An essay on the foundations of geometry · Bertrand Russell
THE AXIOMS OF METRICAL GEOMETRY.
Let us now examine in detail the prerequisites of spatial measurement. We shall find three axioms, without which such measurement would be impossible, but with which it is adequate to decide, empirically and approximately, the Euclidean or non-Euclidean nature of our actual space. We shall find, further, that these three axioms can be deduced from the conception of a form of externality, and owe nothing to the evidence of intuition. They are, therefore, like their equivalents the axioms of projective Geometry, _à priori_, and deducible from the conditions of spatial experience. This experience, accordingly, can never disprove them, since its very existence presupposes them. I. _The Axiom of Free Mobility._ =143.= Metrical Geometry, to begin with, may be defined as the science which deals with the comparison and relations of spatial magnitudes. The conception of magnitude, therefore, is necessary from the start. Some of Euclid's axioms, accordingly, have been classed as arithmetical, and have been supposed to have nothing particular to do with space. Such are the axioms that equals added to or subtracted from equals give equals, and that things which are equal to the same thing are equal to one another. These axioms, it is said, are purely arithmetical, and do not, like the others, ascribe an adjective to space. As regards their use in arithmetic, this is of course true. But if an arithmetical axiom is to be applied to spatial magnitudes, it must have some spatial import[147], and thus even this class is not, in Geometry, _merely_ arithmetical. Fortunately, the geometrical element is the same in all the axioms of this class--we can see at once, in fact, that it can amount to no more than a definition of spatial magnitude[148]. Again, since the space with which Geometry deals is infinitely divisible, a definition of spatial magnitude reduces itself to a definition of spatial equality, for, as soon as we have this last, we can compare two spatial magnitudes by dividing each into a number of equal units, and counting the number of such units in each[149]. The ratio of the number of units is, of course, the ratio of the two magnitudes.
The above axiom is the axiom of Free Mobility[151]. I propose to prove (1) that the denial of this axiom would involve logical and philosophical absurdities, so that it must be classed as wholly _à priori_; (2) that metrical Geometry, if it refused this axiom, would be unable, without a logical absurdity, to establish the notion of spatial magnitude at all. The conclusion will be, that the axiom cannot be proved or disproved by experience, but is an _à priori_ condition of metrical Geometry. As I shall thus be maintaining a position which has been much controverted, especially by Helmholtz and Erdmann, I shall have to enter into the arguments at some length. =145.= A. _Philosophical Argument._ The denial of the axiom involves absolute position, and an action of mere space, _per se_, on things. For the axiom does not assert that real bodies, as a matter of empirical fact, never change their shape in any way during their passage from place to place: on the contrary, we know that such changes do occur, sometimes in a very noticeable degree, and always to some extent. But such changes are attributed, not to the change of place as such, but to physical causes: changes of temperature, pressure, etc. What our axiom has to deal with is not actual material bodies, but geometrical figures[152], and it asserts that a figure which is possible in any one position in space is possible in every other. Its meaning will become clearer by reference to a case where it does not hold, say the space formed by the surface of an egg. Here, a triangle drawn near the equator cannot be moved without distortion to the point, as it would no longer fit the greater curvature of the new position: a triangle drawn near the point cannot be fitted on to the flatter end, and so on.
Thus the method of superposition, such as Euclid employs in Book I. Prop. IV., becomes impossible; figures cannot be freely moved about, indeed, given any figure, we can determine a certain series of possible positions for it on the egg, outside which it becomes impossible. What I assert is, then, that there is a philosophic absurdity in supposing space in general to be of this nature. On the egg we have marked points, such as the two ends; the space formed by its surface is not homogeneous, and if things are moved about in it, it must of itself exercise a distorting effect upon them, quite independently of physical causes; if it did not exercise such an effect, the things could not be moved. Thus such a space would not be homogeneous, but would have marked points, by reference to which bodies would have absolute position, quite independently of any other bodies. Space would no longer be passive, but would exercise a definite effect upon things, and we should have to accommodate ourselves to the notion of marked points in empty space; these points being marked, not by the bodies which occupied them, but by their effects on any bodies which might from time to time occupy them. This want of homogeneity and passivity is, however, absurd; space must, since it is a form of externality, allow only of relative, not of absolute, position, and must be completely homogeneous throughout. To suppose it otherwise, is to give it a thinghood which no form of externality can possibly possess. We must, then, on purely philosophical grounds, admit that a geometrical figure which is possible anywhere is possible everywhere, which is the axiom of Free Mobility.
