Front Page Titles (by Subject) APPENDIX TO PURE NATURAL SCIENCE. Of the System of the Categories. - Kant’s Prolegomena and Metaphysical Foundations of Natural Science.
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APPENDIX TO PURE NATURAL SCIENCE. Of the System of the Categories. - Immanuel Kant, Kant’s Prolegomena and Metaphysical Foundations of Natural Science. 
Kant’s Prolegomena and Metaphysical Foundations of Natural Science, trans. with a Biography and Introduction by Ernest Belfort Bax (2nd revised edition) (London: George Bell and Sons, 1891).
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APPENDIX TO PURE NATURAL SCIENCE.
|The world has a beginning (boundary) in time and space.||The world is infinite in time and space.|
|Everything in the world consists of simple [parts].||There is nothing simple, but everything is composite.|
|There are in the world causes through freedom.||There is no freedom, but all is Nature.|
|In the series of world-causes there exists a necessary being.||There is nothing necessary, but in this series all is contingent.|
The above is the most remarkable phenomenon of the human Reason, of which no instance can be shown in any other sphere. If, as generally happens, we regard the phenomena of the world of sense as things in themselves; if we assume the principles of their connection as universal of things in themselves and not merely as principles valid of experience, as is usual and indeed unavoidable without our Critique; then an unexpected conflict arises, never to be quelled in the ordinary dogmatic way, because both theses and antitheses can be demonstrated by equally evident, clear and irresistible proofs—for I pledge myself as to the correctness of all these proofs—and the Reason thus sees itself at issue with itself, a state over which the sceptic rejoices, but which must plunge the critical philosopher into reflection and disquiet.
One may bungle in metaphysics in many ways, without any danger of being detected in fallacy. For if we only do not contradict ourselves, which is quite possible in synthetic propositions, even though they may be purely invented, we can never in such cases (the conceptions we connect, being mere ideas, which as to their whole content can never be given in experience) be refuted by experience. For how should we decide by experience whether the world exists from eternity, or has a beginning? or whether matter is infinitely divisible, or consists of simple parts? Such conceptions cannot be given in any, even the largest possible experience, and therefore the fallacy of the propositions maintained or denied cannot be discovered by that test.
The only possible case in which the Reason could reveal against its will its secret dialectic, fallaciously given out by it as dogmatic, would be, if it grounded an assertion on a universally admitted axiom, and from another, equally conceded, drew a precisely opposite conclusion, with the greatest logical accuracy. This case is here realised, and indeed in respect of four natural ideas of the Reason whence four assertions on the one hand, and just as many counter-assertions on the other, arise, each as a correct consequence from universally admitted premises, and thereby reveal the dialectical illusion of the pure Reason in the use of these principles, which must otherwise have been for ever hidden.
Here then is a decisive attempt, which must necessarily disclose to us the fallacy lying hidden in the assumptions of the Reason.1 Of two mutually contradictory propositions, both cannot be false, unless the conception at their basis be itself contradictory. For instance, two propositions, a square circle is round and a square circle is not round, are both false. For as regards the first, it is false that the [figure] mentioned is round, because it is square, but it is also false that it is not round, or that it is square, because it is a circle. For in this consists the logical mark of the impossibility of a conception, that under the same assumption two contradictory propositions would be equally false; in other words, because no middle can be conceived between them, nothing at all is cogitated by that conception.
Now, a contradictory conception like the foregoing lies at the basis of the two first antinomies, which I call mathematical, because they are concerned with the addition or division of things similar in Nature; and thence I explain how it happens that thesis and antithesis are alike false.
When I speak of objects in time and space, I do not speak of things in themselves, because of these I know nothing, but only of things in the phenomenon, in other words, of experience, as the special mode of the cognition of objects, which is alone vouchsafed to man. I must not say that what I think in space or in time exists in itself in space and time apart from this my thought; for I should then contradict myself, because space and time, together with the phenomena in them, are nothing existing in themselves and apart from my presentations, but are themselves only modes of presentations, and it is obviously contradictory to say that a mere mode of our presentation exists outside our presentation. The objects of sense exist then only in experience; and to give them a special substantive existence for themselves, apart from or before the latter, is equivalent to imagining that experience can be present without or before experience.
Now, when I inquire as to the size of the world in space and time, it is for all my conceptions just as impossible to say, it is infinite, as it is finite. For neither of them can be contained in experience, because experience is neither possible respecting an infinite space, or an infinite time, or the boundary of the world by an empty space or a previous empty time; these [things] are only ideas. Hence as regards either one or the other kind of determinate quantity, it must lie in the world itself, separate from all experience. But this contradicts the conception of a world of sense, which is only a content of experience, whose reality and connection takes place in presentation, namely, in experience, because it is not a thing in itself, but is itself nothing but a mode of presentation. It follows from the above, that, as the conception of a self-existent world is in itself contradictory, the solution of the problem as to its size will be always fallacious, no matter whether it be affirmatively or negatively attempted.
The same applies to the second antinomy, which concerns the division of phenomena. For these are mere presentations, and the parts exist merely in their presentation, and therefore in their division; in other words, in a possible experience in which they are given, and they only extend as far as the latter reaches. To assume that a phenomenon, for instance, that of body, contains all parts in itself, before all experience, to which nought but possible experience can ever attain, is equal to giving to a mere appearance, which can exist only in experience, a special existence preceding experience, or to say that mere presentations are there before they are met with in the faculty of presentation, which contradicts itself; and so, consequently, does every solution of this misunderstood problem, whether it be maintained that bodies consist of infinitely many parts, or of a finite number of simple parts.
In the first class of antinomy (the mathematical), the fallacy of the assumption consisted in that what is self-contradictory (namely, phenomenon and thing in itself) was represented as capable of union in one idea. But as regards the second, or dynamical class of antinomy, the fallacy of the assumption consists in that what is capable of union is represented as contradictory, and consequently, as in the first case, both contradictory assertions were false; so here, where they are opposed to one another merely through misunderstanding, both may be true.
The mathematical connection necessarily presupposes homogeneity in the connected (in the conception of quantity), while the dynamical by no means requires this. Where the quantity of the extended is concerned, all the parts must be homogeneous, both with each other and with the whole; whereas in the connection of cause and effect, although homogeneity may also be met with, it is not necessary. For the conception of causality, by means of which a thing is posited by something quite distinct therefrom, at least does not require it. If the objects of the sense-world were taken for things in themselves, and the above-cited laws of Nature for laws of things in themselves, the contradiction would be un avoidable. In the same way, if the subject of freedom were presented like other objects as mere appearance, the contradiction would be equally unavoidable; for the same thing would be at once affirmed and denied of the same kind of object in the same sense. But if natural necessity be referred merely to phenomena, and freedom merely to things in themselves, no contradiction arises, in assuming or admitting both kinds of causality, however difficult or impossible it may be to render the latter kind comprehensible.
In the phenomenon, every effect is an event, or something that happens in time; a determination of the causality of its cause (a state of the same), must precede it, upon which it follows according to a uniform law. But this determination of the cause to causality must also be something that takes place, or happens. The cause must have begun to act, otherwise between it and the effect, no succession in time could be conceived. The effect would always have existed, as well as the causality of the cause. Thus, among phenomena, the determination of the cause to the effect must also have arisen, and therefore be just as much as its effect, an event which, in its turn, must have a cause, and so on; and consequently, necessity must be the condition according to which the efficient causes are determined. If, on the other hand, freedom be a characteristic of certain causes of phenomena, it must, as regards the latter as events, be a faculty of beginning them from itself (sponte), i.e., without the causality of the causes themselves having begun, and hence another ground would be necessary to determine its beginning. In that case, however, the cause, as to its causality, must not be subject to time determinations of its state; that is, it must not be phenomenon, but it must be regarded as a thing in itself, and its effects only, as phenomena.1 If one can conceive such an influence of the essences of the understanding on phenomena without contradiction, though necessity would attach to all connection of cause and effect in the sense-world, yet of the cause which is itself no phenomenon, although it lies at the foundation of the latter, freedom would be admitted. Thus Nature and Freedom can be attributed without contradiction to the same thing, at one time as phenomenon, at another, as thing in itself.
We have a faculty within us, not only standing in connection with its subjective determining grounds, which are the natural causes of its actions, and in so far the faculty of a being, belonging to phenomena, but also referable to objective grounds, though these are merely ideas, in so far as they can determine this faculty; and this connection is expressed by ought. The above faculty is termed Reason, and when we contemplate a being (man) simply according to this subjectively determining Reason, it cannot be regarded as an essence of sense, but the quality thought of is the quality of a thing in itself, of the possibility of which, namely, the ought of that which has never happened, and yet the activity of which can be the determination and cause of actions, whose effect is phenomenal in the sense-world, of this we can form no conception whatever. At the same time, the causality of the Reason as concerns its effects in the sense-world would be freedom, so far as objective grounds, which are themselves ideas, are regarded as determining these effects. For its action would then depend not on subjective, and therefore on time-conditions, nor on natural laws, serving to determine these, since grounds of the Reason in general would furnish the rule for actions according to principles, without the influence of circumstances, time, or place.
What I adduce here, is merely meant as an instance for the sake of intelligibility, and does not necessarily belong to our question, which must be decided from mere conceptions, independently of the qualities we meet with in the real world.
I can say now without contradiction, that all actions of rational beings, inasmuch as they are phenomena, met with in any experience, are subject to necessity; but precisely the same actions, with reference to the rational subject, and its capacity of acting according to mere Reason, are free. For what is demanded by necessity? Nothing more than the determinability of every event in the sense-world according to uniform laws; in other words, a reference to Cause in the phenomenon, whereby the thing in itself, lying at its foundation, and its causality, remains unknown. But I say: the natural law subsists alike, whether the rational being [acting] from Reason, and hence through freedom, be the cause of the effects in the sense-world, or whether these are determined by other grounds than those of Reason. For in the first case, the action happens according to maxims, whose effect in the phenomenon will be always in accordance with uniform laws; in the second case, if the action does not happen according to principles of the Reason, it is subordinated to the empirical laws of the sensibility, and in both cases the effects are connected according to uniform laws; more than this we do not require to [constitute] natural necessity, nay, more we do not know respecting it. But in the first case, Reason is the cause of these natural laws, and is hence free; in the second case, the effects follow the mere natural laws of the sensibility, because the Reason exercises no influence upon them; the Reason, however, is not on this account itself determined by the sensibility (which is impossible), and is consequently in this case also free. The freedom does not hinder the natural law of the phenomena, any more than the latter interferes with the freedom of the practical use of the Reason, which stands in connection with things in themselves as determining grounds.
In this way, the practical freedom, namely, that by which the Reason has causality, according to objective determining grounds, is saved, without natural necessity being curtailed in the least, in respect of the same effects as phenomena. The above may also be serviceable as an explanation of what we had to say regarding transcendental freedom, and its union with natural necessity (in the same subject, but not taken in the same connection). For as to this, every beginning of the action of a being, from objective causes, so far as its determining grounds are concerned, is always a first beginning, although the same action in the series of phenomena is only a subaltern beginning, necessarily preceded by a state of the cause determining it, and itself determined by a [state] immediately preceding; so that without falling into contradiction with the laws of Nature, we may conceive of a faculty in rational beings, or in beings generally, in so far as their causality is determined in them, as things in themselves, by which a series of states is begun of themselves. For the relation of the action to objective grounds of the Reason is no relation in time; here, what determines the causality does not precede the action according to time, because such determining grounds [as these] do not present a reference of the objects to sense, or, in other words, to causes in the phenomenon, but to determining causes, as things in themselves, which are not subordinated to time-conditions. Hence, the action may be viewed with regard to the causality of the Reason as a first beginning, but at the same time, as regards the series of the phenomena, as a merely subordinate beginning, and without contradiction, in the former aspect as free, and in the latter, inasmuch as it is merely phenomenon, as subordinate to natural necessity.
As concerns the fourth antinomy, it is solved in the same manner as is the conflict of the Reason with itself, in the third. For if the cause in the phenomenon be only distinguished from the cause of the phenomena, so far as they can be considered as things in themselves, both propositions can subsist beside one another, namely, that no cause takes place anywhere in the sense-world (according to similar laws of causality) whose existence is absolutely necessary; while, on the other hand, this world may be connected with a necessary being as its cause, though of another kind, and according to other laws; the incompatibility of the above two propositions simply resting on the misunderstanding by which what is merely valid of phenomena is extended to things in themselves, both being mixed up in one conception.
This is the arrangement and solution of the whole antinomy, in which the Reason finds itself involved, in the application of its principles to the sense-world, and of which even this (the mere arrangement) would be itself a considerable service to the knowledge of the human Reason, even though the solution of the conflict should not fully satisfy the reader, who has here a natural illusion to combat, which has only recently been presented to him as such, and which he has previously regarded as true. For one consequence of this is inevitable, namely, that seeing it is quite impossible to get free of this conflict of the Reason with itself, so long as the objects of the sense-world are taken for things in themselves, and not for what they are in reality, namely, mere phenomena, the reader is necessitated thereby again to undertake the deduction of all our knowledge à priori, and its examination as given by me, in order to come to a decision in the matter. I do not require more [than this] at present; for if he has but first penetrated deeply enough into the nature of the pure Reason, the conceptions by which the solution of this conflict of the Reason is alone possible, will be already familiar to him, without which circumstance I cannot expect full credit even from the most attentive reader.
III. Theological Idea (Critique, p. 350).
The third transcendental idea, which furnishes material to the most important, but, when merely conducted speculatively, to the exaggerated (transcendent) and thereby dialectical use of the Reason, is the ideal of the pure Reason. The Reason does not here, as with the psychological and cosmological ideas, start from experience, and is not, by a [progressive] raising (Steigerung) of the grounds, misled into an endeavour to contemplate the series in absolute completeness, but wholly breaks therewith, and from mere conceptions of what would constitute the absolute completeness of a thing in general, and consequently by means of the idea of a most perfect original being, descends to the determination of the possibility, and thereby also to the reality, of all other things. For this reason, the mere assumption of a being, which although not given in the series of experience, is nevertheless conceived for the sake of experience, to render comprehensible the connection, order, and unity of the latter, that is, the Idea is more easily distinguishable from the conceptions of experience [in the present] than in the foregoing cases. The dialectical illusion therefore arising from our holding the subjective conditions of our thought for the objective conditions of things themselves, and a necessary hypothesis for the satisfaction of our Reason for a dogma, may be easily exposed to view; and hence I have nothing further to recall on the assumptions of the transcendental theology, for what the Critique has said on this point is comprehensible, clear, and decisive.
General Remark on the Transcendental Ideas.
The objects given us through experience are in many respects incomprehensible, and there are many problems to which the natural law leads us, when it is carried to a certain height, (though always in accordance with these laws,) which can never be solved; as for instance, how it is that substances attract one another. But, if we entirely leave Nature, or in the progress of its connection overstep all possible experience, and thereby immerse ourselves in mere ideas, we cannot then say that the object is incomprehensible, and that the nature of things places insoluble problems before us; for we have in that case, nothing whatever to do with Nature or with given objects, but merely with conceptions, having their origin simply in our Reason, and with mere essences of thought, in respect of which all problems arising from the conception of the same, can be solved, because the Reason can and must certainly give a complete account of its own procedure.1 As the psychological, cosmological, and theological ideas, are simply conceptions of the Reason, not capable of being given in any experience, so the problems which the Reason in respect thereof places before us, are not propounded by the objects, but by mere maxims of the Reason for its own satisfaction, and must be capable of being adequately answered in their totality, which is effected by showing them to be principles [designed] to bring the use of our understanding to thorough agreement, completeness and synthetic unity, and which are in so far valid merely of experience, but of the whole of the latter.
Now, although an absolute whole of experience is impossible, the idea of a whole of knowledge according to principles in general, is what alone can procure a particular kind of unity, namely, that of a system, without which our knowledge is nothing but a patchwork, and cannot be used for the highest end (which is always the system of all ends); by this I understand not merely the practical, but also the highest end of the speculative use of the Reason.
The transcendental ideas express, then, the specific destiny of the Reason, namely, as being a principle of the systematic unity of the use of the understanding. But when this unity of the mode of cognition be viewed as though it depended upon the object of cognition; when we hold that which is merely regulative for constitutive, and persuade ourselves that we can extend our cognition by means of these ideas, far beyond all possible experience in a transcendent manner, notwithstanding that they merely serve to bring experience as nearly as possible to completeness, i.e., to limit its progress by nothing which cannot belong to experience—then this is a simple misunderstanding in judging the special destiny of our Reason and its principles, and a dialectic, partly confusing the use of the Reason in experience, and partly making the Reason to be at issue with itself.
On the determination of the boundary of the Pure Reason.
After all the very clear proofs we have above given, it would be absurd for us to expect to cognise more in any object than what belongs to its possible experience, or to lay claim to the least knowledge of anything whatever which would determine its constitution in itself, unless we assume it to be an object of possible experience. For wherewith shall we effect this determination, inasmuch as time, space, and all the conceptions of the understanding, and still more the conceptions derived from empirical intuition or perception in the sense-world would neither have nor could have any other use than merely to make experience possible, and when if we leave out this condition from the pure conceptions of the understanding, they determine no object whatever, and have no significance anywhere [?].
But it would be a still greater absurdity for us not to admit things in themselves at all, or to wish to give out our experience for the only possible mode of the cognition of objects, in other words, our intuition in space and time for the only possible intuition, and our discursive understanding for the model of every possible understanding, thereby wishing principles of the possibility of experience to be held for the universal conditions of things in themselves.
Our principles, which limit the use of the Reason to possible experience, might accordingly become transcendent, and the limits of our Reason be given out for the limits of things themselves, of which Hume’s Dialogues may serve as an example, if a careful Critique of the boundaries of our Reason did not keep watch on its empirical use, and set a limit to its pretensions. Scepticism originally arose from metaphysics and its anarchical (Polizeilosen) dialectic. At first, to favour the empirical use of the understanding, it might well give out for nugatory and deceptive all that exceeded this; but gradually, as it became evident that the very same principles which we make use of in experience are à priori, and that they led unobserved, and as it seemed with the same right, still farther than experience reaches, a doubt began to be thrown on the principles of experience themselves. Now as to these there is no danger, for herein a healthy understanding will always assert its rights; but there arose a special confusion in science, which could not determine how far, and why only thus far and no farther, the Reason is to be trusted; but this confusion can only be got rid of, and any future relapse prevented, by a formal limitation of the use of our Reason, derived from principles. It is true we cannot form any definite conception of what things in themselves, beyond all possible experience, may be. But we are nevertheless not free to withdraw ourselves wholly from the inquiry as to these; for experience never fully suffices for the Reason; it thrusts us ever farther and farther back for the answer to this question, and leaves us as regards its complete solution dissatisfied; as any one can see from the dialectic of the pure Reason, which on this account has its valid subjective ground. Who can tolerate [the circumstance] that by the nature of our soul we can attain to the clear consciousness of the subject, and to the conviction that its phenomena cannot be explained materialistically without asking what the soul really is, and if no empirical conception suffices [to explain] this, at least assuming a conception of the Reason (of a simple immaterial essence) merely for the above purpose, although we cannot demonstrate its objective reality in any way? Who can satisfy himself in all cosmological questions, as to the size and duration of the world, of freedom or natural necessity, with mere empirical knowledge, since, begin it as we will, every answer given according to the fundamental laws of experience, gives birth to a new question, just as much requiring an answer, and thereby clearly exposing the inadequacy of all physical modes of explanation for the satisfaction of the Reason? Finally, who in the face of the thoroughgoing contingency and dependence of all that he can assume and think according to empirical principles, does not see the impossibility of taking his stand on these, and does not feel himself necessarily impelled, in spite of all prohibition against losing himself in transcendent ideas, to seek rest and satisfaction beyond all conceptions he can verify by experience, in that of a Being, of whom the possibility of the idea in itself cannot indeed be apprehended, but which cannot be refuted, because it is a mere being [essence] of the understanding, and without which the Reason must remain for ever unsatisfied.
Boundaries (with extended beings) always presuppose a space, met with, outside a certain definite place, and enclosing it. Limits do not require this, being mere negations affecting a quantity, so far as it has no absolute completeness. Our Reason, however, sees around it as it were a space for the cognition of things in themselves, although it can never have definite conceptions of them, being limited to phenomena.
As long as the cognition of the Reason is homogeneous, no definite boundaries can be conceived therein. In mathematics and natural science the human Reason recognises indeed limits but no boundaries, i.e., [it recognises] that something exists outside itself, to which it can never attain, but not that it can itself be anywhere terminated in its inner progress. The extension of our views in mathematics and the possibility of new inventions reaches to infinity; and the same can be said of the discovery of new qualities in Nature, and of new forces and laws, through continued experience and the union of the same by the Reason. But, at the same time, it cannot be mistaken that there are limits here, for mathematics refers only to phenomena, and what cannot be an object of sensuous intuition, such as the conceptions of metaphysics and morals, lies wholly outside its sphere, [in a region] to which it can never lead, and which does not at all require it. There is, then, a continuous progress and approach to these sciences, and as it were a point or line of contact. Natural science will never discover for us the inner [nature] of things, namely, that which is not phenomenon, but which can still serve as the highest ground of the explanation of phenomena. But it does not require this for its physical explanations; nay, if such were offered it from another source (e.g., the influence of immaterial beings), it ought to reject it, and on no account to bring it into the course of its explanations, but invariably to base these on that which pertains to experience as object of sense, and which can be brought into connection with our real perceptions, and empirical laws.
But metaphysics leads us to boundaries in the dialectical attempts of the pure Reason (which are not commenced arbitrarily or rashly, but to which the nature of the Reason itself urges us), and the transcendental ideas, as we cannot have intercourse with them, and as they will never allow themselves to be realised, serve, not only to show us the actual boundaries of the use of the pure Reason, but also the way to determine them. And this is also the end and use of this natural disposition of our Reason, which has given birth to metaphysics as its pet child, whose generation, like that of everything else in the world, is not to be ascribed to chance, but to an original germ, wisely organised for great ends. For metaphysics is, perhaps more than any other science, rooted in us in its fundamental features by Nature herself, and can by no means be regarded as the product of a voluntary choice or as chance extension in the progress of experiences (from which it is wholly divided).
The Reason, though all its conceptions and laws of the understanding are adequate in the sense-world, does not find any satisfaction for itself in them, for it is deprived of all hope of a complete solution by questions recurring ad infinitum. The transcendental ideas which have this completion for an object are such problems of the Reason. It sees clearly that the sense-world cannot contain the completeness [required], and therefore just as little can those conceptions which serve simply to the understanding of the same, namely, space and time, and all that we have adduced under the name of pure conceptions of the understanding. The sense-world is nothing but a chain of phenomena, connected according to universal laws, and has therefore no subsistence for itself, being not properly the thing in itself, and only being necessarily referable to that which contains the ground of this phenomenon, to essences that cannot be cognised merely as phenomena but as things in themselves. Only in the cognition of these can Reason hope to see its desire for completeness in the progress from the conditioned to its conditions, once for all satisfied.
We have above (§§ 33, 34) assigned the limits of the Reason in respect of all cognition of mere essences of thought. Now, as the transcendental ideas make the progress up to these necessary, and have thus led us, as it were, to the contact of the full space (of experience) with the void of which we know nothing (to the noumena), we can determine the boundaries of the pure Reason. For in all boundaries there is something positive (for instance surface is the boundary of corporeal space and yet is itself a space; line, a space which is the boundary of the surface; point, the boundary of the line, but still [occupying] a position in space), while, on the other hand, limits contain mere negations. The limits assigned in the paragraph cited, are not sufficient, after we have found that something lies beyond them (although we can never know what this may be in itself). For the question is now, what is the attitude of our Reason in this connection of that which we know, with that which we do not know, and never can know? Here is a real connection of the known with a wholly unknown (and something that will always remain unknown), and even if in this the unknown should not become in the least [degree] more known—which is indeed not to be expected—the conception of this connection must be able, notwithstanding, to be determined and reduced to distinctness.
We are obliged, then, to think of an immaterial essence, an intelligible world, and a highest of all beings (mere noumena), because only in these, as things in themselves, does the Reason meet with the completeness and satisfaction it can never hope for from the derivation of phenomena from their homogeneous ground, because they really refer to something distinct from the latter (and therefore wholly heterogeneous), inasmuch as phenomena always presuppose a thing in itself, and indicate this, [it matters not] whether we may know it more closely or not.
But as we can never know these essences of the understanding as to what they may be in themselves, that is, determinately, but are obliged nevertheless to assume such in relation to the sense-world, and to connect them with it through the Reason, we shall be at least able to cogitate this connection by means of such conceptions as express its relation to the sense-world. For if we cogitate the essence of the understanding, through nothing but pure conceptions of the understanding, we really cogitate thereby nothing definite, and our conception is consequently without meaning; if we cogitate it through qualities borrowed from the sense-world, then it is no longer an essence of the understanding, but is conceived as one of the phenomena, and belongs to the sense-world. We will take an instance from the conception of the Supreme Being.
The deistic conception is an entirely pure conception of the Reason, which, however, only represents a thing containing all reality, without our being able to determine a single one of its [qualities], because for this an instance would have to be borrowed from the sense-world, in which case I should always have to do with an object of sense, and not with something completely heterogeneous, and which cannot be an object of sense. For instance, I attribute understanding to it; but I have no conception whatever of any understanding but of one like my own, namely, of one to which intuitions must be given through the senses, and which occupies itself with reducing these under rules of the unity of the consciousness. But then the elements of my conception would always lie in the phenomenon; yet I was necessitated by the inadequacy of the phenomena to pass beyond this, to the conception of a being in no way dependent on phenomena, or bound up with them, as conditions of its determination. If, however, I sever the understanding from the sensibility in order to have a pure understanding, nothing remains over but the mere form of thought without intuition, by means of which I can cognise nothing determinate as object. For this purpose I should have to conceive another understanding which intuited objects, but of which I have not the least conception, because the human understanding is discursive and can only cognise through universal conceptions. But I am also involved in contradiction if I attribute will to the Supreme Being. For I have this conception only in so far as I derive it from my inner experience, and thereby from the dependence of my satisfaction from objects whose existence we require; but at the foundation of this lies sensibility, which wholly contradicts the pure conception of the Supreme Being. The objections of Hume to Deism are weak, touching no more than the proofs, and never the proposition of the deistic assertion itself. But as regards Theism, which must be arrived at by a closer determination of our, there [viz., in Deism], merely transcendent conception of the Supreme Being, they are very strong, and, according as the conception is constructed, in certain (indeed in all ordinary) cases are irrefragable. Hume always insists, that through the mere conception of an original being, to whom we can attribute none but ontological predicates (eternity, omnipresence, omnipotence) we really think nothing definite, but that qualities expressing an object in concreto must be superadded. It is not enough to say it is Cause, but [we must also say] what is the nature of its causality, as, whether [it operates] through understanding and will; and at this point his attacks on the thing itself, namely, on Theism, commence, whereas before he had only stormed the grounds of proof of Deism, which does not carry any especial danger with it. His dangerous arguments refer entirely to anthropomorphism, which he holds to be inseparable from Theism, and to make it contradictory in itself; while if this be left out, [Theism itself] would also fall, and nothing would remain but a Deism wherewith nothing could be done, which could not avail us for anything, and could not serve as a foundation for religion and morals. If this inevitability of anthropomorphism were certain, the proofs of the existence of a Supreme Being might be what one liked, and all conceded, yet the conception of this Being would never be able to be determined by us, without involving ourselves in contradictions.
But if with the injunction to avoid all transcendent judgments of the pure Reason, we connect the apparently contradictory injunction to proceed to conceptions lying outside the field of its immanent (empirical) use, we shall be aware that both may subsist together, but only on the exact boundary of all admissible use of the Reason; for this belongs as much to the field of experience as to that of essences of thought, and we shall be taught thereby, at the same time, how the above remarkable ideas serve simply, for the determination of the boundaries of the human Reason; namely, on the one hand not to extend cognition of experience in an unbounded manner, so that nothing but mere world remains for us to cognise, and on the other hand not to pass beyond the boundaries of experience, or to seek to judge of things outside the latter as things in themselves.
But we keep to this boundary when we limit our judgment to the relation the world may have to a Being, whose conception itself lies outside all the cognition of which we are capable within the world. For in this case, we do not attribute to the Supreme Being any of the qualities in themselves by which we cogitate objects of experience, and thereby avoid the dogmatic anthropomorphism; but we apply the relations of the same to the world, and thereby allow ourselves a symbolical anthropomorphism, which as a matter of fact only concerns the language and not the object.
When I say we are obliged to regard the world as though it were the work of a supreme understanding and will, I do not really say more than—as a watch, a ship, a regiment is related to the artisan, shipbuilder or general, so is the sense-world (or all that which constitutes the foundation of this sum-total of phenomena) [related] to the unknown, that I cognise, not indeed according to what it is in itself, but according to what it is for me, namely, in respect of the world, of which I am a part.
Such a cognition as this is one according to analogy, which does not signify an imperfect resemblance of two things, as the word is commonly taken [to mean], but a perfect resemblance of two relations between totally dissimilar things.1 By means of this analogy a, for us, adequately defined conception of the Supreme Being remains, although we have left out everything that could determine it simply, and in itself; for we define it as regards the world, and therefore as regards ourselves, and more is not necessary for us. The attacks Hume makes on those who would define this conception absolutely, in that they borrow the materials from themselves and from the world, do not affect us; and moreover he cannot reproach us that there remains nothing over, after the objective anthropomorphism of the conception of the Supreme Being is taken away.
