A Comparative Note on Einstein’s Special-Relativity Field Theory and Lonergan’s Macroeconomic Field Theory

Special Relativity and Functional Macroeconomic Dynamics are field theories.  (Click here and here and here)  And we wish to gain further appreciation of FMD as a field theory by juxtaposing it with Special Relativity.

… 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]

Below we display four field-theoretic equations of Special Relativity.  [Cutnell & Johnson 2004, Two-Volume Soft Cover, Sixth Edition, 855-80]  We will examine the first equation, which explains the dilation of time.  On a vehicle moving at constant velocity along the x axis of a Cartesian-coordinate inertial frame of reference, a beam of light is emitted vertically and reflected back to its source by an observer at rest in the vehicle.  A second observer is fixed at rest on the ground in a Cartesian-coordinate frame of reference.   Δt represents the time interval of the observer fixed on the ground outside the passing vehicle.  Δt0 represents the time interval of the observer at rest with respect to the vehicle on which the beam of light is emitted and reflected.  v represents the velocity of the vehicle; and c represents the velocity of light, which is a constant in every inertial frame of reference:

The equation (1) expressing the dilation of time is unitary.  Its terms and relations are grasped by a single act of understanding.  All the concepts of the insight – duration, length,  and velocity – tumble out together.  The terms are implicitly defined by their relations among themselves, not by their relation to an exogenous shock or by their relation to us.  A different value of one of the three interrelated variables implies, by the unitary nature, a different value in all three variables.

The formula itself is an invariant;  it is general and relevant in any instance.  The variables themselves in the formula may change in magnitude contemporaneously with one another, but the formula itself – consisting of constant or variable terms and the collocation of the formula’s operations – implicitly defining the variables is by its invariance and universal relevance applicable in any instance.  The meaning of the terms of the formula is given by the relations in the formula, not by any nominal definition.  The terms are abstract, they are removed from relativity to the senses of an observer.  The formula is purely relational.

As we have said in The IS_LM, AD-AS Models and the Phillips Curve Correlation,and in A Contrast: Understanding Pricing in Macrostatic DSGE and in Macrodynamic FMD, and in Field Theory in Physics and Macroeconomics,  Lonergan’s Macroeconomic Dynamics is also a field theory.  It consists of primary, field-theoretic relativities – related purely among themselves and secondary determinations from the non-systematic manifold.

In [d”inverno, 1992] under the heading of “Chapter 13, The Structure of the Field Equations”, Ray d’Inverno reads Einstein’s implicit field equations of general relativity from right to left, left to right, and back and forth.  The left and right sides of the equation are mutual to one another.  He states:

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

Gab = 8πTab

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

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

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

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

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.

Revenues and costs contend and compromise with one another.  They mutually determine one another.

Thus, there is a sense in which supply always equals demand.  A good or service is under process until it is finally and actually sold, at which moment it is concomitantly supplied rather than merely under process or sitting in finished-goods inventory.  And a good or service is actually demanded when it is actually and finally purchased, so as to exit the process and no longer be under process.

Explanation, as distinguished from description, is constituted by terms related to one another.  Insight provides terms coherently related to one another.

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

The relations define the terms and the terms define the relations; and insight fixes both.  The terms are implicitly defined by their relations among themselves.  Because science is explanation by terms related to one another, science is  based upon and constituted by those insights which yield explanatory terms interlocked with one another in a definite formulation.  By the formulation’s constants, variables, and collocation of its operators, its form is isomorphic  with the patterns of the data of the phenomena.

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]

… an entirely new type of definition was introduced by Hilbert in his formulation of geometry.[4]  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.

…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)

Thus, masses might be defined as the correlatives implicit in Newton’s law of inverse squares.[1]  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.  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.[2] [CWL 3, 80/102-03]

… 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 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.  It follows that, as long as the symbolic pattern of a mathematical expression is unchanged, it’s mathematical meaning is unchanged.  Further, it follows that if a symbolic pattern is unchanged by any substitutions of a determinate group, then the mathematical meaning of the pattern is independent of the meaning of the substitutions. [CWL 3, 18-19/43]

The relating of terms in Hilbert’s plane and solid geometry bears a resemblance to the relating of explanatory aggregates in economics.  In both fields of inquiry the terms define and explain one another by their intelligible relations.  In the science of macroeconomics, the intelligible relations define what is named the formal cause.  “The causes are formal causes; it is only applied science that is concerned with agents and ends.”  Pure science is concerned with immanent intelligibility and with equations expressing that objective, purely-relational, immanent intelligibility in a formulation which is isomorphic with the patterns in the data. Thus, the meaning in Special Relativity of both space (length and distance) and time, and the meaning of pretty-quantital demand and supply in Functional Macroeconomic dynamics is fixed by the relations expressed in the equation. In both SR and FMD, the field of intelligible relations implicitly defines the objects.

