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objectively valid implies that we can make true or false judgments
about them as well as about spatiotemporal objects. In connecting
empirical reality with objective validity, Kant relativizes the notions
of an object and objective truth. Empirical realism entails that what
counts as an object for us, and therefore what counts as objective truth
for us, is relative to our cognitive capacities. The Aesthetic establishes
those conditions from the side of human sensibility. In the Transcen-
dental Analytic Kant examines the contribution of the understanding
to the objective conditions of cognition.
As a result of his “transcendental turn,” in the rest of the Critique
Kant generally uses the term “object” to refer to objects of knowledge
or appearances, and he typically reserves the term “thing” for things
in themselves. There are passages, of course, where Kant ignores this
distinction, such as at A30/B45, where he says that “objects in them-
selves are not known to us at all.” But generally he uses these terms
The Transcendental Aesthetic 61
in accordance with his conclusion that objects of experience are only
appearances.
We can now appreciate the peculiar sense in which space and time
are subjective, and the connection between transcendental subjectiv-
ity and the necessity of synthetic a priori judgments. At A28“9 and
A28/B44, Kant contrasts the subjectivity of space and time with the
subjectivity of secondary qualities, or experiences of color, sound, and
hot and cold:
Besides space, however, there is no other subjective representation related
to something external that could be called a priori objective. For one cannot
derive synthetic a priori propositions from any such representation, as one
can from intuition in space (§3). Strictly speaking, therefore, ideality does
not pertain to them, although they coincide with the representation of space
in belonging only to the subjective constitution of the kind of sense, e.g.,
of sight, hearing, and feeling, through the sensations of colors, sounds, and
warmth, which, however, since they are merely sensations and not intuitions,
do not in themselves allow any object to be cognized, least of all a priori.
(A28/B44; see also A28“9)
For scienti¬c realists like Descartes and Locke, secondary qualities
such as color, taste, heat, and so on are merely effects caused in per-
ceivers by contact with the primary qualities of physical objects. Sec-
ondary qualities are subjective in the sense that they can vary from
perceiver to perceiver, since they depend on the individual™s sense
organs as well as the conditions of perception. A color-blind person,
for example, will not see the full range of colors seen by someone
who is not color-blind. Now Kant assumes that all human perceivers
share the same forms of intuition. Thus the subjectivity of space
and time differs from the subjectivity of secondary qualities in two
ways. First, space and time are universally or species-subjective, since
they are forms of all human intuition. And this implies, second, that
space and time are necessary rather than contingent features of expe-
rience. By contrast, secondary qualities are not necessary features of
appearances, and so cannot provide a foundation for synthetic a priori
cognition. In addition, these qualities do not yield direct cognition
of objects, although scienti¬c realists assume there are correlations
between secondary qualities and the real properties causing the expe-
riences. That space and time are pure forms of intuition, however,
The Transcendental Aesthetic
62
can account for synthetic a priori knowledge of mathematics and
mechanics. The transcendental subjectivity of space and time means
that they are universal to humans and the ground of necessary fea-
tures of appearance. This is not to be confused with the empirical
subjectivity of contingent sensible qualities that vary from individual
to individual.
This contrast between transcendental and empirical subjectivity
is echoed in the distinction between transcendental and empirical
notions of appearance at A45/B62“3. As Allison explains, the oppo-
sition between the subjective or ideal and the objective or real marks
a division between what is in the mind and what is independent
of the mind.16 But this distinction can be drawn on both the tran-
scendental and empirical levels. Considered transcendentally, “the
mind” refers to all human subjects; empirically it designates only
individual subjects. Accordingly, there are both transcendental and
empirical versions of the distinction between appearances and things
in themselves:
We ordinarily distinguish quite well between that which is essentially
