An essay on the foundations of geometry

by Bertrand Russell

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.

133. The above argument, it may be urged, has overlooked a possibility. It has used a transcendental argument, so an opponent may contend, without sufficiently proving that knowledge about externality must be possible without reference to the matters external to each other. The definition of a position may be impossible, so long as we neglect the matter which fills the form, but may become possible when this matter is taken into account. Such an objection can, I think, be successfully met, by a reference to the passivity and homogeneity of our form. For any dependence of the definition of a position on the particular matter filling that position, would involve some kind of interaction between the matter and its position, some effect of the diverse content on the homogeneous form. But since the form is totally destitute of thinghood, perfectly impassive, and perfectly void of differences between its parts, any such effect is inconceivable. An effect on a position would have to alter it in some way, but how could it be altered? It has no qualities except those which make it the position it is, as opposed to other positions; it cannot change, therefore, without becoming a different position. But such a change contradicts the law of identity. Hence it is not the position which has changed, but the content which has moved in the form. Thus it must be possible, if knowledge of our form can be obtained at all, to obtain this knowledge in logical independence of the particular matter which fills it. The above argument, therefore, granted the possibility of knowledge in the department in question, shows the necessity of a finite integral number of dimensions.

134. Let us repeat our original argument in the light of this elucidation. A position is completely defined when, and only when, enough relations are known to enable us to determine its relation to any fresh known position. Only by relations within the form of externality, as we have just seen, and never by relations which involve a reference to the particular matter filling the form, can such a definition be effected. But the possibility of such a definition follows from the Law of Excluded Middle, when this law is interpreted to mean, as Bosanquet makes it mean, that "Reality ... is a system of reciprocally determinate parts[141]." For this implies that, given the relations of a part A to other parts B, C ..., a sufficient wealth of such relations throws light on the relations of B to C, etc. If this were not the case, the parts A, B, C ... could not be said to form such a system; for in such a system, to define A is to define, at the same time, all the other members, and to give an adjective to A, is to give an adjective to B and C. But the relations between positions are, when we restore the matter from which the positions were abstracted, relations between the things occupying those positions, and these relations, we have seen, can be studied without reference to the particular nature, in other respects, of the related things. It follows that, when we apply the general principle of systematic unity to these relations in particular, we find these relations to be dependent on each other, since they are not dependent, for their definition, on anything else. This gives the axiom of dimensions, in the above general form, as the result, on our abstract geometrical level, of the relativity of position and the law of excluded middle.

135. Before proceeding further, it is necessary to discuss the important special case where a form of externality has only one dimension. Of the two such forms, given in experience, one, namely time, presents an instance of this special case. But it may be shown, I think, that the function, in constituting the possibility of experience, which we demand of such forms, could not be accomplished by a one-dimensional form alone. For in a one-dimensional form, the various contents may be arranged in a series, and cannot, without interpenetration, change the order of contents in the series. But interpenetration is impossible, since a form of externality is the mere expression of diversity among things, from which it follows that things cannot occupy the same position in a form, unless there is another form by which to differentiate them. For without externality, there is no diversity[142]. Thus two bodies may occupy the same space, but only at different times: two things may exist simultaneously, but only at different places. A form of one dimension, therefore, could not, by itself, allow that change of the relations of externality, by which alone a varied world of interrelated things can be brought into consciousness. In a one-dimensional space, for example, only a single object, which must appear as a point, or two objects at most, one in front and one behind, could ever be perceived. Thus two or more dimensions seem an essential condition of anything worth calling an experience of interrelated things.

136. It may be objected, to this argument, that its validity depends upon the assumption that the change of a relation of externality must be continuous. Both to make and to meet this objection, in a manner which shall not imply time, seems almost impossible. For we cannot speak of change, whether continuous or discrete, without imagining time. Let us, therefore, allow time to be known, and discuss whether the temporal change, in any other form of externality, is necessarily continuous[143]. We must reply, I think, that continuity is necessary. The change of relation, in our non-temporal form, may be safely described as motion, and the law of Causality—since we have already assumed time—may be applied to this motion. It then follows that discrete motion would involve a finite effect from an infinitesimal cause, for a cause acting only for a moment of time would be infinitesimal. It involves, also, a validity in the point of time, whereas what is valid in any form of externality is not, as we have already seen, the infinitesimal and self-contradictory element resulting from infinite division, but the finite relation which mathematics analyzes into vanishing elements. Hence change must be continuous, and the possibility of serial arrangement holds good.

In a one-dimensional form other than time, the same argument must hold. For something analogous to Causality would be necessary to experience, and the relativity of the form would still necessarily hold. Hence, since only these two properties of time have been assumed, the above contention would remain valid of any second form whose relations were correlated with those of the first, as the analogue of Causality would require them to be.

137. The next step in the argument, which assumes two or more dimensions, is concerned with the general analogues of straight lines and planes, i.e. with figures—which may be regarded either as relations between positions or as series of positions—uniquely determined by two or by three positions. If this step can be successfully taken, our deduction of the above projective axioms will be complete, and descriptive Geometry will be established as the abstract à priori doctrine of forms of externality.

To prove this contention, consider of what nature the relations can be by which positions are defined. We have seen already that our form is purely relational and infinitely divisible, and that positions (points) are the self-contradictory outcome of the search for something other than relations. What we really mean, therefore, by the relations defining a position, is, when we undo our previous abstraction, the relations of externality by which some thing is related to other things. But how, when we remain in the abstract form, must such relations appear?

138. We have to prove that two positions must have a relation independent of any reference to other positions. To prove this, let us recur to what was said, in connection with dimensions, as to the passivity and homogeneity of our form. Since positions are defined only by relations, there must be relations, within the form, between positions. But if there are such relations, there must be a relation which is intrinsic to two positions. For to suppose the contrary, is to attribute an interaction or causal connection, of some kind, between those two positions and other positions—a supposition which the perfect homogeneity of our form renders absurd, since all positions are qualitatively similar, and cannot be changed without losing their identity. We may put this argument thus: since positions are only defined by their relations, such definition could never begin, unless it began with a relation between only two positions. For suppose three positions A, B, C were necessary, and gave rise to the relation abc between the three. Then there would remain no means of defining the different pairs BC, CA, AB, since the only relation defining them would be one common to all three pairs. Nothing would be gained, in this case, by reference to fresh points, for it follows, from the homogeneity and passivity of the form, that these fresh points could not affect the internal relations of our triad, which relations, if they can give definiteness at all, must give it without the aid of external reference. Two positions must, therefore, if definition is to be possible, have some relation which they by themselves suffice to define. Precisely the same argument applies to three positions, or to four; the argument loses its scope only when we have exhausted the dimensions of the form considered. Thus, in three dimensions, five positions have no fresh relation, not deducible from those already known, for by the definition of dimensions, all the relations involved can be deduced from those of the fourth point to the first three, together with those of the fifth to the first three.