On the Subject of Metaphor
by Lyndon H. LaRouche, Jr
During the twenty-five-odd year reign of today's "New Age" cult, an ominous crippling of the U.S. individual's cognitive functions has been abuilding. This loss of mental capacity is presently affecting a growing majority among the under-fifty generations. Much of this damage is attributable directly to the multi-faceted influence of a modernist dogma which usually parades under such various names as "systems analysis," "linguistics," and "information theory."
Today, for example, rarely are pupils guided to reproduce, within their own minds, the Socratic experience of re-living the original discovery of crucial principles of scientific knowledge. Lacking the benefits of such once-traditional forms of secondary school learning in the subject matter of rigorous formal and synthetic geometries, for example, today's student would virtually never be able to attain an intelligible comprehension of even the bare fundamentals of physical science. Thus, today's modernist classrooms have been turned away from what is too often reviled as "authoritarian" teaching of conceptions; more and more, the modernist's "democratic" classroom and sterile textbook merely "provide information."
Similarly, a generation has passed since the time it was still fashionable to assess a pupil's progress in terms of that student's ability to apply prior learning to the effect of discovering, promptly, appropriate constructions of relevant solutions to unfamiliar problems. More and more, schools employ the "more efficient" practice, of degrading education to the rehearsing of pupils for passing computer-scorable forms of multiple-choice questionnaires.
These, and other enumerable applications of the pathological information theory doctrine, have brought upon us much of that widespread collapse of the individual victim's attention span which has occurred lately, accompanied by a correlated loss of the potential for those qualities of rationality which are associated with achievement in science and technology. That loss of scientific rationality is linked functionally to a parallel loss of personal capacity for comprehension and enjoyment of such once-respected fine arts as great music or the classical tragedies of Aeschylos, Cervantes,1 Shakespeare, and Schiller.
Such observations pose the question: What makes an ostensibly innocent technical doctrine, such as information theory, so wickedly pathological in its social effects? The most efficient tack for exposing the answer to that question, is a more rigorous, Socratic definition of the fine arts term, metaphor. We signify "metaphor" as William Empson's Seven Types of Ambiguity2for onehas identified it, as a phenomenon customarily associated with classical forms of poetry and drama. However, by "more rigorous," we should also show metaphor as the crucial feature of those thought-processes bearing upon the geometrical fundamentals of physical science.
That sets the task before us. So, without more foreword, to work.
What Is Metaphor?
In the case a literary construction points directly toward one object of attention, the ostensible subject, while uttering direct or implied reference to a different object, we have literary irony. Usually, to put the matter in its simplest terms, such irony is expressed usually in one of three forms: comparison, hyperbole, or metaphor. We may summarize fairly the most widely accepted academic view of this kind of ambiguity in the following terms.
The substitution of the name of another object for the customary name of the object in view, is traditionally considered in such academic climes as a matter of symbolism; that interpretation of these devices of irony is mistaken. Exposing this mistake of the academics leads us, in the relatively most direct way, to recognize that pathological fallacy of composition upon which Professor Norbert Wiener's information theory dogma is premised.
For this purpose, reference the domain of elementary geometry.
At an appropriate place in the secondary curriculum, the traditionalist teacher of secondary school geometry introduced the Pythagorean Theorem. The pupils of that class were guided to re-experience the mental act of original discovery by Pythagoras himself, thus to reconstruct a copy of that aspect of Pythagoras' creative mental processes within the mind of each of the pupils. This new existence within the pupil's own mind is itself an object of a special kind, a thought-object identified by the metaphorical name "Pythagorean Theorem."
The crux of this example is the fact, that the thought-object associated with the metaphorical name "Pythagorean Theorem," is neither an object of the outward senses, nor an object which can exist explicitly within any medium of communication.3
In this location, our primary argument is focused upon another example from the realm of synthetic ("constructive") geometry, Nicolaus of Cusa's revolutionary insight into the paradoxical Archimedes' Theorems which treat the subject of squaring the circle.4 This example shall serve us here, henceforth, as the model reference for an initial, more rigorous definition of classical metaphor. It is also a point of reference, therefore, for treating Wiener's fundamental fallacy.
Cusa proved, early in his adult life, that no curved line can be generated by means of joining together many very small straight lines. This proof led directly to the seventeenth-century discovery of the principle of physical least action, that all physical functions are of a species termed "non-algebraic" (or, "transcendental"), rather than arithmetic or algebraic. This, Cusa's referenced discovery, has the positive relevance, of being the continuing point of origin, and the mathematical cornerstone of the past five hundred years' birth and development of modern physical science.
The additional consideration to be stressed, is that this particular discovery by Cusa typifies all cases of creative forms of fundamental discovery in both science and the fine arts. That is to emphasize: solutions to real problems for the case that there exists no solution solely by means of deductive methods of argument. Those non-deductive solutions, solutions by methods which cannot be represented explicitly by any linear medium, such as communications media, typify the class of thought-objects to which belong the pupil's reliving of Pythagoras' discovery and of Cusa's discovery of an isoperimetric species of circular action absolutely distinct from the species of all possible linear functions.
It is thought-objects of that class which are the center of our attention here. It is the use of communicable arrays of names to identify members of that special class (species) of thought-objects, which we hold forth here as the proper form of illustration of the principle of metaphor.
We shall return to the case of metaphor in fine arts practice, after we have explored the definition of metaphor in the practice of physical science.
Squaring the Circle
Cusa reworked the four theorems of Archimedes on the subject of squaring the circle, by constructing a square whose area is equal to, and derived by construction from the preceding construction of a given circle. This assignment might be interpreted in two alternative ways. The student of algebra would wish to construct a square whose area (a2), differs by no more than a negligible amount from that of a given circle, Pi r2. The student of constructive geometry would demand that we accomplish this algebraic result by no means other than a strict, explicitly, exclusively geometric argument. Cusa focused upon the latter, geometric requirement.
From the secondary geometry classroom: the method for estimating the area of a square approximately equal to that of a given circle, is this. Simultaneously inscribe and circumscribe a pair of regular triangles, or squares (see Figure 1). Next, by halving angles, by construction, repeatedly double the number of sides to, for the squares, some number equal to 2 n. Take the average of the areas of the two polygons; estimate the value of Pi , the ratio of the circle's perimeter to its diameter, by dividing the average area of the two polygons by the factor of r2 (the square of the radius). Thus, for n = 8, Pi is estimated at approximately 3.1416321; for n = 16, the estimate for Pi is a much better approximation, 3.1415927.
Nonetheless, there is a very stubborn and profound paradox in this apparent algebraic success. This leads us to Cusa's discovery, and, from that point of origin, to the seventeenth-century discovery of a differential calculus of non-algebraic, least-action functions, by Leibniz and the Bernoullis.5
Admittedly, by the indicated method of estimated averages of the two regular polygons, we could estimate the square area of the given circle to any decimal position, according to the given algorithm. Ask the question: Does the perimeter of the inscribed polygon become ultimately congruent with the bounding circumference of the circle? With that question, a devastating paradox confronts us. Take the illustrative case, that n = 16; look at a region of the circular circumference of one minuteone-sixtieth (1/60) of a degree. There are slightly more than 182 angles of the inscribed polygon within each degree of measurement of the circle's perimeterslightly more than three per minute (see Figure 2). At the far extreme of n = 256, there would be approximately 3.216 × 1074 angles of the polygon for each degree of circumference. At a mere n = 112, for a circle of 1-centimeter radius, the distance along the circle's circumference between angles would be approximately 1.21009 × 10-33 centimeters, approximately the limit of a Planck distance in quantum microphysics.
