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Imagine that we proceed along a line through the middle of a disk that is divided into two precisely symmetrical segments, one of which is red, the other green, and that we pass continuously from the red to the green segment. What happens as we pass the boundary between the two? Do we pass through a last point p ₁ that is red and a first point p ₂ that is green? Clearly not, given the density of every continuum; for then we should have to admit an indefinite number of further points between p ₁ and p ₂ which would somehow have no colour. To acknowledge one of p ₁ and p ₂ but not the other, however, as is dictated by the Dedekindian treatment of the continuum, would be to countenance a peculiar privileging of one of the two segments over the other, and an unmotivated asymmetry of this sort we can surely reject as a contravention of the principle of sufficient reason. Perhaps, then, the line becomes colourless at the point where it crosses the segmentary divide, so that the red and green segments would be analogous, topologically, to open regions. One might seek support for this idea by reflecting that extensionless points are not in any case the sorts of things that can be coloured, since colour properly applies only to what is spatially extended. Imagine, however, a perfectly homogeneous red surface. Are the points and lines within the interior of this surface not then also red?

DOI: 10.1007/978-94-015-9446-2_13

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