No doubt we make only the first steps in the direction of the extended, even when we let ourselves go as much as we can. But suppose for a moment that matter consists in this very movement pushed further, and that physics is simply psychics inverted. We shall now understand why the mind feels at its ease, moves about naturally in space, when matter suggests the more distinct idea of it. This space it already possessed as an implicit idea in its own eventual _detension_, that is to say, of its own possible _extension_. The mind finds space in things, but could have got it without them if it had had imagination strong enough to push the inversion of its own natural movement to the end. On the other hand, we are able to explain how matter accentuates still more its materiality, when viewed by the mind. Matter, at first, aided mind to run down its own incline; it gave the impulsion. But, the impulsion once received, mind continues its course. The idea that it forms of _pure_ space is only the _schema_ of the limit at which this movement would end. Once in possession of the form of space, mind uses it like a net with meshes that can be made and unmade at will, which, thrown over matter, divides it as the needs of our action demand. Thus, the space of our geometry and the spatiality of things are mutually engendered by the reciprocal action and reaction of two terms which are essentially the same, but which move each in the direction inverse of the other. Neither is space so foreign to our nature as we imagine, nor is matter as completely extended in space as our senses and intellect represent it.
We have treated of the first point elsewhere. As to the second, we will limit ourselves to pointing out that perfect spatiality would consist in a perfect externality of parts in their relation to one another, that is to say, in a complete reciprocal independence. Now, there is no material point that does not act on every other material point. When we observe that a thing really is there where it _acts_, we shall be led to say (as Faraday[79] was) that all the atoms interpenetrate and that each of them fills the world. On such a hypothesis, the atom or, more generally, the material point, becomes simply a view of the mind, a view which we come to take when we continue far enough the work (wholly relative to our faculty of acting) by which we subdivide matter into bodies. Yet it is undeniable that matter lends itself to this subdivision, and that, in supposing it breakable into parts external to one another, we are constructing a science sufficiently representative of the real. It is undeniable that if there be no entirely isolated system, yet science finds means of cutting up the universe into systems relatively independent of each other, and commits no appreciable error in doing so. What else can this mean but that matter _extends_ itself in space without being absolutely _extended_ therein, and that in regarding matter as decomposable into isolated systems, in attributing to it quite distinct elements which change in relation to each other without changing in themselves (which are "displaced," shall we say, without being "altered"), in short, in conferring on matter the properties of pure space, we are transporting ourselves to the terminal point of the movement of which matter simply indicates the direction?
What the _Transcendental Aesthetic_ of Kant appears to have established once for all is that extension is not a material attribute of the same kind as others. We cannot reason indefinitely on the notions of heat, color, or weight: in order to know the modalities of weight or of heat, we must have recourse to experience. Not so of the notion of space. Supposing even that it is given empirically by sight and touch (and Kant has not questioned the fact) there is this about it that is remarkable that our mind, speculating on it with its own powers alone, cuts out in it, _a priori_, figures whose properties we determine _a priori_: experience, with which we have not kept in touch, yet follows us through the infinite complications of our reasonings and invariably justifies them. That is the fact. Kant has set it in clear light. But the explanation of the fact, we believe, must be sought in a different direction to that which Kant followed.
Intelligence, as Kant represents it to us, is bathed in an atmosphere of spatiality to which it is as inseparably united as the living body to the air it breathes. Our perceptions reach us only after having passed through this atmosphere. They have been impregnated in advance by our geometry, so that our faculty of thinking only finds again in matter the mathematical properties which our faculty of perceiving has already deposed there. We are assured, therefore, of seeing matter yield itself with docility to our reasonings; but this matter, in all that it has that is intelligible, is our own work; of the reality "in itself" we know nothing and never shall know anything, since we only get its refraction through the forms of our faculty of perceiving. So that if we claim to affirm something of it, at once there rises the contrary affirmation, equally demonstrable, equally plausible. The ideality of space is proved directly by the analysis of knowledge indirectly by the antinomies to which the opposite theory leads. Such is the governing idea of the Kantian criticism. It has inspired Kant with a peremptory refutation of "empiricist" theories of knowledge. It is, in our opinion, definitive in what it denies. But, in what it affirms, does it give us the solution of the problem?
