Implicit Definition by Functional Relation; Scientific Significance of Terms

A.             Implicit Definition by Functional Relation; Scientific Significance of Terms Implicitly Defined by Their Functional Relations

We are doing science – in our case scientific macroeconomics.

Science is explanation in the form of terms related to one another, not in the form of terms related to us.

An understanding and appreciation of implicit definition is critical to understanding and appreciation of Lonergan’s scientific functional macroeconomic dynamics.

Implicit definition yields a set of terms interconnected with, and functionally related to, one another. And, since science is explanation in the form of terms related to one another, a set of terms-in-relation may, therefore, serve to be of scientific and explanatory significance.

Implicit Functions: In calculus, an explicit function is one in which one variable is expressed explicitly in terms of another variable(s), e.g. y = cos x. An implicit function, on the other hand, is an expression wherein terms are mutually responsive to, and mutually defined by, one another, e.g. x2 + y3 = 63.

Implicit Definition: In the search for complete explanation in functional macroeconomics, Lonergan employs the technique of implicit definition, wherein economic functionings are defined implicitly by their functional relations to other functionings. And to be implicitly defined is to be mutually defined.

Implicit definition is the mutual defining of terms by their relation to one another. The relation determines the meaning of the terms, and the meaning of the terms determines the relation between them.

This technique (implicit definition) has been used to great effect by David Hilbert in his Foundations of Geometry in which, for example, the meaning of a point and a straight line is fixed by the relation that two, and only two points, determine a line. [Gibbons, 1987, 313]

Note in the following excerpt: implicit definition is just the expression of the relational element; it picks out what is of scientific significance; it introduces us to complete generality. (D)ealing purely with relations, you have moved on to a further stage of abstraction, i.e. from an inadequate level of abstraction to an adequate level of abstraction. Implicit definitions are simply relational structures, and the terms of the relations can be left indeterminate, but can be filled with the elements of a particular field of inquiry to provide an explanatory theory.

Implicit definition, then, prescinds from the matter. … It is just the expression of the relational element. And it picks out what is of scientific significance, introducing us to complete generality. To use explanatory definitions which presuppose nominal definitions is to tie down (i.e. unfavorably restrict or truncate) your science to what you were thinking about at the start. If you use implicit definitions, you have (beneficially) thrown yourself open to all possible isomorphic cases. In other words, where the same implicit definition holds with regard to materially distinct things, you have an isomorphism. For example, with regard to points and lines in the Euclidean sense and points and lines in the mathematical sense, there is no similarity between the dot and the ordered pair of numbers, but every relation that holds between points and lines in the one sense holds between points and lines in the other sense. When you base your geometry on implicit definitions, you are dealing purely with relations; you have moved on to a further stage of abstraction. Implicit definitions are simply relational structures, and the terms of the relations are left indeterminate. When definitions are implicit, the concrete meaning can be anything that will satisfy them. [CWL 5, 46-7]

Lonergan brings to economics his background in mathematics, physics, and scientific method, and he concentrates on “the relational element”, dealing purely with relations”, and with “simply relational structures.” He is searching for one among many possible relational structures of “complete generality”, which one will accurately explain the interrelations of functionings which constitute the concrete economic process, such that we will have a general theory of equilibria and disequilibria. This general theory will be universally applicable to and explanatory of all configurations of functionings.

Thus, both equilibria and disequilibria are explained by a single cluster of formulations covering both the normative pure functioning and the variational stresses and strains requiring systematically necessary corrections.

In functional macroeconomic dynamics, the terms representing productive and monetary activities will be implicitly defined by the purely functional relations in which they stand to one another. The terms are mutually definitive and mutually exclusive of one another so as to achieve unambiguous distinctions between mutually conditioning production functions and monetary functions. The terms are defined by the functional “relations in which they stand.”

In brief, “basic” will implicitly define “surplus”, and vice versa; and “ordinary surplus” will implicitly define “pure surplus”, and vice versa. And each function’s meaning is based on the functional “relations in which they stand” vis a vis each other. Further, there is no overlap or commonality; the terms are mutually exclusive. Further, there are no gaps or missing terms; the terms are exhaustive of all possible cases. Thus the differential or difference-differential equations relating the velocities capture the instantaneous or the interval’s relationships among the most fundamental, interdependent, but mutually exclusive, velocitous functionings. As such, these differential equations will be comprehensive and explanatory of the objective economic process.

