Conventionality of Topological Structure
Conventionality of Topological Structure (Sklar, pp. 312-317)
Sklar's discussion of "anomalous topological features" in a curved spacetime has its origin in Hans Reichenbach's Philosophy of Space and Time (1958). In the section 12 of this book, Reichenbach presented a figure such as our Figure 1.
This is a torus (like a doughnut) on which several circles are located. Unlike on a Euclidean plane, on this surface it is impossible to determine which circle is inside of which, since if you go from the black circle to the blue, to the red, and to the grey, you can continuously come back to the initial black, and likewise if you go from the black to the grey, to the red, and to the blue, you can also come back to the black.
Reichenbach then invites us to consider a 3-dimensional case (spheres instead of circles).
Figure 8 [replaced by our Figure 2] is to be conceived three-dimensionally, the circles being cross-sections of spherical shells in the plane of the drawing. A man is climbing about on the huge spherical surface 1; by measurements with rigid rods he recognizes it as a spherical shell, i.e. he finds the geometry of the surface of a sphere. Since the third dimension is at his disposal, he goes to spherical shell 2. Does the second shell lie inside the first one, or does it enclose the first shell? He can answer this question by measuring 2. Assume that he finds 2 to be the smaller surface; he will say that 2 is situated inside of 1. He goes now to 3 and finds that 3 is as large as 1.
How is this possible? Should 3 not be smaller than 2? ...
He goes on to the next shell and finds that 4 is larger than 3, and thus larger than 1. ... 5 he finds to be as large as 3 and 1.
But here he makes a strange observation. He finds that in 5 everything is familiar to him; he even recognizes his own room which was built into shell 1 at a certain point. This correspondence manifests itself in every detail; ... He is quite dumbfounded since he is certain that he is separated from surface 1 by the intervening shells. He must assume that two identical worlds exist, and that every event on surface 1 happens in an identical manner on surface 5. (Reichenbach 1958, 63-64)
Reichenbach's argument is that our interpretation of these facts in the Euclidean world demands a causal anomaly, the "preestablished harmony" between sphere 1 and 5. But if we interpret these facts in terms of a non-Eulidean geometry (a "torus space"), the preestablished harmony is unnecessary, and the sphere 1 and sphere 5 become identical. Now which interpretation should we choose?
Sklar repeats a similar argument in terms of a closed time axis (the same phenomena repeat periodically). And Sklar's point is that here is a room for conventionality of topological structure of the world.
Last modified March 27, 2003. (c) Soshichi Uchii