80 Amit Hagar

BERICHTE UND DISKUSSIONEN

Kant and non-Euclidean Geometry

by Amit Hagar, Indiana University, Bloomington

Introduction

It is occasionally claimed that the important work of philosophers, physicists,and mathematicians in the nineteenth and in the early twentieth centuries madeKant’s critical philosophy of geometry look somewhat unattractive. Indeed, fromthe wider perspective of the discovery of non-Euclidean geometries, the replacementof Newtonian physics with Einstein’s theories of relativity, and the rise of quantifi-cational logic, Kant’s philosophy seems “quaint at best and silly at worst”.1 Whilethere is no doubt that Kant’s transcendental project involves his own conceptions ofNewtonian physics, Euclidean geometry and Aristotelian logic, the issue at stake iswhether the replacement of these conceptions collapses Kant’s philosophy into anunfortunate embarrassment.2 Thus, in evaluating the debate over the contemporaryrelevance of Kant’s philosophical project one is faced with the following two ques-tions: (1) Are there any contradictions between the scientific developments of ourera and Kant’s philosophy? (2) What is left from the Kantian legacy in light of ourmodern conceptions of logic, geometry and physics? Within this broad context, thispaper aims to evaluate the Kantian project vis à vis the discovery and application ofnon-Euclidean geometries.

Many important philosophers have evaluated Kant’s philosophy of geometrythroughout the last century,3 but opinions with regard to the impact of non-Euclid-ean geometries on it diverge. In the beginning of the century there was a consensusthat the Euclidean character of space should be considered as a consequence of theKantian project, i.e., of the metaphysical view of space and of the synthetic a prioricharacter of geometry. The impact of non-Euclidean geometries was then thoughtas undermining the Kantian project since it implied, according to positivists such

1 Friedman 1992, 55. Friedman aims to demonstrate that such a view is fundamentally unfairto Kant.

2 See for example Brittan 1978, 68: “Kant and Aristotle are, in my view, the two greatestwestern philosophers. They are also the only two philosophers, to my knowledge, whoseviews often seem to have been decisively refuted by development in science”.

3 See, e.g., Broad 1941, Carnap 1958, Beck 1965, Bennett 1966, Brittan 1978; 1986, Kitcher1975, Parsons 1983, and Friedman 1992.

Kant-Studien 99. Jahrg., S. 80–98 DOI 10.1515/KANT.2008.006© Walter de Gruyter 2008ISSN 0022-8877

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as Reichenbach and Carnap, that geometry is not synthetic a priori after all. Lateron it was shown that if one detached the Euclidean character of space from theKantian project, the positivists’ attack could be turned on its head, and that theexistence of non-Euclidean geometries, far from undermining Kant’s project, servesonly to justify it. It was Michael Friedman, among others, who pointed out thatthis defence on Kant’s behalf is misguided since it relies on anachronistic conceptswhich were foreign to Kant, and on a tacit interpretation of “intuition” as a psy-chological ability to discern the metric of the phenomenal world. Friedman then in-sisted that the Euclidean character of space is indeed a consequence of the Kantianproject, inasmuch as non-Euclidean geometries are logically impossible for Kant.This paper is intended as a contribution to this debate, aiming to show that Fried-man’s move can also be turned on its head in such a way that the existence of non-Euclidean geometries can be thought again to undermine the entire Kantian pro-ject.

Kant’s ideas of geometry, as they unfold in the Inaugural Dissertation (1770) andin the Critique’s “Transcendental Aesthetic” (1787), are the subject of section 1.Section 2 surveys both the objections that were raised against Kant’s philosophy ofgeometry, and their refutation by contemporary commentators on Kant’s behalf. Insection 3 I review the impact of non-Euclidean geometries on Kant’s legacy, focusingon the idea that one’s attitude toward this impact depends on one’s insistence on alogical relation between Euclidean geometry and Kant’s transcendental philosophy.I then show how, if one regards Kant’s project as implying the truth of Euclideangeometry, the non-uniqueness of the Euclidean metric in the phenomenal worldmilitates against formal idealism with respect to space. I conclude in section 4 withpossible responses on behalf of the Kantian.

1. Geometry as a synthetic a priori science

Kant’s doctrine concerning space and geometry, as developed in the InauguralDissertation4 and in the “Transcendental Aesthetic” of the Critique, is threefold: (1)space is the a priori form of pure intuition; (2) geometrical judgements are a prioriand synthetic; (3) the metric of humanly intuited space is Euclidean and the prop-ositions of Euclidean geometry are synthetic and are known a priori. The criticismthat is raised against Kant as a consequence of the discovery of non-Euclidean ge-ometries hinges upon the assumption that there is a logical relation between thesethree doctrines, i.e., that (1) and (2) imply (3). In order to evaluate it we must in-vestigate this assumption. We start with an exposition of Kant’s views on space andgeometry.

4 De mundi sensibilis atque intelligibilis forma et principiis; MSI, AA 02: 385–419.

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1.1 Space5

The context in which Kant’s ideas on space are examined here is ‘the clash of titans’in the seventeenth century: the debate between Newton and Leibniz on absolutespace. Kant’s argument from incongruent counterparts6 can be seen as an objectionto Leibnizian relationalism, but most commentators agree that in his later worksKant did not see it as a vindication of the Newtonian idea that space is metaphysicallyreal.7 The latter was indeed one of Kant’s conclusions in 1768,8 but in the Disser-tation and later in the Prolegomena Kant uses the idea of incongruent counterpartsto illustrate (and not to prove) the intuitive character of spatial knowledge [MSI, §15C, D], and to confirm the contention that space is metaphysically ideal [Prol, §13].Thus, although in the Dissertation Kant believes that space has intrinsic formalproperties, he does not see the perspectives of Leibniz and Newton as exhaustive.

In section 15 of the Dissertation Kant objects to the empiricists’ notion of spaceas abstracted from outer sensations [MSI, §15 A] and develops the idea that space isa precondition for the existence of any experience [MSI, §15 B]. Kant then leads hisreader to the conclusion that the concept of space is known, or given in, pure intui-tion; it is the form of all our outer sensations [MSI, §15 C]. The two primeexamples that support the statement that space is known intuitively are (a) the ideathat propositions of geometry are not deducible from an abstract concept of spacebut require instead constructive methods, i.e., reference to concrete examples, fortheir demonstration; and (b) the idea that incongruent counterparts can only be ap-prehended intuitively, i.e., one must observe an example of incongruent counter-parts in order to comprehend this notion (ibid.).

