  However, the choice of a specific O that selects observables Sx2, Sy2, Sz2 at the same time selects three orthogonal rays in physical space, namely by fixing a coordinate system ±x, ±y, ±z (which defines along which orthogonal rays the squared spin components are to be measured) in physical space. So now, by choice of an observable O, there is a direct connection of directions in space with directions in H3: orthogonality in H3 now does correspond to orthogonality in physical space. The same holds for R3, if, in order to give an argument for H3, we consider R3. Orthogonality in R3 now corresponds to orthogonality in physical space. It is important to notice that this correspondence is not necessary to give the argument, even if we insist that the pure mathematical facts should be supplemented by a physical interpretation — since we have, just before, seen an example without any correspondence. The point is only that we can devise an example such that there is a correspondence. In particular, we can now follow the proof in R3 and all along imagine a system sitting in physical space, namely a spin-1 particle, returning three values upon measurement of three physical magnitudes, associated directly with orthogonal directions in physical space, namely v(Sx2), v(Sy2), v(Sz2), for arbitrary choices of x, y, z. The KS proof then shows that it is impossible (given its premises, of course) to assign to the spin-1 particle values for all these arbitrary choices. That is, the KS argument shows that (given the premises) a spin 1 particle cannot possess all the properties at once which it displays in different measurement arrangements.