But it is likely to be objected that the power of beauty is incapable of directly manifesting its action and changing the characteristics of reality; that this force must pass through vital and conscious refractions before it affects the experience that is taken into account from the outside; and finally, that we do not know the complex processes and ways in which the action of this force will go. — We will not go into these arguments to the depths, but for now we will only answer the following. We know nothing at all of the processes and ways in which the action of any force, mechanical or physical, is manifested, and we always observe only a ready-made, unknown action; Rather, those actions that pass through our consciousness or our self-perception should be recognized as less incomprehensible. Further, we know nothing of the immediacy of the action of mechanical and physical forces, and any attempt to give a model of this kind of action leads precisely to numerous means. Finally, if we do not speak, and do not consider it necessary, to speak of the psychic and biological moment, i.e., of the internal reaction, in regard to mechanical and physical forces, this is not at all from the knowledge that there are none, but from the inability to clarify this aspect of the matter for oneself. For this reason we omit everything that is between the beginning and the end, and are content to establish the fact of the connection between this beginning and this end. If, on the other hand, we know something of the middle in relation to living beings and ourselves, then this mean must in no way be an obstacle, and in this case phenomenologically connect the beginning with the end, leaving a certain vague knowledge of the mean to those who have a special interest in it.

Summing up, then, let us note again: everything that is capable of acting by producing changes in the characteristics of reality, i.e., imparting a certain acceleration to their uniform and inexorable passage in time, all this can rightly be called a stand. The forces of reality are many and varied, and the activity of each of them manifests itself only with the corresponding susceptibility of the objects of influence, and if there is none, it remains unknown. There is no dividing line between the different forces, on one side of which there is the objective, and on the other the subjective: everything objective has its inner side, just as everything subjective is revealed. There is nothing secret that has not been revealed, just as, on the contrary, everything that is manifest has a secret in it.

The space of Euclidean geometry has as its physical basis a group of mechanical motions of an absolutely rigid body; At the same time, the absence of force fields and the presence of a divine embrace of the entire space by consciousness, apart from the physical means and conditions of knowledge, are tacitly assumed. The group of movement is partly joined by the still poorly thought out and inaccurately taken space of visual perceptions. Usually little thought is given to the much wider use of space, although the auditory and tactile sensations clearly require space. But further: smell, taste, then various mystical experiences, thoughts and even feelings have spatial characteristics and mutual coordination, which makes us assert their location in space as well. Extrapositionality, i.e., the presence of one or another of the separates outside of each other, is the basic characteristic of spatiality. Since there is a multiplicity, then its elements are separate, or, as Riemann put it, are images of isolation; thus, they are outside of each other. This is already a sufficient sign of their presence in the corresponding space, because, although we do not confuse, we do not consider them separately, but together, as something coherent, and coordinate them with each other. The possibility of thinking and imagining them as a multitude, not coherent, necessarily leads to the assertion that there is also a condition for the possibility of this coherence and coordination of the set. And in so far as we recognize the image as the objective content of thought and space, we are also recognized as such, i.e., objectively prescient to thought, the condition of the possibility of diversity. This condition is space. We do not confuse it with the images themselves: the reasoning here is the same as that of Georg Cantor, when he proves the existence of an actually infinite line on the basis of the fact of the existence of an infinite line potentially. A segment can increase indefinitely, exceeding any given magnitude: this is therefore the very possibility of infinite growth, existing actually, entirely, ready. This possibility, i.e., this line, can no longer be finite, it transcends any finite magnitude and, consequently, is a line actually infinitely large. In the same way, the presence of corresponding images of isolation presupposes the condition of the possibility of correlation, and this is precisely the space of images of a given perception. There should be many of these spaces, according to the kind of perceptions and images of isolation; there is no data in advance to consider these spaces identical, although it is natural to expect the kinship of some of them, and moreover, different in different cases. Some spaces are very far from each other, others can be very close; a common feature of all is the extraposition of the images of isolation contained in them and united by them.

Each image of isolation is a certain power center. As indicated by Riemann in the above quotation, this space is connected and defined by these centers; Being a condition for the coordination and communication of these power centers, it has a property corresponding to the nature of the activity of these centers. In other words, the curvature of this space at each point is determined by the forces acting in it, of course, in relation to the action of these forces, these and not any others that are manifested in any other kind of receptivity. If, however, we turn to another receptivity, even if it belongs to the same object of perception, then, according to the evaluation of the actions of other forces on this other receptivity, the curvature of space will be different at a given point, and the whole space will have a different structure. This is self-evident: space is the principle which unites the centres of force, i.e., which enables the field of force to unfold. This means that it must contain forces, or have capacity. There is no reason to expect the same capacity when the forces correlated with a given space change.

