Theoretical Breakthroughs of Euclid, Newton, Hilbert, Einstein, and Lonergan

To help the reader gain an appreciation of Lonergan’s achievement of Modern Macroeconomic Field Theory we will, in each section, print leading excerpts, then highlight the key concepts of those excerpts. We will comment on the historically-significant advances in geometry of Euclid and Hilbert, in physics of Newton and Einstein, and in macroeconomics of Lonergan.

  • Euclid’s great achievement was his rigorous deduction of geometry.
  • Hilbert’s great achievement was his employment of implicit definition to reorder Euclid’s geometry.
  • Newton’s two great achievements were unifying the isolated insights of Galileo and Kepler into a unified system of mechanics and his invention of the calculus.
  • One of the great achievements of Einstein was the invention of the field theories of Special Relativity, General Relativity, and Gravitation.
  • One of Lonergan’s several great achievements was his systematization of macroeconomic phenomena in his Modern Macroeconomic Field Theory.  He combined the technique of implicit definition introduced by Hilbert and the concept of a field theory developed by Faraday and Einstein, and he discovered an explanatory macroeconomics which is general, invariant, and relevant in any instance.

Contents

  • .a. Implicit Definition: Euclid, Hilbert, and Lonergan
  • .b. Implicit Definition of Constituent Explanatory Functions in Macroeconomics
  • .c. Determinist vs. Statistical Theories: Newtonian Mechanics vs. Special Relativity and Quantum Mechanics
  • .d. Efficient-Cause Theory vs. Modern Field Theory: Newton’s Mechanics vs. Purely-Relational, Immanent-Intelligibility Theory
  • .e. The Background Medium Done Away

.a. Implicit Definition: Euclid, Hilbert, and Lonergan

Key Excerpts:

… an entirely new type of definition was introduced by Hilbert in his formulation of geometry.[1]  He called it implicit definition.  An implicit definition drops the common matter to express only a relational form…  The significance of implicit definition is that it does not pin down the meaning of the words ’point’ and ‘line’ to anything. Point, in Hilbert’s expression of geometry, can be a Euclidean position without magnitude, and a line can be a length without breadth or thickness lying evenly between its extremes.  But a point can also be an ordered pair of numbers, where (a,b) is not the Cartesian notation for a Euclidean point, but just that ordered pair.  And a straight line can be a first-degree equation: y = mx + c is determined by two ordered pairs, and two ordered pairs will determine a first-degree equation.  Hilbert can mean by point and line the imaginable Euclidean point or line, the Cartesian algebraic expression for point and line, or anything else that will satisfy the relation “two of one determines one of the other,” no matter what they are.  The definitions are in terms of relational form, with no attention to any common matter.  The relational form selects any common matter that will be thought relevant.  Implicit definition is a more abstract type of thinking that omits even the common matter. (CWL 10, 126)

In (Adler, 1958, 199-205)  We note that Euclidean geometry begins with nominal definitions of point and line:

  • a point is that which has no part
  • a line is breadthless length
  • the extremities of a line are points
  • a straight line is a line which lies evenly with the points on itself

In contrast, Hilbert’s purely relational geometry, point and line are classified as “undefined quantities.”

Hilbert’s Axioms of Undefined Quantities:

  • a point is an undefined element
  • a line is an undefined element

However, in Hilbert’s Axioms of Incidence and Axioms of Order, points and lines come to be defined implicitly, rather than nominal or descriptively, by their immanent relations to one anotherThe terms fix the relations and the relations fix the terms.  Hilbert’s concepts are purely relational.

Hilbert’s Axioms of Incidence:

  • Given any two points, A, B, there exists a line a lying on A and B
  • Given A, B, there exists at most one line a lying on A, B
  • There are at least two points which lie on a given line

Hilbert’s Axioms of Order:

  • If B is between A and C, then A, B, C are three different point on a line and B is between C and A.

[Adler, 1958, 199-205]

Key Concepts

… a science emerges when thinking in a given field moves to the level of system. . Prior to Euclid there were many geometrical theorems that had been established.  The most notable example is Pythagoras’ theorem on the hypotenuse of the right-angled triangle, which occurs at the end of  book 1 of Euclid’s Elements.  Euclid’s achievement was to bring together all these scattered theorems by setting up a unitary basis that would handle all of them and a great number of others as well. … [CWL 10, 241-42]

Hilbert’s geometry and Lonergan’s Functional Macroeconomic Dynamics employ implicit definition.  Implicit definition expresses only a relational form. … The significance of implicit definition is that it is purely relational.  The relational form may be applicable to any imaginable common matter or process with which it is isomorphic.  Implicit definition by a purely-relational form is a more abstract type of thinking.

