Читать книгу An essay on the foundations of geometry - Bertrand Russell - Страница 8

form a singly infinite series, since they contain one parameter, namely λ′: λ″. Such a series of geodesics, therefore, must form a two-dimensional manifold, with a measure of curvature in the ordinary Gaussian sense. This measure of curvature can be determined from the above formula for the elementary arc, by the help of Gauss's general formula alluded to above. We thus obtain an infinite number of measures of curvature at a point, but from n.(n–1) 2 of these, the rest can be deduced (Riemann, Gesammelte Werke, p. 262). When all the measures of curvature at a point are constant, and equal to all the measures of curvature at any other point, we get what Riemann calls a manifold of constant curvature. In such a manifold free mobility is possible, and positions do not differ intrinsically from one another. If a be the measure of curvature, the formula for the arc becomes, in this case,

ds² = Σdx² _/ (1 + a 4 Σx²)².

In this case only, as I pointed out above, can the term "measure of curvature" be properly applied to space without reference to a higher dimension, since free mobility is logically indispensable to the existence of quantitative or metrical Geometry.

23. The mathematical result of Riemann's dissertation may be summed up as follows. Assuming it possible to apply magnitude to space, i.e. to determine its elements and figures by means of algebraical quantities, it follows that space can be brought under the conception of a manifold, as a system of quantitatively determinable elements. Owing, however, to the peculiar nature of spatial measurement, the quantitative determination of space demands that magnitudes shall be independent of place—in so far as this is not the case, our measurement will be necessarily inaccurate. If we now assume, as the quantitative relation of distance between two elements, the square root of a quadratic function of the coordinates—a formula subsequently proved by Helmholtz and Lie—then it follows, since magnitudes are to be independent of place, that space must, within the limits of observation, have a constant measure of curvature, or must, in other words, be homogeneous in all its parts. In the infinitesimal, Riemann says (p. 267), observation could not detect a departure from constancy on the part of the measure of curvature; but he makes no attempt to show how Geometry could remain possible under such circumstances, and the only Geometry he has constructed is based entirely on Free Mobility. I shall endeavour to prove, in Chapter III., that any metrical Geometry, which should endeavour to dispense with this axiom, would be logically impossible. At present I will only point out that Riemann, in spite of his desire to prove that all the axioms can be dispensed with, has nevertheless, in his mathematical work, retained three fundamental axioms, namely, Free Mobility, the finite integral number of dimensions, and the axiom that two points have a unique relation, namely distance. These, as we shall see hereafter, are retained, in actual mathematical work, by all metrical Metageometers, even when they believe, like Riemann and Helmholtz, that no axioms are philosophically indispensable.

24. Helmholtz, the historically nearest follower of Riemann, was guided by a similar empirical philosophy, and arrived independently at a very similar method of formulating the axioms. Although Helmholtz published nothing on the subject until after Riemann's death, he had then only just seen Riemann's dissertation (which was published posthumously), and had worked out his results, so far as they were then completed, in entire independence both of Riemann and of Lobatchewsky. Helmholtz is by far the most widely read of all writers on Metageometry, and his writings, almost alone, represent to philosophers the modern mathematical standpoint on this subject. But his importance is much greater, in this domain, as a philosopher than as a mathematician; almost his only original mathematical result, as regards Geometry, is his proof of Riemann's formula for the infinitesimal arc, and even this proof was far from rigid, until Lie reformed it by his method of continuous groups. In this chapter, therefore, only two of his writings need occupy us, namely the two articles in the Wissenschaftliche Abhandlungen, Vol. II., entitled respectively "Ueber die thatsächlichen Grundlagen der Geometrie," 1866 (p. 610 ff.), and "Ueber die Thatsachen, die der Geometrie zum Grunde liegen," 1868 (p. 618 ff.).

25. In the first of these, which is chiefly philosophical, Helmholtz gives hints of his then uncompleted mathematical work, but in the main contents himself with a statement of results. He announces that he will prove Riemann's quadratic formula for the infinitesimal arc; but for this purpose, he says, we have to start with Congruence, since without it spatial measurement is impossible. Nevertheless, he maintains that Congruence is proved by experience. How we could, without the help of measurement, discover lapses from Congruence, is a point which he leaves undiscussed. He then enunciates the four axioms which he considers essential to Geometry, as follows:

(1) As regards continuity and dimensions. In a space of n dimensions, a point is uniquely determined by the measurement of n continuous variables (coordinates).

(2) As regards the existence of moveable rigid bodies. Between the 2n coordinates of any point-pair of a rigid body, there exists an equation which is the same for all congruent point-pairs. By considering a sufficient number of point-pairs, we get more equations than unknown quantities: this gives us a method of determining the form of these equations, so as to make it possible for them all to be satisfied.

(3) As regards free mobility. Every point can pass freely and continuously from one position to another. From (2) and (3) it follows, that if two systems A and B can be brought into congruence in any one position, this is also possible in every other position.

(4) As regards independence of rotation in rigid bodies (Monodromy). If (n–1) points of a body remain fixed, so that every other point can only describe a certain curve, then that curve is closed.

