CWL 3, Chapter II:  “Heuristic Structure of Empirical Method”

Introductory

This entry focuses on a) the general form of explanatory classical science, and b) on scientific explanation vs. non-explanatory common sense. It is constituted largely by excerpts and paraphrases relevant to empirical science, and therefore, also to what constitutes the empirical science of macroeconomics. Though we are herein confined to excerpts and commentary, we recommend strongly that the serious self-educating student read CWL 3, Chapter II, in its entirety.

[CWL 3] Lonergan, Bernard J. F. (1957 ) Insight, A Study of Human Understanding, Longmans, Green and Co. Ltd., London; and (1997) Toronto: University of Toronto Press [CWL 3, 1957 edition/1997 edition] 

First, here are eight introductory pointers:

.1) Empirical inquiry has been conceived as a process from description to explanation.  We begin from things as related to our senses.  We end with things as related to one another.  Initial classifications are based upon sensible similarities.  But as correlations, laws, theories, systems are developed, initial classifications undergo a revision. Sensible similarity has ceased to be significant, and definitions consist of technical terms that have been invented as a consequence of scientific advance…. The basic notions of physics are a mass that is distinct from weight, a temperature that differs from the intensity of the feeling of heat, and the electromagnetic vector fields.¶ Now the principal technique in effecting the trqnsition from description to explanation is measurement. We move away from colours as seen, sounds as heard, from heat and pressure as felt. In their place, we determine the numbers named measurements. In virtue of this substitution, we are able to turn from the relations of sensible terms, which are correlative to our senses, to the relations of numbers, which are correlative to one another.  Such is the fundamental significance and function of measurement. (CWL 3, 164-5)

.2) Our direct understanding abstracts from the empirical residue. (CWL 3, 516/540)

.3) The empirical residue consists of individuality, the space-time continuum, particular times and places, particular incidental quantities and prices, constant velocity, non-systematic divergence of actual frequencies (See CWL 3, Index under “empirical residue”)

.4)  … inasmuch as we are understanding, we are grasping the universal apart from its instances, the limit apart from the continuum, the invariant apart from particular places and times (or particular quantities and prices), the ideal frequency apart from the non-systematic divergences of actual frequencies. (CWL 3, 516/540)

.5) The universal can be thought but cannot be without the instance; the limit can be thought but cannot be without the continuum; the invariant can be considered but does not exist apart from particular places and times (or particular quantities and prices); ideal frequencies can be formulated but cannot be verified apart from actual frequencies.  The empirical residue, then, is at once what spiritual intelligibility excludes and what material intelligibility includes. (CWL 3, 516/540)

.6) Explanation: A distinction has been drawn between description and explanation.  Description deals with things as related to us.  Explanation deals with the same things as related among themselves.  … description supplies, as it were, the tweezers by which we hold things while explanations are being discovered or verified, applied or revised. … [CWL 3, 291/316]

.7) And in General Relativity the significance of measurement with respect to synchronicity mandates the use of tensors for their transformation properties for understanding and expression.  So, the relativity of the tensors to one another is itself the invariant. (self)

8) On the present analysis then, the difference between the anticipations represented respectively by General and by Special Relativity is that , while both expect invariant mathematical expression to result from the abstractness of principles and laws, General Relativity implements this expectation by invoking a direct insight into the significance of measurements but Special Relativity implements it by invoking an inverse insight into the insignificance of constant velocity. [CWL 3, 42/66]

Actual excerpt #1:

Whenever the scientist is seeking to determine some indeterminate function, he is relating things to one another.  And that is just what common sense does not do.  It understands things in their relations to us.  Thus we have Whitehead’s two worlds.  Eddington said that he had two tables in his room: there was a brown table, made of oak, solid, that had a certain shape, and then there was the scientific table that consisted of electrons bounding about and so on.  Most of it was empty space.  Where do the two tables come from?  They come from two approaches.  Common sense understands the table in its relations to us: a table is something you can lean on, something you do not bump into, something you can use for writing; it has a certain visible appearance, certain tactile qualities, and so on.  The table is integrated into the flow, the interests, the Sorge, the concern, of the subject.  But science relates measurements to one another; and it does not have to go very far along that route to discover that it is introducing an entirely new world…. Common sense, like grammar, is egocentric; it concerns the intelligibility of things for me.  In grammar, time and tense relate to my time, my present.  The meaning of fundamental adverbs like ‘here’ and ‘there’ is related to me.  The first person is the point of reference. … The scientific procedure of relating things to one another builds up maps and clocks that leave the whole commonsense approach to things out of the picture. (CWL 10, 140) (Click here re common sense)