Thus, the more nearly perfect our estimate of the circle's square area, the greater the degree of ontological difference between the circumference of the circle, as a geometric species of action, and the perimeter of our developing 2 n polygon, as a second species. The more nearly the polygon's perimeter approaches the trajectory of the circle's circumference, the greater the frequency of discontinuities in the polygonal perimeter, and, therefore, the greater the difference in species of geometric form between the circular and polygonal perimeters. This is true beyond all presently imaginable physical degrees of smallness.6
We have drawn the paradox out to the limits of n = 112 and n = 256, to impart a relevant emotional sense of the intensity of that paradox. Does a square area of the polygon approach approximation of the circular area? Of course it does. Does the perimeter of the polygon thereby converge asymptotically upon geometrical congruence with the circular circumference? No, quite the contrary.
The paradox so adduced from Archimedes' theorem, is also exemplary of the proper posing of the problem underlying all among those scientific discoveries which have more than a merely crucial significance for existing scientific knowledge. The solution to this paradox has what is best termed a unique quality of fundamental importance for all facets of scientific knowledge in general.
These paradoxes are all of the type exemplified by Plato's Parmenides dialogue, on the all-encompassing topic of "the One and the Many."7 Leading subsumed cases of unique discovery include each and all of the successive treatments of the "Platonic solids" by Plato,8 Luca Pacioli and Leonardo da Vinci,9 and Johannes Kepler.10 Similarly, the discovery of a universal principle of least action, by Fermat, Huygens, Leibniz, and the Bernoullis,11 is derived from preceding discoveries including both the isoperimetric principle and the implications of the Platonic solids. Examine the following crucial features of those interconnections.
The Isoperimetric Principle
The application to the squaring of the circle of that method of addressing such a paradox which is exemplified by Plato's Parmenides dialogue, yields essential results which are the common feature of each and all of the solutions for a series of the most fundamental scientific discoveries of the period from c.1440 a.d. through c.1700 a.d. . For reasons to be considered, these features are all presented from a negative standpoint:
1. Circular action is a distinct geometrical species of action in space-time, the which cannot be derived from any species of linear construction. No positive definition of circular action may be employed, if that definition specifies in any part a required point or piece of straight line (such as a radius).
2. Circular action is defined simply (negatively) as the least action of closed perimetric displacement required to subtend the relatively largest area. (Thus, the Fermat-Huygens-Leibniz-Bernoulli principle of least action is already implicit, "hereditarily," in Cusa's discovery.)
3. Circular action, because closed, (see Figure 3) is a form of continuous extension (continuous manifold) which contains its own metrical characteristic: counting in cycles and parts of cycles. A linear continuous manifold contains no inherent metrical quality which is not supplied to it by the external bounding imposed by a higher geometrical species of continuum.
4. Circular action bounds externally, and thus determines all linear species of constructions.
This is underlined by the paradoxical features of the stated case for the relative uniqueness of the five Platonic solids as stated by Plato, Pacioli, Leonardo, and Kepler. That is made general by the development of treatments of the cycloids (see Figure 4), from the work of Christiaan Huygens onward: all physical and arithmetic functions are properly stated in nothing less than terms of those non-algebraic functions which are derived "hereditarily" from the germ of the cycloid, and from the least-action principle embedded in the cycloid functions (see Figure 5). This is first demonstrated in physics, from Leonardo da Vinci through the work of the Bernoullis, for light (propagation of electromagnetic radiation) and hydrodynamics.
5. The additional crucial feature of circular action, is that it defines our universe in terms of both negative and positive curvatures, with the demonstration that negative curvature predominates. This point is summed up rather neatly in Johannes Kepler's 1611 booklet, On the Six-Cornered Snowflake.12 The snowflake is a non-living process determined by the function of positive curvature in determining the close packing of spherical bubbles.13 The negative curvature of the interior of each and all bubbles determines structures "hereditarily" cohering with the five Platonic solids, and, thus with the harmonic orderings cohering with the Golden Section of the circumscribing sphere's great circle.14
The universe can be considered as everywhere superdensely packed with spherical bubbles of all imaginable radii, as the unique, bounding characteristic of generalized "non-algebraic" function shows this to be necessarily the case. By the close of the seventeenth century, it was implicitly demonstrated (see Figure 6), that this bubbly universality of the least-action principle is otherwise characterized by the combined notions of electromagnetic least action and hydrodynamic forms of such action. Thus, frequency of radiation is associated with a corresponding resonant set of bubblese.g., of corresponding radii.15
Each of these discoveries is associated with a special kind of paradox, which might be termed "a true paradox." In the instance of squaring the circle, the paradox is, that the more successfully we estimated the square area of the circle, the more extremely we proved the non-congruence of the polygonal perimeter with the circular circumference. "The more we appear to succeed, the more we truly fail," might be a fair image of "a true paradox."
So, in the case of the five Platonic solids, the more we attempt to circumvent the limitation identified by Plato, as did Archimedes, Pacioli, and so on, the more we understand the germinal uniqueness of the dodecahedron, and of the Golden Section of that great circle's negative curvature.
By the close of the seventeenth century, the successive work of Huygens, Leibniz, and the Bernoullis on the tautochrone/brachistochrone problem for isochronism and for light, had shown implicitly that all possible action in our universe must conform to multiply interacting circular action upon circular action, not straight line interaction between points considered pairwise. Thus, the accumulation of paradoxical, negative considerations, delimited acceptable alternatives to such merely negative, or paradoxical considerations. A leap of consciousness was required to discover the alternative to such a concatenation of merely negative considerations. Plato's Parmenides dialogue is a model for the nature of this problem.
So, an apparent solution leaps into the mind of the successful discoverer. That solution, as a thought-object, cannot be directly depicted in terms of communications media available. Thus, if it cannot be communicated explicitly, how might we know whether the newborn thought-object were valid, or not? There are two conditions which prompt us to recognize such a thought-object as valid. First, it satisfies all of the negative conditions associated with the relevant paradox. Second, it goes beyond those negative requirements, to enable us to generate efficient hypotheses, reaching by these means into realms which were unattainable for us without the aid of these new thought-objects.
By signaling both the negative preconditions of a hypothesis, and also, similarly, describing efficient new constructions derived from the new thought-objects, we communicate to our own and other consciousness the formal proofs of the thought-object's validity. Thus, we may be relatively certain, that the thought-objects so generated by different, communicating intellects are congruent thought-objects.
Therefore, by citing the name of the thought-object among those who share its possession, we may communicate the efficient sharing of consciousness of the thought-object which, by its nature, may be neither explicitly portrayed as a sensuous object, nor be depicted in terms of a medium of formal communication, formal mathematical communications sharing this defect.
The Necessity of Metaphor
So far, we have described the thought-object as the demonstrable solution to those unique paradoxes which are akin in Type to Plato's Parmenides paradox. We have indicated that these thought-objects occur as relatively absolute discontinuities with respect to the characteristics of the medium of communication in which the relevant problem has been stated negatively. Thus, we have indicated, the thought-object itself cannot be depicted explicitly within the domain of the communication medium. However, the reference to such a thought-object can be recognized by a hearer whose mind contains the sibling of that same thought-object.
