Kant on the Imagination and Geometrical Certainty Mary Domski

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Kant on the Imaginationand GeometricalCertainty

Mary DomskiUniversity of New Mexico

In what follows, I aim to develop our understanding of the role the imagina-tion plays in Kant’s Critical account of geometry and suggest, in particular,that the peculiar interpretation of the imagination that Kant forwards in theFirst Critique helps secure the certainty of geometrical knowledge. To makemy case, I ªrst consider the account of geometrical reasoning Kant presentsin his 1764 Prize Essay, a text in which geometrical certainty is tied tothe power that the mind (or understanding) has to perceive the geometricalobjects presented before it. I highlight the problems that emerge from this pre-Critical account of geometrical certainty in order to make better sense of whyKant, in 1787, fashions the imagination as he does and speciªcally, why hedissolves any connection between the imagination’s power of perception—whathe terms our “degree of sensibility”—and the certainty of geometrical knowl-edge. In general, as I hope my treatment brings to light, paying attention tothe transition in Kant’s thinking about geometrical reasoning grants usadded perspective on the role the imagination is intended to serve in the Criti-cal account of geometrical cognition.

I. IntroductionMy goal in this paper is to develop our understanding of the role theimagination plays in Kant’s Critical account of geometry, and I do so byattending to how the imagination factors into the method of reasoningKant assigns the geometer in the First Critique. Such an approach is notunto itself novel. Recent commentators, such as Friedman (1992) andYoung (1992), have taken a careful look at the constructions of the pro-ductive imagination in pure intuition and highlighted the importance ofthe imagination’s activity for securing the universality of geometryknowledge. Speciªcally, as their respective examinations bring to light, it

Perspectives on Science 2010, vol. 18, no. 4©2010 by The Massachusetts Institute of Technology

is only with due attention to the imagination that we can make sense ofhow a Kantian geometer can legitimately make claims about a general classof objects based on the construction of a particular object of that class.

The appeal to particular instances of geometrical objects is emphasizedquite strongly in an example that Kant offers in the Doctrine of Method.In this case, the Kantian geometer carries out a series of constructions on aparticular triangle in order to demonstrate the general equality betweenthe angles of a triangle and two right angles.1 According to Kant, theequality that is demonstrated can be legitimately carried over from thesingle triangle under investigation to all triangles, because when we con-struct any particular triangle, we take “account only of the action of con-structing the concept, to which many determinations, e.g., those of themagnitude of the sides and the angles, are entirely indifferent” (A714/B742). Any particular triangle presented in intuition thus “serves to ex-press the concept [of triangle] without any damage to its universality” invirtue of its rules for construction (A714/B742), making it possible toreach a general solution to a problem by appeal to a particular geometricalobject. And since these rules, or schemata, are expressions of the syntheticprocedures of the productive imagination in pure intuition,2 we ªnd that,as Friedman and Young have emphasized, the imagination helps groundthe universality of geometrical knowledge.3

In what follows I want to expand on this account of the imagination’scontribution to geometrical knowledge and show that, in the First Cri-tique, there is also an important sense in which the imagination helps se-cure the certainty, or objective necessity, of our geometrical propositions.Speciªcally, I urge a reading of the imagination according to which itsdesignated role in geometrical constructions guarantees that our claimsabout space and spatial objects accurately capture the features of our pureform of spatial intuition and the objects presented therein. To reveal thislink between the imagination and geometrical certainty in the Criticalphilosophy, I ªrst turn our sights back to the 1764 Inquiry concerning thedistinctness of the principles of natural theology and morality, also known as

410 Kant on the Imagination and Geometrical Certainty

1. Kant details the steps of the construction at A716/B744–A717/B745. All referencesto the First Critique are taken from Kant 1999 and follow the standard A/B pagination,which refers to the ªrst (1781) and second (1787) editions, respectively.

2. See especially the Schematism chapter of the First Critique, where Kant claims thatthe schema of a concept is a “representation of a general procedure of the imagination forproviding a concept with its image” (A140/B179–180). I will treat the synthetic proce-dures of the imagination in greater detail in Section III below.

3. A related concern of Friedman’s treatment is the role of the imagination in securingthe applicability of geometrical claims to the empirical state of affairs. See especially Fried-man 1992, Chapter 2, Section IV.

the Essay.4 As in the First Critique, particular instances of spatialobjects—what Kant terms signs in concreto—are the focus of geometricalreasoning; but in contrast to the view developed in the Critical period,Kant appeals to our cognitive ability to perceive what is presented inspace as a means of explaining the certainty of geometry in this pre-Critical work. As a result, and as I emphasize in my treatment of the PrizeEssay, the alleged truths of geometry are rendered contingent on what wecan and cannot mentally perceive, which leaves Kant with pressing ques-tions regarding the accuracy of our geometrical propositions—questionsthat place the certainty of geometry in jeopardy and that go unansweredin the Prize Essay. In Section III, I turn to Kant’s use of the imagination inthe Critical philosophy, and aim to reveal the ways in which the imagina-tion helps Kant address the problems that lingered in 1764. Speciªcally, Iclaim that Kant replaces the mental perception of the pre-Critical periodwith the constructive procedures of the imagination come 1781/1787 sothat he might place the certainty of geometry on more solid ground.

II. The Prize Essay

A. The Methods and Certainty of GeometryIn the Prize Essay, Kant offers his response to the 1761 prize questionposed by the Berlin Academy of Sciences, which invited authors to explorethe nature and degree of certainty attributable to metaphysics (speciªcally,to natural theology and morality) by comparing metaphysical certaintywith the certainty characteristic of geometrical truths.5 Kant’s generalstrategy is to distinguish the method of mathematics from the method ofphilosophy and then to elaborate on the certainty afforded by each.6 He

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4. The full title of the Prize Essay is Inquiry concerning the distinctness of the principles ofnatural theology and morality, being an answer to the question posed for consideration by the BerlinRoyal Academy of Sciences for the year 1763. Citations to the essay refer to the Academy ver-sion of the text, the translation of which is found in Kant 1992. I should note here that thestrategy of using the Prize Essay as a springboard for understanding Kant’s Critical accountof geometry is also not itself novel. Emily Carson adopts this very approach in Carson1999. However, the centerpiece of Carson’s excellent essay is the role of the pure form ofspatial intuition in the Critical period, not the role of the imagination.

5. The precise question posed by the Academy was as follows: “One wishes to knowwhether the metaphysical truths in general, and the ªrst principles of Theologiae naturalisand morality in particular, admit of distinct proofs to the same degree as geometricaltruths; and if they are not capable of such proofs, one wishes to know what the genuinenature of their certainty is, to what degree the said certainty can be brought, and whetherthis degree is sufªcient for complete conviction” (Kant 1992, Editors’ Introduction: lxii).

