Remark 2: The Kantian Antinomy of the Limitation and Nonlimitation of the World in Time and Space

§ 518

It was remarked above that the Kantian antinomies are expositions of the opposition of finite and infinite in a more concrete shape, applied to more specific substrata of conception. The antinomy there considered contained the opposition of qualitative finitude and infinitude. In another, the first of the four cosmological antinomics, it is the conflict arising rather from the quantitative limit which is considered. I shall therefore proceed to examine this antinomy here.

§ 519

It concerns the limitation or non-limitation of the world in time and space. This antithesis could be considered equally well with reference to time and space themselves, for whether time and space are relations of things themselves or are only forms of intuition, the antinomy based on limitation or non-limitation in them is not affected thereby.

§ 520

The detailed analysis of this antinomy will likewise show that both statements and equally their proofs (which, like those already considered, are conducted apagogically) amount to nothing more than the two simple opposite assertions: (1) there is a limit, and (2) the limit must be transcended.

The thesis is:

‘The world has a beginning in time and is also enclosed within spatial limits.’

That part of the proof which concerns time assumes the opposite:

‘The world has no beginning in time; therefore, up to any given point of time, an eternity has elapsed and consequently an infinite series of successive states of things in the world has passed away. Now the infinity of a series consists precisely in the impossibility of ever completing it by successive synthesis. Therefore an infinite world series which has passed away is impossible and consequently a beginning of the world is a necessary condition of its existence — which was to be proved.’

§ 521

The other part of the proof which concerns space is based on time. To comprehend a spatially infinite world would require an infinite time and this time must be regarded as having already elapsed in so far as the world in space is to be regarded not as gradually coming to be but as completely given. But it was shown of time in the first part of the proof that it is impossible to assume an infinite time as elapsed.

§ 522

But it is at once evident that it was unnecessary to make the proof apagogical, or even to carry out a proof at all, since the basis of the proof itself is the direct assertion of what was to be proved. Namely, there is assumed some or any given point of time up to which an eternity has elapsed (eternity here has only the trivial meaning of a simply endless time). Now a given point of time means nothing else than a definite limit in time. In the proof therefore, a limit of time is presupposed as actual; but that is just what was to be proved. For the thesis is, that the world has a beginning in time.

§ 523

There is only this difference, that the assumed limit of time is a now as end of the time already elapsed, but the limit which is to be proved is a now as beginning of a future. But this difference is immaterial. The now is taken as the point in which an infinite series of successive states of things in the world is supposed to have passed away, therefore as end, a qualitative limit. If this now were considered to be merely a quantitative limit which flows on and which not only must be transcended but is only as the transcending of itself, then the infinite time series would not have passed away in it, but would continue to flow on, and so the argument of the proof would vanish. On the other hand, if the point of time is assumed as a qualitative limit for the past, in which case it is also a beginning for the future (for each point of time is in itself the connection of the past and the future), then it is also an absolute, that is, abstract beginning for the future — and it was this that was to be proved. The fact that its future and this its beginning is already preceded by a past does not affect the argument; because this point of time is a qualitative limit — and that it is to be taken as qualitative is implied in the description of it as completed, elapsed, and therefore as not continuing — therefore in it time is broken off and the past lacks a connection with this time which could only be called future with reference to that past and, consequently, without such connection is only time as such, which has an absolute beginning. But if — as is, then, the case — it were related to the past through the now, the given point of time, and were thus determined as a future, then this point of time, too, regarded from the other side, would not be a limit; the infinite time series would continue itself in what was called future and would not be, as was assumed, completed.

§ 524

In truth, time is pure quantity; the point of time in which it is supposed to be interrupted, which is employed in the proof, is really only the self-sublating being-for-self of the now. All that the proof does is to represent the absolute limit of time asserted in the thesis as a given point of time, and then straightway to assume it as a completed, that is, abstract point — a popular determination which sensuous conception readily lets pass as a limit, thus allowing as an assumption in the proof what had been put forward as the thing to be proved.

§ 525

The antithesis runs:

‘The world has no beginning and no limits in space but is infinite with reference both to time and space.’