Thus, particular combinations of pricings and quantities are explained as constituting functional “cost” flows or “profit” flows. Pricings and quantities, rather than being taken as singularly fundamental and mistakenly used to construct a supposedly general explanatory macrodynamics, are generally understood in the light of first-order and second-order differential equations relating functional monetary flows in a primary and ultimate relativity. And these secondary determinations – pricings and quantities in combination – thus serve as particular boundary conditions or secondary determinate elements rather than as primary relational elements.

One might be reminded here of a parallel in hydrodynamics: if what is at issue is a general specification of the dynamics of free water waves, a premature introduction of general boundary conditions or worse, specific channel conditions, botches the analytic possibilities….the Robinson-Eatwell analysis is hampered by their building the (microeconomic) economic priora quoad nos of (corporate) profits, wages, prices, etc., into explanation, when in fact the priora quoad nos[1] (of corporate) profits, wages, prices, etc.) are last in analysis: they require explanation. [McShane, 1980, 124][2]

The movement in (Adam) Smith can be identified as a heretical enthusiasm for the priora quoad nos of price, leading to a reliance for salvation through price analysis which fathered Walras. [McShane, 1980, 109]

The real nature of aggregate cost flows and aggregate profit flows is revealed and defined by their interdependent functional relations in the general differential equations capturing the abstract intelligibility of the overall dynamic functioning called the concrete economic process.

Explanatory conjugates in functional macroeconomic dynamics – analogously to charge, electrical intensity, and heat in electromagnetics and thermodynamics – are not defined by Webster or an by an Accounting Standards Board. They are defined as functionings by their functional relations. This intelligibility of functional pricings-quantities flows is a relational intelligibility; it is achieved by a more adequate level of abstraction, and a more adequate generalization.

While an implicit definition is general in the sense that it can be invoked to try to explain any isomorphic cases in any field of inquiry, the significance of Lonergan’s implicit definition by functional relation is a) its complete generality in macroeconomics, b) its scientific significance, and c) its explanation of both equilibria and disequilibria in terms of divergences.

In brief Lonergan is looking for an explanation in which the terms are defined by the relations in which they stand, that is, by a process of implicit definition. …. “The significance of implicit definition is its complete generality. The omission of nominal definition is the omission of a restriction to objects which, in the first instance, one happens to be thinking about. The exclusive use of explanatory or postulational elements concentrates attention upon the set of relationships in which the whole scientific significance is contained.” [Gibbons, 1987, 313]

“Functional” is for Lonergan a technical term pertaining to the realm of explanation, analysis, theory; it does not mean “who does what” in come commonsense realm of activity. … Lonergan illustrates his basic meaning of ‘explanation’ by referring to D. Hilbert’s method of implicit definition: 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. ‘Thus the meaning of both point and straight line is fixed by the relation that two and only two points determine a straight line. … ‘In terms of the foregoing analysis, one may say that implicit definition consists in explanatory definition without nominal definition’ (Insight 12/36-37). … Lonergan went on to identify 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” (insight 37-38/61-62)…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 dynamic (accelerative) network of functional, mutually conditioning, and interdependent relationships of these rates to one another. [CWL 15 26-27 ftnt 27]

The variables discovered in a search for explanatory terms and relations of interconnection and interdependence are not the variables of corporate accounting.

Lonergan’s critique (shows that) by using the technique of implicit definition, the emphasis shifts from trying to define the relevant variables to searching heuristically for the maximum extent of interconnections and interdependence; and that the variables discovered in this way might not resemble very much the objects (or the aggregates) which, in the first instance, one was thinking about.   [Gibbons, 1987]

An ‘accountant’s unity’: that is a category used in (conventional) accounting. For Lonergan, (conventional) accounting generally denotes an enterprise within common sense which uses descriptive, as contrasted with explanatory terms (on these terms see Insight 37-38/61-62, 178-79/201-3, 247-48/272-73). Insofar as that is true, the accountant’s unity is not an adequate index for the normative, explanatory analysis of the productive process. [CWL 15, 26, ftnt. 26]

