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Dialogue of Cultures


Leibniz From Riemann’s Standpoint

Lyndon H. LaRouche, Jr.

July 14, 1996
Part I

This article is reprinted from the Fall, 1996 issue of FIDELIO Magazine. Footnotes for this section are at the bottom of this page. Footnotes for Section 2 is at the bottom of Part II

Part I

Part I Footnotes

Part II

Part II Footnotes

Related Articles

Vol 5, No 3

Leibniz From Riemanns' Standpoint1
by Lyndon H. LaRouche, Jr.
July 14, 1996

One who has not merely learned, but knows relevant features of the work of Johannes Kepler, Gottfried Leibniz, Carl Gauss, and Bernhard Riemann, must be appalled by the unbridgeable gulf between the actual work of those exemplary, leading figures of modern European science, and what most of today's relevant academic specialists misrepresent crucial elements of that work to have been. Such has been the present writer's cumulative experience, over those sixty-odd years, since he began systematic studies of the putatively leading European philosophers from the Seventeenth and Eighteenth centuries.

Bernard Riemann
Gottried Leibniz

During most of those decades, the writer has wrestled with relevant, published scholarly andother misrepresentations, in his verbal and oral exchanges with relevant professors and students of philosophy, with ordinary laymen, and with practitioners of mathematical science. With rare exceptions, whenever any among these crucial issues of principle is addressed, nearly all among the professional opinions encountered, are not merely mistaken, but are uttered with shameless unconcern for truthfulness.

If one applies the method of Socratic dialogue, seeking to smoke out the underlying, axiomatic roots of these differences, two causes for the widespread academic, and popular misrepresentation of Kepler, Leibniz, and Riemann, are brought to the surface. First, that the standpoint of most of those commentators, is that of Aristotle, or the empiricists. Second, when the core of the difference is chased back to its relevant epistemological rabbit-hole, any reference to the fact, that the issue is rooted in opposition to the principles underlying the scientific method of Kepler, Leibniz, and Riemann, evokes their modern opponents' implicitly hysterical effort to deny the fact, that their own, contrary, judgments are derived from such differences in axiomatic assumptions.

Typically, the hysteria expressed on the second count, is of the same form as Isaac Newton's absurd literary outburst: ... et hypotheses non fingo!. The Newtonian system rests upon a very precisely defined hypothesis, which Newton denies to exist.2 On the subject of Kepler, Leibniz, or Riemann,3 the argument of most putative scholarly authorities, is analogous to Newton's denial of the existence of his own hypothesis. Rather than acknowledging the difference between their own and their subject's axiomatic assumptions, Newton et al. have insisted, that they themselves have no such assumptions to be contested. That hysterical behavior by Newton, et al., might remind us, of the startled, wild-eyed boy (probably the local schoolyard bully) caught by his mother at the moment he has his hand in the cookie-jar, with inculpatory crumbs all around his mouth, shrieking at his mother: "What cookie-jar!"

As we shall show in the course of this paper, those writers against which we complain thus, have not relived the Socratic experience of the fundamental discoveries achieved by any among these three crucial figures of modern science. We shall show, that, for that reason, however much they might claim to have learned, they have no direct mental experience of the relevant acts of discovery of principle involved. Thus, however much they have merely learned, they know relatively nothing of crucial importance about those types of subject-matters of science, in which the principal variables to be considered, are differences in underlying (e.g., axiomatic) assumptions.

Thus, one might recognize, as in the manner indicated above, that the seemingly characteristic trait among today's roster of putatively authoritative commentaries, is that each and all are governed much less by a passion for truth, than by blind zeal. We observe that that zeal is commonly mustered in defense of some philosophical standpoint contrary to that of any and all among of such targets of their muddled commentaries, as those four whom we have listed at the outset of this paper. In general, it may be said, that most such commentators are fairly classed, either as Aristoteleans, or philosophical empiricists. All seek to deny, that any influential principle of mathematics or physics (for example) might have been achieved by a scientific method contrary to their own.4 Above all, they reject that fundamental principle of Socratic method, Plato's method of hypothesis, by means of which all of the crucial discoveries of Kepler, Leibniz, and Riemann (for example) were generated.

For that, and related reasons, no competent representation of the central conceptions underlying Leibniz's work can be presented in the terms of scholarship which have, unfortunately, become conventional in qualifying doctoral candidates, or, more generally, in the production of related, putatively "scholarly" theses. In the case, such as this topic, in which most among the putative authorities are distinguished almost as much by their incompetence (or intellectual dishonesty), as their scholarship, one must emulate that most estimable Franciscan, Fran[c]ois Rabelais, to reject, as ridiculous, the suggestion, that consensus among a representative body of putative scholarly authorities, such as our modern Suckfists and Kissbreeches of science, might be the relevant approach to the issues at hand. One must reconstruct the relevant principles, as if from the ground up. To this end, as we have said above, one must follow the map of Plato's method of negation of axiomatically misguided, but official, or other generally held opinion; we must employ the Socratic method of hypothesis.

Today, the most efficient standpoint from which to present, to a modern, literate audience, the axiomatic basis for Leibniz's scientific work, is the case of the fundamental discovery, respecting the principle of hypothesis, which Bernhard Riemann applied to mathematical physics, in his 1854 habilitation dissertation.5 This present writer's discoveries within the domain of Leibniz's science of physical economy, provides the best vantage-point from which to demonstrate this specific connection of Leibniz to Riemann. We summarize that approach to the conceptions; we, thus, avoid the wide, textbook-paved road to Hell, and follow the Classical humanist method, instead. The latter, is the method of re-experiencing, at least in outline of the crucial points, the mental processes of one or more among the relevant original discoverers. The relevant case here, is the present writer's re-enactment of Riemann's discovery, but from a fresh standpoint. This serves, in turn, as our vantage-point for pointing out some characteristic features of Leibniz's method.

Three points are considered below. First, what the present writer came to recognize as the deeper significance of Riemann's habilitation dissertation. Second, how the writer's own discovery in physical economy imparts to Riemann's discovery, an otherwise overlooked authority. Finally, how we are forced, by considering Riemann's and the writer's own discoveries, to adopt a deeper appreciation of some among the more celebrated writings of Leibniz.

The Principle of
'Universal Characteristics'

During the interval from his own fourteenth through eighteenth birthdays, this writer became a follower of Gottfried Wilhelm Leibniz. His acquaintance with Leibniz came through English editions of some of Leibniz's noted books, obtained, chiefly, either from the family household's library, or the Lynn, Massachusetts Public Library. This came as part of a project begun the summer preceding the writer's thirteenth birthday, and continued through his eighteenth year: a comparative study of the relatively most popular titles from leading English, French, and German philosophers of the Seventeenth and Eighteenth centuries, taking each in chronological order.

The writer began with writings of Francis Bacon, turned next to Thomas Hobbes, René Descartes, John Locke, Leibniz, Hume, Berkeley, Rousseau, taking up English translations of Immanuel Kant's Critique of Pure Reason and Prolegomena to Any Future Metaphyics about two and a half years later. The Leibniz writings featured in this series (and read, over and over again), were the Monadology, Theodicee, and Clarke-Leibniz Correspondence.6 At that time, the writer then found the empiricists trivial in content, relative to Leibniz, although foes of some importance respecting their obvious influence on the world as viewed from 1930's Massachusetts. It was the defense of Leibniz against the central argument of Kant's Critique of Pure Reason, which proved itself a more worthy and profitable challenge, back then. Although this writer did not turn to a systematic study of Plato's writings until the mid-1950's, he had already been steeped in Plato's method of hypothesis, through studying and defending certain among the leading published writings of Leibniz.

Obviously, as for any person, many childhood and youthful experiences converged to shape the present writer's character. However, in retrospect, the importance of working through a pro-Leibniz counter-attack upon Kant, was, without doubt, the most crucial of these formative experiences. This influence was hewn into a practical form by his most significant post-war experience, the encounters with, first, Norbert Wiener's Cybernetics,7 and, also, those notions of "operations research" and "systems analysis" converging upon the work of Bertrand Russell's devotee, John Von Neumann. The earlier wrestling against Kant, provided the standpoint from which to identify the kernel of evil implicit in Wiener's statistical definition of "information theory."

As reported in various locations, by the beginning of the 1950's, the writer's original discoveries, effected in the course of refuting "information theory," impelled him to undertake a careful rereading of Riemann's habilitation dissertation. The crucial importance of that rereading, lay in Riemann's addressing the subject of the determining function of Plato's method of hypothesis, in defining any competent form of mathematical physics.8 Once we have considered the implications of Riemann's work, we are able to see his most famous predecessors within modern science in a fresh way: Gauss, Leibniz, and Leibniz's crucial predecessors, Kepler, Leonardo da Vinci, and da Vinci's crucial predecessor, Nicolaus of Cusa. Consider the relevant, central implications of Riemann's habilitation dissertation, and then the significance of Riemann's discovery, when it, in turn, is situated within the context provided by this writer's own original discoveries in physical economy.

Briefly, the significance of Riemann's discovery, is this. Consider the form of algebra introduced to the Seventeenth century by the founder of the "Enlightenment," the atheistic Servite monk, and follower of William of Ockham, Paolo Sarpi. Consider the expression of this in the work of such Sarpi lackeys and followers as Galileo Galilei, Thomas Hobbes, and René Descartes. The proximate source of the Enlightenment forms of algebra, employed by René Descartes, Isaac Newton, and their devotees, is derived from an "Ockhamite" reading of what is most widely recognizable as that modern classroom parody of Euclid's geometry embedded in the mathematics curricula generally, as presented, still, in secondary and higher education during the time of this writer's youth, and earlier.

The fallacies of this algebra, are the starting point of Riemann's dissertation. His point of departure there, is that in the form of algebra derived hereditarily from the work of Galileo, Descartes, Newton, et al.: Discrete events, and their associated movements, are situated within a Cartesian form of idealized space-time. This point has been presented by the present author in numerous earlier locations, but, on pedagogical grounds, it must be stated again here, this time in a choice of setting appropriate to the connection we are exposing, between the ideas of Riemann and his predecessor Leibniz.

