The Critique of Pure Reason

Transcendental Dialectic (suite)

In neither case—the regressus in infinitum, nor the regressus in indefinitum, is the series of conditions to be considered as actually infinite in the object itself. This might be true of things in themselves, but it cannot be asserted of phenomena, which, as conditions of each other, are only given in the empirical regress itself. Hence, the question no longer is, “What is the quantity of this series of conditions in itself—is it finite or infinite?” for it is nothing in itself; but, “How is the empirical regress to be commenced, and how far ought we to proceed with it?” And here a signal distinction in the application of this rule becomes apparent. If the whole is given empirically, it is possible to recede in the series of its internal conditions to infinity. But if the whole is not given, and can only be given by and through the empirical regress, I can only say: “It is possible to infinity, to proceed to still higher conditions in the series.” In the first case, I am justified in asserting that more members are empirically given in the object than I attain to in the regress (of decomposition). In the second case, I am justified only in saying, that I can always proceed further in the regress, because no member of the series is given as absolutely conditioned, and thus a higher member is possible, and an inquiry with regard to it is necessary. In the one case it is necessary to find other members of the series, in the other it is necessary to inquire for others, inasmuch as experience presents no absolute limitation of the regress. For, either you do not possess a perception which absolutely limits your empirical regress, and in this case the regress cannot be regarded as complete; or, you do possess such a limitative perception, in which case it is not a part of your series (for that which limits must be distinct from that which is limited by it), and it is incumbent on you to continue your regress up to this condition, and so on.

These remarks will be placed in their proper light by their application in the following section.

Section IX. Of the Empirical Use of the Regulative Principle of Reason with regard to the Cosmological Ideas

We have shown that no transcendental use can be made either of the conceptions of reason or of understanding. We have shown, likewise, that the demand of absolute totality in the series of conditions in the world of sense arises from a transcendental employment of reason, resting on the opinion that phenomena are to be regarded as things in themselves. It follows that we are not required to answer the question respecting the absolute quantity of a series—whether it is in itself limited or unlimited. We are only called upon to determine how far we must proceed in the empirical regress from condition to condition, in order to discover, in conformity with the rule of reason, a full and correct answer to the questions proposed by reason itself.

This principle of reason is hence valid only as a rule for the extension of a possible experience—its invalidity as a principle constitutive of phenomena in themselves having been sufficiently demonstrated. And thus, too, the antinomial conflict of reason with itself is completely put an end to; inasmuch as we have not only presented a critical solution of the fallacy lurking in the opposite statements of reason, but have shown the true meaning of the ideas which gave rise to these statements. The dialectical principle of reason has, therefore, been changed into a doctrinal principle. But in fact, if this principle, in the subjective signification which we have shown to be its only true sense, may be guaranteed as a principle of the unceasing extension of the employment of our understanding, its influence and value are just as great as if it were an axiom for the à priori determination of objects. For such an axiom could not exert a stronger influence on the extension and rectification of our knowledge, otherwise than by procuring for the principles of the understanding the most widely expanded employment in the field of experience.

I. Solution of the Cosmological Idea of the Totality of the Composition of Phenomena in the Universe

Here, as well as in the case of the other cosmological problems, the ground of the regulative principle of reason is the proposition that in our empirical regress no experience of an absolute limit, and consequently no experience of a condition, which is itself absolutely unconditioned, is discoverable. And the truth of this proposition itself rests upon the consideration that such an experience must represent to us phenomena as limited by nothing or the mere void, on which our continued regress by means of perception must abut—which is impossible.

Now this proposition, which declares that every condition attained in the empirical regress must itself be considered empirically conditioned, contains the rule in terminis, which requires me, to whatever extent I may have proceeded in the ascending series, always to look for some higher member in the series—whether this member is to become known to me through experience, or not.

Nothing further is necessary, then, for the solution of the first cosmological problem, than to decide, whether, in the regress to the unconditioned quantity of the universe (as regards space and time), this never limited ascent ought to be called a regressus in infinitum or indefinitum.

The general representation which we form in our minds of the series of all past states or conditions of the world, or of all the things which at present exist in it, is itself nothing more than a possible empirical regress, which is cogitated—although in an undetermined manner—in the mind, and which gives rise to the conception of a series of conditions for a given object.[60] Now I have a conception of the universe, but not an intuition—that is, not an intuition of it as a whole. Thus I cannot infer the magnitude of the regress from the quantity or magnitude of the world, and determine the former by means of the latter; on the contrary, I must first of all form a conception of the quantity or magnitude of the world from the magnitude of the empirical regress. But of this regress I know nothing more than that I ought to proceed from every given member of the series of conditions to one still higher. But the quantity of the universe is not thereby determined, and we cannot affirm that this regress proceeds in infinitum. Such an affirmation would anticipate the members of the series which have not yet been reached, and represent the number of them as beyond the grasp of any empirical synthesis; it would consequently determine the cosmical quantity prior to the regress (although only in a negative manner)—which is impossible. For the world is not given in its totality in any intuition: consequently, its quantity cannot be given prior to the regress. It follows that we are unable to make any declaration respecting the cosmical quantity in itself—not even that the regress in it is a regress in infinitum; we must only endeavour to attain to a conception of the quantity of the universe, in conformity with the rule which determines the empirical regress in it. But this rule merely requires us never to admit an absolute limit to our series—how far soever we may have proceeded in it, but always, on the contrary, to subordinate every phenomenon to some other as its condition, and consequently to proceed to this higher phenomenon. Such a regress is, therefore, the regressus in indefinitum, which, as not determining a quantity in the object, is clearly distinguishable from the regressus in infinitum.

