Norman Kemp Smith.

A commentary to Kant's 'Critique of pure reason,' online

. (page 58 of 72)
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or even to provoke the conflict, in the hope that we may
finally be able to detect some misunderstanding, and so to
resolve the conflict to the satisfaction of both the litigants.
Such is Kant's description of what he entitles his " sceptical
method." 3

Without here attempting a full discussion of the subject,
it seems advisable to point out at the very start what Kant's
exposition seriously obscures, namely, the real character of
the evidence upon which the theses and the antitheses respect-
ively rest. The latter are not correctly stated as transcend-
ing experience, and as therefore incapable of confirmation by
it. The proofs which . Kant offers of them are, indeed, of a
non-empirical a priori character. They are formulated in
terms of the dogmatic rationalism of the Leibnizian position,
with a constant appeal to abstract principles. But, as a
matter of fact, they can be much more adequately established
in so far as they can be established at all through analysis
of the spatial and temporal conditions of material existence.

1 Cf. per contra A 486 = 6 514.

2 The limitation of Kant's discussion to space, time, and causality is, of course,
due to his acceptance of the current view that the concepts of infinity and con-
tinuity are derived from our intuitions of space and time. As we have already
noted in discussing his intuitional theory of mathematical reasoning (above, pp.
40-1, 117 ff., 128 ff.), he fails to extend to mathematical concepts his own "tran-
scendental " view of the categories, namely, as conditioning the possibility of
intuitional experience. Such concepts as order, plurality, whole and part, con-
tinuity, infinity, are prior to time and space in the logical order of thought ; and
to be adequately treated must be considered in their widest application.

3 Cf. A 507 = B 535, and above, p. 431 ff. ; below, pp. 501, 545-6.

2 I


As space and time are continuous and homogeneous, any
assertion which is true of a space or time however small is
likewise true of a space or time however large. Any space
consists of spaces, and must be regarded as itself part of a
larger whole. 1 Any time consists of parts which are them-
selves times, and is apprehensible only as following upon pre-
ceding times. It is by such considerations as these that we
are led to regard the material world as unlimited, as infinitely
divisible, and as having no first state.

Kant's method of demonstrating the theses that the
world is limited, is finitely divisible, and has a first state is no
less misleading. Here again his rationalistic arguments con-
ceal the basis upon which the various theses really rest.
Their true determining ground is the demand of Reason for
some more satisfactory form of unconditionedness than that
which is found in the actual infinite. It is this demand which
has led philosophers to look around for proofs in support of
the theses, and to elaborate those rationalistic arguments
which Kant here reproduces. Thus the grounds of the anti-
theses are altogether different from those of the theses ; and
in neither case are they properly represented by the argu-
ments which Kant employs. 2

The reasons why Kant in his detailed statement of the
-antinomies has omitted, or at least subordinated, the above
considerations, are complex and various. In the first place,
this doctrine of antinomy was in several of its main features
already formulated prior to his development of the Critical
philosophy. It forms part of his Dissertation of 1770; and
at that time Kant was still largely in fundamental sympathy
with the Leibnizian ontology. Secondly, Kant is here pro-
fessing to criticise the science of rational cosmology, and is
therefore bound to expound it in more or less current form.
And in the third place, he teaches that the antinomies exist as
antinomies only when viewed from the false standpoint of dog-
matic rationalism. Had he eliminated the rationalistic proofs,

1 Cf. Kant's posthumously published Transition from the Metaphysical First
Principles of Natural Science to Physics {Altpreussische Monatsschrift, 1882), pp.
279-80 : " If we take in regard to space, not its definition, but only an a priori
proposition, e.g. that space is a whole which must be thought only as part of a still
greater whole, it is clear . . . that it is an irrational magnitude, measurable
indeed, but in its comparison with unity transcending all number." " If space is
something objectively existent, it is a magnitude which can exist only as part of
another given magnitude."

