It remains to say a word on the subject of dimensions. We have not so far said anything explicit on this subject, though our original introduction of co-punctuality as a five-term relation could only prove satisfactory in a four-dimensional manifold. The most suitable definition of dimensions from our point of view is that of Poincaré, which is inductive. He defines a space as one-dimensional if, given any two points , , there is an isolated set of points such that no connected part of -not- contains both and . And he defines a space as -dimensional if, given any two points , , there is an ()-dimensional set of points such that no connected part of -not- contains both and . Using this definition, or any other which is purely topological, we set up the axiom that our topological space-time is to be four-dimensional.[67] This completes the material required for the topological treatment of space-time. FOOTNOTES: [66] For a geometry based on "neighbourhood," see Hausdorff, Grundzüge der Mengenlehre (Leipzig, 1914), chaps, VII. and VIII. [67] For an account of the modern theory of dimensions, see Karl Menger, Bericht über die Dimensionstheorie, Jahresbericht der deutschen Mathematiker-Vereinigung, 35, pp. 113-150 (1926). [Pg 313] CHAPTER XXX CAUSAL LINES THE notion of causality has been greatly modified by the substitution of space-time for space and time. We may define causality in its broadest sense as embracing all laws which connect events at different times, or, to adapt our phraseology to modern needs, events the intervals between which are time-like. Now owing to the fact that the formula for is formally the same for time-like and for space-like intervals, there is no longer the difference that formerly existed between causal and geometrical relations. Geodesics are geometrical, but they are also the paths of material particles. It is hardly correct to say that a particle moves in a geodesic; it is more correct to say that a particle is a geodesic (though not all geodesics are particles). To say that a particle moves in a geodesic is to use language appropriate to the conception of a space which persists through time, involving the notion of a position which may be occupied either at one time or at another. We think, for example, that it is possible to move from to or from to ; but such a view is incompatible with the theory of space-time. According to that theory, every position of a body has a date, and it is impossible to occupy the same position at another date, since the date is one of the co-ordinates of the position. When we travel from to , the date is continually advancing; the return journey, having different dates, does not cover the same route. Thus geometry and causation become inextricably intertwined.