In a classical humanist form of secondary school education, most emphatically, the emphasis is on presenting the pupils with the most important among the unique and other relatively elementary discoveries in the entire historical sweep of the advancement of civilized knowledge. It is desirable that original sources be used whenever they are both available, and in a form suited to that stage in maturation of the pupil's powers of comprehension. Otherwise, only if such suitable primary sources are not available, we should rely upon paraphrases which effectively and fairly state the true paradox associated with that original discovery.
This form of classroom introduction to such original sources has a required order, as the ordering of Euclid's Elements of geometry illustrates, from a formalist standpoint, the notion of a choice of such an order. Secondly, the ordering is determined by the consideration, that mastery of one discovery is virtually prerequisite for the comprehension of a successor in that series. The sound secondary curriculum teaches geometry and the plastic arts, as the domain of visual experience, as, in parallel, the student concurrently learns language, literature, and musicthe domain of hearing. The historical order internal to the sciences of vision and hearing provides a virtually indispensable concomitant to the study of the rise of the European (Christian Humanist) Renaissance of the fifteenth century, out of ancient and medieval history, and upon that foundation, the study of global post-Renaissance history.
Several most important effects are fostered by such a classical humanist form of secondary education.
In each case, first of all, the pupil replicates an original discovery. Within the student's own intellect, there is approximately a replication of the mental processes of that creative discovery which was experienced earlier by the original discoverer. Later, the pupil experiences another such crucial discovery, by an original source who depended, in turn, as the student does, upon the prior of these two original sources considered. So, it continues. So, in respect to mathematics and physical science, for example, the pupil's mind is populated, in effect, by a growing number of such past historical personalities of science, to the effect that the pupil not merely imagines these persons, as if they were merely characters in some story, but knows each as a living, thinking person, through the replication of some of the creative processes of each within the pupil's own mental processes.
Functions of 'Discontinuity'
In that illustrative case from geometry which we have treated thus far, the Platonic form of paradox embedded within an Archimedean estimate for the squaring of the circle, it is shown, that even far, far beyond the already logically meaningless case of an hypothetical regular polygon of 2256 sides, there remains a distinct, intelligibly measurable gap between the relatively lesser area of each and all inscribed regular polygons and the marginally greater area of the relevant circle. The persistence of the discreteness of that gap, persisting beyond all limit of such extension, is a model for a simple type of mathematical discontinuity. It is not the magnitude of the gap, which is this discontinuity; the discontinuity is the fact of the persisting, transfinite discreteness of this gap, however tiny that persisting gap were to become.16
Examine that class of simple types of discontinuity from a subsuming vantage-point. Explore, in this way, the nature of those mental existences which we have identified as thought-objects. Consider formal theorem-lattices.17
The short definition of a deductive theorem-lattice is provided in the following three, complementary statements. Given, any constant, integral set of deductive axioms and postulates:
1. No consistent theorem derived from that set of axioms and postulates states anything which was not already implicit in that set of fixed underlying assumptions.
2. Any theorem of this lattice, which is constructed to represent an experience, will project upon such representation nothing but the ideas of the ontological qualities and behavioral potentialities already implicit in the latter's underlying, integral set of axiomatic and postulational assumptions.
3. Any demonstration which refutes a single deductively consistent theorem of such a theorem-lattice, refutes axiomatically the choice of underlying integral set of axioms and postulates upon which each and all possible hypotheses or theorems of that theorem-lattice depend.
Thus, for example, to the degree which the intended development of the mathematical physics of a Descartes and Newton is intended implicitly to perfect itself as a deductive theorem-lattice, the development of that physics has the combined form of expanding the number of theorems, while perfecting the deductive consistency of the expanding theorem-lattice as a whole. When nature itself manifestly denies, even in a single instance, what is shown to be a consistent theorem of that lattice, the entire lattice's underlying an integral set of axioms and postulates must be altered. The alteration must remedy the disagreement with nature in that one crucial instance, but without producing experimentally invalid forms of other theorem types.
Let us recognize that principle of axiomatic consistency of a deductive theorem-lattice by the sometimes employed term, "hereditary principle." Let us represent a successful, generalized, successive, step-wise improvement over deductive theorem-lattice A, by the series A, B, C, D, E, ....18 The difference between any two adjacent terms of this series, is some change in the underlying, integral set of axioms and postulates of the predecessor term. Thus, for reason of that change, no deductive consistency exists between any one term of that series, and each and all other terms of that same series. This gap separating each term of that series from each among all the other terms, and doing this with deductive absoluteness, is a discrete discontinuity in the same broad sense as the gap separating the linear generation of the constructions of a regular polygonal perimeter from the different, isoperimetric quality of the relevant, inscribing circular action.
In the simple case of squaring the circle, we address a single object, that circle. The germ of so-called "non-algebraic," or "transcendental" functions is already there, in Cusa's treatment of this paradox; but, we must take additional steps to see it clearly. We must recognize a principle, integral to all competent mathematics, intrinsic to the notion of an isoperimetric circular action; we must recognize the pervasiveness of a non-algebraic principle, which Gottfried Leibniz et al. named analysis situs. The cycloid is the best vantage-point for a secondary classroom treatment of these matters.
Roll a relatively very small circle (r) along the outside perimeter of a relatively extremely large circle (R). The result is that the perimeter of the very large circle, R, appears relatively almost a straight line.19 At the start of this roll, the perimeter of circle r touches the perimeter of the larger circle at point P0; to this corresponds point p on the smaller circle's circumference (see Figure 7). Roll circle r clockwise, making a series of points of tangency of p on circle R each time the rotating point p again touches the perimeter of R. Thus, between points P0 and P1, the trajectory of p forms a curved line, a cycloid, approximately that of Roberval20 or Christiaan Huygens21 (see Figure 8).
Extending the class' study of cycloids a bit, we meet the fact, that there is a different result if circle r is rolled upon the inside, as opposed to the outside of the perimeter of R. Take the cases that R = 2r, and R = 3r, and R = 4r, and R = 5r (see Figure 9). These constructions draw our attention to the fact that there is an important functional difference between the positive curvature of the exterior of a circular perimeter, and the negative curvature of the interior of that perimeter.
Then, we follow Huygens through his treatments of the tautochrone and involute-evolute constructions (see Figure 10).22 Together with Huygens, Leibniz, the Bernoullis, et al., we discover several things which are crucial for all valid developments in mathematical physics, from approximately 1700 a.d. onward, to date, things bearing directly upon these principles of metaphor.
No student should be graduated from any secondary school, unless he or she has assimilated the treatments of cycloid, tautochrone, and involute-evolute relationships as put forth in Huygens' work on these subjects. Without that, and without the mastery of the tautochronic principle of least action for refraction of light as Leibniz and the Bernoullis set this forth during the 1690's (see Figure 6), there could be no competent grounding of the student in the barest prerequisites of as much as uttering the term "modern physical science." (How many science and engineering professionals today have met that requirement?)
