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 deter- mine, 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 our- selves fall back little by little, we get the feeling of ex- tension: we have an extension of the self into recollections that arc 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 conscious- ness 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 b> side enter into each other, each of them undergoing tht 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 withio us in its nascent state. We hold, therefore, the two ends of the chain, though we do not succeed in seizing the inter- mediate links. Will they always escape us? We must remember that philosophy, as we define it, has not yet be- come completely conscious of itself. Physics understands its rdle when it pushes matter in the direction of spatiality; but has metaphysics understood its rfile 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, 80 to speak, a reversed psychology? All that which seems ■posUive 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 l<?ads to the materiality and spatial- ity of its object. The more comjjlexity the intellect puts into its object by analyzing it, the more complex is the order it finds there. And this order and this complexity necea- satily appear to the intellect as a positive reality, since
in.]
The Geometrical Order
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, hve 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 eoimds 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 them- selves 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 mar- velous 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 ex- tension 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 hovei over
210 Creative Evolution
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 sub- traction arises once the two numbers are posited.' But, in the one case as in tlie other, the infinite complexity of the parts and their perfect coordination among them- selves are created ut 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 fend to geometry, as to the goal where tliey find their perfect fulfilment.
' Our KimpariBon does no more than develop the content of the term XiffOB, as Plotinua understaoda it. For while the )i6j-oa of this phi- losopher IB a generating and informing power, an aspect or a fragment of the (puj^ij, on the other hand Flotiniis sometimes spealia of it as of a ditcouTSe. More generQlIj, the relation that we establish in the present chapter between "extension" and "detension" resembles in some aspects that which PlotinUB supposes (some developments of which must have inspired M. Ravaiason) when he makes extension not indeed an inversion of original Being, but an enfceblement of its essence, one of the last stages of the procesmon, (see in particular, Enn. IV. iii. 9-11, and III. vi. 17-lS). Yet ancient philosophy did not see wnai con- sequences would result from this for mathematics, for Plotinu8, ake Plato, erected mathematical essences into al)solulo reahties. Above sU, it BufTored itself to be deceived by the purely superficial analogy of duration wilh extension. It treated the one as it treated the other, regaiding change as a degradation of immutability, the sensible aa a fall from the inlelliEiblo. Whence, as we shall show in the next chapter, K philosophy which fails to reeognize the real (unction and scope of Ibe
The Geometrical Order
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 Ls 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 de- duction 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 constnictible 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 tri- angle, 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, hkely to have been formed on the model of the first, and to borrow their force from the fact that, behind quahty, 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 in-
[OOAF.
telligeDce externalized in action resolves even before reflective intelligence has appeared. The savage under- stands 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 eo to return in a straight line to his starting-point.' If the animal does not deduce explicitly, if he does not form explicit concepts, neither does he form the idea of a homo- geneous space. You cannot present this space to your- eelf without introducing, in the same act, a virtual geometry which will, of itself, degrade itself into logic. All the re- pugnance 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 spirituahty 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 efl'ort, is in itself a letting go. And while, from the point of view of the intellect, there is a petitio pnncipii tn 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 delension, 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 consequence* can here be drawn only up to a certain point, only in a <B&ftL»D, Th* Brain a* an. Organ qf the Mind, pp. 214-16.
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