10.13135/2036-542X/8140



The paper considers Kant’s presentation of the antimony as an indirect proof of transcendental idealism. By examining the two mathematical antinomic conflicts, it purports to show that Kant’s proofs of the respective theses and antitheses turn out to be inconclusive for the alleged dogmatist. Not only does the latter not fall into an inevitable contradiction, he is also able to solve it in advance, as illustrated by Leibniz’s theory of the creation of the world and Kant’s pre–critical physical monadology. Since the source of the antinomy, it is claimed, lies rather in the conflicting requirements for the possibility of experience as exposed by Kant, it is the transcendental idealist who is haunted by an inherent contradiction. This has, it is argued, important consequences for Kant’s system, as it invalidates the principle of complete determination and in general leads to a peculiar top–down gappy ontology



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                                                            DOI 10.4399/978882552179511 – pag. 169-190 (dicembre 2018)




                     Exit from the self–incurred contradiction
                        On Kant’s mathematical antinomy and its consequences
                                                                                        Zdravko Kobe*




Abstract: The paper considers Kant’s presentation of the antimony as an indirect proof
of transcendental idealism. By examining the two mathematical antinomic conﬂicts, it
purports to show that Kant’s proofs of the respective theses and antitheses turn out to be
inconclusive for the alleged dogmatist. Not only does the latter not fall into an inevitable
contradiction, he is also able to solve it in advance, as illustrated by Leibniz’s theory of
the creation of the world and Kant’s pre–critical physical monadology. Since the source of
the antinomy, it is claimed, lies rather in the conﬂicting requirements for the possibility
of experience as exposed by Kant, it is the transcendental idealist who is haunted by an
inherent contradiction. This has, it is argued, important consequences for Kant’s system,
as it invalidates the principle of complete determination and in general leads to a peculiar
top–down gappy ontology.

Keywords: transcendental idealism, transcendental dialectics, antinomy, physical monadol-
ogy, complete determination.



The antinomy occupies a privileged place in the genesis and structure of
the Critique of Pure Reason. From the very beginning of his academic career,
Kant displayed a marked conceptual interest in the phenomenon of error.1
According to one of the two canonic accounts given by him in a late letter
to Garve, it was precisely this “scandal of ostensible contradiction of reason
with itself ” that ﬁrst aroused him from his dogmatic slumber.2 Similarly,
    ∗
        University of Ljubljana; zdravko.kobe@guest.arnes.si.
     1. See, for instance, R 3706 (1760–64?; AA 17: 242): “This does not mean to philosophize, if one
solely seeks to establish that something is a delusion, a deception of the understanding, but rather
one must also learn to have insight into how such a deception would be possible.” Kant’s writings
will be cited by volume and page number of the reference edition, the so–called Akademie–Ausgabe
(AA; see Kant 1900–), save for the Critique of Pure Reason (KrV), which will be cited according to the
second (B) and ﬁrst editions (A). The so–called Reﬂexionen will be referred to as R followed by a
number and the approximate dating according to Adickes. The translations, where available, are
taken from the Cambridge Edition.
     2. Letter to Garve, 1798, September 21, 1798 (AA 12: 257–258). See also the Letter from Hartmann,
September 1774 (AA 10: 169); R 4275 (1770–71; AA 17: 492); and, in relation to the breakthrough of ’69,
R 5037 (1774–76; AA 18: 69).


                                                                                                                169
170   Zdravko Kobe


when Mendelssohn called him “alles zermalmender Kant,” or when Hegel
deplored the disappearance of metaphysics, they both essentially referred
to Kant’s treatment of antinomy. The eﬀect in question was largely brought
about due to the skillful deployment of theatrical dramaturgy within a theo-
retical argument. For what in Kant’s view represents a general consequence
of ﬂying beyond the boundaries of possible experience — a variety of meta-
physical schools ﬁercely ﬁghting one another without ever being able to
obtain a decisive victory — was portrayed in the Critique as a natural, in
a sense even necessary product of human cognitive machinery. Moreover,
this “strange phenomenon” oﬀered Kant an opportunity to formulate an
“indirect proof ” of his system (see KrV, B534/A506). For if he can show
that reason in the end inevitably contradicts itself unless we subscribe to
the standpoint of transcendental idealism, while from his standpoint the
contradiction disappears, then transcendental idealism is not only valid, it is
the only valid philosophical system.
    In our paper, we are going to assume this point of view to investigate
Kant’s treatment of the mathematical antinomies. In a sense, we propose
to employ the skeptical method used by Kant in the antinomy chapter
and apply it to his very treatment of them. In this way, we would like
to scrutinize the validity of Kant’s indirect proof and examine the real
consequences of the antinomy for both conﬂicting doctrines.
    But before we start, we have to remind ourselves that in order to be
conclusive, Kant’s indirect argument must meet at least two requirements.
First, the opposition between the two doctrines has to be exhaustive and
well–deﬁned. Kant often implies that a dogmatist is a transcendental realist
who does not admit of the ideality of space and time, taking appearances
to be things in themselves. For our purpose, however, we are going to
understand dogmatism in a negative reference to the highest principle of
all synthetic judgments, saying that “the conditions of the possibility of
experience in general are at the same time conditions of possibility of the
objects of experience” (KrV, B197/A158). As this deﬁnition singles out Kant’s
standpoint against all previous philosophical systems, it seems to comply
better with his self–understanding. It is important to note, though, that
according to this reading, the mere ideality of space and time does not yet
suﬃce for something to pertain to transcendental idealism. What is decisive
is rather the question whether the object under consideration is, or is not,
determined the way it is independently of any cognition. In this sense, for
instance, Leibniz’s conception of space is still dogmatic: although space is
a mere epiphenomenon of relations that hold among the real, essentially
non–spatial substances, for Leibniz, this structure is completely explainable
in objective terms, without any reference to the subject of cognition.
                                                   Exit from the self–incurred contradiction        171


    And second, the argument has to be compelling for the dogmatist. In the
antinomy chapter, Kant is not addressing the transcendental idealist, who
has presumably already been convinced, he is talking to dogmatists. As a
consequence, Kant insisted that in conceiving his proofs he had “not sought
semblances in order to present . . . a lawyer’s proof ”; on the contrary, each
of these proofs was said to be “drawn from the nature of the case” (KrV, B
458/A430). He went even further in the Prolegomena, where he solemnly
declared that he would personally “vouch for the correctness of all these
proofs” (AA 4: 340), and even incited the reader to ﬁnd eventual fault in
these, as he called them, “equally evident, clear, and incontestable proofs.”3
This discursive strategy is sound, as it were. For if his proofs turned out to
be deﬁcient, if a contradiction happened to arise from some neglect in the
use of reason, he would be unable to bring his indirect point home.4 Again,
Kant is supposed to convince the dogmatist, operating on her terrain and
using her assumptions.
    This is how we propose to read Kant’s presentation of the mathematical
antinomies. In the ﬁrst step we are therefore going to look into what
the antinomic dispute between exemplar dogmatists looks like according
to Kant, and then examine the consequences thereof for transcendental
idealism. We are going to play Kant’s game, but not with Kant’s intentions.
We are going to play by his rules to see if Kant himself complied with
them. In the end, these rules boil down to one simple prescription: It is
not allowed to presuppose the validity of transcendental idealism. To play
Kant’s game is, in brief, to play by the rules of the dogmatist.


