An Essay on the Foundations of Geometry

Helmholtzian deus ex machina, Mechanics. For Mechanics proves—so Erdmann confidently continues—that rigidity must hold, not merely as to ratios, in the above restricted geometrical sense, but as to absolute magnitudes (p. 62). Hence we get at last true Congruence, empirical as Mechanics is empirical, and impossible to prove apart from Mechanics. I have already criticized Helmholtz's view of the dependence of Geometry on Mechanics, and need not here speak of it at length. It is a pity that Erdmann has in no way specified the procedure by which Mechanics decides the geometrical alternatives—indeed he seems to rely on the ipse dixit of Helmholtz. How, if Geometry would be totally unable to discover a change in dimensions of the kind suggested, the Laws of Motion, which throughout depend on Geometry, should be able to discover it if it existed, I am wholly at a loss to understand. Uniform motion in a straight line, for example, presupposes geometrical measurement; if this measurement is mistaken, what Mechanics imagines to be uniform motion is not really such, but Mechanics can never discover the discrepancy. If the Laws of Motion had been regarded as à priori, Geometry might possibly have been reinforced by them; but so long as they are empirical, they presuppose geometrical measurement, and cannot therefore condition or affect it. Erdmann's conclusion, in the second chapter, is that Congruence is probable, but cannot be verified in the infinitesimal; that its truth involves the actual existence of rigid bodies (though, by the way, we know these to be, strictly speaking, non-existent), that rigid bodies are freely moveable, and do not alter their size in rotation (Helmholtz's Monodromy); that the axiom of three dimensions is certain, since small errors are impossible; and that the remaining axioms of Euclid—those of the straight line and of parallels—are approximately, if not accurately, true of our actual space (pp. 78, 83). He does not discuss how Congruence, on the above view, is compatible with the atomic theory, or even with the observed deformations of approximately rigid bodies; nor how, if space, as he assumes, is homogeneous, rigid bodies can fail to be freely moveable through space. The axioms are all lumped together as empirical, and it appears, in the following chapters, that Erdmann regards their empirical nature as sufficiently proved by their applicability to empirical material (cf. pp. 159, 165)—a strange criterion, which would prove the same conclusion, with equal facility, of Arithmetic and of the laws of thought. 83. The third chapter, on the philosophical consequences of Metageometry, need not be discussed at length, since it deals rather with space than with Geometry. At the same time, it will be worth while to treat briefly of Erdmann's criterion of apriority. On this subject it is very difficult to discover his meaning, since it seems to vary with the topic he is discussing. Thus at one time (p. 147) he rejects most emphatically the Kantian connection of the à priori and the subjective[97], and yet at another time (p. 96) he regards every presentation of external things as partly à priori, partly empirical, merely because such a presentation is due to an interaction between ourselves and things, and is therefore partly due to subjective activity, partly due to outside objects. Hence, he says, the distinction is not between different presentations, but between different aspects of one and the same presentation. This seems to return wholly to the Kantian psychological criterion of subjectivity, with the added disadvantage that it makes the distinction, like that of analytic and synthetic, epistemologically worthless. And yet he never hesitates to pronounce every piece of knowledge in turn empirical. The fact seems to be, that where he wants a more logical criterion, he adopts a modification of Helmholtz's criterion for sensations. If space be an à priori form, he says, no experience could possibly change it (p. 108); but this Metageometry has proved not to be the case, since we can intuit the perceptions which non-Euclidean space would give us (p. 115). I have criticised this argument in discussing Helmholtz; at present we are concerned with Erdmann's criterion of apriority. The subjectivity-criterion—though he certainly uses it in discussing the apriority of space, and solemnly decides, by its means, that space is both à priori and empirical since a change either in us or in the outer world could change it (p. 97)—would seem, like several of his other tests, to be a lapse on his part: the criterion which he means to use is Helmholtz's. This criterion, I think, with a slight change of wording, might be accepted; it seems to me a necessary, but not a sufficient condition. The à priori, we may say, is not only that which no experience can change, but that without which experience would become impossible. It is the omission to discuss the conditions which render geometrical (and mechanical) experience possible, to my mind, which vitiates the empirical conclusions of Helmholtz and Erdmann. Why certain conditions should be necessary for experience—whether on account of the constitution of the mind, or for some other reason—is a further question, which introduces the relation of the à priori to the subjective. But in discussing the question as to what knowledge is à priori, as opposed to the question concerning the further consequences of apriority, it is well to keep to the purely logical criterion, and so preserve our independence of psychological controversies. The fact, if it be a fact, that the world might be such as to defy our attempts to know it, will not, with the above criterion, invalidate the conclusion that certain elements in knowledge are à priori; for whether fulfilled or not, they remain necessary conditions for the existence of any knowledge at all. 84. With this caution as to the meaning of apriority, we shall find, I think, that the conclusions of Erdmann's final chapter, on the principles of a theory of Geometry, are largely invalidated by the diversity and inadequacy of his tests of the à priori. He begins by asserting, in