[60] The cosmical series can neither be greater nor smaller than the possible empirical regress, upon which its conception is based. And as this regress cannot be a determinate infinite regress, still less a determinate finite (absolutely limited), it is evident that we cannot regard the world as either finite or infinite, because the regress, which gives us the representation of the world, is neither finite nor infinite.

It follows from what we have said that we are not justified in declaring the world to be infinite in space, or as regards past time. For this conception of an infinite given quantity is empirical; but we cannot apply the conception of an infinite quantity to the world as an object of the senses. I cannot say, “The regress from a given perception to everything limited either in space or time, proceeds in infinitum,” for this presupposes an infinite cosmical quantity; neither can I say, “It is finite,” for an absolute limit is likewise impossible in experience. It follows that I am not entitled to make any assertion at all respecting the whole object of experience—the world of sense; I must limit my declarations to the rule according to which experience or empirical knowledge is to be attained.

To the question, therefore, respecting the cosmical quantity, the first and negative answer is: “The world has no beginning in time, and no absolute limit in space.”

For, in the contrary case, it would be limited by a void time on the one hand, and by a void space on the other. Now, since the world, as a phenomenon, cannot be thus limited in itself for a phenomenon is not a thing in itself; it must be possible for us to have a perception of this limitation by a void time and a void space. But such a perception—such an experience is impossible; because it has no content. Consequently, an absolute cosmical limit is empirically, and therefore absolutely, impossible.[61]

[61] The reader will remark that the proof presented above is very different from the dogmatical demonstration given in the antithesis of the first antinomy. In that demonstration, it was taken for granted that the world is a thing in itself—given in its totality prior to all regress, and a determined position in space and time was denied to it—if it was not considered as occupying all time and all space. Hence our conclusion differed from that given above; for we inferred in the antithesis the actual infinity of the world.

From this follows the affirmative answer: “The regress in the series of phenomena—as a determination of the cosmical quantity, proceeds in indefinitum.” This is equivalent to saying: “The world of sense has no absolute quantity, but the empirical regress (through which alone the world of sense is presented to us on the side of its conditions) rests upon a rule, which requires it to proceed from every member of the series, as conditioned, to one still more remote (whether through personal experience, or by means of history, or the chain of cause and effect), and not to cease at any point in this extension of the possible empirical employment of the understanding.” And this is the proper and only use which reason can make of its principles.

The above rule does not prescribe an unceasing regress in one kind of phenomena. It does not, for example, forbid us, in our ascent from an individual human being through the line of his ancestors, to expect that we shall discover at some point of the regress a primeval pair, or to admit, in the series of heavenly bodies, a sun at the farthest possible distance from some centre. All that it demands is a perpetual progress from phenomena to phenomena, even although an actual perception is not presented by them (as in the case of our perceptions being so weak as that we are unable to become conscious of them), since they, nevertheless, belong to possible experience.

Every beginning is in time, and all limits to extension are in space. But space and time are in the world of sense. Consequently phenomena in the world are conditionally limited, but the world itself is not limited, either conditionally or unconditionally.

For this reason, and because neither the world nor the cosmical series of conditions to a given conditioned can be completely given, our conception of the cosmical quantity is given only in and through the regress and not prior to it—in a collective intuition. But the regress itself is really nothing more than the determining of the cosmical quantity, and cannot therefore give us any determined conception of it—still less a conception of a quantity which is, in relation to a certain standard, infinite. The regress does not, therefore, proceed to infinity (an infinity given), but only to an indefinite extent, for or the of presenting to us a quantity—realized only in and through the regress itself.

