An essay on the foundations of geometry · Bertrand Russell
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.
Every judgment--so modern logic contends--is both synthetic and analytic; it combines parts into a whole, and analyses a whole into parts[73]. 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, _i.e._ their contradictories are not self-contradictory; they combine a subject and predicate which cannot, in any purely logical way, be shewn to have any connection, and yet these judgments have apodeictic certainty. But concerning such judgments, Kant is generally careful not to rely upon the mere subjective conviction that they are undeniable: he proves, with every precaution, that without them experience would be impossible. Experience consists in the combination of terms which formal logic leaves apart, and presupposes, therefore, certain judgments by which a framework is made for bringing such terms together. Without these judgments--so Kant contends--all synthesis and all experience would be impossible. If, therefore, the detail of the Kantian reasoning be sound, his results may be obtained by the principle of contradiction _plus_ the possibility of experience, as well as by his distinction of synthetic and analytic judgments.
Logic, at the present day, arrogates to itself at once a wider and a narrower sphere than Kant allowed to it. Wider, because it believes itself capable of condemning any false principle or postulate; narrower, because it believes that its law of contradiction, without a given whole or a given hypothesis, is powerless, and that two terms, _per se_, though they may be different, cannot be contradictories, but acquire this relation only by combination in a whole about which something is known, or by connection with a postulate which, for some reason, must be preserved. Thus no judgment, _per se_, is either analytic or synthetic, for the severance of a judgment from its context robs it of its vitality, and makes it not truly a judgment at all. But in its proper context it is neither purely synthetic nor purely analytic; for while it is the further determination of a given whole, and thus in so far analytic, it also involves the emergence of _new_ relations within this whole, and is so far synthetic.
It is to be observed that this use of the term is at once more rationalistic and less precise than that of Kant. Kant would seem to have supposed himself immediately aware, by inspection, that some knowledge was apodeictic, and its subject-matter, therefore, _à priori_: but he did not always deduce its apriority from any further principle. Here, however, it is to be shown, before admitting apriority, that the falsehood of the judgment in question would not be effected by a mere change in the _matter_ of experience, but only by a change which should render some branch of experience formally impossible, _i.e._ inaccessible to our methods of cognition. The above use is also less precise, for it varies according to the specialization of the experience we are assuming possible, and with every progress of knowledge some new connection is perceived, two previously isolated judgments are brought into logical relation, and the _à priori_ may thus, at any moment, enlarge its sphere, as more is found deducible from fundamental postulates.
The second of Kant's arguments, that we can imagine empty space, though not the absence of space, is false if it means a space without matter anywhere, and irrelevant if it merely means a space between matters and regarded as empty[75]. 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[76]; and in the second place, things will appear mutually external if I have the presentation of _any_ form of externality, whether Euclidean or non-Euclidean. Whatever may be true of the _psychological_ scope of this argument--whose validity is here irrelevant--the _logical_ scope extends, not to Euclidean space, but only to any form of externality which could exist intuitively, and permit knowledge, in beings with our laws of thought, of a world of diverse but interrelated things. Moreover externality, to render the scope of the argument wholly logical, must not be left with a sensational or intuitional meaning, though it must be supposed given in sensation or intuition. It must mean, in this argument, the fact of Otherness[77], 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[78]; that this element, in its isolation may be called a form of externality; and that those properties of this form, if any such be found, which can be deduced from its mere function of rendering experience of interrelated diversity possible, are to be regarded as _à priori_. What these properties are, and how the various lines of argument here suggested converge to a single result, we shall see in Chapters III. and IV.