Kant concludes this section of the Dissertation with the claim that space is meta-physically ideal, but empirically real [MSI, §15 D, E]. Thus, while mocking New-ton’s notion of absolute space as a fairytale of a ‘container’ devoid of all substancehe also dismisses Leibniz’s relationalism. Kant’s alternative to Newton and Leibniz iscategorically different. Space is the precondition to all phenomena; a formal prin-ciple of our knowledge of the sensible world, or, as Friedman suggests in his dis-cussion of Newton’s theory of gravitation,9 the idea of reason with which we fur-nish the world with a truly privileged frame of reference, the “forever unreachablecommon centre of gravity for all matter”.

Kant’s ideas of space propounded in the Dissertation are refined and restated inthe Critique. Space is viewed as an a priori representation: every outer conception,

5 Here I shall concentrate only on space although he foregoing discussion applies also to time.6 Kant’s contribution to the debate with his argument of incongruent counterparts is thor-

oughly discussed in Van Cleve/Frederick 1991.7 Newton’s view of absolute space can be reconstructed today as a claim about an unobserved

theoretical entity that must exist in order to account for observed phenomena, i.e. inertiaand absolute rotation.

8 Kant 1768, 25–28; GUGR, AA 02: 375–383.9 Friedman 1992, 149.

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and a fortiori the conception of one’s self, is spatial, and hence presupposesspace.10 The relational doctrine of space as abstracted from the relations betweenobjects contradicts the fact that the terms that designate such relations presupposespace, and hence spatial knowledge cannot be acquired through experience.11 Inshowing that space is an a priori condition of human sensuous awareness Kant es-tablishes the necessity of space relative to it. While human awareness of particularappearances is merely contingent, awareness of space is not. This marks an impor-tant development in Kant’s thought: in the Dissertation it was claimed that thea priority of space was sufficient for establishing the necessity of space relativeto empirical knowledge. In the Critique Kant wants to establish that human cog-nition is limited to what can actually be intuited, and this can be achieved only ifall awareness is subject to the formal conditions of sensibility, that is, if spacecannot be “thought away”. The metaphysical exposition of space is then followedby the transcendental exposition which aims to establish that the view of space asan a priori form of intuition must hold in order for synthetic a priori knowledgeto be possible. Thus, the transcendental exposition in the Critique is concernedwith showing that the particular metaphysics of space provides the necessarycondition for a certain genus of knowledge which consists of particular kinds ofjudgements.12

1.2 Geometry

Geometry for Kant is a synthetic a priori science, i.e., it is an example of a body ofknowledge which applies to the empirical world, but is not justified by empiricalfacts. Contrary to the Leibnizian legacy, Kant blurs the distinction across the threecommon types of judgements (the ontological, epistemological, and semantic) andclaims that there can be judgements which are synthetic, i.e., judgements whichapply to the empirical world and are informative, inasmuch as their predicates are

10 “Vermittelst des äußeren Sinnes (einer Eigenschaft unsres Gemüths) stellen wir uns Gegen-stände als außer uns und diese insgesammt im Raume vor.” / “By means of outer sense, aproperty of our mind, we represent to ourselves objects as outside us, and all without ex-ception in space.” (B 37)

11 “Demnach kann die Vorstellung des Raumes nicht aus den Verhältnissen der äußern Er-scheinung durch Erfahrung erborgt sein, sondern diese Erfahrung ist selbst nur durch ge-dachte Vorstellung allererst möglich.” / “The representation of space cannot, therefore, beempirically obtained from the relations of outer appearance. On the contrary, this outer ex-perience is itself possible at all only through that representation.” (B 38)

12 There is a difference in the presentation of Kant’s ideas in the Critique and in the Prole-gomena, but in both cases the metaphysical doctrine of space allows one to understand howsynthetic a priori science such as geometry is possible: although its existence can be grantedindependently, its nature would have been different if space were different (Prol, § 11). Simi-larly, Kant believes that neither the Newtonian nor the Leibnizian can account for both thenecessity and the truth of geometrical propositions (B 57).

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not part of their subject, yet, nevertheless a priori, i.e., judgements that can be jus-tified without an appeal to experience and hence are necessary and universal.13 Theidea of synthetic a priori knowledge is the Kantian response to the long-standing de-bate between rationalists and empiricists. Kant changes the rules of the game: thereis a special kind of knowledge that does not originate in experience, yet applies toexperience, inasmuch it is a precondition of any experience.14 This resolution alsoindicates an unbridgeable gap between the “things-in-themselves” and the way weperceive them; between noumena and phenomena. Noumena are inaccessible to usinasmuch as we can never know them. Thus, if our knowledge is bound to appear-ances, it can be justified with an appeal to the way we organize these appearances:it can be objective, necessary and universal as long as we remember that its scope isrestricted to the phenomenal world.15

Kant ascribes a special role both to Euclidean geometry and to the constructivefeature of geometrical proofs. According to him geometrical reasoning cannot pro-ceed purely logically, i.e. through analysis of concepts, but requires a further activ-ity called “construction in pure intuition”.16 Friedman (1992, 58) notes that thespatio-temporal17 character of this construction enables Kant to give a philosophi-cal foundation for both Euclidean geometry and Newtonian physics. Indeed, as Sha-bel (1998, 618) adds, Kant explains both the difference between mathematical andphilosophical reasoning and the syntheticity of mathematical judgements by therole played by construction in pure intuition [B 741; B 287]. Moreover, almost all ofKant’s examples for the construction of either mathematical or geometrical proofsrely on Euclidean geometry and the use of its postulates for constructing geometri-cal figures. Although these examples do not serve as an argument in his metaphysicsof space, in mentioning them Kant seems to commit himself to the truth of Euclid-ean geometry.

Returning to the opening question of this section regarding Kant’s threefold doc-trine of space and the existence of a logical relation between its constituents: (1) themetaphysical character of space (2) the possibility of synthetic a priori geometryand (3) the Euclidean nature of our appearances, we can see now that (1) and (2) areindeed logically related: Kant’s metaphysics of space ensures the certainty and

13 Kant alternates between the epistemological interpretation of a priori as necessary, or evi-dent, and the ontological interpretation as universal.

14 Compare Parsons 1983, 118: “[K]ant started from the idea that geometry was a body ofnecessary truths with evident foundations. That the axioms of geometry should be empiri-cally verified is contrary to their necessity; that they should be some sort of high-level hy-pothesis is contrary to their evidence.”