1. The ﬁrst antinomy

The ﬁrst antinomic conﬂict deals with the world (i.e., the whole that is not
itself a part) as an extensive whole; more precisely, it refers to the question
whether the material world is limited in space and time or not, whether it is
ﬁnite or inﬁnite. The thesis reads as follows:
    The world has a beginning in time, and in space it is also enclosed in boundaries.
    (KrV, B454/A426)

   The thesis consists of two parts, treating space and time separately. The
same duality is then also reflected in Kant’s formulation of proofs, as the argu-
ment for spatial boundedness relies entirely on the boundedness of events in

    3. See Prolegomena (AA 4: 341): “I therefore desire the critical reader concern himself mainly
with this antinomy . . . I promise to answer for each proof I have given of both thesis and antithesis.”
    4. See KrV, B512/A484: “Thus the dogmatic solution is not merely uncertain, but impossible.”
172      Zdravko Kobe


time, which has allegedly been proved already. By adopting such a procedure,
one might observe that the proof was “infected with transcendental philos-
ophy” from the very beginning. Be that as it may, we are going to limit our
consideration to the spatial part of the thesis. The relevant proof reads:

      For if one assumes that the world has no beginning in time, then up to every given
      in time an eternity has elapsed, and hence an inﬁnite series of states of things in
      the world, each following another, has passed away. But now the inﬁnity of a series
      consists precisely in the fact that it can never be completed through a successive
      synthesis. Therefore an inﬁnitely elapsed world–series is impossible, so a beginning
      of the world is a necessary condition of its existence. (KrV, B 454/A426)

    The argument is dogmatic indeed. It is even traditional, since it was in-
vented back in the antiquity, and was later used on a regular basis in medieval
disputes under the heading de aeternitate mundi. Its thrust is that one cannot
produce infinity by successive synthesis, in the mode of counting, while by
affirming that the world is eternal we say precisely this. To illustrate the point,
let us consider counting the Earth’s revolutions around the Sun (or, rather, the
Sun’s revolutions around the Earth); if the world was eternal, then by now we
would have come to an infinite number, for in the presupposed infinite past
the Earth would have already made infinitely many revolutions.
    This is at least how the traditional argument goes, and Kant is here
content to simply repeat it. What should surprise us, however, is that this
old problem has long had its more or less generally recognized solution.5
It is true that in the event of the eternity of the world we should by now
have counted to inﬁnity; this is, let us assume, contradictory. However, the
problem lies not in eternity, but in the very idea of counting. This idea is
inappropriate insofar as it assumes that somewhere, at a certain point in the
past, we began to count. It therefore assumes that a certain revolution was
ﬁrst; it assumes that at some point the Earth began to revolve; it presupposes,
in short, that the world has a beginning in time. But since this is exactly
what we wanted to prove, the argument is circular.6 Without this arbitrary
assumption, the only thing Kant could prove with this argument is that if
the world has no beginning in time, then it has always already existed for
an inﬁnite time, as was noted for example by Russell (see 2009: 126–127),
to name but one. No matter how far back into the past we go, we will
inevitably ﬁnd that for every ﬁnite past there is always an inﬁnite past lying
before it. And this is deﬁnitely no contradiction.

     5. For a short presentation of some major positions in the dispute, for instance those of Philo-
ponus or St Thomas, see Wood (2010).
    6. This is also Hegel’s objection in the Science of Logic; see GW 21: 229. Interestingly, Hegel
posits the presupposed limit not at the beginning of time, but rather at its end, at the present time.
“But,” Hegel adds, “this diﬀerence is inessential.”
                                                       Exit from the self–incurred contradiction             173


    But if, from the dogmatist’s standpoint, Kant’s argument is evidently invalid,
we have to ask ourselves what made Kant formulate it in the first place.
How was his error possible? The reason is, perhaps, that the eternity of the
world relates not only to the problem of infinity but also to that of the whole.
The concept of the world is the concept of the whole. It is well known,
however, that for Kant there were two types of the whole: totum analitycum
and totum syntheticum, or totum and compositum. In the first case, the whole
is a condition of its parts, which can exist only within it; in the second, the
whole is constituted through a gradual aggregation of pre–existing parts7. If,
therefore, in the analytic whole the idea of infinity is readily acceptable since
its parts are given together with it regardless of whether their number is finite
or infinite, the synthetic whole, on the contrary, cannot be infinite because
infinity cannot be produced through a gradual synthesis.
    The question is, then, why Kant treats the temporal extension of the world
as a synthetic whole. For Kant, space and time are rather all–encompassing
analytic wholes, and in the Transcendental Aesthetic he accordingly explicitly
described them as infinite.8 In the Transcendental Analytic, however, Kant
presented it as an axiom of intuition that all appearances, that is, all materially
occupied parts of this infinitely extended intuitive space, are for their part
extensive magnitudes generated through a successive synthesis of their parts.9
For Kant, every cognized object in space and — because of the conceptual
parallelism — time consequently constitutes a synthetic whole.
    Thus we can imagine that Kant considered the world as a synthetic
whole on the basis of the requirements developed in the Transcendental
Analytic. Indeed, it turns out that for Kant the idea of an inﬁnite past series is
impossible, since according to his view time in a sense runs backwards. To see
why this is so, it suﬃces to take a brief look at Kant’s treatment of actuality, or
Wirklichkeit. Although it does imply a reference to perception, a thing does
not need to be perceived directly in order to be considered actual. Following
the second postulate of empirical thought, Kant declares that “everything
is actual that stands in one context with a perception in accordance with
the laws of empirical progression” (KrV, B521/A493). This subtle distinction
proves to be relevant for those things in particular that, due to the speciﬁc
constitution of our senses, cannot be objects of immediate experience, such
as “magnetic matter” (KrV, B273/A226), things that are simply located too
far from us to be perceived immediately, such as the “inhabitants of the