As regards the first class, on the contrary, we shall see that the relativity of position--by which space is distinguished from all other known manifolds, except time--leads logically to the necessity of three of the most distinctive axioms of Geometry, and yet this relativity cannot be called a deduction from conceptions of magnitude. In analytical Geometry, owing to the fact that coordinate systems start from points, and hence build up lines and surfaces, it is easy to suppose that points can be given independently of lines and of each other, and thus the relativity of position is lost sight of. The error thus suggested by mathematics was probably reinforced by Herbart's theory of space, which, by its serial character, as we have seen, appeared to him to facilitate a construction out of successive points, and to which Riemann acknowledges his indebtedness both in his Dissertation and elsewhere. The same error reappears in Helmholtz, in whom it is probably due wholly to the methods of analytical Geometry. It is a striking fact that, throughout the writings of these two men, there is not, so far as I know, one allusion to the relativity of position, that property of space from which, as our next chapter will shew, the richest quarry of consequences can be extracted. This is not a result of any conception of magnitude, but follows from the nature of our space-intuition; yet no one, surely, could call it empirical, since it is bound up in the very possibility of locating things _there_ as opposed to _here_.
A colour is fixed by its position in the spectrum: two lines in the same spectrum cannot be superposed, and two lines in different spectra need not be--their positions in their respective spectra suffice, or even, roughly, their immediate sense-quality. The fact is, Riemann had space in his mind from the start, and many of the properties, which he enunciates as belonging to all manifolds, belong, as a matter of fact, only to space. It is far from clear what the magnitudes are which the various determinations make possible. Do these magnitudes measure the elements of the manifold, or the relations between elements? This is surely a very fundamental point, but it is one which Riemann never touches on. In the former case, the superposition which he speaks of becomes unnecessary, since the magnitude is inherent in the element considered. We do not require superposition to measure quantities corresponding to different tones or colours; these can be discovered by analysis of single tones or colours. With space, on the other hand, if we seek for elements, we can find none except points, and no analysis of a point will find magnitudes inherent in it--such magnitudes are a fiction of coordinate Geometry. The magnitudes which space deals with, as we shall see in Chapter III., are relations between points, and it is for this reason that superposition is essential to space-measurement. There is no inherent quality in a single point, as there is in a single colour, by which it can be quantitatively distinguished from another. Thus the conception of a manifold, as defined by Riemann, either does not include colours, or does not involve superposition as the only means of measurement. From this dilemma there is no escape.
Now since every point of the colour manifold is completely determined by three magnitudes, which are given in fact, and cannot be arbitrarily chosen, it is plain that measurement by superposition--involving, as it does, motion, and therefore change in these determining magnitudes--is totally out of the question. The superposition of one colour on another, as a means of measurement, is sheer nonsense. And yet measurement is possible in the colour-manifold, by means of Helmholtz's law of mixture (_Mischungsgesetz_); but the measurement is of every separate element, not of the relations between elements, and is thus radically different from space-measurement[81]. The elements are not, like points in space, qualitatively alike, and distinguished by the mere fact of their mutual externality. What we have, in colours, is three fundamental qualitatively distinct elements, out of certain proportions of which we can build up all the other elements of the manifold--each of the resulting elements having the same combination of qualitative diversity and similarity as the three original elements. But in space, what could we make of such a procedure? Given three points, how are we to combine them in certain proportions? The phrase is meaningless. If some one makes the obvious retort, that we have to combine lines, not points, my rejoinder is equally obvious. To begin with, lines are not elements.
Metaphysically, space has _no_ elements, being, as the sequel will show, mere relations between non-spatial elements. Mathematically, this fact exhibits itself in the self-contradictory notion of the point, or zero magnitude in space, as the limit in our vain search for spatial elements. But even if we allow the line to pass as the spatial element, what does the combination of three lines in definite proportions give us? It gives us, simply, the coordinates of a _point_. Here again we see a great difference between the colour and space-manifolds. In colours, the combination of magnitudes gives a new magnitude of the same kind; in space, it defines, not a magnitude at all, but a would-be element of a different kind from the defining magnitudes. In the tone-manifold, we should find still different conditions. Here, no one of the measuring magnitudes can vanish without the tone vanishing too, and all three are so bound up together, in the single resulting sensation, that none can exist without a finite quantity of the others. They are all qualitatively different, both from each other, and from any possible tone, being constituents of it, as mass and velocity are constituents of momentum. All these different conditions require to be examined, before a manifold can be completely defined; and until we have conducted such an examination in detail, we cannot pronounce as to the _à priori_ or empirical nature of the laws of the manifold. As regards space, I have attempted such an examination in the third and fourth chapters of this Essay.