Riemann opens his dissertation, with two prefatory observations. First, that, until that time (1854), "from Euclid through Legendre," it was generally presumed that geometry, as well as the principles for constructions in space, was premised upon a priori axiomatic assumptions, whose origins, mutual relations, and justification remained obscure. The second general point of his plan of investigation, which he restates in the conclusion of the dissertation, is that no rational construction of the principles of geometry could be derived from purely mathematical considerations, but only from experience.9 He concludes his dissertation: "We enter the realm of another science, the domain of physics, which the subject of today's occasion [mathematics] does not permit us to enter." Riemann, thus, refutes the presumption on which a Newton devotee, of Prussia's Frederick II, Leonhard Euler, depended absolutely, for the entirety of his attack on Leibniz's Monadology.10

On grounds of the principles of Classical humanist, or cognitive pedagogy,11 the prudent course of action, now, is to reconstruct the conceptions at issue from the initial standpoint of simple, deductive theorem-lattices. This pedagogical approach leads us by the most direct route, to the central issue of Riemann's discovery: the validation of an axiomatic-revolutionary quality of discovery of universal principle, by reason of which we are obliged to construct a new mathematical physics, to supersede that erroneous one previously in vogue. Later, continuing that process of construction, to the point of examining the writer's own original discovery in physical-economy, we identify the cognizable feature of the individual person's mental life, in which we may then locate the significance of Riemann's revolution in mathematical physics.

Riemann's Principle of Hypothesis

The pedagogical reference-point throughout this paper, is the contrast between that Platonic principle of change,12 on which both Riemann's and the writer's own discoveries were premised, and the sterile formalism of the Aristotelean or quasi-Aristotelean models of an ordinary, deductive form of theorem-lattice. In all cases considered here, the notion of theorem-lattice is defined, and examined from the standpoint of Plato's Socratic method, by the so-called method of hypothesis.

A simple, deductive form of theorem-lattice, is defined by a process of successive approximations, as follows. Given, any set of theorems which are assumed to be not-inconsistent with one another. This presumes that the Socratic method of Plato would be able to adduce certain minimal, but sufficient, underlying assumptions, the which these theorems share in common. If so, these assumptions then constitute a set of interdependent terms, in the form of axioms, postulates, and definitions, none of which are deductively inconsistent with any among the previously given, mutually not-inconsistent theorems. Implicitly, therefore, there might exist an indefinite number of other theorems, none of which is inconsistent, deductively, with the same set of axioms, postulates, and definitions. The combined set of all such theorems, both known and possible, constitutes a simple theorem-lattice.

For the purpose of defining essential terms: The set of underlying, interdependent axioms, postulates, and definitions, underlying any such theorem-lattice, is the elementary, deductive form of an hypothesis. That is the definition of "hypothesis" employed by Plato, Leibniz, Riemann, and the present author.

If, then, there exists some stubbornly real condition or event, which were not consistent with that hypothesis, then there is no proposition based upon that condition or event, the which could be the basis for a theorem of any theorem-lattice corresponding to that hypothesis. However, if, nonetheless, all of the theorems of the first theorem-lattice correspond to actually existing conditions or events, then, there exists a new hypothesis, which defines a new theorem-lattice, for which a proposition corresponding to the newly discovered condition or event, is a valid theorem. However, no theorem of the new theorem-lattice is consistent with any theorem of the first theorem-lattice.

The discovery of the change in hypothesis, which enables the leap from the old, failed theorem-lattice, to the new, is, thus, conveniently described as the discovery of a valid, axiomatic-revolutionary principle.

There is a crucial, corollary point to be taken into account, in reading, and rereading the highly significant, immediately preceding paragraphs. The proposition which we might construct, as our conscious representation of a condition, or event, is not the condition, or event, which may, in our opinion, have prompted the relevant proposition. This is a scientific matter, but one which is also brought to our attention by some relatively common, non-scientific, experiences of the layman's daily life.

For example. On this account, we must become uneasy in our seats, when some typical, philosophically illiterate person insists, that he, or she, is, in the words of Hollywood's "Sergeant Friday," insisting upon "Just the facts, Ma'am." For example, what the attorneys and judges, in a legal proceeding, insist are "facts," are not reality per se, but merely a special kind of subjective assessment, which might, or might not, have relevant correspondence to the reality to which the proceeding is putatively addressed.

To this point: Even if we might be persuaded, that we have overcome the hurdles of sincerity, in assessing a witness's report, the fact that the witness might be presumed to be speaking sincerely, and in his or her best judgment, does not rise to the standard for presuming, that the witness is also speaking competently of what that witness imagines himself, or herself to have experienced. Usually, the most favorable assumption which might be suggested, in the case of virtually any witness, is that the significance of a truthful effort to state a fact, or facts of a matter, is, that it represents the present limits of the subject's competence to interpret what the subject believes to have been the experience of his, or her senses.

"Truthful," when employed, carelessly, as a synonym for "sincerity," does not mean "real." What may qualify as a "fact," or "evidence," by extant legal or other professionals' standards, does not necessarily signify "true," "truthful," or "real," even if the relevant utterance is the most sincere which the subject might utter on the matter of the event being considered.13

In the language of simple theorem-lattices: In the case, that some evidence forces us to abandon one hypothesis, for another, only the valid evidence prompting the theorems of the first theorem-lattice, but not the theorems themselves, are carried forward as evidence addressed by theorems of the second lattice. Virtually none of the theorems of the old lattice are incorporated in the new; virtually all of the theorems which, in the first lattice, were associated with the carried-forward experimental evidence, are abandoned by the second lattice, as inconsistent with truth.

Truthfulness, in science, or in ordinary testimony, lies not in what the witness believes he, or she has seen, heard, touched, felt, tasted, or smelled; truthfulness lies in the choice of hypothesis, which underlies those subjective things, called propositions, which the witness has constructed as much, or more, from his, or her prejudices, as from the relevant experience. This is to be said in the same sense, as to argue, that where a member of an illiterate culture recognizes no more than "rock," a representative of a literate culture recognizes "ore." Or, to say, that the representative of the illiterate culture sees the stars moving about us; whereas, the representative of the literate culture, such as that of Plato's Academy of Athens, sees the moon orbitting the Earth, and the Earth rotating, while orbitting the sun.14

Riemann makes clear, in his referenced dissertation, that his emphasis upon experience, does not signify the popular delusion of the illiterate persons: The delusion that what we know as factual, is what we believe that we have experienced through our senses. Rather, the point of his argument there, is that the truthfulness of our opinions respecting actual experiences, depends, absolutely, upon the validity of the axiomatic assumptions which govern the way in which we form propositions and theorems in response to promptings of experience. It is on this point that Riemann focuses his devastating refutation of both Aristoteleanism and empiricism.

Riemann's exposure of the fraud embedded in the taught geometry and physics of both the Aristoteleans and empiricists, renders transparent the issues listed above.

The simple space-time employed by Galileo, Descartes, Hobbes, Hooke, Newton, et al., was based on certain, a priori, axiomatic assumptions respecting extension in four, mutually independent senses of direction, three of extension in space, and one in time: a "quadruply-extended space-time manifold." It was assumed, a priori, that space is extended without limit, and in perfectly uninterrupted continuity: backward-forward, up-down, side-to-side. It was assumed, a priori, that time is extended, similarly, backward and forward. It was assumed, a priori, that place, size, and movements of events can be situated mathematically, as though these were something plopped into what were otherwise an empty, continuous, space-time void.15

To these arbitrary, a priori assumptions, other assumptions of a physical nature were similarly attached. Those persons who might be classed as "materialists," presumed, not only that these assumptions about space-time were products of the senses, but that the relevant features of sense-perceptions were mirror-images of the real world external to our senses. Others, such as the empiricist followers of Sarpi, Galileo, Hobbes, et al., did not presume that sense-perceptions were necessarily mirror-images of the world outside our skins; however, from the standpoint of the pervasive fallacy intrinsic to popular misconceptions of physical space-time, still today, Riemann's dissertation applies equally to all among the Aristoteleans, materialists, and empiricists.

Riemann's argument against that view of physical space-time, is predominantly twofold. First, that the referenced assumptions of Galileo, Descartes, Newton, et al., were merely arbitrary assumptions. Second, that these assumptions were demonstrably false. The proof of these two arguments lay in the principle set forth by the founder of modern science, Nicolaus of Cusa, in his De Docta Ignorantia: the principle of measurement.

Given the topic under which this paper is subsumed, which is the retrospective view of Leibniz from the standpoint of Riemann's discoveries: The most convenient illustration of the way the principle of measurement applies, is the instance of the use which Jean Bernoulli and Leibniz made of the intersecting subjects of isochronicity (a phenomenon of gravitation) and the brachystochrone problem (refraction of light at a measurable, "constant speed"). Both of these were treated by Bernoulli and Leibniz, as arising out of the work of Christiaan Huyghens.16 In this connection, lay the physical basis for Leibniz's insistence upon replacing the "algebraic" methods of Galileo, Descartes, and Newton, by a "non-algebraic" (transcendental) form of mathematical physics.17

Riemann's dissertation introduces explicitly, a conception already implicit in the work of Leibniz and others, earlier: he establishes there the replacement of Newtonian physics in space-time, by the notion of physical space-time.18 He excludes the recklessly gratuitous, a priori assumptions of limitless extension, and perfectly continuous extension. He then attributes the principle of extension to every physical principle whose validity has been demonstrated by experimental measurement, as Ole R[o]mer, in 1676, had reported his astrophysical measurement of the estimated "speed of light," and as Jean Bernoulli, twenty years later, reported the coincidence of refraction of that light and Huyghens' representation of isochronicity within the gravitational field. Thus, every validated physical principle is to be added to dimensions of space and time, as an independent dimension of a physical space-time manifold of "n dimensions." This arrangement excludes, axiomatically, any toleration of the Euler-Cauchy-Clausius-Helmholtz, et al. notion of "linearization of physical space-time in the very small."

