Re pagerank: This looks a lot like eigenvalues / eigenvectors, which show up a bunch in physics. (Eigenvector with high eigenvalue is like a “self-ratifying / stable generalized state”.)
Quantum mechanics involves topology of course. In operator algebra theory (Gelfand duality). And in TQFT. However the article seems to be making a leap relating quantum topology to minds and phenomenal binding. This is skipping many levels of abstraction!
I will now summarize a quantum topology approach, relevant to observers, which is not skipping nearly as many abstraction levels. We start by a quantum operator space as a C* algebra. This algebra is in general non-commutative. However, it has commutative sub-algebras. (Sub-algebra here is similar to ‘sub-group’, ‘sub-monoid’, ‘category theoretic sub-object’; has a precise characterization). These form a meet-semilattice (indicating: ‘intersection’ of commutative sub-algebras is commutative; ‘union’ is not in general commutative). The meet-semilattice structure reflects complementarity, Heisenberg uncertainty, Kochen-Specker, and so on. In PVM/POVM terms, different Hermitian operators commute when they each have a “diagonalization” (or infinite-dimensional equivalent) in a compatible basis; this is not always the case.
So we have a meet-semilattice of commutative C* sub-algebras. Then each, by Gelfand duality, is iso to a C* algebra of continuous functions S→C for a compact Hausdorff space S. Accordingly, there is a category-theoretic contravariant isomorphism between the category of commutative C* algebras and the category of compact Hausdorff spaces and continuous maps between them.
This is of course highly topological. A quantum operator algebra implies multiple classical contexts, and in general there’s no classical context containing all information from all of them. The contexts are at varying levels of fine-ness and coarse-ness. Some coarsening is necessary to get commutativity (and classicality, under the ‘commutative C* sub-algebras as classical’ interpretation.)
The sub-algebraic picture suggests that ‘high-level computations’ can be really instantiated as sub-algebras. Where sub-algebras also relate to topology through locale theory; categorical sub-objects in a category such as the category of compact regular frames. (Compare: if a group has a sub-group isomorphic to Z under addition, then the group operation implements integer addition; this is a hard mathematical constraint, not merely an interpretation.)
There is some ‘reality to wholes’ here, through sub-algebras (corresponding with quotient spaces in topology through locale-theoretic duality, Isbell). There is some ‘objectivity’ here (or perhaps ‘pre-conditions for objectivity’), in that which coarsenings form valid commutative algebras depends on the physical system in question.
The conflation to avoid making here is that a classical context (given by a compact Hausdorff space corresponding to a commutative C* sub-algebra) is ‘a mind’ or ‘a person’ or that sort of thing. It can be much more detailed than that. It is more like a virtual world simulation that doesn’t exactly have a reductionist lowest level to it; some details are simply coarsened. A given classical context can contain multiple minds (as classical reductionists expected prior to quantum mechanics). If anything, the classical ‘mind vs matter’ distinction is in a frame that makes classical assumptions; a classical context is more like a pre-requisite for ‘mind vs matter’ to be a sensible distinction. (Materialism != physicalism)
This is more my own philosophical spin than something directly implied by quantum topology, but: The ‘phenomena’ here are more like the phenomena of Kant than the phenomena of Chalmers: spatially three-dimensional, multi-personal. See also Wilfrid Sellars on phenomena; his “Empiricism and the Philosophy of Mind” is of course important background, but his “Phenomenalism” addresses multi-personal phenomenal contexts more directly.
(See Bohrification and a review for more technical details on this overall picture.)
Re pagerank: This looks a lot like eigenvalues / eigenvectors, which show up a bunch in physics. (Eigenvector with high eigenvalue is like a “self-ratifying / stable generalized state”.)
Quantum mechanics involves topology of course. In operator algebra theory (Gelfand duality). And in TQFT. However the article seems to be making a leap relating quantum topology to minds and phenomenal binding. This is skipping many levels of abstraction!
I will now summarize a quantum topology approach, relevant to observers, which is not skipping nearly as many abstraction levels. We start by a quantum operator space as a C* algebra. This algebra is in general non-commutative. However, it has commutative sub-algebras. (Sub-algebra here is similar to ‘sub-group’, ‘sub-monoid’, ‘category theoretic sub-object’; has a precise characterization). These form a meet-semilattice (indicating: ‘intersection’ of commutative sub-algebras is commutative; ‘union’ is not in general commutative). The meet-semilattice structure reflects complementarity, Heisenberg uncertainty, Kochen-Specker, and so on. In PVM/POVM terms, different Hermitian operators commute when they each have a “diagonalization” (or infinite-dimensional equivalent) in a compatible basis; this is not always the case.
So we have a meet-semilattice of commutative C* sub-algebras. Then each, by Gelfand duality, is iso to a C* algebra of continuous functions S→C for a compact Hausdorff space S. Accordingly, there is a category-theoretic contravariant isomorphism between the category of commutative C* algebras and the category of compact Hausdorff spaces and continuous maps between them.
This is of course highly topological. A quantum operator algebra implies multiple classical contexts, and in general there’s no classical context containing all information from all of them. The contexts are at varying levels of fine-ness and coarse-ness. Some coarsening is necessary to get commutativity (and classicality, under the ‘commutative C* sub-algebras as classical’ interpretation.)
The sub-algebraic picture suggests that ‘high-level computations’ can be really instantiated as sub-algebras. Where sub-algebras also relate to topology through locale theory; categorical sub-objects in a category such as the category of compact regular frames. (Compare: if a group has a sub-group isomorphic to Z under addition, then the group operation implements integer addition; this is a hard mathematical constraint, not merely an interpretation.)
There is some ‘reality to wholes’ here, through sub-algebras (corresponding with quotient spaces in topology through locale-theoretic duality, Isbell). There is some ‘objectivity’ here (or perhaps ‘pre-conditions for objectivity’), in that which coarsenings form valid commutative algebras depends on the physical system in question.
The conflation to avoid making here is that a classical context (given by a compact Hausdorff space corresponding to a commutative C* sub-algebra) is ‘a mind’ or ‘a person’ or that sort of thing. It can be much more detailed than that. It is more like a virtual world simulation that doesn’t exactly have a reductionist lowest level to it; some details are simply coarsened. A given classical context can contain multiple minds (as classical reductionists expected prior to quantum mechanics). If anything, the classical ‘mind vs matter’ distinction is in a frame that makes classical assumptions; a classical context is more like a pre-requisite for ‘mind vs matter’ to be a sensible distinction. (Materialism != physicalism)
This is more my own philosophical spin than something directly implied by quantum topology, but: The ‘phenomena’ here are more like the phenomena of Kant than the phenomena of Chalmers: spatially three-dimensional, multi-personal. See also Wilfrid Sellars on phenomena; his “Empiricism and the Philosophy of Mind” is of course important background, but his “Phenomenalism” addresses multi-personal phenomenal contexts more directly.
(See Bohrification and a review for more technical details on this overall picture.)