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being” of the Antinomies, unlike God, are located in the sensible
world.
Kant adopts a “skeptical” resolution of the Antinomies, as opposed
to the “critical” solution of the Paralogisms. The antinomial disputes
draw apparently contradictory conclusions. The skeptical method
resolves the debates by showing that the con¬‚ict is “dialectical,”
that the conclusions are only apparent contradictories. Following the
distinction between mathematical and dynamical categories, Kant
adopts one mode of resolution for the ¬rst two “mathematical” Anti-
nomies and another for the “dynamical” Antinomies. For the former,
Kant adopts a “both false” solution, claiming that the conclusions are
actually contraries. For the latter his solution takes a modi¬ed “both
true” position, with the thesis possibly true of things in themselves,
and the antithesis necessarily true of appearances.
Given this skeptical resolution, Kant claims that the ¬rst two Anti-
nomies yield indirect support for transcendental idealism. He could
not do this with the Paralogisms, because the critical method assumes
that things in themselves or objects in general are unknowable. In the
Antinomies, Kant argues that if transcendental realism were true, then
the disjunctions at issue would have to be true. That is, the world of
appearances must be either in¬nite or ¬nite in space and time, and
the real would have to be either in¬nitely divisible or composed of
ultimate indivisible parts. From the realist standpoint the conclusions
are clearly contradictories and must have opposing truth values. But
Kant also believes the con¬‚icting conclusions are each supported by
a valid argument. Thus contradictory conclusions would apply to
the world in itself. Reasoning by modus tollens, since no object can
have contradictory properties, the world of appearance cannot be the
world in itself. Hence transcendental realism is false, and transcen-
dental idealism is true.
Some commentators reject this reasoning, interpreting the argu-
ments as depending on “veri¬cationist” claims concerning what can
be known.7 If this were correct, the arguments could not support
7 For commentators reading at least some of Kant™s arguments this way see Strawson, The
Bounds of Sense, 155“61 and 200; Posy, “Dancing to the Antinomy,” 83ff; Allison, Kant™s
Transcendental Idealism, 46“7, 312“13; and Guyer, Kant and the Claims of Knowledge, 407. I
address this issue below.
Transcendental illusion II 231
transcendental idealism, for the following reason. If the premises
concerned only what we could know about appearances, then they
could not establish what the conditions of appearance must in fact be.
The conclusions, then, would not be realist in nature, but only epis-
temological. Now inherent contradictions in reason could support
transcendental idealism only if they follow from realist claims. From
Kant™s point of view, the arguments cannot be veri¬cationist in nature.
A veri¬cationist reading appears plausible because the premises
refer to the empirical regress involved in synthesis. And we have seen
that Kant™s theory of synthesis explains cognition of spatiotemporal
objects. I agree with several commentators, however, that this is mis-
leading.8 I will argue that claims about synthesis here refer to the
intellectual procedure for thinking the world-whole, which presup-
poses Kant™s distinction between “analytic” and “synthetic” wholes.
Kant believes realists must assume that things in themselves are syn-
thetic wholes, composed of independently existing parts. If statements
about synthesis refer to manner of thinking the totality of appearances
rather than knowing them, then the arguments are not veri¬cationist
in nature.

2 . the a rgu ments of th e a nt in om ie s
The Antinomies concern what must be true of the world of appear-
ances as a whole in itself. Transcendental realists assume that this
world is “given” or exists independently of the process of knowing or
thinking it. All the arguments employ the reductio method, claiming
that the truth of the opposing view leads to a contradiction. This
is an effective way of highlighting the internal con¬‚ict of reason. As
Sebastian Gardner points out, if transcendental realism were true,
exactly one of the contradictory conclusions must be true. But since
both arguments are valid, even if we knew that a thesis were true, we
could not see “how it is possible for the antithesis to be false.”9 Here
I follow Kant™s order, discussing the arguments ¬rst and then their
resolutions.
8 Commentators who reject the veri¬cationist reading include Melnick, Space, Time, and
Thought, Grier, Kant™s Doctrine of Transcendental Illusion, and Gardner, Routledge Philosophy
Guidebook to Kant.
9 See Gardner, Routledge Philosophy Guidebook to Kant, 251.
Transcendental illusion II
232
A. The First Antinomy: the composition of the world in time and space
The First Antinomy concerns whether the world is ¬nite or in¬nite
in time and space. The thesis argues for ¬nitude: that there was a ¬rst
state of the world in in¬nite time, and that the world is bounded in
in¬nite space. The antithesis denies both conclusions, maintaining
that the world extends in¬nitely in both time and space. Kant offers
separate arguments on each side for the temporal and spatial nature
of the world. Here are the thesis arguments.