These axioms, says Helmholtz, suffice to give, with the axiom of three dimensions, the Euclidean and non-Euclidean systems as the only alternatives. That they suffice, mathematically, cannot be denied, but they seem, in some respects, to go too far. In the first place, there is no necessity to make the axiom of Congruence apply to actual rigid bodies—on this subject I have enlarged in Chapter II.[29] Again, Free Mobility, as distinct from Congruence, hardly needs to be specially formulated: what barrier could empty space offer to a point's progress? The axiom is involved in the homogeneity of space, which is the same thing as the axiom of Congruence. Monodromy, also, has been severely criticized; not only is it evident that it might have been included in Congruence, but even from the purely analytical point of view, Sophus Lie has proved it to be superfluous[30]. Thus the axiom of Congruence, rightly formulated, includes Helmholtz's third and fourth axioms and part of his second axiom. All the four, or rather, as much of them as is relevant to Geometry, are consequences, as we shall see hereafter, of the one fundamental principle of the relativity of position.

26. The second article, which is mainly mathematical, supplies the promised proof of the arc-formula, which is Helmholtz's most important contribution to Geometry. Riemann had assumed this formula, as the simplest of a number of alternatives: Helmholtz proved it to be a necessary consequence of his axioms. The present paper begins with a short repetition of the first, including the statement of the axioms, to which, at the end of the paper, two more are added, (5) that space has three dimensions, and (6) that space is infinite. It is supposed in the text, as also in the first paper, that the measure of curvature cannot be negative, and, consequently, that an infinite space must be Euclidean. This error in both papers is corrected in notes, added after the appearance of Beltrami's paper on negative curvature. It is a sample of the slightly unprofessional nature of Helmholtz's mathematical work on this subject, which elicits from Klein the following remarks[31]: "Helmholtz is not a mathematician by profession, but a physicist and physiologist. … From this non-mathematical quality of Helmholtz, it follows naturally that he does not treat the mathematical portion of his work with the thoroughness which one would demand of a mathematician by trade (von Fach)." He tells us himself that it was the physiological study of vision which led him to the question of the axioms, and it is as a physicist that he makes his axioms refer to actual rigid bodies. Accordingly, we find errors in his mathematics, such as the axiom of Monodromy, and the assumption that the measure of curvature must be positive. Nevertheless, the proof of Riemann's arc-formula is extremely able, and has, on the whole, been substantiated by Lie's more thorough investigations.

27. Helmholtz's other writings on Geometry are almost wholly philosophical, and will be discussed at length in Chapter II. For the present, we may pass to the only other important writer of the second period, Beltrami. As his work is purely mathematical, and contains few controverted points, it need not, despite its great importance, detain us long.

The "Saggio di Interpretazione della Geometria non-Euclidea[32]," which is principally confined to two dimensions, interprets Lobatchewsky's results by the characteristic method of the second period. It shows, by a development of the work of Gauss and Minding[33], that all the propositions in plane Geometry, which Lobatchewsky had set forth, hold, within ordinary Euclidean space, on surfaces of constant negative curvature. It is strange, as Klein points out[34], that this interpretation, which was known to Riemann and perhaps even to Gauss, should have remained so long without explicit statement. This is the more strange, as Lobatchewsky's "Géométrie Imaginaire" had appeared in Crelle, Vol. XVII.[35], and Minding's article, from which the interpretation follows at once, had appeared in Crelle, Vol. XIX. Minding had shewn that the Geometry of surfaces of constant negative curvature, in particular as regards geodesic triangles, could be deduced from that of the sphere by giving the radius a purely imaginary value ia[36]. This result, as we have seen, had also been obtained by Lobatchewsky for his Geometry, and yet it took thirty years for the connection to be brought to general notice.

28. In Beltrami's Saggio, straight lines are, of course, replaced by geodesics; his coordinates are obtained through a point-by-point correspondence with an auxiliary plane, in which straight lines correspond to geodesics on the surface. Thus geodesics have linear equations, and are always uniquely determined by two points. Distances on the surface, however, are not equal to distances on the plane; thus while the surface is infinite, the corresponding portion of the plane is contained within a certain finite circle. The distance of two points on the surface is a certain function of the coordinates, not the ordinary function of elementary Geometry. These relations of plane and surface are important in connection with Cayley's theory of distance, which we shall have to consider next. If we were to define distance on the plane as that function of the coordinates which gives the corresponding distance on the surface, we should obtain what Klein calls "a plane with a hyperbolic system of measurement (Massbestimmung)" in which Cayley's theory of distance would hold. It is evident, however, that the ordinary notion of distance has been presupposed in setting up the coordinate system, so that we do not really get alternative Geometries on one and the same plane. The bearing of these remarks will appear more fully when we come to consider Cayley and Klein.

29. The value of Beltrami's Saggio, in his own eyes, lies in the intelligible Euclidean sense which it gives to Lobatchewsky's planimetry: the corresponding system of Solid Geometry, since it has no meaning for Euclidean space, is barely mentioned in this work. In a second paper[37], however, almost contemporaneous with the first, he proceeds to consider the general theory of n-dimensional manifolds of constant negative curvature. This paper is greatly influenced by Riemann's dissertation; it begins with the formula for the linear element, and proves from this first, that Congruence holds for such spaces, and next, that they have, according to Riemann's definition, a constant negative measure of curvature. (It is instructive to observe, that both in this and in the former Essay, great stress is laid on the necessity of the Axiom of Congruence.)

This work has less philosophical interest than the former, since it does little more than repeat, in a general form, the results which the Saggio had obtained for two dimensions—results which sink, when extended to n dimensions, to the level of mere mathematical constructions. Nevertheless, the paper is important, both as a restoration of negative curvature, which had been overlooked by Helmholtz, and as an analytical treatment of Lobatchewsky's results—a treatment which, together with the Saggio, at last restored to them the prominence they deserved.