If we have a measuring rod with which we travel along the line and measure successive infinitesimal arcs, the measuring rod will change its size as we move, so that an arc which appears by the measure to be _ds_ will really be _f(s).ds_, where _s_ is the previously traversed distance. If, on the other hand, we move our line _Op_ slowly through the origin, and measure each piece as it passes through, our measure, it is true, will not alter, but now we have no means of discovering the law by which any element has changed its length in coming to the origin. Hence, until we assume our function _f(p)_, we have no means of determining _p_, for we have just seen that distances from the origin can only be estimated by means of the law _f(p)_. It follows that experience can neither prove nor disprove the constancy of shapes throughout motion, since, if shapes were not constant, we should have to _assume_ a law of their variation before measurement became possible, and therefore measurement could not itself reveal that variation to us[154]. Nevertheless, such an arbitrarily assumed law _does_, at first sight, give a mathematically possible Geometry. The fundamental proposition, that two magnitudes which can be superposed in any one position can be superposed in any other, still holds. For two infinitesimal arcs, whose lengths in the standard position are _ds{1}_ and _ds{2}_, would, in any other position _p_, have lengths _f(p).ds{1}_ and _f(p).ds{2}_, so that their ratio would be unaltered. From this constancy of ratio, as we know through Riemann and Helmholtz, the above proposition follows. Hence all that Geometry requires, it would seem, as a basis for measurement, is an axiom that the alteration of shapes during motion follows a definite known law, such as that assumed above.
To suppose a difference of volume, then, would be to ascribe an effect to mere position, which we saw to be inadmissible while discussing Free Mobility. Except as regards position, they are geometrically indiscernible, and we may call to our aid the Identity of Indiscernibles to establish their agreement in the one remaining geometrical property of volume. This may seem rather a strange principle to use in Mathematics, and for Geometry their equality is, perhaps, best regarded as a definition; but if we demand a philosophical ground for this definition, it is, I believe, only to be found in the Identity of Indiscernibles. We can, without error, make our _definition_ of three-dimensional equality rest on two-dimensional congruence. For since direct comparison as to volume is impossible, we are at liberty to _define_ two volumes as equal, when all their various lines, surfaces, angles and solid angles are congruent, since there remains, in such a case, no _measurable_ difference between the figures composing the two volumes. Of course, as soon as we have established this one case of equality of volumes, the rest of the theory follows; as appears from the ordinary method of integrating volumes, by dividing them into small cubes.
Thus congruence _helps_ to establish three-dimensional equality, though it cannot directly _prove_ such equality; and the same philosophical principle, of the homogeneity of space, by which congruence was proved, comes to our rescue here. But how about right-handed and left-handed screws? Here we can no longer apply the Identity of Indiscernibles, for the two are very well discernible. But as with solids, so here, Free Mobility can help us much. It can enable us, by ordinary measurement, to show that the internal relations of both screws are the same, and that the difference lies only in their relation to other things in space. Knowing these internal relations, we can calculate, by the Geometry which Free Mobility has rendered possible, all the geometrical properties of these screws--radius, pitch, etc.--and can show them to be severally equal in both. But this is all we require. Mediate comparison is possible, though immediate comparison is not. Both can, for instance, be compared with the cylinder on which both would fit, and thus their equality can be proved. A precisely similar proof holds, of course, for the other cases, right and left hands, spherical triangles, etc. On the whole, these cases confirm my argument; for they show, as Kant intended them to show[158], the essential relativity of space.
Such an assumption would still keep them as nearly synchronous as before, and would give an equally possible, though more complex, system of Mechanics; instead of the first law of motion, we should have the following: A particle perseveres in its state of rest, or of rectilinear motion with velocity varying as _f(t)_, except in so far as it is compelled to alter that state by the action of external forces. Such a hypothesis _is_ mathematically possible, but, like the similar one for space, it is excluded logically by the comparative nature of the judgment of quantity, and philosophically by the fact that it involves absolute time, as a determining agent in change, whereas time can never, philosophically, be anything but a passive form, abstracted from change. I have introduced this parallel from time, not as directly bearing on the argument, but as a simpler case which may serve to illustrate my reasoning in the more complex case of space. For since time, in mathematics, is one-dimensional, the mathematical difficulties are simpler than in Geometry; and although nothing accurately corresponds to congruence, there is a very similar mixture of mathematical and philosophical necessity, giving, finally, a thoroughly definite axiom as the basis of time-measurement, corresponding to congruence as the basis of space-measurement[159].
Even if we waive this fundamental objection, however, others remain. To begin with, such judgments of equality are only very rough approximations, and cannot be applied to lines of more than a certain length, if only for the reason that such lines cannot well be seen together. Thus this method can only give us any security in our own immediate neighbourhood, and could in no wise warrant such operations as would be required for the construction of maps &c., much less the measurement of astronomical distances. They might just enable us to say that some lines were longer than others, but they would leave Geometry in a position no better than that of the Hedonical Calculus, in which we depend on a purely subjective measure. So inaccurate, in fact, is such a method acknowledged to be, that the foot-rule is as much a need of daily life as of science. Besides, no one would trust such immediate judgments, but for the fact that the stricter test of congruence to some extent confirms them; if we could not apply this test, we should have no ground for trusting them even as much as we do. Thus we should have, here, no real escape from our absolute dependence upon the axiom of Free Mobility.