6. Kant adopts a broader view than required by the question in his response, and ratherthan focus on the comparative necessity of geometry and metaphysics, he addresses the dis-

initiates this discussion, in the First Reºection, with an account of the dif-ferent methods of deªnition used in mathematics and in philosophy.

As Kant has it, the mathematician creates her objects of investigationby using a synthetic method of deªnition whereby the mathematicianelectively combines concepts.7 For instance, the mathematician deªnes atrapezium by ªrst electing to think of “four straight lines bounding aplane surface so that the opposite sides are not parallel to each other” andthen choosing to call the ªgure a trapezium (2: 276). In so deªning theobject, the mathematician also thereby brings the object into existence,for, according to Kant, the “concept which [she deªnes] is not given priorto the deªnition itself” (2:276). In philosophy, in contrast, deªnitions arearrived at by a process of analysis, and in this case, the deªnition does notproduce the object deªned; rather, the deªnition sought is the deªnitionof an already given concept. For instance, in their investigation of time,philosophers accept the concept of time as given and then attempt to pin-point those characteristic marks that belong to its deªnition, that is, theyseek to isolate the marks [Merkmale], or properties, that together ade-quately deªne the concept (2: 277–278).8 So whereas mathematicians be-gin their investigations by electively deªning their objects and therebybringing their objects into existence, philosophers begin their investiga-tions with a given concept, and the deªnition of the concept is the goalrather than starting-point of their inquiry.9


tinction between mathematics in general and philosophy in general. While Kant’s accountof mathematics in the Prize Essay includes explicit discussion of arithmetic, in what fol-lows I will focus on how his claims regarding mathematical reasoning are related to thecase of geometry. Cf. Rechter 2006 and the citations included therein for treatments ofarithmetic in the Prize Essay.

7. Following Sutherland 2010, I use “elective” as the translation of “willkürlich.”Friedman uses “optional” (Friedman 1992, p. 20), which also captures Kant’s meaning inthe text more accurately than “arbitrary,” which is used by the editors of Kant 1992. Theiruse of “arbitrary” is misleading, because Kant’s point in the Prize Essay is not that wechoose the features of our mathematical objects arbitrarily or at random, but that we vol-untarily and by a product of our own choosing stipulate those features that belong to themathematical objects we deªne. However, I do not agree with Sutherland’s claim thatthere are no constraints on the deªnitions we stipulate (Sutherland 2010, Note 40). As Iwill discuss below, the context of the Prize Essay suggests that our choice of which featureswe attribute to geometrical objects is restricted by a concept of space learned from experi-ence.

8. Kant concedes the difªculty of this analytic process: He claims that “no realdeªnition has ever been given of time” and also references Augustine’s claim that “I knowperfectly well what time is, but if someone asks me what it is I do not know” (2: 283–4).

9. The terminology Kant invokes here clearly resonates with the analytic-syntheticjudgment distinction he will later propose in the First Critique (1781/1787). However,there is no indication that Kant is anticipating that distinction at this early juncture of his

Due to the geometer’s use of a synthetic method of deªnition, Kantclaims that geometry has a higher degree of “objective certainty” thanphilosophy.10 For having deªned their objects of study, geometers have attheir disposal completely adequate deªnitions that allow them to “saywith certainty that what [they] did not intend to represent in the objectby means of the deªnition is not contained in the object” (2: 291). Inother words, because the geometer has brought geometrical objects intoexistence by deªning them, i.e., by stipulating their essential features, sheknows with certainty what the essential features of these objects are. Inphilosophy, on the other hand, where the deªnition is sought rather thanstipulated, the objective certainty is of lesser degree, because the philoso-pher can fail to notice a characteristic mark that belongs to an adequatedeªnition of a given concept (2: 291).

Given Kant’s account of the synthetic method of deªnition, it followsthat our geometrical propositions regarding the essential features of geo-metrical objects—the features included in the deªnition of an object—areobjectively certain. However, there is a different kind of certainty atplay—what Kant terms, in the Third Reºection, a “subjective certainty”—when we consider propositions concerning the non-essential features of geo-metrical objects; because, as he explains, there is a different method thatthe geometer must use when the goal is to draw inferences from “givenconcepts of magnitudes, which are clear and certain” (2: 278). In suchcases, the geometer does not appeal to stipulated deªnitions, or any wordsat all11; she instead appeals to signs in concreto, to particular sensible in-stances of the objects under investigation. For instance, “to discover theproperties of all circles,”

one circle is drawn; and in this one circle, instead of drawing allpossible lines which could intersect each other within it, two linesonly are drawn. The relations which hold between these two linesare proved; and the universal rule, which governs the relations


career, since, in the Prize Essay, he invokes the analytic-synthetic terminology to distin-guish different methods for deªning objects. In the Critical period, in contrast, he invokesthe terminology to characterize the sorts of judgments we make about given concepts. Formore on this issue, see Friedman 1992, p. 21, Note 33.

10. Speaking of certainty in general, Kant writes, “One is certain if one knows that it isimpossible that a cognition is false” (2:290). This general deªnition informs his more spe-ciªc treatments of “objective” and “subjective” certainty, which I discuss immediately be-low.

11. It is the philosopher in fact who appeals to words, what Kant terms signs in ab-stracto, in her investigation of “given” concepts such as time (2: 284). As Kant emphasizesin Section 2 of the First Reºection, the philosopher’s use of these abstract signs offers an-other important distinction between the methods of philosophy and mathematics.

holding between intersecting lines in all circles whatever, is consid-ered in these two lines in concreto. (2: 278)12

A similar account of geometrical reasoning is offered in Kant’s discussionof how we establish the inªnite divisibility of space. As in the case of thecircle, we appeal to sensible signs in concreto, and speciªcally, the geometerwill take “a straight line standing vertically between two parallel lines;from a point on one of these parallel lines he will draw lines to intersectthe other two lines. [Kant points out that] By means of this symbol [thegeometer] recognizes with the greatest certainty that the division can becarried out ad inªnitum” (2: 279).13

Notice that Kant’s description of how the geometer extracts informa-tion from sensible signs in concreto, that is, how the geometer draws infer-ences from the circle and the lines drawn in space, points to a peculiarform of geometrical cognition. In particular, these examples suggest thatto recognize that space is inªnitely divisible and to prove the relations be-tween the lines drawn in the circle requires that the geometer, in somesense, see the properties that hold between the particular signs that are un-der investigation. Understanding Kant’s account of geometrical cognitionin these terms is suggested more strongly by the analogy he draws be-tween the knowledge gained from our mathematical cognition and theknowledge gained from our sense of sight:

. . . since signs in mathematics are sensible means to cognition, itfollows that one can know that no concept has been overlooked, andthat each particular comparison has been drawn in accordance witheasily observed rules, etc. And these things can be known with thedegree of assurance characteristic of seeing something with one’sown eyes. (2: 291)


12. On my reading of this passage, and more generally, of the relationship betweenKant’s pre-Critical and Critical views of geometrical objects, I take it that the “universalrule, which governs the relations between intersecting lines in all circles whatever,” is not arule for construction. That is, it is a not a rule that communicates how we generate thelines or the circle, but instead communicates a relationship between objects. In this re-spect, that the angles of a triangle sum to two right angles would also be considered a uni-versal rule, one that “governs the relations” between the triangle and right angles. Not un-til we reach the Critical period do we ªnd rules for construction as a centerpiece of Kant’sgeometry. I thank Daniel Sutherland for urging me to clarify this point.