#1 Paraphrase and/or commentary:

Whenever the economist is seeking to discover the immanent explanatory intelligibility of the current, purely dynamic process, he is relating abstractly correlated functionings to one another.  And that is just what bookkeeping and National Income accounting do not do.  They understand things in their relations to us. … But scientific macroeconomics relates measurements of explanatory functional flows to one another; and it does not have to go very far along that route to discover that it is introducing an entirely new world…. It revises the whole commonsense approach of bookkeeping and National Income accounting to introduce an entirely new world. (from CWL 10, 140)

Actual, then paraphrased, excerpts mainly from

CWL 3, Chapter II: “Heuristic Structure of Empirical Method”

Subsection II.2.2.3; “Classification and Correlation”;  (CWL 3, 37-8/61-2)

Actual excerpt #2:

Now sensible similarities, which occur in the relations of things to our senses, may be known before ‘the nature of …’ has been discovered.  They form the basis of preliminary classifications.  They specify the ‘nature of …’, so that one states that one is seeking the nature of colour, the nature of heat, the nature of change, the nature of life. … On the other hand, similarities that reside in the relations of things to one another are the proximate materials of insight into nature.  Hence, the empirical inquirer, to emphasize this fact, will say that his objective is not merely ‘the nature of …’ but more precisely, the unspecified correlation to be specified, the undetermined function to be determined. … The second step in the generalization is, then, that just as the mathematician states that he seeks an x which has such and such properties, so too the empirical inquirer states that he seeks a ‘nature  of …’ where the nature antecedently is specified by a classification based on sensible similarity and consequently will be known when some indeterminate function is determined.  [CWL 3, 37-8/62]

#2 Paraphrase and/or commentary:

Now sensible similarities, which occur in the relations of things to our senses, may be known before ‘the nature of …’ has been discovered.  They form the basis of preliminary classifications.  They specify the ‘nature of …’, so that one states that one is seeking the nature of colour, the nature of heat, the nature of change, the nature of life. … On the other hand, similarities that reside in the relations of things to one another are the proximate materials of insight into nature.  Hence, the empirical economist-inquirer, to emphasize this fact, will say that his objective is not merely ‘the nature of’ the economic process, but more precisely, the unspecified explanatory correlation to be specified, the undetermined function or mathematical expression to be determined whose pattern of variables, operators, and interrelations is isomorphic with the patterns immanent in the measured data. … The second step in the generalization is, then, that just as the mathematician states that he seeks an x which has such and such properties, so too the empirical economist-inquirer states that he seeks a ‘nature  of …’ where the nature antecedently is specified by a classification based on sensible similarity and consequently will be known when some indeterminate, classical, explanatory function, or set of coherent explanatory functions, is determined.  [CWL 3, 37-8/62]

Subsection .II.2.2.4 “Differential Equations”;  (CWL 3, 38-9/62-4)

Actual excerpt #3:

… besides individuality, the continuum also pertains to the empirical residue and, as well, that just as the universal is reached by abstracting from the individual, so also the techniques of the infinitesimal calculus deal with the intelligibility reached by abstracting from the non-countable infinity of the continuum.  ¶ The third step, then, in our generalization is the observation that, where the mathematician says, Let x be the required number, the empirical inquirer can say, Let some indeterminate function, f(x, y, … z, = 0, be the required function …, so too the empirical inquirer can move towards the determination of his indeterminate function by writing down differential equations which it must satisfy.  The procedure is named by Lindsay and Margenau in their Foundations of Physics, the“Method of Elementary Abstraction”. (CWL 3, 38-9/62-3)

#3 Paraphrase and/or commentary:

