234/B287 (compare notes 8 and 23 above).See also the passage from the controversy with Eberhard quoted in the paragraph to which note 16 above isappended. Kant there says, first, that metaphysical space (because it is singular) can be brought under noconcept, which would be capable of a construction, but [it] still contains the ground of the construction ofall possible geometrical concepts, and, second, that in this form of intuition, as singular representation,the possibility of all spaces, which proceeds to infinity, is given. Thus, Kant here makes explicit the rela-tionships among the singularity of space as the (all-encompassing) pure form of outer intuition, the pluralityof its parts (bounded spatial regions), and geometrical constructionrelationships that are only implicitin the third argument of the Metaphysical Exposition (compare note 28 above). He therefore clarifies thesense in which the characteristic unity of metaphysical space precedes all geometrical concepts.

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and singularity.31 The original synthesis responsible for this unity does not expressthe schema of any particular category, but rather what we might call the schema ofthe transcendental unity of apperception itself.32 Therefore, although it does representa determination of sensibility by the understanding, the unity of this a priori intui-tion indeed belongs to space and time, and not to the concept of the understanding(emphasis added). The unity in question is indeed intellectual, but it is nonethelesscharacteristic of an intuitive rather conceptual representation.33

31 Kant introduces the notion of the schema of a pure concept of the understanding as follows(A138139/B177178): The concept of the understanding contains pure synthetic unity of the mani-fold in general. Time, as the formal condition of the manifold of inner sense, and thus of the connectionof all representations, contains an a priori manifold in pure intuition. Now a transcendental determinationof time is homogenous with the category (which constitutes the unity of this determination) in so far asit is universal and rests on a rule. But it is homogeneous with the appearance, on the other side, in sofar as time is contained in every empirical representation of the manifold. Therefore, an application of thecategory to appearances becomes possible by means of the transcendental determination of time, which, asthe schema of the concept of the understanding, mediates the subsumption of the latter under the former.Since this passage characterizes time as a formal condition, and emphasizes that, as such, it contains an apriori manifold in pure intuition, comparison with the crucial argument at B160161 quoted above (in theparagraph following the one to which note 26 is appended), suggests that the time in question is not merelythe form of (inner) intuition, but also the (singular) intuition itself (formal intuition), represented withthe determination of the unity of this manifold. This explanation of the schema of a pure concept of theunderstanding therefore appears to presuppose that the determinate unification of time by the transcendentalsynthesis of the imagination has already taken place. (Note, however, that what are sometimes called thepure or unschematized categoriesunlike pure sensible conceptsstill have a meaning independentlyof their schematization, although this meaning is of no use at all in the cognition of phenomena: only theschematized categories have what Kant, at A238249/B297299, calls an empirical use.)32 Compare Kants discussion of the schema of the category (or categories) of quantity or magnitude atA142143/B182: The pure image of all magnitudes (quantorum) for outer sense is space; that of all objectsof the senses in general, however, is time. But the pure schema of magnitude (quantitatis), as a concept ofthe understanding, is number, which is a representation comprising the successive addition of One to One(homogeneous [units]). Therefore, number is nothing other than the unity of the synthesis of the manifoldof a homogeneous intuition in general, in so far as I generate time itself in the apprehension of the intuition.From this passage we see that the schema (rule for the determination of time) associated with the category(or categories) of quantity or magnitude is not the representation of singular space or singular time butthe representation of number. We also see that the representations of singular space and singular time areimages (as opposed to schemata) corresponding to the category (or categories) in question. Since Kantalso says that the schema of a pure concept of the understanding is something that can be brought intono image at all (A142/B181), these images cannot be the product of the schema of any category. Theyare rather the products of the original transcendental synthesis of the imaginationwhich results in spacegiven or presented as what I have called perspectival space, and time given or presented under the imageof a line, in so far as we draw it, without which mode of presentation we could in no way cognize the unityof its measure or dimension [Einheit ihrer Abmessung] (B156). For a subtle and illuminating discussion ofthe difficult passage at A142143/B182 (which, however, I am not entirely following here) see Sutherland(2004, III).33 In particular, the asymmetries Kant emphasizes between intuitive and conceptual representation in thethird and fourth arguments of the Metaphysical Exposition of Space (see again note 27 above) are allretained. And, on this basis, I can now indicate where the present account goes beyond and corrects my ear-lier discussions in Friedman (2000) and (2003). With respect to the former (2000, pp. 198199), it is crucialto distinguish (as we have seen) between the claim that the characteristic (singular) unity of space and time isor involves an intellectual unity (in so far as it is the result of an action of the understanding on sensibility)and the claim that it is or involves a conceptual unity (depending on the characteristic unity or generality ofa concept). That the intuitive unity in question depends directly on the unity of consciousness does notentail that it is a conceptual unity (contrary to p. 198). With respect to the latter (2003, pp. 3941),it is simply a mistake to claim (contrary to p. 40) that the reference to the Transcendental Aesthetic