     7. See, for instance, R 3789 (1764–66; AA 17: 293). Kant adds: “Spatium and tempus are tota analytica,
bodies synthetica”.
     8. “Space is represented as an infinite given magnitude . . . (for all the parts of space, even to infinity,
are simultaneous).” (KrV, B39–40)
     9. “Every appearance as intuition is an extensive magnitude, as it can be only cognized trough a
successive synthesis (from part to part) in apprehension.” (KrV, B203/A163)
174       Zdravko Kobe


moon” (KrV, B521/A492), and ﬁnally, and most interestingly, “the real things
in the past” (KrV, B523/A495), so dear to Meillassoux, which existed in a
past perhaps so remote that no ﬁnite rational subject could have possibly
observed them directly. In this respect, Kant claims that those events are
nonetheless actual if only I can “represent to myself that, in accordance with
empirical laws, or in other words, the course of the world, a regressive series
of possible perceptions. . . leads to a time–series that has elapsed” (ibid.).
That is to say, for Kant, the past can be perfectly actual as a reconstructed past,
a past that in a sense runs backwards, “starting with the present perception,
upward to the conditions that determine it in time” (ibid.). However, there
is a limit to the existence of such a past: since the reconstructive synthesis
can never reach to inﬁnity, no inﬁnite past is possible. In Kant, therefore,
there is no inﬁnitely remote past.
     It turns out that for Kant the proof of the thesis is thus conclusive after
all. But this only makes it clear that, in contravention of the rules of the
game, it holds only on condition that one accepts transcendental idealism.
     The relevant antithesis reads: «The world has no beginning and no bounds
in space, but is infinite with regard to both time and space». (KrV, B455/A427).
In accordance with the apagogical method, the proof of the temporal part
starts by assuming the opposite, that is, that the world did begin at a certain
moment in time. The argument rests on the contention that, following the
definition of “beginning,” this would imply the existence of an empty time
before the beginning of the world. Surprisingly enough, Kant does not seem
to object to it, he merely adds that in this event the world would not have
emerged at all. In such an empty time, all temporal parts would be completely
alike, none having any specific mark that would distinguish it from the others,
which means that no single moment could provide a reason why the world
should begin in one specific moment rather than another. It follows that either
the world has always already existed or else it could have not begun to exist at
all. But the world does exist, and consequently it must have existed infinitely.
     Once again, however, this proof does not have the required argumenta-
tive force. We may start by objecting that Kant’s deﬁnition of the beginning
as an event that is preceded by a time in which something was not, already
seems to preclude the answer. Since the world is the total sum of all things,
not just some parts of the world, the real question should rather be whether
it is possible to speak of the existence of time before the beginning of the world
at all. In this respect, Leibniz would simply aﬃrm that the world was cre-
ated in the ﬁrst moment of time, that is, that it was only with the existence
of the world that time started to exist, too. In this sense, the proof is not
apagogical, but simply aﬃrms what it is supposed to prove.10

      10. This is also Hegel’s assessment in the Science of Logic; see GW 21: 231.
                                                   Exit from the self–incurred contradiction        175


    In addition, one might directly challenge Kant’s objection against the
possibility of the arising in an empty time. It is true that if we assume the
existence of absolute time, no distinctive feature would provide a suﬃcient
reason why the world should begin in one moment rather than another.
However, even Kant did not consider to be an analytical truth that every
event has a reason or a cause; and besides, a follower of Newton could
always claim that the ultimate reason lies in the will of God, who, as a
matter of fact and for no reason, decided to create the world in that very
moment. Once again, the proof would clearly fail to impress the dogmatist.
    It seems, on the other hand, that it is Kant who is obliged to think of a
temporal event in a way that would make the beginning of the world con-
tradictory. In eﬀect, in the ﬁrst analogy of experience, Kant demonstrated
that, as a necessary condition of possible experience, the very notion of
change presupposes the identity of what changes,11 and that, consequently,
there must be something in the world that persists through all time. This
something that persists, usually called substance, thus provides a suﬃcient
proof that Kant’s world could not have had a beginning. In a similar way, in
the second analogy, Kant demonstrated that each event receives its temporal
location only in relation to some preceding event, according to the law of
causal connection. It is for him that no event becomes part of the world
without being determined by its logically and temporally preceding cause.
But again, since for Kant this is something that can be established only with
transcendental idealism, it is quite clear that it will not convince dogmatists.
    In the end, our brief examination of the ﬁrst antinomic conﬂict leads to
a result that is in complete disagreement with Kant’s initial assurances. The
relevant proofs are apagogical in appearance only; they are unable to estab-
lish that the dogmatist will inevitably contradict himself, and they are not
necessarily conclusive for the dogmatist either. These are strong statements.
It is therefore fortunate that we can provide them with additional support
from history. Incidentally, Kant’s presentation can be read as an a priori
reconstruction of a confrontation that actually took place between Clarke
and Leibniz. In their famous exchange of letters they not only pretended to
use the apagogical method, on closer examination it soon becomes evident
that they actually argued past each other rather then against each other,
reading one’s own presuppositions into the other’s theory, especially with



    11. “Arising and perishing are not alterations of that which arises or perishes. Alteration is a way
of existing that succeeds another way of existing of the very same object. Hence everything that is
altered is lasting, and only its state changes.” (KrV, B 230/A187) See also R LXXVIII (AA 23: 30): “All
arising and perishing is only the alteration of that which endures (the substance), and this does not
arise and perish (thus the world also does not).”
176       Zdravko Kobe


regard to space and time.12 In this way they succeeded in showing that
under such presuppositions, the other’s theory involved a contradiction; but
as for themselves, both remained perfectly consistent. For instance, being
good Christians, they both aﬃrmed that the world began at a certain time:
regarding the possibility of an empty time before the creation of the world,
which necessarily constituted a “non–entity” for Kant (KrV, B461/A433),
Clarke simply attached it to God, while Leibniz called it merely imaginary.
    It is Leibniz in particular who is of special interest to us. Not only does he
prove that the dogmatist’s position is free from contradiction, he aptly illustrates
how to think thesis and antithesis together without being forced to accept
transcendental idealism. Leibniz advocated relational theory, claiming, basically,
that space and time are abstract orders arising out of relations sustained by
actual simple substances. According to this doctrine, space and time are con-
tingent with respect to both their existence and their characteristic properties.
For instance, if the law governing the force of attraction were different, then,
perhaps, our space would have more than three dimensions.13 It has been regu-
larly argued against Leibniz that in this conception there could be no empty
space and time, which would necessarily render them materially infinite. The
objection is wrong, however (with the exception that it does hold for Kant).
While it is true that, in Leibniz, there would be no space without a plurality
of substances, the properties of the order of coexistence pertain not only to
actually existing substances, but “to possible ones as well, as if they existed.”
Something similar also holds true for time (though one–dimensionality does
complicate the matter). In Paragraph 55 of the Fifth Letter, Leibniz writes:

