Subspace

“ Were I a Circle, you could discern my intestines, but I am a Being composed, as I told you before, of many Circles, the Many in the One, called in this country a Sphere. And, just as the outside of a Cube is a Square, so the outside of a Sphere presents the appearance of a Circle."
Bewildered though I was by my Teacher's enigmatic utterance, I no longer chafed against it, but worshipped him in silent adoration. He continued, with more mildness in his voice: "Distress not yourself if you cannot at first understand the deeper mysteries of Spaceland. By degrees they will dawn upon you. ”

“ Space that was not Space: I was myself, and not myself. When I could find voice, I shrieked aloud in agony, “Either this is madness or it is Hell.” “It is neither,” calmly replied the voice of the Sphere, “it is Knowledge; it is Three Dimensions: open your eye once again and try to look steadily.” ”

“ And we know that a Circle is a higher Being than a Straight Line, in so far as knowledge and wisdom are more to be esteemed than mere affection.
Sphere. It is not for me to classify human faculties according to merit. Yet many of the best and wisest in Spaceland think more of the affections than of the understanding, more of your despised Straight Lines than of your belauded Circles. ”

“ Position is the relation of one extension to another. Space is founded on extension, for if a particle had no extension it could not be an element of space; a plurality of particles, each having extension, constitutes space. ”

“ In addition to the private spaces belonging to the private worlds of different percipients, there is, however, another space, in which one whole private world counts as a point, or at least as a spatial unit. This might be described as the space of points of view, since each private world may be regarded as the appearance which the universe presents from a certain point of view. ”

“ If this is so, and if, as is generally assumed, position in space is purely relative, it follows that the space of one man's objects and the space of another man's objects have no place in common, that they are in fact different spaces, and not merely different parts of one space. I mean by this that such immediate spatial relations as are perceived to hold between the different parts of the sensible space perceived by one man, do not hold between parts of sensible spaces perceived by different men. ”

“ Thus the whole space of your sensible world with all its percepts counts as one tiny region from the point of view of physics.
There is no direct spatial relation between what one person sees and what another sees, because no two ever see exactly the same object. Each person carries about a private space of his own, which can be located in physical space by indirect methods, but which contains no place in common with another person’s private space. This shows how entirely physical space is a matter of inference and construction. ”

“ The proposition, according to which a space occupied by one body cannot be occupied by another, implies the assumption that space is an absolute, self-measuring entity—an assumption which I may have occasion to examine hereafter—and the further assumption that there is a least space which a given body will absolutely fill so as to exclude any other body. ”

“ When I say that there is space in our percepts, I mean nothing at all difficult to understand. I mean—to take the sense of sight, which is the most important in this connection—that in what we see at one time there is up and down, right and left, inside and outside. ”

“ And this is how and why we construct the qualities of our ideal geometrical space. We make it one and identical by correlating our sense-spaces, by fusing the multitude of fields of vision and by refusing to recognise the spaces of our dream experiences, in order that we may have a common standard to which we can refer all our space-perceptions. We make it empty and invariable by abstracting from that which fills it and changes in it, in order that nothing may distract us from the contemplation of its pure form. ”

“ For the sake of clearness and precision, let us designate the primitive concrete data, the result of perception, as extension, and the concept, the result of abstraction, as space—properly so called.
I. At the outset what is given us by intuition is extension under a concrete form. What first becomes known to us is not space but a limited and determined extension—what the child can hold in its hand, reach by a movement of its arms, later on the room which it crosses with uncertain steps ”

“ If, therefore, they mean by body and extension the same that other people do, viz. by body something that is solid and extended, whose parts are separable and movable different ways ; and by extension, only the space that lies between the extremities of those solid coherent parts, and which is possessed by them, – they confound very different ideas one with another ; for I appeal to every man’s own thoughts whether the idea of space be not as distinct from that of solidity, as it is from the idea of scarlet colour ? ”

“ How does it deal with infinite space as something singular which comprehends both picture and spectator as mere centres of gravity of a spatial dynamic? The full Faustian life-feeling, the passion of the third dimension, takes hold of the form of the picture, the painted plane, and transforms it in an unheard-of way. ”

“ With cubic existence (and its correlative three-dimensional space) , with weight and texture we have therefore got from the contemplated shape to a thought alien to that shape and its contemplation. The thought, to which life and its needs and dangers has given precedence over every other: What Thing is behind this shape, what qualities must be inferred from this aspect? After the possibility of occupying so much space, the most important quality which things can have for our hopes and fears, is the possibility of altering their occupation of space; not our locomotion, but theirs. ”

“ If space is something purely subjective, a condition of the sensibility and of the possibility of experience, it follows that the mind instead of receiving any thing from the object, creates whatever is in the object, or rather, whatever we consider as in it. ”

“ So far as the subdivisions of a sense-space are to be measured exactly against each other, objective forms occupying one subdivision must directly or indirectly be superposed upon the other, and the mind must get the immediate feeling of an outstanding plus. ”

“ In the first place it seems to me that space as conceived by this theory—in fact every conception of space as a quantity consisting of points, and for that reason also the current definition of space of pure geometry—degenerates into a logical circle and is therefore worthless as a definition—through failing to realise that nothing can be a relation and a relatum at the same time. ”