attached to the intuition of appearances, and is valid for every human sense
in general, and that which pertains to them only contingently because it is
not valid for the relation of sensibility in general but only for a particular
situation or organization of this or that sense. And thus one calls the ¬rst
cognition one that represents the object in itself, but the second one only its
appearance. This distinction, however, is only empirical. (A45/B62“3)
In other words, within experience we often call features such as colors
and tastes, which depend on the individual perceiver, mere appear-
ances. And we contrast those with the real physical properties of the
object, which we consider the thing in itself. Kant uses the example of
a rainbow: “we would certainly call a rainbow a mere appearance in a
sun-shower, but would call this rain the thing in itself, and this is cor-
rect, as long as we understand the latter concept in a merely physical
sense” (A45/B63). From the empirical standpoint, physical objects
are real, and the secondary qualities they appear to have are ideal
or mere appearances. At the empirical level the real or objective has
universal validity for all humans, is publicly available, and expresses
the relation between a perception and an object. The empirically
16 Kant™s Transcendental Idealism, 6“8.
The Transcendental Aesthetic 63
ideal or subjective varies among humans, represents a private experi-
ence, and thus expresses a relation merely between perceptions. From
the transcendental standpoint, however, empirically real objects are
themselves mere appearances. On this level the subjective or ideal
consists in necessary conditions of experience, which are valid for
all human subjects. The transcendentally real are things in them-
selves, which Kant believes we cannot know. Later, in the deduction
of the categories, Kant criticizes Hume for trying to account for tran-
scendentally ideal features of experience in terms of the empirically
ideal.
Kant™s notion of transcendental subjectivity is the key to the neces-
sity of synthetic a priori judgments. Earlier I called this an “epistemic”
necessity, since it is grounded in human cognitive capacities. Kant
reminds us repeatedly that it is logically possible for other subjects
to have other forms of intuition. It is just a brute fact about humans
that space and time are our forms of outer and inner sense. Although
we cannot explain this fact, it does explain why human experience
must have certain features. So space and time are necessary features
of objects of experience, although the fact that they are our forms of
intuition is not necessary. In the Transcendental Analytic Kant will
give a similar analysis of pure concepts of the understanding, deriving
their necessity from the logical forms by which humans judge. The
epistemic necessity of synthetic a priori judgments, then, is weaker
than either logical or absolute metaphysical necessity.
Before turning to some issues raised by the Aesthetic, we should
note Kant™s criticism of Leibniz™s analysis of the sensibility at A43“
4/B60“2. There Kant points out that Leibniz and his disciple Wolff
analyzed sensory representations as confused intellectual representa-
tions. But the metaphysical exposition shows that space and time,
and all the sensible data received in them, originate in the capacity
for intuition, which is distinct from the understanding. As Kant says,
The Leibnizian-Wolf¬an philosophy has therefore directed all investiga-
tions of the nature and origin of our cognitions to an entirely unjust point
of view in considering the difference between sensibility and the intellectual
as merely logical, since it is obviously transcendental, and does not concern
merely the form of distinctness or indistinctness, but its origin and content,
so that through sensibility we do not cognize the constitution of things in
themselves merely indistinctly, but rather not at all. (A44/B61“2)
The Transcendental Aesthetic
64
Here Kant classi¬es the difference between clear and confused repre-
sentation as “logical.” As he says later at B415n, “a representation is
clear if the consciousness in it is suf¬cient for a consciousness of the
difference between it and others.” Now degree of clarity is not what
distinguishes sensory from intellectual representations. Instead they
differ in kind “ both in their relation to the object and their content
as particular or general. In criticizing the Leibnizians, Kant carries
out one prong of his attack on reductionistic theories of ideas. In the
Transcendental Analytic, he will reject empiricism for an opposing
error, claiming that all ideas originate in sensory impressions.