      But yet absolutely speaking one may conceive that an universe began sooner than
      it actually did. Let us suppose our universe or any other to be represented by the
      Figure AF, and that the ordinate AB represents its first state and that the ordinates CD
      and EF its following states; I say one may conceive that such a world began sooner
      by conceiving the figure prolonged backwards, and by adding to it SRABS. For thus,
      things being increased, time will be also increased. (Leibniz & Clarke 2000: 50)




    12. For instance, they asked each other whether God could have created the world at some other
time or in some other place—say, several feet to the left or right. For a closer examination, see Vailati
(1997).
    13. Such was at least Kant’s conjecture in his early book on the living forces; see AA 1: 24.
                                          Exit from the self–incurred contradiction   177


    Although the world was presumably created at a certain point in time, we
could, relying on the lawful regularities instantiated in the actually existing
world, nevertheless conceive of a time before the beginning of the world: to
this eﬀect, we would simply need to prolong the past backwards according
to the rules of actual temporal succession. This time would turn out to be
unreal, merely imaginary, because there really was no world and no time
before the creation; still, this imaginary past would prove to be completely
determined, marked by all the usual properties, and in general allowing
us to speak of it in exactly the same way as we speak of the really existing
time.
    By using this distinction between real and imaginary time, Leibniz
would ﬁnally bring thesis and antithesis together. He would aﬃrm that,
from an absolute point of view, the world is indeed bounded in time, but
that beyond this temporal border there is an imaginary time extending
into the inﬁnite past. To illustrate his solution, let us suppose—as some
calculations made upon the Bible seem to suggest—that the world was
actually created around 3500 BC. In this case, there was no time and no
world before that date. However, on the basis of the initial state of the world
(which includes fossils), and on the basis of the presumably stable order
governing the things of the world after 3500 BC, for us, the world would
have stretched back inﬁnitely. Researching the history of the natural and the
human world, we would thus be right to speak of human evolution before
that time, or of the formation of the Solar system. Seen from the outside,
the world would then indeed possess its absolute beginning; however, for
us, being inside, it would be impossible to observe that in going back we have,
at a certain point, crossed the border between the real and the imaginary,
and entered into the imaginary past.
    For Leibniz, there was obviously no contradiction involved here, quite
the contrary. And we can safely assume that Kant knew it. Not only was
he familiar with the Leibniz–Clarke controversy,14 a similar conception
was explicitly expounded in the booklet De aeternitate mundi impossibili,
published by Martin Knutzen, Kant’s teacher.


2. The second antinomy

The second antinomic conﬂict deals with the question whether the compos-
ite substance, or matter, is inﬁnitely divisible, that is, whether it ultimately
consists of some fundamental parts, which are not capable of further divi-
sion. The thesis will posit a bottom limit of the division, and therefore the

  14. See R 4767 (1775–77; AA 17: 700).
178      Zdravko Kobe


existence of simple substances, while the antithesis will insist on the inﬁnity
of the process of division. The thesis, then, reads:

      Every composite substance in the world consists of simple parts, and nothing exists
      anywhere except the simple or what is composed of simples. (KrV, B462/A434).

    Again, the proof starts with the apagogic assumption that composite sub-
stances do not consist of simple parts. To illustrate the absurd consequences
such a hypothesis would imply, Kant invites us to perform a thought ex-
periment. If we removed in thought all composition, he claims, then in the
absence of simples nothing would be left over. Apparently, Kant is unable
to accept such a consequence, presumably since we started with something
substantial. Therefore, he argues further, it is either impossible to remove
all composition in thought or else something simple must be left over after
its removal. But the ﬁrst horn of the dilemma is supposed to be contradic-
tory: in this case, “the composite would once again not consist of substances
(because with substances composition is only a contingent relation, apart
from which, as being persistent by themselves, they must subsist)” (ibid.).
Therefore, concludes Kant, only the second case is left: the last elements of
composite substances are simple.
    The argument is strange. Even on a sympathetic reading, it is hard to
see what constitutes its premise and what its consequence.15 Once stripped
of its—“redundant,” as Hegel would say—proof form, it seems to revolve
around two things: ﬁrst, the factual question whether the initial object,
presumably spatial matter, is to be considered substantial or not; and sec-
ond, the deﬁnitional question whether the substantiality of a composite
substance should be thought of as deriving from the substantiality of (some
of ) its parts or not. Read in this way, the alternatives should be clear and easy.
If we assume that we started with a composite substance, and furthermore
assume that, for its parts, being in relation is by deﬁnition something acci-
dental, then it necessarily follows that the last elements of such a composite
whole are simple.
    But let us ask ourselves what prevents us to accept the other outcome of
Kant’s thought experiment, i.e., that after the removal of all composition
nothing would be left over. The answer is, supposedly, that in that case
either we could not assign substantiality to the initial object or we would
have to allow for the possibility of composite substances that are essentially
relational; that is, we would either have no true substance to start with or

