 In a letter to Birkhoff from 1935, von Neumann says: “I would like to make a confession which may seem immoral: I do not believe in Hilbert space anymore”; the letter is published in von Neumann (2005). The confession is indeed startling since it comes from the champion of the separable Hilbert space formulation of quantum mechanics and it is issued just three years after the publication of his famous treatise, the definitive work on the subject. The irony is compounded by the fact that less than two years after his confession to Birkhoff, his mathematical theorizing about the abstract mathematical structure that was to supersede the separable Hilbert space, continuous geometries with a transition probability, turned out not to provide a generalization of the separable Hilbert space framework. It is compounded again with interest in that subsequent developments in mathematical physics initiated and developed by von Neumann ultimately served to strengthen the entrenchment of the separable Hilbert space framework in mathematical physics (especially with regards to quantum theory). These matters are explained in more detail in Section 4.1.