This article is reprinted from the Spring 1994 issue of FIDELIO Magazine.

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The adequate solution to the paradox of negentropy lies within the domain of a theory of knowledge, an epistemology. We proceed to that as follows.

It is useful now to introduce the relevant, subsidiary argument, that perhaps the most notable feature of my work in this field is that these discoveries were not already established standard as textbook knowledge long prior to my initial, 1948-1952 work in this area. The shocking fact is, that such properly obvious consequences of Riemann's and Cantor's combined contributions were left to be adduced by one of my then modest qualifications in mathematics. Situate this point in the appropriate terms of reference: If one takes into account the most recent 550 years of science, especially the indispensable internal political history of science, the irony of my discoveries is crucially, and most instructively anomalous; it is not rightly considered to be mysterious.

Similar anomalies have appeared in the history of science in the circumstance that the discovery in question has been implicitly forbidden by some more or less intimidating imposition of false axiomatic assumptions upon established institutions of learning, such as commonplace classroom opinion. In my own case, the root of such false, but commonplace opinion is, of course, ultimately traceable to the Venetian neo-Aristotelians of the late-fifteenth and sixteenth centuries; but, the circumstance bearing directly upon the irony of my successes are to be traced to the more recent, special U.S.A. conditions arising in mid-twentieth-century teaching since around the close of the nineteenth century.

To illustrate the kind of argument required: The combination of London-directed,⁴¹ French Jacobin lunacy, and, later, conditions imposed by the 1814 Congress of Vienna, ended France's more than two centuries of supremacy in science and technology.⁴² Similarly, Anglo-Saxon empiricism's subjugation of both the U.S.A. and continental European classrooms came about chiefly through the political hegemony institutionalized under the Versailles and later Yalta-Potsdam peace agreements. The same political logic applies to changes in Twentieth Century scientific opinion within the United States.

Until the close of the nineteenth century, at first French, and then, later, German world-leadership in science had been the standard of leading educational and governmental institutions. The cases of Bache⁴³ and Agassiz⁴⁴ are illustrative of the influence of Gauss in particular.⁴⁵ At the turn of this century there occurred the onset of a sweeping change, toward radical empiricism in the cultural paradigms of relevant U.S. institutions. The concurrence of President Eliot at Harvard University, of Jim Crow law, and the nearly successive U.S. presidencies of Confederacy admirers Theodore Roosevelt and Woodrow Wilson, were all cut from the same piece of treasonous political cloth. The patriotic, economic-protectionist tradition of Washington, Monroe, Adams, and Lincoln was supplanted once again by the “free trade” and related dogmas of those presidents upon whom Britain's villainous Lord Palmerston had most relied, Pierce and Buchanan. At the onset of the century, William James and the British Fabian Society's John Dewey had been unleashed to ruin U.S. public education. Gradually, scientists in the Bache tradition, such as Chicago's Harkins,⁴⁶ were supplanted, at least in large degree, by a dominant role of increasingly radical expressions of empiricism.

These changes in culture fostered corresponding effects in the teaching and practice of science, of political economy, of philosophy, and of history within the world's increasingly hegemonic, Anglo-Saxon Establishment institutions. That politically aversive indoctrination of most among the elites of the world's nations trickled down to its effects upon the opinion-shaping in the classrooms, and among the populations generally. The specific relevance of this for the case at hand is signalled by comparing this twentieth century imperial rise of empiricism to a related pogrom against Georg Cantor by the cronies of Leopold Kronecker.⁴⁷ That shameful political lynching of Cantor was a correlative of the same empiricist mob's malice shown so prominently by Bertrand Russell and other members of the Cambridge Apostles in their continuation of the earlier efforts of Kelvin, Helmholz, Maxwell, and Rayleigh to bury the principal achievements of Riemann, Weber, and Weierstrass.⁴⁸

But for such specific historical circumstances, all that which is in my original contributions would have been well established knowledge long before my initial work of 1948-1952. Consequently, my role has resembled that of the rude little boy in Hans Christian Andersen's celebrated tale of “The Emperor's New Suit of Clothes.” Beginning 1948-1952, I worked to fill a vacuum which had been created almost solely through a pervasive, political corruption of prevailing classroom opinion.

In this circumstance, looking at that retrospectively today, what I did was to extend what I had learned from the hand of Leibniz, to meet the challenge of refuting Wiener's “information theory.” By aid of rereading Riemann's dissertation through the transfinite eyes of Cantor, I developed a fresh overview of the theory of knowledge. This fresh overview, on which I report now, was required to resolve the remaining paradoxes posed by my locating of negentropy elementarily within the higher domain of the ontologically transfinite.

