An essay on the foundations of geometry · Bertrand Russell
THE AXIOMS OF PROJECTIVE GEOMETRY.
"_To cut by a fixed straight line s_ (a _transversal_) a figure _αβγδ_ ... composed of planes, is to construct the points _sα_, _sβ_, _sγ_.... In this way a new figure is obtained, composed of points all lying on the fixed transversal _s_. "If a figure is composed of straight lines _a_, _b_, _c_ ... which all pass through a fixed point or _centre S_, it can be _projected_ from a straight line or _axis s_ passing through _S_; the result is a figure composed of planes _sa_, _sb_, _sc_.... "If a figure is composed of straight lines _a_, _b_, _c_ ... all lying in a fixed plane, it may be cut by a straight line (transversal) _s_ lying in the same plane; the figure which results is formed by the points _sa_, _sb_, _sc_...." =110.= The successive application, to any figure, of two reciprocal operations of projection and section, is regarded as producing a figure protectively indistinguishable from the first, provided only that the dimensions of the original figure were the same as those of the resulting figure, that, for example, if the second operation be section by a plane, the original figure shall have been a plane figure. The figures obtained from a given figure, by projection or section alone, are related to that figure by the principle of duality, of which we shall have to speak later on.
Two sets of four points each are defined as having the same anharmonic ratio, when (1) each set of four lies in one straight line, and (2) corresponding points of different sets lie two by two on four straight lines through a single point, or when both sets have this relation to any third set[123]. And reciprocally: Two sets of four straight lines are defined as having the same anharmonic ratio when (1) each set of four passes through a single point, and (2) corresponding lines of different sets pass, two by two, through four points in one straight line, or when both sets have this relation to any third set. Two sets of points or of lines, which have the same anharmonic ratio, are treated by projective Geometry as equivalent: this qualitative equivalence replaces the quantitative equality of metrical Geometry, and is obviously included, by its definition, in the above account of projective transformations in general.
To prove this, let _DRQ_ meet _OA_ in _T_, and draw _AR_, meeting _OD_ in _S_. Then a projective transformation of _A_, _B_, _C_, _D_ from _R_ on to _OD_ gives the points _S_, _P_, _O_, _D_, which, projected from _A_ on to _DQ_, give _R_, _Q_, _T_, _D_. But these again, projected from _O_ on to _ABD_, give _C_, _B_, _A_, _D_. Hence _A_, _B_, _C_, _D_ can be projectively transformed into _C_, _B_, _A_, _D_, and therefore form a harmonic range. From this point, the proof that the construction is unique and general follows simply[125]. The introduction of numbers, by this construction, offers no difficulties of principle--except, indeed, those which always attend the application of number to continua--and may be studied satisfactorily in Klein's Nicht-Euklid (I. p. 337 ff.). The principle of it is, to assign the numbers 0, 1, ∞ to _A_, _B_, _D_ and therefore the number 2 to _C_, in order that the differences _AB_, _AC_, _AD_ may be in harmonic progression. By taking _B_, _C_, _D_ as a new triad corresponding to _A_, _B_, _D_, we find a point harmonic to _B_ with respect to _C_, _D_ and assign to it the number 3, and so on. In this way, we can obtain any number of points, and we are sure of having no number and no point twice over, so that our coordinates have the essential property of a unique correspondence with the points they denote, and _vice versa_.
This duality suggests that any definition of points must be effected by means of the straight line, and any definition of the straight line must be effected by means of points. When we take the third dimension into account, it is true, the duality is no longer so simple; we have now to take account also of the plane, but this only introduces a circle of three terms, which is scarcely preferable to a circle of two terms. We now say: Three points, or a line and a point, determine a plane: but conversely, three planes, or a line and plane, determine a point. We may regard the straight line as a relation between two of its points, but we may also regard the point as a relation between two straight lines through it. We may regard the plane as a relation between three points, or between a point and a line, but we may also regard the point as a relation between three planes, or between a line and a plane, which meet in it. =116.= How are we to get outside this circle? The fact is that, in pure Geometry, we cannot get outside it. For space, as we shall see more fully hereafter, is nothing but relations; if, therefore, we take any spatial figure, and seek for the terms between which it is a relation, we are compelled, in Geometry, to seek these terms within space, since we have nowhere else to seek them, but we are doomed, since anything purely spatial is a mere relation, to find our terms melting away as we grasp them.
Thus the relativity of space, while it is the essence of the principle of duality, at the same time renders impossible the expression of that principle, or of any other principle of pure Geometry, in a manner which shall be free from contradictions. Nevertheless, if we are to advance at all with our analysis of geometrical reasoning and with our definitions of lines and points, we must, for a while, ignore this contradiction; we must argue as though it did not exist, so as to free our science from any contradictions which are not inevitable. =117.= In accordance with this procedure, then, let us define our points as the terms of spatial relations, regarding whatever is not a point as a relation between points. What, on this view, must our points be taken to be? Obviously, if extension is mere relativity, they must be taken to contain no extension; but if they are to supply the terms for spatial relations, _e.g._ for straight lines, these relations must exhibit them as the terms of the figures they relate. In other words, since what can really be taken, without contradiction, as the term of a spatial relation, is unextended, we must take, as the term to be used in Geometry, where we cannot go outside space, the least spatial thing which Geometry can deal with, the thing which, though _in_ space, _contains_ no space; and this thing we define as the point[127].