Functional Macroeconomic Dynamics uses implicit definition and links terms defined by their functional relations.

Functional Macroeconomic Dynamics is a set of intelligible relations linking what is implicitly defined by the relations themselves; it is a set of relational formsThe form of any element is known through its relations to all other elements.  CWL 10, 154]

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]

In Functional Macroeconomic Dynamics, a function is an abstract term specifying “things in their relations to one another”.

“Functional” is for Lonergan a technical term pertaining to the realm of explanation, analysis, theory;  … 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” … [CWL 15  26-27  ftnt 27]

I would add that the aims and limitations of macroeconomics (that is, the circulation analysis presented here) make the use of a diagram particularly helpful, …  For its basic terms are defined by their functional relations.  [CWL 15, 54]

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]

In order to explain by applied mathematics the current, purely-dynamic, concrete, functional economic process, Lonergan searched among purely-relational forms for the terms and their relations which might be isomorphic with the functional interrelations of macroeconomics and, thus, might explain the isolated, but unsystematized, insights of macroeconomists, and in particular, the observation of Schumpeter that the level of capital spending seems to be related to cyclical excesses and deficiencies called booms and slumps.  Lonergan used abstract forms of a) velocitous point-to-point functionings (“basic functionings”), b) velocitous point-to-indeterminate-future-line functionings (“surplus functionings”), c) difference (temporal), d) summational rectilinear accumulation, and e) velocitous circular movements connected by velocitous crossover movements.  These forms are, by themselves, merely isolated abstract forms; they exist prior to any consideration of their possible application to any particular, common-matter process.  But they might be found to be applicable in combination to explain completely certain concrete schemes of recurrent circulations, whether in hydrodynamics, thermodynamics, electric circuits, survival activities, social relations, or economics.  Lonergan found these abstractions to be isomorphic with, applicable to, and explanatory of productive and monetary processes constituting the field of macroeconomics.

the correspondence between elements in the productive process and elements in the standard of living may be a point-to-point, or a point-to-line, or a point to surface, or even some higher correspondence. (CWL 15, 23 )

Further, since the forms are abstract and the functional movements are defined solely by their relations to one another – in the manner of Hilbert – rather than defined by terms related to us in our concrete experience, they qualify as possible explanatory conjugates to explain an isomorphic process.

Algebraic functions of the first degree capture point-to-point relations.  As in Hilbert’s axioms stated above, two points determine a line, and a line determines two points.

By “algebraic functions of the first degree” is meant functions involving no exponential powers of the variables other than 1. … To give a simple example, if the number of loaves of bread is represented by the variable y, and the number of bushels of wheat by the variable x, then a relevant algebraic function might be  where b represents the number of loaves of bread which can be obtained from one bushel of wheat under current technological means, and c represents any fixed loss or gain which is independent of the number of bushels. [CWL 15, 23 ftnt. 25]

Thus, Hilbert’s purely-relational “two points determine a line” finds application in Lonergan’s precise analytical specification of the correspondence of factors of production with factors exiting the process!  Again, as in Part I,

… an entirely new type of definition was introduced by Hilbert in his formulation of geometry.[2]  He called it implicit definition.  An implicit definition drops the common matter to express only a relational form…  The significance of implicit definition is that it does not pin down the meaning of the words ’point’ and ‘line’ to anything. Point, in Hilbert’s expression of geometry, can be a Euclidean position without magnitude, and a line can be a length without breadth or thickness lying evenly between its extremes.  But a point can also be an ordered pair of numbers, where (a,b) is not the Cartesian notation for a Euclidean point, but just that ordered pair.  And a straight line can be a first-degree equation: y = mx + c is determined by two ordered pairs, and two ordered pairs will determine a first-degree equation.  Hilbert can mean by point and line the imaginable Euclidean point or line, the Cartesian algebraic expression for point and line, or anything else that will satisfy the relation “two of one determines one of the other,” no matter what they are.  The definitions are in terms of relational form, with no attention to any common matter.  The relational form selects any common matter that will be thought relevant.  Implicit definition is a more abstract type of thinking that omits even the common matter. (CWL 10,126)

There exists, then, a point-to-point correspondence between bushels of wheat and loaves of bread, between head of cattle and pounds of meat, between bales of cotton and cotton dresses, between tons of steel and motorcars.  In each case the elements in the standard of living are algebraic functions of the first degree with respect to elements in the productive process….(there is) an inexorable law of limitation. [CWL 15, 23]