… just as the universal is reached by abstracting from the individual, so also the techniques of the infinitesimal calculus, as applied to the temporal continuum of quantities being semi-continuously produced and sold at prices, deal with the intelligibility of the economic process reached by abstracting correlations of correspondences among the continuous activities identified by precise analytical distinctions as point-to-point, point-to-line, and higher.  ¶ The third step, then, in our generalization is the observation that, where the mathematician says, Let x be the required number, the empirical economist-inquirer seeking the abstract, explanatory formulation of macroeconomics can say, Let some indeterminate, but implicit, function, f(x, y, z, …) = 0, be the required function …, so too the macroeconomist, employing the technique of implicit definition, which yields the concomitances among flows, abstracting correlatives from the measured data, and observing the canon of complete explanation, can move towards the determination of his indeterminate function by observing data, tabulating the data, discovering by insight abstract explanatory correlations in the data, and writing down differential equations isomorphic with the patterns in the data.  ¶ The procedure is named by Lindsay and Margenau in their Foundations of Physics, the ‘Method of Elementary Abstraction’.  They illustrate it by examining the general features of a fluid in motion. (CWL 3, 38-9/62-3)

Subsection .II.2.2.5 “Invariance”; (CWL 3, 39b-43/64-7)

Actual excerpt #4:

… 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.

However, to determine under which group of transformations invariance is to be achieved, some further principle has to be invoked and, in fact, in different scientific theories different principles are invoked.  Of these the most general is the principle of equivalence which asserts that physical principles and laws are the same for all observers. (CWL 3, 40/64-5)

#4 Paraphrase and/or commentary:

… the meaning of invariance is that

1. all empirical scientists of Functional Macroeconomic Dynamics expect their correlations and laws to be independent of merely pretio-quantital  differences
2. scientific economists are confronted with a special difficulty inasmuch as they have to use reference frames as to velocitous and accelerative, interdependent, functional magnitudes, price indexes, currency exchange rates, and degrees of complexity.
3. economists surmount their peculiar difficulty by a) generalizing based on precise analytical distinctions among the distinct correspondences of point-to-determinate point, point-to-indeterminate line, etc., b) expressing their general principles and laws in mathematical equations – constituted by abstract, explanatory correlative conjugates of functional fows – which general principles and laws remain invariant under transformations of frames of reference as to magnitudes, currency price, and degree of complexity.

See under Table of Topics in the menu bar “Lonergan’s Goal: Generalization and Practical Precepts for Free People.”  Click here.

Lonergan’s Goal: Generalization and Practical Precepts for Free People

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Lonergan’s Goal: Generalization and Practical Precepts for Free People

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Actual excerpt #5:

… not only are scientific discoveries independent of the particular place and time of their origin but also they can claim to be equally and uniformly valid irrespective of merely spatio-temporal differences. (CWL 3, 40/64)

#5 Paraphrase and/or commentary:

… not only are the explanatory economic discoveries independent of the particular quantities and prices of their origin but also, the scientific economist can claim that the discoveries – – because the explanatory relations among interdependent, correlated, functional, velocitous and accelerative conjugates are implicitly defined by their functional relations with one another in an organic, unitary whole – – their mathematical expression is equally, uniformly, and universally valid.

Actual excerpt #6:

… unless a special effort is made, change in the choice of reference frame may result in the change in the statement of principle or law.  On the other hand, when a special effort is made, the mathematical expression of physical principles and laws undergoes no change in form despite changes in spatio-temporal standpoint and then the mathematical expression is said to be invariant under some specified group of transformations. (CWL 3, 40/64)

#6 Paraphrase and/or commentary:

 All science begins from particular correlations, but the key discovery is the interdependence of the whole.…its basic terms are defined by their functional relations. [CWL 15, 53, 54, and 177]

Let us say, then, that for every basic insight there is a circle of terms and relations, such that the terms fix the relations, the relations fix the terms, and the insight fixes both.  If one grasps the necessary and sufficient conditions for the perfect roundness of this imagined plane curve, then one grasps not only the circle but also the point, the line, the circumference, the radii, the plane, and equality.  All the concepts tumble out together, because all are needed to express adequately a single insight.  All are coherent, for coherence basically means that all hang together from a single insight.  [CWL 3, 12/36]

… unless a special effort is made by the employment of implicit definition at an adequate level of abstraction, change in the choice of reference frame may result in the change in the statement of principle or law.  On the other hand, when a special effort is made in the development of the transformation equations, the mathematical expression of macroeconomic principles and laws undergoes no change in form despite changes in pretio-quantital standpoint; and then, the mathematical expression is said to be invariant under any properly specified group of transformations.

