Grit or Gunk: Implications of the Banach-Tarski Paradox

This paper concerns the structure of any spatially extended things, including regions of space or spacetime. I shall use intuitions about the quantity (measure) of extended things to argue for a dichotomy: either a given finite extended thing is point-free gunk, that is, it has no points as parts, or it is made of grit, that is there are only finitely many points.This Grit or Gunk dichotomy excludes what I call the orthodoxy, namely that: (1) there are points; and (2) not merely are points represented by coordinate triples; but (3) every set of triples of reals represents a region of space. (1) It does not, however, exclude the trivial grit thesis, "Nihilism," that there are no extended things because the only located things are point-like (points or point particles or point instances of fields) and, it is said, these points do not have mereological sums. (2) So we have not a dichotomy but a trichotomy: Nihilism, Grit or Gunk.The Grit or Gunk dichotomy applies to other extended things as well as regions, but with slight complications. Fields will be discussed at the end of the paper, but something needs to be said about extended material objects, which I shall assume are constituted out of finitely or countably many particles. (If not, then presumably they are constituted by fields or by portions of spacetime itself, in which case Grit or Gunk applies to these constituents.) Grit or Gunk applies trivially to point particles, but in a more controversial way to particles which are themselves extended.