Philosophical Theories of Geometry

An Essay on the Foundations of Geometry, by Bertrand Russell is part of HackerNoon’s Book Blog Post series. The Table of Links for this book can be found here. Chapter II: Critical Account of Some Previous Philosophical Theories of Geometry

CHAPTER II. CRITICAL ACCOUNT OF SOME PREVIOUS PHILOSOPHICAL THEORIES OF GEOMETRY.

51.; 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.. 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, , that it would demand little attention. If this be so, the distinction of analysis and synthesis, whatever may be its importance in pure Logic, can have no value in Epistemology. But such a doctrine, it must be observed, allows full scope to the principle of contradiction: this criterion, since all judgments, in one aspect at least, are analytic, is applicable to all judgments alike. On the other hand, the whole which is analysed must be supposed already given, before the parts can be mutually contradictory: for only by connection in a given whole can two parts or adjectives be incompatible. Thus the principle of contradiction remains barren until we already have some judgments, and even some inference: for the parts may be regarded, to some extent, as an inference from the whole, or vice versâ. When once the arch of knowledge is constructed, the parts support one another, and the principle of contradiction is the keystone: but until the arch is built, the keystone remains suspended, unsupported and unsupporting, in the empty air. In other words, knowledge once existent can be analysed, but knowledge which should have to win every inch of the way against a critical scepticism, could never begin, and could never attain that circular condition in which alone it can stand.

But Kant's doctrine, if true, is designed to restrain a critical scepticism even where it might be effective. Certain fundamental propositions, he says, are not deducible from logic, . The only argument of importance, then, is the first argument. But I must insist, at the outset, that our problem is purely logical, and that all psychological implications must be excluded to the utmost possible extent. Moreover, as will be proved in Chapter IV., the proper function of space is to distinguish between different presented things, not between the Self and the object of sensation or perception.

The argument then becomes the following: consciousness of a world of mutually external things demands, in presentations, a cognitive but non-inferential element leading to the discrimination of the objects presented. This element must be non-inferential, for from whatever number or combination of presentations, which did not of themselves demand diversity in their objects, I could never be led to infer the mutual externality of their objects. Kant says: "In order that sensations may be ascribed to something external to me ... and similarly in order that I may be able to present them as outside and beside one another, ... the presentation of space must be already present." But this goes rather too far: in the first place, the question should be only as to the mutual externality of presented things, not as to their externality to the Self; and in the second place, things will appear mutually external if I have the presentation of any form of externality, whether Euclidean , the fact of being different from some other thing: it must involve the distinction between different things, and must be that element, in a cognitive state, which leads us to discriminate constituent parts in its object. So much, then, would appear to result from Kant's argument, that experience of diverse but interrelated things demands, as a necessary prerequisite, some sensational or intuitional element, in perception, by which we are led to attribute complexity to objects of perception. The elements are not, like points in space, qualitatively. Riemann has failed to observe, what I have endeavoured to prove in the next chapter, that, unless space had a strictly constant measure of curvature, Geometry would become impossible; also that the absence of constant measure of curvature involves absolute position, which is an absurdity. Hence he is led to the conclusion that all geometrical axioms are empirical, and may not hold in the infinitesimal, where observation is impossible.

Thus he says (p. 267): "Now the empirical conceptions, on which spatial measurements are based, the conceptions of the rigid body and the light-ray, appear to lose their validity in the infinitesimal: it is therefore quite conceivable that the relations of spatial magnitudes in the infinitesimal do not correspond to the presuppositions of Geometry, and this would, in fact, have to be assumed, as soon as it would enable us to explain the phenomena more simply." From this conclusion I must entirely dissent. In very large spaces, there might be a departure from Euclid; for they depend upon the axiom of, however, it would appear that he regards as empirical whatever applies to empirical matter. For he there maintains, that even if space were an à priori form, yet any Geometry, which aimed at an application to Physics, would, since the actual places of bodies are not known à priori, be necessarily empirical. It seems the more probable that he regards this as a possible criterion, as it is adopted, in several passages, by his disciple Erdmann, and so strange a test could hardly be accepted by a philosopher, unless he had found it in his master. I have called this a strange test, because it seems to me completely to ignore the work of the Critical Philosophy. For if there is one thing which, one might have hoped, had been made sufficiently clear by Kant's Critique, it is this, that knowledge which is à priori, being itself the condition of possible experience, applies—and in Kant's view, applies only—to empirical matter.

Helmholtz and Erdmann, therefore, in setting up this test without discussion, simply ignore the existence of Kant and the possibility of a transcendental argument. Helmholtz assumes always that empirical knowledge must be wholly empirical, that there can be no à priori conditions of the experience in question, that experience will always be possible, and may give any kind of result. Thus in discussing "physical" Geometry, he assumes that the possibility of empirical measurement involves no à priori axioms, and that no à priori element can be contained in the process. This assumption, as we shall see in Chapter III., is quite unwarrantable: certain properties of space, in fact, are involved in the possibility of measuring matter. In spite of the fact, therefore, that we apply measurement to empirical matter, and that our results are therefore empirical, there . I have already criticized this line of argument in the beginning of the present chapter. Helmholtz's criticism, however, was different: admitting the internal consistency of the argument, he denied one of its premisses. We can imagine non-Euclidean spaces,.

