ABSTRACT
Underlying all measurement is the notion that an object remains constant in size throughout any change in position. The movement of an object appears as a congruent transformation of spatial shapes; the transformation is congruent because the length AB of an object which is moved remains identical with itself. Topological and projective notions, like those of affinity and similarity, are not enough to bring about this conservation. When children evolve topological nesting series by reuniting parts and reforming the original whole (A + A? = B, etc.), they realize that a collection of elements remains the same collection even after its parts have been re-arranged. An example would be changing the segments of a length of string or elastic. These segments always make up the same object, although it can be tied in a variety of knots, or stretched, or contracted (cf. C.C.S., ch. V). But this conservation of wholes does not imply that of length or distance. The perceptual and intuitive universe of little children is subject to constant deformation and is thus much closer to the elastic and contractile space of topology than it is to the invariants of Euclidean space. Later, the coordination of perspectives from different points of view enables the subject to reconstruct the order of parts in any direction, e.g. from left to right or from right to left. However, even now, the apparent length of these parts varies continually in the process. How, then, do children attain the concept of invariance of size, which is essential to metric space?