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In any case, it is precisely the argument of the Transcendental Deduction (asexpressed in 26 of the second edition) which now puts Kant in a position to claimthat pure mathematical geometry is necessarily applicable to all possible objects ofempirical perceptionso that [t]he synthesis of spaces and times, as the essentialform of all intuition, is that which, at the same time, makes possible the apprehensionof appearance, and thus every outer experience, [and] therefore all cognition of theobjects thereof; and what mathematics in its pure employment demonstrates of theformer necessarily holds also of the latter (A165166/B206). And it is precisely thisargument which underwrites Kants later explanation of the same claim (A224/B272):[T]hat space is a formal a priori condition of outer experiences; that precisely thesame image-forming [bildene] synthesis by which we construct a triangle in the imag-ination is completely identical with that which we exercise in the apprehension of anappearance, in order to make for ourselves an empirical concept of itit is this alonethat connects this concept [of a triangle] with the [real] possibility of such a thing.34

The urgent need to establish such a result places Kant in a completely different intel-lectual environment from the Ancient Greek schools of Plato and Aristotle, against thebackground of which Euclids Elements was formulated. For it is characteristic of thenew view of mathematics arising in the seventeenth century that pure mathematicalgeometry is taken to be the foundation for all knowledge of physical reality. Puremathematical geometry, beginning with Descartes, is taken to describe, in principle,the most fundamental properties and interactions of matter; and, in this sense, physicalspace and geometrical space (that is, Euclidean space) are now taken to be identical.Kants own understanding of this idea, as I have suggested, is framed by the contro-versy between Newton and Leibnizwhere both took the geometrization of nature tobe a now established fact, but they reacted to this fact in radically different ways. New-ton understood the situation quite literally: mathematical space and timetrueor absolute space and timeconstitute the fundamental ontological framework ofall reality. Even God himself is necessarily spatial and temporal (existing always andeverywhere), and all physical or material objects are then created and moved, asNewton puts it, within Gods boundless and uniform Sensorium.35

Footnote 33 continuedat B160161 is to the Transcendental as opposed to the Metaphysical Expositions of Space and Time. Onthe contrary, the reference is indeed, first and foremost, to the Metaphysical Expositions (especially to thethird argument in the case of space); and the synthetic a priori knowledge whose possibility is explained inthe Transcendental Expositions (geometry and the general doctrine of motion respectively) is groundedin or explained precisely by the prior (singular and unitary) structures of space and time articulated in theMetaphysical Expositions. This brings my position even closer to that defended in Carson (1997), althoughit is still essentially different from her account (see note 28 above).34 See note 14 above, together with the paragraph to which it is appended.35 Newtons famous discussion of absolute, true, and mathematical space and time occurs in theScholium to the Definitions of the Principia. This, together with other relevant texts, can be found in New-ton (2004). In particular, Newton develops his metaphysical conception of space and its relation to Godmost fully in the manuscript De Gravitatione, where, under the influence of the Cambridge Platonism ofHenry More, he says that space is neither a substance nor an accident but rather an emanative effect of Godand an affection of every kind of being (2004, p. 21). In the General Scholium to the Principia Newtonwrites (2004, p. 91): [God] endures always and is present everywhere, and by existing always and every-where he constitutes duration and space. And in Query 31 of the Opticks Newton argues (2004, p. 138):[These natural phenomena] can be the effect of nothing else than the Wisdom and Skill of a powerful