At the outset of his dissertation, Riemann already defends what is to appear as his construction of a multiply extended physical space-time manifold. This defense rests chiefly on two general premises. First, each discovered principle validated by experimental measurement, has, consequently, the manifest quality of extension. Second, each such principle has the quality of a dimension, in the respect of the same rule of mutual independence among dimensions, which any Euclidean form of geometry attributes to mutually independent senses of direction of dimensions of space and time.

Yet, this construction poses problems which can not be resolved within either the confines of a formal mathematics, or any extant formal mathematical physics. To resolve these further problems, one must depart the domain of mathematics, to enter the domain of experimental physics. One must enter Nicolaus of Cusa's domain of measurement.

There must be some experimental proof, which demonstrates, in a measurable way, that a certain crucial-experimental occurrence requires us to construct one kind of mathematical physics, rather than some other. This demonstration must have such unique significance. Riemann points to three hints, on which he has relied for elaborating the general quality of "yardstick" we require for that kind of measurement. Two hints are taken from the work of Riemann's patron, Professor Carl F. Gauss: Gauss's work on bi-quadratic residues,19 and general theory of curved surfaces.20 The third is borrowed from Riemann's own work, the concept of Geistesmassen which he outlined in his posthumously published Zur Psychologie und Metaphysik.21

To be considered validated, the new physical principle must correspond to some measurable difference in the characteristic action "connecting any two points" within the reality corresponding to the choice of mathematical-physics manifold being tested. The notion of this measurable difference, is suggested by the attempt to determine whether the very large surface on which one is travelling is a plane, or a curved surface.22 In terms of a physical space-time manifold of "n dimensions," it is the relative curvature of the "surface," which the crucial experiment must measure. Hence, the importance, for Riemann, of the hints supplied by Gauss's work on biquadratic residues and general theory of curved surfaces.

For Riemann's physics, one such yardstick is required. The present writer's discoveries demonstrate that two yardsticks, rather than one, are required. We shall come to that in due course, below. First, we must locate the place where Riemann's notion of Geistesmassen fits in; this touches the most crucial distinction of Riemann's physics, and also the unique feature from which the unique, crucial superiority of the present writer's work in economics has been derived. To that purpose, we now restate what we have just described, this time, explicitly referencing, as Riemann does, Plato's—and Leibniz's—method of hypothesis.

In place of the words "dimension," substitute such words as "axiom, postulate, definition." That is to say, recognize the equivalence of a Riemann multiply-extended, physical space-time manifold, to Plato's, Leibniz's, Riemann's, and the present author's notion of "hypothesis." The connection is highlighted by reference to Leibniz's notion of necessary and sufficient reason, a notion which is Leibniz's refined treatment of the notion of reason as this appeared in the work of that Johannes Kepler, whose specified requirements for the development of a calculus were satisfied by Leibniz's work.

Proceed to that end, thus. As we proceed, now, bear in mind the following: Think of "dimension, axiom, postulate, definition," and "hypothesis," as representative of a common quality termed, alternately, either "formal discontinuity," or "singularity." Physically, each, as in the case of adding a new degree of independent dimension, signifies some break in the continuum extant prior to the introduction of such a singularity.

Consider the proposition: What is a sufficiency of properly selected, axiomatic assumptions, respecting the task of assessing the significance of a particular event, when that event is considered primarily as a change in the state of the universe in which it occurs? Select, as such an event, the equivalence which Jean Bernoulli demonstrated, between Huyghens' notion of the cycloid path as one of isochronicity (tautochrone) in Kepler's "gravitational field,"23 and the fact that the variable feature of refraction describes the same tautochronic pathway.24 What are the necessary and sufficient features of an hypothesis, which hypothesis defines a physical space-time in which these phenomena and their coincidence must occur? That hypothesis, whatever it may prove to be, constitutes "necessary and sufficient reason."

That reflects Leibniz's refinement of Kepler's use of the notion of Reason. This function of Reason(Kepler), or necessary and sufficient reason(Leibniz), is the alternative to the use of the percussive notion of "causality," as a geometrically degenerate parody of the notion of Reason, in the work of materialists, or empiricists such as Galileo, Newton, et al.

This leads to Riemann's notion of unique events, as those experimental events which force us to reconsider whatever has passed, until now, for a notion of necessary and sufficient reason, that hypothesis heretofore considered as established. The general use of "crucial experiment," as ostensibly a substitute for "unique," does not rise to the functional significance of our use of "unique" here.

Implicitly, every event is, potentially, a unique experimental event. In some circumstance, any event must implicitly overthrow the presumptions of someone's hypothesis. Obviously, we, like Riemann, Leibniz before him, and so on, are situating these and related matters within an historically specific, task-oriented setting, the interdependency between mankind's progressive mastery of the universe, and the internal development of Classical forms of art and science. Therefore, we employ "unique" to designate those events which have pivotal, historic significance for the discovery of valid, axiomatic-revolutionary principles of our universe. E.g., the critical experimental, or analogous events, which correspond to the singularities of a never-perfectly continuous extension of scientific and artistic progress.

In Riemann, this overview of scientific progress is typified by progress from a relatively valid physical space-time of "n dimensions," to a more powerful conception, a superior, relatively valid physical space-time of "n+1 dimensions." In other words, from one, relatively valid hypothesis, to a superior valid hypothesis.

This central implication of the habilitation dissertation, leads us, implicitly, to reconsider the so-called "ontological paradox" of Plato's Parmenides.25 Resituate the notion of a Riemann series (e.g., of surfaces of differing Gaussian curvature), of the topological type (n+1)/n, as implicitly defined by the habilitation dissertation. This presents us a series of hypothesis, n = 4, ... , i, i+1, i+2, ... . What is the ordering principle of such a series? The answer is, first: some principle of valid successive discovery of hypotheses: a higher type of hypothesis, which underlies a series of hypotheses, as an ordinary, relatively valid hypothesis underlies the series of theorems represented by a theorem-lattice. Plato identifies this higher type of hypothesis, simply, as an "higher hypothesis." Hence, the title of Riemann's Platonist dissertation: "The Hypotheses Which Underlie Geometry."

As we depart one hypothesis of that series, to approach its proper supersessor, we must depart the domain of mathematical formalism, for the domain of either experimental physics, or something functionally equivalent to such a physics. These domains are to be found, relative to formalism, within transinfinitesimally small, mathematical discontinuities, the existence of which the followers of Newton, Euler, Bertrand Russell, et al., each and all, fraudulently deny.26 Each valid, axiomatic-revolutionary discovery of principle (e.g., a formal axiom, a dimension, an hypothesis), is a singularity, which, discovered, fills the place defined by a transinfinitesimally small formal discontinuity in the fabric of the mathematical-physics being superseded.

The process by which that valid singularity is generated, can never be detailed at the proverbial "blackboard." Nonetheless, that process exists; its existence is provable, not by mathematics, but according to the principle of measurement.27 The form in which that existence impinges upon knowledge, is the same quality of true metaphor, which is the distinguishing activity of all successful Classical forms of artistic compositions. The activity is known, otherwise, as "creative reason," or, "cognition," when either term is employed to signify the quality of non-deductive mental activity typified by an original valid, axiomatic-revolutionary discovery of a principle of nature. In physical science, this activity is typified by the successful generation of a valid new hypothesis. Riemann approaches the conceptualization of this activity of creative reason, with his use of the term Geistesmassen. This implication of the same principle of hypothesis, which underlies Riemann's dissertation, is the focus of Leibniz's Monadology.

'Psychology & Metaphysics'

That mental activity, through which principles of nature are discovered (and, recognized), and, through which artistic metaphor is generated (and, recognized), is not a subject for deductive methods. In that sense, the validation of an axiomatic-revolutionary principle can not be represented mathematically, either at the blackboard, or in kindred modes.28 Nonetheless, like those discovered, and empirically validated principles of science themselves, the non-deductive mental activity of creative reason (cognition) can be known as clearly as any object presented to our minds by sense-perception. If education is based, not on the stultifying, textbook drill-and-grill mode, of indoctrination in a secularist catechism, but, rather, upon the student's reenacting the original discoverer's act of discovery within the student's own, sovereign cognitive processes, the repeated experience of coming to know these discoveries in this way, enables the pupil to come to recognize the common form of that mental action of change, which is the common feature of the progress of the pupil's mind, from one hypothesis to the next.29

This brings us to the matter of agape: the emotional quality, contrasted to erotic impulses, which is characteristic of what we term here, alternately, "creative mentation," or "cognition."

In Plato, the term agape arises as "love for justice," "love for truth." The Latin translation of Plato's notion of agape, where the Greek term appears in the Christian New Testament, is the caritas which is translated as "charity" in the King James Version's English translation of the Latin edition of Paul's Epistles.30 There are some well-known, if absurd, but clinically foreseeable, capriolically pornographic renderings of the term, from among devotees of the Oxbridge glosses on Plato; despite such sick minds, the intention, "love for justice and truth," is the only accurate rendering of "Platonic love." This quality of emotion, agape, is associated only with a category of objects of thought which belong strictly to the category of "Platonic ideas."

The antonym for agape is eros, the latter the quality of emotion peculiar to either objects of sense-perception, or to those words, methods, and procedures, the which are induced in individual behavior through the anti-cognitive, "sing for your supper," modes of "drill and grill."31

To make clear the significance of the term "Platonic ideas," the present author prefers the example of Eratosthenes' fair estimate for the length of the Earth's meridian. By aid of an ingenious, but mathematically simple experimental procedure, Eratosthenes estimated the polar diameter of the Earth within a margin of error of about fifty miles, and did this more than two thousand years before any person had seen the curvature of our planet. The several Classical Greek estimates of the distance from the Earth to the moon, including that of Eratosthenes, have the same relevance. We can not see, as objects, the actual astrophysical distances from Earth to the moon, sun, or neighoring planets; virtually all of astrophysics, and the entire domain of microphysics address objects which are not defined directly by our senses. Those matters of knowledge which lie outside simple sense-perception, fall within the category of "Platonic ideas."32

The distinction between living and non-living processes, and the distinction between the cognitive processes of the human individual, and the behavior of all lower forms of life, are also subject-matters which are not defined directly by our sense-perceptions. Similarly, neither "justice" and "truth," nor any validated discovery of a principle of nature, are objects defined as sense-perceptions. All of these distinctions of physical processes, which we can not define as matters of direct, simple sense-perception, but which we are able to know to be true in other ways, belong to the catgeory of "Platonic ideas."33

We summarize here, once again, the way in which the case of Eratosthenes' estimate of the length of the Earth's meridian presents the central role of Platonic ideas in science [see Figure 1].