4. cr it ic ism s of ka nt ™s t h e ory of s pace a nd tim e
The most common objections against the theory of the Aesthetic are
to the conclusions that things in themselves are non-spatial and non-
temporal (henceforth NST), and that geometry and arithmetic are
synthetic a priori. While this discussion will undoubtedly not settle
any of these issues, I hope to identify the signi¬cant issues presented
in the literature.

A. NST and the unknowability of things in themselves
From Kant™s time up to the present, critics have made two charges
against his conclusions on space and time. First, they have argued that
he does not adequately support NST. And second, they have pointed
out that both NST and the underlying presupposition that things in
themselves exist are apparently incompatible with the unknowability
thesis (UT). Here we will ¬rst examine whether NST is justi¬ed. In
my concluding remarks at the end of this book I return to UT and
the coherence of Kant™s idealism.
The criticism typically raised against NST is called the “neglected
alternative” view. This position was debated extensively by the
nineteenth-century German commentators Adolf Trendelenburg and
Kuno Fischer; the debates are discussed fully in Hans Vaihinger™s Com-
mentar zu Kants Kritik der reinen Vernunft.17 Trendelenburg pointed
out that even if one agrees that the space and time of our experience

17 Vaihinger, Commentar, 1:134“50.
The Transcendental Aesthetic 65
are subjective forms of intuition, it is still possible that things in
themselves are also spatiotemporal, although their space-time would
be numerically distinct from ours. Consequently, Kant™s arguments
do not preclude the possibility that appearances correspond to things
in themselves, even if we could never know the nature of the corre-
spondence.
In Kant™s Transcendental Idealism, Allison defends NST. His ¬rst
argument misses the mark, since it misconstrues the neglected alterna-
tive position, as maintaining that the numerically same spatiotempo-
ral frameworks are both subjective forms of intuition and also systems
relating things in themselves.18 He then considers the relevant thesis,
that there could be a correspondence between our forms of intuition
and spatiotemporal relations among things in themselves. He argues
that if this version avoids the charge of incoherence, “it does so only
by virtue of its utter vacuity” (320). While he may be right, it does not
explain why Kant thought he was justi¬ed in drawing the strong con-
clusion that things in themselves could not be spatiotemporal, rather
than taking an agnostic stand on the question. As Paton remarks, it
seems “we are entitled to say of things-in-themselves only that we do
not and cannot know them to be in space and time. Since we do not
know them at all, we cannot say what they are not.”19
In Space and Incongruence, I defend NST based on the incon-
gruent counterparts arguments, which Kant set out from 1768 up
through the critical period. Although Kant uses the arguments to
develop his distinction between the sensibility and the intellect, as
well as his theory that space and time are pure forms of intuition,
in his ¬nal versions in the Prolegomena of 1783 and the Metaphysical
Foundations of Natural Science of 1786, he claims the phenomenon
supports NST.20 Although the arguments are too complex to explain
here, by 1781 Kant took the existence of counterparts such as left and
right hands to demonstrate that the kinds of relations exhibited in
the space of our experience could not obtain among things in them-
selves. Although the argument itself does not appear in the Critique,
the theory of relations on which NST is based does. One part of the
18 See my Space and Incongruence, 93“9, for a discussion of Allison™s views.
19 Paton, Kant™s Metaphysics of Experience, 1:180.
20 See Space and Incongruence, chapters 3“5. A condensed version appears in Buroker, “The
Role of Incongruent Counterparts in Kant™s Transcendental Idealism.”
The Transcendental Aesthetic
66
theory is the metaphysical exposition views that space and time are
wholes which are prior to their parts, as well as independent of the
items located in them. The remainder occurs in the Transcenden-
tal Analytic section titled the Amphiboly of Concepts of Re¬‚ection.
Here Kant agrees with Leibniz that, as understood by reason, a sys-
tem of relations always presupposes independently existing relata.
Kant expresses this in terms of the distinction between the “inner”
or intrinsic determinations, and the “outer” or relational determina-
tions of existing things: “Through mere concepts, of course, I cannot
think of something external without anything inner, for the very rea-
son that relational concepts absolutely presuppose given things and
are not possible without these.” He goes on to remark that the space
of our intuition “consists of purely formal or also real relations,” with-
out presupposing something “absolutely inner” (A284/B340). When
Kant refers to what is thought through “mere concepts” he means the
logical conception of a relation. Here he agrees with Leibniz that rela-
tions among things in themselves logically presuppose independently
existing relata. But our intuition of space is of a system of relations that
is prior to and independent of the things occupying it. Accordingly,
in Space and Incongruence I argue that the existence of incongruent
counterparts convinced Kant that space and time are incompatible
with the kinds of relations that could obtain among things in them-
selves, as represented by mere thought. This incompatibility licenses
the strong conclusion that things in themselves could not be spatial
or temporal.
This interpretation stimulated considerable discussion in the lit-
erature.21 Recently Falkenstein has defended a “mitigated” version of
NST based primarily on the arguments of the Aesthetic. With respect
to the neglected alternative, his version maintains that if things in
themselves stood in spatiotemporal relations, those relations could not
correspond in any important way to our forms of intuition. Although
Falkenstein does not characterize his interpretation this way, it seems
to me an extension of my defense based on the theory of relations.
But he delves more deeply into Kant™s assumptions about orders and
relations, as well as the various versions of NST. For these reasons his