Neglecting, then, the fundamental contradiction in this definition, the rest of our definitions follow without difficulty. The straight line is the relation between two points, and the plane is the relation between three. These definitions will be argued and defended at length in section B of this Chapter[128], where we can discuss at the same time the alternative metrical definitions; for our present purpose, it is sufficient to observe that projective Geometry, from the first, regards the straight line as determined by two points, and the plane as determined by three, from which it follows, if we take points as possible terms for spatial relations, that the straight line and the plane may be regarded as relations between two and three points respectively. If we agree on these definitions, we can proceed to discuss the fundamental principle of projective Geometry, and to analyse the axioms implicated in its truth. =118.= Projective Geometry, we have seen, does not deal with quantity, and therefore recognizes no difference where the difference is purely quantitative. Now quantitative comparison depends on a recognized identity of quality; the recognition of qualitative identity, therefore, is logically prior to quantity, and presupposed by every judgment of quantity. Hence all figures, whose differences can be exhaustively described by quantity, _i.e._ by pure measurement, must have an identity of quality, and this must be recognizable without appeal to quantity. It follows that, by defining the word quality in geometrical matters, we shall discover what sets of figures are projectively indiscernible. If our definition is correct, it ought to yield the general projective principle with which we set out.
This statement of the axioms is not intended to have any exclusive precision: other statements equally valid could easily be made. For all these axioms, as we shall see hereafter, are philosophically interdependent, and may, therefore, be enunciated in many ways. The above statement, however, includes, if I am not mistaken, everything essential to projective Geometry, and everything required to prove the principle of projective transformation. Before discussing the apriority of these axioms, let us once more briefly recapitulate the ends which they are intended to attain. =123.= From the exclusively mathematical standpoint, as we have seen, projective Geometry discusses only what figures can be obtained from each other by projective transformations, _i.e._ by the operations of projection and section. These operations, in all their forms, presuppose the point, straight line, and plane[131], whose necessity for projective Geometry, from the purely mathematical point of view, is thus self-evident from the start. But philosophically, projective Geometry has, as we saw, a wider aim. This wider aim, which gives, to the investigation of projectively equivalent figures, its chief importance, consists in the determination of qualitative spatial similarity, in the determination, that is, of all the figures which, when any one figure is given, can be distinguished from the given figure, so long as quantity is excluded, only by the mere fact that they are external to it.
From this point of view, the controversy between Kantians and anti-Kantians becomes wholly irrelevant, since the distinction between pure and mixed mathematics does not lie in the distinction between the subjective and the objective, but between the purely intellectual on the one hand, and everything else on the other. Now Kant had contended, with great emphasis, that space was not an intellectual construction, but a subjective intuition. Geometry, therefore, with Grassmann's distinction, belongs to mixed mathematics as much on Kant's view as on that of his opponents. And Grassmann's distinction, I contend, is the more important for Epistemology, and the one to be adopted in distinguishing the _à priori_ from the empirical. For what is merely intuitional can change, without upsetting the laws of thought, without making knowledge formally impossible: but what is purely intellectual cannot change, unless the laws of thought should change, and all our knowledge simultaneously collapse. I shall therefore follow Grassmann's distinction in constructing an _à priori_ and purely conceptual form of externality.
The same argument may also be stated as follows: If we abstract the conception of externality, and endeavour to deal with it _per se_, it is evident that we must obtain an object alike destitute of elements and of totality. For we have abstracted from the diverse matter which filled our form, while any element, or any whole, would retain some of the qualities of a matter. Either an element or a whole, in fact, would have to be a thing not external to itself, and would thus contain something not pure externality. Hence arise infinite divisibility, with the self-contradictory notion of the point, in the search for elements, and unbounded extension, with the contradiction of an infinite regress or a vicious circle, in the search for a completed whole. Thus again, our form contains neither elements nor totality, but only endless relations--the terms of these relations being excluded by our abstraction from the matter which fills our form. =130.= In like manner we can deduce the homogeneity of our form. The diversity of content, which was possible only within the form of externality, has been abstracted from, leaving nothing but the bare possibility of diversity, the bare principle of differentiation, itself uniform and undifferentiated. For if diversity presupposes such a form, the form cannot, unless it were contained in a fresh form, be itself diverse or differentiated.
