There is a programme of inconsistent mathematics which revives and completes Hilberts logicist programme by defanging Russells Paradox and the incompleteness theorems of Godels.
Can inconsistency/paraconsistency provide insights into Physics as well? And what would this mean for the philosophy of Physics? Has there been any research done along these lines?
The SEP/IEP has nothing about the possible physical ramifications of inconsistency. Though given the symbiotic relationship between mathematics and physics one expects there to be some.
It does mention a couple of geometric incarnations of paraconsistency that may have fruitful physical consequences:
a. gluing of geometries along boundaries - the boundary belongs to the first & not the second and not to the first but to the second (and to both?). Gluing is important in toy theories of QG such as TQFTs.
b. Open set logic is intuitionistic; its dual closed set logic is paraconsistent.
Perhaps when a state collapses phenomenologically when provoked by a measurement (but ontologically not) this could instead be instead be interpreted as a state not having a specific value and having a specific value - which gets rid of the phenomenology - perhaps, as this is all pure speculation.
Andreas Doring, who works on Topos interpretation of Physics as initiated by Chris Isham has the following paper where he explains:
The main thrust of this article is to replace the standard algebraic representation of quantum logic...by a better behaved form based on bi-Heyting algebras. Instead of having a non-distributive orthomodular lattice of projections, which comes with a host of well-known conceptual and interpretational problems, one can consider a complete bi-Heyting algebra of propositions. In particular, this provides a distributive form of quantum logic. Roughly speaking, a non-distributive lattice with an orthocomplement has been traded for a distributive one with two diﬀerent negations.
Now what this means is that this 'new' logic is both intuitionistic & paraconsistent. Of course this is to a large extent a formal result. Its Physical meaning of paraconsistency here remains tentative.