For at the outset, let the deistic conception of an original Being be conceded us as a necessary hypothesis (as Hume does in his Dialogues, in the person of Philo against Cleanthes), in which the original Being is conceived through purely ontological predicates, of substance, cause, &c. This must be done, because the Reason is impelled in the sense-world by mere conditions, which are themselves again conditioned, without the possibility of any satisfaction; it can also be very well done, without lapsing into anthropomorphism, which transfers predicates from the sense-world to a Being quite distinct from the world, inasmuch as these predicates [in our case] are mere categories, affording no definite [conception at all], and hence no conception of it limited to conditions of the sensibility. Nothing can hinder us, therefore, from predicating of this Being a causality through Reason in respect of the world, and so from passing over to Theism without being obliged to attribute to it this Reason, as a quality attaching to it in itself. For as regards the first point, the only possible way of pursuing the use of the Reason in respect of all possible experience in the sense-world, to its highest extent and in thorough agreement with itself, is when a supreme Reason is assumed as a cause of all connections in the world. Such a principle must be throughout advantageous to it, and can never injure it in its natural use. But secondly, the Reason is not transferred as a quality to the original Being in itself, but only in its relation to the sense-world, and thus anthropomorphism is altogether avoided. For here, only the cause of the form of Reason everywhere met with in the world is considered, and to the Supreme Being, so far as it is the ground of this form of Reason in the world, Reason is attributed, but only on the principle of analogy, i.e., in so far as this expression [viz., Reason] indicates what the, to us, unknown ultimate cause of the world has wherewith to determine all things therein, in the highest degree, in accordance with Reason. In this we take care to make use of the quality of Reason, not by its means to conceive God, but [rather] the world, as it is necessary to have the greatest possible use of the Reason in respect of the latter [determined] according to a principle. We confess thereby that the Supreme Being, as to what it may be in itself, is entirely impenetrable to us, and is even unthinkable in a definite manner, and hence we are prevented from making any transcendent use of our conceptions, derived from the Reason as an efficient cause (by means of the will), for determining the divine nature, by qualities that are only borrowed from human nature, and thus from losing ourselves in gross or chimerical conceptions; but, on the other hand, [we are prevented] from inundating the view of the world, [attained] by our conceptions of the human Reason as transferred to God, with hyperphysical modes of explanation, and thus from degrading it, from its proper destination according to which it ought to be a study of mere Nature through the Reason, and not a presumptuous derivation of its phenomena from a supreme Reason. The expression suited to our feeble conceptions will be that we conceive the world as though it orginated from a supreme Reason, as to its reality and as to its inward determination, by which we partly recognise the constitution belonging to it, the world itself, though without presuming to wish to define its cause in itself; and partly, on the other hand, place the ground of this constitution in the relation of the supreme Cause to the world ([viz.] to the form of Reason in the world), without finding the world adequate for this purpose by itself.1
In this way the difficulties seeming to oppose Theism vanish, in that to the principle of Hume, not to push the use of the Reason dogmatically beyond all possible experience, another principle is united, completely overlooked by Hume, namely, not to mistake the field of possible experience for that which bounds itself in the eye of our Reason. Critique of Reason here signifies the true middle path between the dogmatism Hume combated, and the scepticism he would have introduced in its place, a middle path which is unlike other middle paths that attempt to determine themselves as it were mechanically (by taking something from one and something from another), and by which no one is taught a better way, but one, such as can be determined accurately, according to principles.
I have made use of the metaphor of a boundary at the commencement of this observation, in order to fix the limits of the Reason in respect of its appropriate use. The sense-world contains merely phenomena, which are not things in themselves, yet the understanding must assume the latter (noumena), for the very reason that it recognises the objects of experience for mere phenomena. In our Reason both are alike included, and the question is: How does the Reason proceed in determining both fields? Experience, which contains all that belongs to the sense-world, is not bounded by itself; it only attains from one conditioned to another conditioned. That which shall bound it must lie wholly outside it, and this is the field of pure essences of the understanding. But this is for us a blank space, in so far as the determination of the nature of these essences of the understanding is concerned, and thus, when we have to do with dogmatically defined conceptions, we cannot pass beyond the field of possible experience. But as a boundary is itself something positive, belonging as much to what is within as to the space without a given content, so it is a really positive cognition, in which the Reason merely participates, by extending itself to this boundary, in such wise, that it does not attempt to go beyond the boundary, because it finds a blank space before it, wherein it can indeed cogitate forms to things, but cannot cogitate things themselves. But the bounding of the field of experience by something otherwise unknown to it, is a cognition remaining to the Reason in this standpoint, whereby it is not enclosed within the sense-world, neither is it left dreaming [schwärmend] outside it, but limits itself, as befits the knowledge of a boundary, to the relation of that which lies outside the same, to that which is within it.
Natural theology is a conception of this nature, at the boundary of the human Reason, inasmuch as it sees itself necessitated to look beyond to the idea of the Supreme Being (and in a practical connection, also, to that of an intelligible world), not in order to determine anything in respect of this mere essence of the understanding, in other words, anything outside the world of sense, but to guide itself for its own use within the latter, according to principles of the greatest possible unity (theoretically as well as practically). And for this purpose it makes use of the reference of the same to an independent Reason as the cause of all these connections, thereby not merely inventing a being, but inasmuch as outside the world something must necessarily exist (anzutreffen sein) which only the understanding cogitates, determining it [viz., this being] in the above manner, although only on the principle of analogy.
In this way our original proposition remains, which is the result of the whole Critique: “that our Reason can never teach more by its principles à priori than simply objects of possible experience, and even of these no more than what can be cognised in experience.” But this limitation does not prevent it from leading us to the objective boundary of experience, namely, the reference to something which is not itself object of experience, but is nevertheless the highest ground of all experience, without however teaching us anything respecting this in itself, but only with reference to its [viz., the Reason’s] own complete use as directed to its highest end, within the field of possible experience. But this is also all the use that can be reasonably expected or even wished, as concerns it, and with this we have cause to be content.
Thus we have fully exhibited metaphysics according to its subjective necessity, as it is really given in the natural disposition of the human Reason, and indeed in what constitutes its essential purpose. We have found in the course of this investigation, that such a merely natural use of such a disposition of our Reason involves us in extravagant dialectical conclusions, partly apparently, and partly really, conflicting [with one another], if no discipline bridles it and keeps it within limits, which is only possible by means of scientific criticism. And, in addition, [we have found] this fallacious metaphysics to be dispensable to the promotion of the knowledge of Nature, and even prejudicial to it. It always remains, notwithstanding, a task worthy of research, to find out the natural ends aimed at by this disposition in our Reason to transcendent conceptions, since everything in Nature must have been originally designed for some useful purpose.
Such an investigation is here out of place; I confess, moreover, that all I here say respecting the primary ends of Nature is only conjecture, but which may be permitted me in this case, as the question does not concern the objective validity of metaphysical judgments, but refers merely to the natural disposition to the latter, and thus lies outside the system of metaphysics, in that of anthropology.
When I compare all transcendental ideas whose content constitutes the special problem of the natural, pure Reason, compelling it to leave the mere contemplation of Nature and to pass beyond all possible experience, and in this endeavour to produce the thing (be it knowledge or nonsense) called metaphysics, I believe myself to have discovered that this natural disposition is intended to free our conceptions from the chains of experience and the limits of the mere contemplation of Nature, in so far that it may at least see a field opened before it, containing mere objects for the pure Reason, which cannot be arrived at by any sensibility. The purpose is not, indeed, to occupy ourselves speculatively with these objects, (because we can find no firm ground for our feet), but because practical principles, without finding such a space before them for their necessary expectation and hope, could not expand themselves to the universality, the Reason indispensably requires, from a moral point of view.
Now, I find that the psychological idea, however little may be the insight I obtain by its means into the pure nature of the human soul, which is raised above all conceptions of experience, at least sufficiently shows me the inadequacy of the latter, and thereby preserves me from materialism as being a psychological conception of no avail for the explanation of Nature, and besides, as narrowing the Reason in its practical aspect. In the same way the cosmological ideas, by the obvious inadequacy of all possible knowledge of Nature to satisfy the Reason in its justifiable inquiries, serve to keep us from the Naturalism which proclaims Nature for self-sufficing. Finally, as all natural necessity in the sense-world is invariably conditioned, inasmuch as it always presupposes dependence of things on one another, and, as unconditioned necessity must be sought for in the unity of a Cause separate from the sense-world, (but the causality of which, if it were mere Nature, could yet never render comprehensible the existence of the contingent as its consequence;) [this being so,] the Reason frees itself by means of the theological idea from fatalism, as well from that of a blind natural necessity in the coherence of Nature, without a first principle, as in the causality of this principle itself, and leads to the conception of a cause through freedom, in other words, a supreme intelligence. Thus the transcendental ideas serve, if not to instruct us positively, at least to do away with the audacious assertions of materialism, naturalism, and fatalism, which narrow the field of the Reason, and thereby to procure a place for moral ideas outside the region of speculation; and this, as it seems to me, will in some measure explain the above natural disposition.
The practical utility a merely speculative science may have, lies outside the boundaries of this science, and hence can be merely viewed as a scholium, and, like all scholia, not as forming a part of the science itself. At the same time, this reference lies at least within the boundaries of philosophy, especially of that which draws from the sources of pure Reason, where the speculative use of the Reason in metaphysics must have a necessary unity with its practical use in morals. Hence the unavoidable dialectic of the pure Reason in metaphysics must be considered as natural disposition—not merely as an illusion requiring to be resolved, but as a natural institution, as concerns its end—deserving, if possible, to be explained, although this task, being supererogatory, cannot in justice be claimed of metaphysics proper.
As a second scholium, more related to the content of metaphysics, the solution of the problems must be regarded which are discussed in the Critique from pp. 410 to 432. For certain principles of Reason are there expounded, determining the order of Nature, or rather the understanding, which is to seek out her laws through experience, à priori. They seem to be constitutive and legislative in respect of experience, whereas they arise from mere Reason, which cannot be regarded like the understanding as a principle of possible experience. Now whether this agreement rests upon the fact that just as Nature is not itself dependent on the phenomena or their source, the sensibility, but only on the relation of the latter to the understanding; so the thorough-going unity of its use, for the sake of a complete possible experience (in a system), can only pertain to this understanding in its relation to the Reason—whether experience, in other words, stand mediately under the legislation of the Reason—[is a question which] may be further considered by those who desire to investigate the nature of the Reason, apart from its use in metaphysics, and to construct a systematic history of Nature upon general principles. This question I have indeed noticed as important in the book itself, although I have not attempted its solution.1
And thus I conclude the analytical solution of the problem I had myself proposed—How is metaphysics at all possible? having proceeded from that in which its use is really given, at least in its consequences, to the grounds of its possibility.
SOLUTION OF THE GENERAL PROBLEM OF THE PROLEGOMENA.
How is Metaphysics possible as Science.
Metaphysics, as a natural disposition of the Reason, is real, but it is also, in itself, dialectical and deceptive (as was proved in the analytical solution of the third main problem). Hence to attempt to draw our principles from it, and in their employment to follow this natural but none the less fallacious illusion, can never produce science, but only an empty dialectical art, in which one school may indeed outdo the other, but none can ever attain a justifiable and lasting success. In order that, as science, it may lay claim not merely to deceptive persuasion, but to insight and conviction, a Critique of the Reason must exhibit in a complete system the whole stock of conceptions à priori, arranged according to their different sources—the Sensibility, the Understanding, and the Reason; it must present a complete table of these conceptions, together with their analysis and all that can be deduced from them, but more especially the possibility of synthetic knowledge à priori by means of their deduction, the principles of its use, and finally, its boundaries. Thus criticism contains, and it alone contains, the whole plan well tested and approved, indeed all the means whereby metaphysics may be perfected as a science; by other ways and means this is impossible. The question now is not, however, how this business is possible, but only how we are to set about it; how good heads are to be turned from their previous mistaken and fruitless path to a non-deceptive treatment, and how such a combination may be best directed towards the common end.
This much is certain: he who has once tried criticism will be sickened for ever of all the dogmatic trash he was compelled to content himself with before, because his Reason, requiring something, could find nothing better for its occupation. Criticism stands to the ordinary school-metaphysics exactly in the same relation as chemistry to alchemy, or as astronomy to fortune-telling astrology. I guarantee that no one who has comprehended and thought out the conclusions of criticism, even in these Prolegomena, will ever return to the old sophistical pseudo-science. He will rather look forward with a kind of pleasure to a metaphysics, certainly now within his power, which requires no more preparatory discoveries, and which alone can procure for the Reason permanent satisfaction. For this is an advantage upon which metaphysics alone can reckon with confidence, among all possible sciences; namely, that it can be brought to completion and to a durable position, as it cannot change any further, nor is it susceptible of any increase through new discoveries. Since the Reason does not here find the sources of its knowledge in objects and in their intuition (which cannot teach it anything), but in itself; so that when the principles of its possibility are presented completely, and without any misunderstanding, nothing remains for pure Reason to know à priori, or even with justice to ask. The certain prospect of so definite and perfect a knowledge has a special attraction about it, even if all its uses (of which I shall hereafter speak) be set aside.
All false art, all empty wisdom, lasts its time; but it destroys itself in the end, and its highest cultivation is at the same time the moment of its decline. That as regards metaphysics this time has now come, is proved by the state to which it has declined among all cultivated nations, notwithstanding the zeal with which every other kind of science is being worked out. The old arrangement of the university studies preserves its outlines still, a single academy of sciences bestirs itself now and then, by holding out prizes to induce another attempt to be made therein; but it is no longer counted among fundamental sciences, and any one may judge for himself how an intellectually-gifted man, to whom the term great metaphysician were applied, would take this well-meant, but scarcely by any one, coveted, compliment.
But although the period of the decline of all dogmatic metaphysics is undoubtedly come, there are many things wanting to enable us to say that the time of its re-birth by means of a thorough and complete Critique of the Reason, has already appeared. All transitional phases from one tendency to its opposite pass through the state of indifference, and this moment is the most dangerous for an author, but, as it seems to me, the most favourable for the science. For when, through the complete dissolution of previous combinations, party spirit is extinguished, men’s minds are in the best mood for listening gradually to proposals for a combination on another plan. If I say that I hope that these Prolegomena will perhaps make research in the field of criticism more active, and will offer to the general spirit of philosophy, which seems to be wanting in nourishment on its speculative side, a new and very promising field for its occupation, I can already foresee that every one who has trodden unwillingly and with vexation the thorny way I have led him in the Critique, will ask me on what I ground this hope. I answer—on the irresistible law of necessity.
That the spirit of man will ever wholly give up metaphysical investigations is just as little to be expected, as that in order not always to be breathing bad air we should stop breathing altogether. Metaphysics will always exist in the world then, and what is more, [exist] with every one, but more especially with reflecting men, who in default of a public standard will each fashion it in his own way. Now, what has hitherto been termed metaphysics, can satisfy no acute mind; but to renounce it entirely is impossible; hence a Critique of the pure Reason itself must be at last attempted, and when obtained must be investigated and subjected to a universal test, because otherwise there are no means of relieving this pressing requirement, which means something more than mere thirst for knowledge.
Since I have known criticism, on closing the perusal of a work on metaphysics, which had entertained as well as instructed me, by the definition of its conceptions, its variety and its orderly arrangement, in conjunction with its easy style, I could not forbear asking—Has this author brought metaphysics one step farther? I beg the learned men for forgiveness, whose works have in other respects been useful to me, and contributed to the cultivation of the intellectual powers, if I confess that neither in their own nor in my small attempts (to which self-love gives the advantage) have I been able to find that thereby the science has been in the least advanced, and this indeed for the very natural reason that the science did not then exist, and could not be brought together piecemeal, but its germ had to be first fully formed in the Critique. In order, however, to avoid all misconception, it must be remembered from what has gone before, that by analytical treatment our conceptions have indeed been very useful to the understanding, but the science (viz., metaphysics) has not been in the least advanced, because these analyses of conceptions are only materials out of which the science has first to be constructed. We may dissect and define the conception of substance and accident as well as possible; this is useful enough as preparation for its future use. But if I cannot know that in everything that exists, substance continues and only the accidents change, the science would not be furthered in the least by all this dissection. Now, metaphysics has not been able to prove either this proposition, à priori and validly, nor that of adequate cause, much less any more complex, as for instance, one belonging to the theory of the soul or to cosmology, and never any synthetic proposition. Thus nothing has been accomplished by all this analysis, nothing created and nothing promoted, and the science, after so much turmoil and noise, remains where it was in Aristotle’s time, although the arrangements to this end, if the clue to synthetic knowledge à priori had been first found, would indisputably have been much more easily discovered than formerly.
Should any one feel himself offended by what is here said, he can very easily refute the accusation if he will only adduce a single synthetic proposition belonging to metaphysics which admits of being demonstrated in a dogmatic manner à priori; for only when he has achieved this shall I allow that he has really advanced the science, even though the proposition in question may be sufficiently confirmed by common experience. No demand can be more moderate, and more fair, and in the event (unquestionably certain) of non-accomplishment, no statement can be juster than that metaphysics as science has not hitherto existed at all.
I must only forbid two things, in case the challenge be accepted: first, the apparatus of probability and conjecture, which just as ill becomes metaphysics as geometry; and secondly, a decision by means of the magic wand of so-called sound common sense, which every one does not wave, but which regulates itself according to personal characteristics. For as regards the first, nothing can be more absurd than in a system of metaphysics, a philosophy of pure Reason, to attempt to base judgments on probability and conjecture. All that can be known à priori is thereby given out as apodictically certain, and must be proved as such. A geometry or arithmetic might just as well be attempted to be founded on conjectures; (for as concerns the calculus probabilium of the latter, it does not contain probable but perfectly certain judgments, on the degree of possibility in certain cases, under given similar conditions, which in the sum of all possible cases must infallibly follow in accordance with the rule—although in respect of any single instance this is not sufficiently determined). Even in empirical natural science conjectures (by means of induction and analogy) can only be permitted, in such a manner that at least the possibility of what I assume must be quite certain.
With the appeal to sound common sense we are still worse off, if possible, when we have to do with conceptions and principles, not so far as they are valid in respect of experience, but when they would be given out as valid outside the conditions of experience. For what is sound sense? It is the common understanding rightly used. And what is the common understanding? It is the faculty of the cognition and employment of rules in concreto in contradistinction to the speculative understanding, which is a faculty for the cognition of rules in abstracto. Thus, the common understanding will hardly comprehend the rule that all which happens is determined by means of its cause, and never be able to view this rule in its universal bearing. Hence it requires an example from experience, and when it hears that it points to nothing else but what it had always thought, when a window-pane was broken or a household utensil lost, it understands the axiom and admits it. Common understanding has no farther use, then, than to be able to see its rules confirmed in experience (although they really pertain to it à priori), and therefore to regard them à priori and independently of experience belongs to the speculative understanding, and lies wholly outside the horizon of the common understanding. But metaphysics is exclusively occupied with the latter kind of knowledge, and it is certainly a bad sign of a sound understanding to appeal to a protector, having no right of judgment here, and which one otherwise only looks at askance, except when one sees oneself pressed, and does not know how to advise or help oneself in a speculation.
A usual resource employed by these false friends of the common human understanding (who sometimes honour it highly, though they generally despise it) is to say: there must be some propositions, immediately certain, and of which one not only requires to give no proof, but no account whatever, as otherwise we should never come to an end of the grounds of our judgments; but in proof of this assertion they can never bring forward anything undoubted, and which they can attribute immediately to the common human understanding (except the axiom of contradiction, which is inadequate to demonstrate the truth of synthetic judgments) and mathematical propositions; as, for instance, that twice two make four, that between two points there is only one straight line, &c. But these are judgments from which those of metaphysics are totally distinct. For in mathematics I can make (construct) all this by my own thinking, representing it to myself as possible through a conception; I gradually add to the one two, the other two, and myself make the number four; or drawing in thought all sorts of lines from one point to another, can only draw one that is similar in all its parts, equal no less than unequal. But I cannot with my whole power of thought bring out from the conception of one thing the conception of something else, the existence of which is necessarily connected with the first, but must call experience to my aid; and although my understanding à priori offers me such a conception, [viz.] causality (though only in reference to possible experience), I cannot present it à priori in intuition, like the conceptions of mathematics, and thus exhibit its possibility à priori, but the conception together with the principles of its use, if it is to be valid à priori (as is required in metaphysics), demands a demon stration and deduction of its possibility, since otherwise we do not know how far it is valid, and whether it can only be used in experience or [may be used] outside [experience]. Hence, in metaphysics as a speculative science of the pure Reason, we can never appeal to the common human understanding, but when we are obliged to leave it, and to renounce all pure speculative cognition, which must be always a branch of knowledge, and therefore under certain circumstances metaphysics itself and its teaching, a reasonable faith will be found alone possible, and indeed sufficient to our needs, and perhaps even better for us than knowledge itself. Then the aspect of the matter is quite altered. Metaphysics must be a science, not alone as a whole, but in all its parts, else it is nothing; because in speculation of the pure Reason, nothing has a standing but universal notions. But, apart from this, probability and healthy human understanding, have their useful and justifiable employment, but on their own special principles, whose validity always depends on their relation to the practical.
This it is which I hold myself justified in demanding of a system of metaphysics, as science.
On what may be done to make Metaphysics real as Science.
Since none of the ways hitherto trodden have attained this end, and since without a previous Critique of the pure Reason it can never be attained, it seems not unfair to expect that the attempt now laid open to view shall undergo an accurate and painstaking investigation, where it is not deemed more advisable to give up all the claims of metaphysics wholly, in which case, if only the intention be loyally adhered to, there is no objection to be made. If the course of things be taken as it really goes, and not as it should go, there are two classes of judgments, a judgment that precedes examination, and this is in our case the one, when the reader forms a judgment on the Critique of the pure Reason from his system of metaphysics (whereas it ought first of all to prove the possibility of the latter); and there is another judgment that follows examination, where the reader ventures to leave on one side for a time the consequences of critical researches, investigations which might somewhat severely clash with his accepted metaphysics, and first of all examines the grounds from which these consequences may be derived. If what the ordinary metaphysics lays down were demonstrably certain (as with geometry), the first mode of judging would answer; for where the consequences of certain principles conflict with demonstrable truths, these principles must be false, and to be rejected without any further investigation. But if it be not the case that metaphysics has a store of incontestably certain synthetic propositions, and perhaps, so much so, that a number of these, as plausible as the best among them, contradict one another in their consequences; and if there be nowhere any absolutely certain criterion of the truth of properly metaphysical (synthetic) propositions, to be found therein; [in this case] the above mode of judging is inadmissible, and an investigation of the principles of the Critique must precede all judgment as to its worth or worthlessness.
Examination of a Judgment on the Critique that precedes Investigation.
This judgment is to be found in the Göttingen Gelehrten Anzeigen, in the supplement to the third division, of January 19, 1782, page 40 et seq.
When an author who is well acquainted with the subject of his work, and diligent in placing his own reflections in its elaboration, falls into the hands of a critic, who is in his turn keen-sighted enough to discern the points on which the worth or worthlessness of his production rests, who does not cling to words, but goes to the heart of the subject, sifting and testing more than the mere principles which the author takes as his point of departure, the severity of the judgment may indeed displease the latter, but the public is indifferent, as it gains thereby; and the author himself may be contented, as he gets the opportunity of correcting or explaining his positions from the timely examination of a competent judge, in such a manner, that if he believes himself fundamentally right, he can remove in time any stumbling block that might in the result prove prejudicial to his work.
I find myself, with my critic, in quite another position. He seems not to see at all the real matter of the investigation with which (successfully or unsuccessfully) I have been occupied. It is either impatience at thinking out a lengthy work, or vexation at a threatened reform of a science in which he believed he had brought everything to perfection long ago, or, what I am unwilling to imagine, real narrow-mindedness, that prevents him from ever carrying his thoughts beyond his school-metaphysics. In short, he passes impatiently in review a long series of propositions, by which, without knowing their premises, we can think nothing, distributes here and there his blame, the reason of which the reader sees just as little, as he understands the propositions against which it is directed; and hence [his criticism] can neither serve the public as a report, nor damage me in the least, in the judgment of competent men. I should, for these reasons, have passed over this judgment altogether, were it not that it may afford me occasion for some explanations which may in some cases preserve the readers of these Prolegomena from misunderstanding. In order, however, that my critic may most easily attain a point of view from which he may see the whole work in a light most disadvantageous to the author, without venturing to trouble himself with any special investigation, he begins and ends by saying: “This work is a system of transcendent (or, as he translates it, of higher) Idealism.”1 A glance at this line soon showed me the sort of criticism likely to ensue, much as though some one who had never seen or heard of geometry, having found a Euclid, and coming upon various figures in turning over its leaves, were to say, on being asked his opinion of it: “The book is a systematic guide to drawing; the author uses a peculiar language, in order to give dark, incomprehensible directions, which in the end teach nothing more than what every one can effect by a fair natural accuracy of eye, &c.”
Let us see, in the meantime, what sort of an idealism it is that goes through my whole work, although it does not by a long way constitute the soul of the system. The dictum of all genuine idealists from the Eleatic school to Bishop Berkeley, is contained in this formula: “All cognition through the senses and experience is nothing but sheer illusion, and only, in the ideas of the pure Understanding and Reason there is truth.” The principle governing and determining my Idealism throughout, is on the other hand: “All cognition of things merely from pure Understanding or pure Reason is nothing but sheer illusion, and only in experience is there truth.”
But this is the direct contrary of idealism proper; how came I then to use this expression for quite an opposite purpose, and how came my critic to see it everywhere?
The solution of this difficulty rests on something that could have been very easily understood from the general bearing of the work, if it had only been desired to do so. Space and time, together with all that they contain, are not things nor qualities in themselves, but belong merely to the appearances of the latter: up to this point I am one in confession with the above idealists. But these, and amongst them more particularly Berkeley, regarded space as a mere empirical presentation that, like the phenomenon it contains, is only known to us by means of experience or perception, together with its determinations. I, on the contrary, prove in the first place, that space (and also time, which Berkeley did not consider) and all its determinations à priori, can be cognised by us, because, no less than time, it inheres in our sensibility as a pure form before all perception or experience and makes all intuition of the same, and therefore all its phenomena, possible. It follows from this, that as truth rests on universal and necessary laws as its criteria, experience, according to Berkeley, can have no criteria of truth, because its phenomena (according to him) have nothing à priori at their foundation; whence it follows, that they are nothing but sheer illusion; whereas with us, space and time (in conjunction with the pure conceptions of the understanding) prescribe their law to all possible experience à priori, and at the same time afford the certain criterion for distinguishing truth from illusion therein.1
My so-called (properly critical) Idealism is of quite a special character, in that it subverts the ordinary [Idealism], and that through it all cognition à priori, even that of geometry, first receives objective reality, which, without my demonstrated ideality of space and time, could not be maintained by the most zealous realists. This being the state of the case, I could have wished, in order to avoid all misunderstanding, to have named this conception of mine otherwise, but to alter it altogether was impossible. It may be permitted me however, in future, as has been above intimated, to term it the formal, or better still, the critical Idealism, to distinguish it from the dogmatic [Idealism] of Berkeley, and from the sceptical [Idealism] of Descartes.
Beyond this, I find nothing further remarkable in the judgment of the book in question. Its author criticises here and there en gros, a mode prudently chosen, since it does not betray one’s own knowledge or ignorance; a single thorough criticism en detail, had it touched the main question, as is only fair, would have exposed, it may be my error, or it may be the critic’s measure of insight into this species of research. It was, moreover, not a badly conceived plan, in order at once to take from readers (who are accustomed to form their conceptions of books from newspaper reports) the desire to read the book itself, to pour out in one breath a number of passages in succession, torn from their connection, and their grounds of proof and explanations, and which must necessarily sound senseless, especially considering how antipathetic they are to all school-metaphysics; to storm the reader’s patience to nauseation, and then, after having made me acquainted with the sensible proposition that persistent illusion is truth, to conclude with the crude paternal moralisation: to what end, then, the quarrel with accepted language, to what end, and whence, the idealistic distinction? A judgment which turns all that is special to my book, which was previously metaphysically heterodox, into a mere novelty in terminology, proves clearly that my would-be judge has understood nothing of [the subject], and in addition, [has not understood] himself.1
My critic speaks like a man who is conscious of important and superior insight which he keeps hidden; for I am aware of nothing recent with respect to metaphysics that could justify such a tone [as he assumes]. But he does very wrong in withholding his discoveries from the world, for there are doubtless many who, like myself, have not been able to find in all the fine things that have for long past been written in this department, anything that has advanced the science by so much as a finger-breadth; we find indeed the filling out of definitions, the supplying of lame proofs with new crutches, the giving to the body of metaphysics fresh outgrowths or a different figure; but all this is not what the world requires. The world is tired of metaphysical assertions; it wants the possibility of the science, the sources from which certainty therein can be derived, and certain criteria by which it may distinguish the dialectical illusion of the pure Reason from the truth. The critic must possess this key, else he would never have spoken out in such a high tone.
But I am driven to the suspicion that no such requirement of the science has ever entered his thoughts, for in that case he would have directed his judgment to this point, and even a mistaken attempt in such an important matter, would have won his respect. If that be the case, we are once more good friends. He may penetrate as deeply as he likes into metaphysics, without any one hindering him; only as concerns that which lies outside metaphysics, its sources, which are to be found in the Reason, he cannot form a judgment. That my suspicion is not without foundation, is proved by the fact that he does not mention a word about the possibility of synthetic knowledge à priori, the special problem upon the solution of which the fate of metaphysics wholly rests, and upon which my Critique (as well as the present Prolegomena) entirely hinges. The Idealism he encountered, and which he hung upon, was only taken up in the doctrine as the sole means of solving the above problem (although it received its confirmation on other grounds), and hence he must have shown either that the above problem does not possess the importance I attribute to it (even in these Prolegomena), or that by my conception of phenomena, it is either not solved at all, or can be better solved in another way; but I do not find a word of this in the criticism. The critic, then, understands nothing of my work, and possibly also nothing of the spirit and essential nature of metaphysics itself; and it is not, what I would rather assume, the hurry of a man incensed at the labour of plodding through so many obstacles, that threw an unfavourable shadow over the work lying before him, and made its fundamental features incomprehensible.
