The Notion of Invariance

the basic postulate of the Special Theory of Relativity … is that the mathematical expression of physical principles and laws is invariant under inertial transformations. [CWL 3, 23/47-8]

Both the man on the passing train and the man watching the train go by measure the same velocity of light, and they understand extensions and durations by the same form.  Their principles and laws are invariant. [See, for example, Cutnell & Johnson 2004, 855 ff.]

If the mathematician’s insights into principles and laws do not vary because of constant velocity or a transformation of axes, then the expression should remain invariant under these transformations.

But his insights do not vary.

Therefore his expressions should remain invariant.

That syllogism attempts to communicate the insight behind Special Relativity. [CWL 5, 50]

Paraphrasing:

If, in the cases of functional flows which are integrally similar but constituted by different price and quantity components, macroeconomist’s insights reach the same formulations, then the expression of functional relations should remain invariant under these transformations.

But the macroeconomist’s insights reach the same formal relations; the insights do not vary.

Therefore his expressions of functional relations should remain invariant in form.

That syllogism attempts to communicate the insight regarding the relativistic, yet invariant functional flows in macroeconomics.

though there is a difference in the spatio-temporal standpoint from which the data are considered (by different observers), still there is no difference in the act of understanding the data, no difference in the general intelligibility grasped in the data, and no difference in the form of the mathematical expression of the intelligibility. [CWL 3, 23/48]

Paraphrasing (replacing 4-dimensional spacetime with 2-dimensional monetary price-quantity flows):

though there may be a difference in the observer’s standpoint from which particular price-quantity flows are viewed and considered, still there is no difference in the act of understanding the data, no difference in the general intelligibility grasped in the data, and no difference in the form of the mathematical expression of the intelligibility.

the transformation from one set of axes to another does not lead to any modification in the form of the mathematical expression of the appropriate physical principles and laws. [CWL 3, 23/48]

Paraphrasing:

the transformation from one pretio-quantital flow to another contemporaneous or successive flow does not lead to any modification in the form of the mathematical expression of the appropriate physical principles and laws of the two flows.

… the meaning of invariance is that

  1. all scientists expect their correlations and laws to be independent of merely spatio-temporal differences,

  2. physicists are confronted with a special difficulty inasmuch as they have to use reference frames, and

  3. physicists surmount their peculiar difficulty by expressing their principles and laws in mathematical equations that remain invariant under transformations of frames of reference. [CWL 3, 40/64]

Both Newton and Einstein contended with a general notion of constant velocity.  Newton dealt with explanation of states of rest and of constant velocity in the motion of heavenly bodies.  Einstein dealt with the explanation of the constant velocity of the speed of light in different inertial reference frames.  Both sought an explanation of physical invariance under transformations of reference frames.  But Einstein achieved a higher generalization based on his postulate of invariance and his empirical hypothesis of the constant speed of light providing a lead into the redefinition of space and time as purely relative to one another in a 4-dimensional space-time.

The contexts of Newton’s work and Einstein’s work were different.

…  First, Special Relativity regards all physical principles and laws, but Newtonian dynamics is concerned primarily with mechanics.  Secondly, Special Relativity is primarily a field theory, that is, it is concerned not with efficient, instrumental, material, or final causes of events, but with the intelligibility immanent in data; but Newtonian dynamics seems primarily a theory of efficient causes, of forces, their action, and the reaction evoked by action.  [3, 43/67  ]

(Also), Special Relativity (with its postulate of invariance) is stated as a methodological doctrine that regards the mathematical expression of physical principles and laws, but Newtonian dynamics is stated as a doctrine about the objects subject to laws. [3, 43/67]

Implicit in Lonergan’s Functional Macroeconomic Dynamics is the acceptance of Einstein’s methodological doctrine that regards invariance in the understanding and the mathematical expression of scientific principles and laws.  To achieve this goal of invariance Lonergan discovers and constructs an explanatory dynamical theory upon a foundation of

