I'm trying to understand the concept of spin in Quantum Mechanics. I'm reading "Road to Reality" by Penrose, which despite not being a textbook, is reputed to give one a deep insight into physical processes.

Let us suppose that we have a spin $\frac{1}{2}$ particle. It has two eigenstates- $|\uparrow\rangle$ and $|\downarrow\rangle$. I would assume that spin $S$ is an operator such that when it acts on the wavefunction $\alpha |\uparrow\rangle+\beta|\downarrow\rangle$, it collapses it to one eigenstate, with an eigenvalue which would be the spin (so $\frac{1}{2}$ here). However, Penrose says that the spin can be thought of as the point $[\alpha:\beta]$ in $\Bbb{C}P^1$. 

Why is this? Why do we not have a collapse to an eigenstate?