Thursday, January 12, 2012

Self-Description in Mathematics

The foundations of mathematics is a strange, strange place - and not only because of philosophical disagreements on constructivism, incompleteness of logical systems, and set-theoretic paradoxes (although all of that is indeed weird). No, one of the strangest things about mathematics is its uncanny ability to describe itself.

Ideally, we like to think of the flow of mathematics almost as a directed acyclic graph: The propositional calculus extends to predicate logic; set theory builds on predicate logic; order theory, algebra, and topology build on set theory; analysis builds on all of these. Where things get weird, however, is when things move backwards through this hierarchy. This can perhaps be seen no more clearly than in order theory, and in particular, lattice theory.

The following relationship exists:
  • A lattice can be described as:
  • Algebraic structures are defined in terms of sets, the axioms of which are defined in terms of predicate and propositional logic.
  • Propositional logic, with its AND and OR operations, form a distributive lattice (specifically a Boolean algebra).
So in other words, algebra and lattices are defined with the help of propositional calculus, lattices can be defined in terms of algebra, and propositional calculus can be defined in terms of algebraic lattices. It is amazing to me how mathematics is continually able to so gracefully "turn around" and, in a sense, define itself. Such self-reference has in some cases resulted in major philosophical difficulties, but in others, such as this one, all appears to work fine. How is it though that mathematics is able to adequately define structures more fundamental than those doing the defining? I personally believe that this exact phenomenon is symptom of mathematics being not a fundamental deductive science which guides the universe, but a careful, consistent, and beautiful language whose precision grants it power far-reaching enough even to predict events which have not yet come to pass (solar eclipses, weather patterns, etc). Interesting stuff, eh?