13. This proof is similar though not identical to the proof for the inªnite divisibility ofspace provided in Proposition III of the Physical Monadology (1: 478–479). Citations to thatessay refer to the Academy version of the text, the translation of which is found in Kant1992. The relevance of this similarity between the Prize Essay and the Physical Monadologywill be touched on below.

On my reading of this passage, to say that mathematical signs are “sensi-ble” is to say that these particular instances can and must be mentally per-ceived; they are presented to and cognized by the understanding—themind’s eye, as it were—just as empirical objects are presented to and seenby the eyes. But to say that that mathematical signs are “sensible means tocognition” is to suggest the stronger point that we can and must extract in-formation about the sensible objects based on our perception. In otherwords, geometrical inferences, and geometrical cognition in general, aregrounded upon what the mind’s eye observes in the objects before it.

This connection between perception and cognition is more prominentin Kant’s general discussion of how we extract information from sensiblesigns in concreto. Take, for instance, his discussion of space. Unlike the geo-metrical objects we bring into existence by means of deªnition, space iscounted among the unanalyzable concepts of geometry, those whose “anal-ysis and deªnition do not belong to this science at all” (2:279).14 The ab-sence of a deªnition is not problematic for the geometer, for she appeals toa “clear and ordinary” representation of space to complete her investiga-tion of spatial objects.15 However, the philosopher, whose task it is to es-tablish the essential properties of space and thereby offer an adequatedeªnition of this given concept,16 must consider space in concreto and seekout “those characteristic marks, which the understanding initially and im-mediately perceives in the object” (2: 281; emphasis added).17 As he explains,

Before I set about the task of [philosophically] deªning what spaceis, I clearly see that, since this concept is given to me, I must ªrstof all, by analyzing it, seek out those characteristic marks which areinitially and immediately thought in that concept. Adopting thisapproach, I notice that there is a manifold in space of which theparts are external to each other; I notice that this manifold is notconstituted by substances, for the cognition I wish to acquire re-lates not to things in space but to space itself; and I notice that


14. Besides space, Kant also considers magnitude in general, unity, and plurality to beunanalyzable in the sense described above.

15. I will return to this characterization of geometrical space below.16. As Kant says, “The mathematician deals with concepts which can often be given a

philosophical deªnition as well. An example is the concept of space in general” (2: 278).17. In the Prize Essay, Kant has not yet proposed the Critical distinction between sensi-

bility and understanding, which ªrst emerges in the 1770 Inaugural Dissertation and, ofcourse, plays a central role in the First Critique. Given the way Kant is using the term “un-derstanding” in 1764, I take him to be referring to a cognitive faculty for perceiving andreasoning, much as Locke used the term “understanding” in his 1689 An Essay Concerningthe Human Understanding.

space can only have three dimensions, etc. Propositions such as thesecan be well explained if they are examined in concreto so that theycome to be cognized intuitively; but they can never be proved. Foron what basis could such a proof be constructed, granted that thesepropositions constitute the ªrst and the simplest thoughts I canhave of my object, when I ªrst call it to mind? (2: 281)

This characterization of how the philosopher pinpoints the fundamen-tal properties of space, along with Kant’s other examples and remarks inthe Prize Essay, indicate that, when reasoning about signs in concreto,the understanding is taken to be a faculty for perceiving. In the speciªccase above, to cognize the essential features of space, such as its three-dimensionality, is to perceive—to intuitively cognize—the particularproperties of space that “can never be proved”; in other words, the philoso-pher accepts these properties as given insofar as they are clearly perceivedby the understanding—the mind’s eye, so to speak.18 And in those caseswhere the geometer attempts to establish the non-essential properties ofgeometrical objects by appeal to signs in concreto, a similar form of mentalperception is at play. As suggested by the examples above, the geometermust rely on her cognitive ability to perceive particular objects (and theirproperties), such that to know that space is inªnitely divisible, for instance,is to clearly and accurately perceive that space has this property based on anappeal to lines that have been drawn.19

Given the method of geometrical inference Kant details, and his appealto mental perception in particular, we can make better sense of why Kantcharacterizes the certainty of our geometrical inferences as subjectiverather than objective. Recall that the objective certainty of geometry restson the fact that the geometer has deªned her objects of investigation andtherefore has complete and perfect knowledge of the essential features ofthese objects. For instance, the geometer is objectively certain that the op-posite sides of a trapezium are not parallel to each other, because this fea-ture is stipulated in the very deªnition of a trapezium (2: 276). However,as the geometer extracts information from sensible signs in concreto in orderto establish the non-essential features of these objects, the certainty of thepropositions is subjective, where this subjective certainty varies in degreesand is tied to what Kant terms the “degree of intuition” associated withour cognition of signs in concreto: “taken subjectively, the degree of cer-


18. My thanks to Jeremy Heis for remarks that motivated me to think more carefullyabout the link between Kant’s notion of mental perception and the “givenness” of geomet-rical objects.

19. My thanks to Lisa Downing for comments that helped me better explain the con-nection between understanding and perception in the Prize Essay.

tainty increases with the degree of intuition to be found in the cognitionof this necessity” (2: 291). In other words, what Kant suggests with thisalternative form of certainty is that when the non-essential features of geo-metrical objects are under consideration, the certainty of our propositionsis determined by the acuity of the mental perception—the “degree ofintuition”—on which we must rely when drawing geometrical inferences.This certainty, as he says, is “found in the cognition.”

B. The Accuracy and Certainty of GeometryThe notion of mental perception that underwrites the subjective certaintyKant attributes to geometrical reasoning motivates important questionsconcerning the accuracy of our geometrical cognition. Speciªcally, whatguarantee do we have that what we perceive corresponds with what is pre-sented in space? How can we be certain, for instance, that the propositionthat space is three-dimensional accurately represents a feature of spacerather than a feature of our cognitive limitations? It could be, for instance,that we accept “space is three-dimensional” as true, not because space ac-tually has this property, but because we cannot infer that space has anygreater number of dimensions based on our perception of space—because,that is, there are limits to our capacity for mental perception which ulti-mately limit which features we can attribute to space and spatial objects.

