I am interested in theoretical physics and limitations of mathematics.
One of the supposed limitations of maths is that it can never fully cope with real world situations. Some suggested computers could never be as intelligent as us because of this but people don't fully cope either. Godel's Proof is a formalized version of this argument. Godel stated that finite logical explanations can never fully describe real things without either being incomplete or inconsistent.
From my studies of modern physics and information theory I suspect an extrapolation of Godel's proof might apply to the real world such that things in the real world may have to be incomplete or inconsistent.
This means that that mathematical logic may dictate that real worlds as we understand them are mathematically impossible or that any set mathematical logics are logically impossible.
In Physics we have a number of separate models of the world that address different areas, but when we try to put them together to make a consistent whole we encounter contradictions or conclusions that defy common sense.
The most famous of this is the attempted melding of Einsteins's Relativity that works well on large things and Quantum Mechanics that operates on much smaller things.
Uniting models such as String Theory and Loop Gravity are as yet mostly unexplored and introduce new problems such as the complexity of the maths is too loose or complicated.
The explorations of such theories and the possible limitations of mathematics are areas now being explored.
