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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 made
Kant’s critical philosophy of geometry look somewhat unattractive. Indeed, from
the wider perspective of the discovery of non-Euclidean geometries, the replacement
of Newtonian physics with Einstein’s theories of relativity, and the rise of quantificational logic, Kant’s philosophy seems “quaint at best and silly at worst”.1 While
there is no doubt that Kant’s transcendental project involves his own conceptions of
Newtonian physics, Euclidean geometry and Aristotelian logic, the issue at stake is
whether the replacement of these conceptions collapses Kant’s philosophy into an
unfortunate embarrassment.2 Thus, in evaluating the debate over the contemporary
relevance of Kant’s philosophical project one is faced with the following two questions: (1) Are there any contradictions between the scientific developments of our
era and Kant’s philosophy? (2) What is left from the Kantian legacy in light of our
modern conceptions of logic, geometry and physics? Within this broad context, this
paper aims to evaluate the Kantian project vis à vis the discovery and application of
non-Euclidean geometries.
Many important philosophers have evaluated Kant’s philosophy of geometry
throughout the last century,3 but opinions with regard to the impact of non-Euclidean geometries on it diverge. In the beginning of the century there was a consensus
that the Euclidean character of space should be considered as a consequence of the
Kantian project, i.e., of the metaphysical view of space and of the synthetic a priori
character of geometry. The impact of non-Euclidean geometries was then thought
as undermining the Kantian project since it implied, according to positivists such
1
2
3
Friedman 1992, 55. Friedman aims to demonstrate that such a view is fundamentally unfair
to Kant.
See for example Brittan 1978, 68: “Kant and Aristotle are, in my view, the two greatest
western philosophers. They are also the only two philosophers, to my knowledge, whose
views often seem to have been decisively refuted by development in science”.
See, e.g., Broad 1941, Carnap 1958, Beck 1965, Bennett 1966, Brittan 1978; 1986, Kitcher
1975, Parsons 1983, and Friedman 1992.
Kant-Studien 99. Jahrg., S. 80–98
© Walter de Gruyter 2008
ISSN 0022-8877
DOI 10.1515/KANT.2008.006
Kant and non-Euclidean Geometry
81
as Reichenbach and Carnap, that geometry is not synthetic a priori after all. Later
on it was shown that if one detached the Euclidean character of space from the
Kantian project, the positivists’ attack could be turned on its head, and that the
existence of non-Euclidean geometries, far from undermining Kant’s project, serves
only to justify it. It was Michael Friedman, among others, who pointed out that
this defence on Kant’s behalf is misguided since it relies on anachronistic concepts
which were foreign to Kant, and on a tacit interpretation of “intuition” as a psychological ability to discern the metric of the phenomenal world. Friedman then insisted that the Euclidean character of space is indeed a consequence of the Kantian
project, inasmuch as non-Euclidean geometries are logically impossible for Kant.
This paper is intended as a contribution to this debate, aiming to show that Friedman’s move can also be turned on its head in such a way that the existence of nonEuclidean geometries can be thought again to undermine the entire Kantian project.
Kant’s ideas of geometry, as they unfold in the Inaugural Dissertation (1770) and
in 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 of
geometry, and their refutation by contemporary commentators on Kant’s behalf. In
section 3 I review the impact of non-Euclidean geometries on Kant’s legacy, focusing
on the idea that one’s attitude toward this impact depends on one’s insistence on a
logical relation between Euclidean geometry and Kant’s transcendental philosophy.
I then show how, if one regards Kant’s project as implying the truth of Euclidean
geometry, the non-uniqueness of the Euclidean metric in the phenomenal world
militates against formal idealism with respect to space. I conclude in section 4 with
possible 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 Inaugural
Dissertation 4 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 priori
and synthetic; (3) the metric of humanly intuited space is Euclidean and the propositions of Euclidean geometry are synthetic and are known a priori. The criticism
that is raised against Kant as a consequence of the discovery of non-Euclidean geometries hinges upon the assumption that there is a logical relation between these
three doctrines, i.e., that (1) and (2) imply (3). In order to evaluate it we must investigate this assumption. We start with an exposition of Kant’s views on space and
geometry.
4
De mundi sensibilis atque intelligibilis forma et principiis; MSI, AA 02: 385–419.
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1.1 Space 5
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 absolute
space. Kant’s argument from incongruent counterparts6 can be seen as an objection
to Leibnizian relationalism, but most commentators agree that in his later works
Kant did not see it as a vindication of the Newtonian idea that space is metaphysically
real.7 The latter was indeed one of Kant’s conclusions in 1768,8 but in the Dissertation and later in the Prolegomena Kant uses the idea of incongruent counterparts
to illustrate (and not to prove) the intuitive character of spatial knowledge [MSI, §15
C, D], and to confirm the contention that space is metaphysically ideal [Prol, §13].
