Dimensions and their poles: A metric and topological theory of opposites.

We explored the nature of 37 spatial dimensions in Italian, such as LUNGO-CORTO (LONG-SHORT), INIZIO-FINE (BEGINNING-END), and CONVERGENTE-DIVERGENTE (CONVERGENT-DIVERGENT). In Study 1 we investigated their metric structure.We asked: (1) Are the extensions of the two poles (P1 and P2) the same? (2) What proportion of each dimension can be said to be neither P1 nor P2? and (3) Is the extension of P1 that can be called neither P1 nor P2, the same as the extension of P2 that can be called neither P1 nor P2? In Study 2 we investigated the topological structure of the dimensions. We asked: (1) Are the poles, points or ranges? (2) Do intermediates (neither P1 nor P2) exist? and (3) If they do, are they points or ranges? Two metric properties explained a considerable proportion of the variation in the responses in the first task: (1) the asymmetry of the extension of the two poles and (2) the extension of the ‘‘neither-nor’’ region between them. The results of the topological task further refined the two-dimensional structure obtained in Study 1 to produce a map of spatial opposites. Our methods and the resulting maps provide a point of departure from which two questions can be investigated: (1) If these methods were used in other languages to study spatial opposites, to what extent would they produce similar maps of opposites? and (2) If these methods were applied to nonspatial opposites and maps analogous to our spatial mapswere generated, would any dense regions in the nonspatial maps coincide with sparse regions in the spatial maps? We discuss the potential importance of these questions.