15 Contrary to Van Cleve (1999, 143–150) I subscribe to the view that the noumenal and phe-nomenal worlds as two aspects of the same world.

16 This claim is explicitly expressed in the “Discipline of Pure Reason”, where Kant confrontsphilosophical with mathematical reasoning. See B 743–745.

17 “Temporal” because it involves the notion of generating a line from an originating point asa process unfolding in space-time. See B 203–204.

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necessity of geometry.18 But whether (3) is implied by either of them is still an openquestion. It is precisely the difference between the kind of knowledge the characterof space grounds and one’s commitment to the truth of this knowledge’s specialcharacter (a difference between coherence and correspondence; validity and truth)which is at stake.19 We have arrived at a crucial point in the debate on Kant’s phi-losophy of geometry. The transcendental exposition in the Critique establishes that(1) geometry is the science of the properties of space and that (2) geometrical truthsare synthetic a priori [B 40]. Kant seems to argue that our inability to know ge-ometry except by means of construction in pure intuition accounts for the syntheticcharacter of geometrical truth. Whether this statement means that (a) pure intuitiondiscerns the Euclidean metric (and the properties) of space, or that (b) the process ofgeometrical reasoning involves the construction of definitions with which we for-mulate geometrical proofs, is the subject of the next section.

2. Pure and Applied Geometry – excerpts from Kant’s Critics and Defenders

The debate on Kant’s philosophy of geometry focuses on two related but differentissues. The first concerns the role of intuition in Kant’s account of mathematicalknowledge; the second – Kant’s arguments for that role. We shall see that whetheror not non-Euclidean geometries undermine Kant’s philosophy depends on one’sviews on both issues, explicitly on the former and in a more subtle way on the latter.The following review, while not exhaustive, aims to distil the issue of the impact ofnon-Euclidean geometries on the Kantian project form the more general debate onKant’s philosophy of geometry.

The most complete account given by Kant of his view of the nature of mathemat-ical reasoning is in the first section of the “Discipline of Pure Reason” in theCritique.20 One of the sharpest critics of Kant’s view is C. D. Broad (1941) who re-constructs Kant’s definition of mathematical knowledge as follows:

It is characteristic of mathematics to start from definitions which are obviously adequate andfrom axioms which are self-evident and to reach conclusions which are rendered intuitively cer-tain by demonstration. (Broad 1941, 1)

18 Yet, this necessity would be affirmed only later on in the transcendental deduction with theintroduction of the unity of apperception.

19 Kant is well aware of this difference in other contexts, such as his discussion of Newton’slaws of motion in “The Metaphysical Foundations of Phenomenology”, when he states (inMAN, AA 04: 554–565) that the question is not the transformation of illusion into truth,but of appearance into experience.

20 B 741–766. As C.D. Broad (1941, p. 1) notes: “It is to be feared that most of us, unless weare professional students of Kant, have rather flagged before reaching p. 570 of KempSmith’s translation and are inclined to give ourselves a holiday on the plea that the rest ofthe book is just “Kant’s architectonic.” It is noteworthy that Kant’s ideas on the nature ofmathematical knowledge appear also in his earlier writings. See Humphrey 1973, p. 487,fn. 10.

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Broad notes that according to Kant there is a unique condition which can ascer-tain the adequacy and certainty of our definitions. This is when we arbitrarily makeup the concept for ourselves. Next, there is a unique condition that ascertains theapplicability of that concept. This is when the concept contains an arbitrary syn-thesis that admits of a priori construction.

This intuitive construction not only enables one to decide whether two math-ematical concepts, such as “straightest” and “shortest” imply each other, an im-possible task in philosophy, but also excludes the possibility of demonstrations out-side mathematics, whence the methodological difference between mathematical andphilosophical knowledge. While Broad agrees with Kant that the geometrical prop-erties of a triangle do not follow logically from the mere definition of a triangle, hepoints out that the former do follows logically from the latter when accompanied bythe concept of the space in which the triangle is imbedded, i.e., the metric signatureof that space. The fact that many of Euclid’s propositions do not follow deductivelyfrom his definitions, axioms and postulates, and that intuition is indeed neededin some of his proofs, is, for Broad “a defect” in Euclid’s geometry, which Kant has“mistaken” for an inherent property of geometry as such (ibid., 6), a view Broadshares with Russell in his Principles of Mathematics (1903/1937).

Next Broad draws a distinction between pure and applied geometry. In pure ge-ometry, when we are merely supposing and not asserting the axioms, intuition is notneeded for guaranteeing the truth of the axioms but for constructing a perceivableinstantiation of them, i.e., a proof of their consistency.21 Broad regards Kant’s insist-ence on intuition being necessary for guaranteeing the truth of Euclid’s propositionsas an “extraordinary view”,22 yet nevertheless he continues dissecting this view bydistinguishing three classes among Euclid’s postulates: (1) Those which express pe-culiar feature of Euclidean space (e.g. the parallel postulate). (2) Those which ex-press features common to all forms of homaloidal space,23 whether Euclidean, ellip-tic, or hyperbolic (e.g. that the existence of a figure with a given size and shape inone region of space implies the possibility of its existence in any other part of space).(3) Those which are common to any kind of geometry (e.g. the shortest line betweentwo points is a straight line). Broad, in his unique style, demonstrates how each ofthese classes admits either an analytic or synthetic a posteriori view of geometry.24

21 Broad acknowledges the psychological importance of intuition in “discovering” the axiomsbut admits it is too weak a notion for what is needed to establish Kant’s claims, inasmuch asnon-Euclidean spaces have been discovered regardless of their visualization.

22 “It would mean that we are able to know with complete certainty, and not just probably bymeans of induction, the fundamental facts about the spatial structure of nature at all placesand at all times.” (9)

23 Homaloidal means Flat; even. It is a term applied to surfaces and to spaces in which the defi-nitions, axioms, and postulates of Euclid are assumed to hold true, i.e., spaces with zero in-trinsic curvature.

24 With respect to (1) Broad notes that the peculiar status of the parallel postulate prevents onefrom seeing it as self evident, but the self-consistency of Euclid’s geometry requires one to re-gard the parallel postulate as empirically, though indirectly, evident. With respect to (2)

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Broad’s view is characteristic of the general criticism of Kant’s philosophy of ge-ometry in the first half of the twentieth century. The lesson it drives home is thateven before one begins to examine the implications of non-Euclidean geometries onKant’s philosophy there is still a room for questioning the role of intuition in theKantian project. Broad concludes that when properly construed pure geometry isanalytic, and applied geometry is in part synthetic but nevertheless a posteriori. Indoing so Broad denies the Kantian hybrid, the synthetic a priori, at least as far as ge-ometry is concerned, and restricts the role of “construction in intuition” to securingthe consistency of a particular geometry.

