SCHILLER INSTITUTE

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

Vol 5, No 3
1996

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.

1.
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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