An Adequate Level of Abstraction and Generalization; A Deeper Explanatory Unity

We frequently make the claim that Functional Macroeconomic Dynamics sublates all other macroeconomic theories by reaching an adequate level of abstraction, or a deeper unity, or a more profound point of view, or a purely relational, fully explanatory, scientific macroeconomics.  This deeper intelligibility fully explains the economic process and eliminates at a stroke much of what is contained in the current popular 700-page textbooks.

Ragnar Frisch was a Norwegian economist and the co-recipient of the first Nobel Memorial Prize in Economic Sciences in 1969.

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]


“Inadequate level of abstraction” mandates an advance to the deepest-and-primary relativity in the economic process.  Reaching and understanding that primary relativity would constitute operating at an adequate level of abstraction.  Does the analyst start with the relations of prices as explanatory, or does he search for the fundamental dynamic interdependencies which explain prices and for which prices are merely secondary determinations?  He searches heuristically for the vital  interconnections and interdependencies among dynamic elements which generally govern the concrete dynamic process and the “role” of prices and quantities within the concrete process.  And his primary variables might not resemble very much the variables chosen by other analysts operating at an inadequate level of abstraction.  He might even discover a new paradigm.

Because insights arise with reference to the concrete, mathematicians need pen and paper, teachers need blackboards, pupils have to perform experiments for themselves, doctors have to see patients, trouble-shooters have to travel to the spot, people with a mechanical bent take things apart to see how they work.  But because the significance and relevance of insight goes beyond any concrete problem or application, men formulate abstract sciences with their numbers and symbols, their technical terms and formulae, their definitions, postulates, and deductions.  Thus, by its very nature, insight is the mediator, the hinge, the pivot.  It is insight intothe concrete world of sense and imagination.  Yet what is known by insight, what insight adds to sensible and imagined presentations, finds its adequate expression only in the abstract and recondite formulations of the sciences. [CWL 3, 6/30]

Science is explanation in the form of terms which mutually define and mutually determine one another.  And the technique of implicit definition yields terms which implicitly define one another.

Lonergan’s critique (shows that) by using the technique of implicit definition, the emphasis shifts from trying to (nominally) 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 functional aggregates) which, in the first instance, one was thinking about.  [Gibbons 1987]

The form of the interdependencies among functional elements of the process constitutes the explanatory intelligibility underlying the ever-fluctuating rhythms of economic functioning.

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 neoclassicaland Keynesianeconomic theory. [Gibbons 1987]

For Newton, past and future values of any planetary motion would be relatively insignificant.  What he sought is the general law governing any and all motion.  The primary relativity was, for Newton, the pure relations in his second Law of Motion and his equation defining mass as a purely relational concept.

F = ma = md2x/dt2

F = m1m2G/d2

Thus, masses might be defined as the correlatives implicit in Newton’s law of inverse squares. Then there would be a pattern of relationships constituted by the verified equation; the pattern of relationships would fix the meaning of the pair of coefficients, m1, m2; and the meaning so determined would be the meaning of the name, mass.  [CWl 3, 80/102-03]

In like manner, heat might be defined implicitly by the first law of thermodynamics (insert) 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]

Thus, in Clerk-Maxwell’s laws of electromagnetism, the electric intensity and the magnetic intensity are implicitly defined by a pattern of relationships, not by verbal description.  And the pattern of relationships is explanatory and of scientific significance. (Invert the delta to get the del)

Δ X E = (-1/c)H’

Δ X H = (+1/c)E’

Δ . ŸH = 0

Δ . ŸE = 0

Similarly, the law of cosines: c2= a2+ b2-2ab(cos C) is both purely relational and more general than the Pythagorean formula; it contains the Pythagorean Theorem as a mere special case in which the angle between sides and happens to equal 90and, therefore, (cos C) happens to equal 0Mere special cases , whether in mechanics or macroeconomics are of little instance to the theorist operating a a sufficiently general level of abstraction.