  • the basic element is the rate of application of a factor of production
  • thus, the basic terms are rates of so much every so often; i.e. velocities of functional activities
  • a principle of reciprocity of demand and supply
  • a principle of composition of an integral product by factoral components
  • a postulate of invariance of macroeconomic principles and laws across all psychopolitical and purely political systems
  • implicit definition, e.purely relational definition, whereby economic functionings are defined by the functional relations in which they stand with one another ( and the discarding of accountants’ unities as nonexplanatory)
  • precise analytical distinctions and relations among basic terms of systematic and scientific significance
  • a postulate of concomitance among interdependent velocitous functionings necessary to achieve continuity and equilibrium
  • the empirical interdependence and mutual conditioning of functional economic flows
  • projective correlation of productive structures and payments flows
  • a general law of credit as a bridging of the time gap between payments and receipts in the factoral composition of a good or service,
  • an normative pure cycle of the motion of interconnected flows, to which human psychology must adapt
  • a canon of complete explanation
  • a canon of statistical residues

Functional Macroeconomic Dynamics’ expression of macroeconomic principles and laws, like the Special Theory of Relativity’s expression of physical principles and laws, is invariant in several respects. It is invariant to all observers; there is no privileged observational point of view or system of reference.  Thus, different political premises, governmental systems, psychological dispositions, or a vague and indeterminate sense of things are irrelevant and completely ignored. FMD’s science of  interdependent functional flows is conceptually prior to, more fundamental than, and thus independent of psychological complexes and subjective political viewpoints.

As the speed of light cannot be measured within the fixed and absolute space and time coordinates of our everyday activities, but rather must be measured by Special Relativity’s objective geometric analytic, so the general principles and laws of Functional Macroeconomic Dynamics must be understood within FMD’s analytic of interdependent pretio-quantital dynamical flows. It is invariant across, and thus eliminates at a stroke, all utilitous, time-preferential, monetaristic, Keynesian, anticipational, behavioral, rational-expectational, psychopolitical, liberal, conservative, communist, socialist, capitalist systems of reference. Thus, a new paradigm is formulated and FMD singly supersedes and sublates the various and shifting, inadequately abstractive fantasies and subjectivisms of various schools of economic thought.  (URL to Sublation)

FMD’s principles and laws are general and universally applicable.

the most powerful of all scientific techniques is generalization. [ CWL 3, 28/53]

All situations, – or rather, configurations of interdependent flows – such as stagflation, boom, slump, or ideal pure cycle are to be understood in the same terms and in the same fashion because of the universality and generality of the theory of Functional Macroeconomic Dynamics.

A generalization makes the same assumption to argue that any other similar situation, X, is to be understood in the same fashion.  In both cases what is at work is the law, immanent and operative in cognitional process, that similars are similarly understood.  Unless there is a significant difference in the data, there cannot be a difference in understanding the data.  This point has already been made in discussing the heuristic procedure of the classical phase of empirical method. [CWL 3, 288/313]

Both Einstein and Lonergan employed a relativistic heuristic that science seeks explanation in the form of the relations of things to one another.  Motions or flows can be defined only relative to other motions and flows.

Einstein’s position …  follows quite plausibly from the premise that empirical science seeks not the relations of things to our senses but their relations to one another.  For, as has been remarked, observations give way to measurements; measurements relate things to one another rather than to our senses; and it is only the more remote relations of measurements to one another that lead to empirical correlations, functions, laws.  Now clearly if laws are reached by eliminating the relations of things to the senses of observers and by arriving at relations between the measured relations of things to one another, then there exists an extremely solid foundation for the affirmation that principles and laws are the same for all observers because they lie simply outside the range of observational activities. [CWL 3, 41/65-6]

Like Newton and Einstein, Lonergan sought and discovered a basic set of generally governing differential equations, which are independent of people’s senses, perceptions, and points of view. “the physicist will have a set of differential equations and a set of measurements, called boundary conditions, and while he is not able to find out the law, still he is able to solve any of his concrete problems.  So, science can be moving along without knowing the law, simply by using these differential equations.” [CWL10, 137-38]:

dI’= Σ(widni+ nidwi+dnidwi)y[CWL 15, 134]

The differential equation specifying how to adjust of the rate of saving to the requirements for consumption vs. investment of the productive phase

d(P’Q’) = d(p’a’Q’)Basic+ d(p”a”Q”)OrdinarySurplus  [CWL 15, 157-58]

Differentials giving acceleration of expended incomes (P’Q’) and “macroeconomic costs”

δJ = δa’ + a”δR + Rδa” [CWL 15, 160]

The differentials of the basic price-spread ratio

d(ΠΚPurely expansionary)= d( π”a”ΚPurely expansionary)

Differentials of the expansion of investment

δf = vδw + wδv   [CWL 15, 148-49]

The differentials of the behavior of the pure-surplus-income ratio

d(ΣFi)= d(vI”)   [CWL 15, 150]

The differentials of pure surplus income

An integral product is a composition of the component factors that went into it.  And the factors are incorporated at rates.  So, it is an invariant that the process of the production of goods and services is a set of rates of application of factors.

