It is supposed that we have a complete theory of points, straight lines, planes, and the order of points on planes--in fact, a complete theory of non-metrical geometry.

We then enquire about congruence and lay down the set of conditions--or axioms as they are called--which this relation satisfies.

It has then been proved that there are alternative relations which satisfy these conditions equally well and that there is nothing intrinsic in the theory of space to lead us to adopt any one of these relations in preference to any other as the relation of congruence which we adopt.

In other words there are alternative metrical geometries which all exist by an equal right so far as the intrinsic theory of space is concerned.
Poincare, the great French mathematician, held that our actual choice among these geometries is guided purely by convention, and that the effect of a change of choice would be simply to alter our expression of the physical laws of nature.

By 'convention' I understand Poincare to mean that there is nothing inherent in nature itself giving any peculiar _role_ to one of these congruence relations, and that the choice of one particular relation is guided by the volitions of the mind at the other end of the sense-awareness.