Can two distinct mathematical theories be homeomorphic? Can homeomorphism exist between two distinct mathematical theories? I know homeomorphism is used in topological spaces but can two theories, say number theory & complex analysis have homeomorphism in the sense that if I pick a problem in number theory & show it is equivalent to some complex analysis problem in an abstract sense? When the details are stripped from the number theory problem, and it is written in some general logical language, and it can be shown to be logically equivalent to a complex analysis problem. Or is there a universal language of logic in which every mathematical statement can be written? I may do it because the problem maybe easier to solve in one system than the other. I am not a mathematician but I took some courses in abstract algebra and other math subjects back in college. I am just curious if homeomorphism can be a larger connection within abstract mathematical structures
 A: Partial answer.
There are many examples in mathematics of a duality. This is when a certain collection of objects are in some kind of ‘nice’ or ‘natural’ one-to-one correspondence with another collection of objects, but in a deep, meaningful way. The only examples I’m familiar with are the Riesz dualities between measures on a topological measure space and functionals on the space of continuous functions, or between an $L^p$ space and functionals on its conjugate $L^q$ space.
These are very important and do serve to essentially translate problems from one perspective into another. A slightly different notion is of categorical duality, which might appeal to you. There are loads of other examples, but they’re unfortunately over my head for the moment:

*

*Gelfand-Niemark duality

*Stone duality

*Pontryagin duality

*…

As for an “isomorphism of theories”, you’d need some way to very precisely mathematically describe what it means: ‘to do number theory’ or: ‘to do complex analysis’. As far as I know, no such thing exists, but category theory is a strong language for mathematics that can sometimes precisely describe a ‘theory’. For example one can generally describe ‘models of an algebraic theory’ by categories which are monadic for a finitary monad over the category of sets.