Or we may deduce the same property from the relativity of position. For any quality in one position, by which it was marked out from another, would be necessarily more or less intrinsic, and would contradict the pure relativity. Hence all positions are qualitatively alike, _i.e._ the form is homogeneous throughout. =131.= From what has been said of homogeneity and relativity, follows one of the strangest properties of a form of externality. This property is, that the relation of externality between any two things is infinitely divisible, and may be regarded, consequently, as made up of an infinite number of the would-be elements of our form, or again as the sum of two relations of externality[137]. To speak of dividing or adding relations may well sound absurd--indeed it reveals the impropriety of the word relation in this connexion. It is difficult, however, to find an expression which shall be less improper. The fact seems to be, that externality is not so much a relation as bare relativity, or the bare possibility of a relation. On this subject, I shall enlarge in Chapter IV.[138] At this point it is only important to realize, what the subsequent argument will assume, that the relation--if we may so call it--of externality between two or more things must, since our form is homogeneous, be capable of continuous alteration, and must, since our infinitely divisible form is constituted by such relations, be capable of infinite division. But the result of infinite division is defined as the element of our form. (Our form has no elements, but we have to imagine elements in order to reason about it, as will be shown more fully in Chapter IV.) Hence it follows, that every relation of externality may be regarded, for scientific purposes, as an infinite congeries of elements, though philosophically, the relations alone are valid, and the elements are a self-contradictory result of hypostatizing the form of externality. This way of regarding relations of externality is important in understanding the meaning of such ideas as three or four collinear points.
As this point is difficult and important, I will repeat, in somewhat greater detail, the explanation of the manner in which straight lines and planes come to be regarded as congeries of points. From the strictly projective standpoint, though all other figures _are_ merely a collection of any required number of points, lines or planes, given by some projective construction, straight lines and planes themselves are given integrally, and are not to be considered as divisible or composed of parts. To say that a point lies on a straight line means, for projective Geometry proper, that the straight line is a relation between this and some other point. Here the points concerned, if our statement is to be freed from contradictions, must be regarded, if I may use such an expression, as _real_ points--_i.e._ as unextended material centres[139]. Straight lines and planes are then relations between these material atoms. They are relations, however, which may undergo a metrical alteration while remaining projectively unchanged. When the projective relation between the two points _A_, _B_ is the same as that between the two points _A_, _C_, while the metrical relation (distance) is different, the three points _A_, _B_, _C_ are said to be collinear.
Now the metrical manner of regarding spatial figures demands that they should be hypostatized, and no longer regarded as mere relations. For when we regard a quantity as extensive, _i.e._ as divisible into parts, we necessarily regard it as more than a mere relation or adjective, since no mere relation or adjective can be divided. For quantitative treatment, therefore, spatial relations must be hypostatized[140]. When this is done, we obtain, as we saw above, a homogeneous and infinitely divisible form of externality. We find now that distance, for example, may be continuously altered without changing the straight line on which it is measured. We thus obtain, on the straight line in question, a continuous series of points, which, since it is continuous, we regard as constituting our straight line. It is thus solely from the hypostatizing of relations, which metrical Geometry requires, that the view of straight lines and planes as _composed_ of points arises, and it is from this hypostatizing that the difficulties of metrical Geometry spring. =132.= The next step, in defining a form of externality, is obtained from the idea of _dimensions_. Positions, we have seen, are defined solely by their relations to other positions. But in order that such definition may be possible, a finite number of relations must suffice, since infinite numbers are philosophically inadmissible. A position must be definable, therefore, if knowledge of our form is to be possible at all, by some finite integral number of relations to other positions. Every relation thus necessary for definition we call a dimension. Hence we obtain the proposition: _Any form of externality must have a finite integral number of dimensions_.
We may give the argument a more concrete, and perhaps a more convincing shape, by considering the matter arranged in our form. If two things are mutually external, they must since they belong to the same world, have some relation of externality; there is, therefore, a relation of externality between two things. But since our form is homogeneous, the same relation of externality may subsist in other parts of the form, _i.e._ while the two things considered alter their relations of externality to other things. The relation of externality between two things is, therefore, independent of other things. Hence, when we return to the abstract language of the form, two positions have a relation determined by those two positions alone, and independent of other positions. Precisely the same argument applies to the relations of three positions, and in each case the relation must appear in the form as not a mere inference from the positions it relates. For relations, as we have seen, actually constitute a form of externality, and are not mere inferences from terms, which are nowhere to be found in the form[144]. To sum up: Since position is relative, two positions must have _some_ relation to each other; and since our form of externality is homogeneous, this relation can be kept unchanged while the two positions change their relations to other positions. Hence their relation is intrinsic, and independent of other positions. Since the form is a mere complex of relations, the relation in question must, if the form is sensuous or intuitive, be itself sensuous or intuitive, and not a mere inference. In this case, a unique relation must be a unique figure--in spatial terms, the straight line joining the two points.
Ideas discussed in this chapter
Text from Project Gutenberg, public domain in the United States.