[66] Cf. Helmholtz, Wiss. Abh. Vol. II. p. 640, note: "Die Bearbeiter der Nicht-Euklidischen Geometrie (haben) deren objective Wahrheit nie behauptet."
CHAPTER II.
CRITICAL ACCOUNT OF SOME PREVIOUS PHILOSOPHICAL THEORIES OF GEOMETRY.
51. We have now traced the mathematical development of the theory of geometrical axioms, from the first revolt against Euclid to the present day. We may hope, therefore, to have at our command the technical knowledge required for the philosophy of the subject. The importance of Geometry, in the theories of knowledge which have arisen in the past, can scarcely be exaggerated. In Descartes, we find the whole theory of method dominated by analytical Geometry, of whose fruitfulness he was justly proud. In Spinoza, the paramount influence of Geometry is too obvious to require comment. Among mathematicians, Newton's belief in absolute space was long supreme, and is still responsible for the current formulation of the laws of motion. Against this belief on the one hand, and against Leibnitz's theory of space on the other, and not, as Caird has pointed out[67], against Hume's empiricism, was directed that keystone of the Critical Philosophy, the Kantian doctrine of space. Thus Geometry has been, throughout, of supreme importance in the theory of knowledge.
But in a criticism of representative modern theories of Geometry, which is designed to be, not a history of the subject, but an introduction to, and defence of, the views of the author, it will not be necessary to discuss any more ancient theory than that of Kant. Kant's views on this subject, true or false, have so dominated subsequent thought, that whether they were accepted or rejected, they seemed equally potent in forming the opinions, and the manner of exposition, of almost all later writers.
Kant.
52. It is not my purpose, in this chapter, to add to the voluminous literature of Kantian criticism, but only to discuss the bearing of Metageometry on the argument of the Transcendental Aesthetic, and the aspect under which this argument must be viewed in a discussion of Geometry[68]. On this point several misunderstandings seem to me to have had wide prevalence, both among friends and foes, and these misunderstandings I shall endeavour, if I can, to remove.
In the first place, what does Kant's doctrine mean for Geometry? Obviously not the aspect of the doctrine which has been attacked by psychologists, the "Kantian machine-shop" as James calls it—at any rate, if this can be clearly separated from the logical aspect. The question whether space is given in sensation, or whether, as Kant maintained, it is given by an intuition to which no external matter corresponds, may for the present be disregarded. If, indeed, we held the view which seems crudely to sum up the standpoint of the Critique, the view that all certain knowledge is self-knowledge, then we should be committed, if we had decided that Geometry was apodeictic, to the view that space is subjective. But even then, the psychological question could only arise when the epistemological question had been solved, and could not, therefore, be taken into account in our first investigation. The question before us is precisely the question whether, or how far, Geometry is apodeictic, and for the moment we have only to investigate this question, without fear of psychological consequences.
53. Now on this question, as on almost all questions in the Aesthetic or the Analytic, Kant's argument is twofold. On the one hand, he says, Geometry is known to have apodeictic certainty: therefore space must be à priori and subjective. On the other hand, it follows, from grounds independent of Geometry, that space is subjective and à priori; therefore Geometry must have apodeictic certainty. These two arguments are not clearly distinguished in the Aesthetic, but a little analysis, I think, will disentangle them. Thus in the first edition, the first two arguments deduce, from non-geometrical grounds, the apriority of space; the third deduces the apodeictic certainty of Geometry, and maintains, conversely, that no other view can account for this certainty[69]; the last two arguments only maintain that space is an intuition, not a concept. In the second edition, the double argument is clearer, the apriority of space being proved independently of Geometry in the metaphysical deduction, and deduced from the certainty of Geometry, as the only possible explanation of this, in the transcendental deduction. In the Prolegomena, the latter argument alone is used, but in the Critique both are employed.
54. Now it must be admitted, I think, that Metageometry has destroyed the legitimacy of the argument from Geometry to space; we can no longer affirm, on purely geometrical grounds, the apodeictic certainty of Euclid. But unless Metageometry has done more than this—unless it has proved, what I believe it alone cannot prove, that Euclid has not apodeictic certainty—then Kant's other line of argument retains what force it may ever have had. The actual space we know, it may say, is admittedly Euclidean, and is proved, without any reference to Geometry, to be à priori; hence Euclid has apodeictic certainty, and non-Euclid stands condemned. To this it is no answer to urge, with the Metageometers, that non-Euclidean systems are logically self-consistent; for Kant is careful to argue that geometrical reasoning, by virtue of our intuition of space, is synthetic, and cannot, though à priori, be upheld by the principle of contradiction alone[70]. Unless non-Euclideans can prove, what they have certainly failed to prove up to the present, that we can frame an intuition of non-Euclidean spaces, Kant's position cannot be upset by Metageometry alone, but must also be attacked, if it is to be successfully attacked, on its purely philosophical side.