There is a good deal to be done before a learned journal, it matters not with what care its writers may be selected, can maintain its otherwise well-merited reputation, in the field of metaphysics as elsewhere. Other sciences and branches of knowledge have their standard. Mathematics has it, in itself; history and theology, in profane or sacred books; natural science and the art of medicine, in mathematics and experience; jurisprudence, in law books; and even matters of taste in the examples of the ancients. But for the judgment of the thing called metaphysics, the standard has yet to be found. I have made an attempt to determine it, as well as its use. What is to be done, then, until it be found, when works of this kind have to be judged of? If they are of a dogmatic character, one may do what one likes; no one will play the master over others here for long, before some one else appears to deal with him in the same manner. If, however, they are critical in their character, not indeed with reference to other works, but to the Reason itself, so that the standard of judgment cannot be assumed but has first of all to be sought for, then, though objection and blame may indeed be permitted, yet tolerance must lie at its foundation, since the need is common to us all, and the lack of the necessary insight makes a judicially decisive attitude out of place.
In order, however, to connect my defence with the interest of the philosophical common weal, I propose a test, to be decisive as to the mode, whereby all metaphysical investigations may be directed to their common purpose. This is nothing more than what mathematicians have done elsewhere, in establishing the advantage of their methods by competition, namely, by challenging my critic to demonstrate, as is only just, on à priori grounds, in his way, a single really metaphysical principle, asserted by him, that is, [a principle] synthetic and cognised à priori from conceptions, even one of the most indispensable, as for instance, the principle of the persistence of substance, or of the necessary determination of events in the world by their causes. If he cannot do this (silence being confession), he must admit, that as metaphysics without apodictic certainty of propositions of this kind is nothing at all, its possibility or impossibility must before all things be established in a Critique of the pure Reason; and thereby he is bound either to confess that my principles in the Critique are correct, or to prove their invalidity. But as I can already foresee, that, confidently as he has hitherto relied on the certainty of his principles, when it comes to a strict test he will not find a single one in the whole range of metaphysics he can bring forward, I will concede to him an advantageous condition, which can only be expected in such a competition, and will relieve him of the onus probandi by laying it on myself.
He finds in these Prolegomena and in my Critique (pp. 266–290) eight propositions, of which two and two contradict one another, but each of which necessarily belongs to metaphysics, which must either accept it or refute it (although there is not one that has not in its time been assumed by some philosopher). Now he has the liberty of seeking out any one of these eight propositions at his pleasure, and accepting it without any proof, of which I shall make him a present, but only one (for waste of time will be just as little serviceable to him as to me), and then of attacking my proof of the opposite proposition. If I can save this one, and at the same time show, that according to principles which every dogmatic metaphysics must necessarily recognise, the opposite of the proposition adopted by him can be just as clearly proved, it is thereby established that metaphysics has an hereditary failing, not to be explained, much less set aside, until we ascend to its birthplace, the pure Reason itself, and thus my Critique must either be accepted or a better one take its place; it must at least be studied, which is the only thing I now require. If, on the other hand, I cannot save my demonstration, a synthetic proposition à priori from dogmatic principles is to be reckoned to the side of my opponent, my impeachment of ordinary metaphysics was unjust, and I pledge myself to recognise his stricture on my Critique as justified (although this would not be the consequence by a long way). But to this end it would be necessary, it seems to me, to step out of the incognito, as I do not see how it could otherwise be avoided, that instead of one problem, I should be honoured or attacked with several, from unknown and unqualified opponents.
Proposals as to an Investigation of the Critique upon which a Judgment may follow.
I am indebted to the honoured public for the silence with which it for a long time favoured my Critique, for this proves at least a postponement of judgment, and some supposition that in a work, leaving all beaten tracks and striking out a new one, in which one cannot at once perhaps so easily find one’s way, something may perchance lie, from which an important but at present dead branch of human knowledge may derive new life and fruitfulness; and hence a guardedness against destroying by a hasty judgment the as yet tender shoot. A test of a judgment, delayed for the above reasons, is now before my eye in the Gottaischen gelehrten Zeitung, the thoroughness of which every reader will himself perceive, from the comprehensible and unperverted presentation of a fragment of one of the first principles of my work, without taking into consideration my own suspicious praise.
And now I propose, since an extensive structure cannot be judged of as a whole from a hurried glance to test it piece by piece from its foundations, and thereby to use the present Prolegomena as a general outline with which the work itself may be compared. This notion, if it were founded on nothing more than my conceit of importance, such as vanity commonly attributes to one’s own productions, would be immodest and would deserve to be repudiated with disgust. But now, the interests of speculative philosophy have arrived at the point of total extinction, while the human Reason hangs upon them with inextinguishable affection, and only after having been ceaselessly deceived does it vainly attempt to change this into indifference.
In our thinking age, it is not to be supposed but that many deserving men would use any good opportunity of working for the common interest of the more and more enlightened Reason, if there were only some hope of attaining the [desired] end. Mathematics, natural science, laws, arts, even morality, &c., do not completely fill the soul; there is always a space left over, cut out for the pure and speculative Reason, whose vacuity forces us to seek for apparent employment and entertainment, which is in reality mere pastime, in nonsense, trifling, or extravagance; in order to deaden the troublesome call of the Reason, which in accordance with its nature requires something that can satisfy itself, and not merely subserve other ends or the interests of the appetites. A consideration, therefore, concerning itself with the range of the Reason subsisting for itself, because in it all other cognitions, and even purposes, must meet and unite themselves in a whole, has as I may reasonably suppose a great fascination for every one who has only attempted to extend his conceptions, and I may even say a greater than any other theoretical branch of knowledge, for which he would not willingly exchange it.
I put these Prolegomena forward, therefore, as a plan and clue for the investigation, and not the work itself, because, although I am even now perfectly satisfied with it as far as content, order, and mode of presentation, and the care that I have expended in weighing and testing every sentence before writing it down, are concerned (for it has taken me years to satisfy myself fully, not only as regards the whole, but in some cases even as to the sources of one particular proposition); yet I am not quite satisfied with my exposition in some sections of the doctrine of elements, as for instance, in the deduction of the conceptions of the Understanding, or in that on the parallogisms of the pure Reason, because a certain diffuseness takes away from their clearness, and in place of them, what is here said in the Prolegomena respecting these sections, may be made the basis of the test.
It is the boast of the Germans that where steady and continuous industry are requisite, they can carry things farther than other nations. If this opinion be well-founded, an opportunity, a business, presents itself whose successful issue we can scarcely doubt, and in which all thinking men can equally take part, though they have hitherto been unsuccessful in accomplishing it and in thus confirming the above good opinion. But this is chiefly because the science in question is of so peculiar a kind, that it can be at once brought to completion and to that enduring state that it will never be able to be brought in the least degree farther or increased by later discoveries, or even changed (adornment by greater clearness in some places, or additional uses, I here leave out of account); and this is an advantage no other science has or can have, because there is none so fully isolated and independent of others, and which is concerned with an unmixed faculty of cognition. And the present moment seems, moreover, not to be unfavourable to my expectation, for just now, in Germany, no one seems to know what to occupy himself with, apart from the so-called useful sciences, which is not mere play, but a business possessing an enduring purpose.
[To decide] how the endeavours of the learned may be united in such a purpose, and to discover the means to this end, I must leave to others. In the meantime, it is not my intention to persuade any one merely to follow my propositions, or even to flatter me with the hope of this; but he may, as it occurs to him, append thereto attacks, repetitions, limitations, or confirmation, completion, and extension. If the matter be but investigated from its foundation, it cannot fail that a structure of doctrine, if not my own, shall be erected, that shall be a possession for the future, for which it may have reason to be thankful.
The kind of metaphysics that may be expected, after [thinkers] are perfected in the principles of criticism, and as a consequence of this, need by no means, because the old false feathers have been pulled out, appear poor and reduced to an insignificant figure, but may be in other ways richly and respectably adorned, although to show this here, would take too long. But there are other and great uses that strike one immediately. The ordinary metaphysics had its uses, in that it sought out the elementary conceptions of the pure Understanding in order to make them clear through analysis, and definite by explanation. In this way it was [a species of] culture for the Reason, in whatever direction it might afterwards find good to turn itself; and thus far what it did was all for the best. But this service it subsequently effaced in favouring conceit by venturesome assertions, sophistry by subtle distinctions and adornment, and shallowness by the ease with which it decided the most difficult problems by means of a little school-wisdom, which is only the more seductive the more it has the choice, on the one hand, of taking something from the language of science, and on the other from that of popular discourse, thus being everything to everybody, but in reality nothing at all. By criticism, on the contrary, a standard is given to our judgment, whereby knowledge may be with certainty distinguished from its counterfeit, and firmly founded, being brought into full practice in metaphysics; a species of thought extending its beneficial influence in the end over every other mode of the Reason’s use, at once infusing into it the true philosophical spirit. But the service also that it performs for theology, by making it independent of the judgment of dogmatic speculation, thereby ensuring it completely against the attacks of all such opponents, is certainly not to be valued lightly. For ordinary metaphysics, although it promised the latter much advantage, could not keep this promise, and moreover, by summoning speculative dogmatics to its assistance, did nothing but arm enemies against itself. Extravagance, which cannot come in a rationalistic age, except when it hides itself behind a system of school-metaphysics, under the protection of which it may venture to rant about the Reason, is driven from this, its last hiding-place, by critical philosophy. And last, but not least, it cannot be otherwise than important to a teacher of metaphysics, to be able to say with universal assent, that what he expounds is at last science, and that thereby genuine services will be rendered to the common weal.
THE METAPHYSICAL FOUNDATIONS OF NATURAL SCIENCE.
If the word Nature be merely taken in its formal signification, there may be as many natural sciences as there are specifically different things (for each must contain the inner principle special to the determinations pertaining to its existence), inasmuch as it [Nature] signifies the primal inner principle of all that belongs to the existence of a thing.1 But Nature, regarded in its material significance, means not a quality, but the sumtotal of all things, in so far as they can be objects of our senses, and therefore of experience; in short, the totality of all phenomena—the sense-world, exclusive of all nonsensuous objects. Now Nature, in this sense of the word, has two main divisions, in accordance with the main distinction of our sensibility, one of which comprises the objects of the outer, the other the object of the inner sense; thus rendering possible a two-fold doctrine of Nature, the doctrine of body and the doctrine of soul, the first dealing with extended, and the second with thinking, Nature.
Every doctrine constituting a system, namely, a whole of cognition, is termed a science; and as its principles may be either axioms of the empirical or rational connection of cognitions in a whole, so natural science, whether it be doctrine of body or doctrine of soul, would have to be divided into historical and rational natural science, were it not that the word nature (as implying the deduction of the manifold pertaining to the existence of things, from its inner principle) necessitates a knowledge through reason of its system, if it is to deserve the name natural science. Hence, doctrine of nature may be better divided into historical doctrine of nature, comprising nothing but systematically-ordered facts respecting natural things (which again would consist of description of nature as a system of classes according to resemblances, and history of nature as a systematic presentation of the same at different times and in different places), and natural science. Natural science, once more, would be either natural science properly or improperly so-called, of which the first would treat its subject wholly according to principles à priori, and the second according to laws derived from experience.
That only can be called science (wissenschaft) proper whose certainty is apodictic: cognition that can merely contain empirical certainty is only improperly called science. A whole of cognition which is systematic is for this reason called science, and, when the connection of cognition in this system is a system of causes and effects, rational science. But when the grounds or principles it contains are in the last resort merely empirical, as, for instance, in chemistry, and the laws from which the reason explains the given facts are merely empirical laws, they then carry no consciousness of their necessity with them (they are not apodictically certain), and thus the whole does not in strictness deserve the name of science; chemistry indeed should be rather termed systematic art than science.
A rational doctrine of nature deserves the name of natural science only when the natural laws at its foundation are cognised à priori, and are not mere laws of experience. A natural cognition of the first kind is called pure, that of the second applied, rational cognition. As the word nature itself carries with it the conception of law, and this again the conception of the necessity of all the determinations of a thing appertaining to its existence, it is easily seen why natural science must deduce the legitimacy of its designation only from a pure part of it, [a part] namely, which contains the principles à priori of all remaining natural explanations, and why only by virtue of this portion it is properly science, in such wise, that, according to the demands of the reason, all natural knowledge must at last turn on natural science and there find its conclusion. This is because the above necessity of law inseparably attaches to the conception of nature, and hence must be thoroughly comprehended. For this reason the most complete explanation of particular phenomena upon chemical principles, invariably leaves an unsatisfactoriness behind it, because from these accidental laws, learnt by mere experience, no grounds à priori can be adduced.
Thus all natural science proper requires a pure portion, upon which the apodictic certainty required of it by the reason can be based; and inasmuch as this is in its principles wholly heterogeneous from those which are merely empirical, it is at once a matter of the utmost importance, indeed in the nature of the case, as regards method of indispensable duty, to expound this part separately and unmixed with the other, and as far as possible in its completeness; in order that we may be able to determine precisely what the reason can accomplish for itself, and where its capacity begins to require the assistance of empirical principles. Pure cognition of the reason from mere conceptions is called pure philosophy or metaphysics, while that which only bases its cognition on the construction of conceptions, by means of the presentation of the object in an à priori intuition, is termed mathematics.
What may be called natural science proper presupposes metaphysics of nature; for laws, i.e. principles of the necessity of that which belongs to the existence of a thing, are occupied with a conception which does not admit of construction, because its existence cannot be presented in any à priori intuition; natural science proper, therefore, presupposes metaphysics. Now this must indeed always contain exclusively principles of a non-empirical origin (for, for this reason it bears the name of metaphysics); but it may be either without reference to any definite object of experience, and therefore undetermined as regards the nature of this or that thing of the sense-world, and treat of the laws rendering possible the conception of nature in general, in which case it is the transcendental portion of the metaphysics of nature; or it may occupy itself with the particular nature of this or that kind of thing, of which an empirical conception is given, in such wise, that except what lies in this conception, no other empirical principle will be required for its cognition. For instance: it lays the empirical conception of a matter, or of a thinking entity, at its foundation, and searches the range of the cognition of which the reason is à priori capable respecting these objects; and thus, though such a science must always be termed a metaphysic of nature (namely, of corporeal or thinking nature), it is then not a universal but a particular metaphysical natural science (physics and psychology), in which the above transcendental principles are applied to the two species of sense-objects. But I maintain that in every special natural doctrine only so much science proper is to be met with as mathematics; for, in accordance with the foregoing, science proper, especially [science] of nature, requires a pure portion, lying at the foundation of the empirical, and based upon an à priori knowledge of natural things. Now to cognise anything à priori is to cognise it from its mere possibility; but the possibility of determinate natural things cannot be known from mere conceptions; for from these the possibility of the thought (that it does not contradict itself) can indeed be known, but not of the object, as natural thing which can be given (as existent) outside the thought. Hence, to the possibility of a determinate natural thing, and therefore to cognise it à priori, is further requisite that the intuition corresponding à priori to the conception should be given; in other words, that the conception should be constructed. But cognition of the reason through construction of conceptions is mathematical. A pure philosophy of nature in general, namely, one that only investigates what constitutes a nature in general, may thus be possible without mathematics; but a pure doctrine of nature respecting determinate natural things (corporeal doctrine and mental doctrine), is only possible by means of mathematics; and as in every natural doctrine only so much science proper is to be met with therein as there is cognition à priori, a doctrine of nature can only contain so much science proper as there is in it of applied mathematics.
So long, therefore as no conception is discovered for the chemical effects of substances on one another, which admits of being constructed, that is, no law of the approach or retreat of the parts can be stated in accordance with which (as, for instance, in proportion to their densities) their motions, together with the consequences of these, can be intuited and presented à priori (a demand that will scarcely ever be fulfilled), chemistry will be nothing more than a systematic art or experimental doctrine, but never science proper, its principles being merely empirical and not admitting of any presentation à priori; as a consequence, the principles of chemical phenomena cannot make their possibility in the least degree conceivable, being incapable of the application of mathematics.
But still farther even than chemistry must empirical psychology be removed from the rank of what may be termed a natural science proper; firstly, because mathematics is inapplicable to the phenomena of the internal sense and its laws, unless indeed we consider merely the law of permanence in the flow of its internal changes; but this would be an extension of cognition, bearing much the same relation to that procured by the mathematics of corporeal knowledge, as the doctrine of the properties of the straight line does to the whole of geometry; for the pure internal intuition in which psychical phenomena are constructed is time, which has only one dimension. But not even as a systematic art of analysis, or experimental doctrine, can it ever approach chemistry, because in it the manifold of internal observation is only separated in thought, but cannot be kept separate and be connected again at pleasure; still less is another thinking subject amenable to investigations of this kind, and even the observation itself, alters and distorts the state of the object observed. It can never therefore be anything more than an historical, and as such, as far as possible systematic natural doctrine of the internal sense, i.e. a natural description of the soul, but not a science of the soul, nor even a psychological experimental doctrine. This is the reason why, in the title of this work, which, properly speaking, contains the axioms of corporeal doctrine, we have employed, in accordance with the usual custom, the general name of natural science, because this designation in the strict sense is applicable to it alone, and hence occasions no ambiguity.
But to render possible the application of mathematics to the doctrine of body, by which alone it can become natural science, principles of the construction of conceptions belonging to the possibility of matter in general must precede. Hence a complete analysis of the conception of a matter in general must be laid at its foundation; this is the business of pure philosophy, which for the purpose makes use of no special experiences, but only of those which it meets with in separate (although in themselves empirical) conceptions, with reference to pure intuitions in space and time (according to laws, essentially depending on the conception of nature in general), thus constituting it a real metaphysic of corporeal nature.
All natural philosophers, who wished to proceed mathematically in their work, have hence invariably (although unknown to themselves) made use of metaphysical principles, and must make use of such, it matters not how energetically they may otherwise repudiate any claim of metaphysics on their science. Without doubt by the latter they understood the illusion of manufacturing possibilities at pleasure, and playing with conceptions, perhaps quite incapable of being presented in intuition, and possessing no other guarantee of their objective reality than that they do not stand in contradiction with themselves. But all true metaphysics is taken from the essential nature of the thinking faculty itself, and therefore in nowise invented, since it is not borrowed from experience, but contains the pure operations of thought, that is, conceptions and principles à priori, which the manifold of empirical presentations first of all brings into legitimate connection, by which it can become empiricalknowledge, i.e. experience. These mathematical physicists were thus quite unable to dispense with such metaphysical principles, and amongst them, not even with that which makes the conception of their own special subject, namely, matter, available à priori, in its application to external experience (as the conception of motion, of the filling of space, of inertia, etc.). But to allow merely empirical principles to obtain in such a question, they rightly held as quite unsuited to the apodictic certainty they desired to give to their natural laws, and hence they preferred to postulate such, without investigating their sources à priori.
But it is of the utmost importance in the progress of the sciences, to sever heterogeneous principles from one another, to bring each into a special system, so that it may constitute a science of its own kind, and thereby to avoid the uncertainty springing from their confusion, owing to our not being able to distinguish to which of the two, on the one hand the limitations, and on the other the mistakes occurring in their use, are to be attributed. For this reason I have regarded it as necessary to present in one system the first principles of the pure portion of natural science (physica generalis) where mathematical constructions traverse one another, and at the same time the principles of the construction of these conceptions; in short, the possibility of a mathematical doctrine of nature itself. This separation, besides the uses already mentioned, has the special charm, which the unity of knowledge brings with it, if we take care that the boundaries of the sciences do not run into one another, but occupy properly their subdivided fields.
It may serve as a second ground for gauging this procedure, that in all that is called metaphysics the absolute completeness of the sciences may be hoped for, in such a manner as can be promised by no other species of knowledge, and therefore, just as in the metaphysics of nature generally, so here also, the completeness of corporeal nature may be confidently expected; the reason being, that in metaphysics the object is considered merely according to the universal laws of thought, but in other sciences as it must be presented according to data of intuition (empirical as well as pure). Hence the former, because the object must be invariably compared with all the necessary laws of thought, must furnish a definite number of cognitions, which can be fully exhausted; but the latter, because it offers an endless multiplicity of intuitions (pure or empirical), and therefore of objects of thought, can never attain to absolute completeness, but can be extended to infinity, as in pure mathematics and empirical natural knowledge. This metaphysical corporeal doctrine I believe myself to have, as far as it reaches, completely exhausted, but do not affect thereby to have achieved any great work.
The scheme for the completeness of a metaphysical system, whether of nature in general, or of corporeal nature in particular, is the table of the categories.1 For there are not any more pure conceptions of the Understanding, which concern the nature of things. Under the four classes of Quantity, Quality, Relation, and finally Modality, all the determinations of the universal conception of a matter in general, and, therefore, of all that can be thought à priori respecting it, that can be presented in mathematical construction, or given in experience as its definite object, must be capable of being brought. There is no more to do in the way of discovery or addition, although certainly, should there be anything lacking in clearness or thoroughness, it may be made better.
Hence the conception of matter had to be carried out through all the four functions of the conceptions of the the understanding (in four divisions), in each of which a new determination of the same was added. The fundamental determination of a something that is to be an object of the external sense, must be motion, for thereby only can this sense be affected. The understanding leads all other predicates pertaining to the nature of matter back to this, and thus natural science is throughout either a pure or an applied doctrine of motion. The metaphysical foundations of natural science may thus be brought under four main divisions, of which the first—motion considered as pure quantum, according to its composition, without any quality of the movable, may be termed Phoronomy; the second, which regards it as belonging to the quality of the matter, under the name of an original moving force, may be called Dynamics; and the third, where matter with this quality is conceived as by its own reciprocal motion in relation, appears under the name of Mechanics; and the fourth, where its motion or rest [is conceived], merely in reference to the mode of presentation or modality, in other words as determined as phenomenon of the external sense, is called Phenomenology.
But besides the above internal necessity, whereby the metaphysical foundations of the doctrine of body are not only to be distinguished from physics, which employs empirical principles, but even from the rational premises of the latter, in which the employment of mathematics is to be met with, there is an external, and, though only accidental, at the same time an important reason, for separating its thorough working-out from the general system of metaphysics, and for presenting it systematically as a special whole. For if it be permissible to indicate the boundaries of a science, not merely according to the construction of its object, and its specific kind of cognition, but also according to the aim that is kept in view as a further use of the science itself, and it is found that metaphysics has engaged so many heads, and will continue to engage them, not in order to extend natural knowledge (which could be done much more easily and certainly by observation, experiment, and the application of mathematics to external phenomena), but in order to attain to a knowledge of that which lies wholly beyond all the boundaries of experience, of God, Freedom, and Immortality; [in this case] one gains in the promotion of this object, if one liberates it from a shoot springing indeed from its own stem, but only detrimental to its regular growth, and plants this [shoot] apart, without thereby mistaking its origination, or ignoring its entire growth from the system of general metaphysics. This does not affect the completeness of the latter, but it facilitates the uniform progress of this science towards its goal, if in all cases where the universal doctrine of body is required, one can call to aid the separate system of such a science, without encumbering it with the larger system [viz. of metaphysics in general]. It is indeed very remarkable (though it cannot here be thoroughly entered into), that universal metaphysics, in all cases where it requires instances (intuitions) to procure significance for its pure conceptions of the understanding, must always take them from the universal doctrine of body; in other words, from the form and principle of external intuition; and if these are not found to hand in their entirety, it gropes uncertainly and tremblingly amid mere empty conceptions. Hence the wellknown disputes, or at least the obscurity in questions, as to the possibility of an opposition of realities, of intensive quantity, &c., by which the understanding is only taught, through instances from corporeal nature, what the conditions are under which the above conceptions can alone have objective reality, that is, significance and truth. And thus a separate metaphysics of corporeal nature does excellent and indispensable service to the universal [metaphysics], in that it procures instances (cases in concreto) in which to realise the conceptions and doctrines of the latter (properly the transcendental philosophy), that is, to give to a mere form of thought sense and meaning.
I have in this treatise followed the mathematical method, if not with all strictness (for which more time would have been necessary than I had to devote to it), at least imitatively, not in order, by a display of profundity, to procure a better reception for it, but because I believe such a system to be quite capable of it, and that perfection may in time be obtained by a cleverer hand, if stimulated by this sketch, mathematical investigators of nature should find it not unimportant to treat the metaphysical portion, which anyway cannot be got rid of, as a special fundamental department of general physics, and to bring it into unison with the mathematical doctrine of motion.
Newton, in the preface to his mathematical principles of natural science (after having remarked that geometry only requires two of the mechanical actions which it postulates, namely, to describe a straight line and a circle) says: geometry is proud of being able to achieve so much while taking so little from extraneous sources.1 One might say of metaphysics, on the other hand: it stands astonished, that with so much offered it by pure mathematics it can effect so little. In the meantime, this little is something which mathematics indispensably requires in its application to natural science, which, inasmuch as it must here necessarily borrow from metaphysics, need not be ashamed to allow itself to be seen in company with the latter.
METAPHYSICAL FOUNDATIONS OF PHORONOMY.
Matter is the movable in space; space, which is itself movable, is termed material or relative space; that in which all motion must in the last resort be conceived (which is therefore itself absolutely immovable), is termed pure or absolute space.
As in Phoronomy nothing is to be discussed but motion, its subject, namely matter, has here no other quality attributed to it than movability. It can therefore itself be valid for one point so far, and in Phoronomy we abstract from all internal construction, hence also, from the quantity of the movable, and concern ourselves only with motion, and what can be regarded as quantity therein (velocity and direction). If the expression body is sometimes used here, it occurs only to anticipate in a measure the application of the principles of Phoronomy to the following more definite conceptions of matter, in order that the exposition may be less abstract and more comprehensible.
If I am to explain the conception of matter not by a predicate, applying to it as object, but only by the relation to the faculty of knowledge, in which the presentation can be primarily given me, matter is every object of the external sense, and this would be its mere metaphysical explanation. But space would be simply the form of all external sensuous intuition (whether this accrued to the external object we call matter in itself, or remained merely in the construction of our sense, a point which does not enter into the present question). Matter, in contradistinction to form, would be that which in external intuition, is an object of feeling, and consequently the properly empirical of sensible and outward intuition, because it cannot be given at all à priori. In all experience something must be felt, and this is the real of sensuous intuition. In consequence, space, in which we are to institute experience respecting motions, must be capable of being felt, that is, of being indicated by that which can be felt, and this, as the sum-total of all objects of experience, and itself an object of the same, is called empirical space. Now this, as material, is itself movable; but a movable space, if its movement is to be able to be perceived, presupposes again an enlarged material space in which it is movable, and this again another, and so on to infinity.
Thus all motion that is an object of experience is merely relative; the space in which it is perceived is a relative space, which again moves itself perhaps in an opposite direction, in a space further enlarged, and therefore the matter moved in reference to the first may be termed at rest in relation to the second; and these alterations of the conception of motion go forward with the alteration of the relative space to infinity. To assume an absolute space, that is, one which, because it is not material, can be no object of experience as given for itself, means assuming something which, neither in itself nor in its consequences (motion in absolute space), can be perceived, for the sake of the possibility of experience, which nevertheless must always exist without it. Absolute space is in itself nothing and no object at all, but signifies merely every other relative space that I can at any time conceive outside the given space, and that I can extend beyond each given space to infinity; one that includes the [given space], and in which I can assume it as moved. But since I have the enlarged, although still material, space only in thought, nothing is known to me of the matter indicating it. I abstract from this, and it is conceived, therefore, as a pure, non-empirical and absolute space, with which I can compare, and in which I can conceive as movable, each empirical space, and therefore, which is itself always regarded as immovable. To constitute it a real thing means confounding the logical universality of any space, with which I can compare each empirical [space] as being included in it with a physical universality of real compass, and misunderstanding the reason in its idea.
I may observe in conclusion that as the movability of an object in space cannot be known à priori and without the teaching of experience, it could not for the same reason be counted in the Critique of pure Reason amongst the pure conceptions of the understanding, and this conception as empirical could only find a place in a natural science, as applied metaphysics, which occupies itself with a conception given through experience, although according to principles à priori.
Motion of a thing is the change of the external relations of the same to a given space.
I have already laid the conception of matter at the basis of the conception of motion; but, as I wished to determine the latter independently of the conception of extension, and thus could consider matter only in one point, I had to admit the use of the common explanation of motion as change of place. Now that the conception of matter is to be explained universally, and therefore as applicable to moved bodies, this definition is inadequate, for the place of every body is a point. If one wishes to determine the distance of the moon from the earth, one wishes to know the distance of their places, and to this end one does not measure from any point of the surface, or of the interior of the earth, to any point of the moon at pleasure, but takes the shortest line from the central point of the one to the central point of the other, and therefore, in each of these bodies there is only one point that constitutes its place. Now a body may move without changing its place, as the earth in turning on its axis; but its relation to external space changes notwithstanding, for it presents for instance its different sides to the moon in the course of the twenty-four hours, from which all kinds of transformative effects result on the earth. Only of a movable, i.e., physical point can one say: motion is always a change of place. It might be objected against this explanation that internal motion (e.g., fermentation) is not included therein; but the thing which one speaks of as in motion must so far be regarded as unity. That matter, as, for instance, a cask of beer, is in motion signifies something different to the beer in the cask being in motion. The motion of a thing is not one and the same with motion in this thing; but the question is here only of the former. The application of this conception to the latter case is afterwards easy.
Motions may be circular (without change of place) or progressive, and these again may either enlarge the space or be motions limited to a given space. Of the first kind are rectilinear, or even non-rectilinear, [motions] that do not return in upon themselves. Of the second are those that return in upon themselves. The latter are again either circular or oscillating motions. The first cover the same space always in the same direction; the second alternatingly in an opposite direction, like a swaying pendulum. To both belong trembling (motus tremulus), which, though not a progressive motion of a body, is nevertheless a reciprecative motion of a matter, which does not change its place on the whole thereby, as the vibrations of a bell that has been struck, or the tremblings of air set in motion by sound. I merely make mention of these different kinds of motion in a Phoronomy, because with all that are not progressive the word velocity is generally used in another sense than with the progressive, as the following observation shows.
In every motioa direction and velocity are the two momenta for consideration, when one abstracts from all other qualities of the movable. I presuppose here the ordinary definition of both; but that of direction has sundry limitations. A body moved in a circle changes its direction continuously, so that, until its return to the point from which it started, all is comprised in a surface of merely possible directions, and yet one says it moves itself always in the same direction, as, for instance, the planet from evening to morning.
