[commonlisp94]

This is not true.

They are incompatible sets of axioms, but they can describe the same set of shapes, although one system is going to be a lot more inconvenient than the other. The differences are logically interesting, but about as incompatible as choosing to make the origin of a euclidean system one corner of the room instead of another.

In contrast physical theories contradict each other by making different predictions which can be falsified.

[ukj]

At that level of pedantry even the halting problem is solvable. Just choose a suitable representation for the Turing machines in question.

Describe/represent the ones that halt using 1; and the ones that don’t halt using 0. This produces the pairs (TM1, 1), (TM2, 1), (TM3, 0) etc.

Using this encoding the problem becomes trivial. It’s all other encodings which are unwieldy, complex and inconvenient.

[consilient]

That's no longer the halting problem: formally, the input to a halting oracle is an index for some fixed choice of admissible numbering of the set of Turing machines, meaning one that can be computed from the standard numbering induced by some universal Turing machine. Your encoding is not admissible.

[commonlisp94]

No, the comment above mine is the one being pedantic. We can absolutely talk about physical correspondence of mathematical concepts, even though there are multiple equally good logical systems for describing them.

[mcguire]

What is the sum of the interior angles of a triangle?

[commonlisp94]

Let's recall your original comment. You said we can't talk about reality correspondence of math because multiple logical systems can describe physical space equally well.

A triangle in a euclidean system is not the same shape as a triangle in spherical, etc. They are only equivalent in that they have an analogous logical definition in their separate systems.

They don't look the same. They don't describe the same physical phenonoma either. In other words, the physical correspondence is not the same, so there is no contradiction.