An essay on the foundations of geometry · Bertrand Russell
CHAPTER II.
(3) For pure Geometry and the abstract study of space, it may be said, Free Mobility, as applied to an abstract geometrical matter, gives a sufficient possibility of quantitative comparison; but the moment we extend our results to mixed mathematics, and apply them to empirically given matter, we require also, as measures, empirically given rigid bodies, or bodies, at least, whose departures from rigidity are empirically known. In this sense, I admit, the proposition is correct[89]. 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.
This argument may be summed up by the following disjunction: If bodies change their shapes in motion--and to some extent, since no body is perfectly rigid, they must all do so--then one of two cases must occur. _Either_ the changes of shape, as bodies move from place to place, follow no geometrical law, are not, for instance, functions of the amount or direction of motion; in which case the law of causation requires that they should not be effects of the change of place, but of some simultaneous non-geometrical change, such as temperature. _Or_ the changes are regular, and the shape _S_ becomes, in a new position _p_, _Sf_(_p_). In this case, the law of concomitant variations leads us to attribute the change of shape to the mere motion, and shape thus becomes a function of absolute position. But this is absurd, for position _means_ merely a relation or set of relations; it is impossible, therefore, that mere position should be able to effect changes in a body. Position is one term in a relation, not a thing _per se_; it cannot, therefore, act on a thing, nor exist by itself, apart from the other terms of the relation. Thus Helmholtz's view, that Congruence depends on the existence of rigid bodies, must, since it involves absolute position, be condemned as a logical fallacy. Congruence, in fact, as I shall prove more fully in Chapter III., is an _à priori_ deduction from the relativity of position.
This conclusion, I believe, is valid of all actual measurement. But the possibility of such empirical and approximate rigidity, I must insist, depends on the _à priori_ law that _mere_ motion, apart from the action of other matter, cannot effect a change of shape. For without this law, the effect of other matter would not be discoverable; the laws of motion would be absurd, and Physics would be impossible. Consider the second law, for example: How could we measure the change of motion, if motion itself produced a change in our measures? Or consider the law of gravitation: How could we establish the inverse square, unless we were able, independently of Dynamics, to measure distances? The whole science of Dynamics, in short, is fundamentally dependent on Geometry, and but for the independent possibility of measuring spatial magnitudes, none of the magnitudes of Dynamics could be measured. Time, force, and mass are alike measured by spatial correlates: these correlates are given, for time, by the first law, for force and mass, by the second and third.
It is true, then, that an empirical element appears unavoidably in all actual measurement, inasmuch as we can only know empirically that a given piece of matter preserves its shape throughout the necessary change of dynamical relations to other matter involved in motion; but it is further true that, for Geometry--which regards matter simply as supplying the necessary breach in the homogeneity of space, and the necessary term for spatial relations, not as the bearer of forces which change the configuration of other material systems--for Geometry, which deals with this abstract and merely kinematical matter, rigidity is _à priori_, in so far as the only changes with which it is cognizant--changes of mere position, namely--are incapable of affecting the shapes of the imaginary and abstract bodies with which it deals. To use a scholastic distinction, we may say that matter is the _causa essendi_ of space, but Geometry is the _causa cognoscendi_ of Physics. Without a Geometry independent of Physics, Physics itself, which necessarily assumes the results of Geometry, could never arise; but when Geometry is used in Physics, it loses some of its _à priori_ certainty, and acquires the empirical and approximate character which belongs to all accounts of actual phenomena.
In order to study space by itself, we reduce the properties of matter to a bare minimum: we avoid entirely the category of causation, so essential to Dynamics, and retain nothing, in our matter, but its spatial adjectives[91]. The kind of rigidity affirmed of this abstract matter--a kind which suffices for the theory of our science, though not for its application to the objects of daily life--is purely geometrical, and asserts no more than this: That since our matter is devoid, _ex hypothesi_, of causal properties, there remains nothing, in mere empty space, which is capable of changing the configuration of any geometrical system. A change of absolute position, it asserts, is nothing; therefore the only real change involved in motion is a change of relation to other matter; but such other matter, for the purposes of our science, is regarded as destitute of causal powers; hence no change can occur, in the configuration of our system, by the mere effect of motion through empty space. The necessity of such a principle may be shown by a simple _reductio ad absurdum_, as follows. A motion of translation of the universe as a whole, with constant direction and velocity, is dynamically negligeable; indeed it is, philosophically, no motion at all, for it involves no change in the condition or mutual relations of the things in the universe. But if our geometrical rigidity were denied, the change in the parameter of space might cause all bodies to change their shapes owing to the mere change of absolute position, which is obviously absurd.