This is, of course, not a point Kant would readily concede, since hewants to say that our geometrical propositions in fact accurately capturefeatures of space and spatial objects. The question, however, is whether hehas supplied the grounds for the alleged accuracy of geometry—for theneatness of ªt between our geometrical cognition and the spatial objectswe are investigating.

To these questions Kant does have at least a partial reply: we can besubjectively certain that our claims regarding space and spatial objects areaccurate, because the degree of intuition that grounds the subjective cer-tainty of geometry depends on the character of the objects about whichgeometers reason. As suggested by the passage above, because the geome-ter’s objects are “sensible means to cognition, it follows that one can knowthat no concept has been overlooked” (2: 291). Kant ºeshes this point outin greater detail in the discussion that follows, and states in particular thatthe objects of geometry, as sensible signs in concreto, allegedly provide aclear impression that heightens the certainty of our geometrical claims.Again contrasting the objects of mathematics from the objects of philoso-phy, Kant writes:

. . . the grounds for supposing that one could not have erred in aphilosophical cognition which was certain can never be as strong as


those which present themselves in mathematics. But apart fromthis, the intuition involved in this cognition is, as far as its exacti-tude is concerned, greater in mathematics than in philosophy. Andthe reason for this is the fact that, in mathematics, the object is con-sidered under sensible signs in concreto, whereas in philosophy theobject is only ever considered in universal abstracted concepts; andthe clarity of the impression made by such abstracted concepts cannever be as great as that made by signs which are sensible in char-acter. (2: 292; italics added)

These remarks indicate that the sensible character of the objects aboutwhich we reason in geometry underwrites the so-called subjective cer-tainty of our geometrical propositions. However, this only covers half thestory. For notice that geometrical cognition involves, on the one hand,clear and exact sensible signs, and on the other hand, our cognitive abilityto perceive these signs and their properties. It is only with both elementsin place that Kant can assert that our geometrical investigations offer uscertain knowledge about the sensible signs we investigate. ConsiderKant’s example of the circle. In this case, the particular circle and the setof lines drawn allegedly offer us a clear impression; but to say, as Kantdoes, that the relations which hold between the circle and lines are provedalso depends on the clarity of our perception of the relations—the fact thatwe clearly “see” the relations that hold between these objects.20 So whilethe objects of geometry may be clear and exact, we must rely on our capac-ity to perceive, or cognize, the geometrical features of these sensible ob-jects to know that they are in fact clear and exact. And unfortunately,Kant does not seem to do much better than simply assume a neatness of ªtbetween the objects of geometry and our cognition of these objects.

At this juncture, one might suggest on Kant’s behalf that the assumedcorrespondence between object and cognition is in fact well-grounded, be-cause he has offered us an account of geometry according to which we havecreated our very objects of investigation. We have used a synthetic methodof deªnition by means of which we have brought into existence the spatialobjects of geometry, and knowing (with objective) certainty the essentialfeatures that belong to these objects, we also have some guarantee that wecan know (with subjective) certainty the non-essential features of thesevery objects. This is the very approach taken by those who subscribe to


20. There is also a problem of generality that Kant faces, namely, a problem of howKant can justify the claim that we can discern a universal rule based on our investigation ofa particular circle and a particular set of lines. See the works by Friedman and Young citedin the Introduction for more on this problem of generality and its relationship the con-structions of the imagination as presented in the First Critique.

some form of “maker’s knowledge,” according to which, in brief, we haveprivileged access to all the properties of geometrical objects—we cancognize these properties with accuracy and thus, with certainty—becausewe have created them.21 While on the face it this approach seems to offer apromising remedy, Kant’s general account of the concept of space in thePrize Essay ultimately prevents him from taking such a stance.

In particular, there is strong evidence that, in the Prize Essay, our repre-sentation of geometrical space is derived from experience. Though Kantdoes not explicitly make this claim22, he does offer several remarks in thePrize Essay that mirror claims made in the 1756 Physical Monadology, atext in which he does assert that the form of space studied by the geometeris derived from the form of physical space. For our purposes, the remarkthat is most revealing is: “the elements of every body ªll their space bymeans of the force of impenetrability” (2: 287). The proposal that bodiesare situated in space in virtue of their external relations to other bodies isforwarded in the earlier Physical Monadology. Taking our cues from thereadings of Friedman (1992) and Laywine (1993), we ªnd that Kant, in1756, maintains that there are simple monadic elements that are essen-tially non-spatial, and he explains their presence in space through their ac-tion on other bodies. Put differently, the simple parts of bodies have nospatial relations, i.e., are not extended in space, until they act on otherbodies through the force of impenetrability, the same view we see pre-sented above in the Prize Essay. On this account, empirical space is ametaphysically real space that (somehow) emerges from the dynamical re-lations among bodies, and as such, empirical space is ontologically prior togeometrical space, for space as studied by the geometer is, put crudely,empirical space with bodies removed.23


21. Such an account has been attributed to Locke. On his score, we create spatial ob-jects by manipulating our simple idea of space and thereby establishing the “real essences”of these objects (much as Kant claims that we bring geometrical objects into existence bystipulating their essential features). Locke then suggests that because we have completeand perfect knowledge of the geometrical objects we have created, we can perceive theagreement and disagreement between our geometrical ideas and thereby establish geomet-rical demonstrations. For more on geometrical “maker’s knowledge” in Locke’s Essay Con-cerning Human Understanding (1689), see especially Cicovacki 1999 as well as Carson 2002and Domski 2006.

22. What Kant explicitly says is that he does not want to trouble himself with ques-tions about the essence of space (2: 287).

23. The claim that bodies are situated in space in virtue of their external relations toother bodies is also forwarded in Dreams of a Spirit-seer (1766), which was penned a coupleyears after the Prize Essay. See especially 2: 323 in Kant 1992, pp. 310–311. In 1768 Kantexplicitly changes his views on space in Directions in Space, where he defends the existenceof Newtonian absolute space. If we assume that he had already changed his view about thecharacter of space in 1764 (though there’s no textual evidence to support this), the essential

Assuming this empirical view of space is still in play in 1764 helps usmake sense of the textual similarities between the Prize Essay and thePhysical Monadology,24 and beyond that, helps us make sense of Kant’s1764 explanation of how the geometer is to reason about space without adeªnition of space at her disposal. There is no deªnition, recall, becausethe concept of space is unanalyzable, that is, its deªnition does not belongto geometry at all. When the geometer thus reasons about space, she ac-cepts the “concept [of space] as given in accordance with [her] clear and or-dinary representation” (2: 278; emphasis in the original), which, on thereading I forward, is to say that she appeals to that representation of spacederived from—or given by—experience.

