“Riemann has shewn that as there are different kinds of lines and surfaces, so there are different kinds of space of three dimensions; and that we can only find out by experience to which of these kinds the space in which we live belongs. In particular, the axioms of plane geometry are true within the limits of experiment on the surface of a sheet of paper, and yet we know that the sheet is really covered with a number of small ridges and furrows, upon which (the total curvature not being zero) these axioms are not true. Similarly, he says although the axioms of solid geometry are true within the limits of experiment for finite portions of our space, yet we have no reason to conclude that they are true for very small portions; and if any help can be got thereby for the explanation of physical phenomena, we may have reason to conclude that they are not true for very small portions of space.”

—  William Kingdon Clifford , On the Space-Theory of Matter

Abstract
On the Space-Theory of Matter (read Feb 21, 1870)

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William Kingdon Clifford 48
English mathematician and philosopher 1845–1879

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“To help us to understand three-dimensional spaces, two-dimensional analogies may be very useful… A two-dimensional space of zero curvature is a plane, say a sheet of paper. The two-dimensional space of positive curvature is a convex surface, such as the shell of an egg. It is bent away from the plane towards the same side in all directions. The curvature of the egg, however, is not constant: it is strongest at the small end. The surface of constant positive curvature is the sphere… The two-dimensional space of negative curvature is a surface that is convex in some directions and concave in others, such as the surface of a saddle or the middle part of an hour glass. Of these two-dimensional surfaces we can form a mental picture because we can view them from outside… But… a being… unable to leave the surface… could only decide of which kind his surface was by studying the properties of geometrical figures drawn on it. …On the sheet of paper the sum of the three angles of a triangle is equal to two right angles, on the egg, or the sphere, it is larger, on the saddle it is smaller. …The spaces of zero and negative curvature are infinite, that of positive curvature is finite. …the inhabitant of the two-dimensional surface could determine its curvature if he were able to study very large triangles or very long straight lines. If the curvature were so minute that the sum of the angles of the largest triangle that he could measure would… differ… by an amount too small to be appreciable… then he would be unable to determine the curvature, unless he had some means of communicating with somebody living in the third dimension…. our case with reference to three-dimensional space is exactly similar. …we must study very large triangles and rays of light coming from very great distances. Thus the decision must necessarily depend on astronomical observations.”

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William Kingdon Clifford photo

“We may… be treating merely as physical variations effects which are really due to changes in the curvature of our space; whether, in fact, some or all of those causes which we term physical may not be due to the geometrical construction of our space. There are three kinds of variation in the curvature of our space which we ought to consider as within the range of possibility.”

William Kingdon Clifford (1845–1879) English mathematician and philosopher

Clifford & Pearson, Ch IV, Position, §19 On the Bending of Space
The Common Sense of the Exact Sciences (1885)
Context: We may... be treating merely as physical variations effects which are really due to changes in the curvature of our space; whether, in fact, some or all of those causes which we term physical may not be due to the geometrical construction of our space. There are three kinds of variation in the curvature of our space which we ought to consider as within the range of possibility.
(i) Our space is perhaps really possessed of a curvature varying from point to point, which we fail to appreciate because we are acquainted with only a small portion of space, or because we disguise its small variations under changes in our physical condition which we do not connect with our change of position. The mind that could recognise this varying curvature might be assumed to know the absolute position of a point. For such a mind the postulate of the relativity of position would cease to have a meaning. It does not seem so hard to conceive such a state of mind as the late Professor Clerk-Maxwell would have had us believe. It would be one capable of distinguishing those so-called physical changes which are really geometrical or due to a change of position in space.
(ii) Our space may be really same (of equal curvature), but its degree of curvature may change as a whole with the time. In this way our geometry based on the sameness of space would still hold good for all parts of space, but the change of curvature might produce in space a succession of apparent physical changes.
(iii) We may conceive our space to have everywhere a nearly uniform curvature, but that slight variations of the curvature may occur from point to point, and themselves vary with the time. These variations of the curvature with the time may produce effects which we not unnaturally attribute to physical causes independent of the geometry of our space. We might even go so far as to assign to this variation of the curvature of space 'what really happens in that phenomenon which we term the motion of matter.' <!--pp. 224-225

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The sphere as a whole has a character different from that of a plane. A spherical surface made from rubber, such as a balloon, can be twisted so that its geometry changes. ...but it cannot be distorted in such a way as that it will cover a plane. All surfaces obtained by distortion of the rubber sphere possess the same holistic properties; they are closed and finite. The plane as a whole has the property of being open; its straight lines are not closed. This feature is mathematically expressed as follows. Every surface can be mapped upon another one by the coordination of each point of one surface to a point of the other surface, as illustrated by the projection of a shadow picture by light rays. For surfaces with the same holistic properties it is possible to carry through this transformation uniquely and continuously in all points. Uniquely means: one and only one point of one surface corresponds to a given point of the other surface, and vice versa. Continuously means: neighborhood relations in infinitesimal domains are preserved; no tearing of the surface or shifting of relative positions of points occur at any place. For surfaces with different holistic properties, such a transformation can be carried through locally, but there is no single transformation for the whole surface.
The Philosophy of Space and Time (1928, tr. 1957)

Hans Reichenbach photo

“The surfaces of three-dimensional space are distinguished from each other not only by their curvature but also by certain more general properties. A spherical surface, for instance, differs from a plane not only by its roundness but also by its finiteness. Finiteness is a holistic property. The sphere as a whole has a character different from that of a plane. A spherical surface made from rubber, such as a balloon, can be twisted so that its geometry changes…. but it cannot be distorted in such a way as that it will cover a plane. All surfaces obtained by distortion of the rubber sphere possess the same holistic properties; they are closed and finite. The plane as a whole has the property of being open; its straight lines are not closed. This feature is mathematically expressed as follows. Every surface can be mapped upon another one by the coordination of each point of one surface to a point of the other surface, as illustrated by the projection of a shadow picture by light rays. For surfaces with the same holistic properties it is possible to carry through this transformation uniquely and continuously in all points. Uniquely means: one and only one point of one surface corresponds to a given point of the other surface, and vice versa. Continuously means: neighborhood relations in infinitesimal domains are preserved; no tearing of the surface or shifting of relative positions of points occur at any place. For surfaces with different holistic properties, such a transformation can be carried through locally, but there is no single transformation for the whole surface.”

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“I hold in fact
(1) That small portions of space are in fact of a nature analogous to little hills on a surface which is on the average flat; namely, that the ordinary laws of geometry are not valid in them.
(2) That this property of being curved or distorted is continually being passed on from one portion of space to another after the manner of a wave.
(3) That this variation of the curvature of space is what really happens in that phenomenon which we call the motion of matter, whether ponderable or etherial.
(4) That in the physical world nothing else takes place but this variation, subject possibly to the law of continuity.”

William Kingdon Clifford On the Space-Theory of Matter

Abstract
On the Space-Theory of Matter (read Feb 21, 1870)

“And we think that this ultimate blessedness differs only in degree from the happy and joyful experiences of our lives. Whereas the truth is that it differs in kind. The joys, not only of the earth, but of any conceivable heaven - which we can conceive only as some happy and fortunate prolongation of our lives in time - are not of the same order as that ultimate blessedness.”

Walter Terence Stace (1886–1967) British civil servant, educator and philosopher.

p. 7.

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