An essay on the foundations of geometry

Delbœuf.

98. M. Delbœuf's four articles in the Revue Philosophique contain much matter that has already been dealt with in the criticism of Lotze, and much that is irrelevant for our present purpose. The only point, which I wish to discuss here, is the question of absolute magnitude, as it is called—the question, that is, whether the possibility of similar but unequal geometrical figures can be known à priori[109].

In discussing this question, it is important, to begin with, to distinguish clearly the sense in which absolute magnitude is required in non-Euclidean Geometry, from another sense, in which it would be absurd to regard any magnitude as absolute. Judgments of magnitude can only result from comparison, and if Metageometry required magnitudes which could be determined without comparison, it would certainly deserve condemnation. But this is not required. All we require is, that it shall be impossible, while the rest of space is unaffected, to alter the magnitude of any figure, as compared with other figures, while leaving the relative internal magnitudes of its parts unchanged. This construction, which is possible in Euclid, is impossible in Metageometry. We have to discuss whether such an impossibility renders non-Euclidean spaces logically faulty.

M. Delbœuf's position on this axiom—which he calls the postulate of homogeneity[110]—is, that all Geometry must presuppose it, and that Metageometry, consequently, though logically sound, is logically subsequent to Euclid, and can only make its constructions within a Euclidean "homogeneous" space (Rev. Phil. Vol. XXXVII., pp. 380–1). He would appear to think, nevertheless, that homogeneity (in his sense) is learnt from experience, though on this point he is not very explicit. (See Vol. XXXVIII., p. 129.) No à priori proof, at any rate, is offered in his articles. As a result of experience, every one would admit, similarity is known to be possible within the limits of observation; but the fact that this possibility extends to Ordnance maps, which deal with a spherical surface, should make us chary of inferring, from such a datum, the certainty of Euclid for large spaces. Moreover if homogeneity be empirical, Metageometry, which dispenses with it, is not necessarily in logical dependence upon Euclid, since homogeneity and isogeneity are logically separable. I shall assume, therefore, as the only contention which can be interesting to our argument, that homogeneity is regarded as à priori, and as logically essential to Geometry.

99. Now we saw, in discussing Erdmann's views of the judgment of quantity, that in non-Euclidean space, as in Euclidean, a change of all spatial magnitudes, in the same ratio, would be no change at all; the ratios of all magnitudes to the space-constant would be unchanged, and the space-constant, as the ultimate standard of comparison, cannot, in any intelligible sense, be said to have any particular magnitude. The absolute magnitudes of Metageometry, therefore, are absolute only as against any other particular magnitude, not as against other magnitudes in general. If this were not the case, the comparative nature of the judgment of magnitude would be contradicted, and metrical Metageometry would become absurd. But as it is, the difference from Euclid consists only in this: that in Metageometry we have, while in Euclid we have not, a standard of comparison involved in the nature of our space as a whole, which we call the space-constant. We have to discuss whether the assertion of such a standard involves an undue reification of space.

I do not believe that this is the case. For an undue reification of space would only arise, if we were no longer able to regard position as wholly relative, and as geometrically definable only by departure from other positions. But the relativity of position, as we have abundantly seen, is preserved by all spaces of constant curvature—in all of these, positions can only be defined, geometrically, by relations to fresh positions[111]. This series of definitions may lead to an infinite regress, but it may also, as in spherical space, form a vicious circle, and return again to the position from which it started. No reification of space, no independent existence of mere relations, seems involved in such a procedure. The whole of Metageometry, in short, is a proof that the relativity of position is compatible with absolute magnitude, in the only sense required by non-Euclidean spaces. We must conclude, therefore, that there is nothing incompatible, in a denial of homogeneity (in Delbœuf's sense), either with the relational nature of space, or with the comparative nature of magnitude. This last à priori objection to Metageometry, therefore, cannot be maintained, and the issue must be decided on empirical grounds alone.

100. The foundations of Geometry have been the subject of much recent speculation in France, and this seems to demand some notice. But in spite of the splendid work which the French have done on the allied question of number and continuous quantity, I cannot persuade myself that they have succeeded in greatly advancing the subject of geometrical philosophy. The chief writers have been, from the mathematical side, Calinon and Poincaré, from the philosophical, Renouvier and Delbœuf; as a mediator between mathematics and philosophy, Lechalas.

