Maybe it would help to think of the turing machine as analogous. Many programming languages are Turing complete, you can express any computation in any of them, but some languages are more expressive than others and let you reach and work with ideas you wouldn't conceive in a less expressive language.
Lots of things in math are similar. Simon Altmann's Icons and Symmetries makes a case that using representations with insufficient symmetry impeded our learning of the laws of magnetism.
Lots of things in math are similar. Simon Altmann's Icons and Symmetries makes a case that using representations with insufficient symmetry impeded our learning of the laws of magnetism.