Quantum theory's Hilbert space apparatus in its finite-dimensional version is
nearly reconstructed from four simple and quantum-mechanically motivated
postulates for a quantum logic. The reconstruction process is not complete,
since it excludes the two-dimensional Hilbert space and still includes the
exceptional Jordan algebras, which are not part of the Hilbert space apparatus.
Options for physically meaningful potential generalizations of the apparatus
are discussed.Comment: 19 page