Actual excerpt #7:

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. it is, for example, not the appearance of colours but the general explanation in terms of wave-lengths that is exactly the same no matter what may be the state of observers’eyes, the lighting by which they see, or the speed with which they may happen to be in relative motion. [CWL 3, 41/65 ]

#7 Paraphrase and/or commentary:

… Clearly if macroeconomic laws are reached by eliminating the relations of things to the senses and inner feelings of observers and by arriving at correlations grasped by insight into the measurements of the data of the process, then there exists an extremely solid foundation for the affirmation that principles and laws of nacroeconomics are the same for all observers because they lie simply and completely outside the range of sense and feeling.  The immanent intelligibility of macroeconomics is not the experience of personal pleasure or pain associated with an increase or reduction of one’s compensation and purchasing power; but rather the immanent intelligibility consists of the general invariant principles and laws which explain – coldly and objectively by implicit, abstract, functional terms and relations – the aggregate data of the whole, organic, economic process.

Actual excerpt #8:

Accordingly, by the principle of equivalence (in Einstein’s Special Relativity), the mathematical expression of physical principles and laws is to be expected to be invariant as long as the transformation equations are continuous functions of real variables.  (CWL 3, 41/65)

(First postulate per Einstein) The laws of physics are the same in every inertial reference frame.  (Serway, 901)

(Second postulate per Einstein) The speed of light has the same value for all observers, independent of their motion or of the motion of the light source. (Serway, 901)

The speed of light in a vacuum, measured in any inertial reference frame always has the same value of c no mater how fast the source of light and the observer are moving relative to each other. (Cutnell and Johnson, 876)

(Also,) as one examines (the special theory of relativity) dispassionately one will probably be inclined to admit that its chief merit – even greater than the success of its detailed applications – is its insistence on the fundamental nature of the criterion of the invariance of physical principles with respect to the transformation of coordinates.  To be sure, in its restriction to inertial systems it goes only part of the way, but even to this extent it contains the germ of the general theory of which it was the natural forerunner.  (Lindsay amd Margenau, 354)

#8 Paraphrase and/or commentary:

Accordingly, by the principles of equivalence in Lonergan’s Special Relativity of normative Functional Macroeconomic Dynamics, the mathematical expression of macroeconomic principles and laws is to be expected to be invariant as long as the transformation equations connecting internal changes (of a) quantities, or b) the distribution of the supply of money circulating in the circuits to accommodate increasing or decreasing transactions) between frames of reference are practically continuous functions of real pretio-quantital variables.

The principles and laws (of the abstract, general, completely explanatory relations themselves, not the particular incidental magnitudes of quantities and prices, as though in an absolute isolation) of Functional Macroeconomic Dynamics remain invariant in any transformations between pretio-quantital reference frames of a closed economy.  

The relations among abstractly correlated functional explanatory conjugates, measured in any pretio-quantital, double-circuited reference frame, always have the same field-theoretic, general expression of normativity and equilibrium.

As one examines dispassionately and masterfully Lonergan’s abstract, explanatory hypothesis (called Functional Macroeconomic Dynamics, or Macroeconomic Field theory), one will become more and more inclined to admit that its chief merit – even greater than the success of its possible detailed applications – is its reliability and validity regarding its assuring by a) an adequate level of abstraction, b)  the principle of concomitance, and c) the technique of implicit definition of interdependent correlations, the invariance of the macroeconomic principles and laws explaining the systematic dynamics of the accurately measured data. The same principles and laws will apply under any transformation of the incidental magnitudes of quantitative or monetary variables among themselves.  The abstract, field-theoretic, general, and complete, explanatory formulation is valid and reliable.  FMD prescinds from human psychology and, thus, rejects protean subjectivity and insists on a purely scientific objectivity.  By its prescinding from human psychology, FMD is able to progress to being the intended objective explanatory theory, which the quarreling world so desperately requires.  

Lonergan’s intention was ‘to formulate the laws of an economic mechanism more remote and, in a sense, more fundamental than the pricing system. The Robinson-Eatwell analysis is hampered by their building the economic priora quoad nos of profits, wages, prices, etc., into explanation, when in fact the priora quoad nos are last in analysis: they require explanation. [McShane 1980, 124]