But what is the side, in this case, towards which the motion is directed? A question related to the one: Upon what does the internal distinction of spirals, otherwise similar and even equal, rest, but of which one species winds to the right, and the other to the left; or the winding of the kidney-bean, and of the hop, of which the one runs round its pole like a corkscrew, or as sailors express it against the sun, and the other with the sun? This is a conception that allows itself to be constructed indeed, but as conception does not admit of being made plain by universal marks in the discursive mode of cognition. In the things themselves (e.g., in those rare cases of the human subject where on dis ection all the parts agree according to physiological rules with other human subjects, only that all the viscera are found displaced, either to the right or the left, against the usual order) there can be no imaginable difference in the internal consequences, and yet there is a real mathematical and indeed internal difference, whereby two circular movements, differing in direction but in all other respects alike, notwithstanding their not being completely identical, nevertheless correspond. I have elsewhere shown1 that as this difference, though it must be given in intuition, does not admit of being brought to clear conceptions, and therefore intelligibly explained (dari, non intelligi), it affords a good substantiating ground of proof for the proposition: that space generally, belongs, not to the qualities or relations of the things in themselves, for this would necessarily have to admit of reduction to objective conceptions, but merely to the subjective form of our sensible intuition of things or relations, which, as to what they may be in themselves, must remain wholly unknown. But this is a deviation from our present business, in which we must necessarily treat space as a quality of the things we have in consideration, namely, corporeal entities, because these themselves are merely phenomena of the external sense, and only require to be explained as such in this place. As concerns the conception of velocity, this expression acquires in use a variable meaning. We say: the earth moves more rapidly on its axis than the sun, because it does so in a shorter time, although the motion of the latter is much more rapid. The circulation of the blood of a small bird is much more rapid than that of a man, although the streaming motion in the former has, without doubt less velocity; and so with the vibrations of elastic matters. The shortness of the time of return, whether of a circulating or oscillating motion, constitutes the ground of this employment, in which, if otherwise misunderstanding be avoided, there is no harm done. For the mere increase in the hurry of return, without increase of spacial velocity, has special and very important effects in nature, of which, in the circulation of the juices of animals, perhaps not enough notice has been taken. In Phoronomy we use the word velocity merely in a spacial signification: C = S/T.1
Rest is the permanent present (præsentia perdurabilis) in the same place; permanent is that which exists throughout a time, i.e. lasts.
A body, which is in motion, is in every point of the line it passes over—a moment. The question remains, whether it rests therein, or moves. Without doubt the latter, one will say; for, only in so far as it moves is it present in this point. But let us assume the motion in this way:
that the body describes the line A B forwards and backwards, from B to A, with uniform velocity in suchwise that, since the moment it is in B is common to both motions, the motion from A to B is described in half a second, that from B to A also in half a second, but both together in a whole second, so that not the smallest portion of time has been expended on the presence of the body in B; in this way, without the least increase of these motions, the latter, which took place in the direction B A, can be changed into that in the direction B a, which lies in a straight line with A B, and hence the body, while it is in B, must be regarded not as at rest, but as moved. It would have therefore also to be considered as moved in the first motion, returning in upon itself in the point B, which is impossible; because, in accordance with what has been assumed, it is only a moment that belongs to the motion A B, and at the same time to the equal motion B A, which is opposed to the former one and conjoined with it in one and the same moment of complete lack of motion; consequently if this constitutes the conception of rest, in the uniform motion A a, rest of the body must also be proved in every point (e.g., in B), which contradicts the above assertion. Again, let the line A B be represented as over the point A perpendicularly, so that a body rising from A to B, after having lost its motion through gravity in the point B, would fall back again from B to A. Now I ask whether the body in B is to be considered as moved or at rest? Without doubt, it will be said, at rest; because all previous motion has been taken from it, after it has reached this point, and a uniform motion back is as yet to follow, consequently is not present, and the lack of motion, it will be added, is rest. In the first case, however, of an assumed uniform motion, the motion B A could not commence otherwise, than by the motion A B having previously ceased, and that from B to A being non-existent, and consequently there being in B a lack of all motion, whereby, according to the usual explanation, rest would have to be assumed; but we may not assume it, because at a given velocity, no body may be conceived as at rest in any point of its uniform motion. Upon what, then, is the assumption of rest based in the second case, since this rising and falling is only separated by a moment? The ground lies in the latter motion not being conceived as uniform with the given velocity, but as being at first uniformly delayed, and afterwards uniformly accelerated, in suchwise that the velocity in point B is not delayed wholly, but only up to a certain degree, smaller than any velocity that can be given, by which, if instead of falling back, the line of its fall B A were placed in the direction B a; in other words, the body were conceived as still rising, it would, as with a mere moment of velocity (the resistance of gravity being set aside), pass over, in any given time, however great, a space smaller than any space that could be given, and therefore its place (for any possible experience) would not change to all eternity. In consequence of this, it assumes a state of lasting presence in the same place, that is, of rest, although owing to the continuous action of gravity, that is, of the change of this state, the latter is immediately abolished. To be in a permanent state and to persist therein (if nothing else shifts it) are two distinct conceptions, of which one does no violence to the other. Thus rest cannot be explained through the lack of motion, which, as = o, does not admit of being constructed at all, but must be explained by permanent presence in the same place, and as this conception is constructed by the presentation of a motion with infinitely small velocity, throughout a finite time, it can be used for the subsequent application of mathematics to natural science.
To Construct the conception of a composite motion means to present à priori in intuition a motion so far as it arises from two or more given [motions] united in one movable.
For the construction of conceptions, it is requisite that the condition of their presentation should not be borrowed from experience, and thus that they should not presuppose certain forces, the existence of which can only be deduced from experience, or, in short, that the condition of the construction should not be itself a conception incapable of being given à priori in intuition; as for instance, that of cause and effect, action and resistance, &c. It is here especially to be observed that Phoronomy is throughout, primarily construction of motions in general as quantities, and that, as it has for its subject, matter merely as something movable, and of which no quantity therefore comes into consideration, it has to determine these motions alone as quantities (as concerns their velocity as well as their direction, and indeed their combination) à priori. For thus much must be established entirely à priori and intuitionally, for the sake of applied mathematics. For the rules of the connection of motions through physical causes, that is forces, never admit of being fundamentally expounded before the principles of their composition generally are previously laid down mathematically as a foundation.
Every motion, as object of a possible experience, may be viewed, at pleasure, as motion of a body in a space that is at rest, or as rest of the body, and motion of the space in the opposite direction with equal velocity.
In order to make an experience of the motion of a body it is requisite that not only the body but also the space in which it moves should be objects of external experience, or in other words, material. An absolute motion, therefore, that is, in reference to a non-material space, is unsuited to any experience whatever, and hence for use, nothing (even if one were willing to admit absolute space to be something in itself). But even in all relative motion the space itself, because it is assumed as material, may again be conceived as resting or moved. The first happens when, beyond the space in reference to which I regard a body as moved, there is no more extended space given, that includes it (as when in the cabin of a ship I see a ball moved on the table); the second, when, outside this space there is another space given, that includes it (as, in the case mentioned, the bank of the river), since I can view the nearest space (the cabin) with respect to the latter as moved and the body itself as at rest. As thus it is absolutely impossible to determine of an empirically given space, it matters not how extended it may be, whether, with respect to a still greater space enclosing it, it be itself moved or not, it must be wholly the same for all experience, and for every consequence drawn from experience, whether I choose to regard a body as moved or at rest, and the space as moved in the opposite direction with an equal velocity. Once more: as absolute space is nothing for any possible experience, the conceptions are the same whether I say a body moves with respect to this given space, in this direction, with this velocity, or whether I conceive it as at rest, and ascribe all this [motion] to the space, but in an opposite direction. For every conception is wholly of the same kind as the latter, of whose distinction from the former no instance is possible, and only with reference to the connection we wish to give it in the understanding is it different.
We are, moreover, not in a position to postulate a fixed point, in any experience, in reference to which it could be defined what motion and rest mean absolutely; for everything given us in this way is material, and hence movable, and (as we know of no extreme boundary of possible experience in space) it may be really moved without our being able to perceive this motion. Of this motion of a body in empirical space I can assign one portion of the given velocity to the body, the other to the space, but in the opposite direction, and the whole possible experience as concerns the consequences of these two combined motions is wholly the same whether conceived of the body alone as moved with the whole velocity or (conceiving it) as at rest, and the space as moved with the same velocity in the opposite direction. I assume here all motions as rectilinear. For as concerns the non-rectilinear it is not in all respects the same, whether I am at liberty to regard the body as moved (e.g., the earth in its daily rotation), and the surrounding space (the starry heaven) as resting, or the latter as moved and the former as resting; but we shall treat of this more particularly in the sequel. Thus in Phoronomy, where I consider the motion of a body only in relation to the space (on the rest or motion of which it has no influence at all), it is quite undetermined and arbitrary whether any or all, or how much, of the velocity of the given motion I attribute to the one or to the other.
Farther on in mechanics where a moved body is to be considered in real relation to other bodies, in the space of its motion, this will not be any longer so entirely indifferent, as will be demonstrated in its proper place.
The composition of motion is the presentation of the motion of a point as bound together in one with two or more motions of the same.
In Phoronomy, as I can cognise the matter by no other property but that of movability, and can consider it itself therefore only as a point, the motion can only be viewed as description of a space, yet so that I do not, merely pay attention to the space described, as in geometry, but also to the time [involved] therein; in other words, to the velocity with which a point describes the space. Phoronomy is thus the pure doctrine of the quantity (mathesis) of motions. The definite conception of a quantity is the conception of the generation of the presentation of an object through the composition of the homogeneous. Now, as motion is nothing homogeneous, but again motion Phoronomy is a doctrine of the composition of the motions of the same point according to its direction and velocity i.e., the presentation of a single motion as one that comprises within it two or perhaps several motions in one, at the same time, in the same point, so far as they together constitute one, that is, are one with this motion, but not in so far as they produce the latter as causes produce their effects. In order to find the motion arising from the composition of several—as many as one likes—one has only, as with the production of all quantities, first to seek out those that are compounded under given conditions, of two; and thereupon combine this with a third, etc. In consequence the doctrine of the composition of all motions is reducible to that of two. But two motions of one and the same point that are present at the same point may be distinguished in a double manner, and as such be combined in a triple way therein. Firstly, they occur at the same time either in one and the same line, or in different lines; the latter are motions enclosing an angle. Those that occur in one and the same line are either contrary to one another in direction or maintain the same direction. As all these motions are contemplated as taking place alone, there results immediately from the relation of the lines, that is, of the spaces of motion described in equal time, the relation of velocity. Thus there are three cases:—1. As two motions (it matters not whether of equal or unequal velocities) combined in one body in the same direction, are to constitute a resultant compound motion; 2. As Two motions of the same point (of equal or unequal velocity), combined in contrary directions, are, through their composition, to constitute a third motion in the same line; 3. Two motions of a point, with equal or unequal velocities, but in different lines, enclosing an angle, are considered as compounded.
The composition of two motions of one and the same point, can only be conceived by one of them being presented in absolute space, but, instead of the other, a motion of an equal velocity in the contrary direction of the relative space [being presented] as identical with it.
First Case.—Two motions in the same line and direction arrive at the same time in one and the same point.
Let two velocities, AB and ab, be presented as contained in one velocity of the motion. Let these velocities be assumed, for the time, as equal, AB = ab; in this case I assert they cannot be presented at once in the same point, in one and the same space (whether absolute or relative). For, because the lines AB and ab, denoting the velocities, are properly spaces, passed over in equal times, the composition of these spaces AB and ab = BC, and, therefore, the line AC, as the sum of the spaces, cannot but express the sum of both velocities. But the parts AB and BC do not, individually, present the velocity = ab; for they are not passed over in the same time as ab. Thus, the double line AC, which is traversed in the same time as the line ab, does not represent the double velocity of the latter, as was required. Hence the composition of two velocities in one direction in the same space does not admit of being sensuously presented.
On the contrary, if the body A be presented as moved in absolute space with the velocity AB, and I give to the relative space, a velocity ab = AB in addition, in the contrary direction ba = CB; this is the same as though I distributed the latter velocity to the body in the direction AB (axiom 1). But the body moves itself, in this case, in the same time through the sum of the lines AB and BC = 2 ab, in which it would have traversed the line ab = AB only, and yet its velocity is conceived as the sum of the two equal velocities AB and ab, which is what was required.
Second Case.—Two motions in exactly contrary directions are united in one and the same point.
Let AB be one of these motions, and AC the other in the opposite direction, the velocity of which we assume here to be equal to that of the first; in this case the very idea of representing two such motions, at the same time, in one and the same space, and in one and the same point, in short, the case of such a composition of motions would itself be impossible, which is contrary to the assumption.
On the other hand, let the motion AB be conceived as in absolute space, and instead of the motion AC in the same absolute space, let the contrary motion CA of the relative space [be conceived] with the same velocity, which (according to axiom 1) is equal to the motion AC, and may thus be entirely substituted for it; in this case two exactly opposite and equal motions of the same point, at the same time, may be very well presented. Now, as the relative space is moved with the same velocity CA = AB in the same direction with the point A, this point, or the body, present therein, does not change its place in respect of the relative space; i.e., a body moved in two exactly contrary directions with equal velocity, rests, or generally expressed, its motion is equal to the difference of the velocities in the direction of the greater (which admits of being easily deduced from what has already been demonstrated).
Third Case.—Two motions of the same point are presented as combined according to directions that enclose an angle.
The two given motions are AB and AC, whose velocity and directions are expressed by these lines, but the angle, enclosed by the latter, by BAC (it matters not whether it be a right angle, as in this case, or any other angle). If these two motions are to occur, at the same time, in the directions AB and AC, and indeed in the same space, they would not be able to occur, at the same time, in both these lines AB and AC, but only in lines running parallel to these. It would have, therefore, to be assumed, that one of these motions effected a change in the other (namely, the deviation from the given course), although the directions remained the same on either side. But this is contrary to the assumption of the proposition, which indicates by the word composition, that both the given motions are contained in a third, and must therefore be one with this, and not that, by one changing the other, a third is produced.
On the other hand, let the motion AC be taken as proceeding in absolute space, but instead of the motion AB, the motion of the relative space in the opposite direction. Let the line AC be divided into three equal parts, AE, EF, FG. Now, while the body A in absolute space passes over the line AE, the relative space, and therewith the point E, passes over the space Ee = MA; while the body passes over the two parts together = AF, the relative space and therewith the point F, describes the line Ff = NA; while, finally, the body passes over the whole line AC, the relative space, and therewith the point C describes the line Cc = BA. All this is the same as though the body A had passed over in these three divisions of time, the lines Em, Fn and CD = AM, AN, AB, and in the whole time in which it passes over AC, had passed over the line CD = AB. It is therefore at the last moment in the point D, and in the whole time gradually in all points of the diagonal line AD, which expresses the direction as well as the velocity of the compound motion.
Geometrical construction demands that one quantity should be identical with the other, or two quantities in composition, with a third, not that they should produce the third as causes, which would be mechanical construction. Complete similarity and equality, in so far as they can only be cognised in intuition, is congruity. All geometrical construction of complete identity rests on congruity. This congruity of two motions combined with a third (in short, the motu composito itself) can never take place, when the two former are presented in one and the same space, i.e. relative [space]. Hence all attempts to demonstrate the above proposition in its three cases, have always been mechanical solutions only, inasmuch, namely, as though moving causes by which a given motion was combined with another, were made to produce a third, the proofs that the former were the same as the latter, and as such, admitted of being presented in pure intuition à priori [were not given].
When, for instance, a velocity AB is termed double, nothing else can be understood thereby, but that it consists of two simple and equal [velocities] AB and BC, (see Fig. 1). But if a double velocity be explained by saying that it is a motion by which a doubly great space is passed over in the same time, something is here assumed which is not necessarily implied, namely, that two equal velocities may be combined in the same way as two equal spaces, for it is not in itself obvious that a given velocity consists of smaller [velocities]; and in the same way that a rapidity consists of slownesses as a space does of smaller [spaces]. For the parts of the velocity are not outside one another, as the parts of the space; and if the former are to be considered as quantity, the conception of their quantity, as it is intensive, must be constructed in a different manner to that of the extensive quantity of space. But this construction is possible in no other way than by the mediate composition of two equal motions, one of which is that of the body, the other that of the relative space in the contrary direction, but which, for this reason, is completely identical with an equal motion of the body in the previous direction. For in the same direction two equal velocities would not admit of being compounded in one body, except through external moving causes; for instance, a ship carrying the body with one of these velocities, while another movable force, immovably bound up with the ship, impresses upon the body the second velocity, which is equal to the previous one. In this it must always be presupposed that the body maintains itself in free motion with the first velocity when the second enters; but this is a natural law of moving forces, which cannot come into consideration when the question is simply how the conception of velocity is constructed as a quantity; so much as to the addition of velocities to one another. But when the question is of the subtraction of one from the other, this latter is easily conceivable, if the possibility of a velocity, as quantity by addition, has once been admitted; yet this conception cannot be so easily constructed, for to this end two contrary motions must be combined in one body; and how is this to happen? Immediately, namely, in respect of the same resting space, it is impossible to conceive of two equal motions in contrary directions in the same body; but the idea of the impossibility of these two motions in one body is not the conception of its rest, but of the impossibility of the construction of this composition of contrary motions, which is nevertheless assumed in the proposition as possible. Now this construction is not otherwise possible, than by the combination of the motion of the body with the motion of the space as has been demonstrated. Finally, as concerns the composition of two motions, whose direction encloses an angle, they do not admit of being conceived in a body, in reference to one and the same space, if one of them be not affected by an external continuous inflowing force (for instance, a vessel bearing the body onward), while the other maintains itself unaltered, or generally [expressed]: one must have as a basis, moving forces, and the production of a third movement from two combined forces, but this, although the mechanical carrying out of that which contains a conception, is not its mathematical construction, which has only to render intuitable what the object is (as quantum), not, how it may be transformed by nature or art, by means of sundry implements and forces. The composition of motions, in order to determine their relation to others as quantity, must take place according to the rules of congruity, which is only possible, in all three cases, by means of the motion of the space that is congruous with one of the two given motions, whereby both are congruous with the compound [motion].
Thus Phoronomy, not as pure doctrine of motion, but as pure doctrine of the quantity of motion, in which matter is conceived by no other quality but that of mere movability, contains nothing but this single proposition, carried out in the three cases adduced, of the composition of motion, and indeed of the possibility of rectilinear motion alone, not of curvilinear; for, because in the latter the motion is continuously changed in direction, a cause of this motion, which cannot be merely space, must be brought to bear. That only the single case in which the directions of the same enclose an angle, is usually understood by the designation compound motion, does some detriment to the principle of the division of a pure philosophical science generally, although not to physics: for, as concerns the latter, all the three cases treated in the above proposition admit of being adequately prsented in the third alone. For when the angle enclosing the two given motions is conceived as infinitely small, it contains the first [case]; but if it be conceived as only divided in an infinitely small degree from a single straight line, it contains the second case; so that, in the proposition alrady stated respecting composite motion, all three cases mentioned by us, are capable of being given as in a universal formula. But in this way one could not learn to comprehend the qualitative doctrine of motion in its parts à priori, which in many respects is also useful.
If any one cares to connect the three parts in question of the universal Phoronomic proposition with the scheme of the subdivision of all pure conceptions of the understanding, here, especially with that of the conception of quantity, he will observe: that, as the conception of a quantity always contains that of the composition of the homogeneous, the doctrine of the composition of motions is at the same time the pure doctrine of quantity therein; and indeed that in all three momenta furnished by space, the unity of line and direction, the plurality of directions in one and the same line, and finally the totality of directions as well as of lines, according to which the motion can take place, it contains the determination of all possible motion as quantum, although its quantity (in a movable point) consists merely in velocity. This observation only has its uses in transcendental philosophy.
METAPHYSICAL FOUNDATIONS OF DYNAMICS.
Matter is the movable, in so far as it fills a space. To fill a space means to resist everything movable, which endeavours by its motion to press into a certain space. A space that is not filled is an empty space.
This is the dynamical explanation of the conception of matter. It presupposes the Phoronomic, but adds thereto a property that is related as cause to an effect, namely, the capacity of resisting a motion within a certain space. This could not come into consideration in the foregoing science, even when we had to do with the motions of one and the same point in opposite directions. This filling of space keeps a certain space free from the intrusion of any other movable when the motion of the latter is directed to any place within this space. On what the resistance of matter on all sides rests, and what it is, now remains to be investigated. But it may be already seen from the above explanation, that matter is not here considered as resisting when it is driven from its place, and thus as itself moved (this case will hereafter come into consideration as mechanical resistance), but only when the mere space of its own extension is to be diminished. The expression is used to occupy space, namely, to be immediately present in all its points, in order to indicate thereby the extension of a thing in space. But inasmuch as it is not defined in this conception, what effect, or whether any effect at all, arises from this presence, whether in resisting others that are attempting to press into it, or whether it signifies merely a space without matter, in so far as it is a sumtotal of several spaces, just as one may say of every geometrical figure, “it occupies a space” (it is extended); or even whether there be something in space necessitating another movable to penetrate deeper into the same (attracting others); because, I say, by the conception of the occupying of a space, all this is undetermined; so, to fill a space is a closer definition of the conception to occupy a space.
Matter fills a space, not by its mere existence, but by a special moving force.
The penetration into a space (in the moment of commencement this is called the endeavour to penetrate) is a motion. The resistance to motion is the cause of its diminution, and also its change into rest. Now nothing can be connected with any motion, as lessening or destroying it but another motion of the same movable in the opposite direction (phoronomic proposition). Thus the resistance offered by a matter in the space which it fills, to all impression of another [matter], is a cause of the motion of the latter in the opposite direction; but the cause of a motion is called moving force. Thus matter fills its space by moving force and not by its mere existence.
Lambert and others called the property of matter, by which it fills a space, solidity (a rather ambiguous expression), and maintained that we must assume it in everything which exists (substance), at least in the outer world of sense. According to their notions, the presence of something real in space, must carry with it this resistance by its very conception, in other words according to the principle of contradiction; and must exclude the coexistence of anything else, in the space of its presence. But the principle of contradiction does not preclude any matter from advancing, in order to penetrate into a space in which another [matter] exists. Only when I attribute to that which occupies a space, a power of repelling everything externally movable which approaches it, do I understand how it involves a contradiction, that in the space which a thing occupies, another [thing] of the same kind should penetrate. Here the mathematician has assumed something as a first datum of the construction of the conception of a matter, which itself does not admit of being further constructed. Now he can begin his construction of a conception from any datum he pleases, without committing himself again to the further explanation of this datum; but he is nevertheless not thereby permitted to explain the former as something wholly incapable of any mathematical construction, in order by this means to prevent a return to the first principles of natural science.
Attractive force is that moving force whereby a matter may be the cause of the approach of others to itself (or, which is the same thing, whereby it opposes the retreat of others from itself).
Repulsive force is that whereby a matter can be the cause of repelling others from itself (or, which is the same thing, whereby it resists the approach of others to itself). The latter we shall also sometimes term driving, and the former, drawing force.
These are the only two moving forces of matter admiting of being conceived. For all motion which one matter can impress upon another, as in this respect each of them is only considered as a point, must always be regarded as distributed in the straight line between two points. But in this straight line only two kinds of motion are possible, one, by which the above points recede from one another, and a second by which they approach one another. But the force which is the cause of the first motion is called repulsive force, and that of the second attractive force. Thus, only these two kinds of forces, as such, to which all the forces of motion in material nature must be reduced, are capable of being conceived.
Matter fills its spaces by the repulsive forces of all its parts, i.e., by its own force of extension, which has a definite degree, beyond which smaller or larger [degrees] can be conceived to infinity.
Matter fills a space only by moving force (proposition 1), this being such as to resist the impression, that is, the approach of others. Now this is a repulsive force (explanation II.). Thus matter fills its space, and indeed all the parts thereof, by repulsive forces only, because otherwise a part of its space would not be filled (against the assumption), but would only be enclosed. But the force of an extended by virtue of the repulsion of all its parts is a force of extension (expansive). Thus matter fills its space by its own force of extension; which was the first point. Beyond every given force a greater must be conceived, for that beyond which there is no greater possible would be one, whereby, in a finite time, an infinite space would be passed over (which is impossible). Further, beyond every given moving force a smaller must be able to be conceived (for the smallest would be that, by the infinite addition of which to itself, throughout any given time, no finite velocity could be generated, but this signifies the lack of all moving force). Thus below every given degree of a moving force, a smaller must always be able to be given; which is the second [point]. The force of extension, therefore, whereby all matter fills its space, has its degree, which is never the greatest or smallest; but beyond which, greater as well as smaller, may be found to infinity.
The expansive force of a matter is termed elasticity. Now as the former is the basis on which the filling of space, as an essential property of all matter, rests, this elasticity must be termed original; seeing that it cannot be derived from any other property of matter. All matter is accordingly originally elastic.
Because beyond every extending force a greater moving force can be found, which might work against it, and would thus diminish the space it is seeking to extend; in which case the latter would be termed a compressive force; so for every matter a compressive force must be able to be found, capable of driving it from every space it fills into a narrower space.
A matter penetrates another in its motion when it completely abolishes the space of its extension by compression.
When, in the sucker of an air-pump that is filled with air, the piston is driven nearer the bottom, the air-matter is compressed. Now if this compression could be carried so far that the piston completely touched the bottom (without the least amount of air escaping), the air-matter would be penetrated; for the matters, between which it is, leaving no superfluous room for it, it would exist between the bottom and the piston, without occupying a space. This penetrability of matter by external compressive forces, if one were willing to assume, or even conceive, such, would be termed mechanical. I have reasons for distinguishing by such a limitation, this penetrability of matter from another [kind], the conception of which is perhaps just as impossible as that of the present, and of which I may hereafter have occasion to make some mention.
Matter can be compressed to infinity, but it can never be penetrated, by a matter, it does not signify how great its pressing force.
An original force, by which a matter seeks to extend itself on all sides over a given space occupied by it, must, enclosed in a smaller space, be greater, and compressed into an infinitely small space, be infinite. Now, for any given extensive force of matter, a greater compressive force may be found that compels it into a smaller space, and so on to infinity; which was the first [point]. But for the penetration of a matter, a compression into an infinitely small space, and therefore an infinitely compressive force, is required, which is impossible. Hence, a matter cannot be penetrated by the compression of any other [matter]; which is the second [point].
I have, at the commencement of this demonstration, assumed that an extending force, the more it is narrowed, must operate so much the more strongly in the opposite [direction]. Now this would not apply to all kinds of elastic forces, [including those] that are merely derivative: but with matter possessing essential elasticity, in so far as it is matter in general, filling a space, it may be postulated. For expansive force exercised from all points towards all sides, constitutes its very conception. But the same quantum of expanding forces, brought into a narrower space, must, in every point of the latter, repel so much the more strongly, in inverse proportion to the smallness of the space in which a given quantum of force diffuses its activity.
The impenetrability of matter, resting on resistance, which increases proportionately to the degree of the compression, I term relative; but that which rests on the assumption that matter, as such, is capable of no compression at all, is termed absolute impenetrability. The filling of space with absolute impenetrability may be termed mathematical; that with merely relative [impenetrability] dynamical filling of space.
According to the mere mathematical conception of impenetrability (which assumes no moving force as originally inherent in the matter), no matter is capable of compression, except in so far as it contains within itself empty spaces. Matter, therefore, as matter, resists all impression unconditionally and by absolute necessity. But according to our explanation of this property, impenetrability rests on a physical basis; for the extensive force renders it primarily possible, as an extended that fills its space. But as this force has a degree that overpowers, and hence diminishes the space of extension, that is, can be impressed upon the same up to a certain degree, by a given compressive force, but only in such wise that the entire penetration, inasmuch as it would require an endless compressive force, is impossible; [therefore] the filling of space must be regarded only as relative impenetrability.
Absolute impenetrability is, indeed, neither more nor less than a qualitas occulta For we ask the cause, why matters in their motion cannot penetrate one another; and receive the answer: because they are impenetrable. The appeal to repulsive force is free from this objection. For although this likewise cannot be explained further, according to its possibility, and hence must be admitted as a fundamental force, it nevertheless gives a conception of an active cause and its laws, in accordance with which the effect, namely, the resistance in the filled space, may be estimated according to its degrees.
Material substance is that in space, which for itself, namely, separated from all else existing outside it in space, is movable. The motion of a part of matter whereby it ceases to be a part, is separation. The separation of the parts of a matter is physical division.
The conception of a substance signifies the ultimate subject of existence, namely, that which does not itself belong, as mere predicate to the existence of another. Now matter is the subject of all that, in space, which can be counted [as belonging] to the existence of things; for outside it, no subject would be able to be conceived, but space itself; and this is not a conception containing anything existent, but merely the necessary conditions of the external relation of possible objects to our sense. Matter then, as the movable in space, is substance therein. But just in the same way are all its parts substances, in so far as one can say of them that they are subjects, and not merely predicates of other matters; and hence must again themselves be termed matter. But they are themselves subjects, if they are something movable existing in space, and hence not in combination with other adjacent parts. The independent motion of matter, then, or any of its parts, is a demonstration at once, that this movable, and every movable part of it, is substance.
Matter is divisable to infinity into parts, of which each is again matter.
Matter is impenetrable by its own original force of extension (proposition 3); but this is only the result of the repulsive forces of each point in a space filled with matter. Now the space that is filled by matter is mathematically divisible to infinity; that is, its parts can be distinguished to infinity, although they cannot be moved, and consequently cannot be separated (according to demonstrations of geometry). But in a space filled with matter, every part contains the same repulsive force, to counteract all other forces, on all sides; in other words, to drive them back, and in the same way to be driven back by them, that is, to be moved to a distance from them. Hence, every part of a space filled with matter is, movable in itself, and consequently separable from those remaining, as material substance, by physical division. So far, then, as the mathematical divisibility of space filled by a matter reaches, thus far does the possibility of the physical division of the substance that fills it, reach. But the mathematical division extends to infinity, and consequently also the physical; that is, all matter, is divisible to infinity, and indeed to parts, of which each is itself again material substance.