The distinction between pure and applied geometry and its supposedly dire con-sequences for Kant’s philosophy of geometry is the common thread shared by manyof Kant’s critics. As Friedman (1992, 55) notes, it is epitomized in Einstein’s famousphrase: “As far as the laws of mathematics refer to reality, they are not certain; andas far as they are certain, they do not refer to reality.” Thus Carnap, in his intro-ductory remarks to the English edition of Reichenbach’s The Philosophy of Spaceand Time (1958, vi) says:

The statements of pure geometry hold logically, but they deal only with abstract structures andthey say nothing about physical space. Physical geometry describes the structures of physicalspace; it is a part of physics. The validity of its statements is to be established empirically – as ithas to be in any other part of physics – after rules of measuring the magnitudes involved, es-pecially length, have been stated […]. In neither of the two branches of science which are called“geometry” do synthetic judgements a priori occur. Thus Kant’s doctrine must be abandoned.

And yet there is a “small” problem with these modern complaints since far fromas undermining Kant’s claim that geometry is synthetic, one can regard the devel-opment of non-Euclidean geometries as serving only to support it.25 Indeed, whenRiemann (1866) discusses the foundations of geometry his conclusions are equivo-cal: space might posses a unique structure but when we try to discern this structureby way of physical measurements, we already presuppose certain hypotheses (re-garding rigid rods and light rays as gauges) and a certain metric (that establishes thebasic gauges). Thus at least on the latter reading of Riemann, geometry is still syn-

Broad notes that these axioms are indeed analytic, at least if one admits a distinction be-tween space and what occupies space. Although Kant might have objected to this distinctionit is still far from clear whether he would have made it the crucial evidence for the role of in-tuition in geometrical reasoning, especially after Newtonian spacetime was given a generalcovariant formulation (Friedman 1983). With respect to (3) things are more complicated. Innon-flat spaces, where straight lines are geodesics, the axioms that the shortest path be-tween two points is a straight line is merely a definition of the term “straight line”. Agreed,in flat spaces the axiom is not a mere definition but it is also not a synthetic a priori pro-position. In order to measure a length of a curve we need first to divide the curve to smallstraight segments, thus the notion of length applies primarily to straight lines, which seemsto reduce the status of the axiom either to being analytic or to being synthetic a posteriori(as it involves some contingent physical facts involving measuring apparatuses).

25 See Brittan 1978, ch. 3, where the positivists’ arguments on the analyticity of geometry arediscussed.

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thetic a priori: when applied to the world it serves as precondition for any physicalexperience of the world. As for the role of intuition in geometrical reasoning, Rie-mann indeed demonstrates that it is unnecessary when he presents his concept ofn-fold extended manifold,26 but note that any representation of either Riemannianor Lubachevskian geometry, as constructed respectively by Reid (1764) or Bel-trami,27 is achieved only within Euclidean space. It is for this reason that Broadaccepts intuition’s role in securing the consistency of these geometries. Yet, theconsistency of non-Euclidean geometries is only relative: they are consistent if Eu-clidean geometry is. But what secures Euclidean geometry? Assuming it is true,there is no alternative but to appeal to its synthetic character, hence pure intuition.It seems we have closed a circle, and the Broad-Carnap attack, far from refutingKant on Euclidean geometry, only demonstrates its coherence.

Contemporary commentators such as Michael Friedman (1992), however, dis-miss the above line of thought as anachronistic, as it relies on conceptions of logic,modality and differential analysis that were unavailable to Kant. Thus, Friedman(1992, ch. 1) claims that the distinction between pure and applied geometry goeshand in hand with quantificational logic and modality of possible worlds. SinceKant relies on monadic syllogistic logic and on the original “constructive” formu-lations of Euclid’s axioms he is bound to represent concepts like “infinity” as an it-erative process of spatial construction.28 Furthermore, deprived as he was of modernanalysis and the Cauchy-Weierstrass concept of limit, Kant represents this construc-tion as a spatio-temporal process: spatial quantities are not composed of points butare generated by the iterative motion of points. This iterative process takes the placeof modern quantification, and is the very reason for geometry being synthetic apriori.29 Thus, according to Friedman, the reason for geometry being synthetic doesnot lie in the syntheticity of Euclid’s axioms – as Beck (1965) and Brittan (1978)argue. In fact, this picture, to which Friedman objects, reduces the role of intuitionto merely supplying a model for discerning the correct geometry from a class ofpossible geometries and uses a notion of possibility foreign to Kant.30 On thecontrary, Friedman insists that non-Euclidean geometries are impossible within the

26 Friedman 1992, 95.27 Gomez 1986.28 This view is propounded also by Parsons 1983, ch. 5.29 Friedman emphasizes the “inductive” and unbounded character of mathematical proofs, in-

volving iterative processes of substitution (in arithmetic) and construction in intuition (ingeometry).

30 Compare also Kitcher 1975, 50: “The problem lies with the picture behind Kant’s theory.The picture presents the mind bringing forth its own creations as the naïve eye of the mindscanning those creations and detecting their properties with absolute accuracy”. Indeed, asFriedman (ibid., 90) notes, although one can solve the problem of how to generalize from aparticular image to a universal property with Kant’s notion of schemata (see also Risjord(1991) and the discussion that follows his paper), our capacities for visualizing figures hasneither the generality nor the precision to make the required distinctions, not to mentionthat on a small scale Euclidean and non-Euclidean geometries are indistinguishable.

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Kantian project. The only way to represent them, according to Kant’s philosophy ofgeometry, is by drawing, or generating them in the space (and time) of pure intui-tion. But this space for Kant is necessarily Euclidean. It follows that there is no wayto draw, and a fortiori represent, say, a non-Euclidean straight line, and the veryidea of non-Euclidean geometry is quite impossible.31 Consequently, nothing fol-lows from the mere formal possibility of non-Euclidean geometry.32

To summarize Friedman’s view, the impossibility of non-Euclidean geometries inKant’s view should be understood not as referring to the realm of the “things-in-themselves” alone (where concepts, considered independently of our sensible intui-tion, are meaningless, or empty), but as referring to the realm of cognition; to“things-in-themselves” plus our pure intuition. These conditions of cognition arethe best approximation to the modern idea of logical possibility.33 According toFriedman, the role of intuition in Kant’s philosophy of geometry is simply to supplyinitial constructive definitions from which we can formulate geometrical proofs, orin other words, pure intuition underwrites the constructive procedures used inmathematical proofs. For this reason, among others, Friedman (ibid., 95) concludesthat rather than being accused of his failure to anticipate modern developments,Kant should be applauded for his insight, that in its classical formulation, Euclideangeometry required for its proofs certain procedures that the traditional logic was in-capable of characterizing.

