Principles of Topological Psychology

In our figures we have used so far a plane, that is a two- dimensional space, for the representation of psychological fields. One can raise the question: is it correct to use such a manifold for this representation? In other words, how many dimensions has the life space?

Mathematics has only within recent years found a way to treat problems of dimension satisfactorily. Differences in dimension are not differences in size or in potency of the space.¹ One can coordinate one to one the set of points of a line to the points of a limited two-dimensional region or of a three- dimensional body. In considering how many dimensions one ought to attribute to the life space, one therefore does not have to take into account the purely quantitative question of the space "available" in the representation.

Mathematics shows that dimension is a property of the "inner structure" of the space, a property which is closely connected with topological characteristics. It is characteristic of a two-dimensional space, for instance a plane, that within it there is no possibility of connecting each of five or more points with each other such that the connecting lines do not intersect. Further, as we have said, in a two-dimensional space it is impossible to connect a point within a circular area with a point outside of it, without intersecting the boundary of the area.

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