    15. It is often suggested that the argument consists in an arbitrary analysis of the notion of
composite substance or that it is circular. For a closer assessment, see especially Engelhard (2005:
158ﬀ ). Falkenburg (1995: 17) speaks of “assuming mutually incompatible additional suppositions
regarding composite substance ‘in the world’ that cannot be fulﬁlled in one–and–the–same–model”.
                                                   Exit from the self–incurred contradiction        179


the substance would dissolve after the removal of all relations. The problem
is, though, that both options are in principle consistent and that both were
actually defended, for instance by Leibniz. Thus he once claimed that matter
was only a phaenomenon bene fundatum, which means that the argument
concerning composite substances does not refer to it, and later, in his letters
to Des Bosses, that matter could be made into something substantial by
adding the so–called vinculum substantiale, in which case we could speak of
a substance consisting of nothing but mere relations16 .
     Even if Kant’s proof thus again fails to convince the dogmatist, it does in-
clude a relevant point. The real question is how to conceive of substantiality,
especially in relation to space and matter. It seems that, in Kant’s view, true
substantiality and spatiality exclude each other since substance is usually
understood as something that is capable of existing by itself, without any
relation to other substances, while space is essentially relational17 . Or, better
yet, Kant thought that this is the way dogmatists were bound to see the
situation, given their concept of substance. He, on the other hand, was able
to avoid the tension in question, he thought, since he disposed with the
new concept of substantia phenomenon, which comprised many features of
space, including its relational structure, yet was still supposed to function as
something persistent and therefore substantial. In this sense, it may be said
that, in the proof of the thesis, Kant actually operated with two diﬀerent
notions of composite substance,18 and that the contradiction appeared only
because he tried to impose his concept of material substance on dogma-
tists as he understood them. He thus only managed to demonstrate the
following point: had Kant been a dogmatist, he would have contradicted
himself.
     Let us now turn to the antithesis. It reads:
    No composite thing in the world19 consists of simple parts, and nowhere in it does
    there exist anything simple. (KrV, B463/A435)

   In the proof, Kant assumes the contrary, namely the existence of simple
parts, and continues:
    16. De Risi and Graber noted a close similarity between Leibniz’s late views on body and Kant’s
critical conception of material substance, so that in this respect Graber even speaks of Leibniz as
“a transcendental philosopher” (2010: 362). De Risi suggests (2007: 314) that the Leibniz–Des Bosses
correspondence was known to Kant; this, however, seems to be speculation only.
    17. “If I remove all composition from it [i.e. space], then nothing, not even a point, might be left
over” (KrV, B466/A438). See also R 5879 (1785–88; AA 18: 375) and R 5299 (1776–78; AA 18: 147).
    18. In Kant’s dilemma, “Either in substances it is impossible to remove all composition or
else after its removal something simple must remain,” both horns of the dilemma are completely
acceptable: the ﬁrst holds for the spatial substance and the second for the non–spatial one.
    19. Signiﬁcantly, in the antithesis Kant speaks of a “thing,” not of a “composite substance” as
before, thereby indicating that the context is spatial from the very beginning.
180      Zdravko Kobe


      Because every external relation between substances, hence every composition of
      them, is possible only in space, there must exist as many parts of space as there are
      parts of the composite occupying it. Now space does not consist of simple parts . . .
      (Ibid.)

    The so–called proof (we will omit the second part relating to the spirits)
consists in simply projecting the properties of space onto substances: because
space does not consist of simple parts, no substance is simple; because space is
infinitely divisible, the substance, too, is divisible to infinity. But the question
is, precisely, whether the substance is really to be taken as essentially spatial.
Once again, Kant simply asserts that which should actually be proved.
    This time, however, he is somewhat aware that he is cheating. In the
adjoining remark, he cites a serious objection, which is all too quickly (and
invalidly) rejected by reference to the senses. That is, he mentions certain
“monadists” who erroneously attempt to make the inﬁnite divisibility of
space compatible with the indivisibility of substances. “If one listens to
them,” Kant comments, “then besides mathematical points, which are
simple, but are boundaries rather than parts of space, one would have to
think of physical points too as being not only simple, but as also having,
as parts of space, the privilege of ﬁlling it” (KrV, B467/A439). To whom is
Kant referring? It would be wrong to think of Boscovich, for his physical
points were located in a preexisting space, while according to Kant, these
“monadists are subtle enough” to rather presuppose “these objects and the
dynamical relation of substances in general as the condition of the possibility
of space” (KrV, B469/A441).
    In fact, this “absurd” monadist is none other than Kant himself ! In 1756,
he published the so–called Physical Monadology, in which he tried to “marry”
metaphysics and mathematics as regards the inﬁnite divisibility of spatial
matter. And what did Kant do in 1756? He—perhaps under the inﬂuence
of Knutzen—accepted the presupposition of metaphysics that matter was
ultimately grounded in simple substances called monads, but avoided the
usual complication by distinguishing between esse in spatio and implere
spatium: “Each simple element of a body, that is to say, each monad, is not
only in space; it also ﬁlls the space, though it does not, for that reason, forfeit
its simplicity” (AA 1: 480). This is not to say that the monad is itself extended.
In Leibniz’s relational explanation, as we have seen, space arises only out of
the relations existing among a certain plurality of monads. But, under such
conditions, this non–extended substance can still have spatial eﬀects20 , it can
be present in a certain space and make it impenetrable. How?

     20. Eueler developed a subtle theory of virtual spatiality of simple substances, which he called
spirits. “Thus my soul does not exist in a certain place, but it acts in a certain place” (Euler 2003: 175).
It is another question whether Kant was right in trying to ground space in unextended elements.
                                                    Exit from the self–incurred contradiction         181


    The monad does not determine the little space of its presence by the plurality of
    its substantial parts, but by the sphere of activity, by means of which it hinders
    the things which are external to it and which are present to it on both sides from
    drawing any closer to each other. (AA 1: 480)

    Every monad is endowed with an original active force that prevents other
monads from entering into its sphera activitatis, thus limiting the sphere
of activity of other, spatially neighboring monads. To counterbalance this
repulsive force, Kant introduces the force of attraction, and even engages
in speculation on the subject of their respective intensities in relation to
distance from the central point. At the end, the consequence of it all is that
this unextended and probably also non–spatial point can still have spatial
eﬀects that make its little space impenetrable for all the other monads and
the space completely ﬁlled. And since the monad is not extended itself, only
its eﬀects are, one can in thought divide the space it ﬁlls without thereby
aﬀecting the simplicity of the monad.
    Indeed, this inﬁnite divisibility is not only imaginary or metaphysical,
but also physical, at least in principle. We may be tempted to portray
such a physical monad as a single solid ball, or as a bubble with a crusted
envelope formed at the distance where the two internal forces cancel each
other out. This picture is dangerously wrong, however, since the monad
in question is by deﬁnition in relation to other monads with forces of their
own. This means that its state is essentially dynamic, depending on the
number and strength of the forces involved, and that consequently, for any
given space that is presently ﬁlled by a certain monad, there is a greater
external force that, when applied to it, would compress it to any desired
dimension. The physical monad is fundamentally elastic. There is a limit to
this elasticity, however: “It is obvious that by no conceivable force can an
element be penetrated completely” (AA 1: 487). But that aside, according
to this conception, material space is physically divisible to inﬁnity, while
the substances still retain their simplicity. Mathematics and metaphysics are
married happily.
    Having this in mind, there seems to be no reason why the dogma-
tist would need to renounce his position. At least in the Critique of Pure
Reason, Kant provides no independent argument against physical mon-
adology—which is strange, again, since he was obviously familiar with
its strengths. It is only in the Metaphysical Foundations of Natural Science,
published in 1786, that Kant ﬁnally developed a dedicated argument against
the system in question. The initial situation here is very close to that in the