A series of measurements is taken, by sun-dials placed at intervals along a measured (paced off) interval, along a South-North line, between Aswan and Alexandria, in Egypt. Each set of these successive series of measurements is taken at noon (as indicated by the sun-dials) on the same day. The angles of the shadow cast are compared. This comparison shows that the Earth's surface is not flat. However, by use of similar figures, it appears that the data fits the case in which the Earth's surface is approximately that of a sphere, with the South-North direction, from Aswan to Alexandria, corresponding to an arc of a meridian. Since the length of that arc had been measured, the method of similar figures gave an estimate for the size, and diameter of the relevant complete circle.34

The crucial point of describing that, in the present location, is, as stressed earlier, that Eratosthenes' defined and measured the curvature of the planet more than two thousands years before man first saw the curvature of the planet. For related reasons, Columbus did not merely suspect that the Earth was a spheroid; almost five centuries before anyone saw the curvature of the planet, Columbus knew it with scientific certainty, through work done by Toscanelli, based upon ancient Greek science, decades prior to Columbus' acquisition of the map of the planet produced by Toscanelli. The size of the planet, estimated by Toscanelli, was accurate to at least the degree of precision of Eratosthenes estimates, about 1,700 years earlier.35 The estimates of the distance to the moon, by Eratosthenes, and Aristarchus' derivation of the demonstration that the Earth orbitted the sun, are examples of the same principle of Platonic ideas.

The archetypical expression of Platonic ideas, is the quality of mental act, by means of which a valid, axiomatic-revolutionary discovery of a principle of nature is generated. The overriding mission of a competent policy in education, is to prompt the pupil to reenact the series of relatively more truthful, valid, axiomatic-revolutionary discoveries of principle underlying the development of both scientific knowledge, and also of forms of plastic and non-plastic art which are consistent with what we shall identify, below, as the Classical principle of composition and performance. The primary mission of a competent educational policy, is the use of teaching of such crucial principles as a "pretext" for fostering the development of the individual person's potential for deploying and recognizing that distinct quality of mental act (cognition) which is the only means by which such discoveries may be either effected as original discoveries, or by one to whom the principle is presented as a challenge for reenacting the mental experience of the original discovery.

This potential for development of the creative powers of cognition, is that distinction between man and beast underlying Genesis 1:26-30: mankind, male and female, made in the image of God: as Nicolaus of Cusa emphasizes, the principles of imago viva dei and capax dei. In its paradigmatic expression, as knowable to the successful student in such a Classical-humanist program of education, this act of cognition is located in the person's experience, as the quality of mental activity through which the validation of an axiomatic-revolutionary discovery of principle, is effected. In other words, the generation of a valid "leap" from a given hypothesis (theorem-lattice) to a relatively superior hypothesis. This paradigmatic act, is, therefore, the experience of higher hypothesis.

That paradigmatic experience has two distinguishable, but inseparable interdependent qualities. The occurrence of the formally validatable discovery itself, and the distinctive quality of emotion associated with that act of discovery. That latter quality of emotion, is agape as Plato defines it, and as I Corinthians 13 also defines it.36 It is through the summoning of the developed quality of agapic emotion, that the thinker is able, willfully, to summon the creative cognitive powers needed to address a challenge.

The kind of deductive reductionism typical of Aristotelean formalism, is erotic, and hatefully anti-agapic, in type, as the psychopathological case of Kant and his philosophical writings, typifies the pathology of personal character inhering in the true follower of Aristotle's philosophy and method. Thus, Friedrich Schiller and his follower Wilhelm von Humboldt, set forth as the primary objective of a Classical-humanist form of education, the fostering of the development of the personal character of the future adult citizen; the efficient principle referenced by Schiller and Humboldt on this account, is rooted in the argument of I Corinthians 13, and it is also the underlying character of Plato's dialogues taken as a whole.

Hypothesis, and higher hypothesis, are each a special kind of object, an object of the form which Plato associates with the good. To introduce this conception, consider, first, the example offered by a very ordinary sort of theorem-lattice, as we defined this earlier, here.

In the simple theorem-lattice, the derivation of theorems has a certain ordering, in the sense that some theorems, once proven, serve as the basis for deriving later theorems. This sense of ordering implies ordering in time. Nonetheless, the hypothesis underlying that lattice undergoes no modification during the time a sequence of theorems unfolds: from beginning, through to the end, the hypothesis remains unchanged; it is the veritable "alpha and omega" of that theorem-lattice. In Plato's method, every hypothesis, including every higher hypothesis, has this same property: it is the unchanging "alpha and omega" of whatever process of lattice-generation it underlies. In all, higher hypothesis is subsumed by God, the unsurpassable "hypothesis," the ultimate Good. Yet, every relatively valid hypothesis also imitates that form, as a lesser good.37

Agape is the motivating state of mind which corresponds to the experience of any valid, or relatively valid such good.

Every person engaged in cognitive concentration, has lived through a relevant experiment: One's mind is working on the problem, up to the point the concentration collapses, as it were a man who suddenly toppled over, and fell asleep during a brisk walk. This might occur when one were exhausted, but we are considering only the type of case in which exhaustion was not determining. The motivation for the cognitive concentration has collapsed, as if the current had suddenly been cut off from an electronic device, as if the "batteries had died." Consider the instance, in which taking a break to participate in working through, or hearing a good performance of J.S. Bach, Haydn, Mozart, Beethoven, Schubert, or Brahms, returns one to one's cognitive undertaking with full powers of concentration restored—"batteries fully recharged." From this vantage-point, we turn our attention to certain identical features of Classical art-forms and valid axiomatic-revolutionary discoveries of physical principle. We are considering a topic which might be entitled: cognitive energy.

In Classical art-forms, the place of a mathematical discontinuity is taken by the ultimate expression of ambiguity, metaphor. During his 1948-1952 project, to refute Wiener's absurd claim, that human communication could be represented by statistical "information theory," the present author adopted the policy, that, although the case against Wiener could be made best from the standpoint of technological progress's increasing the productive powers of labor, it would be necessary to show that what was true for physical science, was also true for the generation and transmission of knowledge in Classical art-forms.

Thus, the study of "information" from the standpoint of technological progress, was parallelled by focus upon three closely related forms of non-plastic Classical media: poetry, drama, and the Classical art-song, the latter centered upon the Classical German lied, of Mozart, Beethoven, Schubert, Schumann, and Brahms, all compared with the Romantic lied of Hugo Wolf and Richard Strauss.

The standpoint in music, from which Classical forms of drama, poetry, and song were examined during that time, was the principle of motivic thorough-composition, as typified by Wolfgang Mozart's K.475 product of his study of the Bach Musical Offering, and the influence of that, and closely related Mozart compositions in later Classical composition. Today, the present author would have written of that approach, that keys and modes are hypotheses underlying the theorem-lattices of Classical forms of musical compositions, and that motivic thorough-composition, as typified by the Mozart K.475, is a prototype for higher hypothesis as the subject of musical composition.38

Thus, effective Classical musical composition, especially since those aspects of the work of J.S. Bach so deeply admired and emulated by Mozart, Beethoven, et al., is an exercise in agape. Similarly, Classical tragedy, and great Classical poetry, which rely upon the implicit bel-canto well-tempering of the well-spoken language, as the medium for speech, embody the developmental principle of the Greek Classical tragedy and Socratic dialogue. This is that cognitive medium of artistic development, which such poetry and drama employ, to instruct musical composition in the principles of musical dialogue, called polyphony, the which is the principle of Classical artistic development.

It is those artistic resolutions of ambiguity which carry the mind from one hypothesis to another, whether in poetry, drama, music, or plastic art-forms, which are the principle of change underlying Classical forms of artistic composition. This is that principle of Reason in art, which the psychosexually impotent Immanuel Kant could not recognize.39 Those ambiguities which can not be resolved (e.g., "explained") deductively, as mere simile, symbolism, or hyperbole, are metaphors. These metaphors, which exist implicitly in the subjunctive mood, are the Geistesmassen of art.40 Hence, during the course of the 1948-1952 study, the present author employed this sense of "metaphor" to embrace the expression of Platonic hypothesis in both physical science and Classical art-forms.

All successful art meeting those standards, evokes the same sense of uplifting agapic beauty we experience otherwise in those activities of the individual mind, through which original, or reenacted, valid, axiomatic-revolutionary discoveries of principle are generated. Such art is an integral part of science, in the broader sense of science. Such art increases the potential productive powers of labor, in the same sense that technological progress does. Such art also "recharges the batteries" of the individual's, and society's exercise of its creative powers of reason.

All too often, in observing discussions of mathematical, or of scientific work, we may be startled to recognize that the discussion we are witnessing, is painted in fresh coats of gray upon gray, proceeding with the implied assumption, that there is no emotional motivation in scientific thought as such, but only in arguments about its conclusions. Poor actor Leonard Nimoy, trapped for eternity in endless sequels of "Star Trek," babbling forever the idiot-savant's: true scientific "logic" is a quality free from emotions!

John Keats' Ode on a Grecian Urn spoke elegantly for Plato: truth is beauty, and beauty is truth. It is the passion of a mind gripped by a prescience of great beauty, which impels the creative thinker to ascend the impossible alp of scientific risks. Well-meaning laymen speak, foolishly, of financial rewards as motives for scientific (or, artistic) work. Feed a scientist, nourish his family, and offer him the opportunity to meet the kind of challenge which inspires him; freed of distracting such matters, his incentive is his passion never to lose that sense of a (Leibnizian) pursuit of happiness, the which is for him, or her, the lure of the scientific (like the Classical artistic) profession. The sense of truth is the source of the sense of overwhelming beauty; the recall of the emotion one associates with that sense of beauty, is the passion which drives one to push forward, one more step, and another, in pursuit of truth. Like Edmund Hillary, the scientist climbs the Everest of science—and Classical art, "because it is there." Keats' Ode is dedicated, passionately, to the triumph of agape over eros.41

Such is "cognitive energy." The composition and performance of the Classical art-form are the mirror-image of valid scientific discovery, on this account. Thus, does art command the power to recharge the batteries of the cognitive process for the scientist. That is a subject which, however curious that might seem, at first hearing, belongs to the department of economics: to the Leibnizian science of physical economy.