While goods and services move in straight lines, money and monetary substitutes of cash and credit move in circles.

if the real flows of goods and services move, as it were, in straight lines from the potentialities of universal nature, on the other hand, the dummy flows of money and monetary substitutes, of cash and credit, move in circles.  The same currency is used over and over; the same accumulation sustains indefinitely a given volume of credit. … The meaning of the term ‘circulation’ is that … movements in various directions have to balance with opposite movements.  There has to be equilibrium (for continuity). (CWL 21, 57-58)

Note that the intelligibility of the macroeconomic process was discovered by Lonergan in the same way as were Euclidean geometry, Hilbertian geometry, and Newtonian mechanics, by rigorous analysis of primary interdependencies and deduction.  Schumpeter’s vague unmathematical conclusion, that booms and slumps were somehow related to capital spending, was for Lonergan a proto-scientific conclusion to be scientifically analyzed, mathematically formulated using implicitly-defined explanatory conjugates, and empirically verified.  It was a proposition about the “common matter” of a) equilibrated expansions, and b) disequilibrated booms and slumps.  It was to be analytically examined, formulated in a mathematical generalization consisting of pure relations, and empirically verified.  Doing such successfully was Lonergan’s great achievement.  He established a Modern Macroeconomic Field Theory  of pure inner relations among n functions by what seemed to be methods parallel to Hilbert in his geometry and by Faraday, Clerk-Maxwell, and Einstein in their electromagnetics and mechanics, i.e. by a) achieving insight yielding terms related to one another, b) applying purely relational forms to concrete economic phenomena, and c) deducing a completely explanatory field theory. 

Now, at the root of classical method there are two heuristic principles.  The first is that similars are understood similarly, ….  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]

[7/18/19] No doubt Keynes was an economist first and a methodologist second but he was none the less very articulate about his theorizing……..Lonergan, for his part, is perhaps a methodologist first and an economist second, but, as we shall see, he was able to push his economic reflections further than Keynes because he had a firmer grasp of the essentials of an effective theory.   [Gibbons, 1987] [3]

.b. Implicit Definition of Constituent Explanatory Functions

Leading Excerpts

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]

In Lonergan’s Functional Macroeconomic Dynamics the basic terms of systematic, field-theoretic significance are velocities of interdependent functionings.

“Functional” is for Lonergan a technical term pertaining to the realm of explanation, analysis, theory;  … 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” … [CWL 15  26-27  ftnt 27]

Now, at the root of classical method there are two heuristic principles.  The first is that similars are understood similarly, … .  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]

Key Concepts

Lonergan made precise analytical distinctions.  He appropriately appropriated the abstract forms of a) point-to-point, b) point-to-line, c) difference (temporal), d) summational accumulation, and e) velocitous circulation.  These few precise forms, their relations, and the combinations of their relations virtually contain the whole explanation of a dynamic process in a field theory characterized by an equilibrium which is normative in any instance.  The complete explanation is an invariant, and this invariant is relevant in any instance.

Lonergan pointed out that this differentiation of economic activities into the production of consumer goods in the standard of living and the production of producer goods that transform the possibilities for future consumer-goods production is discussed by traditional economists such as S. M. Longfield (1802-1884), John Rae (1796-1872), Nassau Senior (1790-1864), Eugen von Bohm-Bawerk (1851-1914), and in the heavily disputed “Ricardo effect.” But Lonergan credits Piero Sraffa (1898-1983) as having clarified it most thoroughly in his famous essay, Production of Commodities by Means of Commodities(1960).  Yet even Sraffa does not use his sophisticated explanation of the “Ricardo effect” and the “roundabout” or “concertina”-like phenomena associated with it in the way Lonergan does. [CWL 15, Editors’ Introduction lxii]

But Functional Macroeconomic Dynamics identifies functional distinctions and explanatory relations among different classes of  production flows which are analytically prior to, and more fundamental than, … price levels and patterns, interest rates and profits, and so forth.

