Time and Physical Geometry
Mathematical representations of physical entities are shaped by the
mathematical tools used to create them. Space, time, and space-time
have traditionally been represented by topological spaces: sets of
points that are knit together, at the most fundamental level, by a
structure of open sets that satisfies the axioms of standard
topology. Notions such as the connectedness of a space, the boundary
of a set, and the continuity of a function are defined by reference to
these open sets. Additional geometrical structure (such as metrical or
affine structure) can be added to a topological space, but the
mathematical representation typically begins with a topological
manifold.
I will argue that standard topology is the wrong
mathematical tool to use for representing the structure of space and
time (or space-time). I will present an alternative mathematical tool,
the Theory of Linear Structures, whose primitive notion is the line
rather than the open set. The Theory of Linear Structures has a wider
field of useful application than topology in that it can be used to
capture the geometry of discrete spaces as well as continua. It
provides alternative, non-equivalent definitions of, e.g.,
connectedness, boundaries, and the continuity of a function. And it
offers a more detailed account of the sub-metrical geometry of a
space: every Linear Structure induces a topology on a space, but many
different Linear Structures give rise to the same topology. Using the
Theory of Linear Structures rather than standard topology to describe
space-time has a powerful ontological payoff: one can show that the
basic organizing principle of a Relativistic space-time (but not a
classical space-time) is time. Contrary to common belief, Relativity
does not "spatialize time", it rather "temporalizes space".