The proof likewise assumes the opposite:

‘The world has a beginning. Since the beginning is an existence preceded by a time in which the thing is not, there must have been a preceding time in which the world was not, that is, an empty time. Now no originating of anything is possible in an empty time; because no part of such a time possesses in itself and in preference to any other, any distinguishing condition of existence or non-existence. In the world, therefore, many groups of things can indeed begin, but the world itself can have no beginning and with respect to past time is infinite.’

§ 526

This apagogical proof, like the others, contains the direct and unproved assertion of what it was supposed to prove. That is, it first assumes a beyond of the existing world, an empty time; but it also equally continues the existence of the world beyond itself into this empty time which is thereby sublated, with the result that the existence of the world is continued into infinity. The world is an existence; the proof presupposes that this existence comes into being and that the coming-to-be has an antecedent condition which is in time. But the antithesis itself consists in the very fact that there is no unconditioned existence, no absolute limit, but that the existence of the world always requires an antecedent condition. Thus, what was to be proved is found as an assumption in the proof. Further, the condition is sought in empty time, which means in effect that it is taken as temporal and therefore as an existence and as limited. Altogether, then, the assumption is made that the world as an existence presupposes another conditioned existence in time, and so on, therefore, to infinity.

§ 527

The proof regarding the infinity of the world in space is the same. Apagogically, the spatial finiteness of the world is assumed; ’this (the world) would therefore exist in an empty unlimited space and would stand in a relation to it; but such a relation of the world to no object is a nullity'.

Here, too, what was supposed to be proved is directly presupposed in the proof. It is directly assumed that the spatially limited world exists in an empty space and is supposed to stand in a relation to it, that is, there must be a movement out beyond it — on the one hand into the void, into the beyond and non-being of the world, but on the other hand, in order that it be in relation with its beyond, that is, continue itself into it, the beyond must be imagined as filled with the existence of the world. The infinity of the world in space which is asserted in the antithesis is nothing else than, on the one hand, empty space, and on the other the relation of the world to it, that is, the continuity of the world in empty space or the filling of space — which contradiction, namely, space as simultaneously empty and also filled, is the infinite progress of existence in space. This very contradiction, the relation of the world to empty space, is directly made the basis of the proof.

§ 528

The thesis and antithesis and their proofs therefore represent nothing but the opposite assertions, that a limit is, and that the limit equally is only a sublated one; that the limit has a beyond, with which however it stands in relation, and beyond which it must pass, but that in doing so there arises another such limit, which is no limit.

§ 529

The solution of these antinomies, as of those previously mentioned, is transcendental, that is, it consists in the assertion of the ideality of space and time as forms of intuition — in the sense that the world is in its own self not self-contradictory, not self-sublating, but that it is only consciousness in its intuition and in the relation of intuition to understanding and reason that is a self-contradictory being. It shows an excessive tenderness for the world to remove contradiction from it and then to transfer the contradiction to spirit, to reason, where it is allowed to remain unresolved. In point of fact it is spirit which is so strong that it can endure contradiction, but it is spirit, too, that knows how to resolve it. But the so-called world (whether it be called an objective, real world or, according to transcendental idealism, a subjective intuition and a sphere of sense determined by the categories of the understanding) is never and nowhere without contradiction, but it is unable to endure it and is, therefore, subject to coming-to-be and ceasing-to-be.

(c) The Infinity of Quantum § 530

1. The infinite quantum as infinitely great or infinitely small is itself implicitly the infinite progress; as great or small it is a quantum and at the same time it is the non-being of quantum. The infinitely great and infinitely small are therefore pictorial conceptions which, when looked at more closely, turn out to be nebulous shadowy nullities. But in the infinite progress, this contradiction is explicitly present and with it that which is the nature of quantum which, as an intensive magnitude, has attained its reality and now in its determinate being is posited as it is in its Notion. It is this identity which we have now to consider.

