The Fabric of Space: Intrinsic vs. Extrinsic Distance Relations

tart with ordinary Euclidean space. Be a realist about points, and about S distances between points. I ask: how are the points interwoven to form the fabric of space? Are there direct ties only between “neighboring points,” so that points at a distance are connected only indirectly through series of such direct ties? Or are there also direct ties between distant points, so that the fabric is reinforced, as it were, by irreducibly global spatial relations? To fix ideas, roughly, try the following thought experiment. Take a scissors and cut along some plane in space, severing the points just to one side of the plane from the points just to the other. Does space thereby fall into two disconnected pieces, so that points on one side now stand at no spatial distance from points on the other? Or does space, being reinforced, retain its shape? If space is maximally reinforced by direct ties of distance, then a distance relation, such as being rwentyfeetfrorn, is intrinsic to the points that stand in it. Whether or not the relation holds depends solely upon the intrinsic nature of the two points, and of the composite of the points. On the other hand, if space is not maximally reinforced by direct ties of distance, then a distance relation will not in general be intrinsic to the points that stand in it, and its holding may depend in part upon features of the surrounding space. What features? To fix ideas, roughly, try the following thought experiment. Start with two points, say, twenty feet apart, and remove some of the space directly between the two points. (I do not mean just the matter or energy occupying the space; I mean the space itself.) I ask: now how far apart are the points? Are they still twenty feet apart, on the grounds that distance, being intrinsic, is indifferent to changes in the intervening space? Or are they now less than twenty feet apart, on the grounds that there is now less space between them? Or are they now more than twenty feet apart, on the grounds that the shortest (continuous) path from one to the other is now more than twenty feet long? We have three competing answers, each, I think, with some intuitive appeal. Which is correct? And how can we tell?