2 Cf. Schopenhauer, World as Will and Idea (Werke^Frauenstddt, ii. pp. 585-6;
Eng. trans, ii. pp. 107-8). " I find and assert that the whole antinomy is
a mere delusion, a sham fight. Only the assertions of the antitheses really rest
upon the forms of our faculty of knowledge, i.e. if we express it objectively, on the
necessary, a priori certain, most universal laws of nature. Their proofs alone are


the conflict of the antinomies, in its strictly logical form, as
the conflict of direct contradictories, would at once have
vanished. The general framework of this division of the
Dialectic demanded a rationalistic treatment of both theses
and antitheses, and Kant believed that the rationalistic proofs
which he propounds in their support are unanswerable, so long
as the dogmatic standpoint of ordinary consciousness and of
Leibnizian ontology is preserved. But even when that im-
portant limitation is kept in view, Kant fails to justify this
interpretation of the conflict, and we must therefore be pre-
pared to find that his proofs, whether of theses or of antitheses ',
are in all cases inconclusive. I shall append to each of his
arguments a statement of the reasons which constrain us to
reject them as unsound. We shall then be in a position to
consider his whole doctrine of antinomy in its broader aspects,
and in its connection with the teaching of the other main
divisions of the Dialectic.


Thesis. (a) The world has a beginning in time, and (^) is
also limited in regard to space.

Thesis a. Proof. If we assume the opposite, namely,
that the world has no beginning in time, and if we define the
infinite as that which can never be completed by means of a
successive synthesis, we must conclude that the world-series
can never complete itself. But the entire series of past events
elapses, i.e. completes itself at each moment. It cannot there-
fore be infinite.

Criticism. This argument gains its plausibility from the
illegitimate use of the term c elapse ' (verfliessen] as equivalent
to ' complete itself.' If it be really correct to define the in-
finite as that which can never be completed, the conclusion to
be drawn is that the temporal series is always actually infinite,
and that no point or event in it is nearer to or further from

therefore drawn from objective grounds. On the other hand, the assertions and
proofs of the theses have no other than a subjective ground, rest solely on the
weakness of the reasoning individual ; for his imagination becomes tired with an
endless regression, and therefore he puts an end to it by arbitrary assumptions,
which he tries to smooth over as well as he can ; and his judgment, moreover, is
in this case paralysed by early and deeply imprinted prejudices. On this account
the proof of the thesis in all the four conflicts is throughout a mere sophism, while
that of the antithesis is a necessary inference of the reason from the laws of the
world as idea known to us a priori. It is, moreover, only with great pains and
skill that Kant is able to sustain the thesis, and make it appear to attack its
opponent, which is endowed with native power. ... I shall show that the proofs
which Kant adduces of the individual theses are sophisms, while those of the anti-
theses are quite fairly and correctly drawn from objective grounds."


either its beginning or its end. 1 We may select any point in
the series as that from which we propose to begin a regress to
the earlier members of the series, but if the series is actually
infinite, it will be a regress without possibility of completion,
and one therefore which removes all justification for asserting
that at the point chosen a series has completed itself. It
has no beginning, and has no completion. What it has
done at each moment of the past it is still doing at each
present moment, namely, coming out of an inexhaustible past
and passing into an equally inexhaustible future. / Time is by
its given nature capable of being interpreted only as actually
infinite, alike in its past and in its future. It cannot complete
itself any more than it can begin itself. The one would be as
gross a violation of its nature as would the other. The present
exists only as a species of transition, unique in itself, but
analogous in nature to the innumerable other times that con-
stitute time past. It is a transition from the infinite through
the infinite to the infinite. That we cannot comprehend how,
from an infinitude that has no beginning, the present should
ever have been reached, is no sufficient reason for denying
what by the very nature of time we are compelled to accept as
a correct description of the situation which is being analysed.
The actual nature of time is such as to rule out from among
the possibilities the thesis which Kant is here professing to be
able to establish ; time, being such as it actually is, can have
no beginning.

What thus holds of time may likewise hold of events in
time. If time is actually infinite, no proof can be derived
from it in support of the assumption that the world has had
a beginning in time.