Whether one is trying to explain the accelerations or the swept-out areas of the planetary process or the accelerations, booms, or slumps of the economic process, one must use a scientific heuristic which points to second-order equations as general rules or laws.  Taking into account past and future values is not the way to proceed.  Particular boundary conditions, “past and future values” are relatively insignificant for the analysis.  What is significant is the Leibnitz-Newtonian shift of context and search for second-order, explanatory patterns.

Taking into account past and (expected) future values does not constitute the creative key transition to dynamics.  Those familiar with elementary statics and dynamics (in physical mechanics) will appreciate the shift in thinking involved in passing from equilibrium analysis (of for example a suspended weight or a steel bridge)…to an analysis where attention is focused on second-order differential equations, on d2θ/dt2, d2x/dt2, d2y/dt2, on a range of related forces, central, friction, whatever.  Particular boundary conditions, “past and future values” are relatively insignificant for the analysis.  What is significant is the Leibnitz-Newtonian shift of context. [McShane, 1980, 127]

Past and future values constitute boundary conditions, not the primary relativity.  Those familiar with the integral calculus will grasp the idea readily.  The integrand function constitutes the general governing law, while the application of specific boundary values to the antiderivative yields a particular solution for a particular case.  In the following excerpt, one is reminded of Lindsay’s and Margenau’s demonstration of the method of elementary abstraction by an analysis of fluid motion. [Lindsay and Margenau 1957, 29-58] A premature introduction of general boundary conditions or worse, specific channel conditions, botches the analytic possibilities, just as a premature introduction of prices and quantities betrays a fundamental disorientation and dooms the analysis of the economic process to uselessness.

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, 1980,124][2]

The basic terms of the sciences …  are defined by their respective relations to one another.  To distinguish between the defining relation and the defined term can be no more than a notional operation; and even then it cannot be carried through, for if one prescinds from the defining relation, one no longer is thinking of the term as defined but of some other term that is mistakenly supposed to be absolute.  Finally, while there are relations other than such defining relations, still they are not adequately distinct from them; for these other relations are concrete; their primary relativity consists in the defining relations; and their secondary determinations are neither relations nor the reality of relations but the contingent concrete differentiations of the primary relativities. [CWL 3, 496/520]

Paraphrasing:

The basic terms of the science of Functional Macroeconomic Dynamics – velocitous and accelerative functionings, in the abstract forms of d2θ/dt2, d2x/dt2, d2y/dt2…  are implicitly defined by their respective (functional) relations to one another.  Thus the current flowings of macroeconomic costs (as we have defined them) function as a limitation of the current flowings of pure surplus incomes (as we have defined them), and vice versa.  They implicitly define one another. To distinguish between the defining functional relation and the defined term can be no more than a notional operation; and even then it cannot be carried through, for if one prescinds from the defining relation (of how the terms define one another by how they mutually relate), one no longer is thinking of the functional phenomena as mutually defined by the functional relation but of some other term that is mistakenly supposed to be absolute. Finally, while there are relations (such as pricings to one another) other than the primary explanatory relativities of analytically distinguished flows to one another, still they (the happenstantial price relations) are neither adequately distinct from them nor adequately abstract and explanatory; for these other relations of prices are concrete; their primary relativity consists in the defining relations; and their secondary determinations – their measures – are neither relations nor the reality of relations but the contingent concrete differentiations of the primary relativities.

Newton sought the general laws governing the motion of heavenly bodies and apples in any and all cases, Recall Schumpeter had pointed out that the expansion of capital and the occurrence of economic cycles are connected:

 Lonergan agreed with Schumpeter on the importance of systematic or analytic framework in order to explain, rather than merely record or describe, the aggregate phenomena of macroeconomics; he agreed with Schumpeter that to be able to explain the booms, slumps, and crashes of the trade or business cycles the economist’s analysis had to be as dynamic as the subject matter under investigation; and he agreed that the economist had to know what are the significant variables in the light of which price changes are to be interpreted.  According to Lonergan, standard economic theory had successfully achieved none of these desiderata. [CWL 15, Editors’ Introduction liii]

Lonergan used the technique of implicit definition in order to construct a theory whose basic terms are purely relational.  Again:

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]

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. [Gibbons 1987]

Lonergan’s Functional Macroeconomic Dynamics explains it all.  It provides a “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. Like Newton, he achieved a general theory of motion which provides norms for adaptation and explains all booms and slumps rather than merely recites their statistics retrospectively.