In the very simplest case, the simple cycloids, non-algebraic functions, represents that class of functions derived, "hereditarily," by performing circular action upon circular action. This is then extended, to indicate circular action upon that result, and so on more or less indefinitely. This is extended, in turn, by Huygens, Leibniz, the Bernoullis, and by Gaspard Monge later,23 to include those derived constructions obtained in the classroom by winding and unwinding taut threads, the so-called evolutes and involutes (see Figure 11). This includes the class known as envelopes.24 We must include, as a matter of strict principle, the notion of analysis situs introduced by Leibniz.25 We must include, retrospectively, Kepler's work on applications of the distinction between positive and negative curvature,26 doing this from the standpoints of both elementary analysis situs and the treatment of negative curvature in the Riemann-Beltrami counter-attack upon the intertwined hoaxes of Clausius-Kelvin, Helmholz, and Maxwell.27
Short of Cantor's aleph transfinites,28 all possible functions in mathematical physics, including problems of number theory, are non-algebraic, essentially geometrical functions of this extended transcendental form. This specific point will bring us soon here to the case of the problems posed by the widespread influence of such related hoaxes as "information theory," "systems analysis," and the "linguistics" of Russell, Korsch, Carnap, Harris, and Chomsky.29
Metaphor and Function
With the consideration of the indicated series, A, B, C, D, E, ..., we define an ordered sequence of these terms. The "variable" of this ordering is not those terms themselves, but, rather, the discontinuities separating each term from each of all the others. These discontinuities are the point of intelligible access to the relative ontological nature of the classes of thought-objects to which we have referred above.
Consider Cantor's alephs. We have Aleph0, Aleph1, Aleph2, and so on. These alephs, so ordered as in any sequence, form a manifold. This manifold is of a Cantor Type; this Type is ontologically of the quality of discontinuity separating each of Aleph0, Aleph1, etc., from all others.
This manifold and its Type cannot be reduced to any notion of function which is consistent with our use of the term "function" to denote a class of geometrical, non-algebraic, or transcendental functions. Yet, the aleph-manifold, with its many alternative orderings, is defined by a typical quality of all such orderings. That implies a notion of "function," although in no conventional sense of a mathematical-physics function. History proves such a higher aleph-manifold quality of functional ordering to exist.
The continued existence of the history of our human species is a unique demonstration, that functions ordering some sequences of discontinuities of this type (A, B, C, D, E, ...), do exist, are efficiently existent. However, as we have just noted, it is also true, as Cantor and Gödel, most notably, have demonstrated, that these functions are not subsumed under the type of non-algebraic functions. Subjectively, these higher-than-transcendental, aleph-type functions exist only within the sovereign boundaries of the individual mind; they cannot be represented explicitly within the linear terms of any medium of communication. Nonetheless, not only do these higher, subjective functions exist; they are demonstrably efficient causal agencies in our physical universe.
The historical fact, that these higher functions are characteristic of successful scientific progress's raising of the physically efficient, per-capita, productive powers of man's command over the universe, shows that the subjective processes, of creative-mental function, address the physical universe in a manner suggesting that such forms of communication between man and the universe as a whole are akin to the communication of such qualities of thought-objects as thought-objects. These functions of fundamental scientific progress, which act above the reach of any formal mathematical physics, are the characteristic of man's historically efficient relationship to scientific mastery of our universe.
That outline of our proposition now given, we examine the set of relationships among names, thought-objects, and our universe. Let us speak of three domains. First, the domain of thought-objects, within the sovereign bounds of the individual's mental-creative life. Second, the relevant plane of sense and communication media. Third, within the physical universe, behind the superficiality of sense-experience, an underlying governing agency of principle, which controls the lawful behavior of the universe, and which will "recognize" certain of our changes in forms of actions with a favorable response.
Reference Figure 12. We have person A, a secondary-school teacher, and also an experimenter. We have person B, a student, and an observer of the experiment being performed. There is the experimental subject, X. A acts upon X. Student B observes X, and also observes A's actions upon X throughout the experiment. A communicates, reciprocally, with B, a communication which precedes and accompanies the experiment, and which continues after the experiment's completion.
A, beginning from a thought-object in his own mind, provokes the replication of that thought-object within the mind of student B. This occurs through the method of Socratic negation, as applicable to a case which meets the requirement to be a true paradox. Consider an example, related to the Cusa isoperimetric paradox, which illustrates this phase of the transactions among A, B, and X, in this illustration; consider the proof of the uniqueness of the five Platonic solids.30
Take three great circles which can be moved about on the surface of a sphere and arranged at any inclination one to another, as if they were hoops having the same radius as the sphere. Experimenting with such hoops, it will be discovered that when they are arranged such that their respective circumferences mutually divide one another into four equal arcs, the surface of the sphere is partitioned into eight equal, regular spherical triangles. The six points of pairwise intersection of the hoops will be found to form the vertices of an octahedron (see Figure 13).
Do the same for four and six hoops. For four hoops, the pairwise intersection occurs at twelve points, coinciding with the twelve vertices of a cuboctahedron (the truncation through midpoints of edges of the cube or octahedron). The surface of the sphere is thus partitioned into eight equal and regular spherical triangles and six equal and regular spherical quadrilaterals. Each great circle is divided by the others into six equal arcs.
Using six hoops, thirty points of pairwise intersection result, forming the vertices of an icosidodecahedron (the truncation through midpoints of edges of the icosahedron or dodecahedron). The surface of the sphere is partitioned into twelve equal and regular spherical pentagons and twenty equal and regular spherical triangles. Each great circle is divided by the others into ten equal arcs.
It can then be proven that there are no other partitions of the sphere resulting in the division of the great circles into equal arcs. From the limiting case of six hoops, which permits the construction of twelve pentagonal faces, is demonstrated the primacy of the dodecahedron, and relative uniqueness of the five Platonic solids.31 From the six-hooped figure containing dodecahedron and icosahedron, the cube, octahedron, and tetrahedron may be readily derived.
The Golden Section may then be conveniently demonstrated as the ratio of radius to chord on the dodecagon formed by inscription in each of the six great circles, or, alternatively, as one of the many well-known internal relationships of the pentagon, formed by projection of the spherical pentagon onto a plane. In either case, the derivation of this ratio from the construction upon the sphere is to be stressed, rather than derivation from a pentagon or pentagonal division of the circle, presumed as given or constructed by algebraic artifice.
This approach has shown several points which are of crucial importance:
1. The necessity of deriving these regular polyhedra from regular spherical triangles, quadrilaterals, and pentagons is shown. This correlates with our earlier study of the paradoxical effort to square the circle. The construction of the polyhedra is bounded externally by spherical action.
2. That, only regular division of the sphere's surface by the factors 3, 4, and 5 succeeds. Thus, the dodecahedron corresponds to the upper limit of construction, since it is derived from fivefold division. No regular polyhedron of hexagonal sides, or larger, is constructible.
3. That all five regular solids are derived from the construction of the pentagonal-sided dodecahedron.
A strong indication of this is the following view of harmonic orderings cohering with the Golden Section.
The customary classroom and related practice, is to explain the construction of the Golden Section as necessary for the construction of the regular pentagon. This seemingly innocent practice has contributed to the circulation of much nonsense, nonsense which is avoided if the Golden Section is situated directly within a proper reading of the simple construction-proof of the uniqueness of the five Platonic solids. Turn, for illustration of the point, to reference again Pacioli's De Divina Proportione.
Pacioli, Leonardo da Vinci, et al., showed that, on the scale of direct sensory observation of ordinary processes, all living processes have an harmonic ordering of growth and morphology of function which coheres, as a Type, with the Golden Section; whereas, all non-living processes, on this scale, have a different Type of characteristic harmonic ordering. This point is later re-stated by Johannes Kepler in various locations, including his Snowflake paper. Modern evidence leaves no doubt of the correctness of that so-qualified observation of Pacioli, Leonardo, Kepler, et al.