With Kant, space is given as a ready-made form of our perceptive faculty--a veritable _deus ex machina_, of which we see neither how it arises, nor why it is what it is rather than anything else. "Things-in-themselves" are also given, of which he claims that we can know nothing: by what right, then, can he affirm their existence, even as "problematic"? If the unknowable reality projects into our perceptive faculty a "sensuous manifold" capable of fitting into it exactly, is it not, by that very fact, in part known? And when we examine this exact fitting, shall we not be led, in one point at least, to suppose a pre-established harmony between things and our mind--an idle hypothesis, which Kant was right in wishing to avoid? At bottom, it is for not having distinguished degrees in spatiality that he has had to take space ready-made as given--whence the question how the "sensuous manifold" is adapted to it. It is for the same reason that he has supposed matter wholly developed into parts absolutely external to one another;--whence antinomies, of which we may plainly see that the thesis and antithesis suppose the perfect coincidence of matter with geometrical space, but which vanish the moment we cease to extend to matter what is true only of pure space. Whence, finally, the conclusion that there are three alternatives, and three only, among which to choose a theory of knowledge: either the mind is determined by things, or things are determined by the mind, or between mind and things we must suppose a mysterious agreement.
But the truth is that there is a fourth, which does not seem to have occurred to Kant--in the first place because he did not think that the mind overflowed the intellect, and in the second place (and this is at bottom the same thing) because he did not attribute to duration an absolute existence, having put time, _a priori_, on the same plane as space. This alternative consists, first of all, in regarding the intellect as a special function of the mind, essentially turned toward inert matter; then in saying that neither does matter determine the form of the intellect, nor does the intellect impose its form on matter, nor have matter and intellect been regulated in regard to one another by we know not what pre-established harmony, but that intellect and matter have progressively adapted themselves one to the other in order to attain at last a common form. _This adaptation has, moreover, been brought about quite naturally, because it is the same inversion of the same movement which creates at once the intellectuality of mind and the materiality of things._ From this point of view the knowledge of matter that our perception on one hand and science on the other give to us appears, no doubt, as approximative, but not as relative. Our perception, whose rôle it is to hold up a light to our actions, works a dividing up of matter that is always too sharply defined, always subordinated to practical needs, consequently always requiring revision. Our science, which aspires to the mathematical form, over-accentuates the spatiality of matter; its formulae are, in general, too precise, and ever need remaking. For a scientific theory to be final, the mind would have to embrace the totality of things in block and place each thing in its exact relation to every other thing; but in reality we are obliged to consider problems one by one, in terms which are, for that very reason, provisional, so that the solution of each problem will have to be corrected indefinitely by the solution that will be given to the problems that will follow: thus, science as a whole is relative to the particular order in which the problems happen to have been put. It is in this meaning, and to this degree, that science must be regarded as conventional. But it is a conventionality of fact so to speak, and not of right. In principle, positive science bears on reality itself, provided it does not overstep the limits of its own domain, which is inert matter.
Scientific knowledge, thus regarded, rises to a higher plane. In return, the theory of knowledge becomes an infinitely difficult enterprise, and which passes the powers of the intellect alone. It is not enough to determine, by careful analysis, the categories of thought; we must engender them. As regards space, we must, by an effort of mind _sui generis_, follow the progression or rather the regression of the extra-spatial degrading itself into spatiality. When we make ourselves self-conscious in the highest possible degree and then let ourselves fall back little by little, we get the feeling of extension: we have an extension of the self into recollections that are fixed and external to one another, in place of the tension it possessed as an indivisible active will. But this is only a beginning. Our consciousness, sketching the movement, shows us its direction and reveals to us the possibility of continuing it to the end; but consciousness itself does not go so far. Now, on the other hand, if we consider matter, which seems to us at first coincident with space, we find that the more our attention is fixed on it, the more the parts which we said were laid side by side enter into each other, each of them undergoing the action of the whole, which is consequently somehow present in it. Thus, although matter stretches itself out in the direction of space, it does not completely attain it; whence we may conclude that it only carries very much further the movement that consciousness is able to sketch within us in its nascent state. We hold, therefore, the two ends of the chain, though we do not succeed in seizing the intermediate links. Will they always escape us? We must remember that philosophy, as we define it, has not yet become completely conscious of itself. Physics understands its rôle when it pushes matter in the direction of spatiality; but has metaphysics understood its rôle when it has simply trodden in the steps of physics, in the chimerical hope of going further in the same direction? Should not its own task be, on the contrary, to remount the incline that physics descends, to bring back matter to its origins, and to build up progressively a cosmology which would be, so to speak, a reversed psychology? All that which seems _positive_ to the physicist and to the geometrician would become, from this new point of view, an interruption or inversion of the true positivity, which would have to be defined in psychological terms.