Returning to the question we raised in the last section, whether or not Kant’s doc-trine of space implies a logical relation between Euclidean geometry and transcen-dental philosophy, i.e., whether the metaphysical view of space and the synthetic apriori character of geometry imply that the metric of humanly intuited space is Eu-clidean, we can see now that according to Friedman Kant survives even if such animplication exists. As Friedman suggests, non-Euclidean geometries pose no prob-lem to Kant since the domain of Euclidean geometry is restricted only to the phe-nomenal world. This conclusion, however, relies on the premise that all appearancesare indeed Euclidean, and it is this premise that we now turn to investigate.

3. What is Space and is it Euclidean?

Where do we stand? It seems that the main theme of the discussion so far is thatKant might have lost a battle, betting on Aristotelian logic, Newtonian physics andEuclidean geometry, but nevertheless won the war: not only did his legacy of tran-scendental philosophy come out of the nineteenth and twentieth centuries intact butit was also reinforced.

31 Friedman, ibid., p. 82.32 Beck and Brittan seem to rely on B 268 where Kant accepts this formal possibility.33 See also B 137, B 147, B 154 where Kant states that to know a line he must draw it, and Prol

§ 13, AA 04: 285–294, where Kant uses an example of triangles drawn on a sphere.

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Non-Euclidean geometries serve to vindicate Kant’s transcendental philosophywhen the latter is considered detached from Euclidean geometry. This is the con-clusion of the line of thought that regards intuition’s role as discerning the correctgeometry of the world from a class of possible geometries. Michael Friedman rejectsthis line of thought and insists that for Kant the role of intuition is rather different:intuition underwrites the constructive procedures used in mathematical and geo-metrical proofs. This interpretation renders non-Euclidean geometries a logical im-possibility: as far as the phenomenal world (the world of the noumena plus pure in-tuition) is concerned, Euclidean geometry becomes an implication of the Kantianproject. The fact that non-Euclidean geometries were nevertheless conceived onlyvindicates Friedman’s view, since their construction relied on, and was restricted to,Euclidean models.

What follows is an argument that turns Friedman’s conclusion on its head. I shallclaim that if one agrees with Friedman that Kant’s project entails Euclidean ge-ometry as a necessary character of the phenomenal world, the impact of non-Eu-clidean geometries on this project is devastating. The argument involves three steps.I start by (1) claiming that appearances are not all Euclidean and offer two-dimen-sional visual space as an example. This claim is intended to lure the Kantian into atrap, since the common reaction it raises is that two-dimensional visual phenomenaare relational or subjective, i.e., they merely represent a point-of-view of three-di-mensional tangible space: while the latter is invariant, the former co-vary with theobserver in a way that preserves an underlying non-arbitrary structure. Havinglured the Kantian to this position I then claim that (2) such a covariance, if it exists,implies – contrary to the Kantian belief – that “things-in-themselves” do have a par-ticular and unique character. Finally, I argue that (3) since this admission involvesexchanging the notion of truth as coherence with truth as correspondence, it mili-tates against Kant’s formal idealism with respect to space.

3.1 Step 1: The Non-Uniqueness of the Euclidean Metric

Is there any ground to the claim that appearances are not uniquely Euclidean?Kant, like Berkeley before him in The New Theory of Vision,34 believed that there isone and only one metric which applies to the phenomenal world: the Euclidean met-ric of tangible space. The reason Berkeley rejects all other metrics, such as the metricof visual space, is that the latter varies with one’s perspective, whereas the tangiblemetric remains invariant.35 I am not certain that contemporary Kantian scholarswould agree to the positive thesis that Kant regards visual space as possessing a Eu-

34 Berkeley 1709/1910.35 See also Bennett 1966, 30.

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clidean metric.36 As we have seen, they are more inclined to contend the negativethesis, namely, that Kant regards a non-Euclidean metric impossible and the Eu-clidean metric unique:

The reason why non-Euclidean figures cannot be constructed is not because we cannot visual-ize or imagine them, but because there is not an appropriate metric for them as there is, no-tably, in the case of Euclidean geometry […]. And there is not an appropriate metric for them,Kant thought, because it is only on the presupposition that a Euclidean metric is supplied by us,a priori, that we can understand how it is that Euclidean geometry applies with perfect preci-sion to the objects of our experience. […] [T]he possibility of alternative metrics would entailthe erroneous conclusion that the form of the world, and our knowledge of it, is merely con-tingent and relative. (Brittan 1986, 65)37

This uniqueness ensures the correspondence between our knowledge and its ob-jects in experience: the truth of the Euclidean propositions, under this account, is se-cured by the coherence of Euclidean geometry as a complete system of knowledgeaccording to which we organize our experience. For this reason, as Brittan argues,the uniqueness of the Euclidean metric is crucial for establishing objective knowl-edge of appearances.

Unfortunately, Kant and his advocates are mistaken on both the positive and thenegative theses: visual appearances are non-Euclidean; the phenomenal Euclideanmetric is not unique. Indeed, the idea that part of appearances, e.g., two-dimen-sional visual space, possesses a non-Euclidean metric was already suggested by Tho-mas Reid;38 propounded later on by Daniels (1972) and Angell (1974); and can beeasily proven when one attempts to add the angles in the ceiling above one’s head.39

Consequently, if the motivation behind the uniqueness of the Euclidean metric isto ensure correspondence between our knowledge and appearances, that is, to en-sure the truth of geometrical propositions by drawing on both the consistency andthe categorical character of the system that entails them, then the existence of othermetrics threatens to collapse this coherence and a fortiori the entire Kantian project:the non-uniqueness of the phenomenal metric would imply the non-categoricalcharacter of Euclidean geometry as a system of knowledge and the existence of non-Euclidean geometries would then serve as a bait; luring the Kantian into the trap ofacknowledging that “things-in-themselves” do have a particular and non-arbitrarycharacter that gives rise to both tangible and visual appearances.