It is relevant here, to consider what might be described as a "structured" feature to agape, a feature presented in the clearest way by considerations of technological attrition.

We have already indicated, that the Riemann topological series of hypotheses, typified, symbolically, by (n+1)/n, corresponds to a series of formal-mathematical discontinuities. Each such discontinuity corresponds to a corresponding singularity, an added "dimension" of the series of manifolds. All of the singularities functionally extant at the time each of the manifolds is in operation (subjectively and in corresponding practice), is efficiently present in every interval of thought-action of the person whose judgment and practice are being directed in accord with that manifold. Thus, we may apply the notion of implicitly enumerable densities of discontinuities, for any arbitrarily selected interval of thought-action, for that manifold's influence, under those general conditions.

The increase of the density of discontinuities, in such modes, has the twofold quality of "tension" and "potential." The "potential" corresponds to the relative increase of power over nature, per capita and per square kilometer of the planet's surface. The "tension" corresponds to a higher development of the internal (subjective) mental state of the relevant person. The increase in potential, corresponds to capacity for effectiveness of action; the increase of "tension," corresponds to an increase in the psychological motivation for action, to an increased sense of agapic, subjective "energy."42

The notion of hypothesis, and higher hypothesis, as of the timeless form of a good, defines these notions as what Kepler defined as Reason, and Leibniz as necessary and sufficient reason. A related term, to the same general effect, is universal characteristics. The significance of the latter term is shown more clearly from the standpoint of the present author's original discoveries in the domain of physical economy.

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 1. Unless otherwise noted, the references to Leibniz's writings cited here, are limited to the following: [Loemker] Gottfried Wilhelm Leibniz, Philosophical Papers and Letters, ed. by Leroy Loemker (Boston: Kluwer Academic Publishers, 1989); [Monadology] G.W. Leibniz, Monadology and Other Philosophical Essays, trans. by Paul and Anne Martin Schrecker (London: McMillan, 1965); [Theodicy] G. W. Leibniz, Theodicy, trans. by E.M. Huggard, ed. by Austin Farrar, 5th printing (Peru, Ill.: Open Court Publishing Co., 1996). The principal reference to the work of Bernhard Riemann, is to Riemann's 1854 habilitation dissertation, Über die Hypothesen, welche der Geometrie zu Grunde liegen ("On The Hypotheses Which Underlie Geometry"), in Bernhard Riemanns Gesammelte Mathematische Werke, ed. by H. Weber, reprint of (Stuttgart: B. G. Teubner Verlag, 1902) [(New York: Dover Publications, 1953) and (Vaduz, Liechtenstein: Saendig Reprint Verlag)], pp. 272-287. Various English translations of this habilitation dissertation are extant, but, for purposes of precision, reference is made to the German. Other references to Riemann's writings are always to the reprint of the Weber edition: Riemann Werke. As a general, recurring reference, see Ralf Schauerhammer and Lyndon H. LaRouche, on Kepler and Riemann [Schauerhammer and LaRouche], respectively, in the "Riemann Refutes Euler" feature, in 21st Science & Technology, Vol. 8, No. 4, Winter 1995-1996, passim.

 2. See Riemann Werke, pp. 525: Die Unterscheidung, welche Newton zwischen Bewegungsgesetzen oder Axiomen macht, scheint mir nicht haltbar. Das Trägheitsgezetz ist die Hypothese: Wenn ein materieller Punkt allein in der Welt vorhanden wäre und sich im Raum mit einer bestimmten Geschwindigkeit bewegte, so würde er diese Geschwindigkeit beständig behalten. An English translation of this is found in the translation of the "Philosophical Fragments" from the Riemann Werke, published in 21st Century Science & Technology, Vol. 8, No. 4, Winter 1995-1996, p.57. More on the hypothetical basis for Newtonian physics, below.

 3. Hereinafter, we focus upon these three figures of the four listed. Our primary focus here, is the retrospective connection of Riemann to Leibniz. Kepler is kept in focus, for reasons to become clear later in the paper. Gauss, the most prolific mind in modern science after Leibniz, represents, together with his collaborator Wilhelm Weber, and protégé, Riemann, a topic deserving of special attention in a location devoted to that connection.

 4. As James C. Maxwell purported to justify his refusal to acknowledge the work of the Gauss, Weber, and Riemann which Maxwell had parodied. He explained, that it was his policy to refuse to recognize the existence of any geometries but "our own."

 5. See footnote 1.

 6. See footnote 1.

 7. Norbert Wiener, Cybernetics (New York: John Wiley & Sons, 1948). The writer's first encounter with Wiener's book occurred during Winter 1948, prior to the Wiley release of the hardbound U.S. edition, in the form of a loan to him of an earlier, Paris, paperbound printing.

 8. Lyndon H. LaRouche, Jr., "On LaRouche's Discovery," Fidelio, Vol. III, No. 1, Spring 1994. The use of the argument supplied in Riemann's habilitation dissertation, enabled the writer to solve the problem of mathematical representation incurred by his own original discovery in the science of physical economy. Hence, because of this relationship of Riemann's discovery to his own, the result came to be identified as "The LaRouche-Riemann Method." On Riemann's habilitation dissertation, see footnote 1.

 9. Loc. cit., footnote 1. On the second point, Riemann writes: ...dass die Sätze der Geometrie sich nicht aus allgemeinen Grössenbegriffen ableiten lassen, sondern dass diejenigen Eigenschaften, durch welche sich der Raum von anderen dreifach augedehnten Grössen underscheidet, nur aus der Erfahrung entnommen werden können. (pp. 272-273.) The concluding sentence of the dissertation restates this point: Es Führt dies hinueber in das Gebiet einer andern Wissenschaft, in das Gebeit der Physik, welches wohl die Natur der heutigen Veranlassung [the subject of mathematics] nicht zu betreten erlaubt. (p. 286).

10. On Euler's attack on Leibniz, see, Lyndon H. LaRouche, Jr., The Science of Christian Economy, (Washington, D.C.: Schiller Institute, 1987), "Appendix XI: Euler's Fallacies," pp. 407-425. Note a typographical error on p. 407; the passage should read "He [Euler] was a proponent of the Newtonian reductionist method in mathematical physics." Euler was a member of an anti-Leibniz salon within the Berlin Academy of Prussia's "Frederick the Great," closely associated with such followers of Newton's patron, Abbé Antonio Conti, and members of Conti's network of salons, as Pierre-Louis Maupertuis, Johann Lambert, Giammaria Ortes (the founder of "Malthusianism"), Voltaire, and Joseph Lagrange. On this attack on Leibniz by Euler, the following history is most notable. A purely geometrical proof for the fact that π is of a higher cardinality than the Plato-Eudoxus-Eratosthenes-Archimedes notion of "irrationals," was discovered by Nicolaus of Cusa (cf., De Docta Ignorantia, 1440). The physical proof, that non-algebraic (i.e., transcendental) functions must supersede the algebraic notions of Descartes and Newton, was demonstrated by Leibniz, Jean Bernoulli, et al., during the 1690's, in respect to the interconnected facts of isochronicity in the gravitational field (Huyghens) and the relativity of a constant "speed of light" with respect to refraction (Roemer, Huyghens, J. Bernoulli). Using the same false premises which he adopted for the attack on the Monadology, Euler presumed that the distinction between algebraic and non-algebraic ("transcendental") functions could be degraded to its relatively degenerate expression, as a subject of infinite series (see Leibniz-Clarke Correspondence on the subject of differential calculus and infinite series). Around this, the Newtonian devotees, following Euler and Lambert, built the myth that the proof of π's transcendental quality, is the proof derived, "hereditarily," from the tautologically fallacious assumptions of Euler's 1761 attack on the Monadology. Hence, the popularization of the myth, that it was Ferdinand Lindemann, in 1882, who first "proved" the transcendental quality of π! (See Lyndon H. LaRouche, Jr., "Kenneth Arrow Runs Out of Ideas, But Not Words," 21st century Science & Technology, Vol. 8, No. 3, Fall 1995; see reference to the π controversy, under the subhead "Axiomatic Method," pp. 43-44. See also, LaRouche reply to a critic of this section of that paper, in Letters, 21st Science & Technology, Vol. 9, No. 2, Summer 1996.

11. The "Classical humanist" method in education has two leading features which might be treated as the definitional distinctions of that method. "Classical" should be understood, in first impression, as implying a foundation in what are identified as the "Classical," as distinct from "Archaic" (for example) plastic and non-plastic art-forms of Classical Greece. In literature, this implies the Homeric epics, and the tragedies of Athen's Golden Age. In science, it implies Plato's Socratic method of hypothesis, as typified by Plato, Eudoxus, Theaetetus, Eratosthenes, and, implicitly, also, Archimedes. Overall, it signifies the struggle of the Ionian city-states and the tradition of Solon of Athens, in combatting both the Babylonian tradition, expressed as the Persian Empire, and, also, the usurious cult of Gaia-Python/Dionysos-Apollo at Delphi (and, later, pagan Rome). In art, science, and history, it implies the principle of agape, as defined by Plato and the Christian apostles, as in the Gospel of John and the Epistles of Paul. The use of these Classical Greek referents, including the Christian New Testament, is the significance of a Classical-humanist secondary education for the relevant medieval European teaching orders, such as the Brothers of the Common Life, the continuation of that standard of literacy among the proponents of the original (anti-Justice Antonin Scalia) intent of the U.S. Federal Constitution, and the reforms of education in Germany designed by Friedrich Schiller and his followers Wilhelm and Alexander von Humboldt. This exemplary significance of that use of the term, "Classical," extends to the principle, that all of those discoveries of principle which have been proven to be valid, as such discoveries, from all currents of humanity, non-European as European, ought to be replicated mental experiences of discovery within the minds of all prospective secondary graduates, as a precondition for citizenship, in a durable form of society. The Classical currents of philology, as those with which the Humboldt brothers were associated in their time, illustrate the manner in which the notion of "Classical" is to be extended in choice of referents, from Classical Greece, to mankind as a whole. It is the emphasis on recreating the experience of the original discovery of principle, within the mind of each pupil, which distinguishes a cognitive education, from the evil of John Dewey and the "New Math," in particular, and from today's more popular textbook, or even worse standards, in general.