Thus, although in the Dissertation Kant believes that space has intrinsic formal
properties, 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 space
as abstracted from outer sensations [MSI, § 15 A] and develops the idea that space is
a precondition for the existence of any experience [MSI, § 15 B]. Kant then leads his
reader to the conclusion that the concept of space is known, or given in, pure intuition; it is the form of all our outer sensations [MSI, § 15 C]. The two prime
examples that support the statement that space is known intuitively are (a) the idea
that propositions of geometry are not deducible from an abstract concept of space
but require instead constructive methods, i.e., reference to concrete examples, for
their demonstration; and (b) the idea that incongruent counterparts can only be apprehended intuitively, i.e., one must observe an example of incongruent counterparts in order to comprehend this notion (ibid.).
Kant concludes this section of the Dissertation with the claim that space is metaphysically ideal, but empirically real [MSI, § 15 D, E]. Thus, while mocking Newton’s notion of absolute space as a fairytale of a ‘container’ devoid of all substance
he also dismisses Leibniz’s relationalism. Kant’s alternative to Newton and Leibniz is
categorically different. Space is the precondition to all phenomena; a formal principle of our knowledge of the sensible world, or, as Friedman suggests in his discussion of Newton’s theory of gravitation,9 the idea of reason with which we furnish the world with a truly privileged frame of reference, the “forever unreachable
common centre of gravity for all matter”.
Kant’s ideas of space propounded in the Dissertation are refined and restated in
the Critique. Space is viewed as an a priori representation: every outer conception,
5
6
7
8
9
Here I shall concentrate only on space although he foregoing discussion applies also to time.
Kant’s contribution to the debate with his argument of incongruent counterparts is thoroughly discussed in Van Cleve/Frederick 1991.
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. inertia
and absolute rotation.
Kant 1768, 25–28; GUGR, AA 02: 375–383.
Friedman 1992, 149.
Kant and non-Euclidean Geometry
83
and a fortiori the conception of one’s self, is spatial, and hence presupposes
space.10 The relational doctrine of space as abstracted from the relations between
objects contradicts the fact that the terms that designate such relations presuppose
space, and hence spatial knowledge cannot be acquired through experience.11 In
showing that space is an a priori condition of human sensuous awareness Kant establishes the necessity of space relative to it. While human awareness of particular
appearances is merely contingent, awareness of space is not. This marks an important development in Kant’s thought: in the Dissertation it was claimed that the
a priority of space was sufficient for establishing the necessity of space relative
to empirical knowledge. In the Critique Kant wants to establish that human cognition is limited to what can actually be intuited, and this can be achieved only if
all awareness is subject to the formal conditions of sensibility, that is, if space
cannot be “thought away”. The metaphysical exposition of space is then followed
by the transcendental exposition which aims to establish that the view of space as
an a priori form of intuition must hold in order for synthetic a priori knowledge
to be possible. Thus, the transcendental exposition in the Critique is concerned
with showing that the particular metaphysics of space provides the necessary
condition for a certain genus of knowledge which consists of particular kinds of
judgements.12
1.2 Geometry
Geometry for Kant is a synthetic a priori science, i.e., it is an example of a body of
knowledge which applies to the empirical world, but is not justified by empirical
facts. Contrary to the Leibnizian legacy, Kant blurs the distinction across the three
common types of judgements (the ontological, epistemological, and semantic) and
claims that there can be judgements which are synthetic, i.e., judgements which
apply to the empirical world and are informative, inasmuch as their predicates are
10
11
12
“Vermittelst des äußeren Sinnes (einer Eigenschaft unsres Gemüths) stellen wir uns Gegenstände als außer uns und diese insgesammt im Raume vor.” / “By means of outer sense, a
property of our mind, we represent to ourselves objects as outside us, and all without exception in space.” (B 37)
“Demnach kann die Vorstellung des Raumes nicht aus den Verhältnissen der äußern Erscheinung durch Erfahrung erborgt sein, sondern diese Erfahrung ist selbst nur durch gedachte Vorstellung allererst möglich.” / “The representation of space cannot, therefore, be
empirically obtained from the relations of outer appearance. On the contrary, this outer experience is itself possible at all only through that representation.” (B 38)
There is a difference in the presentation of Kant’s ideas in the Critique and in the Prolegomena, but in both cases the metaphysical doctrine of space allows one to understand how
synthetic a priori science such as geometry is possible: although its existence can be granted
independently, its nature would have been different if space were different (Prol, § 11). Similarly, Kant believes that neither the Newtonian nor the Leibnizian can account for both the
necessity 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 justified without an appeal to experience and hence are necessary and universal.13 The
idea of synthetic a priori knowledge is the Kantian response to the long-standing debate between rationalists and empiricists. Kant changes the rules of the game: there
is a special kind of knowledge that does not originate in experience, yet applies to
experience, inasmuch it is a precondition of any experience.14 This resolution also
indicates an unbridgeable gap between the “things-in-themselves” and the way we
perceive them; between noumena and phenomena. Noumena are inaccessible to us
inasmuch as we can never know them. Thus, if our knowledge is bound to appearances, 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 is
restricted to the phenomenal world.15
Kant ascribes a special role both to Euclidean geometry and to the constructive
feature of geometrical proofs. According to him geometrical reasoning cannot proceed purely logically, i.e. through analysis of concepts, but requires a further activity called “construction in pure intuition”.16 Friedman (1992, 58) notes that the
spatio-temporal17 character of this construction enables Kant to give a philosophical foundation for both Euclidean geometry and Newtonian physics. Indeed, as Shabel (1998, 618) adds, Kant explains both the difference between mathematical and
philosophical reasoning and the syntheticity of mathematical judgements by the
role played by construction in pure intuition [B 741; B 287]. Moreover, almost all of
Kant’s examples for the construction of either mathematical or geometrical proofs
rely on Euclidean geometry and the use of its postulates for constructing geometrical figures. Although these examples do not serve as an argument in his metaphysics
of space, in mentioning them Kant seems to commit himself to the truth of Euclidean geometry.