… modern science is not simply an addition to what was known before.  It is the perfecting of the very notion of science itself, of knowing things by their causes, by analysis and synthesis.  What are the causes?  The field of intelligible relations that implicitly define the objects.  The objects with which a science deals are whatever is defined by its field of intelligible relations, whatever falls into that field.  The causes are formal causes; it is only applied science that is concerned with agents and ends.[3] [CWL 10, 155]

And it is only applied scientific macroeconomics – applied with respect to the field theory’s pure cycle, rather than to DSGE’s macrostatic models – that is concerned with human adaptation to the field-theoretical norms of the 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 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.  It follows that, as long as the symbolic pattern of a mathematical expression is unchanged, it’s mathematical meaning is unchanged.  Further, it follows that if a symbolic pattern is unchanged by any substitutions of a determinate group, then the mathematical meaning of the pattern is independent of the meaning of the substitutions. [CWL 3, 18-19/43]

And

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

And in the science of macroeconomics … such a property as invariance is a property, not of any incidental macroeconomic process, but of a set of expressions explaining all macroeconomic processes. (Click here and here)  The equation is a that unity of terms and relations which express the immanent intelligibility or formal cause of the process.

In the “public time” of macroeconomics, a scientifically-significant equation is a whole representing by isomorphism the theoretical concomitance and simultaneousity of elements.  Within the equation – even when the equation specifies a lag or the past or the future – the unity which is the intelligibility itself is all at once, atemporal, general, and universally relevant.  The mathematical variables within the equation – even though they may represent some aspect of public time –  are purely relational and implicitly defined by the empirically-verified, universally true, general relations in which they stand with one another.  The abstract terms fix the relations, and the relations define the abstract terms.  Generally, universally, and unitarily a change in one variable is – by the force of the equals sign functioning as the ”eternal is” of the relation – an atemporal change in another variable – simultaneously, not sequentially.  It is a matter of an atemporal implicit mutuality rather than of action and reaction.  All abstract explanatory terms are endogenous to the explanatory formulation.  No individual variable is an exogenous shock or efficient cause, as in Newton’s mechanics, or Keynes’s General Theory, or DSGE’s models.  So, we can say, Functional Macroeconomic Field Theory is stated as a methodological doctrine that regards the atemporal mathematical expression of the physical principles and laws relevant in any instance.

… 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]

Similar remarks may be made about Clerk-Maxwell’s relativistic electromagnetics.  His equations are general, purely relational, and universally relevant.  In his laws of electromagnetism, the electric intensity and the magnetic intensity are implicitly defined by a universally relevant pattern of dynamic interrelationships among the terms, 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 Ÿ H = 0
  • Δ Ÿ E = 0

In a familiar example of generality, the law of cosines, c2 = a2 + b2 -2ab(cos C), is both purely relational and more general than the less general or particular Pythagorean formula; the law of cosines contains the Pythagorean Theorem as a mere incidental case in which the angle C between sides a and b happens to equal 90o  and, therefore, (cos C) happens to equal 0.  Mere special cases, whether in trigonometry, mechanics or macroeconomics are of little interest to the theorist generalizing and explaining at a sufficiently general level of abstraction.

Now, let’s turn from the space-time of physics, the mutualities of electric and magnetic phenomena, and the identities of geometry and trigonometry towards economics’ public time of clocks and calendars and the field-theoretic relativity of prices and quantities in the explanation of a closed system of pretio-quantital flows.  The explanatory terms of scientific macroeconomics are implicitly defined by the functional relations in which they stand.  Functional Macroeconomic Dynamics links terms defined by their functional relations.  The functions are implicitly defined by the relations themselves. (See Explanatory Macroeconomics, Relevant in any Instance, pp 2-4)

Paraphrasing:  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.  (See CWL 10, 154]

In Lonergan’s Macroeconomic Field Theory, a function has two meanings:  It is initially described as a simple activity performed at a velocity; but then the utterance ”function” becomes an abstract term specifying “things in their relations to one another” in the explanation of the economic process.  Thus a function is an abstract term specifying the relations to one another of functional activities.  As an abstract term – both defining and defined by – it resembles the axiom of Hilbert’s two points and a line specified by the relation itself.

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]

Again, implicit definition implies simultaneous mutuality.  The mutuality of Modern Macroeconomic Field Theory always is a simultaneous mutuality of purely-current, purely dynamic implicitly and mutually defining velocities and accelerations, not a matter of describing or of action and the reaction evoked by the action, as in DSGE’s AD-AS and IS-LM models and the Phillips Curve correlation.

In a treatment of the relations constituting Macroeconomic Field Theory, one must be careful not to get side-tracked by the incidental, time-consuming, frictions of money’s circulation between, say, payday and the purchase of groceries in the subsequent week; one need not quibble over brief lags, because a) credit is available, and b) interest payments are functional constituents easily classified as initial or final payments in a particular circuit.

