Category Archives: Theoretical Physics

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”)

Prediction is Impossible in the General Case

In his book, FREEFALL (2009, Penguin Books), Joseph Eugene Stiglitz, a professor at Columbia University and a recipient of the Nobel Memorial Prize in Economic Sciences (2001) and the John Bates Clark Medal (1979), states that economics is a predictive science. Now, one must distinguish between predicting a) planetary motion in its scheme of recurrence, and b) this afternoon’s weather vs. next month’s weather, or this afternoon’s prices and quantities vs. next year’s prices and quantities, all subject to to conditions diverging in space and time.   Continue reading)

 

 

The IS-LM, AD-AS, and Phillips Curve Models

In this section, we are contrasting familiar textbook models of macrostatic equilibrium, with Lonergan’s explanatory theory of macrodynamic equilibrium.  We are contrasting a macrostatic toolkit with a purely relational field theory of macroeconomic dynamics. Lonergan discovered  a theory which is more fundamental than the traditional wisdom based upon human psychology and purported endogenous reactions to external forces.  His Functional Macroeconomic Dynamics is a set of relationships between n objects, a set of intelligible relations linking what is implicitly defined by the relations themselves, a set of relational forms wherein the form of any element is known through its relations to all other elements.  His field theory is a single explanatory unity; it is purely relational, completely general, and universally applicable to every configuration in any instance. (Continue reading)

 

 

The Emergence of Science

Lonergan, like Euclid, Newton, and Mendeleyev, moved through his field of inquiry to the level of system.

(Given the failure to implement the basic expansion,) the systematic requirement of a rate of losses will result in a series of contractions and liquidations. … [CWL 15, 155]

… a science emerges when thinking in a given field moves to the level of system. Prior to Euclid there were many geometrical theorems that had been established.  The most notable example is Pythagoras’ theorem on the hypotenuse of the right-angled triangle, which occurs at the end of  book 1 of Euclid’s Elements.  Euclid’s achievement was to bring together all these scattered theorems by setting up a unitary basis that would handle all of them and a great number of others as well. … Similarly, mechanics became a system with Newton.  Prior to Newton, Galileo’s law of the free fall and Kepler’s three laws of planetary motion were known.  But these were isolated laws.  Galileo’s prescription was that the system was to be a geometry’; so there was something functioning as a system.  But the system really emerged with Newton.  This is what gave Newton his tremendous influence upon the enlightenment.  He laid down a set of basic, definitions, and axioms, and proceeded to demonstrate and conclude from general principles and laws that had been established empirically by his predecessors.  Mechanics became a science in the full sense at that point where it became an organized system. … Again, a great deal of chemistry was known prior to Mendeleev. But his discovery of the periodic table selected a set of basic chemical elements and selected them in such a way that further additions could be made to the basic elements.  Since that time chemistry has been one single organized subject with a basic set of elements accounting for incredibly vast numbers of compounds.  In other words, there is a point in the history of any science when it comes of age, when it has a determinate systematic structure to which corresponds a determinate field. [CWL 14, Method, 1971, 241-42]

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