Lonergan is alone in using this difference in economic activities to specify the significant variables in his dynamic analysis… a considers the functional distinctions between different kinds of (rhythmic production flows) prior to, and more fundamental than, … price levels and patterns, … interest and profits, and so forth….only Lonergan analyzes booms and slumps in terms of how their (explanatory) velocities, accelerations, and decelerations are or are not equilibrated in relation to the events, movements, and changes in two distinct monetary circuits of production and exchange as considered both in themselves (with circulatory, sequential dependence) and in relation to each other by means of crossover payments. [CWL 15, Editors’ Introduction, lxii]

And … such a property as invariance is a property, not of any particular macroeconomic process, but of a set of expressions explaining all macroeconomic processes. (Click here and here)

… such a property as invariance is a property, not of a geometrical entity, but of an expression regarding geometrical entities. [CWL 3, 148/171]

The process has an exigence for an equilibrated pure cycle of expansion.  Lonergan discovered patterns of functional interdependencies in which always there is 1) a normative, universally-relevant equilibrium, and 2) a path of central tendency, determined in general by the pure cycle and particularly by the current particular technical coefficients in the interdependencies among functionings.  There is a systematic necessity for adaptation in accord with the primary relativities grounded in the fundamental analytical distinctions.  These primary relativities constitute the abstract laws which are universally relevant to any particular concrete expansion’s serial path through component phases.  And, any concrete economic evolution can be explained by the combination of these primary relativities plus the secondary determinations from the non-systematic manifold of prices and quantities.

Economic science is of the general.  “From millions of exchanges one advances to precisely defined aggregate functions, relatively few in number, and hence easy to follow up and handle.”  The explanatory interrelations of these velocitous aggregates may be represented in a diagram, into which one can have insight so as to achieve a frame of reference.

All science begins from particular correlations, but the key discovery is the interdependence of the whole. [CWL 15, 53-4 and 177]

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]

it will be well at once to draw attention to J.A. Schumpeter’s insistence on the merits of the diagram as a tool. (Schumpeter, History 240-43, on the Cantillon-Quesnay tableau.) … First, there is the tremendous simplification it effects.  From millions of exchanges one advances to precise aggregates, relatively few in number, and hence easy to follow up and handle. … Next come the possibilities of advancing to numerical theory.  In this respect, despite profound differences in their respective achievements, the contemporary work of Leontieff may be viewed as a revival of Francois Quesnay’s tableau economique. Most important is the fact that this procedure was the first to make explicit the concept of economic equilibrium.  All science begins from particular correlations, but the key discovery is the interdependence of the whole. … While it is true that a tableau or diagram cannot establish the uniqueness of a system or rigorously ground its universal relevance, it remains that the diagram (of the interconnections of a few precise aggregates) has compensating features that Quesnay’s system of simultaneous equations may imply but does not manifest. … There is the tremendous simplification (a diagram) effects. … the aims and limitations of macroeconomics 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 (distinct process 1) is attributed to a basic process, an ongoing sequence of instances of so much every so often.  The maintenance and acceleration(distinct process 2) of this basic process is brought about by a sequence of surplus stages, in which each lower stageis maintained and accelerated by the next higher.  Finally, transactions that do no more than transfer titles to ownership (distinct process 3) are concentrated in a redistributive function, whence may be derived changes 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, 53-4 and 177]

I would add that the aims and limitations of macroeconomics (that is, the circulation analysis presented here) make the use of a diagram particularly helpful, …  For its basic terms are defined by their functional relations.  [CWL 15, 54]

Functional Macroeconomic Dynamics is a set of intelligible relations linking the functions which are implicitly defined by the relations themselves; it is a set of relational formsThe form of any element is known through its relations to all other elements.  (CWL 10, 154)

Functional is for Lonergan a technical term pertaining to the realm of explanation, analysis, theory;  … 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” … [CWL 15  26-27  ftnt 27]

Scientific explanation – whether in physics or macroeconomics – is to be distinguished from description.  “Description and explanation envisage things in fundamentally different manners.  The relations of things among themselves are, in general, a different field from the relations of things to us.”

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.  The two are not totally independent, for they deal with the same things and, as we have seen, 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]

… despite their intimate connection, it remains that description and explanation envisage things in fundamentally different manners.  The relations of things among themselves are, in general, a different field from the relations of things to us. [CWL 3, 291/316]

We are not going to discuss wealth or value, … capital and labor, interest and profits, production, distribution, and consumption. Because we are not, it certainly will be objected that our discussion has nothing to do with economic science, for economics is precisely the study of wealth and value … .  The answer is as follows.  The discussion moves on a more general plane to terminate in a more general conclusion.  Because the general includes the particular, a generalized economics cannot but include the particular economics. [CWL 21, 8]

Careful distinctions, precise implicit definitions, and formulations of functional relations results in a frame of reference.

