I am a bit bewildered when I try to think about empty space. Does it make sense to think about space insofar as it is space? What sort of existence, if any, does it have? Is it nothing?
There are two major views
There are two major views about space, and they give different answers to your question. One view is "substantivalism." On this view, space really is a thing of a certain sort—a substance. Space would exist even if nothing else did. Needless to say, space it not like things as we usually think of them, but it has its own sort of reality. For Newton space was, among other things, a system of absolute positions. Newton believed that there was an absolute distinction between rest and motion, and that called for a corresponding system of positions. However, the points of space were otherwise indistinguishable; one point was intrinsically like any other. In contemporary physics space and time are deeply intertwined, and we talk about space-time. Space time in general relativity is mathematically like a field (think of the electromagnetic field), and unlike Newtonian space, the points of space-time aren't all alike. This goes with the idea that space-time itself is curved. Roughly, the curvature at two...
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