The invariant expression of the composition of factors is

qi  =  ΣΣqijk     [CWL 15, 30]

And it is an invariant and precise analytic distinction that the factoral elements of the basic productive process are in a point-to-point algebraic correspondence with completed factoral integrals exiting through sale into the standard of living. And it is an invariant that the factoral elements of the surplus productive process are in an algebraic correspondence with completed, point-to-point factoral integrals transitioning from being under point-to-point process to being part of the serial productive process. And these point-to-point “surplus” products impart a point-to-series acceleration to the overall productive process.  Thus, a Porphyrean distinction: either basic or surplus, but not both.

The general law of accelerator and accelerated, called the lagged technical accelerator, expresses the temporal, i.e. dynamic, relationship between basic and surplus production.

kn[f’n(t-a)-Bn] = f”n-1(t) – An-1   [CWL 15, 37]

The lagged technical accelerator allowing for slack and depreciation in a Schumpeterian creative-destruction economy

So, in the productive order we have – across all psyches, isms and political systems – the invariant relations of a) composition of factors, b) the correspondence of elements currently constituting the productive process with elements currently being completed by sale, and c) accelerator and accelerated flows.  With these invariant relations of the productive structure projected into the monetary order, we have the invariant relations of a) the composition of payrolls, b) the identity of expenditures and “macroeconomic costs,” and c) the lagged relation of investment and consumption.

The Special Theory of Relativity employs Pythagoras’ trigonometry and vectors to determine the spatio-temporal relations in the invariant ds.

ds2= dx2+ dy2+ dz2– c2dt2

Functional Macroeconomic dynamics employs trigonometry, vectors, and a metric tensor for the price-quantity relations in the calculation of the pretio-quantital expenditures, P’Q’, with which costs, p’a’Q’ and p”a”Q”, are equated.  Rather than merely repeat the clear vectoral analysis of CWL 15, we simply direct the reader to CWL 15, pp. 107-113.  Here we shall only list without commentary selected equations from that section so that the reader can preview the trigonometry of differing flows, which is universally applicable to all frames of reference and, therefore, general and invariant.

Z = Σpiqi                                                                                       [for expenditures]                                     (13)

Z + DZ = Σ(piqi+ pidqi+ qidpi+dpidqi)            [for expenditures]                         (14)

Therefore, Z = Σ(pi)(aqi)                                                                       [for costs]                                 

Z + DZ = Σ(piaqi+ pidaqi+aqidpi+dpidaqi)                 [for costs]                                  

Z = Σpiq= P.Q  = PQ cosA                                                                             (17)

Z + DZ = (P+dP)(Q + dQ) cos(A + dA)                                                             (20)

Therefore, Σ(piqi+ pidqi+ qidpi+dpidqi) =  (P+dP)(Q + dQ) cos(A + dA)       

DZ = PQ[(dP/P + dQ/Q + dPdQ/PQ) cos (A + dA) – 2 sin(dA/2) sin(A + dA/2)]  (21)

So, as the trigonometry of Special Relativity is general and applicable in all frames of reference in all instances of inertial transformations, so the trigonometry of FMD is general and invariant in all combinations and configurations of differing price-quantity flows.

If one would understand , not men’s notions of space of Space and Time, but the intelligibility immanent in Space and Time, then one must advance from reference frames to the geometrical principles and laws whose expression is invariant under transformations. Moreover, the geometry to be reached will coincide with the geometry determined by physicists in securing invariant expression for physical principles and laws. … However, such a geometry is abstract… in the sense that it consists in a set of abstract propositions and invariant expressions … [CWL 3, 171/194-5]

Paraphrasing:

If one would understand , not men’s notions of Price and Quantity, but the intelligibility immanent in pricings and quantities, then one must advance from conventional textbook static reference frames to the reference frames of the dynamical principles and laws whose expression is invariant under transformations.  Moreover, the trigonometry and geometry to be reached will coincide with the trigonometry and geometry determined by FMD in securing invariant expression for the principles and laws of the concrete, purely dynamic, economic process of production and exchange. … , such a geometry is abstract… in the sense that it consists in a set of abstract propositions and invariant expressions… .