Geometry affords intrinsic evidence of the truth of my division of the axiom of dimensions into an _à priori_ and empirical portion. For while the extension of the number of dimensions to four, or to _n_, alters nothing in plane and solid Geometry, but only adds a new branch which interferes in no way with the old, _some_ definite number of dimensions is assumed in all Geometries, nor is it possible to conceive of a Geometry which should be free from this assumption[167]. =160.= Let us, since the point seems of some interest, repeat our proof of the apriority of this axiom from a slightly different point of view. We will begin, this time, from the most abstract conception of space, such as we find in Riemann's dissertation, or in Erdmann's extents. We have here, an ordered manifold, infinitely divisible and allowing of Free Mobility[168]. Free Mobility involves, as we saw, the power of passing continuously from any one point to any other, by any course which may seem pleasant to us; it involves, also, that, in such a course, no changes occur except changes of mere position, _i.e._, positions do not differ from one another in any qualitative way. (This absence of qualitative difference is the distinguishing mark of space as opposed to other manifolds, such as the colour- and tone-systems: in these, every element has a definite qualitative sensational value, whereas in space, the sensational value of a position depends wholly on its spatial relation to our own body, and is thus not intrinsic, but relative.) From the absence of qualitative differences among positions, it follows logically that positions exist only by virtue of other positions; one position differs from another just because they are two, not because of anything intrinsic in either. Position is thus defined simply and solely by relation to other positions. Any position, therefore, is completely defined when, and only when, enough such relations have been given to enable us to determine its relation to any new position, this new position being defined by the same number of relations. Now, in order that such definition may be at all possible, a finite number of relations must suffice. But every such relation constitutes a dimension. Therefore, if Geometry is to be possible, it is _à priori_ necessary that space should have a finite integral number of dimensions.
Now to begin with, two points must, if Geometry is to be possible, have _some_ relation to each other, for we have seen that such relations alone constitute position or localization. But if two points have a relation to each other, this must be an intrinsic relation. For it follows, from the axiom of Free Mobility, that two points, forming a figure congruent with the given pair, can be constructed in any part of space. If this were not possible, we have seen that metrical Geometry could not exist. But both the figures may be regarded as composed of two points and their relation; if the two figures are congruent, therefore, it follows that the relation is quantitatively the same for both figures, since congruence is the test of spatial equality. Hence the two points have a quantitative relation, which is such that they can traverse all space in a combined motion without in any way altering that relation. But in such a general motion, any external relation of the two points, any relation involving other points or figures in space, must be altered[170]. Hence the relation between the two points, being unaltered, must be an intrinsic relation, a relation involving no other point or figure in space; and this intrinsic relation we call distance[171]. =164.= It might be objected, to the above argument, that it involves a _petitio principii_. For it has been assumed that the two points and their relation form a figure, to which other figures can be congruent. Now if two points have no intrinsic relation, it would seem that they cannot form such a figure. The argument, therefore, apparently assumes what it had to prove. Why, it may be asked, should not three points be required, before we obtain any relation, which Free Mobility allows us to construct afresh in other parts of space?
The answer to this, as to the corresponding question in the first section of this chapter, lies, I think, in the passivity of space, or the mutual independence of its parts. For it follows, from this independence, that any figure, or any assemblage of points, may be discussed without reference to other figures or points. This principle is the basis of infinite divisibility, of the use of quantity in Geometry, and of all possibility of isolating particular figures for discussion. It follows that two points cannot be dependent, as to their relation, on any other points or figures, for if they were so dependent, we should have to suppose some action of such points or figures on the two points considered, which would contradict the mutual independence of different positions. To illustrate by an example: the relation of two given points does not depend on the other points of the straight line on which the given points lie. For only through their relation, _i.e._ through the straight line which they determine, can the other points of the straight line be known to have any peculiar connection with the given pair.
