Functions Are Not Seen, But Must Be Understood

Functions are not seen, but must be understood. (Catherine Blanche King, private communication)

A systematic explanation, then, requires a normative theoretical framework.  The basic terms and relations of such a framework would specify the distinctions and correlations that articulate the causes, which are not necessarily visible, of events that are apparent to all.  (CWL 15,  Editors’ Introduction, lv)

The economic process is not understood by considering how it affects one’s own wealth and well-being, or by tallying Gross Domestic Product and National Income, or by searching for what might be the visible external cause, or by taking a look through special lenses at how things appear to be going out there; rather the immanent intelligibility of the economic functioning is gained by insights which yield abstract explanatory formulations, whose terms represent the process’ constituent functions as implicitly-defined by the relations in which they stand interdependently with other functions immanently in the dynamic process.

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.“ [CWL 15,  26-27  ftnt 27]

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, first, by the advance of the sciences and, secondly, by full information on concrete situations. [CWL 3, 492/516]

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 conceptual terms and relations yielded by insight are not seen; they are invisible products of mind; they cannot be seen, touched, heard, smelled, or tasted.   The abstract terms functionally related to one another in an insight implicitly define one another by their functional relations to one another. The relations, and the terms which they implicitly define, are not seen by the naked eye; they are yielded by an act of understanding which we call “insight”.  The explanation of the overall economic functioning is not seen or felt; it consists of terms and relations in a formulation which is isomorphic with the form of the economic functioning.

…  The mathematical meaning of an expression resides in the distinction between constants and variables and in the sign or collocations that dictate operations of combining, multiplying, summing, differentiating, integrating, and so forth. [CWL 3, 18-19/43]

Per the immanent intelligibility of thermodynamics, “heat” is not felt; it is defined by the relations of a law.  And by the immanent intelligibility of electromagnetics, magnetic and electric field intensities are not seen or felt; they are defined by the relations of the equations of Maxwell.

(Re invisible definitions in other sciences:) heat might be defined implicitly by the first law of thermodynamics, and the electric and magnetic field intensities, E and H, might be regarded as vector quantities defined by Maxwell’s equations of the electromagnetic field. [CWl 3, 80/102-03]

ΔUsystem = Q – W (first law of thermodynamics)

where ΔUsystem ,is the change of energy in the system, Q (sometimes symbolized as ΔQ) is the heat added to the system and W (sometimes symbolized as ΔW) is the work done by the system .

  • ∇ X E = (-1/c)H’
  • ∇ X H = (+1/c)E’
  • ∇ •Ÿ H = 0
  • ∇ •Ÿ = 0
  • Clerk-Maxwell’s equations

In the science of macroeconomics, explanatory functionings are implicitly defined by empirically verified correlations; and we call such terms “explanatory conjugates”.

… verified correlations necessarily involve the verification of terms implicitly defined by the correlations; …what is verified accurately is not this or that particular proposition but the general and abstract proposition on which ranges of ranges of particular propositions converge. … there is a fundamental heuristic structure that leads to the determination of conjugates, that is, of terms defined implicitly by their empirically verified and explanatory relations.  Such terms as related are known by understanding, and so they are forms. Let us name them conjugate forms.[quoted more fully below from CWL 3, 435/460]

What we have called the intelligibility immanent in sensible data and residing in the relations of things to one another might be named more briefly formal causality … [CWL  3, 78/101-102]

Again, abstract conceptual relations and terms of explanatory formulations cannot be seen, touched, heard, smelled, or tasted; nor are they mere mental images.  They are invisible products of acts of understanding.