The phrase "by means of a successive synthesis" gives
a needlessly subjectivist colouring to Kant's method of proof.
The antinomy is professedly being stated from the realist
standpoint, and ought not therefore to be complicated by any
such reference. This objection applies, as we shall find, still
more strongly to Kant's proof of the second part of the thesis.
The latter proof depends upon this subjectivist reference ; the
present proof does not.

Kant limits his problem to the past infinitude of time.
The reason for this lies, of course, in the fact that he is
concerned with the problem of creation. The limitation is,
however, misleading.

Thesis b. The world is limited in regard to space.

Proof. Assume the opposite, namely, that the world is

1 Cf. F. Erhardt's Kritik der Kantischen Antinomienlehre (1888), a brief but
excellent analysis of this section of the Critique.


an infinite, given whole of coexisting parts. A magnitude
not given within the determinate limits of an intuition can
only be thought through the synthesis of its parts, and its
totality through their completed synthesis. In order, there-
fore, that we may be able to think as a single whole the world
which fills all space, the successive synthesis of the parts
of an infinite world must be regarded as completed, i.e. an
infinite must be regarded as having elapsed in the enumera-
tion of all coexisting things. This, however, is impossible.
An infinite aggregate of actual things cannot therefore be
viewed as a given whole, nor as being given as coexistent.
Consequently the world of spatial existences must be regarded
as finite.

Criticism. From the impossibility of traversing infinite
space in thought by the successive addition of part to part,
Kant here argues that " an infinite aggregate of actual things
cannot be viewed as a given whole," and consequently that
the world cannot be infinitely extended in space. That is,
from a subjective impossibility of apprehension he infers an
objective impossibility of existence. But Kant has himself
defined the infinite as involving this subjective impossibility ;
for in the proof of thesis a he has stated that the infinitude
of a series consists in the very fact that it can never be com-
pleted through successive synthesis. Kant is therefore pro-
pounding against the existence., of the infinite the very feature
which by definition constitutes its infinitude.^ The implication
would seem to be that the concept of the infinite is the concept
of that which ex definitione cannot exist, and that there is
therefore a contradiction in the very idea of the actual infinite.

Deferring for a moment the further objections to which
such procedure lies open, we may observe that Kant, in
arguing from a subjective to an objective impossibility,
commits the fallacy of ignoratio elenchi. For when the con-
ditions of objective existence are recognised in their distinction
from those of mental apprehension, the supposed contradiction
vanishes, and the argument ceases to have any cogency.
The use of the words * given ' and * whole ' is misleading. If
space is infinite, it is without bounds, and cannot therefore
exist as a whole in any usual meaning of that term. For
the same reason it must be incapable of being given as a
whole. Its infinitude is a presupposition which analysis of
actually given portions of it constrains us to postulate, and
has to be conceived in terms of the definition employed in
thesis a. The given must always be conceived as involving
what is not itself given and what is not even capable of
complete construction. In terms of this presupposition an


actual infinite, not given and not capable of construction, can
be represented with entire consistency.

But to return to the main assumption upon which Kant's
proof would seem to rest : it is all-important to observe that
Kant does not, either in the Critique or in any other of his
writings, assert that the concept of the actual infinite is
inherently self-contradictory. This is a matter in regard to
which many of Kant's critics have misrepresented his teaching.
Kant's argument may, as we have just maintained, be found on
examination to involve the above assertion ; but this, if clearly
established, so far from commending the argument to Kant,
would have led him to reject it as invalid. The passage in
the Dissertation^- of 1770, which contains his most definite
utterance on this point, represents the view from which he
never afterwards departed. It may be quoted in full.