What is now to be said here may be read in part as parallel to Leibniz's 1695 Système Nouveau de la Nature.⁴⁹ The neo-Aristotelian system of deductive sense-certainty, as introduced to the sixteenth century by the gnostic Venetian associates of Gasparo Contarini,⁵⁰ is self-obliged by its own formalities to reduce everything to some smallest, discrete, finite, elementary particles. This system regards sense-impressions as virtually mirror-images of a reality outside our skins. Within such a linear materialist system, as for Aristotle himself, neither an intelligible notion of creation, nor of living processes, is logically possible; entropy rules always, everywhere. Formally, for Aristotle, his own existence is, speaking formally, like Newton's “Clock-Winder” universe, a logical-mathematical impossibility. If, according to his own system, the historical Aristotle ever existed, that would be sufficient proof that his system had no right to exist. If the prescribed system of knowledge implicitly prohibits the existence of the knower, that system has no right to exist.

The remedy for this fallacy of Aristotle's system was already defined by Plato before the completion of Aristotle's own studies at the Athens School of Rhetoric, the latter headed by the Sophist Isocrates. Negatively, in the sense of Plato's dialectical method of Socratic negation, we can demonstrate rigorously the necessity for the ontological elementarity of negentropy, i.e., for the Platonic elementarity of Heraclitus' notion of universal change. We can also represent this by means of a rigorously Platonic approach to use of constructive geometry, as Cusa thus treated the paradox of Archimedean quadrature. However, we cannot show this positively by means of any among today's generally accepted forms of classroom mathematics; this difficulty is, once again, an echo of Newton's “Clock-Winder” paradox.

We cannot render this notion of negentropic elementarity intelligible from the standpoint of sense-certainty. That is key to the formal fallacy permeating that Boltzmann theorem employed by Norbert Wiener's “information theory:” that is also the form of the sundry kindred blunders of John von Neumann, on economy and the human mind. By means of what faculty can we overcome such paradoxes? Plato provided the general approach needed, but an adequate solution can be achieved only from the standpoint of the Leibniz science of physical economy. The contributions of Cantor, Riemann, and so on, were indispensable, Platonic steps toward my solution of the crucial, relevant issues of an intelligible theory of knowledge; but, until these preliminary results were situated within the domain of physical economy, no adequate proof of the principles of knowledge is accessible.

The form of this required solution is indicated by treating this issue in first approximation in its aspect as a problem in physics. A valid axiomatic-revolutionary discovery in natural philosophy is expressed, as customary, in the form of one or more crucial-experimental designs, experiments which demonstrate the principle of the discovery, each in a crucial way. Each such successful design, adequately refined, supplies a new principle to be incorporated usefully in either sundry machine-tool designs, or some similar use. The application of such designs, accompanied by the transmission of the corresponding new knowledge, expressed as use of improved tools of production, improved products, and so on, results in an increase in the physical productive powers of labor, per capita and per square kilometer. In other words, an increase in the potential population density of mankind.

So, the continued successful existence of mankind⁵¹ relies upon the mental processes which generate and replicate valid, newly-discovered, axiomatic-revolutionary changes in scientific and related knowledge. It is by adopting such manifestly creative states of mind, instead of naive sense-certainties, as the subject of conscious reflection, that we may access the pathway leading to the required theory of knowledge. This policy was the pivotal conception which emerged during my inquiries of the 1948-1952 interval, guiding me to my conclusions, through the pathways of Cantor and Riemann.

This emerging overview of the most crucial problem to be solved, prompted me to turn my earlier notions of geometry upside-down. Rather than build up a geometry, by extension, from primitive, linear sorts of axiomatic formal and ontological assumptions, take the reverse course. That which efficiently bounds externally as the relative macrocosm, is to be seen as the relatively elementary. It is the whole so defined which determines the part. This supplied me a corrected notion of the statement: “The whole is always greater than the sum of its parts.” This view of the axiomatic structure of geometry-in-general freed my conscience from any further reliance upon accepted forms of classroom mathematics.

The realization that, axiomatically, none of the relevant epistemological paradoxes I was facing could find a model representation in terms of any presently accepted notion of a theory of functions, forced me to focus upon the internal history of mathematical physics, in search of some notion of an ordering-principle among axiomatic-revolutionary discoveries. The obvious place to begin a first attempt is the discovery addressed inclusively, and crucially, in Riemann's habilitation dissertation, the famous ubiquitous theorem of Pythagoras. After all, obviously, the thirteen books of the Elements⁵² bring the student from reconstructing that theorem, through, step by step, to Plato's five regular solids inscribed within a sphere. Give up those ordinary notions of denumerable ordering central to all algebraic and transcendental functions; seek a more modest notion of necessary ordering. For every axiomatic-revolutionary discovery, certain other such discoveries are necessary predecessor, and every valid such discovery is a necessary successor of others. Every professionally qualified teacher of mathematical physics employs that guiding notion in constructing efficient lesson-plans. This approach to, implicitly, teaching mathematics and physics, shifts the focus from learning theorems and their formal proofs, to replicating in the student's mind the experience of each crucial, original axiomatic-revolutionary discovery as this occurred, in essence, in the original case, in the mind of the putatively original discoverer. Instead of treating theorems as the principal subject, make the subject the process of axiomatic-revolutionary discovery as replicably experienced by the student in each case. Make that moment of Platonic hypothesis-formation the subject.

