Not if the definition of computing and uncomputable are from two different models of computation. For example, there are functions uncomputable by finite state

Not if the definition of computing and uncomputable are from two different models of computation. For example, there are functions uncomputable by finite state machine, but that can be computed by a Turing machine. One can imagine that someone will find a model of computation that allows to compute more functions than a Turing machine. It's extremely unlikely, but nobody has proven that it is not possible. It may be that our universe is one such model.