36 As it carries a misleading sense of psychological capacity à la Helmholtz (1868/1977). Seealso Craig 1969, 122; 130.

37 Brittan refers here to B 268.38 Reid 1764/1997, ch. 6. Here Reid constructs a model for non-Euclidean geometry and de-

fines a metric for visual space.39 It is true that trying to draw this idea on paper we are led directly to Euclidean geometry, but

only because the paper itself already possesses a Euclidean metric.

92 Amit Hagar

3.2 Step 2: The Existence of an Underlying Structure of Space

The trap is simple. If appearances such as visible space are non-Euclidean, andother such as tangible space are Euclidean, then in order to restore consistency tophenomenal knowledge the Kantian must claim that both tangible and visible spaceare a product of an underlying space with a particular and non-arbitrary structure:while the tangible metric remains invariant, the visual metric co-varies with per-spective while preserving this structure. In so doing the Kantian must trade his no-tion of truth as coherence for truth as correspondence. This admission, however,carries the seeds of its own destruction since it now behoves the Kantian to make adirect reference to “things-in-themselves”; a reference he is reluctant to make.

The demarcation between the “things-in-themselves” and appearances is a keydistinction in the Kantian project: it allows synthetic a priori knowledge; it solvesthe problem of Humean scepticism; it solves the long-standing antinomies; in fact, itis one of the core ideas of the so-called Copernican revolution of transcendentalphilosophy. Yet throughout the Critique Kant is cautious not to assign any propertyto the “things-in-themselves” apart from mere existence.40 In contrast to appear-ances, which are the object of our knowledge and experience, the “things-in-them-selves” are just “out there”, inaccessible in principle, and as Friedman (1992, 94)puts it, because they are “independent of our sensible intuition, any attempt to de-scribe them remains empty, lacking both sense and meaning”.

Among Kant’s contemporaries Thomas Reid is probably the most lucid advocateof the contrary view, namely, the idea that nature has a definite structure, and thatscience’s role is to discover that structure, and not to invent it:

The objects of sense we have hitherto considered are qualities. But qualities must have a sub-ject. I perceive in a billiard ball, figure, color and motion; but the ball is not figure, nor is itcolor, nor motion, nor all of these taken together; It is something that has figure and color andmotion. This is a dictate of nature and a belief of all mankind. As to the nature of this some-thing, I am afraid we can give little account of it, but that it has the qualities which our sensesdiscover. (Reid 1785/1975, Essay II, Ch. 19, § 257)

In his Essays (1785/1975) Reid elaborates on the distinction – made earlier in hisInquiry (1764/1997) – between the metrics of tangible and visible space and offers itas an argument for the existence of an underlying structure in nature. On Reid’sview visible phenomena are two-dimensional non-Euclidean, as opposed to tangiblephenomena which are three-dimensional Euclidean. Both are qualities of the exter-nal world – Reid’s commonsensical phrase for the Kantian’s “things-in-themselves” –whose real nature should be discovered by science. But since, according to Reid, thedirect objects of perception are neither sensations nor physical objects but thequalities which the latter arouse in us with the help of the former, space itself,

40 Kant was at pains to detach himself from Berkelian idealism (see ‘Refutation of Idealism’B 274–B279). This led some commentators to view Kant as a metaphysical realist (VanCleve 1999, ch. 13), or at least as a formal idealist. See B 519.

Kant and non-Euclidean Geometry 93

whether tangible or visible, is not a proper object of perception. As Reid himselfputs it, space is “a necessary concomitant of the objects both of sight and touch[…]” and “[…] the visible and the tangible are different conceptions of the samespace”.41 On this view the external world does posses an underlying structure whichgives rise to the Euclidean character of tangible space and to the non-Euclideancharacter of visible space, and, more important, to the fact that the latter is re-lational, i.e., that the latter co-varies with perspective.

The idea that nature has a definite structure, and that science’s role is to discoverthat structure, and not to invent it, goes hand in hand with an eternal scepticismwith respect to our knowledge of nature, and one of Kant’s motivations is to abolishthis scepticism. Yet, by securing the objectivity of knowledge with the transcenden-tal deduction Kant prevents any correspondence whatsoever between phenomenaand the stripped noumena. Indeed, Bertrand Russell would try to argue for thepossibility of such a correspondence:

For example, it is often said that space and time are subjective, but they have objective counter-parts; or that phenomena are subjective, but are caused by things in themselves, which musthave differences inter se corresponding with the differences in the phenomena to which theygive rise. Where such hypotheses are made, it is generally supposed that we can know very littleabout the objective counterparts. In actual fact, however, if the hypotheses as stated were cor-rect, the objective counterparts would form a world having the same structure as the phenom-enal world, and allowing us to infer from phenomena the truth of all propositions […]. If thephenomenal world has three dimensions, so must the world behind it be; if the phenomenalworld is Euclidean, so must the other be; and so on. (Russell 1919, 61)

But Russell would later acknowledge42 what was already evident to Kant,43

namely, that it is impossible to prove the existence of a non-trivial correspondencebetween appearances and “things-in-themselves” if one accepts the Kantian reluc-tance to attribute to the latter any structure whatsoever.44

By now things get complicated for the Kantian: if tangible space and visual spacehave different characteristics, and these characteristics are a product of an existentnon-arbitrary underlying structure, then the Kantian silence with respect to “things-in-themselves” becomes even more puzzling. Of course the Kantian can alwaysclaim that even if such structure does exist, it is still inaccessible to us. Yet this movebrings us to the third and final step of my argument since it already indicates theabandonment of the notion of truth as coherence in favour of truth as correspon-dence; a move which is tantamount to giving up the metaphysical thesis of space.

41 Reid 1785/1975, Essay II, Ch. 19, § 264.42 Russell 1967, 176, in a letter to M. Newman, 1928: “My statements to the effect that no-

thing is known about the physical world except its structure are either false or trivial. […] Iam somewhat ashamed at not having noticed the point myself. […] had not really intendedto say what in fact I did say […]”.

43 See for example A 358.44 Russell’s idea was soon proved impossible by Newman (1928). On the Russell-Newman af-

fair see Demopoulos and Friedman (1985).