Hitherto we have spoken of force centers, considering the curvature of space to be something secondary, posited by them; but, as has already been explained, we can consider the structure of space as the starting point and the starting point, and see in the centers of force something secondary, the foci of curvature, which here becomes very great, or infinitely large, in the positive or negative sense. With equal right, one can resort to one of the two methods of description and say either "the force field curves and thereby organizes space" or "space by its organization, i.e., curvature, determines a certain set of force centers." Thus, living on the surface, we would bypass a certain center of repulsion that stands in our way and curve our path: this would be the straightest one, if we were guided by the feeling of fatigue, and we could then speak of curvature in this area of our plane, with a certain focus of curvature. But it is also possible to imagine the curvature of the plane, in the usual sense of the word, i.e., a certain special point on our surface; meeting it on the way, we would also begin to bypass this sharpening and also consider our crooked path as the straightest. Then we would have the right to deny the local curvature of the surface, but we would have to talk about the force center of repulsion. It is up to us to choose one or the other method of description, but one of the two must be chosen so that reality itself is not distorted. There is, however, another way of describing it by means of the variable characteristics of the medium, for example, in this case by means of an incompressible fluid spreading from a point. But this third method is very close to the introduction of force centers and rather serves as a model for the latter.

XXII

All culture can be interpreted as the activity of organizing space. In one case, it is the space of our life relations, and then the corresponding activity is called technique. In other cases, this space is a conceivable space, a mental model of reality, and the reality of its organization is called science and philosophy. Finally, the third class of cases lies between the first two. Its space or spaces are visual, as the spaces of technology, and do not admit of vital interference, as the spaces of science and philosophy. The organization of such spaces is called art.

Of course, it will not be possible to distinguish these three kinds of activity unconditionally, as well as the spaces they organize: each of the activities also contains the beginnings of other activities subordinate to it, and each of the spaces is to a certain extent not alien to spaces of the other kind. Thus, in technology there is certainly a certain artistry, which is not necessary in order to achieve the goal set by technology, as well as a certain philosophical and scientific thought that enriches the theoretical attitude to the world. In philosophy and science it is always possible to discover a certain artistry and vital applicability, i.e., the technical side. In the same way, a work of art contains, to one degree or another, both vital utility, something technical, and one or another technical relation to reality. In every activity there is everything, and every space has an affinity with the others. And it could not be otherwise, since culture is one and serves one subject, and spaces, no matter how diverse they are, are still called by one word – space.

Nevertheless, these types of activity can be distinguished according to their prevailing meaning. And despite this distinction, fundamentally they do the same thing: they change reality in order to rearrange space. The force field deployed by them can be interpreted as a producer of the curvature of space. But it is possible, and logically more expedient, to say that the required space is evoked by the force field, manifested, in the photographic sense of the word. The gesture forms space, causing tension in it and thus distorting it. This is one approach to the change in reality that has been produced. But another approach is possible and more appropriate, when the tensions from the gesture indicate a special curvature of space in a given place. She was already there, preceding the gesture with his force field. But this invisible and inaccessible to sensory experience curvature of space became noticeable to us when it manifested itself as a force field, which, in turn, considered a gesture. If a piece of cardboard is superimposed on the poles of the magnet, then the surface of the cardboard seems to the eye to be no different from the same cardboard that does not lie on the magnet. This space therefore appears to us to be homogeneous, and in small areas Euclidean. But this does not mean that it is really so, and will always be perceived as such, but only that we are insensitive to the forces at work here, or to the curvature inherent in space.

By sprinkling cardboard with iron particles, we manifest a force field or spatial curvature for our perception. At the same time, we can interpret the picture of the force field as produced by a magnet and producing, in turn, the curvature of space, or, on the contrary, we can say that the preceding curvature of space at this place (meaning the place in the sense of a place, an event, i.e., as determined by the coordinate of time) determines the force field, which in turn posits a magnet with its poles. In the same way, in the realities of culture, the change in reality produced can be interpreted both as a cause of the organization of space and as a consequence of the already existing organization. In this case, the images of the isolation of reality are places of special curvatures of space, its irregularities, nodes, folds, etc., and force fields are areas of constant approach to these highest or lowest values of curvature. Further, then, those visual images which art posits, or those devices which the technician builds, or, finally, those mental models which the scientist or philosopher puts in word, are all only signs of these folds and distortions in general, together with the areas of approach to these places. A cultural worker places boundary posts, draws boundaries and, finally, draws the shortest paths in this space, together with systems of lines of equal effort, isopotentials. This matter is necessary for the organization of space to reach our consciousness. But this activity reveals what exists, and is not posited by human arbitrariness:

In vain, artist, you think that you are the creator of your own creations.

They were forever hovering over the earth, invisible to the eye...

There are many invisible forms and inaudible sounds in space,

There are many wonderful combinations of word and light in it,