Conflicting Intuitions about Space

My purpose in this chapter is to argue that we have inconsistent intuitions about the structure of space (or spacetime, or some stuff, the aether, that fills space or spacetime). I obtain a contradiction from eleven premises, each of which is either directly intuitive or supported by intuitions. The use of so many premises results from the desire to exhibit as clearly as possible the places where readers might decide some intuition is to be undermined. Nonetheless the argument for inconsistency is, in outline, straightforward: our intuitions support the existence of a 'supersponge', namely a region of less than the total volume but not disjoint from any connected part of space of positive diameter. But the complement of a supersponge is an intuitively impossible region. Yet, intuitively a region of less than maximal volume must have a complement. Some have complained that the technical aspects of this chapter require more effort on the part of readers than the result warrants. I have three things to say about that complaint: First, I am claiming that the eleven premises are literally jointly inconsistent. If this is wrong I need correcting, but that I turn out to be wrong, if I am, will not be controversial. So the reader who does not want to invest much time on this project may simply leave the task of checking to others, and provisionally concede the inconsistency, pondering which are the less firm intuitions. Second, the inconsistency result is not a mere curiosity, for we may generate a family of hypotheses by abandoning just one of the least firm intuitions. These hypotheses should be taken seriously in spite of the clash with the intuition in question.