DOC. 148 S P A C E - T I M E 269 SPACE-TIME 6 0 9 for such an assertion. This blind faith in evidence and in the immediately real meaning of the concepts and propositions of geometry became uncertain only after non-Euclidean geometry had been introduced. Reference to the Earth.— If we start from the view that all spatial concepts are related to contact-experiences of solid bod- ies, it is easy to understand how the concept " space” originated, namely, how a thing independent of bodies and yet embodying their position-possibilities (Lagerungsmöglichkeiten) was posited. If we have a system of bodies in contact and at rest relatively to one another, some can be replaced by others. This property of allowing substitution is interpreted as “available space.” Space denotes the property in virtue of which rigid bodies can occupy different positions. The view that space is something with a unity of its own is perhaps due to the circumstance that in pre- scientific thought all positions of bodies were referred to one body (reference body), namely the earth. In scientific thought the earth is represented by the co-ordinate system. The assertion that it would be possible to place an unlimited number of bodies next to one another denotes that space is infinite. In pre- scientific thought the concepts “ space” and “time” and “ body of reference” are scarcely differentiated at all. A place or point in space is always taken to mean a material point on a body of reference. Euclidean Geometry.— If we consider Euclidean geometry we clearly discern that it refers to the laws regulating the positions of rigid bodies. It turns to account the ingenious thought of tracing back all relations concerning bodies and their relative positions to the very simple concept “distance” (Strecke). Distance denotes a rigid body on which two material points (marks) have been specified. The concept of the equality of distances (and angles) refers to experiments involving coin- cidences the same remarks apply to the theorems on congruence. Now, Euclidean geometry, in the form in which it has been handed down to us from Euclid, uses the fundamental concepts " straight line” and “ plane” which do not appear to corre- spond, or at any rate, not so directly, with experiences concern- ing the position of rigid bodies. On this it must be remarked that the concept of the straight line may be reduced to that of the distance.1 Moreover, geometricians were less concerned with bringing out the relation of their fundamental concepts to experience than with deducing logically the geometrical proposi- tions from a few axioms enunciated at the outset. Let us outline briefly how perhaps the basis of Euclidean geometry may be gained from the concept of distance. We start from the equality of distances (axiom of the equality of distances). Suppose that of two unequal distances one is always greater than the other. The same axioms are to hold for the inequality of distances as hold for the inequality of numbers. Three distances ABl, BC1, CA1 may, if CA1 be suitably chosen, have their marks BB1, CC1, AA1 superposed on one another in such a way that a triangle ABC results. The distance CA1 has an upper limit for which this construction is still just possible. The points A, (BB´) and C then lie in a “ straight line” (definition). This leads to the concepts: producing a distance by an amount equal to itself dividing a distance into equal parts expressing a distance in terms of a number by means of a measuring-rod (definition of the space-interval between two points). When the concept of the interval between two points or the length of a distance has been gained in this way we require only the following axiom (Pythagoras’ theorem) in order to arrive at Euclidean geometry analytically. To every point of space (body of reference) three numbers (co-ordinates) x, y, z may be assigned— and conversely— in such a way that for each pair of points A (x1, y1, z1,) and B (x2, y2, z2) the theorem holds: ______________________ measure-number AB = V (x*—xi)* + (y*—yO* + (z*—Zi)* 1 A hint of this is contained in the theorem: “the straight line is the shortest connection between two points." This theorem served well as a definition of the straight line, although the definition played no part in the logical texture of the deductions. All further concepts and propositions of Euclidean geometry can then be built up purely logically on this basis, in particular also the propositions about the straight line and the plane. These remarks are not, of course, intended to replace the strictly axiomatic construction of Euclidean geometry. We merely wish to indicate plausibly how all conceptions of geometry may be traced back to that of distance. We might equally well have epitomised the whole basis of Euclidean geometry in the last theorem above. The relation to the foundations of experi- ence would then be furnished by means of a supplementary theorem. The co-ordinate may and must be chosen so that two pairs of points separated by equal intervals, as calculated by the help of Pythagoras’ theorem, may be made to coincide with one and the same suitably chosen distance (on a solid). The concepts and propositions of Euclidean geometry may be derived from Pythagoras’ proposition without the introduction of rigid bodies but these concepts and propositions would not then have contents that could be tested. They are not “true” propositions but only logically correct propositions of purely formal content. Difficulties.— A serious difficulty is encountered in the above represented interpretation of geometry in that the rigid body of experience does not correspond exactly with the geometrical body. In stating this I am thinking less of the fact that there are no absolutely definite marks than that temperature, pressure and other circumstances modify the laws relating to position. It is also to be recollected that the structural constituents of matter (such as atom and electron, q.v.) assumed by physics are not in principle commensurate with rigid bodies, but that never- theless the concepts of geometry are applied to them and to their parts. For this reason consistent thinkers have been disinclined to allow real contents of facts (reale Tatsachenbestände) to corre- spond to geometry alone. They considered it preferable to allow the content of experience (Erfahrungsbestdnde) to correspond to geometry and physics conjointly. This view is certainly less open to attack than the one repre- sented above as opposed to the atomic theory it is the only one that can be consistently carried through. Nevertheless, in the opinion of the author it would not be advisable to give up the first view, from which geometry derives its origin. This con- nection is essentially founded on the belief that the ideal rigid body is an abstraction that is well rooted in the laws of nature. Foundations of Geometry.— We come now to the question: what is a priori certain or necessary, respectively in geometry (doctrine of space) or its foundations? Formerly we thought everything— yes, everything nowadays we think— nothing. Already the distance-concept is logically arbitrary there need be no things that correspond to it, even approximately. Some- thing similar may be said of the concepts straight line, plane, of three-dimensionality and of the validity of Pythagoras’ theorem. Nay, even the continuum-doctrine is in no wise given with the nature of human thought, so that from the epistomological point of view no greater authority attaches to the purely topological relations than to the others. Earlier Physical Concepts— We have yet to deal with those modifications in the space-concept, which have accompanied the advent of the theory of relativity. For this purpose we must consider the space-concept of the earlier physics from a point of view different from that above. If we apply the theorem of Pythagoras to infinitely near points, it reads ds*= dx*+dy, +dz* where ds denotes the measurable interval between them. For an empirically-given ds the co-ordinate system is not yet fully determined for every combination of points by this equation. Besides being translated, a co-ordinate system may also be rotated.2 This signifies analytically: the relations of Euclidean geometry are covariant with respect to linear orthogonal trans- formations of the co-ordinates. 2 Change of direction of the co-ordinate axes while their orthog- onality is preserved. [5] [6] [7] [8] [9] [10] [11]