By the demonstration of the infinite divisibility of space, that of matter has not, by a long way, been proved, if it has not previously been established, that in every part of space material substance exists, that is, that parts in themselves movable are to be met with. For if a monadologist wished to assume that matter consisted of physical points, each of which (for this reason) had no movable parts, but nevertheless, filled a space by mere repulsive force, he would still be able to admit that this space, although not the substance acting in it (in other words, the sphere of the latter’s activity, though not the acting movable subject itself), could be divided by the division of its spaces. He would thus compound matter of physical by indivisible parts, and yet allow it to occupy space in a dynamical manner.
But by the above demonstration, the monadologist is entirely deprived of this resort. For, thereby it is clear, that in a filled space there can be no point that does not itself resist repulsion on all sides in the same way as it is repelled; in other words, as a reacting subject, in itself movable, existing outside every other repulsive point; and hence that the hypothesis of a point filling a space by its mere driving force, and not by means of other equal repulsive forces, is impossible. In order to make this, and thereby also the demonstration of the previous proposition apparent, one must assume that A is the place of a monad in space, that ab is the diameter of the sphere of its repulsive force, and therefore that aA is its semi-diameter; so between a, where the impression of an external monad in space, occupying the sphere in question, is understood, and the central point of the latter [viz., the sphere], A, a point c is possible to be indicated (in accordance with the infinite divisibility of space). Now, if A resist that which seeks to impress itself on a, c must resist both the points A and a. For if this were not so, they would approach one another with impunity; consequently A and a would meet in the point c, i.e. the space would be penetrated. Something must thus exist in c that resists the impression of A and a, and thus repels the monad A as much as it is repelled by it. As now, repulsion is a movement, c is something movable in space; in other words, matter, and the space between A and a, could not be filled by the sphere of the activity of a single monad, neither could the space between c and A, and so on to infinity.
When mathematicians conceive the repulsive forces of the parts of elastic matters in their greater or lesser compression, as increasing or diminishing in a certain proportion to their distances from one another (for instance, that the smallest parts of the air repel each other in inverse proportion to their distances from one another, because their elasticity stands in inverse proportion to the spaces in which they are compressed), one would wholly mistake their meaning and misapply their language were one to attribute to the conception in the object itself, what [nevertheless] necessarily belongs to the process of the construction of a conception. For, according to the above, all contact can be conceived as an infinitely small distance, which, moreover, must necessarily happen in distance where a larger or smaller space is to be conceived as entirely filled by the same quantity of matter, that is, by an identical quantum of repulsive forces. By an infinitely divisible [thing], therefore, no real distance of parts, which, with all extension of the space of the whole, always constitute a continuum, may be assumed, although the possibility of this extension can only be made comprehensible under the idea of an infinitely small distance.
Mathematics can indeed, in its internal employment, be quite indifferent to the chicane of a mistaken metaphysics, and rest in the certain possession of its evident assertions of the infinite divisibility of space, no matter what objections a sophistry, clinging to mere conceptions, may throw in its way; but in the application of its propositions, which apply to space, to substance, which fills it, it must rely on a test according to mere conceptions; in other words, on metaphysics. The above proposition is itself a proof of this. For it does not follow necessarily that matter is physically divisible to infinity, although it is so in a mathematical connection, every part of space being again a space, and hence always including within itself parts external to one another; but this cannot prove that in every possible part of this filled space, there is substance, which, consequently, separated from all the rest, exists as in itself, movable; something has been wanting then hitherto, to the mathematical demonstration, without which it can have no certain application to Natural Science, and this defect has been obviated in the proposition above given. But as concerns the remaining attacks of metaphysics on the at present physical proposition, of the infinite divisibility of matter, the mathematician must entirely resign himself to the philosopher, who, apart from this, through these objections, betakes himself into a labyrinth, out of which it is difficult for him to find his way, even in questions immediately concerning him, and hence has enough to do on his own account, without the mathematician mixing himself up in the business. If, namely, matter be infinitely divisible, then (concludes the dogmatic metaphysician), it consists of an infinite number of parts; for a whole must originally contain within itself all the parts into which it can be divided, in their entirety. But the latter proposition is also indubitably certain of every whole as a thing in itself, and, therefore, although one cannot admit matter, or even space, to consist of infinitely many parts (inasmuch as it is a contradiction to think of an infinite number, the conception of which itself implies that it can never be conceived as fully ended), one must resolve either to defy the geometrician by saying space is not infinitely divisible, or to irritate the metaphysician [by saying], space is no property of a thing in itself, and hence, matter is no thing in itself, but the mere phenomenon of our external sense generally, just as space is its essential form.
The philosopher now finds himself in a strait between the horns of a dangerous dilemma. To deny the first proposition, that space is divisible to infinity, is a vain undertaking, for mathematics does not admit of being reasoned away; but yet to regard matter as a thing in itself, in other words, space as property of the thing in itself, and to deny the above proposition, is one and the same thing. He sees himself thus necessitated to depart from this assertion, however common and suited to the common understanding it may be; but of course only under the condition, that in the event of his reducing matter and space to the phenomenon (hence the latter [viz. space] to the form of our external sensuous intuition, and so [constituting] both, not things in themselves, but only subjective modes of the presentation to us, of objects in themselves unknown), he should be helped out of the difficulty as to the infinite divisibility of matter, while it yet does not consist of infinitely many parts. This latter easily admits of being conceived by the Reason, although impossible to construct and render intuitable. For of that which is only real by its being given in presentation, there is not more given than is met with in the presentation, that is, so far as the progressus of presentations reaches. Thus we can only say of phenomena, the division of which goes on to infinity, that there exist so many of the parts of the phenomenon, as we give of them, that is, as far as we can ever subdivide. For the parts, as belonging to the existence of a phenomenon exist only in thought, namely, in their division itself. Now though the division proceeds to infinity, it is never given as infinite, and hence it does not follow that the divisible contains an infinite number of parts in itself and outside our presentation merely because its division is infinite. For it is not the thing, but only its presentation, whose division could be continued to infinity, and in the object that is unknown in itself, which has also a cause, and yet can be never completed and consequently fully given, it proves no real infinite number, for this would be an express contradiction. A great man who has perhaps contributed more than any one else to maintain the reputation of mathematics in Germany, has more than once turned aside metaphysical claims to upset the propositions of geometry relative to the infinite divisibility of space with the well-grounded observation, that space only belongs to the phenomenon of external things; but he has not been understood. The proposition was taken as though he meant: space appears to us, otherwise it is a thing or relation of things in themselves, but the mathematician considers it only as it appears. Instead of this he ought to have been understood [as meaning] that space is no quality appertaining to anything outside our senses, but only to the subjective form of our sensibility, under which objects of our external sense, unknown to us as to their construction in themselves, appear to us, this appearance being termed matter. By the foregoing misunderstanding, space was always conceived as a quality [existing] independently, outside our faculty of presentation, but which the mathematician only thought of according to common conceptions, that is, confusedly (for so appearance [phenomenon] is commonly explained); it ascribed the mathematical proposition of the infinite divisibility of matter, a proposition presupposing the highest clearness in the conception of space, to a confused presentation of space, which the geometrician laid at his foundation. In this way, it remained open to the metaphysician to compound space of points, and matter of simple parts, and thus in his opinion to bring clearness into the conception. The ground of the confusion lies in a misunderstood monadology, which does not belong to the explanation of natural phenomena, but is a platonic conception of the world, carried out by Leibnitz. This is correct in itself, in so far as it [the world] is regarded, not as object of sense, but as thing in itself; but is nevertheless a mere object of the understanding, though it lies at the foundation of the phenomena of sense. The composite of things in themselves must consist in the simple; for the parts must here be given before all composition. But the composite in the phenomenon consists not of the simple, because in the phenomenon, which can never be given otherwise than as composite (extended), the parts can only be given through division, and thus not before the composite, but in it. Hence Leibnitz’s opinion, so far as I understand, [did not consist] in explaining space by the arrangement of simple entities side by side, but rather in [regarding it] as corresponding to a merely intelligible, for us unknown, world by its side, and maintained nothing more than what has elsewhere been shown, namely, that space, together with matter of which it is the form, comprises, not the world of things in themselves, but only the phenomenon of this [world], and is itself only the form of our sensuous intuition.
The possibility of matter requires a force of attraction, as its second essential fundamental force.
Impenetrability, as the fundamental quality of matter, whereby it first reveals itself as something real in the space of our external senses, is nothing but the capacity of extension in matter (proposition). Now an essentially moving force, by which parts of matter recede from one another, cannot, firstly, be limited by itself, because matter is rather impelled thereby to extend the space it fills continuously; secondly, it cannot be fixed by space alone, at a certain boundary of extension—for though space may contain the ground of [the fact] that with the increase of the volume of a matter extending itself, the extending force will become weaker in inverse proportion—yet, inasmuch as smaller degrees of every moving force are possible to infinity, it cannot contain the ground for their ever ceasing. Matter then, by its repulsive force alone (which contains the ground of its impenetrability), and if no other opposing force contradicted this, would be held within no boundaries of extension, that is, would dissipate itself to infinity, and no assignable quantity of matter would be met with in any assignable space. With merely repulsive forces of matter, all spaces would consequently be empty, in other words no matter would properly speaking exist at all. To the existence of all matters, forces opposed to the extending [forces], in other words, compressive forces, are requisite. But these again cannot be sought for originally, in the opposition of another matter, for it requires, in order that it may be matter, itself a compressive force. An original force of matter, working in an opposite direction to the repulsive, in other words [a force] of approach, that is, an attractive force must be assumed. Now as this attractive force belongs to the possibility of a matter, as matter generally, consequently precedes all distinctions of the same, it must not be ascribed merely to a special species [of matter], but to every matter generally and originally. An original attraction then belongs to all matter as a fundamental force pertaining to its essence.
With this transition from one property of matter to another specifically different from it, which yet equally belongs to the conception of matter, although it is not contained therein, the attitude of our understanding must be more closely considered. If attractive force be itself originally requisite to the possibility of matter, why do we not equally make use of it with impenetrability as the primary sign of a matter? why is the last immediately given with the conception of a matter, while the first is not thought in the conception, but only attributed to it, by inference? That our senses do not allow us to perceive attraction so immediately as repulsion and the resistance of impenetrability, does not sufficiently solve the difficulty. For if we had such a faculty, it is easy to comprehend that our understanding would none the less choose the filling of space, in order to indicate thereby the substance in space, namely, matter, just as in this filling, or, as it is otherwise called, solidity, the characteristic of matter as a thing distinct from space, is posited. Attraction, it matters not how well we might feel it, could never reveal to us a matter of definite volume and figure, nor anything beyond the endeavour of our organ to approach a point outside us (the central point of the attracting body). For the attractive force of all parts of the earth can affect us, neither more nor otherwise, than if it were wholly concentrated in its central point, and it were this alone that influenced our sense; similarly with the attraction of a mountain, and of every stone, &c. We should acquire thereby no definite conception of any object in space, as neither figure nor size, nor even the place where it exists, could fall within our senses. The mere direction of the attraction would be able to be perceived as in weight; the attracting point would be unknown, and I do not see how it could be arrived at, through conclusions, without the perception of matter, in so far as it fills space. It is hence clear, that the first application of our conceptions of quantity to matter, by which it is primarily possible for us to transform our external perceptions into the experiential conception of a matter as object generally, is only founded on its property of filling space, which by means of the sense of feeling, procures for us the size and figure of an extended, and therewith a conception of a definite object in space which must be laid at the foundation of all else that one can predicate of any [particular] thing. This is undoubtedly the reason why, with what are the clearest proofs otherwise, that attraction must belong to the fundamental forces of matter, equally as much as repulsion, one is so unwilling to admit it, or to concede any other moving forces but those of impact and pressure (both by means of impenetrability). For that whereby space is filled is substance, it is said, and this is correct enough. But as substance only reveals its existence to us by sense, whereby we perceive its impenetrability, namely by feeling—and therefore only in reference to contact, whose beginning (in the approach of one matter to another) is termed impact, but its continuation pressure—it seems as though the immediate effect of one matter on another could never be anything else but pressure or impact, the only two influences we can immediately feel; while on the other hand attraction, which can give us either no feeling at all, or at least no definite object of it, becomes difficult for us to conceive as fundamental force.
By mere attraction, without repulsion, no matter is possible.
Attractive force is the moving force of matter, whereby it compels another [matter] to approach it; consequently, when it is met with, between all parts of matter, the matter seeks by means of it to diminish the distance of its parts from one another, and therefore the space that they together occupy. Now nothing can hinder the effect of a moving force, except another moving force opposed thereto, but this [force] that is opposed to it is repulsive force. Thus, without repulsive forces, and by mere approach, all parts of matter would approach one another without hindrance and diminish the space that they occupy. As now, in the case assumed, there is no distance of parts, in which a greater approach through attraction is rendered impossible by a repulsive force, they would move towards one another until no distance existed between them; that is, they would coalesce in a mathematical point, and the space would be empty; in other words, without any matter. Matter is accordingly impossible by mere attractive forces, without repulsive.
That property, on which the inner possibility of a thing rests as its condition, is an essential element therein. Hence repulsive force belongs just as much to the essence of matter as attractive force; and the one cannot be separated from the other in the conception of matter.
As no more than two moving forces in space, repulsion and attraction, can ever be conceived, it was previously necessary—to prove the union of both in the conception of a matter generally à priori—that each should be considered separately, in order to see what taken singly they could achieve in the presentation of a matter. It is evident now that as well when we lay neither of them at the basis, as when we assume merely one of them, space always remains empty, and no matter exists therein.
Contact in the physical sense is the immediate action and reaction of impenetrability. The action of one matter upon another outside contact is action at a distance (actio in distans). This action at a distance, which is also possible without a medium between matters lying within oneanother, is called immediate action at a distance, or the action of matter on another [matter] through empty space.
Contact, in a mathematical signification, is a common boundary of two spaces, and is hence neither within the one nor the other space. Straight lines therefore cannot touch one another, but when they have a point in common, it belongs as much within the one as the other of these lines, when they are produced, that is, cut one another. But circle and straight line, circle and circle, touch each other in a point, surfaces in a line, and bodies in surfaces. Mathematical contact therefore is laid at the basis of the physical, but does not alone constitute it; in order that the latter may arise, a dynamical relation must be superadded in thought, and that, not of the attractive, but of the repulsive forces, namely, those of impenetrability. Hence physical contact is the reciprocal action of repulsive forces in the common boundary of two matters.
The attraction essential to all matter is an immediate effect of it on other matter, through empty space.
The original attractive force itself contains the ground of the possibility of matter as that thing which fills a space in a definite degree, in other words of the very possibility of a physical contact. Hence, it must precede this, and its effect must consequently be independent of the condition of the contact. Now, the effect of a moving force is independent of all contact—independent even of the filling of space between the moving and the moved, that is, it must take place without the space between them being filled up, and, therefore, as an effect through empty space. The original and essential attraction of all matter is then an immediate effect of the same upon another [matter] through empty space.
That the possibility of fundamental forces should be made conceivable is a quite impossible demand: for they are called fundamental forces, precisely because they cannot be deduced from any other, that is, cannot be conceived. But the original attractive force is not one whit more inconceivable than the original repulsion. It does not so immediately obtrude itself on the senses as impenetrability, in affording us conceptions of definite objects in space. Hence, while it is not felt, but only to be inferred, it has the appearance of a deduced force, just as though it were only a hidden play of moving forces [produced by] repulsion. More closely considered, [however,] we see that it cannot be further deduced from any source, least of all from the moving force of matters, through their impenetrability, as its effect is precisely the opposite of the latter. The commonest objection to immediate effect at a distance is, that a matter cannot directly operate where it is not. If the earth directly influences the moon to approach it, the earth acts upon a thing many thousand miles removed from it, and nevertheless [acts] immediately, even though the space between it and the moon were regarded as entirely empty. For, although matter may exist between two bodies, this does not affect the attraction. It acts, therefore, directly, in a place where it is not; something, to all appearance, contradictory. But it is so far from being contradictory, that one might rather say: everything in space acts on another [thing] in a place where the acting [thing] is not. For if it acted in the place where it was itself, the thing on which it acted would not be outside it; for outside signifies presence in a place, where the other is not. If earth and moon touched one another, the point of contact would be a place where neither earth nor moon existed, for they would be removed from one another by the sum of their diameters. In the point of contact, moreover, no portion, either of the earth or of the moon would exist, for this point lies at the boundary of either filled space, which constitutes no portion either of the one or of the other. Thus, that matters cannot act upon each other at a distance is as much as to say they cannot act immediately upon one another, without the intervention of the forces of impenetrability. Now this would be as much as though I were to assert, that the repulsive forces were the only ones by means of which matters could be operative, or they were at least the necessary conditions under which alone matters could act upon one another, which would declare the force of attraction either wholly impossible or always dependent on the action of repulsive forces; but both are assertions without any foundation. The confusion of the mathematical contact of spaces and physical [contact] through repulsive forces constitutes the ground of this misunderstanding. To attract immediately outside contact, means to approach one another according to a constant law, without the force of repulsion containing the condition thereto, which must admit of being conceived just as well as directly to repel one another, that is to fly from one another according to a constant law, without the attractive force having any share therein. For the two moving forces are wholly different in kind, and there is not the least reason for making one dependent on the other, or denying its possibility without the intervention of the other.
Except from attraction, no motion can arise on contact, for contact is the reciprocal action of impenetrability, which restrains all motion. Some immediate attraction must thus be found apart from contact, in other words, at a distance: for otherwise, even the pressing and impulsive forces, which produce the effort to approach, as they act in an opposite manner to the repulsive force of matter, could have no cause at least originally inherent in the nature of matter. That attraction which takes place without the intervention of repulsive forces may be termed the true attraction, that which proceeds in the other manner the apparent. For properly, the body which another is striving to approach, exercises no attractive force whatever on the latter, because this has been driven towards it from elsewhere by impact. But even these apparent attractions must, at last, have a true one at their basis, because matter made up only of pressure or impact, instead of attraction, would not even be matter without attractive forces (proposition 5), and consequently the mode of explaining all phenomena of approach by merely apparent attraction moves in a circle. It is commonly held that Newton did not find it necessary to his system to assume an immediate attraction of matters, but with the strictest abstinence of pure mathematics, left the physicists perfect freedom, in this particular, to explain its possibility as they might find good, without mixing up his propositions with their play of hypotheses. But how could he base the proposition that the universal attraction of bodies, exercised by them equidistantly on every side is proportioned to the quantity of their matter, if he did not assume that all matter exercised this force of motion simply as matter, and by its essential property? For although, indeed, between two bodies, whether homogeneous or not, as to matter, if one draws the other, the mutual approach (according to the law of the equality of reciprocal action) must always occur in inverse proportion to the quantity of the matter, this law only constitutes a principle of mechanics, but not of dynamics, i.e., it is a law of motions, following from attractive forces, not the proportion of attractive forces themselves, and applying generally, to all moving forces. If, therefore, a magnet be attracted by another similar magnet, and again by the same magnet enclosed in a wooden box double its weight, in the latter case this will impart more relative motion to the first [magnet] than in the former, although the wood, which increases the quantity of its matter, adds nothing to its attractive power, and proves no magnetic attraction of the box. Newton says (cor. 2, prop. 6, lib. III., Princip. Phil. Nat.): “If the æther or any other body existed without weight, it would, inasmuch as it differs from any other matter in nothing but in form, be capable of being transformed little by little through a gradual change of this form into a matter of the same kind as that which has the greatest weight; and conversely, this latter, by a gradual change of its form, might lose all its weight, which is contrary to experience,” etc. Thus he did not even exclude the æther (much less other matters) from the law of attraction. What kind of matter, then, could remain for him, by the mere impact of which the approach of bodies to one another could be regarded as merely apparent attraction? One cannot, therefore, adduce the great founder of the theory of attraction as our precursor, if one takes the liberty of substituting for the true attraction which he maintained, a false one, and for assuming the necessity of an impulse through impact, in order to explain the phenomena of approach. He justly made abstraction of all hypotheses, in solving the problem, as to the cause of the universal attraction of matter; for this problem is physical or metaphysical, but not mathematical, and although in the preface to the second edition of his Optics, he says: ne quis gravitatem inter essentiales corporum proprietates me habere existimet, quæstionem unam de ejus causa investiganda subjeci, one can easily see that the dislike his contemporaries, and perhaps he himself, had to the conception of an original attraction, made him at issue with himself. For he could not say, unconditionally, that the attractive forces of two planets—for instance, Jupiter and Saturn—which they show in the equal distances of their satellites (whose mass is unknown), is proportioned to the quantity of the matter of these heavenly bodies, if he did not assume that they attracted other matter merely as matter—in other words, according to a universal property of the same.
A moving force, by which matters can directly act upon one another only in the common surface of contact, I call a superficial force; but that whereby one matter can directly act on the parts of the other beyond the surface of contact, a penetrative force.
The repulsive force, by means of which matter fills a space, is a merely superficial force. For the parts touching each other mutually limit each other’s sphere of action, and the repulsive force cannot move any more distant part, except by means of those lying between, and an immediate effect of a matter, passing straight through these, on another, by means of the forces of extension, is impossible. An attractive force, on the contrary, by means of which a matter occupies a space, without filling it, by which therefore it acts on other distant [matters] through empty space, and whose action thus posits no matter intervening [would have] no1 limits. Now it is thus that the original attraction which makes matter itself possible, must be conceived, and which is hence a penetrative force, and for this reason alone always proportioned to the quantity of the matter.
The original attractive force, on which the possibility of matter itself as such rests, extends itself directly throughout the universe to infinity, from every part of the same to every other part.
Because the original attractive force pertains to the essence of matter, it belongs to every part of the same, to act directly at a distance. Now let it be granted, there is a distance beyond which it does not extend, this limitation of the sphere of its activity would rest either on the matter lying within this sphere, or merely on the size of the space, in which the influence was extended. The first does not take place; for this attraction is a penetrative force, and acts directly at a distance, in spite of all intervening matters, through each space as an empty space. The second, in the same way, does not take place. For inasmuch as every attraction is a moving force, having a cause, beyond which smaller can be conceived to infinity; so, in the greater distance, a cause would indeed lie, for diminishing the degree of attraction in inverse proportion, to the amount of the diffusion of the force but never for completely destroying it. As then there is nothing that anywhere limits the sphere of the activity of the original attraction of any part of matter, it extends itself beyond all assignable limits to every other matter, in other words, [extends itself] throughout the universe, to infinity.
From this original attractive force, as a penetrative [force] exercised by all matter upon all other matter—and therefore in proportion to the quantity of the same, extending to all possible regions of its activity—in combination with its opposite, namely, repulsive force, the limitation of the latter, in other words, the possibility of a space filled in a definite degree, can be deduced; and thus the dynamic conception of matter as the movable, filling its space can (in a definite degree) be constructed. But to this, one requires a law of relation, as well of the original attraction as of repulsion at different distances of matter, and of its parts from one another, which, as it rests simply on the difference of direction of these two forces (since a point is driven either to approach others or to recede from them), and on the size of the space, in which these forces diffuse themselves at different distances, is a task belonging to pure mathematics, and with which metaphysics is no longer concerned, not even as regards the responsibility of constructing the conception of matter in this way, in the event of its non-success. For it is responsible only for the correctness of the elements of construction vouchsafed to our cognition of pure Reason, but for the inadequacy and the limits of our Reason, in its working out, it is not responsible.
As all given matter must fill its space with a definite degree of repulsive force, in order to constitute a definite material thing, only an original attraction in conflict with the original repulsion can make a definite degree of the filling of space, in other words, matter, possible. This is so, whether the former results from the proper attraction of the parts of the compressed matter amongst each other, or from their union with the attraction of all matter.
The original attraction is proportional to the quantity of the matter, and extends to infinity. Thus the filling of a space by matter, definite as to amount, can in the end only be effected by the infinitely extending attraction of the same, and every matter [must be] distributed according to the amount of its repulsive force.
The effect of the universal attraction, which all matter exercises directly upon all [matter] and at all distances, is termed gravitation; the endeavour to move itself in the direction of the greater gravitation is weight. The effect of the thorough-going repulsive force of the parts of each given matter is termed its original elasticity. This and weight therefore, constitute the only discoverable à priori universal characteristics of matter, the former in internal, the latter in external relations; for on their mutual bases the possibility of matter itself, rests; cohesion (zusammenhang), when explained as the reciprocal attraction of matter, limited simply to the condition of contact, does not belong to the possibility of matter in general, and cannot therefore be cognised as bound up with it à priori. This characteristic would hence not be metaphysical but physical, and thus would not belong to the present subject of consideration.
I cannot forbear adding a small preliminary observation, for the sake of any attempt that may perhaps be made toward such a possible construction.
1. It may be said of every force, immediately working at different distances, and which is limited in respect of the degree whereby it exercises moving force, on every given point at a certain distance, only by the size of the space over which it has to diffuse itself in order to act upon this point; that in all spaces over which it is diffused, however small or great they may be, it always constitutes an equal quantum; but that the degree of its effect on the particular point in this space always stands in inverse proportion to the space in which it has had to diffuse itself, in order to act upon it [viz. the point]. So, for instance, light diffuses itself from a luminous point on all sides, in discs that increase with the square of the distance, and the quantum of the luminosity is in all these infinitely increasing discs on the whole the same; whence follows, that an equal part assumed in these discs, must be, in point of degree, so much the less luminous as the surface diffusion of the same quantity of light is greater; and so with all other forces, according to the laws of which they must diffuse themselves either in superficial or corporeal space, in order to act according to their nature on distant objects. It is better to represent the diffusion of a moving force from one point at all distances in the ordinary way, [not?] for instance [as?] in optics, by rays diverging in a circle from a central point. For as lines drawn in this way can never fill the space through which they pass, nor therefore the surface which they touch, it matters not how many of them may be drawn or supposed—this being the inevitable consequence of their divergence—they give occasion to troublesome inferences, and these to hypotheses, which can easily be avoided if merely the size of the whole disc be taken into consideration, as uniformly illumined by the same quantity of light, and of course the degree of its luminosity, in every place, as assuming an inverse proportion to the size of the whole; and similarly with every other diffusion of a force, through spaces of different sizes.
2. If the force be an immediate attraction at a distance, the direction of the attraction must still less be represented as rays going out from the attracting point, but rather as coalescing from all points of the surrounding disc (the diameter of which is the given distance) at the attracting point. For the line of direction of the movement to this point, which is its cause and goal, assigns the terminus a quo, whence the lines must begin, namely from all points of the surface, from which they take their direction to the attracting middle-point, and not conversely; for the size of the surface alone determines the number of lines; the middle point leaves them undetermined.1
3 If the force be an immediate repulsion, so that a point (in merely mathematical presentation) fills a space dynamically, and the question is, according to what law of infinitely small distances (here equivalent to contact) an original repulsive force (the limitation of which consequently rests merely with the space in which it is diffused) acts at different distances, this force can still less he rendered apparent by divergent repulsive rays from the assumed repellant points, although the direction of the motion has it for a terminus a quo, because the space in which the force must be diffused, in order to act at a distance, is a corporeal space, which is to be conceived as filled. The manner in which this is done, how, namely a point can fill a space corporeally by moving force, that is dynamically, is certainly capable of no further mathematical demonstration, but, it is impossible for rays diverging from a point to render conceivable the repelling force of a corporeally filled space. The repulsion, at various infinitely small distances, of these mutually repelling points, we could simply estimate in inverse proportion to the corporeal spaces which fill each of these points dynamically; in other words, as the cube of their distances from one another, without our being able to construct them.
4. Thus the original attraction of matter would act in inverse proportion to the square of the distance at all distances, the original repulsion in inverse proportion to the cube at infinitely small distances, and by such an action and reaction of both fundamental forces, matter as a definite degree of the filling of space would be possible; for, insomuch as the repulsion increases in greater degree with approach of the parts than the attraction, the limits of approach beyond which by given attraction no greater is possible, in other words the degree of compression which constitutes the amount of the intensive filling of space, is also determined.
I readily see the difficulty of this mode of explaining the possibility of a matter in general, which consists in that, if a point cannot directly drive another by its repulsive force, without at the same time filling the whole corporeal space, up to the given distance by its force, this, as it seems to follow, must contain several repulsive points, which contradicts the assumption, and was above refuted (proposition 4) under the name of a sphere of repulsion of the simple in space. But there is a distinction to be made between the conception of a real space, that can be given, and the mere idea of a space, simply conceived for the determination of the relations of given spaces, but which is in reality no space. In the case cited of a supposed physical monadology, there ought to be real spaces, to be filled from a point dynamically, namely, by repulsion, for they [the monads] existed as points, before any possible generation of matter from them, and defined by the proper sphere of their activity, the portion of the space to be filled, which could belong to them. In the hypothesis in question, therefore, the matter cannot be regarded as infinitely divisible and as quantum continuum; for the parts, directly repelling one another, have notwithstanding a determinate distance from one another (the sum of the diameter of the sphere of their repulsion) [while] on the contrary, when we, as really happens, think of matter as continuous quantity, no distance whatever of the directly repelling parts obtains, and consequently, no increasing or diminishing sphere of its immediate activity. Matters however can be expanded or compressed (like the air), and in this case we conceive a distance of their nearest parts as capable of increasing or diminishing. But because the nearest parts of a continuous matter touch one another, whether they are farther expanded or compressed, the distances from one another are conceived as infinitely small, and this infinitely small space, as filled in a greater or less degree by its force of repulsion. The infinitely small mediate space is not however distinguishable from contact, and thus it is only the idea of space, which serves to render intuitable the expansion of matter as continuous quality, but whether it is really thus cannot be conceived. When, therefore, it is said: the repulsive forces of the parts of matter immediately driving one another, stand in inverse proportion to the cube of their distances, this only signifies that they stand in inverse proportion to the corporeal spaces that are conceived between parts immediately touching one another notwithstanding, and where distance must for this reason be termed infinitely small, in order that it may be distinguished from all real distance. Hence we must not from the difficulties of the construction of a conception, or rather, from its misapplication, cast any slur on the conception itself; for in that case it would touch the mathematical presentation of the proportion, with which the attraction occurs at different distances, no less than that whereby each point in an expanding or compressed whole of matter, directly repels the other. The universal law of dynamics would in either case be this: the effect of the moving force, exercised from one point upon every other outside it, is in inverse proportion to the space in which the same quantity of moving force has had to expand itself, in order to act directly upon this point at the determinate distance.