Returning to the opening question of this section regarding Kant’s threefold doctrine of space and the existence of a logical relation between its constituents: (1) the
metaphysical character of space (2) the possibility of synthetic a priori geometry
and (3) the Euclidean nature of our appearances, we can see now that (1) and (2) are
indeed logically related: Kant’s metaphysics of space ensures the certainty and
13
14
15
16
17
Kant alternates between the epistemological interpretation of a priori as necessary, or evident, and the ontological interpretation as universal.
Compare Parsons 1983, 118: “[K]ant started from the idea that geometry was a body of
necessary truths with evident foundations. That the axioms of geometry should be empirically verified is contrary to their necessity; that they should be some sort of high-level hypothesis is contrary to their evidence.”
Contrary to Van Cleve (1999, 143–150) I subscribe to the view that the noumenal and phenomenal worlds as two aspects of the same world.
This claim is explicitly expressed in the “Discipline of Pure Reason”, where Kant confronts
philosophical with mathematical reasoning. See B 743–745.
“Temporal” because it involves the notion of generating a line from an originating point as
a process unfolding in space-time. See B 203–204.
Kant and non-Euclidean Geometry
85
necessity of geometry.18 But whether (3) is implied by either of them is still an open
question. It is precisely the difference between the kind of knowledge the character
of space grounds and one’s commitment to the truth of this knowledge’s special
character (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 philosophy of geometry. The transcendental exposition in the Critique establishes that
(1) geometry is the science of the properties of space and that (2) geometrical truths
are synthetic a priori [B 40]. Kant seems to argue that our inability to know geometry except by means of construction in pure intuition accounts for the synthetic
character of geometrical truth. Whether this statement means that (a) pure intuition
discerns the Euclidean metric (and the properties) of space, or that (b) the process of
geometrical reasoning involves the construction of definitions with which we formulate 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 different
issues. The first concerns the role of intuition in Kant’s account of mathematical
knowledge; the second – Kant’s arguments for that role. We shall see that whether
or not non-Euclidean geometries undermine Kant’s philosophy depends on one’s
views 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 of
non-Euclidean geometries on the Kantian project form the more general debate on
Kant’s philosophy of geometry.
The most complete account given by Kant of his view of the nature of mathematical reasoning is in the first section of the “Discipline of Pure Reason” in the
Critique.20 One of the sharpest critics of Kant’s view is C. D. Broad (1941) who reconstructs Kant’s definition of mathematical knowledge as follows:
It is characteristic of mathematics to start from definitions which are obviously adequate and
from axioms which are self-evident and to reach conclusions which are rendered intuitively certain by demonstration. (Broad 1941, 1)
18
19
20
Yet, this necessity would be affirmed only later on in the transcendental deduction with the
introduction of the unity of apperception.
Kant is well aware of this difference in other contexts, such as his discussion of Newton’s
laws of motion in “The Metaphysical Foundations of Phenomenology”, when he states (in
MAN, AA 04: 554–565) that the question is not the transformation of illusion into truth,
but of appearance into experience.
B 741–766. As C.D. Broad (1941, p. 1) notes: “It is to be feared that most of us, unless we
are professional students of Kant, have rather flagged before reaching p. 570 of Kemp
Smith’s translation and are inclined to give ourselves a holiday on the plea that the rest of
the book is just “Kant’s architectonic.” It is noteworthy that Kant’s ideas on the nature of
mathematical 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 ascertain the adequacy and certainty of our definitions. This is when we arbitrarily make
up the concept for ourselves. Next, there is a unique condition that ascertains the
applicability of that concept. This is when the concept contains an arbitrary synthesis that admits of a priori construction.
