The self-representing Universe
Mathematical Structures of the Universe
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We revisit our approach to Fourier and observer-observed duality as the origin of the structure of physical reality. We explain how Fourier duality on the exterior algebra recovers Hodge duality. We describe recent joint work on duality of quantum differential structures on quantum groups and speculate on the interpretation of de Morgan duality and bi-Heyting algebras in this context. We describe our recent work on the algebraic description of a classical Riemannian geometry as a certain type of Batalin-Vilkovisky algebra. Using this we explain how classical Riemannian geometry emerges out of nothing but the Leibniz rule, as a kind of 2-cocycle that governs the possible ways that a noncommutative differential algebra can extend a given classical manifold. This explains how classical Riemannian geometry emerges out of noncommutative algebra under very minimal assumptions and in some sense explains why there is Riemannian geometry in the first place. These matters are discussed within our general philosophy of `Relative Realism' whereby physical reality is in some sense created by the assumptions of Science.
- Mathematics