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. It is true that that sense is not at all an accountant’s sense of costs; … But however remote from the accountant’s meaning of the term “costs,” it remains that there is an aggregate and functional sense in which the fraction… is an index of costs. For 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]

Common sense never aspires to universally valid knowledge and it never attempts exhaustive communication. Its concern is the concrete and particular.. … Its procedure is to reach an incomplete set of insights … It would be an error for common sense to attempt to formulate its incomplete set of insights in definitions and postulates and to work out their presuppositions and implications. … Equally it would be an error for common sense to attempt a systematic formulation of its completed set of insights in some particular case; for every systematic formulation envisages the universal; and every concrete situation is particular. … so too, the plane of reality envisaged by common-sense meaning is quite distinct from the plane that the sciences explore. … It is clear that common sense is not concerned with the relations of things to one another. [CWL 3, 175-78/198-202]

Frisch’s failure was typical, both then and now.[3]

Frisch’s failure to develop a significant theory typifies the failure of economists who search for a dynamic heuristic. As well as a fundamental disorientation of approach there is also a tendency to shift to an inadequate level of abstraction with a premature introduction of boundary conditions in a determinate set of differential and difference equations. [McShane 1980, 114]

In order to construct the whole economic process, we deal with implicit definition and mutual exclusivities,. We are reminded of Porphyry. Nothing is left out. “As long as you proceed by dichotomies, you are constructing the whole universe.”

we move to a more detailed knowledge of the whole by a process of dichotomy. Porphyry’s tree divides being into material and nonmaterial. Nothing in the universe is left out when there is a division by contradictories. The material divides into the living and the nonliving, the living into the sentient and the nonsentient, and the sentient into the rational and the nonrational. As long as you proceed by dichotomies, you are constructing the whole universe. [CWL 10, 151]

Lonergan bypassed the terms of ordinary business parlance. As a scientist he sought explanation by purely intelligible relations. His Diagram of Rates of Flow[4] represents the dynamic interdependence of flows rather than accounting balances. The diagram is brilliantly explanatory.

Lonergan was seeking the explanatory intelligibility underlying the ever-fluctuating rhythms of economic functioning. To that end he worked out a set of terms and relations that ‘implicitly defined’ that intelligible pattern. When all was said and done the relations, and the terms they implicitly defined, were markedly different from either the terms of ordinary business parlance or the terms of neoclassical and Keynesian economic theory. Moreover, not only did Lonergan’s terms differ, but he also indicated that these aforementioned (neoclassical and Keynesian) terms were permeated, as were the terms of Newton’s theory of gravitation, with descriptive, nonexplanatory residues. Hence, just as a mathematical equation may be said to be the most adequate expression of purely intelligible relations among explanatory terms in certain instances – for example, Einstein’s gravitational field tensor equations – something closely akin to Lonergan’s diagram seems necessary for the realm of dynamic economic functioning. So, for example, the existence and manner of dynamic mutual interdependence of the two circuits of payment, basic and surplus, is not adequately expressed either by descriptive terms (since this pattern does not directly relate to the senses of anyone operating in a common-sense way in a concretely functioning economy) nor by the series of (simultaneous) equations that do not explicitly manifest the interchanging of ‘flows.’ [CWL 15, 179]

In the case of economic activities called economic functionings, the implicit definition by functional relation of one economic function-activity to another gives the definition of those two function-activities. Thus, velocitous functions implicitly define one another by the functional relations in which they stand vis a vis one another.

Speaking negatively, the meanings of the terms (the functionings) are not descriptive or nominal; the meanings are not determined by sensation, perception, Dr. Samuel Johnson, or Noah Webster. Thus the definitions are not merely nominal and scientifically useless; they are scientifically useful.

