Well, the OPs reinvention looks like something more high level
Well, there may not be immediate advantages, but in math you never know. There are several hard problems in one domain that are trivial in another domain, for example.
> There are several hard problems in one domain that are trivial in another domain, for example.
Certainly, and this is pretty much what OP did, invent a new domain to solve a problem – Hamilton aka Lord Kelvin merely reformulated the problem slightly, and while I personally love Hamiltonian mechanics, I don’t think it is comparable to ‘inter-universal geometry’ or replacing first order logic with something else.
So, yes, a different field may provide a different perspective and hence easier solution, but if you want to replace first order logic, you’re not looking at a different/new field in maths, you’re looking at rebuilding maths.
Well, there may not be immediate advantages, but in math you never know. There are several hard problems in one domain that are trivial in another domain, for example.
An example from physics: http://en.wikipedia.org/wiki/Hamiltonian_mechanics