The Foundations of Science: Science and Hypothesis, The Value of Science, Science and Method. Henri Poincare

      What, first of all, are the properties of space, properly so called? I mean of that space which is the object of geometry and which I shall call geometric space.

      The following are some of the most essential:

      1º It is continuous;

      2º It is infinite;

      3º It has three dimensions;

      4º It is homogeneous, that is to say, all its points are identical one with another;

      5º It is isotropic, that is to say, all the straights which pass through the same point are identical one with another.

      Compare it now to the frame of our representations and our sensations, which I may call perceptual space.

      Visual Space.—Consider first a purely visual impression, due to an image formed on the bottom of the retina.

      A cursory analysis shows us this image as continuous, but as possessing only two dimensions; this already distinguishes from geometric space what we may call pure visual space.

      Besides, this image is enclosed in a limited frame.

      Finally, there is another difference not less important: this pure visual space is not homogeneous. All the points of the retina, aside from the images which may there be formed, do not play the same rôle. The yellow spot can in no way be regarded as identical with a point on the border of the retina. In fact, not only does the same object produce there much more vivid impressions, but in every limited frame the point occupying the center of the frame will never appear as equivalent to a point near one of the borders.

      No doubt a more profound analysis would show us that this continuity of visual space and its two dimensions are only an illusion; it would separate it therefore still more from geometric space, but we shall not dwell on this remark.

      Sight, however, enables us to judge of distances and consequently to perceive a third dimension. But every one knows that this perception of the third dimension reduces itself to the sensation of the effort at accommodation it is necessary to make, and to that of the convergence which must be given to the two eyes, to perceive an object distinctly.

      These are muscular sensations altogether different from the visual sensations which have given us the notion of the first two dimensions. The third dimension therefore will not appear to us as playing the same rôle as the other two. What may be called complete visual space is therefore not an isotropic space.

      It has, it is true, precisely three dimensions, which means that the elements of our visual sensations (those at least which combine to form the notion of extension) will be completely defined when three of them are known; to use the language of mathematics, they will be functions of three independent variables.

      But examine the matter a little more closely. The third dimension is revealed to us in two different ways: by the effort of accommodation and by the convergence of the eyes.

      No doubt these two indications are always concordant, there is a constant relation between them, or, in mathematical terms, the two variables which measure these two muscular sensations do not appear to us as independent; or again, to avoid an appeal to mathematical notions already rather refined, we may go back to the language of the preceding chapter and enunciate the same fact as follows: If two sensations of convergence, A and B, are indistinguishable, the two sensations of accommodation, A´ and B´, which respectively accompany them, will be equally indistinguishable.

      But here we have, so to speak, an experimental fact; a priori nothing prevents our supposing the contrary, and if the contrary takes place, if these two muscular sensations vary independently of one another, we shall have to take account of one more independent variable, and 'complete visual space' will appear to us as a physical continuum of four dimensions.

      We have here even, I will add, a fact of external experience. Nothing prevents our supposing that a being with a mind like ours, having the same sense organs that we have, may be placed in a world where light would only reach him after having traversed reflecting media of complicated form. The two indications which serve us in judging distances would cease to be connected by a constant relation. A being who should achieve in such a world the education of his senses would no doubt attribute four dimensions to complete visual space.

      Tactile Space and Motor Space.—'Tactile space' is still more complicated than visual space and farther removed from geometric space. It is superfluous to repeat for touch the discussion I have given for sight.

      But apart from the data of sight and touch, there are other sensations which contribute as much and more than they to the genesis of the notion of space. These are known to every one; they accompany all our movements, and are usually called muscular sensations.

      The corresponding frame constitutes what may be called motor space.

      Each muscle gives rise to a special sensation capable of augmenting or of diminishing, so that the totality of our muscular sensations will depend upon as many variables as we have muscles. From this point of view, motor space would have as many dimensions as we have muscles.

      I know it will be said that if the muscular sensations contribute to form the notion of space, it is because we have the sense of the direction of each movement and that it makes an integrant part of the sensation. If this were so, if a muscular sensation could not arise except accompanied by this geometric sense of direction, geometric space would indeed be a form imposed upon our sensibility.

      But I perceive nothing at all of this when I analyze my sensations.

      What I do see is that the sensations which correspond to movements in the same direction are connected in my mind by a mere association of ideas. It is to this association that what we call 'the sense of direction' is reducible. This feeling therefore can not be found in a single sensation.

      This association is extremely complex, for the contraction of the same muscle may correspond, according to the position of the limbs, to movements of very different direction.

      Besides, it is evidently acquired; it is, like all associations of ideas, the result of a habit; this habit itself results from very numerous experiences; without any doubt, if the education of our senses had been accomplished in a different environment, where we should have been subjected to different impressions, contrary habits would have arisen and our muscular sensations would have been associated according to other laws.

      Characteristics of Perceptual Space.—Thus perceptual space, under its triple form, visual, tactile and motor, is essentially different from geometric space.

      It is neither homogeneous, nor isotropic; one can not even say that it has three dimensions.

      It is often said that we 'project' into geometric space the objects of our external perception; that we 'localize' them.

      Has this a meaning, and if so what?

      Does it mean that we represent to ourselves external objects in geometric space?

      Our representations are only the reproduction of our sensations; they can therefore be ranged only in the same frame as these, that is to say, in perceptual space.

      It is as impossible for us to represent to ourselves external bodies in geometric space, as it is for a painter to paint on a plane canvas objects with their three dimensions.

      Perceptual space is only an image of geometric space, an image altered in shape by a sort of perspective, and we can represent to ourselves objects only by bringing them under the laws of this perspective.

      Therefore we do not represent to ourselves external bodies in geometric space, but we reason on these bodies as if they were situated in geometric space.

      When it is said then that we 'localize' such and such an object at such and such a point of space, what does it mean?

      It simply means that we represent to ourselves the movements