In this paper, we analyze axiomatic issues of unconventional computations from a methodological and philosophical point of view. We explain how the new models of algorithms changed the algorithmic universe, making it open and allowing increased flexibility and creativity. However, the greater power of new types of algorithms also brought the greater complexity of the algorithmic universe, demanding new tools for its study. That is why we analyze new powerful tools brought forth by the axiomatic theory of algorithms, automata and computation.