21 See Van Cleve and Frederick, eds., The Philosophy of Right and Left, for various viewpoints
and a detailed bibliography.
The Transcendental Aesthetic 67
account is the most thorough and charitable offered to date. Here I
shall sketch its outlines.22
In considering the neglected alternative, Falkenstein divides the rel-
evant possibilities into two: ¬rst, that space and time are themselves
substances; and second, the relationist view that they are constructed
from properties or relations of things in themselves. He ¬nds Kant™s
argument against the substantival view in the proof of the thesis of
the Second Antinomy, as the following reductio ad absurdum. If com-
posite self-subsisting things were not made up of simple parts, and
all composition were removed “in thought,” no composite or simple
part would remain. And therefore “no substance would be given.”
Thus self-substantial things must ultimately be composed of sim-
ple parts (A434/B462).23 Now space and time are not composed of
simple parts because they are in¬nitely divisible. Consequently they
could not correspond to any conceivable substantival things in them-
selves. This explains Kant™s remarks that were time self-subsistent, “it
would be something that was actual yet without an actual object”
(A32“3/B49), and that the absolute theorists have to admit “two eter-
nal and in¬nite self-subsisting non-entities (space and time), which
exist (yet without there being anything real)” (A39“40/B56). This dis-
poses of the substantival version of the claim that things in themselves
could be spatiotemporal.
Falkenstein thinks Kant™s strongest defense against the relational
version of the neglected alternative is based on an analysis of different
types of orders. Recall the ¬rst metaphysical exposition assumption
that the spatiotemporal positions of appearances are not determined
by the empirical contents. It follows that the spatiotemporal order
of appearances could not possibly be based on (intrinsic) proper-
ties of things in themselves. Falkenstein contrasts a “comparative
order” of things based on their intrinsic qualities, with the “presen-
tational” order of spatiotemporal locations: “In a comparative order,
the locations of the ordered elements are determined by some scalable
quality in the elements themselves. The order of colors in terms of
their brightness, saturation, and hue, or of sounds in terms of pitch
and volume is an example of a comparative order.”24 Thus we can
22 The arguments outlined here are contained in chapter 9 of Kant™s Intuitionism, 289“309.
23 I discuss this argument in chapter 9, section 2.
24 Falkenstein, Kant™s Intuitionism, 184. See 183“5 for this analysis.
The Transcendental Aesthetic
68
“locate” hues by their positions on a spectrum. But in this “color
space” the positions of the hues are ¬xed: green will always appear
between blue and yellow. The ¬rst metaphysical exposition shows,
however, that the spatiotemporal positions of appearances are com-
pletely independent of their intrinsic (scalar) qualities: the fact that a
color patch is green determines nothing about where or when it will
appear. This provides the desired support for NST, for “even if there
were a sense in which things in themselves might be in space or time,
it would have to be a very different sense from that in which, accord-
ing to the metaphysical expositions, the matters of appearance are
in space and time” (303). The incompatibility allowing Kant to rule
out a relationist alternative is that between the independent presen-
tational order of our spatiotemporal experience, and a comparative
order based on intrinsic features of things in themselves.