94 Amit Hagar

3.3 Step 3: Trading Formal Idealism for Non-Naïve Realism

In order to appreciate how the admission in a definite-yet-inaccessible structureof “things-in-themselves” undermines Kant’s metaphysical thesis of space we canput matters in terms of mapping: idealism with respect to space means that the latteris inexistent. But Kant was reluctant to attach himself to the Berkelian camp andpreferred to maintain what he called “formal idealism” with respect to space, claim-ing that the latter is amorphous, which means that space has no geometry, neitherEuclidean nor non-Euclidean, or in other word, that there exists no mapping what-soever from appearances to “things-in-themselves” because space is nothing but apure form of our intuition:

We have sufficiently proved in the Transcendental Aesthetic that everything intuited in space ortime, and therefore all objects of any experience possible to us, are nothing but appearances,that is, mere representations, which, in the manner in which they are represented, as extendedbeings, or as series of alterations, have no independent existence outside our thoughts. Thisdoctrine I entitle transcendental idealism.*) The realist, in the transcendental meaning of thisterm, treats these modifications of our sensibility as self-subsistent things, that is, treats mererepresentations as things in themselves.

*) I have also, elsewhere, sometimes entitled it formal idealism, to distinguish it from materialidealism, that is, from the usual type of idealism which doubts or denies the existence of outerthings themselves. (B 519)45

As argued by Reid, the relational and non-Euclidean character of two-dimen-sional visual space over and above the Euclidean character of tangible space indi-cates that space possesses an underlying structure. But the attribution of an existent,yet inaccessible, structure to space marks the departure from formal idealism, since,as Russell painfully acknowledges, it is impossible to be formal idealist with respectto space and also to claim that there is a unique non-trivial correspondence betweenappearances and “things-in-themselves”.

Agreed, such an attribution does not lead to full-blown naïve realism, which canbe described as a ‘one-to-one’ mapping between appearances and “things-in-them-selves”. One can subscribe to a mapping of ‘many-to-one’ between appearances and“things-in-themselves”, and in so doing secure the correspondence between the two‘up to an isomorphism’. This is exactly Reid’s point when he stresses the relation be-

45 “Wir haben in der transcendentalen Ästhetik hinreichend bewiesen: daß alles, was imRaume oder der Zeit angeschauet wird, mithin alle Gegenstande einer uns möglichen Er-fahrung nichts als Erscheinungen, d. i. bloße Vorstellungen, sind, die so, wie sie vorgestelltwerden, als ausgedehnte Wesen oder Reihen von Veränderungen, außer unseren Gedankenkeine an sich gegründete Existenz haben. Diesen Lehrbegriff nenne ich den transcendentalenIdealism.*) Der Realist in transcendentaler Bedeutung macht aus diesen Modificationen un-serer Sinnlichkeit an sich subsistirende Dinge und daher bloße Vorstellungen zu Sachen ansich selbst.*) Ich habe ihn auch sonst bisweilen den formalen Idealism genannt, um ihn von dem ma-terialen, d. i., dem gemeinen, der die Existenz äußerer Dinge selbst bezweifelt oder leugnet,zu unterscheiden […].”

Kant and non-Euclidean Geometry 95

tween tangible and visible phenomena as different aspects of the external world.Reid’s Non-naïve realism, however, is far from idealism.46

It turns out that in attempting to secure the truth of Euclidean geometry in light ofthe non-uniqueness of the Euclidean metric, the Kantian finds himself in a positionwhich is tantamount to renouncing his transcendental idealism with regards tospace. Although some try to describe Kant as a metaphysical realist,47 the case ofnon-Euclidean geometries illustrates how subtle one must be when one regards thetruth of Euclidean geometry as entailed by transcendental idealism and the synthetica priori character of geometry.48

4. A Kantian Response?

The gist behind the threefold argument just presented is that the possibility ofnon-Euclidean phenomena manifest in the non-Euclidean metric of two-dimen-sional visual space militates against Kant’s formal idealism with respect to spacesince it signifies a departure from a coherence theory of knowledge in favour of analternative based on correspondence. Here are two possible responses to this argu-ment on behalf of the Kantian.

First, the Kantian might claim, as Berkeley did, that visual appearances are notobjective in the sense that they are “unreal”; they represent nothing but a two-di-mensional perspective of “real” three-dimensional tangible phenomena, and sincethe metric projected from three-dimensional tangible space onto two-dimensionalvisible space is Euclidean, visual appearances, whether Euclidean or not, cut no icein the debate.

This scepticism with respect to the reality of two-dimensional visual space, how-ever, must be judged, as the whole of Kantian philosophy, with reference to the de-velopments in Geometry. Kant’s contemporaries may indeed discard non-Euclideantwo-dimensional visual space as nothing but a projection of three-dimensional tan-gible space and regard visual space as having no intrinsic properties, but such criti-cism becomes unavailable to one who – after Riemann and Gauss – is equipped withthe distinction between intrinsic and extrinsic curvature.

A space can possess intrinsic non-flat geometry yet contain lines that will bestraight according to any form of measurement intrinsic to that space. A line iscalled straight in relation to its own manifold. Euclidean straightness thus charac-terizes lines in a three dimensional space with no intrinsic curvature, and flat spacesof more than three dimensions may be called Euclidean because of their lack of cur-

46 Note that even in Einstein’s theories of relativity, where many physical properties becomerelational, or foliation-dependent, there still exists an invariant structure that underlies theirco-variance: the spacetime interval, or the metric.

47 Van Cleve 1999, ch. 13.48 Dummett 1982, 248–249. Dummett claims that Frege’s realist account turns geometrical

knowledge to either a posteriori or groundless, thus vindicating his own anti-realism.

96 Amit Hagar

vature. What we now call extrinsic curvature presupposes a higher dimension inwhich curved objects are embedded: for a line or a plane or a space to be curvedit must occupy a space of higher dimension, e.g., a curved line requires a plane,a curved plane a volume, a curved volume some fourth dimension, etc. Yet intrinsiccurvature has nothing to do with any higher dimension. Confusion regarding thispoint arose because we can model non-Euclidean planes as extrinsically curved sur-faces within Euclidean space. Thus, the surface of a sphere is the classic model ofa two-dimensional, positively curved Riemannian space, as demonstrated by Reid(1764/1997). But while great circles are straight lines according to the intrinsicproperties of that surface, we see the surface itself curved into the third dimensionof Euclidean space.

Reconstructing the Kantian objection to the reality of visual space in terms of cur-vature, one can say that the Kantian claim amounts to viewing the non-Euclidean(non-flat) curvature of two-dimensional visual space as extrinsic rather than intrin-sic: visual space is a ‘second-order’ phenomena; a by-product of the phenomenalworld – three-dimensional Euclidean tangible space – and not of the noumenal one.But what other ground than the transcendental project itself has the Kantian tooffer for this claim? Such a move would situate the Kantian closer to Berkeley thanhe would have liked to admit. Moreover, if one understands, as Reid did, the non-Euclidean character of visible space and the Euclidean character of tangible space ascomplementary conceptions of the external world, our cognitive ability to translatethe tangible into the visible and vice versa is just another proof that visible space isas real as tangible space, and that both spaces are no more than different qualities ofthe same external world; a world which – the Kantian indifference to it notwith-standing – does posses a unique structure.