In Insight a distinction is made of three types of definition: nominal, explanatory, and implicit. Nominal definition supposes insight into the use of words. For example you can nominally define a circle as a perfectly round plane curve, and you can go on to use the word ‘circle’ correctly in the light of that definition. Again, you can define a straight line as a line that lies evenly between its extremes; and that gives you a good rule for using the words ‘straight line.’ ¶An explanatory definition adds a further element which, if not added in the definition, would have to be added by way of a postulate. Euclid defined a straight line as a line that lies evenly between its extremes, but he had to add the postulate, All right angles are equal. He defined the circle as the locus of coplanar points equidistant from a center, and consequently he did not have to add the postulate, All radii of the same circle are equal. You can put your postulates in the definitions or separate from them. If they are separate, the definitions are merely nominal; if they are in the definition, the definitions are explanatory. ¶Finally the postulational element of the explanatory definition can be used alone, and then the definition is implicit. Implicit definition is of far greater generality In Hilbert;s Foundations of Geometry, points and straight lines are defined by the postulate, A straight line is determined by two and only two points. Two points and a straight line are set in correlation; if we have two points, that will determine what we mean by a straight line, and if we have a straight line, that will determine what is meant by a point. The only thing that is settled is the relation between the two points and a straight line. [CWL 5, 45-46]

Feynman understood the difference between nominal definition and implicit, scientific definition. In remarks about good education, he stated:

“Have you got science? No! You have only told what a word means in terms of other words. You haven’t told them anything about nature – what crystals produce light when you crush them, why they produce light … “ An examination question would read,”What are the four types of telescope?” (Newtonian, Cassegrainian, …) Students could answer, and yet, Feynman said, the real telescope was lost: the instrument that helped begin the scientific revolution, that showed humanity the humbling vastness of the stars … [Gleick, 284]

Speaking positively, the meanings of the purely relational terms – whether geometry’s center, radius, circumference, plane, and equality of a circle or macroeconomic’s basic, surplus, costs, and pure surplus income – are fixed by the insight grasping their functional interrelations in a coherent whole.

“All of these terms can be defined by their relations to one another; they can all be implicitly defined.”

So, one may wonder Why, in treatment of economic phenomena, is there all this talk about insight? What’s the point? Well, insight, which supplies one with implicitly defined terms and relations upon which Lonergan, in Insight builds a science of human knowing and, in CWL 15 and 21, builds a science of macroeconomic dynamics is the primal source of the science. Insight is not extraneous. It constitutes our doing science. Insight supplies terms of scientific significance.

(In our analysis of human knowing) we are dealing with the set of fundamental terms: presentations, inquiry, insight, and conception. … we will have occasion to deal with further terms, reflection, reflective insight, and judgment. All of these terms can be defined by their relations to one another; they can all be implicitly defined. Presentations are what one inquires about and has insights into, so as to be able to formulate what is essential, significant, relevant, in the presentations. [CWL 5, 59]

In a complete explanation, all concepts hang together. The equations are consistent with and complementary to one another. They cohere. A science is coherent.

(In) 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]

… (The)point-to-point basic process and the point-to-line surplus process are invariants across all political systems and, within the exchange process, among all possible production and sale of goods and services. [CWL 15, ?]

Insight into macoeconomics’ functional interrelations finds its adequate expression in an abstract formulation. The terms and relations can be set in a mathematical form which is isomorphic with and faithfully represents the interrelations of the concrete economic process.

Again, to take perhaps a simpler and more familiar example, if someone is doing physics and you open his book, what do you find? You find just mathematical equations. He is solving problems, and what is it? It is more mathematics.
Why do you say he is doing physics? He seems to be doing mathematics all the time. It is because there are regions of mathematics that are isomorphic with physical reality. There is the same relational structure between a given mathematical theory or system as there is between events that can be observed.   This is another case, a big case, of isomorphism: on the one hand, mathematical expressions, and on the other hand, physical events. There is the same relational structure. But in the mathematical case, the relational structure links symbolic expressions, or mathematical concepts, with one another, while in the physical case what are related are concrete physical events, wave lengths that you observe through a machine and so on. ¶ So there is an isomorphism of geometry, algebra, physics; the same relational structure can be found in all three. Consequently, one’ symbolism can be given a geometrical interpretation, or an algebraic interpretation, or a physical interpretation. [CWL 18, 32-33]

The fundamental terms of macroeconomic dynamics, because the process is dynamic, will be functional velocities and accelerations defined by the functional relations in which they stand with one another in their constitution of the current, purely-dynamic economic process. And these relations of functionings to one another can be given mathematical formulation which explain the process.