Key equations in Macroeconomic Field Theory are

• R’ = E’     (CWL 15, 54)
• R” = E”      (CWL 15, 54)
• I’ = O’ +M’      (CWL 15, 54)
• I” = O” +M”     (CWL 15, 54)
• G = c”O” –i’O’   (CWL 15, 54)
• G = c”O” –i’O’ = 0     the condition of dynamic equilibrium   (CWL 16, 50)
• M’ = (S’ – s’O’) + (D’ – s’I’) + G   (CWL 15, 54)
• M” = (S” – s”O”) + (D” – s”I”) – G   (CWL 15, 54)
• (S’-s’O’) = ΔT’ + (O’ – R’) + ΔR’   (CWL 16, 67)
• (S”- s”O”) = ΔT” + (O” – R”) + ΔR”   (CWL 16, 67)

Thus for example, assuming G = 0, and by the principle of concomitanceand by the functional role of credit to bridge time gaps, we have for the basic circuit the relations of functional flows among themselves:

• R’ = E’ = I’ = O’ +M’
• R’ = E’ = I’ = O’ +[S’ – s’O’] + [D’ – s’I’]
• R’ = E’ = I’ = O’ + [ΔT’ + (O’ – R’) + ΔR’] + [D’ – s’I’]

One can develop parallel equations for the surplus circuit, and then combine both circuits to calculate Gross Domestic Functional Flows, GDFF, to replace Gross domestic Product, GDP, as the primary reference for management of the economy.

Actual excerpt #9:

To implement this conclusion (re invariance), which is no more than a general anticipation based on cognitional theory, two further steps are required. First, the broad invariance that we have described has to be conceived precisely in terms of tensors.  Secondly, appropriate empirical hypotheses have to be formulated and verified.  But by those steps there are reached the General Theory of Relativity and the Generalized Theory of Gravitation, and incidentallly it may not be amiss to note that our remote anticipation offers a simple explanation for certain aspects of those theories.  For what was anticipated was a non-relatedness of abstract laws to observers.  It follows that the consequences of the anticipation should not be verified

1. if the laws lose their abstract character through particularization,*or
2. if investigation concentrates on the frequencies of concrete events accessible to the observers as seems to be the case in Quantum Mechanics. (CWL 3, 41-2/65-6)

#9 Paraphrase and/or commentary:

To implement this conclusion (of the invariance of macroeconomic principles and laws), which is no more than a general anticipation based on cognitional theory, two further steps are required. First, the broad invariance that we have described has to be conceived precisely in terms of vectors and tensors and probabilities of divergence necessitating systematic correction.  Secondly, appropriate empirical hypotheses have to be formulated and verified. But by those steps there are reached and verified the General Theory of Macroeconomic Dynamics. … For what was anticipated using a higher level of abstraction was a generalized explanatory dynamics of an objective, general, unitary dynamic process.  It follows that the consequences of the anticipation should not be verified if the laws lose their abstract, objective, general character through particularization, or by reconstitution in terms of non-systematic, human psychology and its subjective preferences and biases.

Generalization comes with Newton, who attacked the general theory of motion, laid down its pure theory, identified Kepler’s and Galileo’s laws by inventing the calculus, and so found himself in a position to account for any corporeal motion known. Aristotle, Ptolemy, Copernicus, Galilei, and Kepler had all been busy with particular classes of moving bodies. Newton dealt in the same way with all. He did so by turning to a field of greater generality, the laws of motion, and by finding a deeper unity in the apparent disparateness of Kepler’s ellipse and Galilei’s time squared. … Similarly the non-Euclidean geometers and Einstein went beyond Euclid and Newton. … The non-Euclideans moved geometry back to premises more remote than Euclid’s axioms, they developed methods of their own quite unlike Euclid’s, and though they did not impugn Euclid’s theorems, neither were they very interested in them; casually and incidentally they turn them up as particular cases in an enlarged and radically different field. … Einstein went beyond Newton by employing the new geometries to make time an independent variable; and as Newton transformed the formulation and interpretation of Kepler’s laws, so Einstein transforms the Newtonian laws of motion. … It is, , a scientific generalization of the old political economy and of modern economics that will yield the new political economy which we need. … Plainly the way out is through a more general field. [CWL 21, 6-7]

Functional Macroeconomic Dynamics takes a higher viewpoint and operates at a more adequate level of abstraction so as to achieve a general and universal field-theoretic explanation of the economic process.

