The context of an operational description is given by the distinction
between what we consider as relevant and what as irrelevant for a
particular experiment or observation. In terms of a mathematically
formulated context-independent fundamental theory, a rigorous description
of a context is possible by the restriction of the domain of the basic
theory and the introduction of a new coarser topology. Such a new topology
is never given by first principles but depends in a crucial way on the
abstractions made by the cognitive apparatus or the pattern recognition
devices used by the experimentalist. A consistent mathematical formulation
of a higher-level theory requires the closure of the restriction of the
basic theory in the new contextual topology. The validity domain of the so
constructed higher-level theory intersects nontrivially with the validity
domain of the basic theory: neither domain is contained in the other. As a
consequence, higher-level theories cannot be totally ordered and theory
reduction is not transitive. The emergence of qualitatively new properties
is a necessary consequence of such a formulation of theory reduction (which
does not correspond to the traditional one). Emergent properties are not
manifest on the level of the basic theory but they can be derived
rigorously by imposing new, contextually selected topologies upon
context-independent first principles.
Most intertheoretical relations are mathematically describable as singular
asymptotic expansions which do not converge in the topology of the primary
theory, or by choosing one of the infinitely many possible, physically
inequivalent representations of the primary theory
(Gelfand-Naimark-Segal-construction of algebraic quantum mechanics). As
examples we discuss the emergence of the properties shadow, inductance or
capacitance from Maxwell's electrodynamics, the emergence of order
parameters in statistical mechanics, the emergence of mass as classical
observable in Galilei-relativistic theories, the emergence of the shape of
molecules in quantum mechanics, the emergence of classical observables and
temperature in algebraic quantum mechanics.