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For Leibniz, by contrast, the entire physical world described by the new mathemat-ical science (including the space in which bodies move) is a secondary appearance orphenomenon of an underlying metaphysical reality of simple substances or monadssubstances which, at this level, are not spatial at all but rather have only purely internalproperties and no external relations. And this point, in turn, is closely connected withthe fact that Leibniz self-consciously adheres to the idea that purely intellectual knowl-edge is essentially logical. For, although Leibniz appears to have envisioned some sortof extension of Aristotelian logic capable of embracing the new algebraic methods ofhis calculus, there is no doubt that the traditional subject-predicate structure of thislogic pervades his monadic metaphysics: it is precisely because ultimate metaphysi-cal reality is essentially intellectual in the logical sense that the entire sensible world,including space, is a merely secondary reality or phenomenon.36 Thus, although Leib-niz, like everyone else in the period, holds that there are mathematical laws governingthe sensible and material world (the phenomenal world), he reintroduces a new kind ofnecessary gapa new kind of Platonic gapbetween reality as known by the intellect(noumenal reality) and this sensible world.

Kants philosophy of transcendental idealism is also based on a fundamental dichot-omy between reality as thought by the pure understanding alone (noumenal reality)and the phenomenal world in space and time given to our senses. But Kant sharplydiffers from Leibniz in two crucial respects. First, mathematical knowledge, for Kant,is sensible rather than purely intellectual: indeed, mathematics is the very paradigm ofrational and objective sensible knowledge, resulting from the schematism of specifi-cally mathematical concepts within our pure forms of sensible intuition. Second, andas a consequence, we can only have theoretical knowledge, for Kant, of precisely thissensible (phenomenal) world: the noumenal reality thought by the pure understandingalone remains forever unknowable from a theoretical point of view, and we can onlyhave purely practical knowledge of its inhabitants (God and the soul) via moral expe-rience.37 Indeed, it is precisely this necessary limitation of all theoretical knowledgeto the sensible or phenomenal world that ultimately results from Kants doctrine ofthe schematism of the pure concepts of the understandingwhich, as Kant sees it,Leibniz entirely missed.38

Footnote 35 continuedever-living Agent, who being in all Places, is more able by his Will to move the Bodies within his boundlessand uniform Sensorium, and thereby to form and reform the Parts of the Universe, than we are by our Willto move the Parts of our own Bodies. For further discussion of Newtons metaphysics of space, in relationto Descartes, Leibniz, and Kant, see Friedman (2009).36 An individual simple substance, for Leibniz, is characterized by a complete concept consisting of aninfinite conjunction of all the marks or partial concepts that are (ever) true of it, and Leibnizs metaphysicsof ultimate simple substances is thus intimately connected with his commitment to the traditional logic ofconcepts. Precisely because he rejects the possibility of such a complete concept (for finite human thinkers),Kant, as we have seen, argues that the representation of space cannot be a concept (note 11 above, togetherwith the paragraph to which it is appended).37 For further discussion of this aspect of Kants view see again Friedman (2009) and also Friedman (2005).38 See A145147/B185187: Thus the schemata of the pure concepts of the understanding are the trueand sole conditions for providing the latter a relation to objects and thus a meaning; and the categoriesare therefore, in the end, of no other than a possible empirical use, in that they serve merely . . . to sub-ject appearances to universal rules of synthesis, and thereby to make them suitable for thoroughgoing