From the law that the parts of matter originally repel one another in inverse cubic proportion to their infinitely small distances, a quite different law of their extension and compression must necessarily follow to that of Mariotte [in respect] of the air; for this proves repulsive forces of its nearest parts, which stand in inverse proportion to their distances, as Newton demonstrates. (Princ. Phil. Lat., Lib. II., Propos. 23, Schol.) But the expansive force of the latter also cannot be regarded as the effect of originally repulsive forces, but rests on heat, which compels the proper constituents [viz. the molecules] of the air (to which moreover real distances from each other may be conceded) to fly from one another, not as a matter interpenetrating them, but, to all appearance through their vibrations. But that these vibrations of the parts nearest one another must communicate a repulsive force, standing in inverse proportion to their distances, may be made readily comprehensible by the laws of the communication of motion through the vibration of elastic matters.
I may explain that I do not wish the present exposition of the law of an original repulsion to be regarded as necessarily belonging to the object of my metaphysical treatment of matter, nor the latter (for which it is enough, to have presented the filling of space as dynamic property) to be mixed up with the disputes and doubts which might affect the former.
General Note to the Dynamics.
If we review all [our] discussions on the above, we shall observe that the following things have been taken into consideration: Firstly, the real in space (otherwise called the solid) in its filling through the force of repulsion; Secondly, what, in respect of the first, as the proper object of our external perception, is negative, namely, the force of attraction, by which, so far as may be, all space is penetrated, [or], in other words, the solid, is wholly abolished; Thirdly, the limitation of the first force by the second, and the thence resulting determination of the degree of a filling of space; [we shall observe] therefore that the quality of matter has been thoroughly dealt with, under the heads of reality, negation, and limitation, in so far as they belong to a metaphysical dynamics.
General Observation on Dynamics.
The universal principle of the Dynamics of material nature, that all [that is] real in the objects of our external sense, that, namely, which is not mere determination of space (place, extension and figure), must be regarded as moving force; by which, therefore, the so-called solid, or absolute impenetrability, is banished from natural science as an empty conception, and in its stead a repulsive force is posited; while the true and immediate attraction is defended against all the sophistries of a metaphysics that misunderstands itself, and is explained as a fundamental force necessary even to the possibility of the conception of matter. Now from this the consequence arises, that space, should it be found necessary, could be assumed as throughout, and at the same time in different degrees, filled even without distributing empty mediate spaces within the matter. For according to the originally varying degree of the repulsive forces on which is founded the first property of matter, namely, that of filling a space, its relation to the original attraction (whether of each matter for itself, or to the united attraction of all matter in the universe) is conceived as infinitely diverse, inasmuch as attraction rests on the mass of matter in a given space while its expansive force [rests] on the degree in which it fills it [viz., the space], which can be specifically very different (as for instance the same quantity of air, in the same volume, exhibits greater or less elasticity, according to its higher or lower temperature). The general ground of this is that by true attraction all parts of matter act directly on all parts of other matter, but through expansive force only those on the surface of contact, owing to which it is the same, whether behind this, much or little of the matter exists. From the above, however, a great advantage for Natural Science arises, by its being relieved of the burden of having to manufacture a world from fullness and emptiness, merely according to fancy, and being able rather to conceive all spaces as full, and yet as filled in varying amount, by which empty space at least loses its necessity, and is relegated to the rank of an hypothesis; whereas otherwise, under the pretext of being a necessary condition to the explanation of the varying degree of the filling of space, it might lay claim to the title of a principle.
With all this the advantage of a methodically-employed metaphysic to the detriment of equally metaphysical principles, but such as have not been subjected to the test of criticism, is apparently only negative. But indirectly, notwithstanding, the field of the investigator of Nature is extended, since the conditions, by which it previously limited itself, and whereby all original forces of motion were philosophised away, now lose their validity. But one must guard against going beyond what the universal conception of a matter in general renders possible, and seeking to explain its particular or specific definition and variety à priori. The conception of matter is reduced to mere moving forces, and this could not be expected to be otherwise, seeing that in space no activity—no change—can be thought of, except as motion. But who can comprehend the possibility of fundamental forces? They can only be assumed, if they inevitably belong to a conception of which it is demonstrable that it is a fundamental conception which cannot be deduced from any other (as that of the filling of space), and of this [nature] is the force of repulsion, and the opposing force of attraction, [considered] generally. We can indeed judge of this, their connection and consequences well enough à priori, whatever their relations among each other may be conceived to be, provided they do not contradict themselves; but [must] not lay claim to assume either of them as real, because to the admissibility of constructing an hypothesis, it is indispensably requisite that the possibility of what is assumed be quite certain, while with fundamental forces, their possibility can never be comprehended. And in this, the mathematico-mechanical mode of explanation has an advantage over the metaphysico-dynamical, which cannot be taken from it—namely, that from a completely homogeneous material, through the manifold form of the parts, by means of empty mediate spaces interspersed, it can accomplish a great specific multiplicity of matters, in density no less than in mode of action (if foreign forces be superadded). For the possibility of the forces, as wel as of the empty mediate spaces, admit of demonstration with mathematical evidence; on the other hand, if the matter itself be transformed into fundamental forces (to define the laws of which, à priori, we are not in a position, and still less to indicate confidently a multiplicity of the same, sufficient for the explanation of the specific variety of matter), all means are wanting for the construction of this conception of matter, and for presenting as possible, in intuition, what we conceived in general. But a mere mathematical physics, pays for the foregoing advantage doubly on the other side, in that it first of all lays at its foundation an empty conception (that is, absolute impenetrability), and secondly that it must give up all the proper forces of matter, in addition to its original configuration of the fundamental matter and interspersion of empty spaces, and, after having called forth the need for explanation, must concede more freedom to the imaginative faculty in the field of philosophy—[and concede it] indeed as legitimate claim—than is consistent with the caution of the latter.
Instead of an adequate explanation of the possibility of matter and its specific variety, from the fundamental forces, which I am unable to furnish, I shall, as I hope, present the momenta to which its specific variety must admit of being reduced, completely in its totality à priori (although [I cannot] conceive its possibility in the same way). The observations inserted between the definitions will explain their application.
1. A body in a physical signification, is a matter between definite boundaries (which therefore has a figure). The space between these boundaries considered as to its size, is the content of space (volume). The degree of the filling of a space of definite content is termed density. Otherwise the expression dense is used absolutely, for that which is not hollow (bladdery, perforated). In this sense there is an absolute density in the system of absolute impenetrability, if a matter contains no empty mediate spaces. According to this conception of the filling of space comparisons are instituted, and one matter containing less emptiness within itself is called denser than another, till at last, that in which no part of the space is empty is termed perfectly dense. The latter expression can only be made use of, on the mere mathematical conception of matter, for in the dynamical system of a simply relative impenetrability there is no maximum or minimum of density, and any matter however thin can equally be termed fully dense if it wholly fill its space, without containing empty mediate spaces; in other words, if it be a continuum and not an interruptum; but it is in comparison with another [matter], less dense in a dynamical sense, if, although it fill its space wholly, it does not do so in an equal degree. Yet even in the latter system, it is awkward to conceive a relation of matters according to their density, unless they are represented as specifically homogeneous among one another, so that one can be generated from the other merely by mutual pressure. As now, the latter does not appear to be absolutely requisite to the nature of all matter in itself, no comparison can properly be made between heterogeneous matters in respect of their density, as for instance, between water and quicksilver, although this is commonly done.
II. Attraction, in so far as it is merely conceived as active in contact, is called cohesion [zusammenhang]. It is demonstrated by very good experiments, that the same force, called cohesion in contact, is found active at a very small distance; but attraction is only called cohesion, in so far as I think of it only in contact, in accordance with common experience by which it is hardly perceived at small distances. Cohesion is commonly assumed as an altogether universal property of matter, not because we are led to it through the mere conception of a matter, but because experience presents it everywhere. But this universality must not be understood collectively, as though every matter, through this kind of attraction, acted at the same time on every other [matter] in the universe—in the same way as gravitation—but merely disjunctively, namely on one or the other, it does not signify what kind of matters they may be, that come in contact with it. For this reason, and since this attraction, as is demonstrable on various grounds, is not a penetrating but only a superficial force, inasmuch as it is not itself regulated on all sides according to the density—since to complete strength of cohesion a preceding state of fluidity of the matters and their subsequent solidification is requisite, and the closest contact of broken but hard matters in the same surfaces, with which they previously firmly cohered (as for instance a looking-glass where there is a crack), do not any longer admit the degree of attraction which they received on solidifying after their fluid [state—for this reason] I hold this attraction in contact to be no fundamental force of matter, but only a derivative one; of which more hereafter. A matter whose parts, notwithstanding their strong cohesion among one another, can be impelled by every moving force—be it never so small—past one another, isfluid.But parts of a matter areimpelledpast one another, if, without diminishing the quantum of contact, they are obliged to change [places] among one another. Parts, in other words, matters, areseparatedif their contact is not merely changed with others but destroyed, or its quantum diminished. Afirm—better asolid—body (corpus rigidum) is that whose parts cannot be impelled past one another by every force, and which consequently resist impulsion with a certain degree of force.
The obstacle to the impulsion of matters past one another isfriction.
The resistance to separation of matters in contact is cohesion. Fluid matters, therefore, suffer no friction in their division; but where this is met with, the matters are assumed as solid, in greater or less degree, of which the smallest is termed adhesiveness (viscositas), at least in its lesser parts. The solid body isbrittle,if its parts cannot be impelled past one another without breaking, in other words when its cohesion cannot be changed without being at the same time destroyed. The distinction between fluid and solid matters is very incorrectly placed in the different degree of the cohesion of their parts. For to call a body fluid does not depend on the degree of its resistance to rupture, but only on [its resistance] to the impulsion of its parts past one another. The former may be as great as one chooses, but the latter is always in a fluid matter = 0. Let us contemplate a drop of water. If a molecule within the same be drawn on one side, by never so great an attraction of the neighbouring parts, touching it, it will be drawn exactly as much toward the opposite side, and as the attractions reciprocally abolish their effects, the molecule is just as easily movable as if it existed in empty space. The force namely, which is to move it, has no cohesion to overcome, but only the so-called inertia which it would have to overcome with all matter, even if it did not cohere at all. A small microscopical animalcule would therefore move itself as easily within this drop as if there were no cohesion to overcome. For in reality it has not any cohesion of the water to abolish, nor to diminish its contact within itself, but only to change it. But conceive this animalcule as wanting to work its way through the outer surface of the drop; it is then first to be observed, that the reciprocal attraction of the parts of this drop of water cause them to move themselves, until they have attained the greatest contact among one another, in other words, the smallest contact with empty space, that is, have constituted a globular form. If now, the said insect be endeavouring to work its way beyond the surface of the drop, it must change this globular form, and consequently effect more contact of the water with the empty space and hence less contact of the parts among one another, that is, diminish its cohesion; and now for the first time the water resists it through its cohesion, though [even now] not within the drop, for here the contact of the parts among one another is in no way lessened, but only changed in their contact with other parts, in other words, not separated, but only shifted. One may therefore, and indeed for similar reasons, apply to this microscopical animalcule, what Newton says of the lightray; that it cannot be repelled through dense matter, but only through empty space. It is thus clear that the increase of the cohesion of the parts of a matter does not in the least affect its fluidity. Water coheres in its parts much more strongly than is commonly believed, when an experiment with a metal plate drawn off from the surface of the water is relied upon, which decides nothing, because the water does not split in the whole surface of the original contact, but from a much smaller surface resulting from the shifting of its parts, just as a stick of soft wax when a weight is suspended at the end, becomes gradually thinner, and is then torn off from a much smaller surface than the original one. What, however, is quite decisive with respect to our conception of fluidity is this, that fluid matters can be explained as those of which every point seeks to move itself in all directions with the same force, with which it is impressed towards any one [in particular]; a property, upon which the first law of hydro-dynamics rests, but which can never be attributed to an aggregation of smooth and at the same time solid particles, as a very slight removal of its pressure according to the laws of composite motion will show, and thereby prove the originality of the property of fluidity. If now the fluid matter should suffer the least hindrance to impulsion, in other words the smallest friction, this would grow with the strength of the pressure with which the parts were pressed against one another, and finally a pressure would obtain, by which the parts of this matter would not admit of impulsion past one another, by every small force. For instance, in a bent tube, [composed] of two pieces, of which the one may be as wide as one chooses, the other as narrow as one chooses, provided it is not a mere hair-tube—if one supposes both pieces to be some hundred feet high, the fluid matter in the narrow one would stand just as high as that in the wide, according to the laws of hydrostatics. But because the pressure on the bottom of the tubes, and hence on the part uniting both these tubes (which stand in communication), can be conceived as in proportion to the heights increasingly greater to infinity, so, if the least friction between the parts of the fluid took place, a height of the tubes must be able to be found, by which a small quantity of water, poured into the narrow one, would not move that in the wide one out of its place, in short, [by which] the column of water in the latter would come to stand higher than that in the former, inasmuch as the lower parts, with such great pressure against one another, would not any longer admit of impulsion, by so small a moving force as the added weight of water—[a cohesion] which is opposed to experience, and even to the conception of the fluid. The same may be said if, instead of pressure by weight, the cohesion of the parts be posited, it matters not how great it may be. The second definition of fluidity cited, upon which the fundamental law of hydrostatics rests, namely, that it is the property of a matter by which every part of the same endeavours to move itself towards all sides with the same force with which it is impressed in a given direction, follows from the first definition, if the fundamental principle of universal dynamics be combined with it, that all matter is originally elastic, since it must endeavour to extend itself—that is (if the parts of a matter admit of being impelled past one another by every force without hindrance, as is actually [the case] with fluids), to move itself—towards all sides of the space in which it is compressed, with the same force with which the pressure in any [given] direction, whichever it may be, is exercised. There are therefore properly only the solid matters (the possibility of which requires another ground of explanation beside the cohesion of the parts), to which friction can be attributed, and the friction already presupposes the property of solidity. But why certain matters, although possessing not a larger, it may be even a smaller, force of cohesion, than fluid [matters], resist notwithstanding so powerfully the shifting of their parts, as not to admit of separation otherwise than by the abolition of the cohesion of all parts at once in a given surface, whereby the appearance of a pre-eminent cohesion is afforded—in short, how rigid bodies are possible—is still an unsolved problem, in spite of the ease with which ordinary natural science believes itself to dispose of it.
3. Elasticity (spring-force) is the capacity of a matter, to reassume its size or shape [which has been] altered by another moving force, on the cessation of the latter. It is either expansive or attractive elasticity; the former in order after compression to assume the previously greater [volume], the latter in order after expansion [to assume] the previously smaller volume. The attractive elasticity, as the expression itself shows, is obviously derived. An iron wire stretched by weights appended, springs, if the connection is cut, back into its [original] volume. By virtue of this attraction, which is the cause of its cohesion (or with fluid matters, [as?] when the heat is suddenly withdrawn from quicksilver), their matter hastens to assume again the previous smaller volume. The elasticity which consists in rehabilitation of the previous figure, is always attractive, as in a bent sword-blade, where the parts on the convex side which are forced back, seek to recover their former proximity, and in the same way a small drop of quicksilver may be called elastic. But the expansive elasticity may be original or it may be derivative. Thus the air has a derivative elasticity, by means of the matter of heat which is most intimately united with it, and the elasticity of which is perhaps original. On the other hand, the fundamental material of the fluid which we term air, must nevertheless as matter generally already have elasticity in itself, which may be called original. Of what kind a perceived elasticity may be, is not possible to decide with certainty in cases as they arise.
4. The effect of moved bodies on one another through the communication of their motion is termedmechanical;but that of matters, in so far as they change the combination of their parts reciprocally by their own forces while at rest, is termedchemical. This chemical influence is termed solution [auflosung] in so far as it has for its effect the separation of the parts of a matter; (mechanical division, as for instance a wedge driven between the parts of a matter, is thus, since the wedge does not act by its own force, entirely different from chemical [division]); but that which has for its effect the severance of two matters resolved by one another, is [chemical] analysis. The solution of specifically distinct matters by one another, in which no part of the one is met with, that is not united with a part of the other specifically distinct from it in the same proportion as the whole, is absolute solution, and may also be termed chemical penetration. Whether the resolving forces really discoverable in nature, are capable of effecting a complete solution may remain undiscussed. Here the question is only whether such admit of being conceived. Now it is obvious that so long as the parts of a resolved matter are still particles (moleculœ), a solution of them is not less possible than of the larger, indeed that this must really proceed, if the resolving force continue, until there is no part left, that is not compounded of the medium of solution and the matter to be resolved in the proportion in which they each stand to one another in the whole. As, then in such a case, there can be no part of the volume of the solution, not containing a part of the resolving medium, this must also, as a continuum, completely fill the volume. In the same way, as there can be no part of this volume of solution, that does not contain a proportional part of resolved matter, this must also, as a continuum, fill the whole space, constituting the volume of the mixture. But when two matters, each of them, entirely fill one and the same place, they penetrate one another; hence a perfect chemical solution would be a penetration of the matter, which nevertheless would be wholly distinguished from the mechanical, inasmuch as by the latter it would be conceivable that with the greater approach of moved matters, the repulsive force of the one might entirely counterbalance that of the other, and one or both reduce its extension to nothing. On the contrary, here, the extension remains, only that the matters [are] not outside, but within one another, i.e. occupy by intersusception (as it is usually termed) together a space equal to the sum of their densities. Against the possibility of this perfect solution, and hence of chemical penetration, it is difficult to allege anything, although it involves a complete division to infinity, for this in the present case contains no contradiction, as the solution takes place continuously throughout time; in other words, through an infinite series of moments, with acceleration; by the division moreover, the sums of the outer surfaces of the matters yet to be divided, grow, and as the resolving force acts continuously, the whole solution may be completed in an assignable time. The incomprehensibility of such a chemical penetration of two matters is to be ascribed to the score of the incomprehensible [nature] of the divisibility to infinity of every continuum, generally. If we depart from this complete solution we must assume it to extend only to certain small particles of the matter to be resolved, which swim in the medium of solution at fixed distances from each other, without our being able to assign the least ground why these particles, as they are still divisible matters, may not in the same way be resolved. For that the medium of solution does not act farther, may always, in nature, so far as experience teaches be true enough; but the question here is of the possibility of a resolving force, which may resolve this particle, and every other that remains over, till the solution is completed. The volume occupied by the solution may be equal to the sum of the spaces occupied by the mutually resolving matters before the mixture, or [it may be] smaller or larger, according to the relation in which the attractive forces stand to the repulsions. They constitute in solution, each for itself and both combined, an elastic medium. This alone, will afford a sufficient reason why the resolved matter does not by its weight separate itself again from the resolving medium. For the attraction of the latter, as it occurs with equal strength toward all sides, abolishes its resistance, and to assume any adhesiveness in the fluid, does not harmonise with the great force exercised by such resolved matters, as for instance, acids diluted with water, on metallic bodies, on which they do not merely rest, as must happen if they simply swam in their medium, but which separate themselves from each other with great attractive force, and diffuse themselves in the whole space of the vehicle. Admitting, moreover, that art has no chemical forces of solution of this kind, capable of effecting a complete solution, in its power, nature might still exhibit them in its vegetal and animal operations and thereby perhaps generate matters, which although indeed mixed, no art could again separate. This chemical penetration might even be met with, where one of the two matters might not be severed by the other, and in a literal sense resolved; as for instance, heat-matter penetrates bodies, since if it only distributed itself in their empty mediate spaces, the solid substance itself would remain cold, since it could not absorb any of it. In the same way, an apparently free passage of certain matters through others could be conceived in such a manner as that of magnetic matter, without preparing for it, to this end, open pores and empty mediate spaces, in all, even the densest matters. But this is not the place to point out hypotheses for special phenomena, but only the principle according to which they are all to be judged. Everything that relieves us of the necessity of having recourse to empty spaces, is a real gain to natural science. For these give far too much freedom to the imagination, to supply the want of accurate knowledge of nature by fancy. Absolute vacuity and absolute density are, in natural science, much the same as blind chance and blind fate in metaphysical science, namely, stumbling-blocks for the investigating reason, by which, either fancy occupies its place, or it is lulled to rest on the pillow of occult qualities.
But as concerns the procedure in natural science in respect of the most important of all its problems, namely, the explanation of a possible specific variety of matters [extending] to infinity, one can only strike out two ways: the mechanical, by the union of the absolutely full with the absolutely empty, or a dynamical way, opposed to it, by explaining all varieties of matters through the mere variety in the combination of the original forces of repulsion and attraction. The first has, as the materials of its deduction, atoms and the void [emptiness]. An atom is a small portion of matter physically indivisible. A matter is physically indivisible, whose parts cohere with a force, capable of being overpowered by no discoverable moving force in Nature. An atom, in so far as it is specifically distinguished from others by its figure, is called a primal body. A body whose moving force depends on its figure is called a machine. The mode of explanation of the specific variety of matters by the construction and composition of their smallest parts as machines is mechanical natural philosophy, but that which derives the specific variety of matter from matters not as machines, that is, mere tools of external moving forces, but from the moving forces of attraction and repulsion originally belonging to them, may be called dynamical natural philosophy. The mechanical mode of explanation, as it is the most available in mathematics, has, under the name of the atomistic or corpuscular philosophy, always retained its reputation and influence on the principles of natural science, with little change from old Demokritos to Descartes, and even our own times. It consists essentially in the presupposition of the absolute impenetrability of the primitive matter, in the absolute homogeneity of this matter, differences only being admitted in the figure, and in the absolute unconquerability of the cohesion of the matter of these fundamental bodies themselves. Such were the materials for the generation of specifically different matters, in order not only to have at hand an unchangeable, and at the same time variously-formed fundamental material for the unchangeableness of species and kinds, but, also from the form of these primal parts, as machines (to which nothing more than an externally impressed force was wanting), to explain the several effects of nature mechanically. The first and most important credential of this system rests, however, on the pretended unavoidable necessity of employing empty spaces for the specific distinction of the density of matters which were assumed as distributed within the matters and between the said particles in [such] proportion as was found necessary, for the sake of some phenomena so large, that the filled part of the volume, even of the densest matter, would be well nigh as nothing, against the empty. In order, now, to introduce a dynamical mode of explanation (which is far more suited and more advantageous to experimental philosophy, inasmuch as it leads directly to the discovery of the proper moving forces of matters and their laws, while it limits the freedom of assuming empty mediate spaces and fundamental bodies of definite figures, neither of which admit of definition or discovery by any experiments) it is by no means necessary to forge new hypotheses, but merely to refute the postulate of the mechanical mode of explanation [namely] that it is impossible to conceive a specific distinction of the density of matters without the intermixture of empty spaces, by the mere citation of a way in which this admits of being conceived without contradiction. For if the postulate in question, on which the mere mechanical mode of explanation stands, be only first declared invalid, as a fundamental principle, it is self-evident that it must not be adopted as a hypothesis in natural science, so long as a possibility remains of conceiving the specific distinction of densities without any mediate spaces. But this necessity rests upon [the fact] that matter does not (as mere mechanical investigators of nature assume) fill its space by absolute impenetrability, but by repulsive force, which has its degree, that may be different in different matters, and as it has nothing in itself, in common with the attractive force, which is regulated by the quantity of the matter, it may be originally different in degree, in different matters with the same attractive force; and consequently the degree of extension of these matters may with the same quantity of matter, and conversely, the quantity of matter with the same volume—i.e., density—admit of very great original specific differences. In this way we should not find it impossible to conceive a matter (as, for instance, the ether is represented), which wholly filled its space, without any void, and yet with incomparably less quantity of matter, at an equal volume, than any bodies which we can subject to our experiments. The repulsive force in ether must, in relation to its proper attractive force, be conceived as incomparably greater than in any other matter known to us. And the only [reason] why we merely assume it, because it can be conceived, is as a foil to a hypothesis (that of empty spaces), which is alone supported by the pretension, that such [viz., matter] does not admit of being conceived without empty spaces. Besides this, no law whatever of the attractive or repulsive force may be risked on à priori conjectures, but everything, even the universal attraction as cause of gravity must, together with its laws, be inferred from data of experience. Still less may such be attempted with chemical affinities, otherwise than by way of experiment. For it lies generally beyond the horizon of our Reason, to comprehend original forces à priori as to their possibility; all natural philosophy consists rather in the reduction of given forces in appearance diverse, to a small number of forces and powers, adequate to the explanation of the effects of the former, but which reduction only extends to fundamental forces, beyond which our Reason cannot proceed. And thus, metaphysical research, behind what lies at the foundation of the empirical conception of matter, is only useful for the purpose of leading natural philosophy so far as is possible to the investigation of dynamical grounds of explanation, as these alone admit the hope of definite laws, and consequently of a true rational coherence of explanations.
This is all that metaphysics can ever accomplish to the construction of the conception of matter—in other words, for the application of mathematics to natural science, in respect of properties whereby matter fills its space in definite amount—namely, to regard these properties as dynamical and not as unconditioned original positions, such for instance, as a mere mathematical treatment would postulate.
The well-known problem as to the admissibility of empty spaces in the world may furnish the conclusion. The possibility of this does not admit of dispute. For to all forces of matter space is requisite, and, as it also contains the conditions of the laws of its diffusion, is necessarily pre-supposed before all matter. Thus, attractive force is attributed to matter, in so far as it occupies a space around itself by attraction, without, at the same time, filling it, which, therefore, even where matter is active, may be conceived as empty, because it is not active by repulsive forces, and hence does not fill it. But, to assume empty spaces as real, no experience, inference from [experience], or hypothesis necessary to its explanation, can justify us. For no experience gives us any but comparatively empty spaces to cognise, which can be perfectly explained, from the property of matter, as filling its space by an expansive force, greater or progressively smaller to infinity, in all possible degrees, without requiring empty spaces.
METAPHYSICAL FOUNDATIONS OF MECHANICS.
Matter is the movable, in so far as it is something having a moving force.
Now this is the third definition of a matter; the mere dynamical conception could also regard matter as in rest; the moving force, which was then taken into consideration, concerned merely the filling of a particular space, without our being permitted to regard the matter which filled it, as itself moved. Repulsion was thus an original moving force to impart motion; in mechanics, on the contrary, the force of a matter, set in motion, is considered as [present] in order to communicate this motion to another. But it is clear that the movable would have no moving force through its motion if it did not possess original moving forces, whereby it is active before all proper motion, in every place in which it exists, and that no matter would impress uniform motion upon another matter, the motion of which lay in the path of the straight line before it, if both did not possess original laws of repulsion; nor that it could compel another by its motion, to follow it in the straight line (that it could drag it after it), if both did not possess attractive forces. Thus, all mechanical laws presuppose dynamical, and a matter as moved can have no moving force, except by means of its repulsion or attraction, upon which, and with which, it acts directly in its motion, and thereby communicates its own motion to another. It will be observed that I do not make further mention here of the communication of motion by attraction—for instance, as if a comet of stronger attractive capacity than the earth, in passing by the latter, should drag it after it—but only of the mediation of repulsive forces, in other words, of pressure (as by means of a distended spring), or by impact, since, without this, the application of the laws of the one to those of the other is only different in the line of direction, but otherwise the same in both cases.
The quantity of the matter is the multitude of the movable in a definite space. This, in so far as all its parts may be considered as at the same time active (moving) in their motion is termed the mass, and it is said a matter acts in mass when all its parts are moved in the same direction, exercising, at the same time, their moving force, outside themselves. A mass of definite figure is called a body in a mechanical sense). The quantity of motion (mechanically estimated) is that which is estimated at once, by the quantity of the moved matter and its velocity; phoronomically it consists merely in the degree of the velocity.
The quantity of the matter may be estimated in comparison, with every other, only by the quantity of motion at a given velocity.
Matter is divisible to infinity; consequently none of its quantity can be determined directly by a multitude of its parts. For if this occur in the comparison of the given matter, with a homogeneous one, in which case the quantity of the matter is proportional to the quantity of the volume, this is opposed to the requirements of the proposition [which says], it is to be estimated in comparison with every other (even specifically different) [matter]. Thus matter can be neither indirectly nor directly estimated in comparison with every other matter, so long as abstraction is made of its own motion. Consequently, no other universally valid measure of it remains, but the quantity of its motion. But in this, the difference of the motion, which rests on the different quantity of the matter, can only be given when the velocity is assumed as equal among the compared matters, therefore, &c.
The quantity of the motion of bodies is in compound proportion to the quantity of its matter and its velocity, i.e., it is the same whether I make the quantity of the matter of a body doubly as great, and retain the velocity, or whether I double the velocity and retain the mass. For the definite conception of a quantity is only possible through the construction of the quantum. But this is, in respect of the conception of the quantity, nothing but the composition of the equivalent; and consequently the construction of the quantity of a motion is the composition of many motions equivalent to each other. Now it is the same thing, according to the phoronomic propositions, whether I impart to a movable a certain degree of velocity, or to many equal movables all the smaller degrees of velocity, produced by the given velocity being divided by the multitude of the movable. Hence arises, at first, an apparently phoronomic conception of the quantity of a motion, as compounded of many motions outside one another, but yet as a whole united in a movable point. If now this point be conceived as something possessing moving force by its motion, there arises the mechanical conception of the quantity of the motion. But in phoronomy it is not practicable to conceive of a motion as compounded of many parts outside one another, because the movable, since it is conceived as without any moving force, gives no distinction in real quantity of the motion, no matter with how many others of its kind it be compounded, beyond that which consists merely in the velocity. As the quantity of the motion of a body to that of another, so is related also the quantity of its effect, the whole effect being understood thereby. Those who assumed merely the size of a space filled with resistance (e.g., the height to which a body can rise with a given velocity against gravitation or the depth to which the same [body] can penetrate into soft matters) as the measure of the whole effect, brought forward another law of moving forces with real motions, namely, that of compound relation, from [the law] of the quantity of the matters and of the squares of their velocities; but they overlooked the quantity of the effect in the given time, in which the body traverses its space with less velocity, and this can alone be the measure of a motion exhausted by a given uniform resistance. Hence no difference can obtain between living and dead forces, if moving forces are considered mechanically, that is, as those such as bodies possess, in so far as they are themselves moved, it matters not whether the velocity of their motion be finite or infinitely small (mere effort towards motion). One might far more suitably indeed call those forces with which matter (even when abstraction is wholly made of its own proper motion, or even effort to move itself), acts on others; in other words, the original moving forces of dynamics, dead forces, and all mechanical [forces], that is, forces moving by their own motion, living forces, regard not being given to the difference of velocity, the degree of which may be infinitely small; always supposing that these designations of dead and living forces deserve to be retained at all.