§ 531

Quantum as degree is unitary, self-related and determinate within itself. Through this unitary nature, the otherness and determinateness in quantum are sublated, so that the determinatensess is external to it; it has it determinateness outside it. This its self-externality is in the first place the abstract non-being of quantum generally, the spurious infinity. But, further, this non-being is also quantitative and this continues itself into its non-being, for it is in its externality that quantum has its determinateness; this its externality is, therefore, itself equally a quantum., this non-being of quantum, infinity, is thus limited, that is, this beyond is sublated, is itself determined as quantum which, therefore, in its negation is with itself.

§ 532

But this is what quantum as such is in itself. For it is itself just by being external to itself; externality constitutes that whereby it is quantum and is with itself. In the infinite progress, therefore, the Notion of quantum is posited.

§ 533

Let us take the progress at first in its abstract determinations as we find them; then in it we have the sublating of quantum, but equally too of its beyond, therefore the negation of quantum as well as the negation of this negation. Its truth is their unity in which they are, but only as moments. It is the resolution of the contradiction of which it is the expression, and its immediate significance is, therefore, the restoration of the Notion of quantity, namely, that quantity is an indifferent or external limit. In the infinite progress as such, the only reflection usually made is that every quantum, however great or small, must be capable of vanishing, of being surpassed; but not that this self-sublating of quantum, the beyond, the spurious infinite itself also vanishes.

§ 534

Even the first sublation, the negation of quality as such whereby quantum is posited, is in principle [an sich] the sublating of the negation — the quantum is sublated qualitative limit, hence sublated negation — but at the same time it is this only in principle; it is posited as a determinate being, and then its negation is fixed as the infinite, as the beyond of quantum, which remains on this side as an immediate; thus the infinite is determined only as a first negation and it appears as such in the infinite progress. But we have seen that in this something more is present, the negation of the negation, or that which the infinite in truth is. We regarded this previously as the restoration of the Notion of quantity; this restoration means in the first place, that its determinate being has received a more precise determination; we now have quantum determined in conformity with its Notion, which is different from quantum in its immediacy; externality is now the opposite of itself, posited as a moment of quantity itself — quantum is posited as having its determinateness in another quantum by means of its non-being, of infinity; that is, it is qualitatively that which it is. However, this comparison of the Notion of quantum with its determinate being belongs more to our reflection, to a relationship which is not yet present here. The immediately following determination is that the quantum has reverted to quality, is from now on qualitatively determined. For its peculiarity, its quality, is the externality, the indifference of the determinateness; and quantum is now posited as being in fact itself in its externality, as self-related therein, in simple unity with itself, that is, qualitatively determined. This qualitative moment is still more closely determined, namely as being-for-itself; for the self-relation to which it has attained has proceeded from mediation, from the negation of the negation. Quantum has infinity, self-determinedness, no longer outside it but within itself.

§ 535

The infinite, which in the infinite progress has only the empty meaning of a non-being, of an unattained but sought beyond, is in fact nothing else than quality. Quantum as an indifferent limit goes out beyond itself to infinity; in doing so it seeks nothing else than to be determined for itself, the qualitative moment, which, however, is thus only an ought-to-be. Its indifference to limit, and hence its lack of an explicit determinateness of its own and its passage away from and beyond itself, is that which makes quantum what it is; this its passage into the beyond is to be negated and quantum is to find in the infinite its absolute determinateness.

§ 536

Quite generally: quantum is sublated quality; but quantum is infinite, goes beyond itself, is the negation of itself. Thus its passage beyond itself is, therefore, in itself the negation of the negated quality, the restoration of it; and thus quantum is explicitly determined as possessing as its own moment, the externality which formerly appeared as a beyond.

§ 537

Quantum is thus posited as repelled from itself, with the result that there are two quanta which, however, are sublated, are only as moments of one unity, and this unity is the determinateness of quantum. Quantum as thus self-related as an indifferent limit in its externality and therefore posited as qualitative, is quantitative ratio. In the ratio, quantum is external to itself, is distinguished from itself; this its externality is the relation of one quantum to another, each of which has meaning only in this its relation to its other; and this relation constitutes the determinateness of the quantum, which is as such a unity. It has in this unity not an indifferent, but a qualitative, determination; in this its externality it has returned into itself, and in it quantum is that which it is.