If this reading of geometrical space is correct and Kant is in fact wed-ded to an empirical account of geometrical space in 1764, the problemssurrounding his use of mental perception to explain geometrical cognitioncan be brought into fuller relief. On the one hand, we have an explanationof geometrical cognition according to which we rely on our cognitive abil-ity to perceive an object (space) that is learned from experience and thus,an object that we have not created. As a result, Kant cannot appeal to thetenets of “maker’s knowledge,” since our creation of geometrical objects isnot completely free; our creation of objects is restricted by the “clear andordinary representation” of space learned from experience.25 Put differ-ently, this account of space prevents Kant from embracing an ‘idealist’ ac-count of geometry, according to which geometry is no more than thestudy of geometrical ideas created and presented before the mind.

On the other hand, Kant’s suggestion that geometrical space is derivedfrom empirical space makes the problems surrounding the contingency ofmental perception more pronounced.26 For now it is not only the case that


relationship between empirical and geometrical space would remain intact. For if space isabsolute in the Newtonian sense, it stands as a metaphysically real background in whichwe perceive bodies, and the form of geometrical space is determined by the form of thismetaphysically real empirical space. In other words, the geometer’s “clear and ordinary”representation of space derives from empirical space, just as it would if empirical spacewere emergent from the dynamical relations between simple substances.

24. As mentioned above (see Note 12), Kant also offers a proof for the inªnitedivisibility of space in the Prize Essay that is strikingly similar to the proof offered in thePhysical Monadology.

25. Here I diverge from the reading offered by Sutherland (2010). See Note 7 above.26. Though in what follows I stress how the role of mental perception bears on the al-

leged certainty of geometrical knowledge, there is also an important sense in which experi-ence itself could change the truths of geometry, as brought to light by (Friedman 1992).For since it is the case in the Physical Monadology, and, I contend, also in the Prize Essay,that Kant explains the existence of a metaphysically real empirical space in terms of the

we must suppose a neatness of ªt between object and cognition when de-termining the non-essential features of spatial objects. Given that the verydeªnitions we employ to bring objects into existence rely on this form ofspace, even the objective certainty of geometry is cast into doubt, becausewhat we deªne as legitimate spatial objects may not be spatially possibleat all. Take, for instance, the cone, whose geometrical (and hence, synthet-ically generated) deªnition is characterized by Kant as “an elective[willkürlich] representation of a right-angled triangle which is rotated onone of its sides” (2:276). Deªning a cone in such a way requires that therotation of the right-angled triangle is possible in space, and in general, todeªne our spatial objects and stipulate their essential properties is to makeclaims about what is in fact possible in geometrical space. But given thatour mental perception of the actual features of space may be inaccurate—given, that is, that the “clear and ordinary” representation on which thegeometer relies may not in fact capture the actual properties of empiricalspace—the objects we deªne may not be legitimate spatial objects at all.To put the point differently, there is a problem applying claims madeabout the form of space we mentally perceive—the form that is “given” tothe understanding—to the form of empirical space from which geometri-cal space allegedly derives.27

In the context of the Prize Essay, then, it seems that the best Kant cansecure for geometry is a “merely subjective necessity.” For, since his ac-count leaves open the possibility that our geometrical claims may not ac-curately capture the features of space, he presents an analysis according towhich our so-called geometrical truths are subject to possible revision.And as a result, the alleged certainty of geometry is itself brought intoquestion. In the Critical philosophy, Kant is famously trying to overcomethe charge that geometry is “merely subjectively necessary,” and as I claim


fundamental forces acting between simple substances, “our knowledge of the essentialproperties of space [are] entirely derivative from our (empirical!) knowledge of the laws ofdynamics” (Friedman 1992, p. 25). Thus, the claims made by the geometer about the gen-eral form and properties of space, which are claims about empirical space emptied of bodiesand forces, are ultimately contingent on the dynamical laws governing the empirical stateof affairs. Consequently, as Friedman so nicely puts it, “it is by replacing Leibniz’s concep-tion of the ideality of space with his own conception of the fundamentally dynamical char-acter of space that Kant himself has ªrst exposed geometry to the threat of empiricaldisconªrmation” (Friedman 1992, p. 27).

27. Carson (1999) also questions Kant’s ability to justify the applicability of our geo-metrical deªnitions and propositions to space in the Prize Essay. However, since she doesnot consider the empirical reading of space I urge above, she offers a reading according towhich the fundamental question for the pre-Critical Kant is whether and in what ways hissystem of geometry is different from the axiomatic metaphysical systems he explicitly crit-icizes.

in the section that follows, it is by replacing mental perception with theconstructive activity of the productive imagination that Kant is able toplace the objective necessity of our geometrical propositions on more solidground.

III. The First CritiqueIn the First Critique, and also in the Prolegomena (1783), Kant famouslyattempts to provide an account of geometry—of its proper object of inves-tigation and its proper method of reasoning—that secures its “objectivenecessity,” that is, that explains its status as a synthetic a priori science,which offers certain knowledge of space and spatial objects. In adoptingthis project, Kant claims to be improving upon empiricist and speciªcally,relationist accounts of geometry, according to which the form of space isderived from the relations between objects outside us. As laid out in theTranscendental Aesthetic in particular, such analyses of space are inade-quate, because they render the propositions of geometry “merely subjec-tively necessary.” That is, they cannot explain the apodictic certainty thatrightly characterizes geometry, and they cannot, as Kant says, because theclaims made of an empirically derived form of space are subject to the con-tingency of perception:28

. . . if this representation of space were a concept acquired a posteri-ori, which was drawn out of general outer experience, the ªrst prin-ciples of mathematical determination would be nothing but percep-tions. They would therefore have all the contingency of perception,and it would not even be necessary that only one straight line liebetween two points, but experience would merely teach us that.(A24)

A similar charge is made in the Prolegomena as Kant contrasts accounts ofgeometry that rely on perception with his account of geometrical reason-ing, one which relies on the “immediate” and a priori intuition of space:

All proofs of the thoroughgoing equality of two given ªgures (thatone can in all parts be put in place of the other) ultimately comesdown to this: that they are congruent with one another; whichplainly is nothing other than a synthetic proposition based uponimmediate intuition; and this intuition must be given pure anda priori, for otherwise that proposition could not be granted apodic-


28. There are, of course, further problems that Kant claims riddle empiricist, relation-ist accounts of space. He argues, for instance, that space cannot be derived from our experi-ence with outer objects, because to have an experience of outer objects requires these ob-jects already be situated in space.

tically certain but would have only empirical certainty. It wouldonly mean: we observe it always to be so and the proposition holdsonly as far as our perception has reached until now. (4: 284)29

