  This article contributes to the debate of the meaning of relationalism and
background independence, which has remained of interest in theoretical physics
from Newton versus Leibniz through to foundational issues for today's leading
candidate theories of quantum gravity. I contrast and compose the substantially
different Leibniz--Mach--Barbour (LMB) and Rovelli--Crane (RC) uses of the word
`relational'. Leibniz advocated primary timelessness and Mach that `time is to
be abstracted from change'. I consider 3 distinct viewpoints on Machian time:
Barbour's, Rovelli's and my own. I provide four expansions on Barbour's taking
configuration space to be primary: to (perhaps a weakened notion of) phase
space, categorizing, perspecting and propositioning. Categorizing means
considering not only object spaces but also the corresponding morphisms and
then functors between such pairs. Perspecting means considering the set of
subsystem perspectives; this is an arena in which the LMB and Rovelli
approaches make contact. By propositioning, I mean considering the set of
propositions about a physical (sub)system. I argue against categorization being
more than a formal pre-requisite for quantization in general; however,
perspecting is a categorical operation, and propositioning leads one to
considering topoi, with Isham and Doering's work represents one possibility for
a mathematically sharp implementation of propositioning. Further applications
of this article are arguing for Ashtekar variables as being relational in LMB
as well as just the usually-ascribed RC sense, relationalism versus
supersymmetry, string theory and M-theory. The question of whether scale is
relational is also considered, with quantum cosmology in mind.