* * * * * When we consider the admirable order of mathematics, the perfect agreement of the objects it deals with, the immanent logic in numbers and figures, our certainty of always getting the same conclusion, however diverse and complex our reasonings on the same subject, we hesitate to see in properties apparently so positive a system of negations, the absence rather than the presence of a true reality. But we must not forget that our intellect, which finds this order and wonders at it, is directed in the same line of movement that leads to the materiality and spatiality of its object. The more complexity the intellect puts into its object by analyzing it, the more complex is the order it finds there. And this order and this complexity necessarily appear to the intellect as a positive reality, since reality and intellectuality are turned in the same direction.
When a poet reads me his verses, I can interest myself enough in him to enter into his thought, put myself into his feelings, live over again the simple state he has broken into phrases and words. I sympathize then with his inspiration, I follow it with a continuous movement which is, like the inspiration itself, an undivided act. Now, I need only relax my attention, let go the tension that there is in me, for the sounds, hitherto swallowed up in the sense, to appear to me distinctly, one by one, in their materiality. For this I have not to do anything; it is enough to withdraw something. In proportion as I let myself go, the successive sounds will become the more individualized; as the phrases were broken into words, so the words will scan in syllables which I shall perceive one after another. Let me go farther still in the direction of dream: the letters themselves will become loose and will be seen to dance along, hand in hand, on some fantastic sheet of paper. I shall then admire the precision of the interweavings, the marvelous order of the procession, the exact insertion of the letters into the syllables, of the syllables into the words and of the words into the sentences. The farther I pursue this quite negative direction of relaxation, the more extension and complexity I shall create; and the more the complexity in its turn increases, the more admirable will seem to be the order which continues to reign, undisturbed, among the elements. Yet this complexity and extension represent nothing positive; they express a deficiency of will. And, on the other hand, the order must grow with the complexity, since it is only an aspect of it. The more we perceive, symbolically, parts in an indivisible whole, the more the number of the relations that the parts have between themselves necessarily increases, since the same undividedness of the real whole continues to hover over the growing multiplicity of the symbolic elements into which the scattering of the attention has decomposed it. A comparison of this kind will enable us to understand, in some measure, how the same suppression of positive reality, the same inversion of a certain original movement, can create at once extension in space and the admirable order which mathematics finds there. There is, of course, this difference between the two cases, that words and letters have been invented by a positive effort of humanity, while space arises automatically, as the remainder of a subtraction arises once the two numbers are posited.[80] But, in the one case as in the other, the infinite complexity of the parts and their perfect coördination among themselves are created at one and the same time by an inversion which is, at bottom, an interruption, that is to say, a diminution of positive reality.
* * * * * All the operations of our intellect tend to geometry, as to the goal where they find their perfect fulfilment. But, as geometry is necessarily prior to them (since these operations have not as their end to construct space and cannot do otherwise than take it as given) it is evident that it is a latent geometry, immanent in our idea of space, which is the main spring of our intellect and the cause of its working. We shall be convinced of this if we consider the two essential functions of intellect, the faculty of deduction and that of induction.