On such a methodological model (i.e. implicit, explanatory definition replacing nominal definition and accountant’s categories)… classes of payments quickly become rates of payment standing in the mutual conditioning of a circulation; to this mutual and, so to speak, internal (monetary) conditioning there is added the external (monetary) 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 the conditioning constituted (in the hierarchical productive order) by productive (and sequential) rhythms of goods and services;[3]and from the foregoing dynamic configuration of conditions during a limited interval of time, there is deduced a catalogue of possible types of change in the configuration over a series of intervals. 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.  Through such a frame of reference one can see and express the mechanism to which classical precepts are only partially adapted; and through it again one can infer the fuller adaptation that has to be attained. [CWL 21, 111]

Lonergan sought the general laws governing the economic process in any and all cases.

Diagram of Rates of Flow 2

Diagram of Rates of Flow

From the premise of (the functional nature of) the process of production and its correlation with payments of money, Lonergan advances to general laws which govern any and all configurations of the current, purely dynamic, concrete economic process.

From the Newton-Leibnitz shift of heuristic and context, we get the general laws of the accelerations of the process at an adequate level of abstraction.

The differential equations that are the equations of accelerations (changes of velocity):

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

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

d(P’Q”) = d(p’a’Q’) + d(p”a”Q”)  [CWL 15, 158]

dΣFi= dvI”  [CWL 15, 150]

The differential equations of the ratios:

df = vdw + wdv  [CWL 15, 148-49]

d(P’/p’) = dJ = da’ + a”dR + R da”  [CWL 15, 158]

And, we may apply boundary conditions to the antiderivatives to get the particular laws of any configuration.

The lower-order equations that are the equations of fundamental velocities (changes of accumulations)

qi= Σqijk  [CWL 15, 30]

P’Q’ = p’a’Q’ + p”a”Q”  [CWL 15, 302]

Σsij  =  Σvirij   [CWL 21, 170]

I’ = Σwiniyi   [CWL 15, 134]

ΣFi= vI”[CWL 15, 150]

The lower-order equations of the ratios

f = vw   [CWL 15, 148]

P’/p’ = a’ + a”(p”Q”/p’Q’)  [CWl 15, 158]

J = a’ + a”R   [CWl 15, 158]

Other images and attendant insights which are essential to Functional Macroeconomic Dynamics are, in CWL 15, Figures 24-4, 24-6, 24-7, and 27-1, and the basic formalisms:

[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.

[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]… In figure 14-1 the reader will notice five circles representing the monetary functions. … I would add that the aims and limitations of macroeconomics (that is, the macroeconomic circulations presented here) make the use of a diagram particularly helpful, …  For its basic terms are defined by their functional relations.  The maintaining of a standard of living is attributed to a basic process(distinct process 1), an ongoing sequence of instances of so much every so often.  The maintenance and acceleration (positive or negative) (distinct process 2) of this basic process is brought about by a sequence of surplus stages, in which each lower stage is maintained and accelerated by the next higher.  Finally, transactions that do no more than transfer titles to ownership are concentrated in a redistributive function, whence may be derived changes (distinct process 3) in the stock of money dictated by the acceleration (positive or negative) in the basic and surplus stages of the process. … So there is to be discerned a threefold process in which a basic stage is maintained and accelerated by a series of surplus stages, while the needed additions to or subtractions from the stock of money in these processes is derived from the redistributive area. … it will be possible to distinguish stable and unstable combinations and sequences of rates in the three main areas and so gain some insight into the long-standing recurrence of crises in the modern expanding economy. CWL 15, 53-54