This intuitive construction not only enables one to decide whether two mathematical concepts, such as “straightest” and “shortest” imply each other, an impossible task in philosophy, but also excludes the possibility of demonstrations outside mathematics, whence the methodological difference between mathematical and
philosophical knowledge. While Broad agrees with Kant that the geometrical properties of a triangle do not follow logically from the mere definition of a triangle, he
points out that the former do follows logically from the latter when accompanied by
the concept of the space in which the triangle is imbedded, i.e., the metric signature
of that space. The fact that many of Euclid’s propositions do not follow deductively
from his definitions, axioms and postulates, and that intuition is indeed needed
in 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 Broad
shares with Russell in his Principles of Mathematics (1903/1937).
Next Broad draws a distinction between pure and applied geometry. In pure geometry, when we are merely supposing and not asserting the axioms, intuition is not
needed for guaranteeing the truth of the axioms but for constructing a perceivable
instantiation of them, i.e., a proof of their consistency.21 Broad regards Kant’s insistence on intuition being necessary for guaranteeing the truth of Euclid’s propositions
as an “extraordinary view”,22 yet nevertheless he continues dissecting this view by
distinguishing three classes among Euclid’s postulates: (1) Those which express peculiar feature of Euclidean space (e.g. the parallel postulate). (2) Those which express features common to all forms of homaloidal space,23 whether Euclidean, elliptic, or hyperbolic (e.g. that the existence of a figure with a given size and shape in
one 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 between
two points is a straight line). Broad, in his unique style, demonstrates how each of
these classes admits either an analytic or synthetic a posteriori view of geometry.24
21
22
23
24
Broad acknowledges the psychological importance of intuition in “discovering” the axioms
but admits it is too weak a notion for what is needed to establish Kant’s claims, inasmuch as
non-Euclidean spaces have been discovered regardless of their visualization.
“It would mean that we are able to know with complete certainty, and not just probably by
means of induction, the fundamental facts about the spatial structure of nature at all places
and at all times.” (9)
Homaloidal means Flat; even. It is a term applied to surfaces and to spaces in which the definitions, axioms, and postulates of Euclid are assumed to hold true, i.e., spaces with zero intrinsic curvature.
With respect to (1) Broad notes that the peculiar status of the parallel postulate prevents one
from seeing it as self evident, but the self-consistency of Euclid’s geometry requires one to regard the parallel postulate as empirically, though indirectly, evident. With respect to (2)
Kant and non-Euclidean Geometry
87
Broad’s view is characteristic of the general criticism of Kant’s philosophy of geometry in the first half of the twentieth century. The lesson it drives home is that
even before one begins to examine the implications of non-Euclidean geometries on
Kant’s philosophy there is still a room for questioning the role of intuition in the
Kantian project. Broad concludes that when properly construed pure geometry is
analytic, and applied geometry is in part synthetic but nevertheless a posteriori. In
doing so Broad denies the Kantian hybrid, the synthetic a priori, at least as far as geometry is concerned, and restricts the role of “construction in intuition” to securing
the consistency of a particular geometry.
The distinction between pure and applied geometry and its supposedly dire consequences for Kant’s philosophy of geometry is the common thread shared by many
of Kant’s critics. As Friedman (1992, 55) notes, it is epitomized in Einstein’s famous
phrase: “As far as the laws of mathematics refer to reality, they are not certain; and
as far as they are certain, they do not refer to reality.” Thus Carnap, in his introductory remarks to the English edition of Reichenbach’s The Philosophy of Space
and Time (1958, vi) says:
The statements of pure geometry hold logically, but they deal only with abstract structures and
they say nothing about physical space. Physical geometry describes the structures of physical
space; it is a part of physics. The validity of its statements is to be established empirically – as it
has to be in any other part of physics – after rules of measuring the magnitudes involved, especially 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 from
as undermining Kant’s claim that geometry is synthetic, one can regard the development of non-Euclidean geometries as serving only to support it.25 Indeed, when
Riemann (1866) discusses the foundations of geometry his conclusions are equivocal: space might posses a unique structure but when we try to discern this structure
by way of physical measurements, we already presuppose certain hypotheses (regarding rigid rods and light rays as gauges) and a certain metric (that establishes the
basic gauges). Thus at least on the latter reading of Riemann, geometry is still syn-
25
Broad notes that these axioms are indeed analytic, at least if one admits a distinction between space and what occupies space. Although Kant might have objected to this distinction
it is still far from clear whether he would have made it the crucial evidence for the role of intuition in geometrical reasoning, especially after Newtonian spacetime was given a general
covariant formulation (Friedman 1983). With respect to (3) things are more complicated. In
non-flat spaces, where straight lines are geodesics, the axioms that the shortest path between 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 proposition. In order to measure a length of a curve we need first to divide the curve to small
straight segments, thus the notion of length applies primarily to straight lines, which seems
to 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).
See Brittan 1978, ch. 3, where the positivists’ arguments on the analyticity of geometry are
discussed.