12. Once one has worked one's way through the sets of later dialogues of Plato, it becomes clear, that his Parmenides serves implicitly as a prologue to all of those dialogues; it poses the crucial, ontological paradox, which the other dialogues address, each in its own respect. For this purpose, the Parmenides should be read as if it were the prefatory chorus of a tragedy, modelled upon the tragic principle characteristic of Aeschylos' work. One might apply Friedrich Schiller's explication of the principles for design of a tragedy: from opening germ, through punctum saliens, to conclusion. In the dialogue taken as a whole, the character Parmenides fails as pitiably as Shakespeare's Hamlet. The character Parmenides, like his real-life image, can not comprehend the notion of change as an efficient principle, just as Hamlet identifies the same cause for his own, oncoming doom, in the famous Act III, Scene 1 soliloquy. This is change as Heraclitus references its definition; so, for Plato, and for Riemann, the elementary form of efficient existence, is not objects akin to the notion of objects of sense-perception, but, rather, the principle of change, which brings such secondary phenomena as mere, apparently fixed objects, into being. Change, so referenced, has the connotation of generate or create. That is key to any competent reading of Plato, of Cusa, of Kepler, of Leibniz, of Riemann, or this writer's own original discoveries of the same efficient principle in physical economy.

13. In the line of discussion being developed here, we have already put to one side the substitution of non-existent conditions or events, for real ones. Three distinct classes of such substitutions are notable among those excluded from consideration in this portion of the text. (A) Simple lies. (B) Sophistries derived, as conclusions, from wishfully altered hypotheses. For a simple example: "I do not like him, therefore, I choose to find plausible anything bad said of him, and profess to consider as incredible, anything which might work to his credit." (C) Fallacies of composition superimposed, like a Procrustean Bed, upon perceived reality, to the purpose of protecting either an hypothesis, or some specific, isolated belief. Illustration: the principal origin of spread of gnosticism within western European Christianity, is the legalization of Christianity, as part of the Roman pagan Pantheon, by the Emperor Constantine. The most important action to this effect, was the later Byzantine emperors' virtual, or actual banning of the Plato who had been the correlative of Christian theology, and the introduction of Plato's adversary and bellwether of oligarchical social order, Aristotle, as authorized replacement. The efforts of the powerful oligarchical families, to defend their feudal and financier-aristocratic privileges, despite Christianity, has been the continuing source of renewal of the corrupting influence, within the clergy and churches, of the gnosticism inherent in Aristotle's philosophy and method. To avoid the embarrassing truth about the origins of gnosticism, the myth was created, that it was the Jews who are chiefly responsible for introducing gnosticism to western Europe, as via "Averroesism." This apology for oligarchism of both the landed and financier oligarchies—and, Aristotle, has been, thus, the most common source of religious anti-semitism. On the other hand, Friedrich Nietzsche, like his follower Adolf Hitler, premised his argument for ridding Europe of Jews, on the charge that it was the Jews whose collective crime had been the establishment of Christianity. Similarly, another illustration of category (C) taken from real life: To defend the Venice-created cult of Isaac Newton, Leonhard Euler, and many other devotees of the Newton cult, were willing to go to any lengths, as did J.C. Maxwell and Hermann Helmholtz, to defend the hypothesis of their cult's demi-god. Or, for a concluding example of this most relevant problem: The babbling fool who insists, that, since Karl Marx approved the idea of a progressively graduated income-tax, in the Communist Manifesto, that a man as fascistic as that "Miniver Cheevy" of the Confederacy's "Lost Cause," Ku Klux Klan fanatic and U.S. President Woodrow Wilson, was a Communist. Under "Lost Cause" devotee J. Edgar Hoover, the FBI was riddled with precisely such fanatical fools of the Roy M. Cohn breed.

14. These elementary considerations respecting solar phenomena, underscore the fact, that any university which tolerates a policy of eliminating, or minimizing the student's requirement for mastery of the work of "dead European males," is clearly guilty of perpetrating a fraud upon both the students, and those institutions of society, including government, to which that university presents its graduates as competently educated. Exemplary is the fairy-tale, repeated by many illiterates with university bachelor and even terminal credentials, who believe in the myth of the "Copernican Revolution," that Mesopotamian lunatic calendars preceded solar calendars, and that the best astronomy, prior to Copernicus, was that of the fraud concocted, for ideological purposes, by Claudius Ptolemy. India's Bal Gangadhar Tilak was only citing already extant astrophysical and scholarly evidence, when he reported, in his Orion, that the Vedic solar astronomical calendars of Central Asia, circa 6,000-4,000 b.c. , were already vastly more advanced scientifically, than any of the lunar calendars later presented in Mesopotamia. A similar case is demonstrated for ancient Egypt's solar astronomy. Aristarchus, long prior to Claudius Ptolemy's concoction of his hoax, had already defined the elementary hypothesis upon which rested the modern solar astronomy of such as the pre-Copernicus (1473-1543) Nicolaus of Cusa (1401-1464). Every competent program of combined secondary and higher education, requires a student's mastery of the work in mathematics, astronomy, and philosophy, by Thales, Plato, Theaetetus, Eudoxus, Euclid, Aristarchus, Eratosthenes, and Archimedes, through the construction, by Cusa's collaborator, Paolo Toscanelli (1397-1482) of the world map, which Christopher Columbus acquired through the Portugal-based executor of Nicolaus of Cusa's estate, and upon which Columbus largely relied, for his planning his first, 1492, voyage to the Americas. Most of the ideas underlying modern science, in every country, are derived chiefly from the original discoveries in geometry and scientific method, which we have inherited, chiefly, from such representatives of the Classical Greece tradition as these. As in astronomy, so, in general, the truthfulness of any report of a condition or event, lies in the hypothesis which has governed the manner the revelant experience has been comprehended by the mind of the witness. "Truth in education" cannot exist, without prompting the student to reenact, in his, or her mind, the act of original discovery by those ancient Greek and other individual minds, to which our civilization is largely indebted for the development of those hypotheses upon which the truthfulness of contemporary judgment depends, without exception.

15. Cf. Riemann, Plan der Untersuchung, Werke, pp. 272-273.

16. See Christiaan Huyghens, The Pendulum Clock, trans. by Richard Blackwell (Ames, Iowa: Iowa State University Press, 1986); , A Treatise on Light (1678), reprint of English translation: (New York: Dover Publications). On Huyghens' relationship to the discovery of the "speed of light," see Poul Rasmussen, "Ole R[o]mer and the Discovery of the Speed of Light," 21st Century Science & Technology, Vol. 6, No. 1, Spring 1993. On the relationship to Jean Bernoulli's solution to the brachystochrone problem, see D.J. Struik, A Source Book in Mathematics, 1200-1800 (Princeton, N.J.: Princeton University Press, 1986), pp. 391-399.

17. This latter transformation became a central issue of the Leibniz-Clarke correspondence: Leibniz's insistence that a competent calculus could not be represented by the relatively degenerate geometry of infinite series.

18. For the purposes of this paper, it should be sufficient merely to note, as we do here, that Riemannian physical space-time does not permit "linearization in the very small." On this, note the conflict between Riemann and Rudolf Clausius. In a related example, also contrast Riemann's notion of physical space-time with that presented by Princeton's Hermann Weyl. For example, in editor H. Weber's appended note to Riemann's Ein Beitrag zur Electrodynamik [Werke, p. 293], Weber reports Rudolf Clausius' attack upon Riemann's function, as follows.

P = − 0t ∑∑ εε′ F (τ− rα, τ) dτ .

Of which, Weber reports Clausius to argue: Die Operation, vermöge deren später dafuer ein nicht verschwindend kleiner Werth gefunden wird, muss daher einen Irrthum enthalten, den Clausius in der Ausführung einer unberechtigten Umkehrung der Integrationsfolge findet. Thus, Clausius demands linearization in the very small. An English translation, by James Cleary, of H. Weber's note, is found in the textbook by Carol White, Energy Potential (New York: Campaigner Publications, 1977), pp. 299-300.

The formal-mathematical aspect of Clausius' argument is to be recognized at once as an "hereditary" influence of the same tautological fallacy on which Euler premised his 1761 attack upon Leibniz's Monadology. Similarly, it is the failure of Euler, Lagrange, Laplace's Augustin Cauchy, Hermann Grassmann, Clausius, Hermann Helmholtz, et al., to recognize Leibniz's argument against Venetian Abbot Antonio Conti's agent, Dr. Samuel Clarke, respecting the implications underlying the incompetency of the mere numerical approximations supplied by use of an infinite series as a substitute for an actual calculus. In the Beitrag, Riemann is referencing work-product of his own collaboration with Wilhelm Weber, of which more is to be learned in a forthcoming issue of 21st century Science & Technology. In short, Clausius' invocation of the notorious "sliding rule," is not only flatly wrong, but, reveals much more about his own, and Grassmann's mathematics, than it does respecting the work of Weber and Riemann.

19. Riemann, op. cit., p. 273: ... Gauss, in der zweiten Abhandlung über die biquadratischen Reste. [Theoria Residuorum Biquadaticorum: Commentatio Secunda (1831), Carl Friedrich Gauss Werke, II (Hildesheim: Georg Olms Verlag, 1981). pp. 93-178. See, also Zur Theorie der Biquadratischen Reste Werke, II, pp. 315-385.]