“Functional” is a technical term pertaining to the realm of explanation, analysis, theory; it does not mean “who does what” in some commonsense realm of activity … Lonergan (identified) the contemporary notion of a “function” as one of the most basic kinds of explanatory, implicit definition – one that specifies “things in their relations to one another”…In Lonergan’s circulation analysis, the basic terms are rates – rates of productive activities and rates of payments.  The objective of the analysis is to discover the underlying intelligible and indeed dynamic (accelerative) network of functional, mutually conditioning, and interdependent relationships of these rates to one another. [CWL 15,  26-27  ftnt 27]

Actual excerpt #10:

A less general anticipation of invariance is contained in the basic postulate of Special Relativity. … On the present analysis then, the difference between the anticipations represented respectively by General and by Special Relativity is that , while both expect invariant mathematical expression to result from the abstractness of principles and laws, General Relativity implements this expectation by invoking a direct insight into the significance of measurements but Special Relativity implements it by invoking an inverse insight into the insignificance of constant velocity. [CWL 3, 42/66]

#10 Paraphrase and/or commentary:

A less general anticipation of invariance in Functional Macroeconomics Dynamics is contained in the basic postulate of macroeconomic relativity – the distinctions among point-to-point, point-to-line, point-to-surface, and higher correspondences. … On the present analysis then, the difference between the anticipations represented respectively by Lonergan’s General Relativity and by Lonergan’s Special Relativity is that , while both expect invariant mathematical expression to result from the abstractness of principles and laws, Lonergan’s General Macroeconomic Relativity implements this expectation by invoking a direct insight into the significance of measurements of the data of the economic process and insight into the abstract, implicitly defined terms and relations among the abstract correlations thereof; but Lonergan’s Macroeconomic Special Relativity implements the invariant mathematical expression by invoking an inverse insight into the insignificance of a) the single circuit, and b) the limited significance of the constant velocity of the stable, non-expansionary phase of a pure cycle of expansion.

There is a sense in which one may speak of the fraction of basic outlay that moves to basic income as the “costs” of basic production. … the greater the fraction that basic income is of total income (or total outlay), the less the remainder which constitutes the aggregate possibility of profit.  But what limits profit may be termed costs.  Hence we propose ….to speak of c’O’ and c”O” as costs of production, having warned the reader that the costs in question are aggregate and functional costs…. [CWL 15 156-57]

P’Q’ = p’a’Q’ + p”a”Q”  [CWL 15,156-62], (44)

P’Q’ – p’a’Q’ + p”a”Q” = 0

P’/p’ = a’ + a”(p”Q”/p’Q’), or  (45a)

J = a’ + a”R, and thus  (45b)

dJ = da’ + Rda” + a”dR ( 45c) (CWL 15, 156-62)

Thus, in Macroeconomic Special Relativity, the insignificance of a) the textbooks’ single circuit, and b) the constant monetary velocity of the stable phase points to the tensorial relativity of price and quantity to one another a) within supply and demand relations and b)  between the point-to-point, point-to-line, etc. circuits, in a pure cycle of expansion. And the set of functional relations among correlated functional flows provides a general explanation and is itself the invariant.  And in Macroeconomic General Relativity the significance of measurement … mandates the use of tensors (of the first degree) and their dot-products for their transformation properties for understanding and expression.  The relativity of the tensors to one another is itself the invariant.  And in FMD an inverse insight into the limited significance of equal-velocity flows of price times quantity, i.e. invariant by virtue of scalar equality, points to possible underlying differences in the scale factors of price and quantity components in a type of pretio-quantital relativity.  So, the pretio-quantital relativity itself of prices and quantities in their combinations with one another is itself the invariant.

II.2.2.6 “Summary”. (CWL 3, 44-6/67-70)

Actual excerpt #11:

So a direct insight into the significance of measurements yields the anticipations of General Relativity; an inverse insight into the insignificance of constant velocity yields the anticipations of Special Relativity; and a restriction of this inverse insight to the context of Newtonian dynamics yields the anticipations that sometimes are named Newtonian relativity. (CWL 3, 45/68)

#11 Paraphrase and/or commentary:

Measurements yield data; insight into the data yields classical abstract correlations and statistical ideal frequencies; these yield the formulations of classical or statistical systems; these constitute the explanatory normative theory upon which actual data converge.

So, a direct insight into the significance of measurements yielding correlations and ideal frequencies yields the anticipations of a multiple-circuited, credit-centered process called General Functional Macroeconomic Dynamics.