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thetic a priori: when applied to the world it serves as precondition for any physical
experience of the world. As for the role of intuition in geometrical reasoning, Riemann indeed demonstrates that it is unnecessary when he presents his concept of
n-fold extended manifold,26 but note that any representation of either Riemannian
or Lubachevskian geometry, as constructed respectively by Reid (1764) or Beltrami,27 is achieved only within Euclidean space. It is for this reason that Broad
accepts intuition’s role in securing the consistency of these geometries. Yet, the
consistency of non-Euclidean geometries is only relative: they are consistent if Euclidean 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 refuting
Kant on Euclidean geometry, only demonstrates its coherence.
Contemporary commentators such as Michael Friedman (1992), however, dismiss 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 goes
hand in hand with quantificational logic and modality of possible worlds. Since
Kant relies on monadic syllogistic logic and on the original “constructive” formulations of Euclid’s axioms he is bound to represent concepts like “infinity” as an iterative process of spatial construction.28 Furthermore, deprived as he was of modern
analysis and the Cauchy-Weierstrass concept of limit, Kant represents this construction as a spatio-temporal process: spatial quantities are not composed of points but
are generated by the iterative motion of points. This iterative process takes the place
of modern quantification, and is the very reason for geometry being synthetic a
priori.29 Thus, according to Friedman, the reason for geometry being synthetic does
not 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 intuition
to merely supplying a model for discerning the correct geometry from a class of
possible geometries and uses a notion of possibility foreign to Kant.30 On the
contrary, Friedman insists that non-Euclidean geometries are impossible within the
26
27
28
29
30
Friedman 1992, 95.
Gomez 1986.
This view is propounded also by Parsons 1983, ch. 5.
Friedman emphasizes the “inductive” and unbounded character of mathematical proofs, involving iterative processes of substitution (in arithmetic) and construction in intuition (in
geometry).
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 mind
scanning those creations and detecting their properties with absolute accuracy”. Indeed, as
Friedman (ibid., 90) notes, although one can solve the problem of how to generalize from a
particular 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 has
neither the generality nor the precision to make the required distinctions, not to mention
that on a small scale Euclidean and non-Euclidean geometries are indistinguishable.
Kant and non-Euclidean Geometry
89
Kantian project. The only way to represent them, according to Kant’s philosophy of
geometry, is by drawing, or generating them in the space (and time) of pure intuition. But this space for Kant is necessarily Euclidean. It follows that there is no way
to draw, and a fortiori represent, say, a non-Euclidean straight line, and the very
idea of non-Euclidean geometry is quite impossible.31 Consequently, nothing follows from the mere formal possibility of non-Euclidean geometry.32
To summarize Friedman’s view, the impossibility of non-Euclidean geometries in
Kant’s view should be understood not as referring to the realm of the “things-inthemselves” alone (where concepts, considered independently of our sensible intuition, are meaningless, or empty), but as referring to the realm of cognition; to
“things-in-themselves” plus our pure intuition. These conditions of cognition are
the best approximation to the modern idea of logical possibility.33 According to
Friedman, the role of intuition in Kant’s philosophy of geometry is simply to supply
initial constructive definitions from which we can formulate geometrical proofs, or
in other words, pure intuition underwrites the constructive procedures used in
mathematical proofs. For this reason, among others, Friedman (ibid., 95) concludes
that rather than being accused of his failure to anticipate modern developments,
Kant should be applauded for his insight, that in its classical formulation, Euclidean
geometry required for its proofs certain procedures that the traditional logic was incapable of characterizing.
Returning to the question we raised in the last section, whether or not Kant’s doctrine of space implies a logical relation between Euclidean geometry and transcendental philosophy, i.e., whether the metaphysical view of space and the synthetic a
priori character of geometry imply that the metric of humanly intuited space is Euclidean, we can see now that according to Friedman Kant survives even if such an
implication exists. As Friedman suggests, non-Euclidean geometries pose no problem to Kant since the domain of Euclidean geometry is restricted only to the phenomenal world. This conclusion, however, relies on the premise that all appearances
are 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 that
Kant might have lost a battle, betting on Aristotelian logic, Newtonian physics and
Euclidean geometry, but nevertheless won the war: not only did his legacy of transcendental philosophy come out of the nineteenth and twentieth centuries intact but
it was also reinforced.
31
32
33
Friedman, ibid., p. 82.
Beck and Brittan seem to rely on B 268 where Kant accepts this formal possibility.