II. Solution of the Cosmological Idea of the Totality of the Division of a Whole given in Intuition

When I divide a whole which is given in intuition, I proceed from a conditioned to its conditions. The division of the parts of the whole (subdivisio or decompositio) is a regress in the series of these conditions. The absolute totality of this series would be actually attained and given to the mind, if the regress could arrive at simple parts. But if all the parts in a continuous decomposition are themselves divisible, the division, that is to say, the regress, proceeds from the conditioned to its conditions in infinitum; because the conditions (the parts) are themselves contained in the conditioned, and, as the latter is given in a limited intuition, the former are all given along with it. This regress cannot, therefore, be called a regressus in indefinitum, as happened in the case of the preceding cosmological idea, the regress in which proceeded from the conditioned to the conditions not given contemporaneously and along with it, but discoverable only through the empirical regress. We are not, however, entitled to affirm of a whole of this kind, which is divisible in infinitum, that it consists of an infinite number of parts. For, although all the parts are contained in the intuition of the whole, the whole division is not contained therein. The division is contained only in the progressing decomposition—in the regress itself, which is the condition of the possibility and actuality of the series. Now, as this regress is infinite, all the members (parts) to which it attains must be contained in the given whole as an aggregate. But the complete series of division is not contained therein. For this series, being infinite in succession and always incomplete, cannot represent an infinite number of members, and still less a composition of these members into a whole.

To apply this remark to space. Every limited part of space presented to intuition is a whole, the parts of which are always spaces—to whatever extent subdivided. Every limited space is hence divisible to infinity.

Let us again apply the remark to an external phenomenon enclosed in limits, that is, a body. The divisibility of a body rests upon the divisibility of space, which is the condition of the possibility of the body as an extended whole. A body is consequently divisible to infinity, though it does not, for that reason, consist of an infinite number of parts.

It certainly seems that, as a body must be cogitated as substance in space, the law of divisibility would not be applicable to it as substance. For we may and ought to grant, in the case of space, that division or decomposition, to any extent, never can utterly annihilate composition (that is to say, the smallest part of space must still consist of spaces); otherwise space would entirely cease to exist—which is impossible. But, the assertion on the other hand that when all composition in matter is annihilated in thought, nothing remains, does not seem to harmonize with the conception of substance, which must be properly the subject of all composition and must remain, even after the conjunction of its attributes in space—which constituted a body—is annihilated in thought. But this is not the case with substance in the phenomenal world, which is not a thing in itself cogitated by the pure category. Phenomenal substance is not an absolute subject; it is merely a permanent sensuous image, and nothing more than an intuition, in which the unconditioned is not to be found.

But, although this rule of progress to infinity is legitimate and applicable to the subdivision of a phenomenon, as a mere occupation or filling of space, it is not applicable to a whole consisting of a number of distinct parts and constituting a quantum discretum—that is to say, an organized body. It cannot be admitted that every part in an organized whole is itself organized, and that, in analysing it to infinity, we must always meet with organized parts; although we may allow that the parts of the matter which we decompose in infinitum, may be organized. For the infinity of the division of a phenomenon in space rests altogether on the fact that the divisibility of a phenomenon is given only in and through this infinity, that is, an undetermined number of parts is given, while the parts themselves are given and determined only in and through the subdivision; in a word, the infinity of the division necessarily presupposes that the whole is not already divided in se. Hence our division determines a number of parts in the whole—a number which extends just as far as the actual regress in the division; while, on the other hand, the very notion of a body organized to infinity represents the whole as already and in itself divided. We expect, therefore, to find in it a determinate, but at the same time, infinite, number of parts—which is self-contradictory. For we should thus have a whole containing a series of members which could not be completed in any regress—which is infinite, and at the same time complete in an organized composite. Infinite divisibility is applicable only to a quantum continuum, and is based entirely on the infinite divisibility of space, But in a quantum discretum the multitude of parts or units is always determined, and hence always equal to some number. To what extent a body may be organized, experience alone can inform us; and although, so far as our experience of this or that body has extended, we may not have discovered any inorganic part, such parts must exist in possible experience. But how far the transcendental division of a phenomenon must extend, we cannot know from experience—it is a question which experience cannot answer; it is answered only by the principle of reason which forbids us to consider the empirical regress, in the analysis of extended body, as ever absolutely complete.

Concluding Remark on the Solution of the Transcendental Mathematical Ideas—and Introductory to the Solution of the Dynamical Ideas.

We presented the antinomy of pure reason in a tabular form, and we endeavoured to show the ground of this self-contradiction on the part of reason, and the only means of bringing it to a conclusion—namely, by declaring both contradictory statements to be false. We represented in these antinomies the conditions of phenomena as belonging to the conditioned according to relations of space and time—which is the usual supposition of the common understanding. In this respect, all dialectical representations of totality, in the series of conditions to a given conditioned, were perfectly homogeneous. The condition was always a member of the series along with the conditioned, and thus the homogeneity of the whole series was assured. In this case the regress could never be cogitated as complete; or, if this was the case, a member really conditioned was falsely regarded as a primal member, consequently as unconditioned. In such an antinomy, therefore, we did not consider the object, that is, the conditioned, but the series of conditions belonging to the object, and the magnitude of that series. And thus arose the difficulty—a difficulty not to be settled by any decision regarding the claims of the two parties, but simply by cutting the knot—by declaring the series proposed by reason to be either too long or too short for the understanding, which could in neither case make its conceptions adequate with the ideas.