Unfortunately, too frequently, those who point to this distinctive Platonic coherence of living processes with the Golden Section, either degrade this connection to a kind of cabalistic speculation, or simply present the Golden Section itself as a section in a circle, without showing necessity, in such popularized terms as to leave the matter of harmonic ordering vulnerable to a false charge of numerological mystification. This latter negligence appears whenever we might misdefine the Golden Section in terms of either, simply, "the Golden Mean," or as simply the derivation of the pentagon, by construction from a given circle.
If the following, restated, preconditions of rigorous treatment are satisfied, in defining the Golden Section, the risk of misleading mystification is avoided.
First, the Golden Section is located as a necessary, (intrinsic) metrical characteristic of negative spherical curvature, as nothing other than the characteristic distinction of the spherical generation of a subsumed, constructed dodecahedron.
Second, the five Platonic solids are recognized as each and all subsumed by the construction of a single one among them, the dodecahedron.
Third, this topic, of spherical determination of the Platonic solids' uniqueness, is referenced from the standpoint of the method we indicated above, for recognizing and solving the deep paradox inhering in Archimedean squaring of the circle. In short, that the spherical action, of a different, higher species than any polyhedron, bounds externally, and thus determines the constructible existence and metrical characteristics of the species of polyhedra in general.
These points are underscored by comparing the paradoxical process of squaring the circle to the way in which harmonic orderings coherent with the Golden Section bound externally the linear Fibonacci series (see Figure 14)32 This may then be compared with Johannes Kepler's distinction between packings contrary to, respectively, negative and positive spherical curvatures (see Figure 15). In short, the Golden Section is a determined, necessary limit of packing of the type illustrated by the Fibonacci "growth" series under the constraint of negative curvature. With that observation, the premises for mystification evaporate.
That material covered by teacher A, the teacher brings the student's attention to the work of Huygens and his successors on the subjects of tautochrone and brachistochrone.33 This leads the student through (a) the elements of the cycloid, (b) the proof, by Huygens, that the cycloid is a tautochrone, and (c) the proof, by Johann Bernoulli et al.,that the tautochrone is also the brachistochrone (see Figure 16).
The teacher, A, then reviews the work which was referenced by Johann Bernoulli, Huygens' Treatise on Light,34 as the next unit of study in B's classroom. In this setting, A includes relevant references to the subject of light and hydrodynamics in the Leonardo da Vinci Codices, in the work of Fermat, and the treatments of a universal principle of least action by Fermat, Leibniz, and the Bernoullis. The geometrical construction employed as proofs, together with the Bernoulli experiment itself, are, combined into one, the experiment X; the Bernoulli experiment itself, is the relevant physical experiment.
This experiment shows implicitly that the universe portrayed by René Descartes and Isaac Newton does not exist. First, the tautochrone/brachistochrone equivalence, for the case of a constant relative speed of light, shows that the notion of physical function in our universe requires that family of non-linear, non-algebraic functions which is derived from the isoperimetric principle. This notion of non-algebraic function supersedes all those notions of arithmetic-algebraic function derived from a notion of pairwise, linear causal interrelationship as primary. Thus the refutation of Descartes and Newton. Whereas, the non-algebraic and algebraic conceptions conflict respecting a notion of causal principle, the algebraic view is shown to be axiomatically false.
This signifies that the Cartesian domain is axiomatically false in conception from the outset. Isaac Newton's case is ultimately the same, but historically of greater ironical interest.
Newton refers to what he admits to be an absurdity of his mathematical-physics scheme, that it represents the universe as "running down," in the sense of a mechanical time-piece.35 This "clock-winder" topic is a featured element within the Leibniz-Clark-Newton correspondence later.36 Later, during the 1850's, Rudolf Clausius, at the prompting of Lord Kelvin, employed the assistance of the mathematician Herman Grassman to codify the so-called "universal entropy" dogma,37 or "Second Law of Thermodynamics," which is nothing but a nineteenth-century version of Newton's seventeenth-century "clock-winder" fallacy. The key reference-point for discussion here, is that the seventeenth-century Newton, unlike the nineteenth-century Clausius, Kelvin, Helmholz, Rayleigh, and Boltzmann, states clearly that the fallacy of "universal entropy" erupts within his physics as a consequence of a defect embedded within his choice of mathematics.
This represents an important challenge for teacher A. B asks, "Does entropy exist?" "Yes," replies A, "but not as a governing principle of the physical universe." B is perplexed by this. A explains, by reference to Kepler, "Remember our studies of Kepler's work?"
"Remember our review of this matter in our study of Kepler's Snowflake paper?" Positive curvature is associated with non-living functions, such as the snowflake, which do exhibit entropy as an included characteristic. However, negative curvature requires a non-entropic ordering cohering with the limiting implications of the Golden Section.
The point here is, that, in a universe super-densely packed with spherical bubbles,38 the envelope of all positive curvatures is a negative curvature. Thus, although some phase-states of our universe are entropic, other phase-states are not. Up to recent decades, we have known that the astrophysical realm, like living processes, is negentropic; we have found, as, for example, so-called "cold fusion" illustrates this, that the extremes of microspace are also characteristically negentropic.
Thus, Newton was correct in blaming his choice of Cartesian algebraic mathematics for the "clock-winder" fallacy "hereditarily" embedded within his Principia as a whole.
The succession of fundamental elementary discoveries shared among persons such as A and B here, all involve significant alterations in the Socratically implicit underlying set of axiom-equivalent and postulate-equivalent assumptions.39 The difference between Leibniz's physics, and the flawed, inferior model of Newton, helps us to recognize some leading features of that system of metaphor which is modern science practice.
Think of the elementary system of stereographic projection. Use this as an analogue of metaphor (see Figure 17). The sphere NS sits upon a flat sheet. Point S, touching the sheet, is termed the South Pole, and the opposite point, N, the North Pole. To trace any figure drawn on the flat sheet onto the surface of the sphere NS, draw a moving ray from the North Pole to the sphere to the outline of the figure on the flat sheet. Where the moving ray cuts through the surface of the sphere, there lies the spherical image of the relevant trace figure upon the flat sheet.
Now reverse the projection, from a figure on the sphere, to a shadow cast by the moving, tracing ray upon the flat sheet. Then, add a third feature to this; that the image on the sphere itself be a projection of some original image in an unknown domain, that real, unseen universe, hidden behind the metaphorical imageries of our sense experience. Let this unseen, real universe be approximated, metaphorically, by the Cantorian Type of the aleph-manifold as a whole. Let the domain of scientific physical functions, in the mind, be represented, metaphorically, in communication, by the analysis-situs-enriched, extended Type of non-algebraic functions in general. Then, thirdly, let the lowest order, the linear world of Aristotelian nominalist sense-certainty, be represented, metaphorically, by the Type of systems of deductive theorem-lattices.
Those three levels, combined so, represent, metaphorically, the domain of metaphor. The notion of a Type which subsumes all the possible relationships among these three, including matters of physical science, but also classical forms of drama, poetry, and music, we define here as the function of metaphor.