As suggested by the remarks above, a key part of Kant’s remedy to themerely subjective necessity and comparative universality attributed to ge-ometry by empirical accounts of space—accounts which are subject to thecontingency of perception and rely on inductive generalizations over whatwe perceive—involves the proposal of a new understanding of space.30 Theform of geometrical space is not, for Kant, derived from experience; rather,the form of geometrical space is determined by our a priori form of spatialintuition. Recall here that according to the framework of Kant’s “tran-scendental idealism,” there are two fundamental sources of human cogni-tion, sensibility and understanding (A50/B74). Sensibility is deªned bytwo forms of intuition—space, which is the form of outer sense, and time,which is the form of inner sense—while the understanding is deªned, as itwere, by twelve categories, or pure concepts. To say, as Kant does, that theform of space studied by the geometer is the form of our pure spatial intu-ition is thus to say that the object of geometrical investigation has an apriori rather than empirical origin. As a result, we have a promising way ofguarding geometry against the comparative universality that is character-istic of empirical knowledge, because now that the object of geometry isitself given a priori, the truths of geometry are not subject to the contin-gencies of sensory perception. In other words, with the form of space nowbelonging to sensibility and thus internal to reason, so to speak, the prop-ositions of geometry are no longer grounded on what experience hastaught us or, in particular, on how far “our perception has reached untilnow.”

However, there still remains a question of whether Kant has completelyrid geometry of perceptual modes of cognition. For, as we saw in the treat-ment of the Prize Essay above, it was Kant’s embrace of a notion of mentalperception that brought the accuracy and the certainty of geometry intodoubt. Thus, if in the framework of the First Critique Kant again claimsthat to cognize is to perceive—that to cognize our a priori form of spacerequires we mentally perceive its properties –, then we’d be left with thesame questions that lingered in 1764. Speciªcally, we’d be left wanting


29. Citations to the Prolegomena refer to the Academy edition of the text, the translationof which is found in (Kant 1997).

30. A further part of the remedy, as seen in the Prolegomena passage above, is the appealof “immediate intuition” in geometrical cognition. I will return to this feature of Kant’sgeometry at the end of the paper, when I connect this notion of immediate and pure intu-ition with my reading of the imagination.

some ground for the alleged neatness of ªt between the a priori form ofspatial intuition and our propositions concerning its features and the ob-jects situated therein. To see then whether and how Kant rids geometry ofthe “contingency of perception,” we must extend our examination beyondKant’s revision of the object of geometry and turn to his characterizationof geometrical reasoning. It is in this context that we can better appreciatehow the imagination helps Kant secure the apodictic certainty of geome-try.

Before turning to the imagination, some preliminary remarks about therole of the understanding in geometrical cognition are in order. Unlikewhat we ªnd in the Prize Essay, it is clearly the case that, come 1781, theunderstanding is not a faculty for perceiving. It is instead presented as afaculty for judging by means of pure concepts, or categories (see especiallyA66/B91–A69/B94), and in the domain of geometry, it is via the applica-tion of the categories to our pure form of spatial intuition that we cognizethe features of space and spatial objects.31 However, as Kant is himselfaware, the story of geometrical cognition cannot end with the claim thatcategories must be applied to intuitions; he must also offer an explanationof how such an application proceeds. For, as presented at the outset of theSchematism, there seems to be a general problem surrounding the applica-tion of concepts to intuitions, because these representations lack “homoge-neity”: intuitions are non-discursive, immediate representations, whereasconcepts are discursive, mediate representations that “can never be en-countered in any intuition” (A137/B176). The thrust of Kant’s solution tothis problem of homogeneity is to propose a “third thing,” the transcen-dental schema, which stands

in homogeneity with the category on the one hand and the appear-ance [in intuition] on the other, and makes possible the applicationof the former to the latter. This mediating representation must bepure (without anything empirical) and yet intellectual on the onehand and sensible on the other. (A138/B177)


31. Emphasizing the necessary role of the categories in all domains of cognition, Kantstates in the Prolegomena, “Even the judgments of pure mathematics in its simplest axiomsare not exempt from this condition. The principle: a straight line is the shortest line be-tween two points, presupposes that the line has been subsumed under the concept of mag-nitude, which certainly is no mere intuition, but has its seat solely in the understandingand serves to determine the intuition (of the line) with respect to such judgments as maybe passed on it as regards the quantity of these judgments, namely plurality (as judiciaplurativa*), since through such judgments it is understood that in a given intuition muchthat is homogenous may be contained” (4: 301–302).

It is here where the importance of the imagination for Kant’s model ofcognition comes to the fore, because as he emphasizes in later portions ofthe Schematism, the imagination is the very faculty that mediates be-tween the understanding and sensibility by means of the schemata. As heputs it, the schema of a concept “is in itself always only a product of theimagination,” and more speciªcally, a “representation of a general proce-dure of the imagination for providing a concept with its image” (A140/B179–180). The suggestion here is that the schemata represent rules forthe construction of a concept in intuition, and, in the case of geometricalcognition, these rules represent how the imagination produces instances ofgeometrical concepts in our pure form of spatial intuition. The schema ofthe triangle, for instance, “can never exist anywhere except in thought,and signiªes a rule of the synthesis of the imagination with regard to pureshapes in space” (A141/B180).32 More generally,

the schema of sensible concepts (such as ªgures in space) is a prod-uct and as it were a monogram of pure a priori imagination,through which and in accordance with which the images ªrst be-come possible, but which must be connected with the concept, towhich they are in themselves never fully congruent, always only bymeans of the schema that they designate. (A141/B181–A142/B181)33

The above remarks emphasize the connection between pure conceptsand the rules for constructing instances, or images, of these concepts.What is perhaps less apparent but certainly no less important is that theserules, as representations that mediate between concepts and intuitions, donot simply communicate how to construct an instance of a concept in gen-eral; they express how an instance must be constructed given the form ofour pure form of spatial intuition. That is, with the imagination as thefaculty situated between sensibility and understanding, its synthetic pro-cedures must be directed by rules for construction that are faithful to thefeatures of space. Take, for instance, Kant’s characterization of the generalrelationship between sensibility, understanding, and the imagination pre-sented in the B-Deduction:


32. In the Schematism, Kant does not set forth the precise rules for construction thatdeªne the schema of a triangle, though in the Axioms of Intuition, he suggests the follow-ing characterization: “With three lines, two of which taken together are greater than thethird, a triangle can be drawn.” This, he says, is a “mere function of the productive imagi-nation” (A164/B205).

33. See the Introduction above for a brief account of how the appeal to rules for con-struction allows Kant a way to explain the universality of geometrical claims.