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]

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 functional interrelations in which they stand. [CWL 21, 112]

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. … If the system is to move into a long-term expansion, this movement has to begin with a surplus quantity acceleration: surplus production has not merely to maintain or renew existing capital equipment but has to reach a level at which it turns out new units of production and maintains or renews a greater number of existing units; this gives the quantity surplus expansion. [CWL 21, 132]

The exchange solution is superior to the bureaucratic solution.

The excellence of the exchange solution becomes even more evident when contrasted with the defects of a bureaucratic solution.  The bureaucrat … (gives the people) what he thinks is good for them, and he gives it in the measure he finds possible or convenient; nor can he do otherwise, for the brains of a bureaucrat are not equal to the task of thinking of everything; only the brains of all men together can even approximate to that. … when a limited liability company has served its day, it goes to bankruptcy court; but when bureaucrats take over power, they intend to stay. … when the pressure of terrorism is needed to oil the wheels of enterprise, then the immediate effect is either an explosion or else servile degeneracy. … the exchange solution is a dynamic equilibrium resting on the equilibria of markets. … every product of the exchange economy must mate through exchange with some other product, and the ratio in which the two mate is the exchange value.  The generality of this equilibrium makes it indifferent to endless complexity and endless change; for it stands on a level above all particular products and all particular modes of production.  While these multiply and vary indefinitely, the general equilibrium of the exchange process continues to answer with precision the complex question, Who, among millions of persons, does what, among millions of tasks, in return for which, among millions of rewards?  Nor is the dynamic solution unaccompanied by a continuous stimulus to better efforts and more delicate ingenuity.  For the uniformity of prices means that the least efficient of those actually producing will at least subsist, while every step above the minimum efficiency yields a proportionately greater return.[4] [CWL 21, 34-35]

Again: 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]

Our immediate task is to work out the correlations that exist between the velocity and accelerator rhythms of production and the corresponding rhythms of income and expenditure.  The set of such correlations constitutes the mechanical structure, a pattern of laws that stand to (purely experiential) economic activity as the laws of mechanics to buildings and machines. [CWL 21, 43]

.c. Determinist vs. Statistical Theories: Newtonian Mechanics vs. Quantum Mechanics

Leading Excerpts

… a science emerges when thinking in a given field moves to the level of system. … Mechanics became a system with Newton.  Prior to Newton, Galileo’s law of the free fall and Kepler’s three laws of planetary motion were known.  But these were isolated laws.  Galileo’s prescription was that the system was to be a geometry; so there was something functioning as a system.  But the system really emerged with Newton.  This is what gave Newton his tremendous influence upon the enlightenment. He laid down a set of basic, definitions, and axioms, and proceeded to demonstrate and conclude from general principles and laws that had been established empirically by his predecessors.  Mechanics became a science in the full sense at that point where it became an organized system. … [CWL 10, 241-42]

… , Newtonian mechanics is constructed in the same way as was Euclidean geometry.  Kepler’s discovery that the planet Mars and the planets in general moved in ellipses was for Newton a conclusion to be demonstrated.  That was Newton’s great achievement.  He established, by what seemed to be methods parallel to Euclid’s, by rigorous deduction, exactly what Kepler had found by empirical correlations.  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 – it is not the only factor. ¶ 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)

(re Groups)  … , there can be extension in the interpretation placed upon the symbols.  Pure mathematics places no interpretation on them, but they can be interpreted as a geometry, as a space-time, as a physics, as a chemistry, and so on.  In the conception of mathematics in terms of a group of operations, there can be combined the greatest concreteness with a full appreciation of abstraction.  The form grasped by insight into phantasm is the form of the group.  If you understand what it is to do arithmetic, you can develop from that insight something that stands to doing arithmetic as the definition of the circle stands to the image of a circle.  It is something much more rigorous and much more systematic than any image.  The group of operations in any particular case is what is represented by a formula, and the formula can be realized simply in the symbols or in a series of isomorphic cases which may be applications or different uses of symbols.  And with this procedure you get new types of definitions, such as implicit definitions that are simply relational forms without the common matter; it is quite possible to add any common matter that one pleases. (CWL 10, 131-32)

Field theory is a matter of the immanent intelligibility of the object. [CWL 10, 154]

Special Relativity is primarily a field theory, that is, it is concerned not with efficient, instrumentalmaterial, or final causes of events, but with the intelligibility immanent in data; but Newtonian dynamics seems primarily a theory of efficient causes, of forces, their action, and the reaction evoked by action. … Special Relativity is stated as a methodological doctrine that regards the mathematical expression of physical principles and laws, but Newtonian dynamics is stated as a doctrine about the objects subject to laws.  [3, 43/67]