20. Ibid., p. 276: ... Zu beidem sind die Grundlagen enhalten in der berühmten Abhandlung des Herrn ... Gauss über die krummen Flächen. See, Disquisitiones Generales Circa Superficies Curvas (1828) Gauss Werke, IV, pp. 217-258. See, Gauss' notice of this paper: pp. 341-347; the crucial issue of mapping is presented on pp. 344-345. See, also, Allgemeine Auflösung der Aufgabe die Theile einer gegebenen Fläche so abzubilden (the famous "Copenhagen Prize Essay") (1822), pp. 189-216. Notable is the issue of mapping of an ellipsoid onto a sphere; the referenced work of Gauss' on this subject was, most immediately, a reflection of his discoveries in geodesy, in the setting of his 1818-1832 triangulation-survey of the territory of the Kingdom of Hanover. However, Gauss' work in "non-Euclidean geometry" dates not only from his earlier discoveries in astronomy, but, according to a Nov. 28, 1846 letter to H.C. Schumacher, to 1792. Notably, it was from this starting-point in the work of Gauss, not the quasi-Kantian Newton devotee and plagiarist of Abel, Augustin Cauchy, that Riemann derived what some wags amuse themselves to describe as the "Cauchy-Riemann" function; the debt to A.M. Legendre is significant, not to Monge's and Legendre's hateful adversary, and Laplace protégé, Cauchy.

21. Ibid., p. 273: ... und einigen philosophischen Untersuchungen Herbart's, durchaus keine Vorarbeiten benutzen konnte. For the relevant text of Riemann's earlier commentary on this, see Werke, pp. 509-520. For an English translation of the latter, see "Riemann's Philosophical Fragments," 21st Century Science & Technology, op. cit., pp. 51-55.

22. As is suggested by Eratosthenes' experimental measurement of the estimated curvature of the Earth's meridian, more than two thousand years before any person had yet seen the Earth's curvature.

23. On this item, no scientifically literate person would introduce, as objection, the somewhat popularized nonsense, of asserting that the original discovery of gravitation was the work of Galileo, Newton, et al. Newton's algebraic representation of gravitation was explicitly derived, as a relatively degenerate representation, from Kepler's formulation for gravitation. For a summary of the way in which Newton's plagiarism of Kepler was constructed, see Lyndon H. LaRouche, Jr., The Science of Christian Economy, op. cit., Chapter VII, Note 8 (see pp. 471-473).

24. D.J. Struik, loc. cit.

25. See Proclus' Commentary on Plato's Parmenides, trans. by Glenn R. Morrow and John M. Dillon (Princeton, N.J.: Princeton University Press, 1987), passim.

26. In every case examined, the argument against the existence of mathematical discontinuities is a parody of the tautological fallacy which Euler deployed in his attempted sodomy of 1761, against Leibniz' Monadology.

27. Cf. B. Riemann, Über die Fortpflanzung ebener Luftwellen von endlicher Schwingungsweite, Werke, pp. 156-175. In this paper, Riemann addressed the implications of the mistaken assumption, that the speed of sound represented an insuperable barrier to movement of a propelled projectile at higher speeds through the air medium. Out of his understanding of the physical significance of discontinuities arising in such functions, not only was the possibility of accelerated transsonic flight indicated, but, more generally, the general principle of isentropic compression. The crucial point illustrated, for our purposes, here, is that Riemann recognized that the appearance of a formal discontinuity, in the mathematical form of the design of his experiment, represented the presence of a singularity, a new principle—isentropic compression—to be entered into the validated physical principles of physical space-time. The problem which Riemann had successfully attacked, was that on which Britain's Lord Rayleigh discredited himself so recklessly on this point. Rayleigh's commentary on Riemann's Fortpflanzung shrieked, to the effect, that, if Riemann were right, then all of the physics of Rayleigh and the pro-Newton faction, were thoroughly bankrupt intellectually. The root of Rayleigh's consternation: the argument against Riemann's method, by such as Clausius, Grassmann, Helmholtz, Maxwell, and Rayleigh, is that the wrong view of gas theory is embedded axiomatically in those notions of percussive causality which Sarpi and his followers had embedded in the Cartesians and British empiricists. Riemann's representation of isentropic compression has important implications within applications of the LaRouche-Riemann method in physical economy. On the latter account, the present writer commissioned a translation of this paper of Riemann's, by Uwe Henke and Steven Bardwell, which appeared in the SUPPLY DATE edition of The International Journal of Fusion Energy (Vol. X, No. X).

28. This is the key to understanding the convoluted argument which underlies such later publications of Immanuel Kant as: Critique of Pure Reason (1781), Prolegomena to a Future Metaphysics (1783), Fundamental Principles of a Metaphysics of Ethics (1785), Critique of Practical Reason (1788), Critique of Judgment (1790), and Perpetual Peace (1795). Kant's argument is the basis for the mysticism of such Nineteenth-century neo-Kantian mystics as (implicit Volksgeist doctrinaire) Johann Fichte, (Weltgeist doctrinaire) G.W. Hegel, (Zeitgeist/Volksgeist doctrinaire, and Hegel ally) F.K. Savigny, and the pathological Franz Liszt. The central feature of Kant's Critiques, and related writings on science, psychology, morals, and aesthetics, centers around the mystical irrationalism of his discussion of synthetic judgment a priori. Unlike his more radical, logical-positivist followers, such as Norbert Wiener of "information theory" notoriety, agnostic Kant is prepared to allow both God and creative reason to exist somewhere, but not to permit them to be known. Although there is a foretaste of Kant's argument in the mystical side of the gnostic René Descartes, in the notion of deus ex machina, the empiricists deny the existence of creative reason altogether. (See relevant writings of the neo-Kantians W. Windelband and E. Cassirer, for insight into the continuing distinctions between neo-Kantianism, on the one side, and empiricism and positivism, on the other.) Similarly, as a reflection of their pro-atheistic, empiricist "mind set," the pseudo-Christian gnostics of Britain deny the existence of a "divine spark of reason" within the individual person, i.e., deny both Genesis 1:26-30, and the Christian principles of imago dei and capax dei. It is for these same "Brutish" varieties of religious motives, that Galileo student Thomas Hobbes decreed the policy, for banning both metaphor and the subjunctive mood (e.g., Leviathan), which is the continuing policy-trend among empiricist and positivist species of modern-language stylists, to the present day. This streak, expressed variously as the atheism axiomatically inherent in empiricism and positivism, and as "agnosticism" among the followers of Kant, is a strictly correct reading of the import of Aristotle's method and writings. In modern Europe, this atheistic current is to be traced chiefly to Cardinal Gasparo Contarini's extremely influential teacher, the Pietro Pomponazzi of Padua, who taught, that, among the followers of Aristotle (and, of Pomponazzi), the human soul could not exist.

29. Cf. Lawrence S. Kubie, "The Fostering of Scientific Creativity," Daedalus, Vol. XX, No. XX, Spring 1962; also, The Neurotic Distortion of the Creative Process (Lawrence:1958).

Although Kubie, a rather celebrated Yale psychoanalyst, was a participant in the Josiah Macy, Jr. Foundation's notorious "Cybernetics" project, he proved himself insightful in his investigation of the reasons why some of those persons nominally among the most highly qualified, and formerly most promising academics, had proven sterile in the field of scientific creativity. Kubie's referenced works were published after the writer's structured, quality-control study of indicated patterns of behavior in formally well-qualified management consultants who tended to fail, consistently; hence, the referenced titles attracted this writer's attention. From the standpoint of the writer's own investigations, Kubie's observations in the 1962 Daedalus piece were on target. In the typical case of the failure-prone management consultant, in this writer's study, and in related cases, it was the case's educational successes which were, arguably, the source of his performance failures as a consultant. In his education, usually, that subject had been the kind of "nerd" who hit the books, learned the subject, passed the examination, whose opinions won the approval of his teachers, all the way to his pre-doctoral orals and written examination. The subject's mind was trapped inside that mere learning as a virtual reality. Clearly, during his education, the subject had employed his cognitive powers sometimes, but had never recognized the distinction between learning and the role cognitive processes contributed to assisting the learning process. Only rarely, would that subject rely upon thinking cognitively "in a pinch." If the subject must have been somewhat creative during the earlier phases of his education, his willingness to continue the learning process in that way would begin to wither away at a point proximate to his completing higher education. As he grew older, the growing maturity of his professional experience was accompanied by an apparent "calcification" of his cognitive potential. Under the pressure of desire for approval from actually present, or possible professional peers, he would fall back into the virtual reality of academically, and bureaucratically induced habits of Pavlovian "academic correctness." In a related type of case, the gifted experimental scientist might go stale, during the moments he is confronted with the prospect of defending mathematically, at the blackboard, or in a paper submitted to referees, what he knows, otherwise, to be his valid experimental discovery. As indicated in later paragraphs of this text, this is not merely a formal problem, but also a psychiatric problem, arising to this form through the victim's substituting the inappropriate, erotic form of intellectual motivation, where the non-erotic, agapic form of behavior is required.

30. The paradigmatic New Testament text is I Corinthians 13. Paul's meaning for the term, is fully consistent with that of Plato.

31. The student, and professional, who approaches his subject-matters like one who "sings no better than he believes necessary to gain his supper," is referenced by Friedrich Schiller as of the category of Brotgelehrten. That has been increasingly the characteristic of the education and standard of adult practice of professionals in general.

32. The empiricist and positivist would argue, that such ideas are "constructs," derived, thus, from sense-perceptions. That empiricist argument, is traced to Padua's Pietro Pomponazzi through Pomponazzi's student, the Venetian Francesco Zorzi (a.k.a, "Giorgi"), who took up residence in England to serve as marriage counsellor to King Henry VIII, and served as the intellectual resource upon which the King relied, together with Venice's agent Thomas Cromwell, et al., in that celebrated Anne Boleyn affair upon which the Church of England was established. Zorzi is otherwise notable in the history of England during that same period, for his direct attack on the influence of Cardinal Nicolaus of Cusa, the crucial organizer in the process leading into 1439-1440 Council of Florence, and, later, mid-Fifteenth-century canon of the Papacy. Zorzi's attack was directed against the influence of the Erasmians, the principal conveyers of the Renaissance heritage into England at that time. Zorzi demanded extirpation of the method of "docta ignorantia," and its replacement by a kind of proto-empiricism. The influence of Pomponazzi and his leading students, apart from the key role they played in orchestrating, as did Gasparo Contarini, the great schism of the early Sixteenth century, was the current of Venice's influence leading into Paolo Sarpi's founding of what we know today as the British empiricism of Bacon, Hobbes, Locke, Bentham, et al. Echoing Zorzi, the Sixteenth through Nineteenth centuries witnessed an hysterical effort by the followers of Hobbes, Locke, and Newton, to eliminate the notion of ideas from science and philosophy, through the establishment of the notion that those ideas were merely "constructs." The issue of infinite series, posed by Leibniz in the Leibniz-Clarke-Newton correspondence, and Euler's lunatic use of a tautological fallacy, to attack Leibniz's Monadology, are bellwether cases of this effort to promote the hoax of the "construct."