Before passing beyond the thought-objects of physical science, to the classical art-forms, we have two final matters to settle respecting physical science. One of these two is, obviously, the query, "If formal, explicitly communicable aspects of physical science are metaphorical, what happens, then, to the idea of an objective mathematical physics?" The other of the two propositions next to be considered is that notion of negentropy which Professor Wiener so crudely abused. We review the special topic of negentropy first, before proceeding to the issue of objectivity of formal physical science in general.
Prior to the referenced work of Pacioli and Leonardo da Vinci, the mathematical representation of growth was given, as we noted above, by Leonardo of Pisa's Fibonacci Series. The Fibonacci series does not represent a principle of growth, but only an attempt to approximate the notion of growth descriptively, using methods analogous to Archimedes' squaring of the circle. We stressed earlier here, that the Golden Section bounds externally the extended Fibonacci series, as the circle bounds externally the 2 n -polygonal perimeter; that which bounds, is of a different, higher species than that which is bounded; the higher cannot be derived "hereditarily" from the lower.
There is another notion of growth, the one corresponding to a simple "compound-interest" function, (1 + x) t (see Figure 18). The characteristics of this growth (arithmetic mean, geometric mean, harmonic mean, arithmetic-geometric mean) are given by elliptic functions of the conical cross-section of the unit cycle of growth (see Figure 19). The relatively higher orders of growth functions are hyperconic ones, which shows us that generation of increasing density of apparent discontinuities which is the observable characteristic of growth per se, or negative entropy (negentropy).
In other words, it is not possible to represent growth of this Type characteristic of living processes by means of a deductive form of mathematics, such as a Fibonacci series, the mathematics of pairwise interactions along straight-line pathways. The attempt to define negentropy, as Wiener does, by means of Ludwig Boltzmann's statistical mechanics ("H-theorem"), is simply outright incompetence from the outset of such an endeavor.40
The characteristics of human scientific progress are, as we have indicated earlier, changes in the axiomatic basis of theorem-lattices which are of a Type of the Cantorian aleph-manifold. Biological evolution is a process of this same formal Type. Kepler's universe is, ultimately, of this same Type. Until the twentieth century, we observed this Type of process in living processes, in manifest results of creative mental discoveries, and in Kepler's implicit ordering of the universe as a whole. Recently, we observe the same underlying pattern of elementary causation as we approach phenomena of physical chemistry on the scale of 10-16 to 10-17 centimeters.
The point here is not simply to refute Wiener and the prejudiced dupes who follow his gnostic teaching. The term "entropy" was given distinct significance by the arguments of such collaborating spokesmen as Clausius, Grassman, Kelvin, Helmholz, Maxwell, and Rayleigh, and also, later, by such continental-science figures as Max Planck, et al. As we have indicated above, from the vantage-points of Leonardo da Vinci, Kepler, et al., and also their famous opponent Isaac Newton, the present-day term "entropy" signifies to the sixteenth and seventeenth centuries' literature, Kepler's distinction between the five-petal flower, determined by negative spherical curvature, and the six-cornered snowflake, the latter determined by close packing of positive spherical curvatures. Nineteenth-century developments in the field of Leibniz's analysis situs only illuminate more brightly that elemental distinction in species between the intrinsically entropic "hereditary" characteristic of positive curvature, and the intrinsically negentropic characteristic of negative spherical curvature.
It should be evident, on these very elementary grounds, that the mathematical schema of Grassman, upon which Clausius and Kelvin's introduction of the so-called "Second Law of Thermodynamics" depends, is a fraud, a fallacy of composition akin to recognizing only one side of the set of terms of Schrödinger's Psi-function.41 Since the work of Pacioli and Leonardo, or, since that of Kepler, positive spherical curvature bounds externally a system of linear inequalities (functions) which are pervasively, "hereditarily," characteristically entropic; but, negative spherical curvature, the externally bounding curvature of universal physical processes, generates processes which are, like life itself, characteristically negentropic.
These "hereditary" distinctions in harmonic orderings, between positive and negative modes of spherical curvature, obviously pertain to the metaphorical domain of extended non-algebraic function, the which is the middle, second of the three levels of a function of metaphor. This apparent negentropy, of negative spherical-curvature harmonics, is, of course, externally bounded, subsumed by the higher Cantorian Type associated with the aleph-manifold. Negentropy does not exist, as a concept of a governing principal process on the level of any single deductive theorem-lattice.42
There is plainly no intrinsic error in employing commonly accepted names as pointers for indicating the respectively appropriate kinds of phenomenon. Absurdity, veering toward insanity, enters if we tolerate the radical nominalist proposal, to base our belief respecting the intrinsic physics of phenomenon upon the dictionary definition of those mere terms. However, communication is not limited to pointing while uttering a noun or nominative phrase. In the civilized efforts to impose literacy upon customary forms of use of a language, we render a literate form of language a method for mapping our communicable representation of both the place of a phenomenon in the universe, and also mapping some of the internal relations within the phenomenon itself.
The most relevant of the characteristics of any literate form of spoken language, must be its adducible implied philosophy, its implicit way of delimiting the manner in which cause-effect relations are defined ontologically, and "mapped." These differences in the use of language for "mapping" what are assumedly cause-effect relationships, may be absolute or merely relative. That is to say, they are absolute if they inhere in the accepted forms of use of that language; they are relative, if they reflect one among several optional forms of use of that language in currency.
Consider the similarity to "mathematical languages."
There are absolute philosophical differences separating the reductionist algebra of a Descartes from the non-algebraic representations of function of a Leibniz. Yet, as long as we limit the use of reductionist algebra to mere description of ordinary kinds of non-crucial phenomena, algebra can be a useful tool. One must recognize there are circumstances under which the intrinsically inferior, philosophically false method of such an algebra must be avoided, and the superior, non-algebraic method is mandatoryin treating topics bearing upon least action, for example.
This consideration brings us to a higher degree of metaphor.
At the beginning here, we have emphasized the simplest aspect of our topic, the metaphorical relationship between a single term and an unutterable, but real, individual thought-object. Now we have to consider a higher order of thought-object; we must consider the point, that entire statements, statements which purport to "map" cause-effect relationships, even entire books sometimes, may also be metaphors for single thought-objects. Turn, now, to an elementary illustration of this point.
In our treatments of some elementary thought-objects of scientific work, thus far, we have considered some crucial thought-objects originally attributed to such authors as Pythagoras, Plato, Archimedes, Euclid, Nicolaus of Cusa, Luca Pacioli, Leonardo da Vinci, Johannes Kepler, Pierre Fermat, Christiaan Huygens, Gottfried Leibniz, and Johann Bernoulli. Associate the original form of that true paradox and its solution as associated with a name, a portrait, and a brief biographical sketch of that author. Those images you now associate with a corresponding memory of your re-experiencing the production of the relevant thought-object originally experienced by them.
How should one order the seating of these discoverers in one's memory? For scientific work, the primary consideration must be, not raw chronology as such, but, rather, the rather obvious principle of "this necessary predecessor" among crucial discoveries as a whole. That ordering principle permits a range of equally valid, but different orderings among the same array of discoverers. Each of those choices of orderings among arrayed thought-objects, is a distinct thought-object, with the included quality of a Cantorian Type, indeed, subsumed by the Type of an aleph-manifold.