Key Concepts

Newton set down laws of motion and proceeded to demonstrate that if a body moves in a field of central force, its trajectory is a conic section.  He set out with a minimal cluster of insights, definitions, postulates, axioms and proceeded to account for the laws that had previously been empirically established, bringing them into a single explanatory unity.  ¶A single insight yields a conception, a definition, an object of thought; but from a cluster of insights, you build up a system of definitions, axioms, postulates, and deductions.  We have to note that a system is quite an achievement; systems are not numerous. [CWL 5, 52]

Explanation in field-theoretic Functional Macroeconomic Dynamics has two components, a primary component and a secondary component.

General laws contain a primary relativity and are applied to the concrete “only through the addition of further determinations, and such further determinations pertain to a non-systematic manifold. … it is not enough to think about the general law; one has to add further determinations that are contingent from the very fact that they have to be obtained from a non-systematic manifold.” [CWL 3, 492/516]

The primary relativity is inseparable from its base and for that reason all change is change in the base and only incidentally and consequently change in the relativity.  The secondary determinations are constitutive neither of the relation nor of its reality as relation but simply of the differentiation of concrete relations; and because the differentiation depends, not on the base alone but on the base and term together, it can vary without variation in the base [CWL 3, 495] (Click here and here and here.)

Macroeconomic Field Theory’s primary relativities are at a higher level of abstraction.  Its secondary determinations – prices and quantities – are coincidental determinations obtained from a non-systematic manifold. In dealing with the non-systematic, MFT deals with what it knows to be processes that cannot suitably be precisely imagined and determined ahead of time, and getting away from anything that can be imagined is connected with the fact that MFT includes the statistical.  All pure cycles have the same universally-relevant principles and laws, the the particular pure cycle will have technical coefficients of money supply, labor supply, material resources, prices and quantities obtained from a non-systematic manifold.

… , Newtonian mechanics is constructed in the same way as was Euclidean geometry.  Kepler’s discovery that the planet Mars and the planets in general moved in ellipses was for Newton a conclusion to be demonstrated.  That was Newton’s great achievement.  He established, by what seemed to be methods parallel to Euclid’s, by rigorous deduction, exactly what Kepler had found by empirical correlations.  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 – it is not the only factor. ¶ 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)

To repeat:

So you can see how even the ideas of definition and abstraction have become much more fluid.  Thinking in (Group Theory), Quantum Theory, and in Functional Macroeconomic Dynamics is much more versatile, much more attentive to all the possibilities of the fundamental act that is insight into sensible data. (CWL 10, 126-27)

While we agree with Schumpeter that Walras’s system implicitly includes the aggregates commonly considered in macroanalysis, (Walras’s system) can hardly be credited with distinctions between basic and surplus expenditure, receipts, outlay, income, and much less with an account of their various dynamic relations.  But until such distinctions are drawn and their dynamic significance understood, the aggregates and relations cannot be contained implicitly in any (explanatory) system. [CWL 15,  91-92]

The exchange solution, sketched in Walrasian IS-LM and AD-AS models, is characterized by complexity and endless change.  It is a process that is non-systematic and defies prediction, though it has definite results.  The exchange solution, somewhat like quantum mechanics whose processes cannot be imagined, and like group theory being a “form” of a group of operations on objects it need not imagine, stands on a level above all particular modes of particular production or processes.

To repeat:

(re Groups)  … , there can be extension in the interpretation placed upon the symbols.  Pure mathematics places no interpretation on them, but they can be interpreted as a geometry, as a space-time, as a physics, as a chemistry, and so on.  In the conception of mathematics in terms of a group of operations, there can be combined the greatest concreteness with a full appreciation of abstraction.  The form grasped by insight into phantasm is the form of the group.  If you understand what it is to do arithmetic, you can develop from that insight something that stands to doing arithmetic as the definition of the circle stands to the image of a circle.  It is something much more rigorous and much more systematic than any image.  The group of operations in any particular case is what is represented by a formula, and the formula can be realized simply in the symbols or in a series of isomorphic cases which may be applications or different uses of symbols.  And with this procedure you get new types of definitions, such as implicit definitions that are simply relational forms without the common matter; it is quite possible to add any common matter that one pleases. (CWL 10, 131-32)

Macroeconomic Field Theory stands to the exchange solution as quantum theory stands to quantum processes.