In functional macroeconomic dynamics one cannot think properly of functional surplus-income flows without thinking properly of functional basic-income flows, and vice versa. They implicitly define one another by their functional relations and constitute a pure Porphyrean dichotomy of the entire process. The definitions are purely relational.

If a current-point-to-a-current-point correspondence (basic ) is distinguished from and related to current-point-to-future-indeterminate-series correspondence (surplus) by their distinct correspondences with points[5] currently exiting into the standard of living, and if these distinct correspondences, named basic and surplus, cover the whole process in a dichotomous Porphyrean arrangement, then one cannot think of basic without thinking of surplus, and vice versa. The intelligibilities of each are given implicitly by the other.

Basic production (point-to-point in an algebraic function of the first degree[6]) is what limits and determines surplus production, and surplus production (point-to-line in an indeterminate series) is what limits and determines basic production. Production is of one or the other.

Productive flows have concomitant dummy money flows. The production of a product or service is completed by its sale. A product is “under process” in the current process until it is sold. Productive flows are correlated with their payments flows.

So, in the monetary order, aggregate macroeconomic costs limit and, thereby, determine aggregate macroeconomic income-for-investment.

  • Investment “cost” (π”α”Κ”, pure surplus income) leaps across the analysis to implicitly define and to be identified as investment revenues (Π”Κ”)[7]
  • Conversely, investment revenues (Π”Κ”) leap across the analysis to implicitly define and to be identified as investment cost (π”α”Κ”)

And, since macroeconomic cost is what limits and determines so-called macroeconomic profit; and so-called macroeconomic profit is what limits and determines macroeconomic costs.

  • Cost-Outlays (p’a’Q’ and p”a”Q”R&M) leap across the analysis to be identified as expenditures E’
  • Profit-outlays (π”α”Κ”) leap across the analysis to be identified as investment E”

By the purely relational definitions of Lonergan’s functional macroeconomic dynamics, basic revenues are identified as macroeconomic costs; not the accountant’s costs, but rather the functional macroeconomic costs. Thus, by the functional macroeconomic identity, it is a truism that basic revenues equal costs used as incomes in the basic circuit, and pure surplus revenues equal investment outlays used as pure surplus income in the surplus circuit.

The whole structure is purely relational. Thus the relations may be formulated purely mathematically with the terms having functional economic content.

The whole structure is relational: one cannot conceive the terms without the relations nor the relations without the terms. Both terms and relations constitute a basic framework to be filled out … by full information on concrete situations. [CWL 3, 492/516]

 

 

[1] The priora quoad nos – first for us – are the things which we notice first because they are related to our sensitive and perceptive selves, e.g. hot and cold, fast and slow, red and green.. The priora quoad se – first among themselves – are the things or terms which are related to each other, e.g. pressure, volume, temperature; space, time, mass; space time, charge; spacetime and energy, etc.

 

[2] More fully, the quote is: One might be reminded here of a parallel in hydrodynamics: if what is at issue is a general specification of the dynamics of free water waves, a premature introduction of general boundary conditions or worse, specific channel conditions, botches the analytic possibilities….the Robinson-Eatwell analysis is hampered, not only by an absence of paradigmatic heuristic thinking in a field whose principles involve ends, but also 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, Philip (1980) Lonergan’s Challenge to the University and the Economy, (Washington, D.C.: University Press of America) P. 124[2]

 

[3] Ragnar Frisch, 1895-1973, Nobel Memorial Prize in Economic Sciences in 1969

[4] CWL 15, 55

[5] The correspondences are of component current factors of production with the components integrated in a completed product.

[6] CWL 15, 28-29

[7] Next, both R’ and R” stand as double summations to activities in basic and surplus industry, respectively. The analysis leaps across the double summations to the initial elements. O’ is the aggregate of initial basic payments during the given interval; O” is the aggregate of initial surplus payments during the same interval. These rates may be named basic outlay and surplus outlay, respectively; they are the payments of wages and salaries, rents and royalties, interest and dividends, and allocations to depreciation, sinking funds, undistributed profits; they are the rewards of the ultimate factors of production in the basic stage and in the surplus stage, respectively, of the productive process. CWL 15, 47