Surfaces and Superposition: Field Notes on some Geometrical Excavations (Lecture Notes)

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Buildings appear to rest on top of the earth's surface, yet the surface is actually permeated by the buildings' foundations-out of view. If a foundation's blueprints are unavailable, as in archaeology, excavation would be needed to discover what actually supports a specific building. Analogously, the fields of geometry and topology have easily observable concepts resting on the surface of theoretical underpinnings that have not been completely discovered, unearthed or understood. Moreover, geometrical and topological principles of superposition provide insight into probing the connections between accessible superstructures and their hidden underpinnings. This book develops and applies these insights broadly, from physics to mathematics to philosophy. Even analogies and abstractions can now be seen as foundational superpositions.

This book examines the dimensionality of surfaces, how superpositions can make stable frameworks, and gives a quasi-Leibnizian account of the relative `spaces' that are defined by these frameworks. Concluding chapters deal with problems concerning the spatio-temporal frameworks of physical theories and implications for theories of visual geometry. The numerous illustrations, while surprisingly simple, are satisfyingly clear.