Kant and Non-Euclidean Geometry, Historia - Logiki

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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 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 quantifi-
cational 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 ques-
tions: (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-Euclid-
ean 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
Friedman 1992, 55. Friedman aims to demonstrate that such a view is fundamentally unfair
to Kant.
2
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”.
3
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
DOI 10.1515/KANT.2008.006
© Walter de Gruyter 2008
ISSN 0022-8877
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 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 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 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) 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 prop-
ositions 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 ge-
ometries 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 in-
vestigate 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.
82
Amit Hagar
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 counterparts
6
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
Disser-
tation
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 intui-
tion; 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 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 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 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 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
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. inertia
and absolute rotation.
8
Kant 1768, 25–28; GUGR, AA 02: 375–383.
9
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 es-
tablishes the necessity of space relative to it. While human awareness of particular
appearances 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 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 cog-
nition 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
“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, a
property 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, be
empirically 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 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). Simi-
larly, Kant believes that neither the Newtonian nor the Leibnizian can account for both the
necessity
and
the truth of geometrical propositions (B 57).
84
Amit Hagar
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
The
idea 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: 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 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 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 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 the
spatio-temporal
17
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 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 geometri-
cal 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 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) 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
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 of
necessary 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 confronts
philosophical with mathematical reasoning. See B 743–745.
17
“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.
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