In order to avoid diffuseness, we will condense the explanation of the preceding three paragraphs into one observation.
That the quantity of the matter can only be conceived as the multitude of the movable (outside one another), as the definition expresses it, is a remarkable and fundamental proposition of universal mechanics. For it is indicated thereby, that matter can have no other quantity than that which consists in the multitude of the manifold outside one another; consequently no degree of moving force with given velocity that would be independent of this multitude, and which could be conceived as merely intensive quantity, which would certainly be the case if the matter consisted of monads, whose reality in every connection must have a degree, that might be greater or smaller, without depending on a multitude of parts external to one another. As to that which concerns the conception of mass in the same explanation it cannot be regarded, as is usually [done], as the same as the quantity. Fluid matters can act by their own motion in mass, and they can also act in flux. In the so-called water-hammer the water in striking acts in mass, that is, with all its parts at the same time; the same occurs in water which has been enclosed in a vessel, and which presses by its weight upon the scale on which it stands. On the other hand, the water of a mill-stream acts on the paddle of the water-wheel that strikes it, not in mass, that is, at the same time with all its parts that rush against it, but only successively. If therefore, in this case, the quantity of the matter that is moved with a certain velocity, and that has moving force, is to be determined, one must first of all seek the body of the water, that is, such quantity of matter, that when it acts in mass with a certain velocity (by its weight) can produce the same effect. Hence by the word mass is generally understood the quantity of the matter of a solid body (the vessel, in which a fluid is enclosed, taking the place of its solidity). Finally, as concerns the proposition, together with the appended note, there is nothing strange that according to the former, the quantity of the matter has to be estimated by the quantity of the motion with given velocity, while according to the latter, on the contrary, the quantity of the motion (of a body, for that of a point, consists only in the degree of the velocity) at the same velocity, by the quantity of the moved matter, though this seems to revolve in a circle, and to promise no definite conception of either the one or the other. This supposed circle would indeed be real if it were a reciprocal deduction of two identical conceptions from one another. It contains, however, on the one side only the explanation of a conception, and on the other its application to experience. The quantity of the movable in space is the quantity of the matter; but this quantity of the matter (the multitude of the movable), demonstrates itself in experience only by the amount of the motion, at equal velocity (e.g. by equilibrium.)
It remains yet to be observed, that the quantity of matter is the quantity of substance in the movable; consequently, not the amount of a given quality of the same (of repulsion or attraction, as has been said in the dynamics), and that the quantum of the substance is here nothing else than what is signified by the multitude of the movable, which constitutes matter. For only this multitude of the moved can with the same velocity give a difference in the amount of the motion. But that the moving force a matter possesses in its own motion can alone prove the quantity of the substance, rests on the conception of the latter as the ultimate subject (that is no further predicate of another) in space, which for this reason can have no other quantity, but that of the multitude of the homogeneous outside one another. But as the proper motion of matter is a predicate which determines its subject (the movable), and in a matter, as a multitude of the movable, indicates the plurality of the moved subjects (at equal velocity in the same kind)—while with dynamical properties, whose quantity may be also the quantity of the effect of a single subject (e.g. a [single] molecule of air may have more or less elasticity), this is not the case—it is clear that the quantity of the substance in a matter can only be estimated mechanically, that is, by the amount of its motion, and not dynamically, by the amount of its original moving forces. In the same way the original attraction, as the cause of universal gravitation can afford a measure of the quantity of matter and its substance (as really happens in the comparison of matters by weighing), although in this case, not proper motion of the attracting matter, but a dynamical measure, namely attractive force, seems to be laid at the foundation. But inasmuch as with this force the effect of a matter occurs with all its parts, directly on all parts of another, and thus (at equal distances) is obviously proportioned to the multitude of the parts, and the attracting body itself thereby imparts a velocity of its own motion (by the resistance of the attracted [body]), which, in similar external circumstances, is exactly proportioned to the multitude of its parts, [for this reason] the estimate takes place here, [also] as a matter of fact, mechanically, although only indirectly so.
First law of mechanics.—With all changes of corporeal nature, the quantity of the matter remains, on the whole, the same, unincreased and undiminished.
(From universal metaphysics the proposition is laid at the foundation, that with all changes of nature, no substance can either arise or be annihilated, and here it is only demonstrated what is substance in matter.) In every matter the movable in space is the ultimate subject of all the accidents inhering in matter, and the multitude of this movable outside one another the quantity of the substance. Thus the amount of the matter as substance, is nothing other than the multitude of the substances of which it consists. Hence the quantity of the matter cannot be increased or diminished except by new substance arising or being annihilated. Now, with all change of matter, substance never arises or is destroyed; thus the quantity of matter is thereby neither increased nor diminished, but remains always the same as a whole, that is, so that somewhere in the world it continues [to exist], although this or that [particular] matter may by the addition or subtraction of its parts be increased or diminished.
The essential, characterising substance in this demonstration, which is only possible in space and according to the conditions of the same, consequently as object of the external sense, is that its amount cannot be increased or diminished, without substance arising or being annihilated; therefore as any quantity of a merely possible object in space must consist of parts outside one another, these, if they are real (something movable) must be necessarily substances. That, on the contrary, which is considered as object of the internal sense may have a quantity as substance, not consisting of parts outside one another, whose parts are therefore not substances, whose origination or annihilation therefore need not be the origination or annihilation of a substance, and hence whose increase or diminution is possible, notwithstanding the principle of the permanence of substance. Thus consciousness, in other words, the clearness of the presentations of my soul, and in consequence of this also, the faculty of consciousness, apperception, and therewith even the substance of the soul, has a degree that may be greater or smaller, without, to this end any substance requiring to arise or to be annihilated. But because with the gradual diminution of this faculty of apperception, a total disappearance of the same could not but finally result, the substance of the soul would still be subjected to a gradual destruction, even were it of simple nature, inasmuch as this disappearance of its fundamental force could not result through division (separation of substance from a composite), but, as it were, by extinction, and even this not in a moment, but by the gradual failing of its degree, from whatever cause arising. The ego, the universal correlate of apperception and itself merely a thought, indicates as a mere prefix, a thing of undefined signification, namely, the subject of all predicates without any condition distinguishing this presentation of the subject from a something generally, in short, substance, of which no conception of what it is [is conveyed] through this expression. On the contrary, the conception of a matter as substance is the conception of the movable in space. It is no wonder therefore, if permanence of substance can be proved of the latter, but not the former, since with matter it follows from its conception, namely, as being the movable, which is only possible in space, that that which possesses quantity in it, contains a plurality of the real outside one another, in other words of substances, and consequently its quantity can only be diminished by division, which is no disappearance, and even the latter would be impossible in this case according to the law of permanence. The thought I is on the contrary, no conception, but only inward perception; from it therefore nothing whatever can be deduced (except the complete distinction of an object of the internal sense from that which is merely conceived as object of external sense), and consequently not the permanence of the soul as substance.
Second law of mechanics.—All change of matter has an external cause. (Every body remains in its state of rest or motion in the same direction and with the same velocity, if not compelled by an external cause to forsake this state.)
(From universal metaphysics the proposition that all change has a cause, is laid at the foundation; here it only remains to be proved of matter, that its change must always have an external cause.) Matter, as mere object of the external sense, has no determinations but those of external relation in space, and hence is subject to no change except through motion. In respect of this, a change of one motion with another, or of the same with rest, and conversely, a cause of the same though this, must be traceable (according to principles of metaphysics). But this cause cannot be internal, for matter has no absolutely internal determinations and grounds of determination. Hence all change of a matter is based upon external causes (i.e., a body continues, &c.).
This mechanical law can only be called the law of inertia (lex inertiæ); the law that every action has an equal reaction opposed to it, cannot bear this name. For the latter says what matter does, but the former, only what it does not do, which is better adapted to the expression inertia. The inertia of matter is and means nothing but its lifelessness, as matter in itself. Life means the capacity of a substance, to act from an internal principle, determining a finite substance to change, and a material substance to rest or motion, as change of its state. Now we know no other internal principle of a substance to change its state but desire, and no other internal activity whatever but thought, with that which depends upon it, feeling of pleasure or pain, and impulse or will. But these grounds of determination and action in no wise belong to the presentations of the external sense, and thus not to the determinations of matter as matter. Thus all matter as such is lifeless. The proposition of inertia says so much and no more. If we seek the cause of any change of matter whatsoever in life, we shall have to seek it at once in another substance, distinct from matter, although bound up with it. For in natural knowledge it is necessary, first of all, to know the laws of matter as such, and to clear them from the admixture of all other efficient causes, before connecting them therewith, in order to distinguish how each acts for itself alone. On the law of inertia (next to that of the permanence of substance) the possibility of a natural science proper entirely rests. The opposite of the first, and therefore the death of all natural philosophy, would be hylozoism. From the same conception of inertia as that of mere lifelessness, it follows of itself, that it does not signify a positive effort to maintain its state. Only living beings can be termed inert in this latter sense, inasmuch as they have a conception of another state, which they dread and strive against with all their might.
Third mechanical law.—In all communication of motion, action and reaction are always equal to one another.
(From universal metaphysics the proposition must be borrowed, that all external action is reciprocal action. In this place it only has to be shown in order to remain within the bounds of mechanics that this reciprocal action (actio mutua) is at the same time reaction (reactio); but, without doing violence to the completeness of the insight, the above metaphysical law of reciprocity nevertheless cannot be left out here. All active relations of matters in space, and all changes of these relations, in so far as they can be causes of certain effects, must always be conceived as reciprocal, that is since all change of the same is motion, no motion of a body, with reference to an absolutely-resting [one] which would be thereby set in motion, can be conceived; but the latter must rather be conceived as only relatively-resting in respect of the space, to which it is referred, but together with this space as moved in the opposite direction with the same quantity of motion in absolute space, as the moved [body] has against it, in the same space. For the change of relation (in other words, the motion) is completely reciprocal between both; by as much as the one body approaches every part of the other, by so much the other approaches every part of the first. And because here the question is not as to the empirical space surrounding both bodies, but only of the line lying between them (inasmuch as these bodies are considered simply in mutual relation, according to the influence, which the motion of the one can have on the change of state of the other, by abstraction of all relation to empirical space), their motion will be regarded as merely determinable in absolute space, in which each of the two bodies must have an equal share of the motion attributed to the one in relative space, since there is no ground for ascribing more to one of them than to the other. On this footing the motion of a body, A, against another, resting, B, with regard to which it may be moving if reduced to absolute space—that is, as the relation of active causes merely referred to one another—is so considered that each has an equal share in the motion, which in the phenomenon is attributed to the body A alone. This cannot occur otherwise, than by the velocity attributed to the body A in the relative space, being distributed between A and B in inverse proportion to the masses, to A only what belongs to it in absolute space, to B, on the other hand, the relative, in addition, in which it rests, in the opposite direction, whereby the same phenomenon of motion is completely retained, the effect in the reciprocity of both bodies being constructed in the following manner:
Let a body A be in motion with a velocity = AB in respect of the relative space towards the body B, which in respect of the same space is resting. Let the velocity AB be divided into two parts, Ac and Bc, which are related to one another inversely as the masses B and A. Conceive A as moved with the velocity Ac, in absolute space, but B with the velocity Bc, in the opposite direction, together with the relative space; both motions are then opposite and equal to one another, and as they reciprocally destroy one another, both bodies are translated with reference to one another, that is, in absolute space, into [a state of] rest. B, however, was in motion with the velocity Bc in the direction BA, which is exactly opposed to that of the body A, namely AB, together with the relative space. If then the motion of the body B is destroyed by impact, the motion of the relative space is not therefore also destroyed. Thus, after the impact, the relative space moves in respect of both bodies A and B (which now rest in absolute space) in the direction BA with the velocity Bc, or, which is the same thing, both bodies move after the impact with equal velocity, Bd = Bc, in the direction of the impacting AB. According to the foregoing, however, the quantity of the motion of the body B in the direction and with the velocity Bc, and hence also that in the direction Bd with the same velocity, is equal to the quantity of the motion of the body A with the velocity and in the direction Ac. Consequently the effect, namely, the motion Bd, which maintains the body B by impact in relative space, and therefore the action of the body A with the velocity Ac, is always equal to the reaction Bc. Since this law (as mathematical mechanics teaches) suffers no alteration, when instead of the impact of a resting, an impact of the same body in the same way on a moved body is assumed; similarly as the communication of motion by impact, is only distinguished from that by traction by the direction in which the matters resist one another in their motion, it follows that in all communication of motion action and reaction are always equal to one another (that no impact can communicate the motion of a body to another except by means of an equal counter-impact, no pressure except by means of an equal counter-pressure, and in the same way no traction except by means of an equal counter-traction).*
From the above there follows, the natural, and for universal mechanics, not unimportant law, that every body, however great its mass may be, must be movable by the impact of every other, however small its mass or velocity may be. For to the motion of A in the direction AB, there corresponds necessarily an equal opposite motion of B in the direction BA. Both motions destroy one another in absolute space by impact. But thereby both bodies retain a velocity Bd = Bc in the direction of the striking [one]; consequently the body B is movable by even the smallest force of impact.
This, then, is the mechanical law of the equality of action and reaction, which is based upon [the fact] that no communication of motion takes place except in so far as a community of these motions is pre-supposed, and thus that no body strikes another, which is at rest in respect of itself, but that if it be so in respect of the space, it is only in so far as together with this space it is moved in equal degree, but in contrary direction to the motion, falling to the relative share of the former, [both together] giving the quantity of the motion to be attributed to the former, in absolute space. For no motion which is [conceived as] moving in respect of another body, can be absolute; but if it be relative in respect of the latter, there is no relation in space that is not reciprocal and equal. But there is yet another, namely, a dynamical law of the action and reaction of matters not in so far as one communicates its motion to another, but imparts it to the latter originally, and by its resistance at the same time produces it in itself. This may be readily demonstrated in a similar way. For if the matter A attract the matter B, it compels the latter to approach it, or, which is the same thing, the former resists the force with which the latter strives to retreat. But inasmuch as it is the same thing whether B retreats from A or A from B, this resistance is at the same time a resistance that the body B exercises against the body A in so far as it strives to retreat, and hence traction and countertraction are equal to one another. In the same way, if A repel the matter B, A resists the approach of B. But it is the same thing whether B approaches A, or A B, for B resists just as much the approach from A, hence pressure and counter-pressure are always equal to one another.
This, then, is the construction of the communication of motion, which at the same time carries with it as its necessary condition the law of the equality of action and reaction, which Newton did not trust himself to prove à priori, but for which we appealed to experience, and for the sake of which others introduced into natural science a special force of matter under the name force of inertia (vis inertiæ) first invented by Kepler, and thus, in the end, also deduced it from experience; while finally others again placed it in the conception of a mere communication of motion which they regarded as a gradual transference of the motion of one body into the other, whereby the moving sacrificed precisely as much as it imparted to the moved until it impressed the latter no longer (when, namely, it had arrived at equality of velocity in the direction of it).* In this way all reaction, that is, all really reacting force of the one struck against the striking [body], (such for instance as would be possible to distend a spring) is abolished; and besides that it fails to prove what is really meant by the law referred to, in nowise explains the communication of motion itself, as to its possibility. For the word transference of motion from one body to another explains nothing, and if one is unwilling to take it, so to speak literally ([as being] opposed to the principle, accidentia non migrant e substantiis in substantias) as though motion were poured from one body into the other, as water from one glass into the other, the problem is, how to make this possibility—the explanation of which rests precisely on the same ground, whence the law of the equality of action and reaction is derived—comprehensible. One cannot conceive how the motion of a body A is necessarily connected with the motion of another B, except that forces are conceived in both, as accruing to them before all motion (dynamically)—as for instance repulsion—and it can be proved, that the motion of the body A through approach towards B, with the approach of B towards A, and if B be regarded as at rest, its motion together with its space towards A, are necessarily connected, in so far as the bodies with their (original) moving forces, are merely considered in motion as relative to one another. This latter can be thereby fully comprehended à priori [viz.] that whether the body B in respect of empirically cognisable space be resting or moved, it must be regarded as necessarily moved in respect of the body A, and [moved] in an opposite direction; since otherwise, no influence thereof on the repulsive force of both would take place, without which no mechanical action whatever of matters on one another, i.e. no communication of motion by impact is possible.
The designation force of inertia (vis inertiæ) must thus, in spite of the eminence of its founder’s name, be entirely banished from natural science,—not only because it carries with it a contradiction in expression, or because the law of inertia (lifelessness) might thereby be easily confounded with the law of reaction in every communicated motion, but principally—because thereby the mistaken conception of those, insufficiently acquainted with the mechanical laws, would be maintained and strengthened according to which the reaction of bodies, of which we are speaking under the name force of inertia, consists in the motion being thereby swallowed up, diminished or destroyed, without the mere communication of motion being effected, in that, namely, the moving body would have to apply a part of its motion to overcoming the inertia of the resting [one] (which would be pure loss), and with the remaining portion only, could set the latter in motion; but if nothing remained, would not be able by its impact to bring the latter into motion on account of its great mass. A motion can resist nothing except opposite motion of another, but, in nowise its rest. Here therefore inertia of matter, that is mere incapacity to move of itself, is not the cause of a resistance. The expression force of inertia used to designate a special and quite peculiar force, merely in order to resist without being able to move a body, would be a word without any significance. The three laws of universal mechanics might be more suitably designated, the law of the subsistence, the inertia, and the reaction of matters (lex subsistentiæ, inertiæ et antagonismi) by all changes of the same. That these, in other words, the entire propositions of the present science, exactly answer to the categories of substance, causality and community, in so far as these conceptions are applied to matter, requires no further elucidation.
General Observation on Mechanics.
The communication of motion only takes place by means of such moving forces, as inhere in a matter at rest (impenetrability and attraction). The action of a moving force on a body in one moment is its solicitation, the velocity acquired by the latter through solicitation, in so far as it increases in equal proportion to the time, is the moment of acceleration. (The moment of acceleration must therefore only contain an infinitely small velocity, as otherwise the bodies would attain through this an infinite velocity in a given time, which is impossible. The possibility of acceleration generally moreover, rests, through a continuous moment of the same, on the law of inertia.) The solicitation of matter through expansive force (e.g., a compressed air that bears a weight) occurs always with a finite velocity; but the velocity impressed thereby on another body (or withdrawn from it) can only be infinitely small; for the former is only a superficial force, or, which is the same thing, the motion of an infinitely small quantum of matter, which must occur consequently with finite velocity in order to be equal to the motion of a body of finite mass with infinitely small velocity (a weight). On the other hand attraction is a penetrating force, by virtue of which, a finite quantum of matter exercises moving force on a similarly finite quantum of another [matter]. The solicitation of attraction must therefore be infinitely small, because it is equal to the moment of acceleration (which must always be infinitely small), while with repulsion, where an infinitely small portion of matter is to impress a moment on a finite [portion] this is not the case. No attraction admits of being conceived with a finite velocity without the matter being obliged to penetrate itself by its own attractive force. For the attraction, which a finite quantity of matter exercises on [another] finite with a finite velocity, must be superior to every finite velocity, whereby matter reacts through its impenetrability, but only with an infinitely small portion of the quantity of its matter, on all points of the compression. If attraction is only a superficial force, as cohesion is conceived, the opposite of this would follow. But it is impossible, so to conceive it, if it is to be true attraction (and not mere external compression).
An absolutely hard body would be one whose parts attracted one another so strongly, that they could not be separated by any weight, nor altered in their position with regard to one another. Now, since the parts of the matter of such a body would have to attract one another with a moment of acceleration, which would be infinite as against that of gravity, but finite as to the mass thereby driven, resistance by impenetrability as expansive force, since it always occurs with an infinitely small quantity of matter, would have to take place with more than finite velocity of solicitation, that is, the matter would seek to extend itself with infinite velocity which is impossible. Thus an absolutely hard body, that is, one which would oppose in one moment a resistance on impact, to a body moved with finite velocity equal to the whole of its force, is impossible. Consequently, a matter exercises by its impenetrability or cohesion only an infinitely small resistance in one moment, to the force of a body in finite motion. Hence follows the mechanical law of continuity (lex continui mechanica), namely: in no body is the state of rest or motion—and in the latter, velocity or direction—changed by impact, in one moment, but only in a certain time, through an infinite series of intermediate states whose difference from one another is smaller than the first and last. A moved body that strikes against a matter, is not brought to rest by its resistance at once, but only by continuous retardations, or that which was at rest only [set in] motion by continuous acceleration, or from one degree of velocity into another according to the same rule. In the same way, the direction of its motion in [a body] that describes an angle, is only changed by means of all possible intermediate directions, that is, by means of motion in a curved line (which law for a similar reason, can be also extended to the change of the state of a body by attraction). This lex continui is based on the law of the inertia of matter, while, on the other hand, the metaphysical law of continuity in all change (internal as well as external) must be extended universally, and hence would be based on the mere conception of a change in general, as quantity, and on the generation of the same (which must necessarily proceed continuously in a certain time, like time itself), and thus has no place here.
METAPHYSICAL FOUNDATIONS OF PHENOMENOLOGY.
Matter is the movable, in so far as it can be an object of experience as such.
Motion, like all that can be presented through sense, is only given as phenomenon. In order that its presentation may become experience, it requires in addition, that something should be conceived through the understanding, namely, as to the way in which the presentation inheres in the subject, not the definition of an object through the same. Thus the movable, as such, is an object of experience, when a certain object (here a material thing) is conceived as defined in respect of the predicate of motion. But motion is change of relation in space. Hence, firstly there are always two correlates here, to one no less than to the other of which, change is attributed in the phenomenon, and either the one or the other can be termed moved inasmuch as it is indifferent to both, or secondly, of which one must, in experience be conceived as moved to the exclusion of the other, or thirdly of which both must necessarily be conceived through Reason as moved at the same time. In the phenomenon, which contains nothing but the relation in motion (as to its change), there are none of these determinations, but when the movable, as such, i.e. as to its motion, is to be conceived as determined, namely, for the sake of a possible experience, it is necessary to indicate the conditions, by which the object (matter) would have to be determined in this or that manner, by the predicate of motion. Here, the question is not of the transformation of illusion into truth, but of phenomenon into experience. For with illusion the understanding is always engaged with its own judgment determining an object—although it is in danger of mistaking the subjective for objective—but in the phenomenon no judgment of the understanding is to be met with; and this is necessary to be remembered, not only here, but in the whole of philosophy, because, otherwise, when we are concerned with phenomena, and this expression is taken as identical in signification with that of illusion, misunderstanding will always arise.
The rectilinear motion of a matter is, in respect of an empirical space, as distinguished from the opposite motion of the space, a merely possible predicate. The same [thing] conceived in no relation to a matter outside it, that is, as absolute motion, is impossible.
Whether [in the case of] a body moved in relative space, this latter be described as resting, or conversely, as moved with equal velocity in an opposite direction, and the former as resting, there is no statement as to what belongs to the object, but only to its relation to the subject, in other words, to the phenomenon and not to experience. For if the spectator place himself in the same space as resting, he terms the body moved; but if he place himself (at least in thought) in another space enclosing this, in respect of which the body is, in the same way, resting, then the relative space is termed “moved.” In experience, therefore (a cognition, determining validly the object for all phenomena), there is no difference whatever between the motion of the body in relative space, or the rest of the body in absolute, and the equal and opposite motion of the relative, space. Now the presentation of an object by one of its two predicates—which, in respect of the object, are equivalent, and only as regards the subject and its mode of presentation distinguished from one another—is not its determination according to a disjunctive, but merely an alternative judgment according to choice (of which the first of two objectively opposed predicates, one with the exclusion of its contrary, but the other of objectively equivalent indeed, but subjectively opposed judgments without excluding the contrary of the object, in other words, by mere choice)—one is assumed for the determination of the same [viz., the object].* This means: by the conception of motion as object of experience, it is in itself undetermined, and therefore equivalent, whether a body is conceived as moved in relative space or the space in respect of the body. Now that which, in respect of two mutually opposed predicates, is in itself undetermined, is so far merely possible. Hence the rectilinear motion of a matter in empirical space, as distinguished from the equal opposite motion of the space, is in experience a merely possible predicate, which was the first [point].
Further, since a relation, in other words a change of the same, namely, motion, can only be an object of experience in so far as both correlates are objects of experience—but pure space, also called, in contradistinction to the relative (empirical), absolute space, is no object of experience and nothing at all—therefore rectilinear motion, without reference to anything empirical—that is, absolute motion—is simply impossible;—which was the second [point.]
This proposition determines the modality of the motion in respect of Phoronomy.
The circular motion of a matter as distinguished from the opposite motion of the space, is a real predicate of the same; while, on the other hand, if the opposite motion of a relative space be taken, instead of the motion of the body, there is no real motion of the latter, but [should it be regarded as such] a mere illusion.
The circular motion is (like every non-rectilinear [motion]) a continuous change of the rectilinear, and as this is itself a continuous change of relation in respect of external space, the circular motion is a change of the change of these external relations in space, and consequently a continuous arising of new motions; since, now, according to the law of inertia, a motion, in so far as it arises, must have an external cause, while the body, in every point of this circle, is endeavouring, according to the same law, to proceed in the straight line touching the circle, which motion works against the above external cause, every body in circular motion demonstrates by its motion a moving force. Now the motion of the space as distinguished from that of the body is merely phoronomic, and has no moving force. As a consequence, the judgment, that here, either the body or the space is moved in an opposite direction, is a disjunctive judgment, by which, if the one member, the motion of the body, be posited, the other, namely, that of the space, is excluded. Hence the circular motion of the body, as distinguished from the motion of the space, is a real motion, and consequently the latter, even though as phenomenon it coincide with the former, nevertheless, in the complex of all phenomena, that is, of possible experience, contradicts it, and hence is nothing but mere illusion.
This proposition determines the modality of motion in respect of Dynamics; for a motion, which cannot take place without the influence of a continuously active external moving force, proves indirectly or directly original moving forces of matter, either of attraction or repulsion. For the rest, Newton’s scholium to the definitions with which he introduces his Princ. Philos. Nat. Math., towards the end, may be referred to, on the present subject, from which it will appear, that the circular motion of two bodies round a common centre (hence, also the motion of the earth on its axis), even in empty space, and thus without any comparison being possible through experience, with external space, may nevertheless be cognised by means of experience, in short, that a motion which is a change of external relation in space can be given empirically, although this space itself is not empirically given, and is no object of experience—a paradox deserving to be solved.
In every motion of a body, whereby it is moving in respect of another, an opposite and equal motion of the latter is necessary.
According to the third law of mechanics (Proposition 4) the communication of the motion of a body is only possible through the community of its original moving forces, and these only through reciprocal and equal motion. The motion of both is then real. But as the reality of this motion does not rest (as in the second proposition) on the influence of external forces, but follows immediately and inevitably from the conception of the relation of the moved in space, to every other [thing] thereby movable, the motion of the latter is necessary.
This proposition determines the modality of motion in respect of mechanics; that, for the rest, these three propositions determine the motion of matter in respect of its possibility, reality, and necessity, in short, in respect of all the three categories of modality, is sufficiently obvious of itself.
General Observation on Phenomenology.
There are thus three conceptions noticeable here, whose employment in universal natural science is unavoidable, and whose exact definition is for this reason necessary, although not so easy and comprehensible: firstly, the conception of motion in relative (movable) space; secondly, the conception of motion in absolute (immovable) space; thirdly, the conception of relative motion generally, as distinguished from absolute [motion.] The conception of absolute space is laid at the foundation of all [these]. But how do we come by this singular conception, and on what rests the necessity of its employment?
It can be no object of experience; for space without matter is no object of perception, and yet is a necessary conception of the Reason, and therefore nothing but a a mere idea. For in order that motion may be given even as phenomenon, an empirical presentation of space in respect of which the movable has to change its relation is required. But space, which is to be perceived, must be material, and therefore in accordance with the conception of a matter generally, itself movable. Now, in order to conceive it as moved, one has only to conceive it as contained in a space of greater compass, and to assume the latter as resting. But this admits of being treated similarly as regards a still more extended space, and so on to infinity, without ever attaining through experience to an immovable (immaterial) space, in respect of which any matter could have absolute motion or rest attributed to it; but the conception of these relational determinations will have to be constantly changed, according as the movable is considered as in relation to one or the other of these spaces. Now, as the condition of regarding anything as resting or moved is always again and again conditioned to infinity in relative space, it thence appears: firstly, that all motion or rest is merely relative, and that neither can be absolute, i.e., that matter can merely be conceived in relation to matter as moved or resting, but not in respect of mere space without matter; in other words, that absolute motion, such, namely, as is conceived without any reference of one matter to another, is simply impossible: secondly, [it will appear] that for this very reason no conception of motion or rest, in relative space, valid for every phenomenon, is possible, but that a space must be conceived, in which the latter itself can be thought of as moved, but whose determination does not depend on any other empirical space, and hence is not again conditioned, that is, an absolute space to which all relative motions may be referred, and in which everything empirical is movable; [and this] in order that all motions of the material in the same can be valid as merely relative to one another, as alternatively-reciprocal,* but none as absolute motion or rest (since, inasmuch as one is called moved, the other, with reference to which our former is moved, may be similarly conceived as absolutely resting). Absolute space is then necessary, not as a conception of a real object, but as a mere idea which is to serve as a rule, for considering all motion therein as merely relative, and all motion and rest must be reduced to absolute space if the phenomenon of the same is to be transformed into a definite conception of experience (which combines all phenomena).