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Kants philosophy of geometryseen against the background of his more gen-eral transcendental idealismcombines central insights of both Leibniz and Newton.For, in the first place, Kants emphasis on the perceptual and intuitive aspects ofgeometry (and mathematics more generally) corresponds to Newtons approach, incontrast to the logico-algebraic approach of Leibniz. And, in the second place, Kantssharp distinction between the faculties of intellect and sensibility, together with hisparallel sharp distinction between logical or discursive and mathematical or intuitivereasoning, arises precisely against the background of the Leibnizean conception of thepure intellect, and it is aimed, more specifically, at Leibnizs view that pure mathemat-ics (including geometry) is, in Kants sense, analyticdepending only on relationsof conceptual containment within the traditional logic of concepts. Nonetheless, Kantaccepts Leibnizs characterization of the pure intellect in terms of the traditional logicof concepts, and Kants point about pure mathematics, against Leibniz, is simply thatthe pure intellect, characterized in this way, is not, after all, adequate to the task.39 It isfor precisely this reason, in Kants view, that the pure understanding must be appliedto, or schematized in terms of, a second rational faculty modelled on Newtonian abso-lute spaceno longer conceived along the lines of Newtons divine sensorium, butrather as a pure form of our human faculty of sensibility.40

In this way, Kants distinctive conception of geometry and spatial intuition addressesthe fundamental intellectual concerns of both Leibniz and Newton, while simulta-neously rejecting their metaphysical and theological ambitions. Kant rejects Leibnizstheological perspective by disavowing his conception of an (infinitary) complete con-cept as inaccessible to our (necessarily finite) human cognition; and it is for preciselythis reason, in Kants view, that space must be a pure form of sensible intuition ratherthan any kind of conceptual representation (note 36 above). Only so, Kant thinks, can

Footnote 38 continuedconnection in an experience . . .. Therefore, the categories, without schemata, represent only functions of theunderstanding for concepts, but do not represent any object. This meaning accrues to them from sensibility,which realizes the understanding by simultaneously restricting it. Kant is here discussing the schemata ofthe pure concepts of the understanding, not those of mathematical concepts (pure sensible concepts). Nev-ertheless, the above analysis of the central argument of the second edition Transcendental Deduction of theCategories implies that the schemata of mathematical concepts are also implicated in the schematism of thecategories, since the original transcendental synthesis of the imagination responsible for the characteristicunity and singularity of space and time thereby grounds the possibility of both the science of geometry andthe general doctrine of motion (see notes 26 and 33 above).39 Kants argument for this, as we have seen, depends on his rejection of the possibility of Leibnizeancomplete concepts (for finite human thinkers), and thus depends, in the end, on the circumstance that Kantunderstands the traditional logic of concepts in a substantially more limited way than does Leibniz himself(compare note 36 above).40 For Newton, God, by his immediate omnipresence throughout all of space, brings it about that all matterobeys the laws of motion by a creative act of his will. For Kant, it is our human understanding (not Gods)that injects itself into our pure forms of sensibility (not Gods), and, at the same time, brings it about(through precisely the schematism of the categories) that material or phenomenal substances necessarilyobey the (Newtonian) laws of motion. For further discussion see again Friedman (2009). This is because theschematism of the categories, as we have said, is intimately connected with both the mathematical scienceof geometry and the new (Newtonian) mathematical physics (see again notes 26 and 38 above). And thispoint, in turn, is connected with the fact that the schemata of the categories are determinations of time (note31 above), and that time, as a pure image, is intuitively presented by the motion of a point in space in thedrawing of a straight line (notes 26 and 32 above).

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our a priori knowledge of the geometrical structure of space be made intelligible.Similarly, Kant rejects Newtons theological perspective by insisting that the mathe-matical structure of nature must ultimately be due to the action of our pure intellect(not Gods) on our pure forms of sensible intuition. The Newtonian conception ofspace as the divine sensorium, for Kant, is completely impossible.41

In a famous passage in the Transcendental Aesthetic, Kant therefore depicts hisconception of space and time as pure forms of our sensible intuition as combining theadvantages of both the Leibnizean and Newtonian conceptions while simultaneouslyavoiding their respective disadvantages:

The [Newtonians] gain this much, that they make the field of appearances freefor mathematical assertions. On the other hand, they confuse themselves verymuch by precisely