Inverse insights into the insignificance of a single circuit and into the limited significance of the constant velocity of a stable phase necessitate the search for the intelligibility of the cyclic phenomena of a double-circuited, changing-velocity, pure cycle of expansion. And the absence of these inverse insights to the context of aggregate microeconomics yields the sparse and incomplete anticipations that sometimes are named “exact but incomplete” Walrasian macrostatics. (CWL 3, 45/)

In conjunction with the reading of the above, please read CWL 3, Chapter V, “Space and Time”, subsections V.4.4.2, pages 164-67 “The Generic Notion of Measurement” and V.4.4.3, pages 167-70 “Differentiations of the Generic Notion.”

Also, we refer the reader to this website’s The Lack of Ultimacy in Price Theory: Prices are Last in the Analysis and Purely Relative and, in particular, to Subsection 4.f entitled The IMMANENT INTELLIGIBILITY of Prices as Real and Relative.  (Click here)

Further,

Per R. d’Inverno, 1992, under the heading of Chapter 13, “The Structure of the Field Equations”, d’Inverno reads Einstein’s implicit field equations of General Relativity from right to left, left to right, and back and forth.  d’Inverno states:

Before attempting to solve the field equations we shall consider some of their important physical and mathematical properties in this chapter.  The full field equations (in relativistic units) are

See d’Inverno, page 87, formula 6.85

G_ab= 8πT_{ab (pages)}

The field equations are differential equations for determining the metric tensor g_ab from a given energy-momentum tensor T_ab. Here we are reading the equations from right to left.  … one specifies a matter distribution and then solves the equations to ascertain the resulting geometry.

The field equations are equations from which the energy-momentum tensor T_ab can be read off corresponding to a given metric tensor g_ab. Here we are reading the equations from left to right.

The field equations consist of ten equations connecting twenty quantities, namely, the ten components of g_ab and the ten components of T_ab. Hence, from this point of view, the field equations are to be viewed as constraints on the simultaneous choice of  g_ab and T_ab.  This approach is used when one can partly specify the geometry and the energy-momentum tensor from physical considerations and then the equations are used to try and determine both quantities completely. [d’Inverno, 1992, 169]

Analogously, focusing on one of Lonergan’s implicit equations, the terms are implicitly defined by the relations in which they stand with one another.  We may read from left to right, right to left, or back and forth between right and left.

P’Q’ = p’a’Q’ + p”a”Q”  [CWL 15,156-62].

John A. Wheeler (Princeton) commented colloquially about Einstein’s general explanatory scheme,

G_ab= 8πT_ab

“Matter tells space how to curve, and space tells matter how to move.” [John A. Wheeler as quoted in Lieber, 2008, 350]

Expanding on Wheeler: On the left, the distribution of matter given by the Einstein tensor G_ab , containing the metric tensor g_{ab ,} determines the 4-dimensional geometric curvature of space-time along which matter can move; and, on the right, the energy-momentum tensor of movement T_ab specifies the general relations among possible, energy-momentum motions.  And one may calculate the quantities of one side from the quantities of the other, or one may proceed forth and back from an initially-partial knowledge on each side to a complete knowledge on both sides.

And, with reference to the Diagram of Rates of Flow, by the law immanent and operative in cognitional process is that similars are similarly understood and by the use of analogy,  we would say colloquially about Lonergan’s scientific, non-colloquial implicit equating of normatively concomitant variations of basic macroeconomic Expenditures, P’Q’, on the left, and basic macroeconomic Costs, p’a’Q’ + p”a”Q”, on the right. 

the selling prices and quantities on the left (basic macroeconomic Expenditures) of what is purchased are implicitly and imperiously normatively defined by, and concomitant with, basic Incomes (macroeconomic costs-Outlays becoming Incomes) how much they must spend; and basic pretio-quantital Incomes from Outlays on the right tell the dualistic pretio-quantitalities, P’Q’ how much they can afford to pay for what is offered for sale.

Also, with reference to Figures 24, 26, and 27, and using as an example the logistic equation

or  alternatively  

we have said colloquially about Lonergan’s general explanatory scheme,

The curvatures in the equations of potential of the distribution of invention, skills, and finite resources (economic matter)”constitute” by “identity” the curvature-geometry of an expanding process; and the two costs-incomes tensors (of one degree) constituting the dualistic flows of purchasing power tell kinetic pretio-quantital production how to implement the basic and surplus expansions; and the normative implementation or effecting of the two normative curvatures by basic and surplus production tells the potential, constituted by capacities and resources how it is kinetically and coordinatedly, and normatively, moving for maximum implementation.