I thought about this some more and the first perspective seems somewhat natural as well.
Consider the MUH’s cousin, the CUH (i.e. all computable mathematical objects are real). Each object can be thought of as an equivalence class (using the first equivalence I mentioned) of the TMs that compute it. If you have a strong belief in the CUH then the first equivalence seems to cut reality at its joints.
On a side note, it’s interesting that TMs are used to define computable mathematical objects and are also computable mathematical objects themselves. Following this train of thought might lead one to new notions of symmetry in this space of objects.
I thought about this some more and the first perspective seems somewhat natural as well.
Consider the MUH’s cousin, the CUH (i.e. all computable mathematical objects are real). Each object can be thought of as an equivalence class (using the first equivalence I mentioned) of the TMs that compute it. If you have a strong belief in the CUH then the first equivalence seems to cut reality at its joints.
On a side note, it’s interesting that TMs are used to define computable mathematical objects and are also computable mathematical objects themselves. Following this train of thought might lead one to new notions of symmetry in this space of objects.