Mea culpa. I was naively thinking of super-imposing the ‘previous’ axes. I hope the underlying worry still stands given the arbitrarily many sets of mathematical objects which could be reversibly mapped onto phenomenological states, but perhaps this betrays a deeper misunderstanding.
I’ll assume you meant isomorphically mapped rather than reversibly mapped, otherwise there’s indeed a lot of random things you can map anything.
I tend to think of isomorphic objects as equivalent in every way that can be mathematically described (and that includes every way I could think of). However, objects can be made of different elements so the equivalence is only after stripping away all information about the elements and seeing them as abstract entities that relate to each other in some way. So you could get {Paris, Rome, London} == {1,2,3}. What Mike is getting at though I think is that the elements also have to be isomorphic all the way down—then I can’t think of a reason to not see such completely isomorphic objects as the same.
If they’re isomorphic, then they really are the same for mathematical purposes. Possibly if you view STV as having a metaphysical component then you incur some dependence on philosophy of mathematics to say what a mathematical structure is, whether isomorphic structures are distinct, etc.
Mea culpa. I was naively thinking of super-imposing the ‘previous’ axes. I hope the underlying worry still stands given the arbitrarily many sets of mathematical objects which could be reversibly mapped onto phenomenological states, but perhaps this betrays a deeper misunderstanding.
I’ll assume you meant isomorphically mapped rather than reversibly mapped, otherwise there’s indeed a lot of random things you can map anything.
I tend to think of isomorphic objects as equivalent in every way that can be mathematically described (and that includes every way I could think of). However, objects can be made of different elements so the equivalence is only after stripping away all information about the elements and seeing them as abstract entities that relate to each other in some way. So you could get {Paris, Rome, London} == {1,2,3}. What Mike is getting at though I think is that the elements also have to be isomorphic all the way down—then I can’t think of a reason to not see such completely isomorphic objects as the same.
If they’re isomorphic, then they really are the same for mathematical purposes. Possibly if you view STV as having a metaphysical component then you incur some dependence on philosophy of mathematics to say what a mathematical structure is, whether isomorphic structures are distinct, etc.