Thesis: “The world has a beginning in time, and in space it is also
enclosed in boundaries” (A426/B454). The ¬rst paragraph argues for
the ¬rst part as follows:
1. Assume the contradictory, that the world has no beginning in time.
2. By hypothesis, at any given time “an eternity has elapsed, and hence
an in¬nite series of states of things in the world . . . has passed.”
3. The idea of an in¬nite series is the idea of a succession that cannot
be completed.
4. By 3, “an in¬nitely elapsed world-series is impossible . . .”
5. By 4, the series of past states of the world in time must be ¬nite.
6. Therefore, by 1, 2, and 5, “a beginning of the world is a necessary
condition of its existence.”
The argument that the world is ¬nite in space follows in the next
paragraph at A427“8/B455“6:
1. Assume the contradictory, that the world is “an in¬nite given whole
of simultaneously existing things” in space.
2. The only way to think “the magnitude of a quantum that is not
given” as bounded in intuition is “through the completed synthesis,
or through the repeated addition of units to each other.”
3. By 2, to think the whole world ¬lling space would require com-
pleting “the successive synthesis of the parts of an in¬nite world,”
which entails that “an in¬nite time would have to be regarded as
having elapsed.”
4. But it is impossible to think of an in¬nite time as having elapsed.
5. Therefore, by 3 and 4, “an in¬nite aggregate of actual things cannot
be regarded as a given whole, hence cannot be regarded as given
simultaneously.”
6. “Consequently, a world is not in¬nite in its extension in space,
but is rather enclosed within its boundaries.”
Transcendental illusion II 233
Clearly the second argument incorporates the ¬rst by translating the
idea of a spatial whole in terms of a temporal series. Both arguments
reject an actually in¬nite world-whole because the idea of a completed
in¬nite series is impossible. Contrary to some interpreters, the alleged
impossibility is not psychological (nor epistemological) but logical.10
The main questions are why an in¬nite spatial world must be thought
through an in¬nite temporal series, and why an in¬nite completed
series is logically impossible. Answers to these questions will explain
why this argument does not apply to Kant™s theory of space and time,
as well as responding to some objections to the argument.
Kant explains his notion of the in¬nite by contrasting his “true
(transcendental) concept of in¬nity” (A432/B460) with the dog-
matist™s “defective concept of the in¬nity of a given magnitude”
(A430/B458). According to the latter, “a magnitude is in¬nite if none
greater than it . . . is possible.” That is, the defective concept of in¬n-
ity represents a maximally great magnitude. But because there is no
limit to the addition of units, there is no greatest multiplicity. The
true or mathematical notion of in¬nity, by contrast, “thinks only of
the relation to an arbitrarily assumed unit, in respect of which it is
greater than any number” (A432/B460). Unlike the defective idea
of a maximum magnitude, the true notion is of a magnitude that
surpasses any ¬nite number. Now this (true) notion is not in itself
logically incoherent. The contradiction in the notion of an in¬nite
whole arises only when it is represented as a completed series: “The
true (transcendental) concept of in¬nity is that the successive synthe-
sis of unity in the traversal of a quantum can never be completed”
(A432/B460). That is, the successive enumeration of an in¬nite series
(such as the natural numbers) can never be completed, because no
matter where one stops, there is always an additional member of the
series to be thought.
As Melnick points out “Kant is here defending the concept of an
actually in¬nite whole (= a whole encompassed by units only if these
units are together ˜greater than all number™).”11 He notes that Kant
made the same distinction between true and defective notions in the
Inaugural Dissertation of 1770. There, in a footnote in section 1,
10 Two commentators who take the impossibility as epistemological or psychological are Guyer,
Kant and the Claims of Knowledge, 407, and Kemp Smith, Commentary, 485.
11 Melnick, Space, Time, and Thought in Kant, 331. I am heavily indebted to Melnick™s inter-
pretation of the mathematical Antinomies.
Transcendental illusion II
234
Kant says a non-human understanding “might distinctly apprehend
a multiplicity at a single glance, without the successive application of
a measure.”12 In short, there is nothing inherently contradictory in
the idea of an actual in¬nite whole; the contradiction is in the idea
of a completed in¬nite series.