Consider an obvious choice of illustration here. We have begun with Nicolaus of Cusa's 1430's discovery of an isoperimetric principle paradoxically underlying theorems of Archimedes. That isoperimetric notion, as elaborated by Cusa,43 set the stage for a range of crucial discoveries by Pacioli, Leonardo da Vinci et al., at the close of the fifteenth century and the first decades of the sixteenth. The treatment of Plato's discovery of the Golden Section's implications, by Leonardo et al., referenced Plato, Archimedes, Euclid, and Cusa (and, probably also the De Musica of St. Augustine).44 This, in turn, set the stage for the most crucial of the discoveries by Kepler. Leonardo da Vinci, on the same basis, developed the crucial discovery of the transverse wave-function for electromagnetic propagation, and the finite speed of light later first measured (approximately) early during the seventeenth century. The work of Leonardo da Vinci, Kepler, Fermat, Desargues, and Pascal, informed the discoveries of Huygens, Leibniz, and the Bernoullis on least-action principles and the related features of non-algebraic functions.
If we examine the arrangement by the rule, that the passage from one or more crucial discoveries to a successor crucial discovery, must always occur in the paradoxical manner we generate, initially, an individual thought-object, the result of constructing this choice of order by that method, is to generate a higher-order thought-object, of that quality which subsumes the successive generation of the constituent thought-objects in that selected array. This higher quality of thought-object is therefore of a distinct Cantorian Type: that array of thought-objects, considered as they might have been generated in that selection and assigned order, generated by a constant principle of difference, forms a manifold, or sub-manifold of this description. The higher quality of thought-object generated by that ordered array is the Type of that manifold or sub-manifold.
From that standpoint, all communication relating to significant ideas is necessarily, intrinsically metaphorical.
The progressive ordering of a succession of thought-objects in this way, according to such a higher quality of thought-object Type, is the phenomenon which corresponds to what we ought to signify by the term negentropy. For classroom purposes, we signify the case in which a series of successively higher states of organization is generated according to a principle which intrinsically orders such a succession.
For example, let us imagine that at a certain point the higher state of matter in our universe is either a population of neutrons or hydrogen atoms. By combining these, through fusion, a periodic table of elements and their isotopes is generated. Where lies the negentropy in this image of fusion development? Is it that lithium might be a "higher state of organization" than hydrogen? Or, is it not that the universe is now populated by neutrons, hydrogen atoms, and also helium and lithium atoms? The point being illustrated, is that higher states of organization of the process as a whole are being generated successively, in accord with a higher, subsuming principle of a periodic table in general: in that latter aspect of the phenomenon lies the true negentropy.
1. Cervantes' Don Quixote is a Platonic form of classical tragedy given in prose form. See Miguel de Cervantes, The Ingenius Gentleman Don Quixote de la Mancha, trans. by Samuel Putnam (New York: Random House, 1949).
2. William Empson, Seven Types of Ambiguity (Middlesex: Penguin Books, 1961).
3. Cf. Bernhard Riemann, "Zur Psychologie und Metaphysik," on Herbart's Göttingen lectures, for Riemann's reference to Geistesmassen, in Mathematische Werke, 2nd. ed. (1892), posthumous papers, ed. by H. Weber in collaboration with R. Dedekind.
4. Archimedes, "Measurement of a Circle" and "Quadrature of the Parabola," in The Works of Archimedes, ed. by T.L. Heath (New York: Dover Publications). Cardinal Nicolaus of Cusa, De Docta Ignorantia (On Learned Ignorance), Book I, trans. by Jasper Hopkins as Nicholas of Cusa on Learned Ignorance (Minneapolis: Arthur M. Banning Press, 1985); also, "De Seculii Quadratura" ("On the Quadrature of the Circle"), trans. into German by Jay Hoffman (Mainz: Felix Meiner Verlag).
5. See G.W. Leibniz, "History and Origin of the Differential Calculus," in The Early Mathematical Manuscripts of Leibniz, trans. by J.M. Child (LaSalle: Open Court Publishing Co, 1920), pp. 22-58.
6. It is always useful in mathematics, to reflect upon the physical implications of one's calculations. At n = 112, two adjacent polygonal angles on the circumference of a circle of 1 centimeter radius are 1.21009 × 10 -33centimeters. For a n = 256 polygon, 5.42626 × 10 -77centimeters. Thus, to increase the distance between two adjacent angles of an n = 256 polygon to a significant 10 -33centimeters, would require a circle 2.23006 × 1043larger than our circle of a 1 centimeter radius. The radius of this larger circle would be 2.236006 × 1038 kilometers, or 2.35717 × 1025light years. Compare this kind of calculation with Archimedes' famous sand-reckoner ("The Sand-Reckoner," in The Works of Archimedes, op. cit.). How old is a universe whose radius is 2.35717 × 1025light years?
7. Plato, Parmenides, Loeb Classical Library (Cambridge: Harvard University Press); Lyndon H. LaRouche, Jr., "Project A," in The Science of Christian Economy and Other Prison Writings (Washington, D.C.: Schiller Institute, 1991).
8. Plato, Timaeus, trans. by R.G. Bury, Loeb Classical Library (Cambridge: Harvard University Press, 1975); see also Plato's Timaeus: The Only Authentic English Transtlation, trans. by associates of Lyndon H. LaRouche, Jr. The Campaigner, Vol. 13, No. 1, Feb. 1979.
9. Luca Pacioli, De Divina Proportione, for which Leonardo da Vinci did the drawings.
10. Johannes Kepler, Harmonice Mundi, (On the Harmony of the Worlds), in Opera Omnia, vol. 5, Frankfurt (1864), of which only Book V has been translated into English, in the Great Books of the Western World series (Chicago: Encyclopaedia Britannica, Inc.); On the Six-Cornered Snowflake, trans. and ed. by Colin Hardie (Oxford: Clarendon Press, 1966), reprinted by 21st Century Science & Technology, 1991. See LaRouche, "In Defense of Common Sense," chap. VIII, and "The Science of Christian Economy," Appendices I-III, V, VI, in Christian Economy, op. cit.; also A Concrete Approach to U.S. Science Policy, chap. II (Washington, D.C.: Schiller Institute, 1992).
11. Pierre de Fermat, Oeuvres Fermat, ed. (1891), epistl. xlii, xliii. Christiaan Huygens, The Pendulum Clock, or Geometrical Demonstrations Concerning the Motion of Pendula as Applied to Clocks, trans. by Richard J. Blackwell (Ames: Iowa State University Press, 1986). Gottfried Wilhelm Leibniz, "Specimen Dynamicum" (1695), in Leibniz Selections, ed. by Philip P. Wiener (New York: C.S. Sons, 1951); also, Système Nouveau de la Nature, 1695. Johann Bernoulli, Acta Eruditorum, May, 1697; trans. in D.E. Smith, A Source Book in Mathematics (New York: Dover, 1959), pp. 648-55. Jakob Bernoulli, Acta Eruditorum, May, 1697; trans. in D.J. Struik, ed., A Source Book in Mathematics, 1200-1800 (Princeton, N.J.: Princeton University Press, 1986), pp. 292-299.
12. Johannes Kepler, Snowflake, op. cit.
13. See LaRouche, U.S. Science Policy, chaps. II and III op. cit.
16. If the argument here appears to parallel that of Georg Cantor on aleph types, and Kurt Gödel's famous proof, the resemblance is intentional and well-grounded. See Georg Cantor, Beiträge zur Begründung der transfiniten Mengenlehre (Contributions to the Founding of the Theory of Transfinite Numbers), trans. by Philip E.B. Jourdain (New York: Dover Publications, 1955). See also Ernest Nagel and James R. Newman, Gödel's Proof (New York: New York University Press, 1958).