conformity with the quantitative bias noticed above, that the question as to the nature of geometrical axioms is completely analogous to the corresponding question of the foundations of pure mathematics (p. 138). This is, I think, a radical error: for the function of the axioms seems to be, to establish that qualitative basis on which, as we saw, all qualitative comparison must rest. But in pure mathematics, this qualitative basis is irrelevant, for we deal there with pure quantity, i.e. with the merely quantitative result of quantitative comparison, wherever it is possible, independently of the qualities underlying the comparison. Geometry, as Grassmann insists[98], ought not to be classed with pure mathematics, for it deals with a matter which is given to the intellect, not created by it. The axioms give the means by which this matter is made amenable to quantity, and cannot, therefore, be themselves deduced from purely quantitative considerations. Leaving this point aside, however, let us return to Erdmann. He distinguishes, within space, a form and a matter: the form is to contain the properties common to all extents, the matter the properties which distinguish space from other extents. This distinction, he says, is purely logical, and does not correspond with Kant's: matter and form, for Erdmann, are alike empirical. The axioms and definitions of Geometry, he says, deal exclusively with the matter of space. It seems a pity, having made this distinction, to put it to so little use: after a few pages, it is dropped, and no epistemological consequences are drawn from it. The reason is, I think, that Erdmann has not perceived how much can be deduced from his definition of an extent, as a manifold in which the dimensions are homogeneous and interchangeable. For this property suffices to prove the complete homogeneity of an extent, and hence—from the absence of qualitative differences among elements—the relativity of position and the axiom of Congruence. This deduction will be made at length in the sequel[99]; at present, I have only to observe that every extent, on this view, possesses all the properties (except the three dimensions) common to Euclidean and non-Euclidean spaces. The axioms which express these properties, therefore, apply to the form of space, and follow from homogeneity alone, which Erdmann allows (p. 171) as the principle of any theory of space. The above distinction of form and matter, therefore, corresponds, when its full consequences are deduced, to the distinction between the axioms which follow from the homogeneity of space and those which do not. Since, then, homogeneity is equivalent to the relativity of position, and the relativity of position is of the very essence of a form of externality, it would seem that his distinction of form and matter can also be made coextensive with the distinction of the à priori and empirical in Geometry. On this subject, I shall have more to say in Chapter III. In the remainder of the chapter, Erdmann insists that the straight line, etc., though not abstracted from experience, which nowhere presents straight lines, must yet, as applicable to admittedly empirical sciences, be empirical (p. 159)—a criterion which he appears to employ only when all other grounds for an empirical opinion fail, and one which, obviously, can never refuse to do its work, since all elements of knowledge are susceptible of employment on some empirical material. He also defines the straight line (p. 155) as a line of constant curvature zero, as though curvature could be measured independently of the straight line. Even the arithmetical axioms are declared empirical (p. 165), since in a world where things were all hopelessly different from one another, these axioms could not be applied. After this reminder of Mill, we are not surprised, a few pages later (p. 172), at a vague appeal to "English logicians" as having proved Geometry to be an inductive science. Nevertheless, Erdmann declares, almost on the last page of his book (p. 173), that Geometry is distinguished from all other sciences by the homogeneity of its material: a principle of which no single application occurs throughout his book, and which, as we shall see in Chapter III., flatly contradicts the philosophical theories advocated throughout his preceding pages. On the whole, then, it cannot be said that Erdmann has done much to strengthen the philosophical position of Riemann and Helmholtz. I have criticized him at length, because his book has the appearance of great thoroughness, and because it is undoubtedly the best defence extant of the position which it takes up. We shall now have the opposite task to perform, in defending Metageometry, on its mathematical side, from the attacks of Lotze and others, and in vindicating for it that measure of philosophical importance—far inferior, indeed, to the hopes of Erdmann—which it seems really to possess. Lotze. 85. Lotze's argument as regards Geometry[100]—which follows a metaphysical argument as to the ontological nature of space, and assumes the results of this argument—consists of two parts: the first discusses the various meanings logically assignable (pp. 233–247) to the proposition that other spaces than Euclid's are possible, and the second criticizes, in detail, the procedure of Metageometry. The first of these questions is very important, and demands considerable care as to the logical import of a judgment of possibility. Although Lotze's discussion is excellent in many respects, I cannot persuade myself that he has hit on the only true sense in which non-Euclidean spaces are possible. I shall endeavour to make good this statement in the following pages. 86. Lotze opens with a somewhat startling statement, which, though philosophically worthy to be true, does not appear to be historically borne out. Euclidean Geometry has been chiefly shaken, he says, by the Kantian notion of the exclusive subjectivity of space—if space is only our private form of intuition, to which there exists no analogue in the objective world, then other beings may have other spaces, without supposing any difference in the world which they arrange in these spaces (p. 233). This certainly