Key ideas in this section include:

  • Explanation
  • Abstraction as the primary component in explanation of the dynamic concrete process.
  • Implicit definition by equations of functional relation
  • Images, necessity, and impossibility:
  • Basic terms as precise functional distinctions
  • Functional velocities as the basic terms of the dynamic macroeconomic process

Because the macroeconomic process is a unitary whole, some key ideas pertain to more than one aspect; they overlap.  Consequently, some excerpts will be quoted more than once.  Also, since Lonergan’s Functional Macroeconomic Dynamics, AKA Macroeconomic Field theory is a unitary whole, whose every aspect can be seen and understood in a single view.  “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]

What follows is largely a gathering of excerpts.  Many of the quotes are from CWL 3, – “the most significant book of the twentieth century” (McShane) – whose ideas on human knowing and on empirical method, underpin Lonergan’s economics.  The excerpts are relevant to the leading  idea that

Functions are not seen, but must be understood. (Catherine Blanche King, private communication)

 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]

Some commonsense everyday utterances are in some instances employed as abstract technical terms of explanation.

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]

Classes of payments quickly become rates of payment standing in the mutual conditioning of a circulation. [CWL 21, 111]

Abstraction:

The concept is a product of a person’s abstracting, understanding, relating mind.  The concept is not “seen.”  It is an element of the “word” which issues from the act of understanding called “insight.”  The insight “produces” primary abstract terms and relations.  Insight’s formulation explains the concrete process by application to secondaryconcrete price and quantity determinations from the non-systematic manifold.

Thus, while FMD’s analysis of the primal productive process yields distinctions of point-to-point and point-to-line, and while its primary field-theoretic relativities are purely abstract, the explanation of the concrete process has two components; it consists of 1) primary abstract relations applied to 2) secondary concrete determinationsfrom the non-systematic manifold. (CWL 3, 494-95/515-16)

Abstraction is the attainment of insight that reveals in the data what is variously named as the significant, the relevant, the important, the essential, the idea, the form. (CWL 3, 88/112) … intelligence concentrates on the significant to abstract from all else. (CWL 3, 356/380)  To abstract is to grasp the essential and to disregard the incidental, to see what is significant and set aside the irrelevant, to recognize the important as important and the negligible as negligible. (CWL 3, 30/55)  Check out 141, 160, 130-31, 175-76

… it is necessary to distinguish in concrete relations between two components, namely, a primary relativity and other secondary determinations.  Thus, if it is true that the size of A is just twice the size of B, then the primary relativity is a proportion and the secondary determinations are the numerical ratio, twice, and the two observable sizes.  Now ‘size’ is a descriptive notion that may be defined as an aspect of things standing in certain relations to our senses, and so it vanishes from an explanatory account of reality.  Again, the numerical ratio, twice, specifies the proportion between A and B, but it does so only at a given time under given conditions; moreover, this ratio may change, and the change will occur in accord with probabilities; but while probabilities will explain why objects like A and B every so often have sizes in the ratio of two to one, they will not explain why A and B are in fact in that relation here and now; and so the numerical ratio, twice, is a non-systematic element in the relation.  However, if we ask what a proportion is, we necessarily introduce the abstract notion of quantity and we make the discovery that quantities and proportions are terms and relations such that the terms fix the relations and the relations fix the terms.  For the notion of quantity is not to be confused with a sensitive or imaginative apprehension of size; a quantity is anything that can serve as a term in a numerical ratio; and, inversely, a proportion, in the present context, is a numerically definable ratio between two quantities. [CWL 3, 491]

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, first, by the advance of the sciences and, secondly, by full information on concrete situations. [CWL 3, 492/516]

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 economic priora quoad nos of profits, wages, prices, etc., into explanation, when in fact the priora quoad nos[1] 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]

…  Newton demonstrated that if there is a central field of force, that is, a force that causes an acceleration according to the law of inverse squares and is concentrated at the center of a field, then any body moving in that field will move along a conic section, such as an ellipse, a circle, a hyperbola, and so on.  You can see that this theorem includes the common matter; there is something that you can imagine, namely the conic section.  And this type of mechanics is determinist: it includes not only the intelligible form, but also an element of the matter.  On the other hand, quantum theory deals with what it knows to be processes that cannot be imagined.  It is a higher level of abstraction, and getting away from anything that can be imagined is connected with the fact that quantum theory is statistical – …  ¶ So you can see how even the ideas of definition and abstraction have become much more fluid.  Scientific thinking is much more versatile, much more attentive to all the possibilities of the fundamental act that is insight into sensible data.  I have given a series of illustrations of this.  (CWL 10, 126-27)