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 – it is not the only factor. ¶ 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)

The excellence of the exchange solution becomes even more evident when contrasted with the defects of a bureaucratic solution.  The bureaucrat … (gives the people) what he thinks is good for them, and he gives it in the measure he finds possible or convenient; nor can he do otherwise, for the brains of a bureaucrat are not equal to the task of thinking of everything; only the brains of all men together can even approximate to that. … when a limited liability company has served its day, it goes to bankruptcy court; but when bureaucrats take over power, they intend to stay. … when the pressure of terrorism is needed to oil the wheels of enterprise, then the immediate effect is either an explosion or else servile degeneracy. … the exchange solution is a dynamic equilibrium resting on the equilibria of markets. … every product of the exchange economy must mate through exchange with some other product, and the ratio in which the two mate is the exchange value.  The generality of this equilibrium makes it indifferent to endless complexity and endless change; for it stands on a level above all particular products and all particular modes of production.  While these multiply and vary indefinitely, the general equilibrium of the exchange process continues to answer with precision the complex question, Who, among millions of persons, does what, among millions of tasks, in return for which, among millions of rewards?  Nor is the dynamic solution unaccompanied by a continuous stimulus to better efforts and more delicate ingenuity.  For the uniformity of prices means that the least efficient of those actually producing will at least subsist, while every step above the minimum efficiency yields a proportionately greater return.[5] [CWL 21, 34-35]

Again, the exchange solution is indifferent to endless complexity and endless change; for it stands on a level above all particular modes of production or processes.  Similar to quantum theory, the exchange solution deals with what it knows to be processes that cannot be imagined.  Its macro functioning is formulated by MFT at a higher level of abstraction, and getting away from anything that can be imagined is connected with the fact that MFT’s primary relativities must be combined with the statistical and probabilistic secondary determinations obtained from the non-systematic manifold. Prediction in the general case is impossible.

Proviso: Our analysis … acknowledged the existence of schemes of recurrence in which a happy combination of abstract laws and concrete circumstances makes typical, further determinations recurrent, and so brings them under the domination of intelligence.  Moreover, it acknowledged that concrete patterns of diverging series of conditions are intelligible; granted both the requisite information and mastery of systematic laws, it is possible in principle to work from any physical event, Z, through as many prior stages of its diverging and scattering conditions as one pleases; and it is this intelligibility of concrete patterns that grounds the conviction of determinists, such as A. Einstein. … However, we agree with the indeterminists inasmuch as they deny in the general case the possibility of deduction and prediction.  For while each concrete pattern of diverging conditions is intelligible, still its intelligibility lies not on the level of the abstract understanding that grasps systems of laws but on the level of the concrete understanding that deals with particular situations.  Moreover such concrete patterns form an enormous manifold that cannot be handled by abstract systematizing intelligence for the excellent reason that their intelligibility in each case is concrete. There results the peculiar type of impossibility that arises from mutual conditioning. Granted complete information on a totality of events, one could work out from knowledge of all laws the concrete pattern in which the laws related the events in the totality.  Again, granted knowledge of the concrete pattern, one could use it as a guide to obtain information on a totality of relevant events.  But the proviso of the first statement is the conclusion of the second; the proviso of the second statement is the conclusion of the first; and so both conclusions are merely theoretical possibilities.  For the concrete patterns form a non-systematic aggregate, and so it is only by appealing to the totality of relevant events that one can select the concrete pattern; on the other hand, the relevant totality of events is scattered, and so they can be selected for observation and measurement only if the relevant pattern is known already  [CWL 3, 650/672-73]

.d. Efficient Cause Theory vs. Purely-Relational Modern Field Theory: Newton’s Mechanics  vs. Purely-Relative Immanent-Intelligibility Dynamics

Leading Excerpt

… , as to the notion of cause, Newton conceived of his forces as efficient causes, and the modern mechanics drops the notion of force; it gets along perfectly well without it.  It thinks in terms of a field theory, the set of relationships between n objects.  The field theory is a set of intelligible relations linking what is implicitly defined by the relations themselves; it is a set of relational forms.  The form of any element is known through its relations to all other elements.  What is a mass?  A mass is anything that satisfies the fundamental equations that regard masses.  Consequently, when you add a new fundamental equation about mass, as Einstein did when he equated mass with energy, you get a new idea of mass.  Field theory is a matter of the immanent intelligibility of the object. [CWL 10, 154]