It is true that, if we exclude infinitesimal radii for the measuring circles, the lengths of the infinitesimal arcs would be determinate, even if the circles varied, but that is only because all the small circular arcs through two consecutive points coincide with the straight line through those two points. Thus, even with the help of the arbitrary restriction to a finite radius, all that happens is that we are brought back to the straight line. If, to mend matters, we take three consecutive points of our curve, and reckon distance by the arc of the circle of curvature, the notion of distance loses its fundamental property of being a relation between _two_ points. For two consecutive points of the arc could not then be said to have any corresponding distance apart--three points would be necessary before the notion of distance became applicable. Thus the circle is not a possible basis for measurement, and similar objections apply, of course, with increased force, to any other curve. All this argument is designed to show, in detail, the logical impossibility of measuring distance by any curve not completely defined by the two points whose distance apart is required. If in the above we had taken distance as measured by circles of _given radius_, we should have introduced into its definition a relation to other points besides the two whose distance was to be measured, which we saw to be a logical fallacy. Moreover, how are we to know that all the circles have equal radii, until we have an independent measure of distance?
"Just as, in the field of quantity, the relation between two numbers is another number, so in the field of space the relations are facts of the same order with the facts they relate.... When we speak of the relation of direction of two points towards each other, we mean simply the sensation of the line that joins the two points together. _The line is the relation...._ The relation of position between the top and bottom points of a vertical line is that line, and nothing else." If I had been willing to use this doctrine at the beginning, I might have avoided all discussion. A unique relation between two points _must_ in this case, involve a unique line between them. But it seemed better to avoid a doctrine not universally accepted, the more so as I was approaching the question from the logical, not the psychological, side. After disposing of the objections, however, it is interesting to find this confirmation of the above theory from so different a standpoint. Indeed, I believe James's doctrine could be proved to be a logical necessity, as well as a psychological fact. For what sort of thing can a spatial relation between two distinct points be? It must be something spatial, and it must, since points are wholly constituted by their relations, be something at least as real and tangible as the points it relates. There seems nothing which can satisfy these requirements, except a line joining them. Hence, once more, a unique relation must involve a unique line. That is, linear magnitude is logically impossible, unless space allows of curves uniquely determined by any two of their points.
We saw, in section A § 119, that the figure formed by two points is projectively indistinguishable from that formed by any two other points in the same straight line; the figure, in both cases, is, from the projective standpoint, simply the straight line on which the two points lie. The difference of relation, in the two cases, is not qualitative, since projective Geometry cannot deal with it; nevertheless, there is some difference of relation. For instance, if one point be kept fixed, while the other moves, there is obviously some change of relation. This change, since all parts of the straight line are qualitatively alike, must be a change of quantity. If two points, therefore, determine a unique figure, there must exist, for the distinction between the various other points of this figure, a unique quantitative relation between the two determining points, and therefore, since these points are arbitrary, between only two points. This relation is _distance_, with which our argument began, and to which it at least returns.
Distance and the straight line, as relations uniquely determined by two points, are thus _à priori_ necessary to metrical Geometry. But further, they are properties which must belong to any form of externality. Since their necessity for Geometry was deduced from homogeneity and the relativity of position, and since these are necessary properties of any form of externality, the same argument proves both conclusions. We thus obtain, as in the case of Free Mobility, a double apriority: The axiom of Distance, and its implication, the axiom of the Straight Line, are, on the one hand, presupposed in the possibility of spatial magnitude, and cannot, therefore, be contradicted by any experience resulting from the measurement of space; while they are consequences, on the other hand, of the necessary properties of any form of externality which is to render possible experience of an external world.
This _à priori_ argument, however, though I believe it to be quite sound, is not likely to carry conviction to any one persuaded of the opposite. Let us, therefore, examine metrical coordinate systems in detail, and show, in each case, their dependence on the straight line. We have already seen that the notion of distance is impossible without the straight line. We cannot, therefore, define our coordinates in any of the ordinary ways, as the distances from three planes, lines, points, spheres, or what not. Polar coordinates are impossible, since,--waiving the straightness of the radius vector--the length of the radius vector becomes unmeaning. Triangular coordinates involve not only angles, which must in the limit be rectilinear, but straight lines, or at any rate some well-defined curves. Now curves can only be metrically defined in two ways: _Either_ by relation to the straight line, as, _e.g._, by the curvature at any point, _or_ by purely analytical equations, which presuppose an intelligible system of metrical coordinates. What methods remain for assigning these arbitrary values to different points? Nay, how are we to get any estimate of the difference--to avoid the more special notion of distance--between two points? The very notion of a point has become illusory. When we have a coordinate system, we may define a point by its three coordinates; in the absence of such a system, we may define the notion of point _in general_ as the intersection of three surfaces or of two curves. Here we take surfaces and curves as notions which intuition makes plain, but if we wish them to give us a precise numerical definition of _particular_ points, we must specify the kind of surface or curve to be used. Now this, as we have seen, is only possible when we presuppose either the straight line, or a coordinate system. It follows that every coordinate system presupposes the straight line, and is logically impossible without it.
Ideas discussed in this chapter
Text from Project Gutenberg, public domain in the United States.