In the third place, it will be best to consider financial operations, that is any exchange in which a sum of money is paid for a sum of money to be received. … fill in …Financial operations are partly redistributive payments and partly payments for services rendered; thus in banking, payments of principal are redistributive, but payments of interest are operative, with interest paid to the banks as a final operative payment and interest paid by the banks to depositors an initial operative payment; again, in insurance the payment of policies is redistributive, but the payment of premiums on policies is partly redistributive and partly operative; it is redistributive to the extent it balances the payment of policies; and it is (a final operative payment) to the extent it pays insurance companies for their services.  (CWL 15, 44-45)

Though they in their use involve time, interest payments, when made, are merely constituents of the always-current pretio-quantital functional flows which explain the process and in the light of which the rental-price of money is to be understood.  Interest payments are intrinsic to the pattern of terms and relations through which the phenomena of the process may be embraced in a single coherent view..

the formal element that makes a treatise a treatise, consists in the pattern of terms and relations through which the materials may be embraced in a single coherent view. [CWL 3, 742]

In contrast, the Dynamic Stochastic General Equilibrium of the textbooks fails to reach a primary, abstract understanding of the eoconomic process and, by default, resorts to the imaginary primacy of unexpected, exogenous shocks from the probabilistic, non-systematic manifold, such as sudden changes in supply or demand.  And to these shocks endogenous variables react in some intrinsically-unpredictable sequence of action and reaction and reaction and reaction during what remains, as it was in the beginning, the probabilistic, non-systematic manifold.  Newton’s a) efficient-causal action of force and reactive effect of acceleration, and b) action and equal and opposite reaction is alive and well in today’s establishment macroeconomics.  And Einstein’s supervening Special-Relativity field theory is alive and gaining notice in today’s Modern Macroeconomic Field Theory.

Elsewhere we have treated primary relations and secondary determinations (Click here and here) , and implicit definition.

Paraphrasing:

Thinking in Functional Macroeconomic Dynamics requires both classical and statistical heuristics and methods.  FMD considers both a) the general, primary, abstract, immanent systematics of the process, and b) the always-current fact of indeterminacy of secondary boundary values in the concrete, non-systematic manifold.  Modern Macroeconomic Field Theory is versatile and attentive to all the possible classical and statistical applications of the fundamental act that is insight into sensible data. (CWL 10, 126-27) (See CWL 3, The Canon of Complete Explanation and The Canon of Statistical Residues, pp. 84-103)

The converse to this difference in the meaning of verification appears in the difference between classical and statistical predictions.  Classical predictions can be exact within assignable limits, because relations between measurements converge on the functional relations that formulate the classical laws.  But because relative actual frequencies differ at random from probabilities, statistical predictions primarily regard the probabilities of events and only secondarily determine the corresponding frequencies that differ at random from probabilities. Hence, even when numbers are very great and probabilities high, as in the kinetic theory of gases, the possibility of exception has to be acknowledged; and when predictions rest on a statistical axiomatic structure, as in quantum mechanics, the structure itself seems to involve a principle of indeterminacy or uncertainty. [CWL 3, 66/ 88-89 ]

As both Aristotle and Lonergan acknowledged, forms are grasped by mind in images.

τα μεν ουν ειδη το νοητικον εν τοις φαντασμασι νοει

[Aristotle, De Anima, III, 7, 431b 2] and [CWL 3, title page]

Forms are grasped by mind in images  [CWL 3, 677]

Under Key Notions we have treated The Necessity of a Diagram , and we have shown Lonergan’s Simple Move.  We encourage the reader to fully appreciate all that is contained in the Diagram of Rates of Flow.  It is an image.  The normative organization of flows, which it represents, must be grasped by mind.

Thus, if we want to have a comprehensive grasp of everything in a unified whole, we shall have to construct a diagram in which are symbolically represented all the various elements along with all the connections between them. [McShane 2014, 11 (quoting CWL 7, 151)]

We emphasize the importance of images in achieving classical laws and, when an image is not possible, resorting to statistical method.  Thus, as in quantum mechanics

… , 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 – …  ¶ 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)

And, finally, we again call attention to one of Lonergan’s key implicit equations in which macroeconomic costs and macroeconomic expenditures implicitly define one another.  The implications of this simple insight are vast.

P’Q’ = p’a’Q’ + p”a”Q”  (CWL 15, 156-59)

We note the truism or absoluteness of supply matching demand and demand matching supply because of their mutuality- they are merely two aspects of a single exchange – in an exchange interaction. A transaction is constituted by the exchange of something, which is thus simultaneously demanded-purchased and sold-supplied.  In the sense that a) what is not yet sold is still under process because it has not yet exited the process, and b) no matter how much something is desired, if it is not bought it has not been supplied, in that sense it is a truism that actual supply and actual demand are conjoined and unitary.

 

[1] F = gm1m2/d2

[2]Δ X E = (-1/c)H’

Δ X H = (+1/c)E’

Δ Ÿ H = 0

Δ Ÿ E = 0

[3] i.e. efficient cause, and final cause or purpose

[4] Foundations of Geometry