Since time is by its nature successive, the true notion of the in¬nite
entails that an in¬nite series of times cannot be thought as completed.
But it is not clear why the spatial parts of the world-whole must be
represented successively. In the last paragraph of the remark, Kant
says to think the totality of a simultaneous in¬nite extension, where
the boundaries are not given in intuition, requires having a concept
that “must establish the possibility of a whole through the successive
synthesis of the parts. Now since this synthesis has to constitute a
series that is never to be completed, one can never think a totality
prior to it and thus also through it” (A432/B460). Here Kant claims
that the idea of a whole composed of parts could arise in only two
ways: either through intuition or through thinking its relation to
its parts. Because we cannot intuit the whole of appearances, that
leaves only the second option, representing the whole by thinking its
relation to the parts. Further, Kant assumes that the thought of the
world in itself must represent the whole as composed of previously
given parts. In Kant™s terms, the world-whole in itself is a synthetic
whole, a totum syntheticum rather than an analytic whole, a totum
analyticum. As Allison puts it, “the concept of a totum syntheticum
is here operationally de¬ned in terms of the intellectual procedure
through which it is conceived . . . The problem, then, is that the
rule or procedure for thinking a totum syntheticum clashes with the
rule or procedure for thinking an in¬nite quantity.”13 Nevertheless,
Allison questions why it is necessary to conceive the series of states of
the universe as a totum syntheticum.
Melnick defends Kant by emphasizing the realist view of the mark-
ing procedures for synthetic wholes.14 Regardless of how the parts are
12 On the Form and Principles of the Sensible and Intelligible World, Theoretical Philosophy,
1755“1770, 379.
13 Other commentators refer to Kant™s distinction between a totum syntheticum and a totum
analyticum. See Kemp Smith, Commentary, 94“7, and Al-Azm, The Origins of Kant™s Argu-
ments in the Antinomies, 11. Allison locates the original terminology in Erdmann, Re¬‚exionen,
393. See Allison, Kant™s Transcendental Idealism, 43 and 338n26.
14 Melnick™s main discussion of the First Antinomy is in chapter 2 of Space, Time, and Thought
in Kant, 329“53.
Transcendental illusion II 235
individuated, a transcendental realist could not conceive the totality
of either temporal or spatial parts of the world as an analytic whole.
Analytic wholes are those in which the whole is prior to the parts. This
means the parts have no real existence in themselves independently of
the whole, but come into existence (as parts) only as constructed by the
marking process. The Aesthetic showed that our space and time are
analytic wholes. Now consider what it would mean to claim that
the world is an analytic whole. For temporal states, the existence of
each state of the world would depend on the existence of all the oth-
ers, entailing that the present depends also on the future as well as the
past. For spatial parts this means, similarly, that no particle of matter
could exist without the existence of all particles of matter. It is hard
to see how a realist could defend such a conception of the world.
Melnick argues that transcendental realists must accept Kant™s view
that the world-whole in space and time is a synthetic whole, precisely
because they represent spatiotemporal things as self-subsistent “by
representing them as all there to be met with by our procedures.”15
Because the world is there to be encountered by the subject, the parts
must be thought as given independently of the constructive process. Since
humans do not intuit the world as a whole, Kant seems justi¬ed in
claiming that the idea of the world-whole arises by synthesis of the
parts. Now since all synthesis is successive for humans, whether the
parts exist successively or simultaneously, the thought of its compo-
sition requires a successive synthesis. And since representing an in¬-
nite series as completed is impossible, “this completion, hence also
its concept, is impossible” (A432/B460). Thus the thesis argument
concludes: “Therefore an in¬nite given magnitude, and hence also
an in¬nite world (regarding either the past series or extension), is
impossible” (emphasis mine; A430/B458).
This reading highlights both strengths and weaknesses of Kant™s
argument. The weak points are the theoretical assumptions that the
idea of the whole must arise by synthesis from the parts, and that
human synthetic thought is successive. On the other hand, this inter-
pretation shows the argument to escape some standard criticisms.
Allison discusses several common objections by different commen-
tators.16 Bertrand Russell raises two of them, one to Kant™s concept

15 Space, Time, and Thought in Kant, 322.
16 See Allison, Kant™s Transcendental Idealism, 40“5.
Transcendental illusion II
236
of in¬nity, and the other to the argument itself. First Russell objects
to introducing the notion of synthesis in the idea of in¬nity, since
by the Cantorian mathematical concept of in¬nity, “classes which are
in¬nite are given all at once by the de¬ning property of their mem-
bers.”17 This objection is misguided, however, for as we have seen,
Kant claims not that the concept of the in¬nite requires synthesis, but
rather that thinking of an in¬nite whole made of given parts requires
a synthesis. Kant has no objection to the mathematical concept of an
in¬nite set of members.