 Frish’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]

Ought there not to be introduced a technical term to denote this type of intelligibility?  … The intelligibility that is neither final nor material nor instrumental nor efficient causality is, of course, formal causality…What we have called the intelligibility immanent in sensible data and residing in the relations of things to one anothermight be named more briefly formal causality … [CWL  3, 78/101-102]

Implicit definition by equations of functional relation:

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.“ [CWL 15,  26-27  ftnt 27]

… the initial procedure of description gradually yielded to definition by relation; …  But definition by this type of relation is explanatory, and so descriptive procedure was superseded by explanatory. (CWL 3, 334/358)

Now, at the root of classical method there are two heuristic principles.  The first is that similars are understood similarly, that a difference of understanding presupposes a significant difference of data.  The second is that similarities, relevant to explanation, lie not in the relations of things to our senses but in their relations to one another.  Next, when these heuristic principles are applied, there  result classifications by sensible similarity, then correlations, and finally the verification of correlations and of systems of correlations.  But verified correlations necessarily involve the verification of terms implicitly defined by the correlations; and they do not involve more than such implicitly defined terms as related, for what is verified accurately is not this or that particular proposition but the general and abstract proposition on which ranges of ranges of particular propositions converge. … there is a fundamental heuristic structure that leads to the determination of conjugates, that is, of terms defined implicitly by their empirically and explanatory relations.  Such terms as related are known by understanding, and so they are forms. Let us name them conjugate forms.[CWL 3, 435/460]

Again,

In like manner, heat might be defined implicitly by the first law of thermodynamics and the electric and magnetic field intensities, E and H, might be regarded as vector quantities defined by Maxwell’s equations of the electromagnetic field. [CWl 3, 80/102-03]

The difficulty here is the absence of the functional classifications, basic and surplus, and the rhythmic ramifications.  It is like trying to have a clear view on fire-hazardous chemicals prior to the emergence of the perspectives of Lavoisier and Mendeleev. [McShane, 2017, 87]

Now as the statistical approach differs from the descriptive, the analytic differs from both.  Out of endless classificatory possibilities it selects not the one sanctioned by ordinary speech nor again the one sanctioned by facility of measurement but the one that most rapidly yields terms which can be defined by the functionalinterrelations in which they stand. [CWL 21, 112]

 Again, (Lonergan) approaches the focus armed with precise analytic distinctions upon which a superstructure of laws, coherent with one another and comprising a complete theory, may be constructed.  Paraphrasing [CWL 3, 80/103]

Both nominal and explanatory definitions suppose insights.  But a nominal definition supposes no more than an insight into the proper use of language.  An explanatory definition, on the other hand, supposes a further insight into the objects to which language refers.  The name, circle, is defined as a perfectly round plane curve, as the name straight line is defined as a line lying evenly between its extremes.  But when one goes on to affirm that all radii are equal or that all right angles are equal, one is no longer talking merely of names.  One is making assertions about the objects which names denote.  (CWL 3, 11/36)

Explanatory conjugates:

… Pure or explanatory conjugates, … are correlatives defined implicitly by empirically established correlations, functions, laws, theories, systems. [CWL 3, 79-80/101-03]

The explanation of data consists in a process from experiential conjugates towards pure conjugates.  Therefore, from extensions and durations as experienced, there must be a process to extensions and durations as implicitly defined by empirically established laws. [CWL 3, 85/108]

Paraphrasing:  The explanation of data consists in a process from experiential conjugates towards pure conjugates.  Therefore, from quantities and prices as experienced in everyday life in a non-systematic manifold, there must be a process to quantities and prices as implicitly defined and interpreted in the light of scientifically significant variables. [CWL 3, 85/108]

As pure conjugates, extension and duration are defined implicitly (in relativity physics) by the postulate that the principles and laws of physics are invariant under inertial or, generally, under continuous transformations. [CWL 3, 84/108]

[Paraphrasing] As pure conjugates, quantities and prices are defined implicitly by the postulate that the principles and laws of macroeconomics are invariant under transformations between fiscal orders, monetary orders, executive or judicial institutions, and political systems.