According to De Risi, Leibniz was indeed capable of showing that “space is actually constituted by
points, even though it is not composed of them” (2007: 311); yet, at least in De Risi’s view, Kant’s early
physical monadology should be “regarded as backward” compared to Leibniz.
182       Zdravko Kobe


second antinomy: in Explication 4 of the Dynamics, Kant similarly seeks to
prove that matter, as the composite substance is called here, is divisible to
inﬁnity. This time, though, he is more thorough:
      The proof of the inﬁnite divisibility of space has not yet come close to proving the
      inﬁnite divisibility of matter, if it has not been previously shown that there is a
      material substance in every part of space that is, that parts movable in themselves
      are to be found there. (AA 4: 504)

   Contrary to what he did in the proof of the second antithesis, Kant now
openly admits that in order to refute the physical monadist, one ﬁrst has to
prove that every part of space is occupied by a substance. Here, too, Kant
starts apagogically, drawing the following diagram:




    Let us then assume that A is the site of a simple substance in space; and
let us further assume that a and b are the points marking the opposite limits
of the sphere of activity of substance A.21 According to the inﬁnite divisibility
of space, Kant notes, it is possible to specify a point c somewhere on the
line between central point A and border point a, i.e., within substance A’s
sphere of activity.
      If now A resists that which strives to penetrate into a, then c must also resist the two
      points A and a. For, if this were not so, they would approach one another without
      hindrance, and thus A and a would meet at the point c, that is, the space would be
      penetrated. Therefore, there must be something at c that resists the penetration of
      A and a . . . (AA 4: 504–505)

    Hence it follows, Kant concludes, that in point c there must be a center
of force repelling monad A, and that, therefore, in this point c a monad
has to be found, too. Otherwise, monad A would not be able to resist
external monads from penetrating into its sphere of activity, and contrary
to the presupposition, its space would not be ﬁlled. That is to say, the space
of substance A is ﬁlled only if at point c there is actually another monad
present. And since the same reasoning applies to the monad at point c,
or indeed at any point in the indeﬁnitely divisible space, there must be a
material substance in any given point of space. The physical monadist is
refuted.
    21. As Engelhard (2009: 331) notes, it is unclear what the circles around the points a, b, and A
stand for. This may not be a trivial concern.
                                                      Exit from the self–incurred contradiction          183


    Is the argument sound? Commentators tend to disagree: many seem
to accept it as valid; others claim that Kant has made certain tacit presup-
positions in the process, for instance regarding the speciﬁc mode in which
repulsive forces operate;22 still others emphasize the complex structure of
argumentation in the Metaphysical Foundations, where Kant conceives matter
merely as the movable in space and insists on the relativity of motion.23 So,
how to proceed? If we simply follow the argument, we see that Kant ﬁrst
presupposes the existence of a simple, non–extended substance with point
A as its center of force, which successfully repels other substances from
penetrating into its sphere of activity. This substance thus dominates over
its little space, which, to outside observers, appears to be perfectly ﬁlled.
Inside this little space Kant now posits an arbitrary point, point c. Up to
this moment, everything is ﬁne. Now, however, Kant simply aﬃrms that
this point c performs a certain activity on its own, that it constitutes a real
center and a subject of force itself. If anything, it is this aﬃrmation that is
contradictory since it was presupposed in the beginning that substance A
ﬁlls its space and thereby prevents any other substance from entering it. This
is its space, after all, that constitutes the sphere of its activity.
    Kant makes his move on the ground that otherwise, that is, if there
were no independent substance in point c, the points A and a would come
together. But this is obviously false. We have a clear–cut dilemma: either
substance A has a repulsive force or not. If it does have it, then, according to
the hypothesis, it ﬁlls the space by itself. This space can always get bigger
or smaller, true, but no matter how big the force applied against it may
be, it will never squeeze it into an unextended point. If, on the other hand,
     22. According to Kauark–Leite (2004: 142) and Schönfeld (2000: 171), for instance, Kant assumed
that the repulsive force was a contact force acting on surfaces only. This is begging the question, it
seems.
     23. In his detailed discussion, Friedman emphasizes that since, according to the principle of
relativity of space and motion, there is no absolute reference frame to determine motion, and since
the exercise of force is considered to be instantaneous motion, “there is just as much reason to
say c is resisting a and A . . . as there is to say that A is resisting a” (Friedman 2013: 151). It may be
true, Friedman admits, that this reasoning no longer applies once the forces and causal relations are
brought into the picture, for in this case (compare AA 4: 547) the privileged frame of reference is
clearly determined by the center of mass. But, Friedman now observes, “although the central point A
is initially taken to be at rest, the proper frame of reference for considering this particular exercise of
repulsive force is actually centered on a point between A and a (and thus at the point c, for example)”
(2013: 153). It is hard to see why this is so. The conclusion would hold, it seems, if point a constituted
an independent source of repulsive force. However, since according to the initial assumption point a
marks only a limit of substance A’s sphere of activity, we have to assume that some other substance
(call it E) is present which strives to penetrate into A’s sphere at this point a (which, incidentally, could
also be called e, since it also marks the limit of E’s sphere of activity). In this case, it seems that the
natural center of mass lies in the point a = e, and to put it anywhere else, an additional argument
would be needed. Our reasoning suggests that Kant’s diagram is (deliberately?) misleading in that it
draws a circle around point a, presumably implying that it constitutes a force center of its own, that
is, a substance.
184   Zdravko Kobe