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 philosophy
when the latter is considered detached from Euclidean geometry. This is the conclusion of the line of thought that regards intuition’s role as discerning the correct
geometry of the world from a class of possible geometries. Michael Friedman rejects
this line of thought and insists that for Kant the role of intuition is rather different:
intuition underwrites the constructive procedures used in mathematical and geometrical proofs. This interpretation renders non-Euclidean geometries a logical impossibility: as far as the phenomenal world (the world of the noumena plus pure intuition) is concerned, Euclidean geometry becomes an implication of the Kantian
project. The fact that non-Euclidean geometries were nevertheless conceived only
vindicates 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 shall
claim that if one agrees with Friedman that Kant’s project entails Euclidean geometry as a necessary character of the phenomenal world, the impact of non-Euclidean 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-dimensional visual space as an example. This claim is intended to lure the Kantian into a
trap, since the common reaction it raises is that two-dimensional visual phenomena
are relational or subjective, i.e., they merely represent a point-of-view of three-dimensional tangible space: while the latter is invariant, the former co-vary with the
observer in a way that preserves an underlying non-arbitrary structure. Having
lured 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 particular and unique character. Finally, I argue that (3) since this admission involves
exchanging the notion of truth as coherence with truth as correspondence, it militates 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 is
one and only one metric which applies to the phenomenal world: the Euclidean metric of tangible space. The reason Berkeley rejects all other metrics, such as the metric
of visual space, is that the latter varies with one’s perspective, whereas the tangible
metric remains invariant.35 I am not certain that contemporary Kantian scholars
would agree to the positive thesis that Kant regards visual space as possessing a Eu-
34
35
Berkeley 1709/1910.
See also Bennett 1966, 30.
Kant and non-Euclidean Geometry
91
clidean metric.36 As we have seen, they are more inclined to contend the negative
thesis, namely, that Kant regards a non-Euclidean metric impossible and the Euclidean metric unique:
The reason why non-Euclidean figures cannot be constructed is not because we cannot visualize or imagine them, but because there is not an appropriate metric for them as there is, notably, 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 precision to the objects of our experience. […] [T]he possibility of alternative metrics would entail
the erroneous conclusion that the form of the world, and our knowledge of it, is merely contingent and relative. (Brittan 1986, 65)37
This uniqueness ensures the correspondence between our knowledge and its objects in experience: the truth of the Euclidean propositions, under this account, is secured by the coherence of Euclidean geometry as a complete system of knowledge
according to which we organize our experience. For this reason, as Brittan argues,
the uniqueness of the Euclidean metric is crucial for establishing objective knowledge of appearances.
Unfortunately, Kant and his advocates are mistaken on both the positive and the
negative theses: visual appearances are non-Euclidean; the phenomenal Euclidean
metric is not unique. Indeed, the idea that part of appearances, e.g., two-dimensional visual space, possesses a non-Euclidean metric was already suggested by Thomas Reid;38 propounded later on by Daniels (1972) and Angell (1974); and can be
easily 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 is
to ensure correspondence between our knowledge and appearances, that is, to ensure the truth of geometrical propositions by drawing on both the consistency and
the categorical character of the system that entails them, then the existence of other
metrics threatens to collapse this coherence and a fortiori the entire Kantian project:
the non-uniqueness of the phenomenal metric would imply the non-categorical
character of Euclidean geometry as a system of knowledge and the existence of nonEuclidean geometries would then serve as a bait; luring the Kantian into the trap of
acknowledging that “things-in-themselves” do have a particular and non-arbitrary
character that gives rise to both tangible and visual appearances.
36
37
38
39
As it carries a misleading sense of psychological capacity à la Helmholtz (1868/1977). See
also Craig 1969, 122; 130.
Brittan refers here to B 268.
Reid 1764/1997, ch. 6. Here Reid constructs a model for non-Euclidean geometry and defines a metric for visual space.
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.
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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, and
other such as tangible space are Euclidean, then in order to restore consistency to
phenomenal knowledge the Kantian must claim that both tangible and visible space
are 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 perspective while preserving this structure. In so doing the Kantian must trade his notion 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 a
direct reference to “things-in-themselves”; a reference he is reluctant to make.
The demarcation between the “things-in-themselves” and appearances is a key
distinction in the Kantian project: it allows synthetic a priori knowledge; it solves
the problem of Humean scepticism; it solves the long-standing antinomies; in fact, it
is one of the core ideas of the so-called Copernican revolution of transcendental
philosophy. Yet throughout the Critique Kant is cautious not to assign any property
to the “things-in-themselves” apart from mere existence.40 In contrast to appearances, which are the object of our knowledge and experience, the “things-in-themselves” 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 describe them remains empty, lacking both sense and meaning”.
Among Kant’s contemporaries Thomas Reid is probably the most lucid advocate
of the contrary view, namely, the idea that nature has a definite structure, and that
science’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 subject. I perceive in a billiard ball, figure, color and motion; but the ball is not figure, nor is it
color, nor motion, nor all of these taken together; It is something that has figure and color and
motion. This is a dictate of nature and a belief of all mankind. As to the nature of this something, I am afraid we can give little account of it, but that it has the qualities which our senses
discover. (Reid 1785/1975, Essay II, Ch. 19, § 257)
In his Essays (1785/1975) Reid elaborates on the distinction – made earlier in his
Inquiry (1764/1997) – between the metrics of tangible and visible space and offers it
as an argument for the existence of an underlying structure in nature. On Reid’s
view visible phenomena are two-dimensional non-Euclidean, as opposed to tangible
phenomena which are three-dimensional Euclidean. Both are qualities of the external 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, the
direct objects of perception are neither sensations nor physical objects but the
qualities 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 (Van
Cleve 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 himself
puts 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 same
space”.41 On this view the external world does posses an underlying structure which
gives rise to the Euclidean character of tangible space and to the non-Euclidean
character of visible space, and, more important, to the fact that the latter is relational, i.e., that the latter co-varies with perspective.