A second route open for the Kantian is to claim that looking at the psychologicalquestion of the structure of visual space may not do justice to the idea that appear-ances are Euclidian, since the question what appearances are is presumably not apsychological question.

But here it is crucial to note that nothing in the threefold argument just presentedhinges on the “visual” versus “tangible” distinction apart from the fact that theformer is two-dimensional and the latter is three-dimensional, hence the distinctionbetween them is exactly the one which is needed in order to embed n-dimensionalnon-Euclidean geometry in n+1 dimensional Euclidean space. Both modes of cogni-tion are available to us in detecting appearances, and there is no reason, other thansecuring the Kantian project itself, to dismiss one in favour of the other.

It is indeed true that we are unable to visualize non-Euclidean geometries inspaces whose dimension is bigger than two, but it is also the case that we cannot vis-ualize Euclidean geometry in spaces whose dimension is bigger than three. Thus theaccusation against the Kantian propounded here is not that he is guilty of purport-ing that the impossibility to visualize certain geometries militates against their exist-ence. It is rather that notwithstanding this psychological impossibility, the fact thatnot all appearances are Euclidean indicates that appearances and “things-in-them-

Kant and non-Euclidean Geometry 97

selves” are so related that the latter must posses a non-arbitrary and definite struc-ture – a structure of which the Kantian is deliberately indifferent, since admitting itwould be tantamount to renouncing his project.

In sum, if one insists on a logical relation between Euclidean geometry and themetaphysical and transcendental expositions of space, i.e., that the latter imply thetruth of the former, then non-Euclidean geometries lead to the demolition of theKantian project. A coherence theory of truth cannot be replaced with a theory ofcorrespondence and also remain idealistic with respect to the form of its truth-makers. Unfortunately, although useful in political situations, there exists no ‘thirdway’ between idealism and realism, no matter how “formal” the former might be.Friedman’s defence of Kant’s philosophy in light of non-Euclidean geometry hasthus led the Kantian into an impasse.

References

Angell, R. B.: ‘The Geometry of Visibles’. In: Nous 8, 1974, 87–134.Beck, L.: Studies in the Philosophy of Kant. Indianapolis 1965.Bennett, J.: Kant’s Analytic. Cambridge (UK): Cambridge University Press 1966.Berkeley, G.: An Essay Towards a New Theory of Vision. New York: Dutton

1709/1910.Brittan, G.: Kant’s Theory of Science. Princeton: Princeton University Press 1978.Brittan, G.: ‘Kant’s Two Grand Hypotheses’. In: Kant’s Philosophy of Physical

Science. Ed. by R. E. Butts. Dordrecht: D. Reidel Pub. Com. 1986.Broad, C. D.: ‘Kant’s Theory of Mathematical and Philosophical Reasoning’. In:

Proceedings of the Aristotelian Society 42, 1941, 1–24.Craig, E. J.: ‘Phenomenal Geometry’. In: British Journal for the Philosophy of

Science 20, 1969, 121–134.Daniels, N.: ‘Thomas Reid’s Discovery of a Non-Euclidean Geometry’. In: Philos-

ophy of Science 39, 1972, 219–234.Dummett, M.: ‘Frege and Kant on Geometry’. In: Inquiry 25, 1982, 233–254.Friedman, M.: Kant and the Exact Sciences. Cambridge: Harvard University Press

1992.Gomez, R. J.: ‘Beltrami’s Kantian View of non-Euclidean Geometry’. In: Kant-Stu-

dien 77, 1986, 102–107.Helmholtz, H.: Epistemological Writings. Dordrecht: D. Reidel Pub. 1868/1977.Humphrey, T.: ‘The Historical and Conceptual Relations between Kant’s Meta-

physics of Space and Philosophy of Geometry’. In: Journal of History of Philos-ophy 11, 1973, 483–512.

Kant, I.: The First Ground of the Distinction of Regions of Space. Trans. by J. Han-dyside. Chicago: The Open Court 1768/1929.

Kant, I.: Inaugural Dissertation. Trans. by J. Handyside. Chicago: The Open Court1770/1929.

98 Amit Hagar

Kant, I.: Prolegomena to Any Future Metaphysics That Can Qualify as a Science.Trans. by P. Carus. Chicago: The Open Court 1783/1902.

Kant, I.: The Critique of Pure Reason. Trans. by N. Kemp Smith. London: Macmil-lan & Co 1787/1958.

Kant, I.: Metaphysical Foundations of Natural Science. Trans. by J. Ellington. NewYork: The Bobbs Merrill Company 1786/1980.

Kitcher, P.: ‘Kant and the Foundations of Mathematics’. In: Philosophical Review84, 1975, 23–50.

Newman, M.: ‘Mr. Russell’s Causal Theory of Perception’. In: Mind 37, 1928,137–148.

Parsons, C.: Mathematics in Philosophy. Ithaca: Cornell University Press 1983.Reichenbach, H.: The Philosophy of Space and Time. New York: Dover Publi-

cations 1958.Reid, T.: An Inquiry Into The Human Mind. Penn. State University Press 1764/1997.Reid, T.: Essays on the Intellectual Powers of Man. Indianapolis: The Bobbs-Merril

Company 1785/1975.Reimann, B.: ‘On the Hypotheses which Lie at the Foundations of Geometry’. 1866.

In: M. Spivak: Differential Geometry. Vol. 2. 1970.Risjord, M. ‘The Sensible Foundation for Mathematics: A Defence of Kant’s View’.

In: Studies in History and Philosophy of Science 21, n. 1, 1991, 123–143.Russell, B.: Principles of Mathematics. London: Allen & Unwin 1903/1937.Russell, B.: Introduction to Mathematical Philosophy. London: Allen & Unwin

1919.Russell, B.: The Autobiography of Bertrand Russell. Vol. 2. London: Allen &

Unwin 1967.Shabel, L. ‘Kant on the ‘Symbolic Construction’ of Mathematical Concepts’. In:

Studies in History and Philosophy of Science 29, n. 4, 1998, 589–621.The Philosophy of Left and Right. Ed. by J. Van Cleve and R. Frederick. Dordrecht:

Kluwer 1991.Van Cleve, J.: Problems from Kant. Oxford: Oxford University Press 1999.