To make quite plain the function of rigid bodies in Geometry, let us suppose a liquid geometer in a liquid world. We cannot suppose the liquid perfectly homogeneous and undifferentiated, in the first place because such a liquid would be indistinguishable from empty space, in the second place because our geometer's body--unless he be a disembodied spirit--will itself constitute a differentiation for him. We may therefore assume "dim beams, Which amid the streams Weave a network of coloured light," and we may suppose this network to form the occasion for our geometer's reflections. Then he will be able to imagine a network in which the lines are straight, or circular, or parabolic, or any other shape, and he will be able to infer that such a network, if it can be woven in one part of the fluid, can be woven in another. This will form sufficient basis for his deductions. The superposition he is concerned with--since not actual equality, but only the formal conditions of equality, are the subject-matter of Geometry--is purely ideal, and is unaffected by the impossibility of congealing any actual network. But in order to apply his Geometry to the exigencies of life, he would need some standard of comparison between actual networks, and here, it is true, he would need either a rigid body, or a knowledge of the conditions under which similar networks arose. Moreover these conditions, being necessarily empirical, could hardly be known apart from previous measurement. Hence for applied, though not for pure Geometry, one rigid body at least seems essential.
Erdmann. =74.= In connection with Riemann and Helmholtz, it is natural to consider Erdmann's philosophical work on their theories[92]. 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 hesitation--to be incompatible with any system but that of Mill: there is apparently no distinction, to him, between the general and the universal, and consequently no concept not embodied in a series of instances. Such a theory of logic, to my mind, vitiates most of his work, as it vitiated Riemann's philosophy[93]. This general criticism will find abundant illustration in the course of our account of Erdmann's views.
A judgment of magnitude is always a judgment of comparison, and what is more, the comparison is never concerned with quality, but only with quantity. Quality, in the judgment of magnitude, is supposed identical, in the object whose magnitude is stated, and in the unit with which it is compared. But quality, except in pure number, and in pure quantity as dealt with by the Calculus, is always present, and is partly absorbed into quantity, partly untouched by the judgment of magnitude. As Bosanquet says (Logic, Vol. I. p. 124); "Quantitative comparison is not _prima facie_ coordinate with qualitative, but rather stands in its place as the _effect of comparison on quality_, which so far as comparable _becomes quantity_, and so far as incomparable furnishes the distinction of parts essential to the quantitative whole" (italics in the original). Thus, if we are to regard space as a magnitude, we must be able to adduce all those series of instances of which Erdmann speaks, and which, for the conception of space, seemed irrelevant. But it remains to be proved that the comparison, which we _can_ institute between various spaces, is capable of expression in a quantitative form.
Rather it would seem that the difference of quality is such as to preclude quantitative comparison between different spaces, and therefore also to preclude all judgments of magnitude about space as a whole. Here an exception might seem to be demanded by non-Euclidean spaces, whose space-constants give a definite magnitude, inherent in space as a whole, and therefore, one might think, characterizing space as a magnitude. But this is a mistake. For the space-constant, in such spaces, is the ultimate unit, the fixed term in all quantitative comparison; it is itself, therefore, destitute of quantity, since there is no independently given magnitude with which to compare it. A non-Euclidean world, in which the space-constant and all lines and figures were suddenly multiplied in a constant ratio, would be wholly unchanged; the lines, as measured against the space-constant, would have the same magnitude as before, and the space-constant itself, having no outside standard of comparison, would be destitute of quantity, and therefore not subject to change of quantity. Such an enlargement of a non-Euclidean world, in other words, is unmeaning; and this proves how inapplicable is the notion of quantity to space as a whole.