Imagination is the faculty for representing an object even withoutits presence in intuition. Now since all of our intuition is sensible,the imagination, on account of the subjective condition underwhich alone it can give a corresponding intuition to the concepts ofthe understanding, belongs to sensibility; but insofar as its synthe-sis is still an exercise of spontaneity, which is determining and not,like sense, merely determinable, and can thus determine the formof sense a priori in accordance with the unity of apperception, theimagination is to this extent a faculty for determining the sensibil-ity a priori, and its synthesis of intuitions, in accordance with thecategories, must be the transcendental synthesis of the imagina-tion, which is an effect of the understanding on sensibility and itsªrst application (and at the same time the ground of all others) toobjects of the intuition that is possible for us. (B151–152; boldfacein the original)34

Notice that the a priori activity of imagination is, on the one hand, re-stricted by the “subjective condition” of our form of intuition, and, on theother hand, directed by the categories of the understanding. To say, then,that the “transcendental synthesis of the imagination” is the ªrst applica-tion of the understanding to sensibility that is possible for us, is to say thatthis application of categories to intuitions by the imagination is restrictedby the general features of our spatial intuition. And thus, the rules forconstruction introduced in the Schematism are to be taken as rules thatindicate what is possible in the form of spatial intuition—the “subjectivecondition”—that characterizes human sensibility. In the Axioms of Intu-ition, Kant puts the point as follows:

On this successive synthesis of the productive imagination, in thegeneration of shapes, is grounded the mathematics of extension (ge-ometry) with its axioms, which express the conditions of sensibleintuition a priori, under which alone the schema of a pure concept


34. Given my emphasis on Kant’s move away from perception in his Critical account ofgeometry, I focus attention on the mediating role assigned to the imagination in the B-edition First Critique, because in the A-edition, and especially in the A-Deduction, ques-tions about the imagination’s capacity to perceive the spatial manifold still linger for Kant.In that earlier version of the text, Kant famously describes the categories (i.e., the pureconcepts of the understanding) as emergent from—i.e., the products of—the imagination’ssynthesis of intuition (see especially A119). As such, one could maintain that the imagina-tion’s perception of the manifold—its ability or inability to perceive what is presented inintuition—plays a foundational role in the Critical framework and especially in the Criti-cal account of geometry. It is thus not until the B-edition, where the categories are pre-sented as expressions of unity that direct the imagination, that the ties between perceivingthe manifold and judging the manifold are adequately divorced.

of outer appearance can come about, e.g., between two points onlyone straight line is possible; two straight lines do not enclose aspace, etc. (A164/B204)

To gain a ªrmer handle on how the rule-directed activity of the imagi-nation helps Kant address the problems of the Prize Essay, we can turn toKant’s own appraisal of how the Critical account of geometrical cognitionimproves upon analyses of geometry that rely on perception. Immediatelyafter critiquing accounts of space that rely on sensory perception, he writesin the Prolegomena,

That full-standing space (a space that is itself not the boundary ofanother space) has three dimensions, and that space in general can-not have more, is built upon the proposition that not more thanthree lines can cut each other at right angles in one point; thisproposition can, however, by no means be proven from concepts,but rests immediately upon intuition and indeed on pure a prioriintuition, because it is apodictically certain; indeed, that we can re-quire a line should be drawn to inªnity (in indeªnitum), or that a se-ries of changes (e.g., space traversed through motion) should becontinued to inªnity, presupposes a representation of space and oftime that can only inhere in intuition, that is, insofar as the latter isnot itself bounded by anything; for this can never be concludedfrom concepts. Therefore pure intuitions a priori indeed actually dounderlie mathematics, and make possible its synthetic and apodic-tically valid propositions. (4: 284–285)

Recall that in the Prize Essay the three-dimensionality of space was “intu-itively cognized” insofar as this feature of space was “given” by our “clearand ordinary representation” of space, that is, insofar as the understandingcould immediately and intuitively perceive the three-dimensionality ofspace (2: 281). In the Critical period, in contrast, where Kant’s emphasis ison the activity of construction, the three-dimensionality of space and theindeªnite extension of a line are grounded on what can possibly be repre-sented in our form of spatial intuition. And importantly, this possibility ofrepresentation rests not on what we mentally perceive is possible, but onwhat the imagination can construct in pure space, that is, on whetherthere are rules for construction corresponding to the concepts of our geo-metrical objects. It is because of the impossibility of establishing suchrules for construction that Kant rejects the two-sided plane ªgure, namely,a biangle, as an object of geometrical cognition. While such a ªgure has acorresponding concept, viz., a ªgure that is enclosed between two straight


lines, there are no rules for constructing this ªgure, precisely because itcannot be presented by the imagination in our form of spatial intuition:

Thus in the concept of a ªgure that is enclosed between twostraight lines there is no contradiction, for the concepts of twostraight lines and their intersection contain no negation of theªgure; rather the impossibility rests not on the concept itself, buton its construction in space, i.e., on the conditions of space and itsdeterminations. (A220–A221/B268)

What I want to emphasize is that the characterization of the imagina-tion and its rules for construction that Kant offers in the Critical periodoffer him a way of addressing the very problems that arose in the contextof the Prize Essay. Whereas in the pre-Critical work Kant could not ade-quately account for the alleged neatness of ªt between our mental percep-tion of space and the actual features of space and spatial objects, in theCritical period no such problem of ªt arises, because the very rules as-signed to the imagination are rules that allow him to collapse the dividebetween object and cognition. With the imagination as the faculty situ-ated between understanding and sensibility and with the rules of con-struction assigned to the imagination as rules that express how a conceptmust be constructed given our form of spatial intuition, what we havefrom Kant is a model of cognition where there is no gap between objectand cognition precisely because the constructive procedures of the produc-tive imagination replace perception as that which allows us to gain syn-thetic knowledge in geometry.35

This reading of the imagination is further supported by remarks Kantmakes in his 1789 correspondence with Reinhold. Responding to criti-cisms of the First Critique leveled by Eberhard, he claims rather stronglythat the imagination’s own power to perceive is negligible in discussionsof what counts as a possible object of geometry:

the question is just whether there can be an exhibition of the ideain a possible intuition, in accordance with our kind of sensibility;the degree—the power of the imagination to grasp the manifold—may be as great or small as it will. Even if something were pre-sented to us as a million-sided ªgure and we were able to spot thelack of a single side at ªrst glance, this representation would stillbe a sensible one; and only the possibility of exhibiting the conceptof a chiliagon in intuition can ground the possibility of this object


35. My thanks to David Hyder for remarks that helped me better understand, andhopefully better explain, the relationship between Kant’s pre-Critical account of percep-tion and his Critical account of the imagination.

itself in mathematics. For then the construction of the object can becompletely prescribed, without our worrying about the size of themeasuring stick that would be required in order to make thisªgure, with all its parts, observable to the eye. (To K.L. Reinhold,May 19, 1789; Zweig, p. 149)

Kant had already in the First Critique presented the understanding as afaculty for judging by means of concepts rather than a faculty for perceiv-ing. In these remarks to Reinhold, Kant makes quite clear that geometri-cal cognition is not in any respect grounded on a capacity for mental per-ception. As he puts it above, our geometrical claims about the features ofspace are not contingent on our imaginative power to perceive space, that is,on the imagination’s power to “grasp the manifold.” They are claims in-stead grounded on the constructive procedures of the imagination, andthese procedures are themselves guided by the actual features of our formof spatial intuition. In Kant’s terms, the synthetic activity of the imagina-tion proceeds solely “in accordance with our kind of sensibility.”