And so, contingent prices are not the given absolute basis of explanation; they are not the given, from which all explanation emanates.  Contingent prices are to be understood in the light of the significant interdependent variables, which are the interdependent and mutually-definitive accelerator flows and accelerated flows of pretio-quantital basic and surplus outlays, expenditures, borrowing, and reverse-borrowing.

Now, at the root of classical method there are two heuristic principles.  The first is that similars are understood similarly, that a difference of understanding presupposes a significant difference of data.  The second is that similarities, relevant to explanation, lie not in the relations of things to our senses but in their relations to one another.  Next, when these heuristic principles are applied, there  result classifications by sensible similarity, then correlations, and finally the verification of correlations and of systems of correlations.  But verified correlations necessarily involve the verification of terms implicitly defined by the correlations; and they do not involve more than such implicitly defined terms as related, for what is verified accurately is not this or that particular proposition but the general and abstract proposition on which ranges of ranges of particular propositions converge. … there is a fundamental heuristic structure that leads to the determination of conjugates, that is, of terms defined implicitly by their empirically and explanatory relations.  Such terms as related are known by understanding, and so they are forms. Let us name them conjugate forms.[CWL 3, 435/460]

For the equations (stating the relations among the explanatory conjugates) are or can be established empirically.  And by definition pure conjugates mean no more than necessarily is implicit in the meaning of such verified equations. [CWL 3, 80/103]

the pure conjugate has its verification, not in contents of experience nor in their actual or potential correlatives, but only in combinations of such contents and correlatives.  … pure conjugates are the minimal correlatives implicit in such functions; and their verification finds its ground, not in experience as such, but only in the combination of combinations, etc., etc., etc., of experiences. [CWL 3, 80/103-04  ]

Images, necessity and impossibility:

The third observation is that the image is necessary for the insight. … Points and lines cannot be imagined.  But neither can necessity or impossibility be imagined.  Yet in approaching the definition of the circle, there occurred some apprehension of necessity and of impossibility.  As we remarked, if all the radii are equal, the curve must be perfectly round; and if any radii are unequal, the curve cannot avoid bumps or dents. … ¶It follows that the image is necessary for the insight.  Inversely, it follows that the insight is the act of catching on to a connection between imagined equal radii and, on the other hand, a curve that is bound to look perfectly round. [CWL 3, 8-9/31-2]

Also note again that :

… on the other hand, quantum theory deals with what it knows to be processes that cannot be imagined.  It is a higher level of abstraction, and getting away from anything that can be imagined is connected with the fact that quantum theory is statistical – …  ¶ So you can see how even the ideas of definition and abstraction have become much more fluid.  Scientific thinking is much more versatile, much more attentive to all the possibilities of the fundamental act that is insight into sensible data.  I have given a series of illustrations of this.  (CWL 10, 126-27)

Basic terms as precise functional distinctions:

A systematic explanation, then, requires a normative theoretical framework.  The basic terms and relationsof such a framework would specify the distinctions and correlations that articulate the causes, which are not necessarily visible, of events that are apparent to all.  (CWL 15,  Editors’ Introduction, lv)

Again, (Lonergan) approaches the focus armed with precise analytic distinctions upon which a superstructure of laws, coherent with one another and comprising a complete theory, may be constructed.  Paraphrasing [CWL 3, 80/103]:

Lonergan makes a precise analytic distinction between point-to-point and point-to-line correspondences.

in the long run, and especially in the very long run, such a correlation exists.  It is that surplus production is the accelerator of basic production.  In other words the correspondence between the two is not a point-to-point but a point-to-line correspondence; … Now such a correspondence, if it is to be expressed not in terms of expectations of the future but in terms of present fact, is a correspondence of accelerator to accelerated. [CWL 21, 132]

