73 research outputs found
On the relationship between plane and solid geometry
Traditional geometry concerns itself with planimetric and stereometric considerations, which are at the root of the division between plane and solid geometry. To raise the issue of the relation between these two areas brings with it a host of different problems that pertain to mathematical practice, epistemology, semantics, ontology, methodology, and logic. In addition, issues of psychology and pedagogy are also important here. To our knowledge there is no single contribution that studies in detail even one of the aforementioned area
The Homological Kähler-De Rham Differential Mechanism part I: Application in General Theory of Relativity
The mechanism of differential geometric calculus is based on the fundamental notion of a connection on a module over a commutative and unital algebra of scalars
defined together with the associated de Rham complex. In this communication,
we demonstrate that the dynamical mechanism of physical fields can be formulated by purely algebraic means, in terms of the homological Kähler-De Rham differential schema, constructed by connection inducing functors and their associated curvatures, independently of any background substratum. In this context, we show explicitly that the application of this mechanism in General Relativity, instantiating the case of gravitational dynamics, is
related with the absolute representability of the theory in the
field of real numbers, a byproduct of which is the fixed background
manifold construct of this theory. Furthermore, the background independence of the homological differential mechanism is of particular importance for the formulation of dynamics
in quantum theory, where the adherence to a fixed manifold substratum is
problematic due to singularities or other topological defects
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