The Single Paragraph For Consideration Today [#41]

[10/9/19] Lonergan brought a deep knowledge of mathematics and physics to his work in Functional Macroeconomic Dynamics.

Now the principles and laws of a geometry are abstract and generally valid propositions.  It follows that the mathematical expression of the principles and laws of a geometry will be invariant under the permissible transformations of that geometry. … Such is the general principle and it admits at least two applications.  In the first application one specifies successive sets of transformation equations, determines the mathematical expressions invariant under those transformations, and concludes that the successive sets of invariants represent the principles and laws of successive geometries.  In this fashion one may differentiate Euclidean, affine, projective and topological geometries. … A second, slightly different application of the general principle occurs in the theory of Riemannian manifolds.  The one basic law governing all such manifolds is given by the equation for the infinitesimal interval, namely,

ds2= Σgijdxidxj           [i, j = 1,2…n]

where dx1, dx2… are differentials of the coordinates, and where in general there are n2products under the summation.  Since this equation defines the infinitesimal interval, it must be invariant under all permissible transformations. However, instead of working out successive sets of transformations, one considers any transformations to be permissible and effects the differentiation of different manifolds by imposing restrictions upon the coefficients.  This is done by appealing to the tensor calculus. …  Thus in the familiar Euclidean instance, gij is unity when equals j; it is zero when does not equal j; and there are three dimensions.  In Minkowski space, the gij is unity or zero as before, but there are four dimensions, and xequals ict. In the General Theory of Relativity, the coefficients are symmetrical, so that gij equals gji; and in the Generalized Theory of Gravitation, the coefficients are anti-symmetrical.   [CWL 3, 146 -147/170-71] [41] (Click here for previous “Single Paragraphs”)

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