B. Is arithmetic analytic or synthetic?
Kant™s view that mathematics is synthetic a priori has also been much
debated by philosophers. The issues are complex, and are related
to three important developments since Kant™s time. These are, in
chronological order, the development of non-Euclidean geometries
in the nineteenth century, the failure of the Frege“Russell program
to reduce mathematics to logic in the early twentieth century, and
¬nally, the assumption in relativity theory that only empirical science
can determine whether space is Euclidean or non-Euclidean. The
¬rst development apparently supports the synthetic nature of geom-
etry, while the third poses a serious challenge to its a priori status.
The failure of the reduction program has the more startling result
of supporting Kant™s view that arithmetic is synthetic. This section
considers whether arithmetic propositions are synthetic, and the fol-
lowing section treats the synthetic a priori nature of geometry.
In claiming that arithmetic is synthetic a priori, Kant rejected Leib-
niz™s view that arithmetic propositions are founded on the principle
of contradiction. According to Leibniz, formulae such as “2 + 2 = 4”
could be demonstrated from de¬nitions of numbers and the analytic
axiom “If equals be substituted for equals, the equality remains.”25
25 Leibniz, second letter to Clarke, Leibniz“Clarke Correspondence, 5. Also the New Essays, book
IV, chapter 7, p. 413. My discussion here relies heavily on Brittan, Kant™s Theory of Science,
43“67.
The Transcendental Aesthetic 69
Leibniz thus held that mathematical truths could be reduced to logical
truths and de¬nitions. Early in the twentieth century both Gottlob
Frege and Bertrand Russell actually attempted the reduction. Frege
never doubted that geometry is synthetic, but he hoped to show
that arithmetic could be reduced to general logical laws and def-
initions.26 As Gordon Brittan explains in Kant™s Theory of Science,
the program would have two steps: the ¬rst would reduce differ-
ent branches of mathematics to arithmetic, and the second would
reduce arithmetic to logic. In their Principia Mathematica, Russell
and Whitehead attempted the second step by giving logical de¬ni-
tions of the arithmetical terms appearing in the ¬ve Peano axioms at
the basis of arithmetic, namely:
A.1: 0 is a number.
A.2: The successor of any number is a number.
A.3: No two numbers have the same successor.
A.4: 0 is not the successor of any number.
A.5: If P is a predicate true of 0, and whenever P is true of a number
n, it is also true of the successor of n, then P is true of every
number.27
The notions needing de¬ning are “0,” “is a number,” and “is the
successor of.” If this could be done successfully in set-theoretic terms,
then presumably all the properties of integers could be derived by
logical proof. Since Frege de¬ned analytic truths as those based on
general laws of logic and de¬nitions, a successful reduction would
show arithmetic to be analytic in his sense.
Now the reduction failed because of Russell™s famous discovery of
the paradox of set theory. As Russell showed, a contradiction arises
concerning the concept “is not a member of itself.” If we have the
class of all such things “ classes that are not members of themselves “
and we ask whether that class is or is not a member of itself, either
way a contradiction arises. If the class is a member of itself, then it
satis¬es the condition of members, so it is not a member of itself.
If it is not a member of itself, then it satis¬es the condition, so
it is a member of itself. Although Russell developed the theory of
types to avoid the paradox, it led Frege to give up his view that

26 See Frege, The Foundations of Arithmetic, 19“20.
27 See Brittan, Kant™s Theory of Science, 48.
The Transcendental Aesthetic
70
arithmetic is analytic. Ultimately he came to the conclusion that
the basis of all mathematics is geometry, which he believed to be
synthetic.28
Independently of the logical paradox, however, the attempt to
de¬ne arithmetical notions in set-theoretic terms would not have
convinced Kant that arithmetic is analytic. This is because, as Brit-
tan points out, the Zermelo“Fraenkel axiomatization of set theory
includes two existential assumptions: the axiom that there exists a
null set (null set axiom), and the axiom that there exists a set contain-
ing at least all natural numbers (axiom of in¬nity).29 Absent these
assumptions one cannot derive all of arithmetic. But for Kant all
existential judgments must be synthetic. In criticizing the ontological
argument he says, “in all fairness you must [concede], that every exis-
tential proposition is synthetic” (A598/B626). These considerations,
then, lend support to Kant™s view that arithmetic is synthetic in his
sense.

C. Is geometry synthetic a priori?
Kant™s view of geometry is less controversial than his view of arith-
metic. From Euclid up to the nineteenth century, philosophers gen-
erally regarded Euclid™s postulates as universally and necessarily true,
but not based on laws of formal logic. With the development of non-
Euclidean geometries in the nineteenth century by N. I. Lobachevsky
and G. F. B. Riemann, it became apparent that the ¬fth postulate of
Euclidean geometry is independent of the others, and thus can be
denied without contradiction. In Lobachevsky™s geometry, this entails
that through a point not on a given line, more than one line can be
drawn parallel to the given line, as well as that the sum of angles of
a triangle is always less than two right angles. Riemann™s geometry
denied both Euclid™s ¬fth postulate and the assumption that a straight
line can be extended to any length. In this geometry space is ¬nite;
through a point not on a given straight line, no straight line can be
drawn parallel to the given line, and the sum of angles of a triangle is
greater than two right angles. When later developments proved that

28 See Brittan, Kant™s Theory of Science, n. 40, 59, for the source.
29 Brittan, Kant™s Theory of Science, 58“9.
The Transcendental Aesthetic 71
both geometries are formally consistent, the question arose: which
geometry is true of our space?
Although the development of non-Euclidean geometries supports
the synthetic nature of geometry, following Hilbert, philosophers dis-

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