Strawson shares Russell™s second objection to the idea that an in¬-
nite series cannot be completed. Russell says, “all that [Kant] has even
conceivably a right to say is that it cannot be completed in a ¬nite
time. Thus what he really proves is, at most, that if the world had
no beginning, it must have already existed for an in¬nite time.”18
That is, either the world begins in time or it has existed in¬nitely.
If Kant is entitled to assume only that an in¬nite series cannot be
completed in a ¬nite time, then the argument proves, contrary to
its purpose, that the ¬rst alternative is false, and that the world is
in¬nite in time. Both Russell and Strawson apparently assume that it
makes sense to say that an in¬nite series can be “completed” in an
in¬nite time. But if the idea of an in¬nite is the idea of a number
greater than any ¬nite number, then, regardless of the time allot-
ted, it appears no such series could be thought (successively) as a
completed whole.19 Both objections miss the target as interpreted
here.

Antithesis: “The world has no beginning and no bounds in space, but
is in¬nite with regard to both time and space” (A427/B455). Again
Kant offers separate proofs for time and space. The ¬rst paragraph
contains the argument for time:

17 Russell, Our Knowledge of the External World, 123.
18 Russell, Our Knowledge of the External World, 123, and Strawson, The Bounds of Sense, 176.
19 The same response can be made to a related objection by G. E. Moore and Jonathan Bennett,
that Kant wrongly inferred the impossibility of an in¬nite series with one bound from the
impossibility of an in¬nite series bounded at both ends. Allison points out the irrelevance,
since Kant “does not claim that a series cannot be in¬nite if it has one end . . . His point
is rather that since, as in¬nite, the series has only one end, it cannot constitute a totality”
(44). The issue is whether it is possible to think a totality as composed of an in¬nity of parts
through a successive synthesis.
Transcendental illusion II 237
1. Suppose the world has a beginning in (in¬nite) time.
2. “Since the beginning is an existence preceded by a time in which
the thing is not, there must be a preceding time in which the world
was not, i.e., an empty time.”
3. But time is homogeneous: no part of time has “any distinguishing
condition of its existence rather than its non-existence.”
4. By 3, “no arising of any sort of thing is possible in an empty time.”
5. “Thus many series of things may begin in the world, but the
world itself cannot have any beginning, and so in past time it is
in¬nite.”
In this case, the antithesis argument for space appears to differ from
the argument for time:
1. Assume the opposite, “namely that the world is ¬nite and bounded
in space; then it exists in an empty space, which is not bounded.”
2. By 1, there would be not only relations “between things in space,
but also a relation of things to space.”
3. But “the world is an absolute whole, besides which there is encoun-
tered no object of intuition,” and therefore nothing else to which
the world could be related.
4. Hence “the relation of the world to empty space would be a relation
of the world to no object. Such a relation, however, and hence
also the boundedness of the world by empty space, is nothing.”
5. Therefore “the world is not bounded at all in space, i.e., in its
extension it is in¬nite.”
These proofs are aimed against the view that the world is ¬nite in
absolute time and absolute space. On this conception, the world
would begin at a time preceded by (in¬nite) empty time, and be
surrounded by (in¬nite) empty space. Both proofs argue that these
are incoherent conceptions. The ¬rst proof explicitly invokes the
Principle of Suf¬cient Reason, claiming that there is no suf¬cient
basis in time for the world to begin at one particular moment rather
than another. Clearly this same argument applies to space: absolute
space provides no suf¬cient basis for locating the world in one region
rather than another. But if the world were temporally and spatially
¬nite, then it would occupy determinate regions of absolute time and
absolute space.