“ We discover that we can also touch the things which we see, on the single condition that we perform the necessary movements. Hence we suppose that we sense distance and size immediately. Space as such cannot be perceived any more than color as such. Which of the two spaces which we actually know— visual or touch space— shall we regard as "absolute" space? ”

“ In all practical experience our subdivisions do consume space. They consume it in our geometrical figures. But for simplicity's sake, in geometry we postulate subdivisions which violate experience and consume none of it. ”

“ According to the ordinary view, therefore, space exists and things exist in it: according to our view, only Things exist, and between them nothing exists, but space exists in them. That is to say, to the individual being the other beings with which it stands, either immediately or mediately, in reciprocal causation, appear arranged in one space according to the kind and magnitude of the effect exercised upon this being by them,—a space which is extended merely within the individual as its intuition, and in which it assigns to itself a definite place. ”

“ So also we imagine space, the place of all motion, with its parts immovable, the field of all separation, with its parts inseparable; but we are treating of division, not of separation. If there is true extension, it is divisible; we conceive space with its parts inseparable, but still divisible; for we measure them, count them, and it is by relation to them that we form an idea of the size, distance, and motion of bodies. ”

“ My dear sir, geometry deals with facts of space, space itself is taken as an ultimate datum. The work of geometry is not to ask what space is in itself, but what the relations are of spatial forms, space itself being postulated. ”

“ So far from one all-embracing space being given, there are several spaces for each person, according to the different senses which give relations that may be called spatial. Experience teaches us to obtain one space from these by correlation, and experience, together with instinctive theorising, teaches us to correlate our spaces with those which we believe to exist in the sensible worlds of other people. ”

“ As long as we do not define point independently of space, space defined as an aggregate of points becomes a relation between things (points) which in their turn are only relations between its different parts, i.e., between themselves; pure geometry and many philosophers are clearly conscious of this circle. ”

“ It is the unperceived ambiguity of the word "place" which, as we shall shortly see, has caused the difficulties to realists and given an undeserved advantage to their opponents. Two "places" of different kinds are involved in every sense-datum, namely the place at which it appears and the place from which it appears. These belong to different spaces, although, as we shall see, it is possible, with certain limitations, to establish a correlation between them. What we call the different appearances of the same thing to different observers are each in a space private to the observer concerned. ”

“ And the act by which we thus create space of three dimensions out of visible extent of length and breadth, is constantly and imperceptibly going on. We are perpetually interpreting in this manner the language of the visible world. From the appearances of things which we directly see, we are constantly inferring that which we cannot directly see,—their distance from us, and the position of their parts. ”

“ If the second part of my system be true, that the idea of space or extension is nothing but the idea of visible or tangible points distributed in a certain order; it follows, that we can form no idea of a vacuum, or space, where there is nothing visible or tangible. ”

“ We should then be in this ampler space like beings confined to the surface of a plane would be in ours. Let 28 us suppose that just as in our space there are centres of attraction whose influence radiates out in every direction, so in this ampler space there are centres of attraction whose influence radiates out in every direction. Is there anything to be observed in nature which would correspond to the effect of a centre of attraction lying out of our space, and acting on all the matter in it? ”

“ Lastly, if they find it surprising that a small space has as many parts as a great one, let them understand also that they are smaller in measure, and let them look at the firmament through a diminishing glass, to familiarize themselves with this knowledge, by seeing every part of the sky in every part of the glass. ”

“ I mean that every Point in you—for you are a Square and will serve the purpose of my illustration—every Point in you, that is to say in what you call your inside, is to pass upwards through Space in such a way that no Point shall pass through the position previously occupied by any other Point ”

“ But if a body, which is moved, be considered as long, and be supposed to be so moved, as that all the several parts of it be understood to make several lines, then the way of every part of that body is called breadth, and the space which is made is called superficies, consisting of two dimensions, one whereof to every several part of the other is applied whole. ”

“ Of course, the walls within have their own beauty, of surface and pervading energy, but this does not differ markedly from that of the walls seen from the outside, and what we have established for the one holds for the other. But the beauty of the inclosed space is something entirely new.
In itself, however, mere volume of space is no more aesthetic than mere bounding line or surface; in order to become beautiful, it must become alive. But how can space—the most abstract thing in the world—become alive? ”

“ Again, must this quasi-perpendicularity be regarded as a mere convenient metaphor, or may it be an actual perpendicularity, the spiritual forces pulling in a space of their own, more or less like our known space? In the latter case, are the two spaces coincident [1] ; or does the spiritual include the other as that includes a given plane or line; or is every line and every plane of one space perpendicular to every line and plane of the other? The third alternative seems untenable; for, admitting it, how could a spiritual force produce even a deflective effect in physical space? ”

“ I think it is obvious—granting the spacial quale to exist in the primitive sensations—that discrimination, association, addition, multiplication, and division, blending into generic images, substitution of similars, selective emphasis, and abstraction from uninteresting details, are quite capable of giving us all the space-perceptions [Pg 203] we have so far studied, without the aid of any mysterious 'mental chemistry' or power of 'synthesis' to create elements absent from the original data of feeling. ”