IV. ConclusionI pointed out in the Introduction that the primary goal of this paper wasto expand our understanding of the role the imagination plays in Kant’sCritical account of geometry. A further goal, as should now be clear, hasbeen to deepen our understanding of how Kant grounds the certainty ofgeometrical knowledge in his Critical framework. While it has long beenstandard in such discussions to focus on Kant’s revision of the object ofgeometrical investigation, viz., on his proposal that the proper object ofgeometry is our pure form of spatial intuition, I hope I have said enoughto show that any comprehensive account of geometrical certainty in theCritical philosophy must take into account the role assigned to the con-structive procedures of the imagination in geometrical cognition.

Certainly, the story of Kant’s Critical geometry does not end with theimagination, for as pointed out above, Kant also appeals to pure and “im-mediate” intuition to distinguish his account of geometrical reasoningfrom those marked by “subjective necessity.”36 This notion of pure intu-ition may in fact suggest that there is still some form of perception at playin Kant’s geometry;37 however, given my reading of the imagination, andspeciªcally, the foundational role played by the imagination in geometri-cal cognition, I take it that our capacity to have “immediate intuitions” ofspatial objects piggy-backs on the a priori constructive procedures of the


36. See in particular the text appended to Note 29 above.37. As has been suggested by Charles Parsons. See, for instance, Chapters 4 and 5 of

Parsons 2005.

imagination. That is, the pure intuition of a triangle or circle or any spa-tial object is possible because these objects have been constructed bythe imagination and rendered objects of geometrical cognition. To put thepoint differently, the a priori constructions of the imagination serve asthe conditions for the possibility of intuiting spatial objects—of whatKant elsewhere terms the “givenness” of these objects.38

While a more detailed exploration this idea will have to wait for an-other time, at the least, I hope that my treatment above has broughtneeded light to the foundational role of the imagination in Kant’s geome-try. As emphasized above, by presenting the constructive rules of theimagination as rules expressing what can possibly be constructed in space,Kant revises his pre-Critical claim that geometrical reasoning rests on per-ceptual modes of cognition precisely by collapsing the divide betweenconcepts and intuitions, i.e., the division between our categories for judg-ing and the objects being judged. In this respect, we have from Kant a“transcendental” explanation for the accuracy of our geometrical proposi-tions, and in turn, a more solid basis on which to ground the objective cer-tainty that characterizes geometrical knowledge.39

ReferencesCarson, Emily. 2002. “Locke’s Account of Certain and Demonstrative

Knowledge.” British Journal for the History of Philosophy 10: 359–78.Carson, Emily. 1999. “Kant on the Method of Mathematics.” Journal of the

History of Philosophy 37 (4): 629–52.Cicovacki, Predrag. 1990. “Locke on Mathematical Knowledge.” Journal

of the History of Philosophy 28: 511–24.


38. My thanks to Daniel Sutherland for pushing me to think more about the connec-tion between “pure intuition” in Kant’s geometry and the reading of the imagination I de-fend above. I regret not having the time or space to consider his points more carefully ordevelop my suggestions further in the current paper.

39. Earlier versions of this paper were presented at the Seventh International History ofPhilosophy of Science (HOPOS) Conference, held in June 2008 at the University of BritishColumbia, and at a workshop on geometrical thinking sponsored by the Ideals of Proofprogram, held in December 2008 at the University of Nancy. I am grateful for the feed-back I received from audience members at these events. I owe special thanks to JeremyHeis, Alison Laywine, Marco Panza, Daniel Sutherland, and Mic Detlefsen for their criticalremarks. Daniel also read the penultimate version of this paper and offered insightful com-ments that helped me see more clearly what is at stake for the reading I offer above. Finally,my thanks to Lisa Downing and David Hyder, each of whom read two different drafts ofthis paper and provided comments and suggestions that guided me as I brought the paperinto its current, more polished state. Of course, the fault for any remaining errors or omis-sions rests with me.

Domski, Mary. 2006. “Construction without Spatial Constraints: A Replyto Emily Carson.” Locke Studies 6: 85–99.

Friedman, Michael. 1992. Kant and the Exact Sciences. Cambridge, MA:Harvard University Press.

Laywine, Alison. 1993. Kant’s Early Metaphysics and the Origins of the Criti-cal Philosophy. NAKS Studies in Philosophy Volume 3. Atascadero, CA:Ridgeview.

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Kant, Immanuel. 1967. Philosophical Correspondence, 1759–99. Edited andtranslated by Arnulf Zweig. Chicago: University of Chicago Press.

Kant, Immanuel. 1992. Theoretical Philosophy, 1755–1770. Translated andedited by David Walford in collaboration with Ralf Meerbote. Cam-bridge: Cambridge University Press.

Kant, Immanuel. [1783] 1997. Prolegomena to Any Future Metaphysics.Translated and edited by Gary Hatªeld. Cambridge: Cambridge Uni-versity Press.

Kant, Immanuel. [1781/1787] 1999. Critique of Pure Reason. Translatedand edited by Paul Guyer and Allen Wood. Cambridge: CambridgeUniversity Press.

Parsons, Charles. [1983] 2005. Mathematics in Philosophy: Selected Essays.Ithaca, NY: Cornell University Press.

Rechter, Ofra. 2006. “The View from 1763: Kant on the ArithmeticalMethod before Intuition.” Pp. 21–46 in Intuition and the AxiomaticMethod. Edited by Emily Carson and Renate Huber. Dordrecht: KluwerAcademic Publishers.

Sutherland, Daniel. 2010. “Philosophy, Geometry, and Logic in Leibniz,Wolff, and the Early Kant.” Pp. 155–192 in Discourse on a New Method:Reinvigorating the Marriage of the History and Philosophy of Science. Editedby Mary Domski and Michael Dickson. LaSalle, IL: Open Court.

Young, J. Michael. 1992. “Construction, Schematism, and the Imagina-tion.” Pp. 159–175 in Kant’s Philosophy of Mathematics. Edited by Carl J.Posy. Dordrecht: Kluwer. Originally published in Topoi (1984).


Kant on the Imagination and Geometrical Certainty Mary Domski

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