Calinon, in an interesting article on the geometrical indeterminateness of the universe, maintains that any Geometry may be applied to the actual world by a suitable hypothesis as to the course of light-rays. For the earth only is known to us otherwise than by Optics, and the earth is an infinitesimal part of the universe. This line of argument has been already discussed in connection with Lotze, but Calinon adds a new suggestion, that the space-constant may perhaps vary with the time. This would involve a causal connection between space and other things, which seems hardly conceivable, and which, if regarded as possible, must surely destroy Geometry, since Geometry depends throughout on the irrelevance of Causation[112]. Moreover, in all operations of measurement, some time is spent; unless we knew that space was unchanging throughout the operation, it is hard to see how our results could be trustworthy, and how, consequently, a change in the parameter could be discovered. The same difficulties would arise, in fact, as those which result from supposing space not homogeneous.

Poincaré maintains that the question, whether Euclid or Metageometry should be accepted, is one of convenience and convention, not of truth; axioms are definitions in disguise, and the choice between definitions is arbitrary. This view has been discussed in Chapter I., in connection with Cayley's theory of distance, on which it depends.

Lechalas is a philosophical disciple of Calinon. He is a rationalist of the pre-Kantian type, but a believer in the validity of Metageometry. He holds that Geometry can dispense with all purely spatial postulates, and work with axioms of magnitude alone[113], which, in his opinion, are purely analytic. The principle of contradiction, to him, is the sole and only test of truth; we make long chains of reasoning from our premisses to see if contradictions will emerge. It might be objected that this view, though it saves general Geometry from being logically empirical, leaves it only empirically logical; this must, in fact, be the fate of every piece of à priori knowledge, if M. Lechalas's were the only test of truth. However, he concludes that general Geometry is apodeictic, while the space of our actual world, like all other phenomena, is contingent.

Delbœuf criticizes non-Euclidean space from an ultra-realist standpoint: he holds that real space is neither homogeneous nor isogeneous, but that conceived space, as abstracted from real space, has both these properties. He offers no justification for his real space, which seems to be maintained in the spirit of naïve realism, nor does he show how he has acquired his intimate knowledge of its constitution[114]. His arguments against Metageometry, in so far as they are not repetitions of Lotze, have been discussed above.

Renouvier, finally, is a pure Kantian, of the most orthodox type. His views as to the importance, for Geometry, of the distinction between synthetic and analytic judgments, have been discussed, in connection with Kant, at the beginning of the present Chapter[115].

101. Before beginning the constructive argument of the next Chapter, let us endeavour briefly to sum up the theories which have been polemically advocated throughout the criticisms we have just concluded. We agreed to accept, with Kant, necessity for any possible experience as the test of the à priori, but we refused, for the present, to discuss the connection of the à priori with the subjective, regarding the purely logical test as sufficient for our immediate purpose. We also refused to attach importance to the distinction of analytic and synthetic, since it seemed to apply, not to different judgments, but only to different aspects of any judgment.

We then discussed Riemann's attempt to identify the empirical element in Geometry with the element not deducible from ideas of magnitude, and we decided that this identification was due to a confusion as to the nature of magnitude. For judgments of magnitude, we said, require always some qualitative basis, which is not quantitatively expressible.

In criticizing Helmholtz, we decided that Mechanics logically presupposes Geometry, though space presupposes matter; but that the matter which space presupposes, and to which Geometry indirectly refers, is a more abstract matter than that of Mechanics, a matter destitute of force and of causal attributes, and possessed only of the purely spatial attributes required for the possibility of spatial figures. But we conceded that Geometry, when applied to mixed mathematics or to daily life, demands more than this, demands, in fact, some means of discovering, in the more concrete matter of Mechanics, either a rigid body, or a body whose departure from rigidity follows some empirically discoverable law. Actual measurement, therefore, we agreed to regard as empirical.

Our conclusions, as regards the empiricism of Riemann and Helmholtz, were reinforced by a criticism of Erdmann. We then had an opposite task to perform, in defending Metageometry against Lotze. Here we saw that there are two senses in which Metageometry is possible. The first concerns our actual space, and asserts that it may have a very small space-constant; the second concerns philosophical theories of space, and asserts a purely logical possibility, which leaves the decision to experience. We saw also that Lotze's mathematical strictures arose from insufficient knowledge of the subject, and could all be refuted by a better acquaintance with Metageometry.