substance A has no relevant force, then no number of such subjects will
manage to ﬁll the space in question, as Kant would have it. Anyhow, it was
assumed that substance A does have the relevant force and that it does
dominate over a certain space. Consequently, no other substance can be
posited at point c within this space. Kant’s proof is inconclusive, arbitrary,
and in the end contradictory.
    As regards the second antinomy, we ﬁnally have to conclude that for the
dogmatist both thesis and antithesis once again remain standing. Leibniz
would thus deny that inﬁnitely divisible matter is anything more than a
phenomenon, or else, relying on his late theory of vinculum substantiale, he
would maintain that the composite substance called matter is essentially
relational. As for our physical monadist, he would obviously continue to
stick to his theory. In any event, all the doctrines mentioned above seem
to be internally consistent, so that the question of their truth is ﬁnally
empirical.
    The situation is diﬀerent for Kant, though. It is for him that, again, both
proofs turn out to be conclusive—at least, that is, had he been a dogmatist.
For in his constitution of an empirical world, Kant introduced a new con-
cept of substance, one that was fundamentally deﬁned in reference to the
determination of time. As such, it remained underdetermined intrinsically.
It is for instance extremely diﬃcult to tell whether in Kant’s world there
is in the last analysis only one continuous substance, as there is only one
time, or if there are inﬁnitely many discrete substances, as there are so many
parts matter can be divided into. It seems that, for various reasons, Kant
was obliged to aﬃrm both, thinking of substance as essentially relational
and essentially self–standing at the same time. Paradoxically, however, it
was not open to him to avoid the contradiction in question by referring to
the standard distinction between the world as it is in itself and its appear-
ance, so readily available to every dogmatist, because for him, the world of
appearances was in a sense the only world there was.


3. The consequences

As we have seen, Kant’s indirect proof has failed completely. Instead of
showing how dogmatism is trapped in contradiction, the antinomy has
exposed the contradictory nature of transcendental idealism itself. But, as
we know, this is not critical for Kant, since he is supposed to provide an
adequate cure for this self–inﬂicted wound.
    According to Kant, the fundamental matrix of dialectical reasoning reads:
“If the conditioned is given, then the whole series of all conditions for it is
also given” (KrV, B525/A497). The dogmatist is justiﬁed in his conclusion,
                                                    Exit from the self–incurred contradiction        185


Kant believes,24 since for her the things are given independently of any
subjective condition and are as such determined by the way they are in
themselves. In the case of the world, this means that, with every element of
it, the complete series of its conditions is given at the same time, be it ﬁnite
or inﬁnite. Here, therefore, one of the two options must be true: either
the world has a ﬁrst condition, which is unconditioned, or the uncondi-
tioned is the entire inﬁnite whole taken together. For the transcendental
idealist, however, the situation is diﬀerent. Since for her the things of the
world are mere appearances, they are never given in themselves, but ﬁrst
have to be constituted by the synthesis of cognition and are thus “given
only in this synthesis” (B527/A499). The synthesis in question is successive,
however. Although its general mechanism makes sure that with every step
reached the synthesis could always continue, it can nonetheless never attain
the inﬁnite whole. Within this framework, the world as an inﬁnite given
whole is therefore a contradiction.25 And according to the predicament of
transcendental idealism, this can only mean that there is no world: the world
simply does not exist!
     The contradictory nature of the world explains why, for Kant, the proofs
on both sides were actually valid. From contradiction, anything can be
inferred. Kant’s general solution to the problem of antinomy would thus
be that the world is a forbidden object nothing can be predicated of.26 Or
alternatively—since the world is indeed spoken of—that it is not a completely
determined object, in particular regarding its magnitude. “If it is said that
the world is either inﬁnite or ﬁnite (non–ﬁnite), then both propositions
could be false,” Kant remarks (KrV, B532/A504). Under the condition of
indeterminacy, it is no longer allowed to reason that if the world is proven
not to be ﬁnite, then it must be inﬁnite, as was the case in the antinomic
dispute.27 Kant consequently concluded, ﬁrst, that the world has no begin-
ning in time, but for every past there is another past open to our exploration,
without us ever being able to reach to inﬁnity; and second, that matter can

    24. See R 5553 (1778–79?; AA 18: 223): “The proposition that if the conditioned is given, the whole
series of all conditions through which the conditioned is determined is also given is, if I abstract
from the objects or take it merely intellectually, correct.”
    25. See R 4525 (1772–75?; AA 17: 582): “The absolute–whole in the appearance is a contradiction.”
    26. See R 5962 (1785–89; AA 18: 402): “The reason why the ﬁrst two antinomies are both false is
that I had to ground them both on a contradictory concept, namely, that of a whole in space and time
that is also supposed to be an absolute whole.”
    27. The ﬁrst antithesis, for instance, reads: “The world has no beginning and no bounds in space,
but is inﬁnite with regard to both time and space.” According to Kant, the ﬁrst part is actually valid
and agrees with his own position. It is, however, invalid to conclude that the world is therefore inﬁnite;
to this eﬀect, it should have ﬁrst been proved that the world is actually determined in this respect.
Here, and this is precisely the point of the so–called inﬁnite judgment, the rule of the excluded third
does not apply.
186     Zdravko Kobe


be decomposed into the inﬁnite without itself consisting of inﬁnitely many
parts, since here, the parts are given only through the process of division,
which can never reach inﬁnity.
    The proposed solution thus manages to pull Kant out of his self–inﬂicted
contradiction. But since we have found that antinomy is inherent to tran-
scendental idealism alone, it is not obvious that we are generally speaking
in a better position than we started with. To conclude, we will therefore
brieﬂy consider where the source of the antinomy lies. In this respect, it
might be helpful to recall that Kant discovered the phenomenon rather
early in his philosophical development, but used to judge it quite diﬀerently
at diﬀerent stages. For instance, in 1769, the year that gave him “a great
light,” Kant interpreted this “evident self–contradiction” as the surest proof
that the metaphysical concepts were merely subjective (AA 17: 357); in the
Dissertation, in 1770, he explained the phenomenon as the “lack of accord
between the sensitive faculty and the faculty of understanding,” (AA 2: 389)
where both logical regimes were considered to be equally correct, each in
its proper sphere; and later, as he restricted the real use of understanding to
the sensible world, he described the opposition in question as the conﬂict
between “the partiality of sensibility and the totality of reason,” declaring
the thesis wrong and the antithesis right.28 Only in the Critique, in 1781, did
Kant return to symmetrical presentation, saying that both are wrong since
they both make an impossible presupposition.29
    In the year preceding the ﬁrst Critique, Kant’s treatment of antimony,
it appears, revolved around the opposition between the abstract and the
concrete, the intellectual and the sensitive, the conceptual and the intu-
itive, and even the total and the partial; he once deplored, for instance, that
understanding often cannot expose its abstract ideas in the concrete and
convert them into intuitions (see AA 2: 389). Considering this historical
dynamism, we would thus propose that, in Kant, the contradiction between
thesis and antithesis manifests the structural tension between the intellec-
tual and the sensitive conditions of cognition. This was, in fact, long ago
proposed by Maas (see Maas 1788: 441). If so, the antinomic conﬂict would
be in principle generated simply by juxtaposing Kant’s statements in the
Transcendental Analytic and the Transcendental Aesthetic. In the Aesthetic,
for instance, Kant claims that space is an inﬁnite given whole, while in the
Analytic he presents it as an axiom that everything in space is an extensive
magnitude brought about by a successive aggregation of given parts. These
two aﬃrmations are clearly in opposition. Now, in Kant, since sensitivity