17. See LaRouche, "In Defense of Common Sense," chaps. 2-4, in Christian Economy, op. cit.
19.Cf. Nicolaus of Cusa, De Docta Ignorantia, op. cit., Book I, chap. XIII, which has the use of a very large circle to approximate a straight line.
20. Gilles de Roberval, "The Cycloid," from Treatise on Indivisibles, trans. Evelyn Walker (New York: Teachers College, 1932); also, quoted in D.J. Struik, op cit.
Huygens, The Pendulum Clock, op. cit.
23. Huygens, The Pendulum Clock, op. cit. Johann and Jakob Bernoulli, op. cit. Gaspard Monge, Application de l'algèbre a la géométrie, 5th ed., ed. by J. Liouville (Paris: Bachelier, 1850).
24. Christiaan Huygens, Treatise on Light (1690), trans. Sylvanus P. Thompson (New York: Dover Publications, 1962). Gaspard Monge, op. cit.
25. G.W. Leibniz, "On Analysis Situs," in Gottfried Wilhelm Leibniz Philosophical Papers and Letters, trans. and ed. by Leroy E. Loemker, (Chicago: University of Chicago Press, 1956), Vol. I, pp. 390-396. Bernhard Riemann, "Lehrsätze aus der Analysis Situs ...," in Mathematische Werke, op. cit., quoted as "On Analysis Situs," in D.E. Smith, op. cit.
26.Johannes Kepler, Snowflake, op. cit. See also LaRouche, U.S. Science Policy, chap. IV, op. cit.
27. Eugenio Beltrami's devastating refutation of the entire theory of elasticity upon which the Maxwll electromagnetic theory is based can be found in "Sulle equazioni generali dell' elasticita" ("On the General Equations of Elasticity"), Annali di Matemàtica pura ed applicata, serie II, tomo X (1880-82), pp. 188-211; trans. by Rick Sanders, 21st Century Science & Tecnlogy, unpublished.
28. Georg Cantor, Theory of Transfinite Numbers, op. cit.; see also, Georg Cantors Gesammelte Abhandlung, ed. by Ernst Zermelo (Hildschein, 1962). Also, Gödel's Proof, op. cit.
29. Like the philosophically allied project, the "Frankfurt School" of Adorno, Horkheimer, Marcuse, Heidegger, Arendt, et al. (see Michael J. Minnicino, "The New Dark Age: The Frankfurt School and 'Political Correctness,' " Fidelio, Vol. 1, No. 1, Winter 1992), modern linguistics was also launched by the 1920's Communist International. The key Communist official was Stalin's collaborator in this project, Germany's Karl Korsch. During the 1930's, Korsch collaborated on this project with Rudolf Carnap, both in turn collaborating with Bertrand Russell and the Russell-Hutchins "Unification of the Sciences" project, in the initial, pre-war sessions held at the University of Pennsylvania. The University of Pennsylvania's Professor Harris adopted this linguistics as his profession, followed by his student, today's Professor Noam Chomsky.
30. See Plato, Timaeus, op. cit.; Euclid, The Thirteen Books of Euclid's Elements, Books 10-13, trans. T.L. Heath (New York: Dover Publications, 1956); Luca Pacioli, De Divina Proportione; Johannes Kepler, Harmonice Mundi, op. cit., chap. II; and also, Leonhard Euler, "Elementa doctrinae solidorum," St. Petersburg Academy of Science (1751).
31. We pass over, for the moment, the additional stellated solids defined by, first, Johannes Kepler, Harmonice Mundi, op. cit., chap. II; and Louis Poinsot, Memoirs sur Les Polygons et Les Polyhedras (Notes on Polygons and Polyhedra), trans. by Laurence Hecht, 21st Century Science & Technology, unpublished.
32. Leonardo of Pisa, Liber Abaci (The Book of the Abacus), as quoted in D.J. Struik, op. cit.
33. See footnote 11 for the relevant works of Huygens, Leibniz, and the Bernoulli brothers.
34. Huygens, Treatise on Light, op. cit. Cf. Johann Bernoulli, "Curvatura Radii," in Diaphanous Nonformabus Acta Eruditorum, May, 1697, as quoted by D.J. Struik, op. cit., pp. 391-399.
35. See LaRouche, U.S. Science Policy, chap. III, op. cit.
36. Ibid. Leibniz's commentary on this view of Newton is in his first letter to Clarke, from 1715: "Sir Isaac Newton and his followers have also a very odd opinion concerning of the work of God. According to their doctrine, God Almighty wants to wind up his watch from time to time; otherwise, it would cease to move." In Clarke's reply, he acknowledges that God "not only composes or puts things together but is himself the author and continual preserver of their original forces or moving powers." Reprinted in Leibniz Philosophical Papers, Vol. II, pp. 1095-1169, op. cit.
37. In 1850, Rudolf Clausius wrote his first article discussing the theory of heat. Clausius' book was without experimental proof, and also without any reference to a "universal law." In 1852, Wiliam Thomson (later Lord Kelvin), wrote an article entitled, "On a universal tendency in nature to the dissipation of mechanical energy." This article consisted of ideological speculations on the experimental work on heat-powered machines of the French scientist Sadi Carnot, in which Thomson had not participated. In that article, Thomson postulated that the universe, since it was nothing but a machine, would one day run down. In 1854, Thomson's friend Helmholtz used the same thesis in his On the Transformation of Natural Forces. Finally, Clausius, in the second (1865) edition of his book, after a meeting with Thomson, concluded the book with the famous two axioms: (1) the energy of the universe is constant; and (2) the entropy of the universe tends toward a maximum. See LaRouche, U.S. Science Policy, chap. III, op. cit.
38. See LaRouche, U.S. Science Policy, chap. III, op. cit.
39. A postulate is distinguished as an unproven assumption, in imitation of a true axiom, introduced to eliminate arbitrarily an otherwise ambiguous or incomplete feature of the theorem-lattice manifold generated by a pre-existing set of axioms and postulates.
40. The great Hilbert is reported to have thrown student Norbert Wiener out of a scientific seminar at Göttingen for persistent methodological incompetence.
41. Cf. Winston Bostick, "The Plasmoid Construction of the Superstring," 21st Century Science & Technology, Vol. 3, No. 4, Winter 1990; also, "How Superstrings Form the Basis of Nuclear Matter," 21st Century Science & Technology, Vol. 3, No. 1, Jan-Feb 1990.
42. Hence, as Philo of Alexandria denounced the gnostic followers of Aristotle among Jewish rabbis of his time, neither Creation nor God could actually exist in a nominalist form of deductive system such as that strictly attributable to Aristotle. See Philo, "On the Account of the World's Creation Given by Moses" ("On the Creation"), in Philo, Vol. I, trans. F.H. Colson and G.H. Whitaker, Loeb Classical Library, (Cambridge: Harvard University Press, 1981).
43. Works in which Cusa emphasized the isoperimetric principle include De Docta Ignorantia, Book I, op. cit., and "On the Quadrature of the Circle," op. cit.
44. St. Augustine, De Musica (On Music), trans. by R. Catesby Taliaferro (Annapolis: St. John's College Bookstore, 1939).
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