  We lay the groundwork for a formal framework that studies scientific theories
and can serve as a unified foundation for the different theories within
physics. We define a scientific theory as a set of verifiable statements,
assertions that can be shown to be true with an experimental test in finite
time. By studying the algebra of such objects, we show that verifiability
already provides severe constraints. In particular, it requires that a set of
physically distinguishable cases is naturally equipped with the mathematical
structures (i.e. second-countable Kolmogorov topologies and $\sigma$-algebras)
that form the foundation of manifold theory, differential geometry, measure
theory, probability theory and all the major branches of mathematics currently
used in physics. This gives a clear physical meaning to those mathematical
structures and provides a strong justification for their use in science. Most
importantly it provides a formal framework to incorporate additional
assumptions and constrain the search space for new physical theories.