33. It is also stressed, in sundry other locations, that scientific knowledge requires uncovering the necessary and sufficient reason underlying the existence of the division of experience among three distinct qualities of scale, and three mutually exclusive categories of characteristic functional distinction. Of scale, we have astrophysical and microphysical, which are beyond the scope of objects perceivable to the senses, and, thus, by elimination, the macrophysical scale. Of characteristic functional distinctions, we have putatively non-living, putatively non-cognitive living, and cognitive processes. The combinations of the two types of distinctions define a simple matrix; a functionally comprehensive definition of all of the relations implicit in that matrix, is science. Thus, science as a whole does not exist outside the domain of Platonic ideas.

34. See Selections Illustrating the History of Greek Mathematics, trans. by Ivor Thomas, Vol. II (Cambridge, Mass.: Harvard University Press, 1980), Loeb Classical Library, pp. 266-273. Note, that Eratosthenes also supplied an estimate for the arc of a great circle passing through Alexandria and Rome. Eratosthenes' estimates are typical of the application of Classical Greek science (from Thales through Eratosthenes' time) to the methods of observation of ancient through early Ptolemaic Egypt. (The fact that Claudius Ptolemy's hoax could be tolerated by his contemporaries, illustrates the significant degeneration in scientific practice which had occurred since the deaths of Aristarchus, Eratosthenes, and Eratosthenes' correspondent Archimedes.) To gauge this, one might wisely take into account, Indo-European culture's knowledge of the long equinoctial solar-sidereal astronomical cycle, shown (by progression of positions of observed stellar constellations) to date from some time between 6,000 and 4,000 b.c. (within Orion), in Central Asia.

35. The conspicuous error in Toscanelli's map, is neither his estimated size of the planet, not the indicated distance to be spanned in crossing the Atlantic. The problem is Venetian lies respecting the distance across Asia to China and Japan, placing the latter in the middle of the United States.

36. The connection stated here is key to understanding the Lawrence Kubie's thesis set forth in his 1962 Daedalus piece, which we have referenced in a note, above. As matured and reflective sports fanatics will concede, "erotic" refers not only to explicitly sexual behavior, but to notions of power to dominate, and submission to power, and, more generally, to ideas associated with sense-perception, as opposed to ideas associated with cognition. This underlies certain more readily recognized connections which come to the surface in forms of sexual abuse, such as rape, sodomy, intra-family violence, or simply the forms of psychosexual impotence in which the sex-act is performed with little more than a "sex-as-power," animalist pleasure-seeking impulse, for domination or submission. In the instance of the "Don Juan," or "Macho" type, this may be expressed as a person who is either emotionally confused by, or even virtually incapable of, a human quality of enduring attachment to merely one woman. "Macho" Don Juan protests, with all the feigned sincerity of indignation such an inveterate confidence man might muster, "Me psycho-sexually impotent?: you have to be kidding!" In healthy states, the "erotic" impulse (eros) is associated with ideas within the domain of sense-perception; whereas, all ideas associated with cognition are associated with the emotional impulse of agape. The neurotically pathological characteristic of philosophical empiricism, neo-Kantian romanticism, and positivism, is typified in the extreme by the sexual history of such empiricists as Francis Bacon, Thomas Hobbes, and Jeremy Bentham. These three typify the neurotically confused state of mind essential to such philosophical currents. All of the ideas which are distinctively characteristic of Plato and of Christianity are within the domain of agape, as I Corinthians 13 denies the quality of "Christian" to any ostensibly worthy act, which is not generated and controlled by agape. Thus, the "Macho" type of neurotic responds to that challenge to his beliefs which is beyond what he senses he might be able to refute, not with reason, but with outbursts of an erotic quality of screaming, shouting, fist-waving rage. The "neurotic distortion of the creative process" which occupied Kubie's attention, is the result of the inappropriateness of the summoning of the erotic quality of emotional impulse, to address a challenge which requires the kind of ideas summonable only by the agapic impulse pecular to Platonic ideas.

37. This definition of the good, is congruent with Leibniz's definitions for the monad. See, notably, Monadology, 9-18, pp. 149-150 [footnote 1].

38. A few points of clarification must be supplied here, respecting the stages of the development, and related indebtednesses, of the author's progress to his present views on the subject of music. First, although the author's knowledge of lattice principles dates from his study of the work of Harvard's Birkhoff, during the late 1940's, he did not employ the theorem-lattice as a pedagogical approach to the principle of hypothesis until a middle 1950's manuscript examining problems of Operations Research from the standpoint of economic principles. In a sense, the author's views on motivic thorough-composition had perhaps a greater role in prompting the author to employ the pedagogy of theorem-lattices, than the other way around. By 1952, the author's views on motivic thorough-composition, were centered upon the traceable influence of Mozart's K.475 on Beethoven, Brahms, et al. This is typified by such matters, as the recognition of Brahms' direct quotation from this Bach-Mozart source in the C-minor (First) Symphony, and the direct quotation from the Adagio Sostenuto (measures 70-85) of Beethoven's Opus 106, as the motivic germ opening Brahms' Fourth (E-minor) Symphony (measures 2-19). During the same interval, 1948-1952, the author had chosen the characteristics of the composition of the German Classical lied, from Mozart through Brahms, as the key to all music, including all Classical instrumental compositions, and had emphasized the origins of music in the singing of ancient Classical poetry, and related principles of irony in Classical drama, especially Classicial tragedy. The next qualitative advance, as contrasted to gradual ones, came through collaboration with immediate associates and others, the others including, most emphatically, his dear friend, Professor Norbert Brainin, former Primarius of the Amadeus Quartet. In the first phase, 1979-1985, the emphasis was upon the implications of tuning from the standpoint of Florentine bel canto modes of voice-training. During that period, beginning 1981, the author projected the compilation of a text on the scientific principles underlying Classical musical composition, which became Book I (On the Human Singing Voice) of A Manual on the Rudiments of Tuning and Registration, ed. by John Sigerson and Kathy Wolfe (Washington, D.C.: Schiller Institute, 1992). In the preparation of the forthcoming Book II (On the motivic thorough-composition and the ensemble), Professor Brainin outlined his own discovery of approximately two decades, respecting the relationship between Joseph Haydn's launching of Motivführung with his own Opus 33 quartets, and the revolution in motivic thorough-composition which Mozart launched, from approximately 1782-1783 onward, in response to Haydn's program (e.g., Mozart's six quartets dedicated to Haydn). See, Lyndon H. LaRouche, Jr.,"Musical memory and thorough-composition," Executive Intelligence Review, Vol. 22, No. 35, Sept. 1, 1995, and the relevant addendum, "Norbert Brainin on Motivführung," Executive Intelligence Review, Vol. 22, No. 38, Sept. 22, 1995.

39. I.e., Critique of Judgment.

40. It is important to stress, that the subjunctive mood is not the grammatical forms with which its employment may, or may not be associated. The subjunctive mood is the mood of hypothesis, the mood of thought taking thought-processes as an object. Its Classical expression is the relevant literature of Greece, such as the Homeric epics, the great tragedies of Athens' Golden Age, and the dialogues of Plato. The type of Classical Greek literature which presents the actuality of the subjunctive mood (as distinct from a mere accident of conventions in grammatical forms) is a trio, of persons from two cities of different cultural heritage, interacting in a common setting, with one or more representatives of the pagan gods of Olympus. The actual events are shared in common, but those propositions, generated in response to the events, lead to theorems which are, respectively, mutally inconsistent. One character's, or the audience's, comparison of the differing mental processes leading to the different reactions, and related ultimate outcomes, is the actuality of the subjunctive mood. Hence, the dialogues of Plato are all written in the subjunctive mood.

41. In music, for example, the difference between a Classical and Romantic style of performance of a Classical composition (e.g., Mozart, Beethoven, Schubert, Schumann, Brahms) is implicit in conductor Wilhelm Furtwängler's instruction, to perform "between the notes." In the simplest degree, this requires that the performer express the counterpoint, rather than present a sensuous array of individual notes. To this end, the emphasis must be upon the motivic implications of the interval as an element of change, avoiding resort to erotic obsession with the utterance of the individual chord or note as such. Ultimately, it requires that each interval be performed with an eye to the hypothesis established by the concluding resolution of that developmental process which is the composition taken in its entirety. This applies not only to recognizing the proper relative tempi among movements, etc., as motivic considerations of the composition as a whole demand this; it prohibits decadently erotic emphasis upon uttering individual tones, in movements performed with exaggerated slowness for this purpose, and, on the contrary, excessive velocity, used to bury the meaninglessof the performance under a sensuous heap of haste. It means a hatred of misrepresenting compositions through resort to readings of portions of a Classical score, such as Schumann, as "passage work" imported to make the composer appeal more erotically to the taste of a decadent Manhattan audience. The same applies to Classical drama and poetry. In good art, there is no symbolism, but, rather, the expression of interdependent empyreal ideas and agapic passions, expressed by metaphor.

42. This is not to be confused with erotic qualities of manic elation. The subjective effect is "calming," directly opposed to manic. The increased capacity for action, is associated, metaphorically, with the notion of serenity and a source of "energy" for action. It suggests the quality of serenity in that great military commander who has achieved the appropriate capacity for what Clausewitz references in use of the term Entschlossenheit.

Part I

Part I Footnotes

Part II

Part II Footnotes

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