Duality and correspondence - Alessandra Palmigiano

Duality and correspondence are useful mathematical tools to explore connections
between often very different areas of mathematics.  Dualities preserve
information while reversing perspective.  Examples are links between topology,
algebra and set theory.  When such a link is established, results can be
translated from one area to the other.  This provides a strategic advantage:
different formalisms have different strengths and weaknesses.

A case in point are the axioms of projective geometry, which can be formalized
in a first order language using variables of two sorts: points and lines.
These points and lines can be interchanged, which yields new theorems.  
An example of correspondence is the correspondence which can be drawn between
sets and boolean algebras, which facilitates a soundness & completeness proof
of modal logic (although this proof is special to modal logic, and
non-canonical).

In logic dualities often involve classes of objects of very different kinds,
eg. from algebras to spaces.  All of their properties can be systematically
translated, including soundness & completeness. Thus dualities can function as
a Rosetta stone for logics.



Word count: 185 
Andreas van Cranenburgh 0440949.