The point-to-line and higher correspondences are based upon the indeterminacy of the relation between certain (surplus) products and the (later ultimate basic) products that (exit the process and eventually) enter into the standard of living. … The analysis that insists on the indeterminacy is the analysis that insists on the present fact: estimates and expectations are proofs of the present indeterminacy and attempts to get round it; and, to come to the main point, an analysis based on such estimates and expectations can never arrive at a criticism of them; it would move in a vicious circle.  It is to avoid that circle that we have divided the process in terms of indeterminate point-to-line and point-to-surface and higher correspondences. [CWL 15, 27-28]

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 terms are defined by the relations in which they stand, that is, by a process of implicit definition. [Gibbons 1987, 313]

Analogously, focusing on one of Lonergan’s implicit equations, the terms are implicitly defined by the relationsin which they stand with one another.

P’Q’ = p’a’Q’ + p”a”Q”

We may read from left to right, right to left, or back and forth between right and left.  From left to right, expenditures-receipts P’Q’ define  and determine concomitant macroeconomic costs, p’a’Q’ and p”a”Q” as they are defined (CWL 15, 156-58)  From right to left,  basic and surplus costs-outlays constitute the incomes which define and determine what is concomitantly spent for basic products.  Travelling back and forth between left and right, the equals sign mandates the reciprocal constraining influence on one another of pretio-quantial expenditures-receipts and pretio-quantital costs-outlays constituting basic incomes.

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, first, by the advance of the sciences and, secondly, by full information on concrete situations. [CWL 3, 492/516]

Back, then, to the parallel with physics.  The introduction of a new distinction places a huge burden on the physics community to shuffle previous data, to take a new and precise view on present and future data.  Is this one of the reasons why the present distinction is unwelcome in economics? …  I am inclined to think that the main difficulty is the subtle absence of the scientific spirit in contemporary economics. … There is nothing wrong with mathematical subtlety or rigor: the rigor mortis belongs to the absence of significant variables coupled regularly with the absence of empirical and pragmatic perspective; a case in point is the sophistications of rational expectations theory.  … The real difficulty is in the scientific perspective that can come to grips with precise functional distinctions. [McShane 2002-2, 21-22]

Functional velocities as basic terms:

On such a methodological model (i.e. implicit definition according to functional relation)… Classes of payments quickly become rates of payment standing in the mutual conditioning of a circulation; to this mutual and, so to speak, internal conditioning there is added the external conditioning that arises out of transfers of money from one circulation to another; in turn this twofold conditioning in the monetary order is correlated with conditioning constituted (in the hierarchical productive order) by productive rhythms of goods and services; … There results a closely knit frame of reference that can envisage any total movement of an economy as a function of variations in rates of payment, and that can define the conditions of desirable movements as well as deduce the causes of breakdowns. … [CWL 21, 111]

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]

An ‘accountant’s unity’ 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 CWL 3, 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]

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]

the terms are defined by the relations in which they stand, that is, by a process of implicit definition. [Gibbons 1987, 313]

A systematic explanation, then, requires a normative theoretical framework.  The basic terms and relations of such a framework would specify the distinctions and correlations that articulate the causes, which are not necessarily visible, of events that are apparent to all.  The framework would thus stand to the ordinary apprehension of the booms and slumps of the trade cycle in much the same way that the explanatory grasp of acceleration as the second derivative of a continuous function of distance and time stands to the ordinary, commonsense grasp of what it is to be going faster.  CWL 15,  lv

All such payments form a class by themselves.  They stand in a network that is congruent with the technical network of the productive process. …above all, their connection with production is immediate: they …  are, so to speak, the immanent manifestation of the productive process as a process of value. [CWL 21, 114]

[1] The priora quoad nos – first for us – are the things which we notice first because they are related to our sensitive selves, e.g. hot and cold, fast, slow.  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, etc.

Thus the system is an overall functioning explained in terms of interdependent correlated monetary functioning.  Functions represent a deeper level of abstraction than do bookkeepers’ entities.  As correlated, these functions are related to each other rather that related to us; they are priora quoad se rather than priora quoad nos; they are explanatory rather than merely descriptive; they are implicitly defined by the co-relations in which they stand with each other.

[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]