Finally, we discussed the question of absolute magnitude, and found in it no logical obstacle to non-Euclidean spaces. Our conclusion, then, in so far as we are as yet entitled to a conclusion, is that all spaces with a space-constant are à priori justifiable, and that the decision between them must be the work of experience. Spaces without a space-constant, on the other hand, spaces, that is, which are not homogeneous throughout, we found logically unsound and impossible to know, and therefore to be condemned à priori. The constructive proof of this thesis will form the argument of the following chapter.

CHAPTER III.

Section A.

THE AXIOMS OF PROJECTIVE GEOMETRY.

102. Projective Geometry proper, as we saw in Chapter I., does not employ the conception of magnitude, and does not, therefore, require those axioms which, in the systems of the second or metrical period, were required solely to render possible the application of magnitude to space. But we saw, also, that Cayley's reduction of metrical to projective properties was purely technical and philosophically irrelevant. Now it is in metrical properties alone—apart from the exception to the axiom of the straight line, which itself, however, presupposes metrical properties[116]—that non-Euclidean and Euclidean spaces differ. The properties dealt with by projective Geometry, therefore, in so far as these are obtained without the use of imaginaries, are properties common to all spaces. Finally, the differences which appear between the Geometries of different spaces of the same curvature—e.g. between the Geometries of the plane and the cylinder—are differences in projective properties[117]. Thus the necessity which arises, in metrical Geometry, for further qualifications besides those of constant curvature, disappears when our general space is defined by purely projective properties.

103. We have good ground for expecting, therefore, that the axioms of projective Geometry will be the simplest and most complete expression of the indispensable requisites of any geometrical reasoning: and this expectation, I hope, will not be disappointed. Projective Geometry, in so far as it deals only with the properties common to all spaces, will be found, if I am not mistaken, to be wholly à priori, to take nothing from experience, and to have, like Arithmetic, a creature of the pure intellect for its object. If this be so, it is that branch of pure mathematics which Grassmann, in his Ausdehnungslehre of 1844, felt to be possible, and endeavoured, in a brilliant failure, to construct without any appeal to the space of intuition.

104. But unfortunately, the task of discovering the axioms of projective Geometry is far from easy. They have, as yet, found no Riemann or Helmholtz to formulate them philosophically. Many geometers have constructed systems, which they intended to be, and which, with sufficient care in interpretation, really are, free from metrical presuppositions. But these presuppositions are so rooted in all the very elements of Geometry, that the task of eliminating them demands a reconstruction of the whole geometrical edifice. Thus Euclid, for example, deals, from the start, with spatial equality—he employs the circle, which is necessarily defined by means of equality, and he bases all his later propositions on the congruence of triangles as discussed in Book I.[118] Before we can use any elementary proposition of Euclid, therefore, even if this expresses a projective property, we have to prove that the property in question can be deduced by projective methods. This has not, in general, been done by projective geometers, who have too often assumed, for example, that the quadrilateral construction—by which, as we saw in Chap. I., they introduce projective coordinates—or anharmonic ratio, which is primâ facie metrical, could be satisfactorily established on their principles. Both these assumptions, however, can be justified, and we may admit, therefore, that the claims of projective Geometry to logical independence of measurement or congruence are valid. Let us see, then, how it proceeds.

105. In the first place, it is important to realize that when coordinates are used, in projective Geometry, they are not coordinates in the ordinary metrical sense, i.e. the numerical measures of certain spatial magnitudes. On the contrary, they are a set of numbers, arbitrarily but systematically assigned to different points, like the numbers of houses in a street, and serving only, from a philosophical standpoint, as convenient designations for points which the investigation wishes to distinguish. But for the brevity of the alphabet, in fact, they might, as in Euclid, be replaced by letters. How they are introduced, and what they mean, has been discussed in Chapter I. Here we have only to repeat a caution, whose neglect has led to much misunderstanding.

106. The distinction between various points, then, is not a result, but a condition, of the projective coordinate system. The coordinate system is a wholly extraneous, and merely convenient, set of marks, which in no way touches the essence of projective Geometry. What we must begin with, in this domain, is the possibility of distinguishing various points from one another. This may be designated, with Veronese, as the first axiom of Geometry[119]. How we are to define a point, and how we distinguish it from other points, is for the moment irrelevant; for here we only wish to discover the nature of projective Geometry, and the kind of properties which it uses and demonstrates. How, and with what justification, it uses and demonstrates them, we will discuss later.

