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Spinors can be constructed on spin manifolds.  If we accept the evidence for a compact four-dimensional universe, then we can say that spinors exist in our universe because it is a spin manifold in the sense that the structure group SO(4) can be lifted without topological obstruction to Spin(4).  Just by the argument for compactness — which uses the CBR bound of 2.7 and heat equation analysis or of four spatial dimensions, which comes from analyzing crystal symmetries of orders 5, 8, 10 and 12 we cannot claim that spinors makes sense in the universe.  If we make the choice of $S^4(1/h)$ then we know spinors exist as a mathematical deduction.
Note that the Dirac operator for riemannian manifolds was originally described not by Dirac but Atiyah and Singer.  Their definition led to an extremely fruitful and deep theory relating topology, analysis and geometry of spin manifolds.  From the viewpoint of geometry I think that $S^4(1/h)$ is a natural choice for the shape of the universe not simply because it is spin but for other reasons having to do quantization of energy and matching the measured cosmological constant.  The importance of spin manifolds in nature can perhaps be gleaned from the fruitfulness of the concept in purely mathematical question.  Thus if one arrives at a conclusion by however means that the actual space of the universe is a spin manifold, that should be re-assuring for those who believe (as did Gauss and Riemann) that mathematics is part of nature.