The idea that nature has a definite structure, and that science’s role is to discover
that structure, and not to invent it, goes hand in hand with an eternal scepticism
with respect to our knowledge of nature, and one of Kant’s motivations is to abolish
this scepticism. Yet, by securing the objectivity of knowledge with the transcendental deduction Kant prevents any correspondence whatsoever between phenomena
and the stripped noumena. Indeed, Bertrand Russell would try to argue for the
possibility of such a correspondence:
For example, it is often said that space and time are subjective, but they have objective counterparts; or that phenomena are subjective, but are caused by things in themselves, which must
have differences inter se corresponding with the differences in the phenomena to which they
give rise. Where such hypotheses are made, it is generally supposed that we can know very little
about the objective counterparts. In actual fact, however, if the hypotheses as stated were correct, the objective counterparts would form a world having the same structure as the phenomenal world, and allowing us to infer from phenomena the truth of all propositions […]. If the
phenomenal world has three dimensions, so must the world behind it be; if the phenomenal
world 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 correspondence
between appearances and “things-in-themselves” if one accepts the Kantian reluctance to attribute to the latter any structure whatsoever.44
By now things get complicated for the Kantian: if tangible space and visual space
have different characteristics, and these characteristics are a product of an existent
non-arbitrary underlying structure, then the Kantian silence with respect to “thingsin-themselves” becomes even more puzzling. Of course the Kantian can always
claim that even if such structure does exist, it is still inaccessible to us. Yet this move
brings us to the third and final step of my argument since it already indicates the
abandonment of the notion of truth as coherence in favour of truth as correspondence; a move which is tantamount to giving up the metaphysical thesis of space.
41
42
43
44
Reid 1785/1975, Essay II, Ch. 19, § 264.
Russell 1967, 176, in a letter to M. Newman, 1928: “My statements to the effect that nothing is known about the physical world except its structure are either false or trivial. […] I
am somewhat ashamed at not having noticed the point myself. […] had not really intended
to say what in fact I did say […]”.
See for example A 358.
Russell’s idea was soon proved impossible by Newman (1928). On the Russell-Newman affair see Demopoulos and Friedman (1985).
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3.3 Step 3: Trading Formal Idealism for Non-Naïve Realism
In order to appreciate how the admission in a definite-yet-inaccessible structure
of “things-in-themselves” undermines Kant’s metaphysical thesis of space we can
put matters in terms of mapping: idealism with respect to space means that the latter
is inexistent. But Kant was reluctant to attach himself to the Berkelian camp and
preferred to maintain what he called “formal idealism” with respect to space, claiming that the latter is amorphous, which means that space has no geometry, neither
Euclidean nor non-Euclidean, or in other word, that there exists no mapping whatsoever from appearances to “things-in-themselves” because space is nothing but a
pure form of our intuition:
We have sufficiently proved in the Transcendental Aesthetic that everything intuited in space or
time, 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 extended
beings, or as series of alterations, have no independent existence outside our thoughts. This
doctrine I entitle transcendental idealism.*) The realist, in the transcendental meaning of this
term, treats these modifications of our sensibility as self-subsistent things, that is, treats mere
representations as things in themselves.
*) I have also, elsewhere, sometimes entitled it formal idealism, to distinguish it from material
idealism, that is, from the usual type of idealism which doubts or denies the existence of outer
things themselves. (B 519)45
As argued by Reid, the relational and non-Euclidean character of two-dimensional visual space over and above the Euclidean character of tangible space indicates 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 respect
to space and also to claim that there is a unique non-trivial correspondence between
appearances and “things-in-themselves”.
Agreed, such an attribution does not lead to full-blown naïve realism, which can
be described as a ‘one-to-one’ mapping between appearances and “things-in-themselves”. 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 be45
“Wir haben in der transcendentalen Ästhetik hinreichend bewiesen: daß alles, was im
Raume oder der Zeit angeschauet wird, mithin alle Gegenstande einer uns möglichen Erfahrung nichts als Erscheinungen, d. i. bloße Vorstellungen, sind, die so, wie sie vorgestellt
werden, als ausgedehnte Wesen oder Reihen von Veränderungen, außer unseren Gedanken
keine an sich gegründete Existenz haben. Diesen Lehrbegriff nenne ich den transcendentalen
Idealism.*) Der Realist in transcendentaler Bedeutung macht aus diesen Modificationen unserer Sinnlichkeit an sich subsistirende Dinge und daher bloße Vorstellungen zu Sachen an
sich selbst.