107. Now it is obvious that a mere collection of points, distinguished one from another, cannot found a Geometry: we must have some idea of the manner in which the points are interrelated, in order to have an adequate subject-matter for discussion. But since all ideas of quantity are excluded, the relations of points cannot be relations of distance in the ordinary sense, nor even, in the sense of ordinary Geometry, anharmonic ratios, for anharmonic ratios are usually defined as the ratios of four distances, or of four sines, and are thus quantitative. But since all quantitative comparison presupposes an identity of quality, we may expect to find, in projective Geometry, the qualitative substrata of the metrical superstructure.

And this, we shall see, is actually the case. We have not distance, but we have the straight line; we have not quantitative anharmonic ratio, but we have the property, in any four points on a line, of being the intersections with the rays of a given pencil. And from this basis, we can build up a qualitative science of abstract externality, which is projective Geometry. How this happens, I shall now proceed to show.

108. All geometrical reasoning is, in the last resort, circular: if we start by assuming points, they can only be defined by the lines or planes which relate them; and if we start by assuming lines or planes, they can only be defined by the points through which they pass. This is an inevitable circle, whose ground of necessity will appear as we proceed. It is, therefore, somewhat arbitrary to start either with points or with lines, as the eminently projective principle of duality mathematically illustrates; nevertheless we will elect, with most geometers, to start with points[120]. We suppose, therefore, as our datum, a set of discrete points, for the moment without regard to their interconnections. But since connections are essential to any reasoning about them as a system, we introduce, to begin with, the axiom of the straight line. Any two of our points, we say, lie on a line which those two points completely define. This line, being determined by the two points, may be regarded as a relation of the two points, or an adjective of the system formed by both together. This is the only purely qualitative adjective—as will be proved later—of a system of two points. Now projective Geometry can only take account of qualitative adjectives, and can distinguish between different points only by their relations to other points, since all points, per se, are qualitatively similar. Hence it comes that, for projective Geometry, when two points only are given, they are qualitatively indistinguishable from any two other points on the same straight line, since any two such other points have the same qualitative relation. Reciprocally, since one straight line is a figure determined by any two of its points, and all points are qualitatively similar, it follows that all straight lines are qualitatively similar. We may regard a point, therefore, as determined by two straight lines which meet in it, and the point, on this view, becomes the only qualitative relation between the two straight lines. Hence, if the point only be regarded as given, the two straight lines are qualitatively indistinguishable from any other pair through the point.

109. The extension of these two reciprocal principles is the essence of all projective transformations, and indeed of all projective Geometry. The fundamental operations, by which figures are projectively transformed, are called projection and section. The various forms of projection and section are defined in Cremona's "Projective Geometry," Chapter I., from which I quote the following account.

"To cut by a fixed plane σ (transversal plane) a figure (αβγδ ... abcd ...) made up of planes and straight lines, is to construct the straight lines or traces σα, σβ, σγ ... and the points or traces σa, σb, σc....[121] By this means we obtain a new figure composed of straight lines and points lying in the plane σ.

"To project from a fixed straight line s (the axis) a figure ABCD composed of points, is to construct the planes sA, sB, sC.... The figure thus obtained is composed of planes which all pass through the axis s.

"To cut by a fixed straight line s (a transversal) a figure αβγδ ... composed of planes, is to construct the points sα, sβ, sγ.... In this way a new figure is obtained, composed of points all lying on the fixed transversal s.

"If a figure is composed of straight lines a, b, c ... which all pass through a fixed point or centre S, it can be projected from a straight line or axis s passing through S; the result is a figure composed of planes sa, sb, sc....

"If a figure is composed of straight lines a, b, c ... all lying in a fixed plane, it may be cut by a straight line (transversal) s lying in the same plane; the figure which results is formed by the points sa, sb, sc...."

110. The successive application, to any figure, of two reciprocal operations of projection and section, is regarded as producing a figure protectively indistinguishable from the first, provided only that the dimensions of the original figure were the same as those of the resulting figure, that, for example, if the second operation be section by a plane, the original figure shall have been a plane figure. The figures obtained from a given figure, by projection or section alone, are related to that figure by the principle of duality, of which we shall have to speak later on.

111. The two mathematically fundamental things in projective Geometry are anharmonic ratio, and the quadrilateral construction. Everything else follows mathematically from these two. Now what is meant, in projective Geometry, by anharmonic ratio?