Essentially, the STV is an argument that much of the apparent complexity of emotional valence is evolutionarily contingent, and if we consider a mathematical object isomorphic to a phenomenological experience, the mathematical property which corresponds to how pleasant it is to be that experience is the object’s symmetry.
I don’t see how this can work given (I think) isomorphism is transitive and there are lots of isomorphisms between sets of mathematical objects which will not preserve symmetry.
Toy example. Say we can map the set of all phenomenological states (P) onto 2D shapes (S), and we hypothesize their valence corresponds to their symmetry along the y=0 plane. Now suppose an arbitrary shear transformation applied to every member of S, giving S!. P (we grant) is isomorphic to S. Yet S! is isomorphic to S, and therefore also isomorphic to P; and the members of S and S! which are symmetrical differ. So which set of shapes should we use?
Trivial objection, but the y=0 axis also gets transformed so the symmetries are preserved. In maths, symmetries aren’t usually thought of as depending on some specific axis. E.g. the symmetry group of a cube is the same as the symmetry group of a rotated version of the cube.
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.
Aside:
I don’t see how this can work given (I think) isomorphism is transitive and there are lots of isomorphisms between sets of mathematical objects which will not preserve symmetry.
Toy example. Say we can map the set of all phenomenological states (P) onto 2D shapes (S), and we hypothesize their valence corresponds to their symmetry along the y=0 plane. Now suppose an arbitrary shear transformation applied to every member of S, giving S!. P (we grant) is isomorphic to S. Yet S! is isomorphic to S, and therefore also isomorphic to P; and the members of S and S! which are symmetrical differ. So which set of shapes should we use?
Trivial objection, but the y=0 axis also gets transformed so the symmetries are preserved. In maths, symmetries aren’t usually thought of as depending on some specific axis. E.g. the symmetry group of a cube is the same as the symmetry group of a rotated version of the cube.
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.