*) Ich habe ihn auch sonst bisweilen den formalen Idealism genannt, um ihn von dem materialen, 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 of
the non-uniqueness of the Euclidean metric, the Kantian finds himself in a position
which is tantamount to renouncing his transcendental idealism with regards to
space. Although some try to describe Kant as a metaphysical realist,47 the case of
non-Euclidean geometries illustrates how subtle one must be when one regards the
truth of Euclidean geometry as entailed by transcendental idealism and the synthetic
a priori character of geometry.48
4. A Kantian Response?
The gist behind the threefold argument just presented is that the possibility of
non-Euclidean phenomena manifest in the non-Euclidean metric of two-dimensional visual space militates against Kant’s formal idealism with respect to space
since it signifies a departure from a coherence theory of knowledge in favour of an
alternative based on correspondence. Here are two possible responses to this argument on behalf of the Kantian.
First, the Kantian might claim, as Berkeley did, that visual appearances are not
objective in the sense that they are “unreal”; they represent nothing but a two-dimensional perspective of “real” three-dimensional tangible phenomena, and since
the metric projected from three-dimensional tangible space onto two-dimensional
visible space is Euclidean, visual appearances, whether Euclidean or not, cut no ice
in the debate.
This scepticism with respect to the reality of two-dimensional visual space, however, must be judged, as the whole of Kantian philosophy, with reference to the developments in Geometry. Kant’s contemporaries may indeed discard non-Euclidean
two-dimensional visual space as nothing but a projection of three-dimensional tangible space and regard visual space as having no intrinsic properties, but such criticism becomes unavailable to one who – after Riemann and Gauss – is equipped with
the distinction between intrinsic and extrinsic curvature.
A space can possess intrinsic non-flat geometry yet contain lines that will be
straight according to any form of measurement intrinsic to that space. A line is
called straight in relation to its own manifold. Euclidean straightness thus characterizes lines in a three dimensional space with no intrinsic curvature, and flat spaces
of more than three dimensions may be called Euclidean because of their lack of cur46
47
48
Note that even in Einstein’s theories of relativity, where many physical properties become
relational, or foliation-dependent, there still exists an invariant structure that underlies their
co-variance: the spacetime interval, or the metric.
Van Cleve 1999, ch. 13.
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.
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vature. What we now call extrinsic curvature presupposes a higher dimension in
which curved objects are embedded: for a line or a plane or a space to be curved
it 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 intrinsic
curvature has nothing to do with any higher dimension. Confusion regarding this
point arose because we can model non-Euclidean planes as extrinsically curved surfaces within Euclidean space. Thus, the surface of a sphere is the classic model of
a two-dimensional, positively curved Riemannian space, as demonstrated by Reid
(1764/1997). But while great circles are straight lines according to the intrinsic
properties of that surface, we see the surface itself curved into the third dimension
of Euclidean space.
Reconstructing the Kantian objection to the reality of visual space in terms of curvature, 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 intrinsic: visual space is a ‘second-order’ phenomena; a by-product of the phenomenal
world – three-dimensional Euclidean tangible space – and not of the noumenal one.
But what other ground than the transcendental project itself has the Kantian to
offer for this claim? Such a move would situate the Kantian closer to Berkeley than
he would have liked to admit. Moreover, if one understands, as Reid did, the nonEuclidean character of visible space and the Euclidean character of tangible space as
complementary conceptions of the external world, our cognitive ability to translate
the tangible into the visible and vice versa is just another proof that visible space is
as real as tangible space, and that both spaces are no more than different qualities of
the same external world; a world which – the Kantian indifference to it notwithstanding – does posses a unique structure.
A second route open for the Kantian is to claim that looking at the psychological
question of the structure of visual space may not do justice to the idea that appearances are Euclidian, since the question what appearances are is presumably not a
psychological question.
But here it is crucial to note that nothing in the threefold argument just presented
hinges on the “visual” versus “tangible” distinction apart from the fact that the
former is two-dimensional and the latter is three-dimensional, hence the distinction
between them is exactly the one which is needed in order to embed n-dimensional
non-Euclidean geometry in n+1 dimensional Euclidean space. Both modes of cognition are available to us in detecting appearances, and there is no reason, other than
securing 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 in
spaces whose dimension is bigger than two, but it is also the case that we cannot visualize Euclidean geometry in spaces whose dimension is bigger than three. Thus the
accusation against the Kantian propounded here is not that he is guilty of purporting that the impossibility to visualize certain geometries militates against their existence. It is rather that notwithstanding this psychological impossibility, the fact that
not 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 structure – a structure of which the Kantian is deliberately indifferent, since admitting it
would be tantamount to renouncing his project.
In sum, if one insists on a logical relation between Euclidean geometry and the
metaphysical and transcendental expositions of space, i.e., that the latter imply the
truth of the former, then non-Euclidean geometries lead to the demolition of the
Kantian project. A coherence theory of truth cannot be replaced with a theory of
correspondence and also remain idealistic with respect to the form of its truthmakers. Unfortunately, although useful in political situations, there exists no ‘third
way’ 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 has
thus led the Kantian into an impasse.
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