Let us begin with deduction. The same movement by which I trace a figure in space engenders its properties: they are visible and tangible in the movement itself; I feel, I see in space the relation of the definition to its consequences, of the premisses to the conclusion. All the other concepts of which experience suggests the idea to me are only in part constructible _a priori_; the definition of them is therefore imperfect, and the deductions into which these concepts enter, however closely the conclusion is linked to the premisses, participate in this imperfection. But when I trace roughly in the sand the base of a triangle, as I begin to form the two angles at the base, I know positively, and understand absolutely, that if these two angles are equal the sides will be equal also, the figure being then able to be turned over on itself without there being any change whatever. I know it before I have learnt geometry. Thus, prior to the science of geometry, there is a natural geometry whose clearness and evidence surpass the clearness and evidence of other deductions. Now, these other deductions bear on qualities, and not on magnitudes purely. They are, then, likely to have been formed on the model of the first, and to borrow their force from the fact that, behind quality, we see magnitude vaguely showing through. We may notice, as a fact, that questions of situation and of magnitude are the first that present themselves to our activity, those which intelligence externalized in action resolves even before reflective intelligence has appeared. The savage understands better than the civilized man how to judge distances, to determine a direction, to retrace by memory the often complicated plan of the road he has traveled, and so to return in a straight line to his starting-point.[81] If the animal does not deduce explicitly, if he does not form explicit concepts, neither does he form the idea of a homogeneous space. You cannot present this space to yourself without introducing, in the same act, a virtual geometry which will, of itself, degrade itself into logic. All the repugnance that philosophers manifest towards this manner of regarding things comes from this, that the logical work of the intellect represents to their eyes a positive spiritual effort. But, if we understand by spirituality a progress to ever new creations, to conclusions incommensurable with the premisses and indeterminable by relation to them, we must say of an idea that moves among relations of necessary determination, through premisses which contain their conclusion in advance, that it follows the inverse direction, that of materiality. What appears, from the point of view of the intellect, as an effort, is in itself a letting go. And while, from the point of view of the intellect, there is a _petitio principii_ in making geometry arise automatically from space, and logic from geometry--on the contrary, if space is the ultimate goal of the mind's movement of _detension_, space cannot be given without positing also logic and geometry, which are along the course of the movement of which pure spatial intuition is the goal.
It has not been enough noticed how feeble is the reach of deduction in the psychological and moral sciences. From a proposition verified by facts, verifiable consequences can here be drawn only up to a certain point, only in a certain measure. Very soon appeal has to be made to common sense, that is to say, to the continuous experience of the real, in order to inflect the consequences deduced and bend them along the sinuosities of life. Deduction succeeds in things moral only metaphorically, so to speak, and just in the measure in which the moral is transposable into the physical, I should say translatable into spatial symbols. The metaphor never goes very far, any more than a curve can long be confused with its tangent. Must we not be struck by this feebleness of deduction as something very strange and even paradoxical? Here is a pure operation of the mind, accomplished solely by the power of the mind. It seems that, if anywhere it should feel at home and evolve at ease, it would be among the things of the mind, in the domain of the mind. Not at all; it is there that it is immediately at the end of its tether. On the contrary, in geometry, in astronomy, in physics, where we have to do with things external to us, deduction is all-powerful! Observation and experience are undoubtedly necessary in these sciences to arrive at the principle, that is, to discover the aspect under which things must be regarded; but, strictly speaking, we might, by good luck, have hit upon it at once; and, as soon as we possess this principle, we may draw from it, at any length, consequences which experience will always verify. Must we not conclude, therefore, that deduction is an operation governed by the properties of matter, molded on the mobile articulations of matter, implicitly given, in fact, with the space that underlies matter? As long as it turns upon space or spatialized time, it has only to let itself go. It is _duration_ that puts spokes in its wheels.