It might be objected that this only proves the absence of quantitative difference between different spaces of positive space-constant, or between those of negative space-constant: the quantitative difference persists, it might be said, between those of positive curvature in general and those of negative curvature in general, or between both together and Euclidean space. This I entirely deny. There is no qualitatively similar unit, in the three kinds of space, by which quantitative comparison could be effected. The straight lines of one space cannot be put into the other: the two straight lines, in one space, whose product is the reciprocal of the measure of curvature, have no corresponding curves in the other space, and the measures of curvature cannot, therefore, be quantitatively compared with each other. That the one may be regarded as positive, the other negative, I admit, but their values are indeterminate, and the units in the two cases are qualitatively different. A debt of £300 may be represented as the asset of -£300, and the height of the Eiffel Tower is +300 metres; but it does not follow that the two are quantitatively comparable. So with space-constants: the space-constant is itself the unit for magnitudes in its own space, and differs qualitatively from the space-constant of another kind of space.
Again, to proceed to a more philosophical argument, two different spaces cannot co-exist in the same world: we may be unable to decide between the alternatives of the disjunction, but they remain, none the less, absolutely incompatible alternatives. Hence we cannot get that coexistence of two spaces which is essential to comparison. The fact seems to be that Erdmann, in his admiration for Riemann and Helmholtz, has fallen in with their mathematical bias, and assumed, as mathematicians naturally tend to assume, that quantity is everywhere and always applicable and adequate, and can deal with more than the mere comparison of things whose qualities are already known as similar[95]. =80.= This suggests the fourth and last of the above points, that the _qualities_ of space, even if space could be successfully regarded as a magnitude, would have to be entirely omitted in such a manner of regarding it, and that, therefore, none of its important or essential properties would emerge from such treatment. For to regard space as a magnitude involves, as we saw, a comparison with something qualitatively similar, and an abstraction from the similar qualities. To some extent and by the help of certain doubtful arguments, such a comparison is instituted by Riemann and Erdmann; but when they have instituted it, they forget all about the common qualities on which its possibility depends. But these are precisely the fundamental properties of space, and those from which, as I shall endeavour to prove in Chapter III., the axioms common to Euclid and Metageometry follow _à priori_. Such are the dangers of the quantitative bias.
This is the more strange, as he actually proves (p. 50) that measurement is impossible, in an extent, unless the parts are independent of their place, and can be carried about unaltered as measures. In spite of this, he proceeds immediately to discuss whether the measure of curvature is constant or variable, without investigating how, in the latter case, Geometry could exist. We cannot know, he says, from geometrical superposition, that geometrical bodies are independent of place, for if their dimensions altered in motion according to any fixed law, two bodies which could be superposed in one place could be superposed in any other. That such a hypothesis involves absolute position, and denies the qualitative similarity of the parts of space, which he declares (p. 171) to be the principle of his theory of Geometry, is nowhere perceived. But what is more, his notion that magnitude is something absolute, independent of comparison, has prevented him from seeing that such a hypothesis is unmeaning. He says himself that, even on this hypothesis, a geometrical body can be defined as one whose points retain constant distances from each other, for, since we have no absolute measure, measurement could not reveal to us the change of absolute magnitude (p. 60). But is not this a _reductio ad absurdum_?
For magnitude is nothing apart from comparison, and the comparison here can only be effected by superposition; if, then, as on the above hypothesis, superposition always gives the same result, by whatever motion it is effected, there is no sense in speaking of magnitudes as no longer equal when separated: absolute magnitude is an absurdity, and the magnitude resulting from comparison does not differ from that which would result if the dimensions of bodies were unchanged in motion. Therefore, since magnitude is only intelligible as the result of comparison, the dimensions of bodies _are_ unchanged in motion, and the suggested hypothesis is unmeaning. On this subject I shall have more to say in Chapter III.[96] =82.= This hypothesis, however, is not introduced for its own sake, but only to usher in the 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.
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.
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.
Ideas discussed in this chapter
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