In this way the rectilinear motion of a body in relative space, is reduced to absolute space, which does not fall within the range of the senses if I conceive the body, as at rest in itself, and this presentation as that which gives precisely the same phenomenon, whereby all possible phenomena of rectilinear motions, which a body may happen at the same time, to possess, are reduced to the conception of experience, which unites them together (namely, to that of merely relative motion and rest).
Circular motion, inasmuch as, according to the second proposition, even without reference to the external empirically given space, it can be given as real motion in experience, seems to be really absolute motion. For the relative in respect of external space (for instance, the motion of the earth on its axis, relative to the heavenly bodies), is a phenomenon, in place of which, the opposite motion of this space (the heavens), in the same time, can be posited as fully equivalent to the former, but which, according to this proposition, can never in experience be put in the place of the former; and therefore the above circular motion cannot be regarded as externally relative, which sounds as though this kind of motion were assumed as absolute.
But it is to be observed that the question is here of the true (real) motion, which does not appear as such—which therefore, were we content to judge according to empirical relations of the space, might be regarded as rest—in other words, the question is of the true motion as distinguished from the illusive, but not of it as absolute motion in contradistinction to the relative; and hence circular motion, although it exhibits in the phenomenon, no change of place, that is, no phoronomic [change] of the relation of the moved to empirical space, exhibits, nevertheless, a continuous dynamic change of the relation of matter in its space, demonstrable by experience; for instance, it shows a constant diminution of the attraction by an effort to retreat, as the effect of circular motion, and thereby decisively indicates its distinction from illusion. For instance, one can conceive the earth as turned about its axis in infinite empty space, and demonstrate this motion by experience, although neither the relation of the parts of the earth among one another, or to the space outside it, is changed phoronomically, i.e., in the phenomenon. For, as regards the first, nothing changes its place upon or in the earth as empirical space; and with reference to the second, which is quite empty, no externally changed relation, and therefore no phenomenon of a motion can take place. But if I suppose a deep cavern tending towards the centre of the earth, and dropping a stone into it, find that although at every distance from the centre, the gravity is always directed thereto, the falling stone nevertheless, continuously reverts from its upright position, from west to east, I conclude that the earth is from evening to morning turned about its axis. Or, if I withdraw the stone from the surface of the earth, and it does not remain over the same point of the surface, but moves itself from east to west, I shall still infer the foregoing motion of the earth on its axis, and both perceptions are a sufficient proof of the reality of this motion, for which the change of relation to external space (the starry heaven) is inadequate as it is mere phenomenon, which may proceed from two actually opposed causes, and which is not a cognition deducible from the ground of explanation of all phenomena of this change, that is, experience. But that this motion, although no change of relation to empirical space, is nevertheless no absolute motion, but continuous change of the relation of matters to one another, and while conceived in absolute space, is really only relative and for this very reason, alone true motion; this rests on the conception of the reciprocally continuous retreat of each part of the earth (outside the axis) from every other [part], situated opposite to it in the diameter, at equal distance from the centre. For this motion is real in absolute space, in that thereby the retreat from the distance in question, when gravity in itself would attract to the body, and indeed without any dynamical repulsive cause (as may be seen from the instances chosen in Newton’s Princ. Phil. Nat., p. 10, Edit. 1711),* is continuously replaced by real motion inclosed within the moved matter (namely, the centre of the same), but not having reference to the external space.
As to the case of the third proposition, it does not require, in order to show the truth of the reciprocally opposed and equal motion of two bodies even without reference to the empirical space, [to exhibit] the active dynamical influence (of gravity or of a distended string) given through experience, which is necessary in the second case, but the mere dynamical possibility of such an influence as property of matter (repulsion or attraction) since the motion of the one carries with it, at the same time, the opposite and equal motion of the other, and indeed from mere conceptions of a relative motion, if it be considered in absolute space, i.e. according to truth; and it is, therefore, like all that is adequately demonstrable from mere conceptions a law of absolutely necessary counter-motion.
There is no absolute motion, even where a body is conceived as moved in respect of another in empty space; the motion of both being here, not relative to the space surrounding them, but only to that between them, which alone determines their external relation to each other, considered as abstract space, and is thus in its turn, only relative. Hence, absolute motion would be only that accruing to a body without relation to any other matter. But such would be the rectilinear motion of the universe, i.e. the system of all matter. For so long as any other matter existed outside of a matter, even though separated by empty space, the motion would still be relative. For this reason every proof of a law of motion, having as its result, that its contrary would necessarily imply a rectilinear motion of the whole universe as its consequence, is an apodictic demonstration of its truth; simply because absolute motion would thence ensue, which is quite impossible. Of this kind is the law of antagonism in all community of matter through motion. For every deviation from the same would move the common centre of gravity of all matters, in short, the whole universe, from its place, while on the contrary this would not happen if one regarded the latter as turned on its axis, a motion always possible to be conceived, although so far as one can see, there would be no use in assuming it.
The different conceptions of empty space also have their reference to the different conceptions of motion and moving forces. Empty space in a phoronomic sense, also termed absolute space, ought not properly to be called empty space; for it is only the idea of a space, in which I abstract from all particular matter, making it an object of experience, in order to conceive therein, the material, or every empirical space, as movable, and the motion not merely as on one side absolute, but as mutually relative predicate. Hence it is nothing belonging to the existence of things, but merely to the determination of the conception, and in so far no empty space exists. Empty space, in a dynamic sense, is that which is not filled, i.e., in which nothing else movable resists the penetration of the movable, consequently in which no repulsive force acts, and it may be either the empty space within the world (vacuum mundanum), or, if the latter be conceived as bounded, empty space outside the world (vacuum extramundanum); the first moreover, either as distributed (vacuum disseminatum), which constitutes only one portion of the volume of the matter, or as continuous empty space (vacuum coacervatum, which separates bodies, for instance, the heavenly bodies, from one another), a distinction which, inasmuch as it rests on the difference of places, assigned to empty space in the universe, is not essential, but is used in various ways; firstly, in order to deduce the specific difference of density, and secondly, in order [to deduce] the possibility of a movement in the universe, free from all external resistance. That empty space in the first sense is not necessary to be assumed, has already been shown in the general remark on dynamics; but that it is impossible can by no means be demonstrated from its conception alone, according to the principle of contradiction. Yet, even if no merely logical ground for its rejection be present, a universal physical ground for banishing it from natural science exists, namely, that of the possibility of the composition of a matter generally, if the latter [question] were only better understood. For if attraction, which is assumed for the explanation of the cohesion of matter, be only apparent, not real, attraction—but as it were the effect of a compression, by external matter (the ether) existing throughout the universe, which is itself brought to this pressure, by a universal and original attraction, namely, gravitation, an opinion supported by many reasons—empty space within matters would then, although not logically, be nevertheless dynamically, and hence physically, impossible, since every matter would expand of itself, in the empty spaces assumed within the same (as nothing would then resist its expansive force), and they would thus be always filled. An empty space outside the world, would, if by this expression be understood all the principal attractive matters (the large heavenly bodies), be impossible, for the same reason, for in proportion as the distance from these increased, the attractive force on the ether (which encloses all the above bodies, and impelled by them maintains in their density by compression), would diminish in inverse proportion, and the latter itself, would diminish in density to infinity, though it would nowhere leave the space entirely empty. Meanwhile, it need surprise no one that in this rejection of empty space, we are proceeding quite hypothetically; for its assumption fares no better. Those who venture to decide this moot question dogmatically, whether they do so affirmatively or negatively, support themselves in the end on mere metaphysical assumptions, as may be seen from the dynamics; but it was at least necessary to show here, that this could not decide in the problem in question. Thirdly, as concerns empty space in a mechanical sense, this is continuous emptiness within the universe, in order to procure free motion for the heavenly bodies. It is easily seen, that the possibility or impossibility of this rests, not on metaphysical grounds, but on the hardly disclosed secrets of nature, as to the way in which matter sets limits to its own expansive force. Notwithstanding this, if that be admitted which has been said in the general observation on dynamics, as to the possibly greater expansion to infinity of specifically different matters, with the same quantity of matter (as regards its weight) an empty space might indeed be then unnecessary to assume, even for the sake of the free and lasting motion of the heavenly bodies, as the resistance, even in entirely filled spaces, might then be assumed to be as small as one liked.
And so ends the metaphysical doctrine of body with emptiness and therefore incomprehensibility, and the reason has the same fortune in all other attempts, where it strives to reach principles of the ultimate grounds of things, inasmuch as its nature is such, that it can never comprehend anything except in so far as it is determined under given conditions; consequently, since it can neither rest at the conditioned nor can make the unconditioned comprehensible, when thirst for knowledge stimulates it, to grasp the absolute totality of all conditions, nothing remains for it but to turn back from objects, upon itself, in order that instead of the ultimate boundaries of things, it may investigate and determine the ultimate boundaries of the capacity pertaining to itself.
[1 ]1, Substantia; 2, Qualitas; 3, Quantitas; 4, Relatio; 5, Actio; 6, Passio; 7, Quando; 8, Ubi; 9, Situs; 10, Habitus (Substance; Property; Quantity; Relation; Action; Passion; When; Where; Position; State.)
[2 ]Oppositum; Prius; Simul; Motus; Habere. (Opposition; Priority; Simultaneity; Motion; Possession.
[1 ]On the table of the categories many ingenious observations may be made; as (1) that the third arises from the combination in one conception of the first and second; (2) that those of quantity and quality are merely a progression from unity to totality, or from something to nothing (for which purpose the categories of quality must stand thus: reality, limitation, complete negation) without correlata or opposita; while, on the other hand, those of relation and modality carry the latter with them; (3) that, as in logic, categorical judgments lie at the foundation of all others, so the category of substance does to all conceptions of real things; (4) that, as modality is no particular predicate in judgments, so also modal conceptions add no determination to things, &c. Such considerations are very useful. If, in addition, all the predicables are counted up, that can be drawn pretty completely from any good Ontology (e.g., Baumgarten’s), and are arranged in classes under the categories—whereby we must not omit, however, to add as complete a dissection of all these conceptions as possible—a purely analytic part of metaphysics will arise, containing, not a single synthetic proposition, which might precede the second (the synthetic), and by its definiteness and completeness be not only useful, but by virtue of its symmetrical character contain a certain beauty.
[1 ]Critique, p. 190 et seq.
[1 ]If it be said that a science is at least real in the idea of all men when it is constituted; that the problems leading to it are put forward by the nature of the human reason in all men, and consequently that many, if faulty, attempts at its solution are at all times unavoidable, we must then say, metaphysics is subjectively (and necessarily) real, and hence we ask with justice, How is it (objectively) possible?
[1 ]In disjunctive judgments we consider all possibility as divided in relation to a particular conception. The ontological principle of the thorough determination of a thing generally (that of all possible opposite predicates one must attach to each thing), which is at the same time the principle of all disjunctive judgments, is based on the content (Inbegriff) of all possibility, in which the possibility of a thing in general is regarded as determined. This serves as a slight explanation of the above proposition, that the act of Reason, in disjunctive conclusions of the Reason, is the same, as regards form, as that whereby it attains to the idea of a content of all reality, embracing in itself the positive of all mutually opposing predicates.
[1 ]Were the presentation of the apperception, the ego, a conception whereby anything whatever was thought, it could also be used as predicate of other things, or it would contain such predicates. It is, really, nothing more than the feeling of a reality without the least conception, but only presentation of that to which all thought stands in relation (relatione accidentis).
[1 ]It is indeed very remarkable that the metaphysicians of all times should have so carelessly passed over the permanence of substances without ever attempting a demonstration of it, doubtless because they saw themselves forsaken by all proofs as soon as they began [to deal] with the conception of substance. Common sense, well aware that without this assumption no union of perceptions in an experience is possible, supplied this deficiency by a postulate; for from experience itself it could never have drawn this axiom; partly because it could not pursue the matters (substances) in all their changes and dissolutions far enough to find the matter for ever undiminished; partly because it contained the axiom of necessity, which is always the sign of an à priori principle. Now they composedly applied this axiom to the conception of the soul as a substance, and inferred its necessary continuance after the death of man, especially as the simplicity of this substance, deduced from the indivisibility of consciousness, assured it against destruction by dissolution. Had they found the real source of this axiom, which, however, demanded much deeper investigations than they were disposed to give to it, they would have seen that the above law, of the permanence of substances, only obtains for the sake of experience, and for things in so far as they are to be cognised and connected with others in experience, and that it can never be valid of things, irrespective of all possible experience, such as the soul after death.
[1 ]Hence I am anxious that the critical reader should especially occupy himself with this antinomy, because Nature herself seems to have set it up, in order to make the Reason stagger in its pretensions, and to force it into self-examination. Each proof that I have given, as well for the thesis as the antithesis, I undertake to guarantee, and thereby to exhibit the certainty of this unavoidable antinomy of the Reason. If the reader is only brought by this singular phenomenon to go back to the examination of the assumption at its foundation, he will feel himself compelled to investigate more deeply with me the primary foundation of all cognition of the pure Reason.
[1 ]The idea of freedom finds a place solely in the relations of the intellectual as cause to the phenomenon as effect. Hence we cannot attribute freedom to matter with regard to the ceaseless action with which it fills its space, although this action results from an internal principle. Just as little can we find any conception of freedom suited to pure essences of the understanding; as, for instance, God, in so far as His action, is immanent; for His action, although independent of external determining causes, is nevertheless determined in His eternal Reason, that is, in the divine nature. Only if an action is to commence something, in other words, if the effect is to be met with in the time-series, and consequently in the sense-world (e.g., the beginning of the world), only then does the question arise whether the causality of the cause itself must commence, or whether the cause can give rise to an effect without its causality itself commencing. In the first case the conception of this causality is a conception of necessity, in the second, of freedom. The reader will see from the above that in explaining freedom to be the faculty of beginning an event spontaneously, I exactly hit the conception constituting the problem of metaphysics.
[1 ]Herr Plattner in his Aphorisms says with acuteness, §§ 728, 729: “If the Reason be a criterion no conception can be possible which is incomprehensible to the human Reason. In the real alone is incomprehensibility to be found. Here the incomprehensibility arises from the insufficiency of the ideas acquired.” It, therefore, only sounds paradoxical and is really not strange to say that in Nature there is much that is incomprehensible (for instance, the faculty of procreation), but that when we rise higher and pass beyond Nature all is again comprehensible; for we then quit the objects that can be given us, and occupy ourselves merely with ideas, by which we may well comprehend the law wherewith the Reason prescribes to the Understanding its use in experience, because it is its own product.
[1 ]Of this nature is an analogy between the juridical relations of human actions and the mechanical relations of moving forces: I can do nothing to another without giving that other the right, under the same conditions, to do the same to me; just as no body can act upon another body with its moving force without causing thereby that other body to react upon itself to the same extent. Here right and moving force are quite dissimilar things, but in their relation there is complete resemblance. Hence, by means of such an analogy as this, I can give a relational conception of things absolutely unknown to me. For instance, how the promotion of the happiness of children is related (= a), to the love of parents (= b), to the welfare of the human race (= c), to the unknown [quality] in God (= x), which we term love, not as though it had the least resemblance to any human affection, but because we can conceive its relation to the world as similar to that which things of the world have among one another. But the relational conception is here a mere category, namely, the conception of cause, which has nothing to do with sensibility.
[1 ]I should say, the causality of the supreme Cause is, in respect of the world, what human Reason is in respect of art-works. The nature of the supreme Cause itself remains unknown throughout. I only compare its effect known to me (the order of the world) and its accordance with Reason, with the known workings of human Reason, and hence call the former a Reason, without thereby attributing to it as its characteristics, what I understand by this expression in men or anything else known to me.
[1 ]It has been my constant design throughout the Critique to omit nothing that could render the investigation into the nature of the pure Reason complete, however deeply hidden it might be. Every one is at liberty afterwards to carry his researches as far as he likes, if it has been only indicated to him what yet remains to be done; for this may be reasonably expected of any one who has made it his business to survey this whole field, in order afterwards to consign it to others for future cultivation and allotment. To this department belong also both the scholia, which by their dryness will scarcely recommend themselves to amateurs, and hence have only been added for specialists.
[1 ]Not certainly higher. High towers, and metaphysically-great men resembling them, round both of which there is commonly much wind, are not for me. My place is the fruitful bathos of experience; and the word transcendental, the meaning of which is so often elucidated by me, but not once grasped by my critic (so carelessly has he regarded everything), does not signify something passing beyond all experience, but something that indeed precedes it à priori, but that is intended simply to make cognition of experience possible. If these conceptions overstep experience, their employment is termed transcendent, which is distinguished from their immanent [employment], that is, their employment as limited to experience. All misunderstandings of this kind have been sufficiently guarded against in the work itself, but the critic found his advantage in misunderstanding.
[1 ]Idealism proper always has a mystical tendency, and can have no other, but mine is solely designed for the understanding of the possibility of our cognition à priori of objects of experience, which is a problem never hitherto solved or even suggested. In this way the whole mystical idealism falls to the ground, for (as may be seen already in Plato) it inferred from our cognitions à priori (even from those of geometry) another intuition to that of the senses (namely, an intellectual intuition), because it never occured to [philosophers] that the senses themselves might intuite à priori.
[1 ]The critic often fights with his own shadow. When I oppose the truth of experience to dream, he never thinks that I am here speaking simply of the well-known somnio objective sumto of the Wolffian philosophy, which is merely formal, and with which the distinction between sleeping and waking is in no way concerned, and in a transcendental philosophy indeed can have no place. For the rest, he calls my deduction of the categories and table of the principles of the understanding, “common well-known axioms of logic and ontology, expressed in an idealistic manner.” The reader need only consult these Prolegomena upon this, to convince himself that a more miserable and historically incorrect, judgment, could hardly be made.
[1 ]Essence is the primal inner principle of all that belongs to the possibility of a thing. Hence one can only predicate an essence, but not a nature of geometrical figures (for nothing is contained in their conception expressive of an existence).
[1 ]I find doubts expressed in the criticism of Professor Ulrich’s Institutiones Logicæ et Metaphysicæ, in the ‘Allgemeine Litteratur Zeitung’ (1785), No. 295, not indeed respecting this table of the pure conceptions of the understanding, but the conclusions drawn therefrom as to the limitation of the whole faculty of the pure Reason, and therefore of all metaphysics, in which the learned critic expresses himself at one with his no less accurate author; doubts which, because they are supposed to touch the foundation-stone of my system, as put forward in the Critique, should be reasons for thinking that the latter did not by far carry that apodictic necessity with it, in respect of its main object, which is indispensable in compelling an unqualified acceptance. This foundation-stone is said to be a deduction expounded partly there, and partly in the Prolegomena, of the pure conceptions of the understanding, which in that part of the Critique, that should have been the clearest, is said to be the most obscure, or indeed, to move in a circle, etc. I direct my answer to these objections, only to their chief point, namely, that without a completely clear and adequate deduction of the categories, the system of the Critique of pure Reason would totter to its foundations. I maintain, on the contrary, that for those who subscribe to my propositions as to the sensibility of all our intuition, and the sufficiency of the table of the categories, as determinations of our consciousness borrowed from the logical functions of judgment in general (as the Reviewer does) the system of the Critique must carry with it apodictic certainty because it is built on the proposition, that the whole speculative use of our Reason never reaches beyond objects of possible experience. For if it can be proved that the categories, of which the Reason must make use in all its cognition, can have no other employment whatever, except merely with reference to objects of experience (in such a way that only in them [viz. the categories] is the form of thought possible), the answer to the question, how they make such possible is indeed important enough, in order, as far as may be to complete this deduction, but in respect of the main object of the system, namely the determination of the boundary of the pure Reason in nowise necessary, but merely desirable. For in this respect, the deduction is already carried far enough, when it shows that the conceived categories are nothing but mere forms of the judgments, in so far as they are applied to intuitions (which are with us always sensuous), by which they first of all become objects and cognitions, because this already suffices to found the whole system of the Critique proper with complete certainty. Thus Newton’s system of universal gravitation is established, although it carries with it the inexplicable difficulty of how attraction at a distance is possible; but difficulties are not doubts. That the foundation remains even without the complete deduction of the categories being established, I can prove, from what is conceded, thus:
Conceded: that the table of the categories contains all the pure conceptions of the understanding complete, as well as all the formal operations of the understanding in judgments, from which they are deduced and differ in nothing, beyond that in the conception of the understanding an object is regarded as defined in respect of one or the other function of judgment (e.g., in the categorical judgment the stone is hard; the stone is employed as subject, and hard as predicate, so that it remains permissible to the understanding to turn the logical function of these conceptions round, and say, something hard is a stone: on the contrary, when I represent it to myself in the object as determined, that the stone (in every possible determination of an object, not of the mere conception) must be conceived only as subject, and the hardness only as predicate, the same logical functions become pure conceptions of the understanding of objects, namely, as substance and accident;)
2, Conceded: that the understanding, by its nature, carries with it synthetic principles à priori, by which it subordinates to the foregoing categories all objects that may be given it; and therefore that there must be also intuitions à priori, containing the requisite conditions for the application of the above pure conceptions of the understanding, because, without intuition there is no object in respect of which the logical function can be determined as category, and hence no cognition of any object; and that without pure intuition, no axiom defining it à priori in this respect can obtain;
3, Conceded: that these pure intuitions can never be anything but mere forms of the phenomena of the external or internal sense (space and time), and consequently only of the objects of possible experience:
It follows, that no employment of the pure Reason can ever refer to anything but objects of experience, and, as in axioms à priori, nothing empirical can be the condition, they can be nothing more than principles of the possibility of experience generally. This alone is the true and adequate foundation of the determination of the boundary of the pure Reason, but not the solution of the problem: how experience is possible by means of these categories and only by means of them. The last problem, although even without it the structure would be firm, has meanwhile great importance, and, as I now see, equally great facility, since it can be solved well-nigh by a single conclusion from the precisely determined definition of a judgment in general (an act by which the given presentations first become cognitions of an object). The obscurity which, in this portion of the deduction attaches to my previous operations, and which I do not disclaim, is attributable to the usual fortune of the understanding in research, the shortest way being commonly not the first it is aware of. I shall, therefore, take the earliest opportunity of supplying this defect (which more concerns the style of exposition than the ground of explanation, which is given correctly enough, even there) without placing my acute critic in the, doubtless, to himself, unpleasant necessity of taking refuge in a pre-established harmony, by reason of the unaccountable agreement of the phenomena with the laws of the understanding notwithstanding that the latter have sources quite distinct from the former—a remedy, by the way, far worse than the evil it is intended to cure, and against which it can really avail nothing at all. For the objective necessity in question, characterising the pure conceptions of the understanding (and the principles of their application to phenomena) cannot come out of this. For instance, in the conception of cause in connection with effect, everything remains merely subjectively necessary, but objectively simply chance combination, just as Hume has it, when he terms it mere illusion through custom. No system in the world can derive this necessity otherwise than from the pure à priori principles lying at the foundation of the possibility of thought itself, whereby alone the cognition of objects whose phenomenon is given us, that is, experience, is possible; and even supposing that the mode, how experience is thereby possible, were never adequately explained, it would remain indisputably certain that it is merely possible through these conceptions, and conversely that these conceptions are capable of no meaning or employment in any other reference than to objects of possible experience.
[1 ]Gloria geometria, quod tam paucis principiis alicunde petitis tam multa praestet.—Newton, Princ. Phil. Nat. Math. Praefat.
[1 ]See Prolegomena.—[Tr.]
[1 ]This formula means: “Velocity (Celeritas: C) is related as the space passed over (Spatium: S) divided by the time consumed therein, (Tempus: T) or: the velocity increases in direct ratio to the space passed over, and in inverse ratio to the time consumed therein.” (Kirchmann, Erläuterungen, p. 25).—[Tr.]
[1 ]The verb is wanting to this sentence in the original.—[Tr.]
[1 ]It is impossible to represent surfaces at given distances as wholly filled by the action of lines spreading out from a point in the form of rays, whether of luminosity or attraction. Thus, by such diverging rays of light, the inferior luminosity of a distant surface would merely rest on the fact that between the luminous there remain non luminous places, and these so much the larger the farther the surfaces are removed. Euler’s hypothesis avoids this inconvenience, but has certainly so much the greater difficulty in rendering the rectilinear motion of the light conceivable. But this difficulty arises from an easily avoidable mathematical conception of light-matter as a mass of globules, which according to their variously oblique arrangement, as regards the direction of the impact, would produce a lateral motion of light; whereas nothing prevents us from conceiving this matter as originally and in every sense fluid, instead of as divided into fixed globules. If the mathematician wishes to render intuitable the diminution of light by increasing distance, he makes use of rays spreading in a circle, in order to exhibit on the disc of its diffusion the size of the space, in which the same quantity of light is to be uniformly diffused between these circle-rays, in short, the diminution of the degree of luminosity; but he does not intend these rays to be regarded as the only [places of] luminosity, as though there were always places devoid of light, to be met with between them, these increasing with the distance. If one wishes to conceive each of these places as throughout luminous, the same quantity of luminosity which covers the smaller must be conceived as in equal proportion in the larger, and therefore, in order to indicate the rectilinear direction, they must be drawn from the surface and all its points to the luminous straight lines. The effect and its quantity must be previously fixed, and the cause indicated in accordance therewith. The same applies to rays of attraction, if one chooses to call them so, and indeed to all directions of forces, which are to fill a space, be it even a corporeal one, from a point.
[* ]In Phoronomy, as the motion of a body in respect of its space, was considered as change of relation in the same, it was quite indifferent whether I sought to ascribe to the body in space—or instead thereof to the relative space—an equal but opposite motion. Both give fully the same phenomenon. The quantity of the motion of the space was merely the velocity, and hence that of the body was similarly nothing but its velocity (for which reason it could be conceived as a mere movable point). But in Mechanics, since a body is conceived as in motion toward another, respecting which it has a causal relation through its motion—namely that of moving itself, inasmuch as either by its approach by the force of impenetrability or its retreat by the force of attraction, it comes into community with it—then it is no longer indifferent, whether I seek to attribute to this body or to the space, an opposite motion. For now another conception of the quantity of motion comes into play, namely not only that merely conceived in respect of the space and only consisting in the velocity, but that whereby at the same time, the quantity of the substance (as moving cause) must be taken into consideration; and it is here no longer optional, but necessary, to assume both bodies as moved, and [moved] with an equal quantity of motion in an opposite direction; but when the one relative in respect of space is at rest, to attribute to it, together with the space, the requisite motion. For one cannot act on the other by its own motion, unless, through approach by means of repulsive force, or at a distance by means of attraction. As now both forces always act equally and reciprocally in opposite directions, no body can act by means of it, through its motion, on another, except precisely in so far as the other reacts with equal quantity of motion. Thus no body can impart motion through its motion to an absolutely resting [body], but this [latter] must be moved (together with the space) in an opposite direction to that which it is to maintain by the motion and in the direction of the former. The reader will easily perceive, that apart from the unusual [character] which this conception of the communication of motion has in itself, it admits of being placed in the clearest light, if one is not afraid of the diffuseness of the exposition.
[* ]The equality of the action with the, in this case, falsely-called reaction, appears just as much, when under the hypothesis of the transfusion of motions, from one body into the other, the moved body A is allowed to transmit its entire motion in one moment to the resting [body], so that it would rest after the impact, a case that would be inevitable, as soon as both bodies were conceived as absolutely hard (a property which must be distinguished from elasticity). But as this law of motion could not be made to coincide in its application either with experience or with itself, nothing else remained to be done but to deny the existence of absolutely hard bodies, which was equivalent to confessing the contingency of this law, inasmuch as it ought to rest on the special quality by which matters move one another. In our presentation of this law, on the other hand, it is quite the same whether bodies that strike one another are considered absolutely hard or not. But how the transfusionists of motion can explain the motion of elastic bodies by impact in their way is quite incomprehensible to me. For it is clear that resting bodies do not, as merely resting, acquire motion, which the striking body sacrifices, but that in the impact real force is exercised in the opposite direction against the striking [body], in order as it were to compress the springiness between both, which to this end from its side demands as much real motion (although in the opposite direction) as the moving body on its side.
[* ]Of this distinction of disjunctive and alternative opposition, more in the general observation to this division.
[* ]In logic the either or always denotes a disjunctive judgment; for if one be true, the other must be false. For instance, a body is either moved or not-moved, that is, at rest. For it is simply the relation of the cognition to the object which is there spoken of. In phenomenal doctrine, where the relation to the subject is referred to, in order therefrom to determine the relation to the object, it is otherwise. For there the proposition: the body is either moved and the space at rest, or conversely, is not a disjunctive proposition in an objective, but only a subjective connection, and both these judgments therein contained are alternatively valid. In the same phenomenology, where the motion is considered not merely phoronomically, but rather dynamically, on the contrary, the disjunctive proposition is to be taken in an objective signification, that is, in place of the turning of a body I cannot assume its rest and the opposite motion of the space. But even where the motion is regarded mechanically (as when a body rushes against another apparently resting) even then, the, as regards form, disjunctive judgment in respect of the object is to be employed distributively, so that the motion must not be attributed either to the one or to the other, but to each an equal share. This distinction of alternative, disjunctive and distributive determinations of a conception as regards mutually opposed predicates has its importance, but cannot be further discussed here.
[* ]He there says: Motus quidem veros corporum singulorum cognoscere et ab apparentibus actu discriminare difficillimum est; propterea quod partis spatii illius immobilis, in quo corpora vere moventur, non incurrunt in sensus. Causa tamen non est prorsa disparata. Thereupon he allows two spheres attached by a thread, to turn about their common centre of gravity in empty space, and shows how the reality of their motion, together with its direction, can nevertheless be found in experience. I have also sought to demonstrate this under somewhat altered circumstances from the earth as moved on its axis.