Transcendental illusion II
238
The second argument makes a different point concerning space: if
the world is ¬nite in absolute space, then it has what Melnick calls
“multiple situatability.”20 That is, if it is bounded at S1 , then it has the
determinate relation of being 10 feet away from some empty space
S2 distinct from S1 , and so on for all its relations to all other empty
spaces. Because space is homogeneous, there is no possible way to
think the difference in that situation, and one in which the world is
shifted exactly 20 feet away from S2 . Yet for the absolutist they must be
distinct. Kant puts the point in terms of the correlates of the relation:
because the world encompasses all that is real in space, there is nothing
real against the world on which to ground its determinate relation
to space. “Such a relation, however, and hence also the boundedness
of the world by empty space, is nothing” (A429/B457). Therefore,
“the world is not bounded at all in space, i.e., in its extension it is
in¬nite.” If these determinate locations are not thinkable, then the
realist conception of a ¬nite world is incoherent.
In his remark on the antithesis Kant considers an alternative
¬nitism, based on a relational theory of space and time. This is the
Leibnizian view discussed earlier in chapter 3, according to which
space and time are not independent of the real, but are derived from
the relations among real things. As Kant explains, a relationist must
think of a ¬nite world abstracted from spatial and temporal limits,
since the boundaries of the world precede its “location” in space and
time: “instead of a ¬rst beginning . . . one thinks of an existence in
general that presupposes no other condition in the world, rather
than the boundary of extension one thinks of the limits of the world-
whole, and thus one gets time and space out of the way” (A433/B461).
In thinking the limits of the world as non-temporal and non-spatial,
the relationist must think “surreptitiously of who knows what intelli-
gible world in place of a world of sense.” But cosmology concerns the
nature of appearances in space and time. Thus a mundus intelligibilis
“is nothing but the concept of a world in general, and in regard to
which, consequently, no synthetic proposition at all, whether af¬r-
mative or negative, is possible” (A433/B461).
These antithesis arguments, based on the Principle of Suf¬cient
Reason, appear stronger than the thesis arguments. First, both

20 Melnick discusses the antithesis of the First Antinomy at Space, Time, and Thought in Kant,
329“44; the discussion of “multiple situatability” begins at 335.
Transcendental illusion II 239
Leibniz and Clarke accept this principle.21 And second, since the
unconditioned is the set of conditions jointly necessary and suf¬cient
for the given, accepting the demand of reason implicitly commits one
to some version of the Principle of Suf¬cient Reason. The proofs also
infer the impossibility of a bounded world from the impossibility of
thinking its location in absolute space and time. Without some logi-
cal basis for giving the world a determinate location in absolute space
and time, the realist would be hard pressed to reject the antithesis
arguments.

B. The Second Antinomy: the nature of substance
Thesis: “Every composite substance in the world consists of simple
parts, and nothing exists anywhere except the simple or what is com-
posed of simples” (A434“6/B462“4).
The ¬rst paragraph contains the argument for the thesis:
1. Assume the opposite, that “composite substances do not consist of
simple parts.”
2. By 1, “if all composition is removed in thought, no composite part,
and (since there are no simple parts) no simple part, thus nothing
at all would be left over.”
3. If nothing at all would be left over, “no substance would be given.”
4. Implied: substance is given.
5. Therefore, either (a) “it is impossible to remove all composition in
thought” or (b) “after its removal something must be left over that
subsists without any composition, i.e., the simple.”
6. For substances, “composition is only a contingent relation, apart
from which, as beings persisting by themselves, they must subsist”
(A435“6/B463“4).
7. Therefore for substances it must be possible to remove all compo-
sition in thought: “the composite would once again not consist of
substances.”
8. By 6 and 7, (a) is impossible.
9. By 5 and 8 it follows that “what is a substantial composite in the
world consists of simple parts.”
In the remark on this argument Kant points out that the conclusion
applies neither to space and time, nor to accidents of substances. First,
21 For Leibniz and Clarke, see Al-Azm, The Origins of Kant™s Arguments in the Antinomies, 30“5.
Transcendental illusion II
240
space and time are not substances. Second, as the Aesthetic showed,
although they are composed of parts, the “parts are possible only in the
whole” (A438/B466). Here he classi¬es them as ideal as opposed to real
composites.22 For space and time “if I remove all composition from it,
then nothing, not even a point, might be left over; for a point is pos-
sible only as the boundary of a space (hence of a composite)” (A438“
40/B466“8). The conclusion also does not apply to states or accidents
of substances, since they “do not subsist by themselves” (A440/B468).
This argument is based on conceiving a substance as a self-
subsistent being and, I suspect, on the view that relations are based
on non-relational properties of things. Despite the various theories of
substance, substances were generally conceived as independent enti-
ties. This implies that where a being is composed of substances, the
existence of the composite depends on the existence of the parts. This

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