We are thus brought back to the point from which we started, namely, the falsity of Riemann's initial disjunction, and the consequent fallacy in his proof of the empirical nature of the axioms. His philosophy is chiefly vitiated, to my mind, by this fallacy, and by the uncritical assumption that a metrical coordinate system can be set up independently of any axioms as to space-measurement[82]. 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 parallels, which is not contained in the axiom of Free Mobility; but in the infinitesimal, departures from Euclid could only be due to the absence of Free Mobility, which, as I hope my third chapter will show, is once for all impossible.
Helmholtz. =66.= Helmholtz, like Riemann, was important both in the mathematics and in the philosophy of Geometry. From the mathematical point of view, his work has been already considered in Chapter I.; the consideration of his philosophy, which must occupy us here, will be a more serious task. Like Riemann, he endeavoured to prove that all the axioms are empirical, and like Riemann, he based his proof chiefly on Metageometry. He had an additional resource, however, in the physiology of the senses, which first led him to reject the Transcendental Aesthetic, and enabled him to attack Kant from the psychological as well as the mathematical side[83]. The principal topics, for a criticism of Helmholtz, are three: First, his criterion of the _à priori_; second, his discussion with Land as to the "imaginability" of non-Euclidean spaces; third--and this is by far the most important of the three--his theory of the dependence of Geometry on Mechanics. Let us discuss these three points successively.
Another and a better criterion, it is true, is also to be found in Helmholtz, and has also been adopted by Erdmann. Whatever might, by a different experience, have been rendered different--so this criterion contends--must itself be dependent on experience, and so empirical. This criterion seems perfectly sound, but Helmholtz's use of it is usually vitiated by his neglecting to prove the possibility of the different experience in question. He says, for example, that if our experience showed us only bodies which changed their shapes in motion, we should not arrive at the axiom of Congruence, which he pronounces accordingly to be empirical. But I shall endeavour to prove, in Chapter III., that without the axiom of Congruence, experience of spatial magnitude would be impossible. If my proof be correct, it follows that no experience can ever reveal spatial magnitudes which contradict this axiom--a possibility which Helmholtz nowhere discusses, in setting up his hypothetical experience. Thus this second criterion, though perfectly sound, requires always an accompanying transcendental argument, as to the conditions of possible experience. But this accompaniment is seldom to be found in Helmholtz.
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 an analogy with heat. The dictum "Nihil est in intellectu quod non fuerit ante in sensu," would unquestionably be true, if for _intellect_ we were to substitute _imagination_; it is vain, therefore, _if_ our actual space be Euclidean, to hope for a power of _imagining_ a non-Euclidean space. What Helmholtz might, I believe with perfect truth, have urged against Land, is that the image we actually have of space is not sufficiently accurate to exclude, in the actual space we know, all possibility of a slight departure from the Euclidean type.
But in maintaining that we cannot imagine, though we can conceive and describe, a space different from that we actually have, Land is, in my opinion, unquestionably in the right. For a pure Kantian, who maintains, with Land, that none of the axioms can be proved, this question is of great importance. But if, as I have maintained, some of the axioms are susceptible of a transcendental proof, while the others can be verified empirically, the question is freed from psychological implications, and the imaginability or non-imaginability of metageometrical spaces becomes unimportant. =69.= We come now to the third and most important question, the relation of Geometry to Mechanics. There are three senses in which Helmholtz's appeal to rigid bodies may be taken: the first, I think, is the sense in which he originally intended it; the second seems to be the sense which he adopted in his defence against Land; while the third is admitted by Land, and will be admitted in the following argument. These three senses are as follows: (1) It may be asserted that the actual meaning of the axiom of Free Mobility lies in the assertion of empirical rigid bodies, and that the two propositions are equivalent to one another. This is certainly false. (2) The axiom of Free Mobility, it may be said, is logically distinguishable from the assertion of rigid bodies, and may even be not empirical; but it is barren, even for pure Geometry, without the aid of measures, which must themselves be empirical rigid bodies. This sense is more plausible than the first, but I believe we can show that, in this sense also, the proposition is false.
Ideas discussed in this chapter
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