" Those who reject the actual mathematical infinite do so in a
very casual manner. For they so construct their definition of the
infinite that they are able to extract a contradiction from it. The
infinite is described by them as a quantity than which none greater is
possible, and the mathematical infinite as a multiplicity of an assign-
able unit than which none greater is possible. Since they thus
substitute maximum for infinitum, and a greatest multiplicity is
impossible, they easily conclude against this infinite which they
have themselves invented. Or, it may be, they entitle an infinite
multiplicity an infinite number, and point out that such a phrase is
meaningless, as is, indeed, perfectly evident. But again they have
fought and overthrown only the figments of their own minds. If,
however, they had conceived the mathematical infinite as a quantity
which, when related to measure, as its unity, is a multiplicity greater
than all number and if furthermore, they had observed that measur-
ability here denotes only the relation [of the infinite] to the standards
of the human intellect, which is not permitted to attain to a definite
conception of multiplicity save by the successive addition of unit to
unit, nor to the sum-total (which is called number] save by completing
this progress in a finite time ; they would have perceived clearly that
what does not conform to the established law of some subject need
not on that account exceed all intellection. An intellect may exist,
though not indeed a human intellect, which perceives a multiplicity
distinctly in one intuition [unv obtutu\ without the successive applica-
tion of a measure."

The concluding sentences of this Dissertation passage may
be taken as Kant's own better and abiding judgment in
regard to the question before us. We must not argue from
the impossibility of mentally traversing the infinite to the
impossibility of its existence. Indeed the essentials of the

1 I n.


ibove passage are restated in the 'Observation' on this
thesis. 1 Thus the concept of the actual infinite is not only,
as a concept, perfectly self-consistent, it is also one which, in
view of the nature of time and of space, we are constrained to
accept as a correct representation of the actually given. The
thesis of this first antinomy runs directly counter to admitted
facts. That Kant is here arguing in respect to the world, and
not merely in respect to space and time, does not essentially
alter the situation. For if space and time are necessarily to
be viewed as infinite, there can be no a priori proof none, at
least, of the kind here attempted that the world-series may
not be so likewise.

Antithesis. (a) The world has no beginning in time ; (fr)
has no limits in space. In both these respects the world is

In these antitheses Kant assumes that space and time
are actually infinite, and from that assumption advances to
the proof that this is likewise true of the world in its spatial
and temporal aspects. This, by itself, ought to be sufficient
evidence that Kant does not regard the actual infinite as
an inherently impossible conception. As the antinomies are
avowedly formulated from the realist, dogmatic standpoint
of ordinary consciousness, Kant is also enabled to assume that
if the world begins to be, it must have an antecedent cause
determining it to exist at that moment rather than at another.

Antithesis a. Proof. Let us assume the opposite, namely,
that the world has a beginning. It will then be preceded by
an empty time in which it was not. But in an empty time
no becoming is possible, since in such a time no part possesses
over any other any distinguishing condition of existence
rather than of non-existence. The world must therefore be
infinite as regards past time.

Criticism. In this argument everything depends upon
what is to be meant by the term 'world.' If Kant means
by it merely the material world, the assumption of its non-
existence does norteave only empty time and space. Other
kinds of existence may be possible, and in these a sufficient
cause of its first beginning may be found. The nature of
creative action will remain mysterious and incomprehensible,
but that is no sufficient reason for denying its possibility. If,
on the other hand, Kant means by the world * all that is/ the
assumption of its non-existence is likewise the assumption of
the non-existence of all its possible causes. That, however,

1 Cf. A 431-2 = 6 460-1 : "... the concept [of the infinite] is not the con-
cept of a maximum ; by it we think only its relation to any assignable unit, in
respect to which it is greater than all number."


is for ordinary consciousness a quite impossible assumption,
since it runs counter to the causal principle which is taken as
universally valid. LFrom this point of view the argument
consists in making an impossible assumption, and in then
pointing out the impossible consequence which must follow.
By such a mode* of argument no conclusion can be reached.
Kant's decision ought rather to have been that, as time is
actually infinite, the world may be so likewise, but that though
reality must in some form be eternally existent, the material
ivorld cannot be proved to be so by any a priori proof of the
kind here given.

