The Church-Turing thesis postulates that essentially all models of computation -Turing machines, Post systems, lambda-definability etc yield the same class of computable functions. Now we can think of a much wider class of models of computation including dynamical systems, DNA and molecular models, quantum computers, chemical reactions, etc.
Can we regard the notion of computation as analogous to that of an abstract group and in analogy to Cayley theorem of group theory (every group can be realized as a subgroup of a permutation group), a computation as a realization in a particular model. More abstractly, the realization of computation in any of the different models is similar to the notion of group representation, in which the same abstract group can be realized variously as a group of linear automorphisms of a chosen linear space. Is this view of computation correct or reasonable?
Why does (or should) a mathematical structure carry a generic computation? Just as symmetries of physical systems give rise to conservation laws in physics (Noether's theorem), are there analogs in this setting?
How do we determine whether a mathematical structure has a rich or poor computational structure?
How is the notion of computation supported in each structure? Is it always carried in the same way? I am looking for ideas in analogy to that of 'configuration space' or phase space in a dynamical system which has the structure of a symplectic manifold and is the cotangent bundle of the dynamical system enabling the Hamiltonian formulation of dynamics (which holds uniformly for classical as well as quantum mechanics)
Are there any possible approaches towards a mathematical proof (or disproof) of the 'Church-Turing thesis'?