CHAPTER V. THE NATURE AND VALIDITY OF THE PARALLEL POSTULATE. We have reached the conclusion that geometrical spaces, merely as such, are all of them abstract conceptions. They are grounded on and grow out of the same general experience which they interpret differently while seeking to simplify and to systematize it by means of the peculiar postulates which define them. We can also see that the quantitative concepts which underlie the different geometries have been chosen somewhat arbitrarily so that when we carry them back to the facts of spatial experience they do not reproduce these facts in any case with absolute precision. Different groups of ideas may therefore serve to express all the facts with equal exactitude within the region accessible to observation. The space-world as we know it is not quantitatively infinite nor does the assumption that it is seem to be a necessary one, although Euclid requires it. If then we refuse to call it infinite in this sense but still accept the present laws of optics and astron omy which presuppose Euclid's validity and if we also admit the postulate of free mobility which is the same as to assume that the parameter of space (K) is a constant quantity, we find that even under these restrictions it is still possible to represent all the facts with equal accuracy by the geometries of Euclid, Lobatchewsky, and Riemann. So far then as experience goes at present, or can ever go for that matter, there is no necessary reason for starting any physical inquiry with the Euclidean assumption that K is infinite. All we need is to take K sufficiently large to make the deviation from Euclid fall within the limits of astronomical observation. As this observation becomes more refined and exact, one of two things must inevitably happen. Either the facts will ultimately appear to go against Euclid or else it will be shown that the actually known quantity of space, though always finite, will transcend successively certain increasingly large amounts which the new approximations to the value of K will give us the right to affirm. But to go so far as to assert even that K is constant seems, on the surface at least, to be an arbitrary matter which is not demanded either by experience or by logic. It means the same thing as to assert an absolutely rigid standard which may be transported unchanged to any part of space. 1 See Chapter II. under the discussion of Riemann's idea of curvature. But we certainly do not meet with any such rigid measures among the objects of experience from which as we have seen the idea of a metrical standard has actually been derived. What we really know is that some of these objects are less subject to quantitative variations than others are, and that consequently a series of them may be arranged whose members as we ascend the scale approximate more and more nearly a quantitative invariability. Hence here as in the case of the straight line the intellect may form the pure abstract conception of a rigid body which, when thought of as independent of position, furnishes also the conception of spatial homogeneity. But between the actual facts of experience and the essential properties of such an intellectual construct there exists, as stated, a gulf which sense-perception of itself cannot bridge. It is then both logically and empirically permissible to assume that the actual parameter of space is a variable quantity which oscillates within certain narrow limits. By assuming this oscillation to occur in accordance with law other geometries compatible with experience could also be obtained. It is such considerations as these that force upon 2 When we assume the possibility of measurement in the exact sense required by geometry, and the facts of experience make this of course a legitimate inference, this assumed variability of K must be ruled out on purely logical grounds, as we shall later attempt to show. us the necessity of making and maintaining a careful distinction between the facts of experience and those intellectual constructs whose formation these facts have suggested. Upon these constructs two conditions are imposed: they must fit the facts to which they relate, and must also meet the logical requirements of mutual non-contradiction. When these conditions are fulfilled, that is, when it is shown that different systems of geometry, Euclidean and non-Euclidean, are equally permissible under both requirements, the question of validity assumes its most interesting and difficult form. We have shown that these geometries within certain limitations already pointed out, are empirically indistinguishable; it therefore remains to consider the other half of the problem. Are these geometries when considered strictly from the logical point of view equally tenable? Fortunately, as our historical chapter has abundantly shown, we already have all that seems to be desired for a satisfactory answer. Granting the groups of assumptions from which they set out, accepting the condition merely that these assumptions be so defined as to be mutually independent and logically consistent, and finally, disregarding the question as to their easy compatibility with the known facts of reality; almost an unlimited number of geometries can be, and very many indeed have been, actually built up |