Although I agree with Helmholtz in thinking the distinction between Euclidean and non-Euclidean spaces empirical, I cannot think his argument on the "imaginability" of the latter a very happy one. The validity of any proof must turn, obviously, on the definition of imaginability. The definition which Helmholtz gives in his answer to Land is as follows: Imaginability requires "die vollständige Vorstellbarkeit derjenigen Sinneseindrücke, welche das betreffende Object in uns nach den bekannten Gesetzen unserer Sinnesorgane unter allen denkbaren Bedingungen der Beobachtung erregen, und wodurch es sich von anderen ähnlichen Objecten unterscheiden würde" (Wiss. Abh. II. p. 644). This definition is not very clear, owing to the ambiguity of the word "Vorstellbarkeit." The following definition seems less ambiguous: "Wenn die Reihe der Sinneseindrücke vollständig und eindeutig angegeben werden kann, muss man m. E. die Sache für anschaulich vorstellbar erklären" (Vorträge und Reden, II. p. 234).

This makes clear, what also appears from his manner of proof, that he regards things as imaginable which can be described in conceptual terms. Such, as Land remarks (Mind, Vol. II. p. 45), "is not the sense required for argumentation in this case." That Land's criticism is just, is shown by Helmholtz's proof for non-Euclidean spaces, for it consists only in an analogy to the volume inside a sphere, which is mathematically obtained thus: We take the symbols representing magnitudes in "pseudo-spherical" (hyperbolic) space, and give them a new Euclidean meaning; thus all our symbolic propositions become capable of two interpretations, one for pseudo-spherical space, and one for the volume inside a sphere. It is, however, sufficiently obvious that this procedure, though it enables us to describe our new space, does not enable us to imagine it, in the sense of calling up images of the way things would look in it. We really derive, from this analogy, no more knowledge than a man born blind may derive, as to light, from.

In discussing these three meanings, I shall not confine myself strictly to the text of Helmholtz or Land: if I endeavoured to do so, I should be met by the difficulty that neither of them defines the à priori, and that each is too much inclined, in my opinion, to test it by psychological criteria. I shall, therefore, take the three meanings in turn, without laying stress on their historical adequacy to the views of Land or Helmholtz.

70.. This is certainly the most important book on the subject which has appeared from the philosophical side, and in spite of the fact that, like the whole theory of Riemann and Helmholtz, it is inapplicable to projective Geometry, it still deserves a very full discussion.

Erdmann agrees throughout with the conclusions of Riemann and Helmholtz, except on a few points of minor importance; and his views, as this agreement would lead one to expect, are ultra-empirical. Indeed his logic seems—though I say this with. This general criticism will find abundant illustration in the course of our account of Erdmann's views.

75..

79..

80.; 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 , or again Helmholtz's belief in the dependence of Geometry on the behaviour of rigid bodies. This belief led to the view that Geometry, like Physics, is an experimental science, in which objective truth can be attained, it is true, but only by empirical methods. However, Lotze's ground for uncertainty about Euclid is a philosophically tenable ground, and it will be instructive to observe the various possibilities which arise from it.

If space is only a subjective form—so Lotze opens his argument—other beings may have a different form. If this corresponds to a different world, the difference, he says, is uninteresting: for our world alone is relevant to any metaphysical discussion. But if this different space corresponds to the same world which we know under the Euclidean form, then, in his opinion, we get a question of genuine philosophic interest. And here he distinguishes two cases: either the relations between things, which are presented to these hypothetical beings under the form of some different space, are relations which do not appear to us, or at any rate do not appear spatial; or they are the same relations which appear to us as figures in Euclidean space (p. 235). The first possibility would be illustrated, he says, by beings to whom the tone or colour-manifolds appeared extended; but we cannot, in his opinion, imagine a manifold, such as is required for this case, to have its dimensions homogeneous and comparable inter se, and therefore the contents of the various presentations constituting such a manifold could not be combined into a single content containing them all. But the possibility of such a combination is of the essence of anything worth calling a space: therefore the first of the above possibilities is unmotived and uninteresting. Lotze's conclusion on this point, I think, is undeniable, but I doubt whether his argument is very cogent. However, as this possibility has no connection with that contemplated by non-Euclideans, it is not worth while to discuss it further.

The second possibility also, Lotze thinks, is not that of Metageometry, but in truth it comes nearer to it than any of the other possibilities discussed. If a non-Euclidean were at the same time a believer in the subjectivity of space, he would have to be an adherent of this view. Let us see more precisely what the view is. In Book II., Chapter I., Lotze has accepted the argument of the Transcendental Aesthetic, but rejected that of the mathematical antinomies: he has decided. This import of Metageometry is denied by Lotze, on the ground that non-Euclidean logic is faulty, a ground which he endeavours, by much detail and through many pages, to make good—with what success, we will now proceed to examine.

90..

93.. But . The logical validity of Metageometry, and the mathematical possibility of three-dimensional non-Euclidean spaces, will therefore be regarded, throughout the remainder of the work, as sufficiently established.

97..

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—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 . This series of definitions may lead to an infinite regress, but it may. 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

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