  28. Compare, for instance, R 4760 (1775–77; AA 17: 711): “Thus there is actually no antinomy.”
  29. For a closer discussion, see Kreimendahl (1990), Guyer (1987), in particular pp. 387–404, and
Kobe (1995), pp. 113–123 and 264–268.
                                        Exit from the self–incurred contradiction   187


and understanding are both necessary for cognition, the antinomy is un-
avoidable due to the very logic of cognition, not merely because of the
trespassing of its boundaries: the contradiction arises here, in the middle of
experience, not somewhere beyond it. Insofar as the logic of intuition and
the logic of concept do not accord with each other, and insofar as cognition
is deﬁned precisely as a conjunction of intuition and concept, the antinomy
is built into the very fabric of cognition.
    However, such a rigid duality of aesthetics and analytics is still inadequate,
as it misrepresents the structure of cognition, and especially as it does not
pay appropriate attention to the adjustments made by Kant to the Critique,
which, even if located outside the antimony chapter, signiﬁcantly aﬀect its
implications. To put it in extremely short terms, in the second edition, Kant
extended the reach of the spontaneity of understanding, cut its reference to
any superior capacity, and made it into understanding that is “speciﬁc” to a
ﬁnite rational being, such as a human. The contradiction now comes, as it
were, from understanding alone. In his 1789 letter to Herz, Kant thus writes
that “the antinomies of pure reason” could provide a good test stone . . . that
one cannot assume human understanding to be of one kind with the divine”
(AA 11: 54). We may say that the antinomy now has its source in the tension
between the synthetic and analytic unities of consciousness, that is, in the
incongruence between the universal and the particular inherent to every
discursive understanding such as ours. This opposition may assume various
guises, usually exploring the gap between the abstract and the concrete.
For instance, it comes to the fore in Kant’s observation: “I have a penny for
every poor, but not for all the poors.”30 But since it arises out of the logical
form of ﬁnite cognition and is built into the very notion of the object that
joins together the unity of a thing with the plurality of its determinations, it
may be said that for the transcendental idealist, every object is ultimately an
antinomy.
    A devoted Kantian would probably protest. For him, antinomies are
strictly attached to the concept of the world and restricted to four, “no more,
no less.” However, when we examine the consequences of Kant’s solution
for the principle of complete determination, the proposition of universal-
ized antinomy may regain its Kantian ﬂavor. In traditional metaphysics, in
Baumgarten for instance, this principle of determinatio omnimoda was held
in high esteem, up to the point of being considered—under the heading of
complementum possibilitatis—virtually synonymous with existence. In gen-
eral, Kant seemed to agree with it. In the opening passages to his assessment
of natural theology, he thus—approvingly, it appears—cites the principle
in question, saying that “everything existing is thoroughly determined” (KrV,

  30. R 4409 (1771?; AA 11: 535).
188     Zdravko Kobe


B601/A573). If Kant refused to consider it equivalent to existence, it was
because he deemed it too weak, in the sense that not everything thoroughly
determined is for that matter existent.31
    Kant was well aware that this principle could never be exhibited in
concreto, since “in order to cognize a thing completely one has to cognize
everything possible and determine the thing through it” (ibid.). He did not
seem to realize, however, at least not to the full extent, that by admitting
this he subscribed to the conclusion that according to his theory—in exactly
the same way as in the antinomies—the existing objects are not completely
determined. Quite the opposite, for Kant, the things of this world are full of
gaps, lacks, and cracks, and for every well–determined face they show us
there is always a bottomless indeterminacy opening underneath them.32
    By examining the consequences of Kant’s presentation of antinomy, we
have ﬁnally ended in a strange world. In a world where due to the logical
form of ﬁnite understanding everything harbors a contradiction, where
things are penetrated by gaps in the very fabric of their being, with an
inﬁnite abyss of indeterminacy gaping beneath them, and where they are
therefore supported in their being only by a self–sustaining web of reasons.
In short, we arrive at a suspended gappy ontology, quite similar to the one
recently attributed to Hegel (see Bowman 2013, Kreines 2015).
    Kant wanted to present the antinomy in the instrumental fashion as an
indirect proof of transcendental idealism. By examining the two mathemat-
ical antinomic conﬂicts we tried to show that his attempt was a multi–layer
failure. What is remarkable, however, is the inner structure of this failure,
its conditions of possibility, and the consequences it brings along. We have
seen, for instance, that Kant must have known perfectly well that the dog-
matist would reject his arguments. We have also seen that it was precisely
the transcendental idealist who found herself under the threat of contra-
diction, as her conception of the object of experience included conﬂicting
requirements. It was her who desperately needed a solution to exit from
this self–inﬂicted wound. But if, consequently, the antinomy has to be read
as a symptom of transcendental idealism as such, then its solution should
also be understood in this general sense, that is, ontologically (especially
since for the transcendental idealist there was ultimately but one world).

    31. “Everything that exists is thoroughly determined; but it is not this thoroughgoing determina-
tion that constitutes the concept of existence, rather that a thing is posited absolutely and not merely
in relation to its concept.” (R 5710; 1780s?; AA 18: 332) Compare also R 6256 (1783–84; AA 18: 533) and R
6322 (1792–94; AA 18: 638). For a closer examination, see Engelhardt (2005), pp. 321–334, and Klimmek
(2005), pp. 165–180.
    32. This is not to say that we could somehow observe their gaps, for by bringing the cognitive
synthesis further, the gap of indeterminacy gets eo ipso ﬁlled. However, in exactly the same way as in
the antinomy, we know a priori that existing things are not completely determined.
                                           Exit from the self–incurred contradiction   189


This is a Hegelian point, obviously. However, the diﬀerence is, we hope,
that it was made here following Kant’s arguments and according to his
criteria.


References

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Baumanns, P., 1997, Kants Philosophie der Erkenntnis, Würzburg, Königshausen &
   Neumann.
Bowman, B., 2013, Hegel and the Metaphysics of Absolute Negativity, Cambridge, Cam-
  bridge University Press.
De Risi, V., 2007, Geometry and Monadology: Leibniz’s Analysis Situs and Philosophy
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