Antithesis b. Proof. Let us assume the opposite, namely,
that the world is finite, existing in an empty limitless space.
There will then be not only a relation of things in space y but
also of things to space. But as the world is a totality outside
of which no object of intuition can be found, the relation of
the world to empty space is a relation to no object. Such a
relation is nothing. Consequently the opposite holds ; the
world must be infinitely extended.

Criticism. That Kant himself felt the inadequacy of this
argument, when taken from the dogmatic standpoint, is indi-
cated by the lengthy note which he has appended to it, and
which develops his own Critical view of space as not a real
independent object, but merely the form of external intuition.
From the standpoint of ordinary consciousness space is a
self-existent entity, and there is no insuperable difficulty in
conceiving a relation as holding between it and its contents.
The introduction of the opposed standpoint of the Aesthetic
therefore runs directly counter to Kant's own intention of
expounding the antinomies from the dogmatic standpoint
which involves this realist view of space, and of showing
that they afford, in independence of the arguments of the
Aesthetic, an indirect proof of the untenableness of that
belief. 1 The conclusion which ought to have been drawn is
analogous to that above suggested for thesis a. As space
is actually infinite, the material world may be so likewise ;
but that it actually is so, cannot be established by an a priori
argument of the kind here attempted.


Thesis. Every composite substance in the world consists
of simple parts, and nothing anywhere exists save the simple
or what is composed of it.

Proof. Let us assume the opposite, namely, that substances

1 Cf. Kant's statement in the Observation to this antithesis, A 431-3 = 6 459-61.


do not consist of simple parts. If all composition be then
removed in thought, no composite part, and (as there are no
simple parts) also no simple part, and therefore nothing what-
soever, will remain. Consequently no substance will be given.
Either, therefore, it is impossible to remove in thought all
composition, or after its removal something that exists without
composition, i.e. the simple, must remain. In the former case
the composite would not itself consist of substances (with
them composition is a merely accidental relation, and they
must, as self-persisting beings, be able to exist independently
of it). As this contradicts our assumption, only the latter
alternative remains, namely, that the substantial compounds
in the world consist of simple parts.

Criticism. Kant here assumes, by his definition of terms,
the point which he professes to establish by argument. The
substance referred to, though never itself mentioned by name,
is extended matter. Kant identifies it with 'composite
substance.' Substance, he further dogmatically decides, is
that which is capable of independent existence, and to which
all relations of composition are therefore merely accidental.
If these assumptions be granted, it at once follows that
composition cannot be essential to matter, and that when
all composition is thought away, its reality will be disclosed
as consisting in simple parts. Kant, however, makes no
attempt to prove that extended matter can be defined in any
such terms. From the dogmatic point of view of ordinary
consciousness, though not from the sophisticated standpoint
of Leibniz, extension is of the very essence of matter ; and, as
Kant himself believed, 1 the continuity of extension is such
as to exclude all possibility of elimination of the composite.
For he maintains that, however far division be carried, the
parts remain no less composite than the whole from which
the regress has started. On any such view the extended and
the composite are not equivalent terms. The opposite of the
composite is the simple ; the opposite of the extended is the
non-extended. Kant is here surreptitiously substituting a
Leibnizian metaphysics in place of the empirical reality which
is supposed to necessitate the argument.

In the Observation on this thesis Kant shows consciousness

1 Kant regarded the point as a limit, i.e. as a boundary (Dissertation, 14, 4 ;
15, C : " The simple in space is not a part but a limit" ; A 169-70 = 6 211) ;
whereas certain modern mathematicians take the point as one of the undefined
elements. When the point is regarded in this latter manner, space may perhaps be
satisfactorily defined as a set of points. In arguing for the antithesis, and in the
passages just cited, Kant also assumes that, in the case of space, the properties
of the class are determined by the properties of its elements. This questionable
assumption is involved in his assertion that space can consist only of spaces.


of the defects of his argument. It does not apply to space,
time, or change.

" We ought not to call space a compositum but a totum, because

Online LibraryNorman Kemp SmithA commentary to Kant's 'Critique of pure reason,' → online text (page 58 of 72)