[2/16/20] Albert Einstein, Bernard Lonergan, and the theory of Riemannian manifolds: The contents of the excerpt below from CWL 3 are pretty standard stuff in an advanced physics course. The ideas are not Lonergan’s discoveries. Nevertheless, it is worthwhile to point out that Lonergan knew his physics and indeed taught physics. He understood and appreciated the difference between Newtonian mechanics and modern field theory. And he brought that thinking to his revolutionary ideas in purely relational Functional Macroeconomic Dynamics. So, to provide an indication of the mind that Lonergan brought to macroeconomics, we print this excerpt.
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 n2 products under the summation. Since this equation defines the infinitesimal interval, it must be invariant under all permissible transformations. … … … Thus in the familiar Euclidean instance, gij is unity when i equals j; it is zero when i 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 x4 equals 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. [#77] [CWL 3, 146 -147/170-71] (Click here for previous “Single Paragraphs” or “Brief Items”)