  A crucial step in the history of General Relativity was Einstein's adoption
of the principle of general covariance which demands a coordinate independent
formulation for our spacetime theories. General covariance helps us to
disentangle a theory's substantive content from its merely representational
artifacts. It is an indispensable tool for a modern understanding of spacetime
theories. Motivated by quantum gravity, one may wish to extend these notions to
quantum spacetime theories (whatever those are). Relatedly, one might want to
extend these notions to discrete spacetime theories (i.e., lattice theories).
This paper delivers such an extension with surprising consequences.
  One's first intuition regarding discrete spacetime theories may be that they
introduce a great deal of fixed background structure (i.e., a lattice) and
thereby limit our theory's possible symmetries down to those which preserve
this fixed structure (i.e., discrete symmetries). However, as I will discuss,
these intuitions are doubly wrong and overhasty. Discrete spacetime theories
can and do have continuous translation and rotation symmetries. Moreover, the
exact same theory can be given a wide variety of lattice structures and can
even be described with no lattice at all. As my discrete analog of general
covariance will reveal: lattice structure is rather less like a fixed
background structure or part of an underlying manifold and rather more like a
coordinate system, i.e., merely a representational artifact. Thus, the world
cannot be "fundamentally set on a square lattice" (or any other lattice) any
more than it could be "fundamentally set in a certain coordinate system". Like
coordinate systems, lattice structures are just not the sort of thing that can
be fundamental; they are both thoroughly representational. Spacetime cannot be
discrete (even when it might be representable as such).