* * * * * Deduction, then, does not work unless there be spatial intuition behind it. But we may say the same of induction. It is not necessary indeed to think geometrically, nor even to think at all, in order to expect from the same conditions a repetition of the same fact. The consciousness of the animal already does this work, and indeed, independently of all consciousness, the living body itself is so constructed that it can extract from the successive situations in which it finds itself the similarities which interest it, and so respond to the stimuli by appropriate reactions. But it is a far cry from a mechanical expectation and reaction of the body, to induction properly so called, which is an intellectual operation. Induction rests on the belief that there are causes and effects, and that the same effects follow the same causes. Now, if we examine this double belief, this is what we find. It implies, in the first place, that reality is decomposable into groups, which can be practically regarded as isolated and independent. If I boil water in a kettle on a stove, the operation and the objects that support it are, in reality, bound up with a multitude of other objects and a multitude of other operations; in the end, I should find that our entire solar system is concerned in what is being done at this particular point of space. But, in a certain measure, and for the special end I am pursuing, I may admit that things happen as if the group _water-kettle-stove_ were an independent microcosm. That is my first affirmation. Now, when I say that this microcosm will always behave in the same way, that the heat will necessarily, at the end of a certain time, cause the boiling of the water, I admit that it is sufficient that a certain number of elements of the system be given in order that the system should be complete; it completes itself automatically, I am not free to complete it in thought as I please. The stove, the kettle and the water being given, with a certain interval of duration, it seems to me that the boiling, which experience showed me yesterday to be the only thing wanting to complete the system, will complete it to-morrow, no matter when to-morrow may be. What is there at the base of this belief? Notice that the belief is more or less assured, according as the case may be, but that it is forced upon the mind as an absolute necessity when the microcosm considered contains only magnitudes. If two numbers be given, I am not free to choose their difference. If two sides of a triangle and the contained angle are given, the third side arises of itself and the triangle completes itself automatically. I can, it matters not where and it matters not when, trace the same two sides containing the same angle: it is evident that the new triangles so formed can be superposed on the first, and that consequently the same third side will come to complete the system. Now, if my certitude is perfect in the case in which I reason on pure space determinations, must I not suppose that, in the other cases, the certitude is greater the nearer it approaches this extreme case? Indeed, may it not be the limiting case which is seen through all the others and which colors them, accordingly as they are more or less transparent, with a more or less pronounced tinge of geometrical necessity?[82] In fact, when I say that the water on the fire will boil to-day as it did yesterday, and that this is an absolute necessity, I feel vaguely that my imagination is placing the stove of yesterday on that of to-day, kettle on kettle, water on water, duration on duration, and it seems then that the rest must coincide also, for the same reason that, when two triangles are superposed and two of their sides coincide, their third sides coincide also. But my imagination acts thus only because it shuts its eyes to two essential points. For the system of to-day actually to be superimposed on that of yesterday, the latter must have waited for the former, time must have halted, and everything become simultaneous: that happens in geometry, but in geometry alone. Induction therefore implies first that, in the world of the physicist as in that of the geometrician, time does not count. But it implies also that qualities can be superposed on each other like magnitudes. If, in imagination, I place the stove and fire of to-day on that of yesterday, I find indeed that the form has remained the same; it suffices, for that, that the surfaces and edges coincide; but what is the coincidence of two qualities, and how can they be superposed one on another in order to ensure that they are identical? Yet I extend to the second order of reality all that applies to the first. The physicist legitimates this operation later on by reducing, as far as possible, differences of quality to differences of magnitude; but, prior to all science, I incline to liken qualities to quantities, as if I perceived behind the qualities, as through a transparency, a geometrical mechanism.[83] The more complete this transparency, the more it seems to me that in the same conditions there must be a repetition of the same fact. Our inductions are certain, to our eyes, in the exact degree in which we make the qualitative differences melt into the homogeneity of the space which subtends them, so that geometry is the ideal limit of our inductions as well as of our deductions. The movement at the end of which is spatiality lays down along its course the faculty of induction as well as that of deduction, in fact, intellectuality entire.
* * * * * It creates them in the mind. But it creates also, in things, the "order" which our induction, aided by deduction, finds there. This order, on which our action leans and in which our intellect recognizes itself, seems to us marvelous. Not only do the same general causes always produce the same general effects, but beneath the visible causes and effects our science discovers an infinity of infinitesimal changes which work more and more exactly into one another, the further we push the analysis: so much so that, at the end of this analysis, matter becomes, it seems to us, geometry itself. Certainly, the intellect is right in admiring here the growing order in the growing complexity; both the one and the other must have a positive reality for it, since it looks upon itself as positive. But things change their aspect when we consider the whole of reality as an undivided advance forward to successive creations. It seems to us, then, that the complexity of the material elements and the mathematical order that binds them together must arise automatically when within the whole a partial interruption or inversion is produced. Moreover, as the intellect itself is cut out of mind by a process of the same kind, it is attuned to this order and complexity, and admires them because it recognizes itself in them. But what is admirable _in itself_, what really deserves to provoke wonder, is the ever-renewed creation which reality, whole and undivided, accomplishes