Special Relativity is primarily a field theory, that is, it is concerned not with efficient, instrumentalmaterial, or final causes of events, but with the intelligibility immanent in data; but Newtonian dynamics seems primarily a theory of efficient causes, of forces, their action, and the reaction evoked by action. … Special Relativity is stated as a methodological doctrine that regards the mathematical expression of physical principles and laws, but Newtonian dynamics is stated as a doctrine about the objects subject to laws.  [3, 43/67]

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, we believe, 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]

Key Concepts

Paraphrasing (CWL 10, 154):  as to the notion of cause, analysis in present-day academe – DSGE using IS-LM and AD-AS models and the Phillips Curve correlation – begins with the conception of prices and their changes as initially-given, external, efficient causes; but the Modern Macroeconomic Field Theory postpones the interpretation of prices; it gets along perfectly well without the notion of external efficient cause until it has achieved the immanent-intelligibility field theory of the process in terms of various dynamic relations.  It thinks in terms of a field theory, the set of internal relationships between n interdependent productive and monetary velocities. Modern Macroeconomic Field Theory is a set of intelligible relations linking what is implicitly defined by the relations among the productive and monetary velocities; it is a set of relational forms.  Like the relations defining Hilbert’s two points and a line, the form of any element is known through its relations to all other elements.  Modern Macroeconomic Field Theory is a matter of the immanent intelligibility among a)  n velocitous point-to-point and point-to-line production functionings, and b) the payments of dummy money correlated with these velocitous production functionings.  MMFT acknowledges “external”, human, efficient-causal agency, but it drops from its theory the notion of external cause ; it gets along perfectly well without it.  It thinks in terms of an immanent-intelligibility field theory, the universally-relevant set of relationships between n objects. [CWL 10, 154]

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]

Lonergan’s basic terms of systematic significance and field-theoretic significance are functional velocities.

Functional is for Lonergan a technical term pertaining to the realm of explanation, analysis, theory;  … 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” … [CWL 15  26-27  ftnt 27]

Now, at the root of classical method there are two heuristic principles.  The first is that similars are understood similarly, … .  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]

Paraphrasing [CWL 3, 43/67]:  Functional Macroeconomic Dynamics is primarily a field theory, that is, it is concerned not with efficient, instrumentalmaterial, or final causes of events, but with the intelligibility immanent in data; but present-day macrostatics and comparative macrostatics seem primarily a theory of efficient causes, of external forces, their action, and the reaction evoked by action. … The theory of FMD can be stated as a methodological doctrine that regards the mathematical expression of macroeconomic principles and laws as universally relevant in any state of politics, technology, and culture; in contrast, present-day DSGE with its IS-LM and AD-AS models and its defect-ridden Phillips Curve is stated as a doctrine about efficient, instrumentalmaterial, or final causes of situations subject to shocks.

.e. The Background Medium Done Away

Leading Excerpt

Maxwell, at first, did not adopt the modern concept of a field as a fundamental quantity that could independently exist. … the situation was resolved by the introduction of the Special Theory of Relativity by Albert Einstein in 1905.  This theory changed the way the viewpoints of modern observers were related to each other.  They became related to each other in such a way that velocity of electromagnetic waves in Maxwell’s theory would be the same for all observers.  By doing away with the need for a background medium, this development opened the way for physicists to start thinking about fields as truly independent entities. [Wikipedia (3), Field (Physics), page 3 of 11]

Key Concepts

Academe has not adopted the modern concept of a macroeconomic field as the immanent intelligibility among n implicitly-defined velocities and accelerations. The situation is to be resolved by the adoption of Lonergan’s Modern Macroeconomic Field theory called Functional Macroeconomic Dynamics.  Doing away with the need for a background medium such as ether and of external forces or shocks ex machina would open the way for macroeconomists to start thinking about fields as universally-relevant explanations of the always current, purely-dynamic functional process. [Wikipedia (3), Field (Physics), page 3 of 11]

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, we believe, 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]

At the end of “A Note on Geometric Possibility,” Lonergan encouraged others to complete and perfect his treatment.  Let me also say

As a final word, may I say that I present this section in the hope that I have noted some worthwhile points and that others will be moved to complete and perfect my treatment. [Collection  1967, 113]

Thank you.

Elements of Analysis

 

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[1] Foundations of Geometry

[2] Foundations of Geometry

[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

[4] CWL 21, 34-35

[5] CWL 21, 34-35

[6] Which is not to be identified with corporate or conventional National Income accounting

[7] Because of the rapidity of turnover, as contrasted with the efficiency of the use of money in transactions