Then, next, find the ordering-principle—the Cantorian equivalence, type—among a series of such successful acts of hypothesis-formation. Determine, according to such an adduced equivalence, the type of ordering of a network-sequence of such hypotheses according to the rule of “necessary predecessor”/“necessary successor.”

The following step must be to render that adduced ordering-principle, that type the intelligible subject of conscious comprehension. This is done, in first approximation, by contrasting this scientific method, as a Platonic method, to Aristotelian formalism. The recognition of the incurable fallacy of all Aristotelian and analogous argument, from this standpoint, is the beginning of a true epistemological insight into the required principles governing a scientific method. That view of the type of ordered hypotheses, is rendering the higher hypothesis an intelligible subject of conscious comprehension, in turn. It is at this stage of the process of inquiry, that the crucial features of my definition of negentropy become adequately intelligible; the essential paradox is thus solved.

Reconsider the steps just described.

In a preliminary way, this pedagogical approach to the internal history of science has a well-established basis in Christian Classical humanist secondary education. The case of Groote's Brothers of the Common Life, and, later the Schiller-Humboldt educational reforms, are obvious references.⁵³ These great Christian humanist educational reforms were reflected also, if in a diluted way, in the later examples of pre-1970, precatastrophe, U.S. secondary education.⁵⁴ In the better schools, as reflected in traditional professional scientific practice still, the student comes to know an axiomatic-revolutionary, or related discovery of principle by both its approximate date of occurrence, and the personal name (plus a short biographical sketch, perhaps) of the discoverer. I emphasize: that discoverer as an individual thinking person, whose discovery today's student can master only by replicating the mental process of discovery which occurred in that historic moment of discovery by the original discoverer. As already noted, a teacher's good lesson-plan must reflect some degree of insight into the matter of arranging topics of principle according to “necessary predecessor”/“necessary successor.” The crucial difference of emphasis proposed, relative to such established classroom precedents, is to shift the emphasis from getting to the accepted proof of the theorem, to concentration upon the internal features of the mental process of formulating the relevant hypothesis.

Thus, to each valid, axiomatic-revolutionary discovery assign the name of hypothesis. As said above, assign to the idea of an equivalence in ordering of necessary successive hypotheses, an higher hypothesis.

In the classroom, and here, too, the notion of hypothesis is brought into clearer focus, by contrasting hypothesis with the theorem-proofs of a formal, deductive theorem-lattice. In the latter case, every provable theorem of that more or less indefinitely expandable array will be deductively consistent with a set of axioms and postulates which underlies the initial germ-kernel of theorems of that lattice.

Let us denote such deductive consistency of formal theorem-lattices by a term borrowed from the customary usage of our adversaries, “hereditary principle.”⁵⁵ Every possible theorem of a consistent theorem-lattice will be nothing but a reflection of the original body of “genetic material,” the underlying set of axioms and postulates. The Platonic hypothesis, generated by the Platonic dialectical method of Socratic negation, overturns one or more of the axioms and postulates of any theorem-lattice of reference.

Thus, for the hereditary form of theorem-lattice, the theorem-proof of deductive consistency is the characteristic mental activity of the student. Once we introduce true discovery, and therefore hypothesis, theorem-proof is submerged; creative mental activity as such is everything. It is in this latter domain of conscious thought, and only here, that my notion of negentropy becomes adequately intelligible. The challenge immediately presented at that juncture in our argument is the following: If we abandon formal theorem-proof, as we must (since we are replacing axioms or postulates), what is the nature of proof of hypothesis? The required proof has two fundamentally distinct aspects, two aspects which ultimately dissolve into one another, but not at first consideration.

For the student, the first kind of proof encountered is study of crucial discoveries from the past. Once that student has adduced a sense of the equivalence (higher hypothesis) of valid past discoveries of an axiomatic-revolutionary quality, the student's first resort, at each confronting of an unfamiliar such discovery, is to test that discovery for its quality of Cantorian equivalence. Later, that student may acquire a second notion of proof, a proof rooted in the Leibnizian notion of a science of physical economy. If an hypothesis satisfies the standard of equivalence, and also increases implicitly humanity's potential population-density, it is relatively valid.

These two proofs merge into one historically. The equivalence among past discoveries (hypotheses) reflects the test of an implicit increase of mankind's potential population-density.

That is the general principle of the relevant theory of human knowledge, but only in one aspect, natural science.

This brings us to the last of the principal issues posed by Wiener's “information theory,” to the subject of communication of ideas. We focus upon the idea of a language in its most general sense of a medium for communicable aspects of ideas. Within that setting, we treat the crucial special case of ideas which, by their nature, cannot be communicated literally. Consider the case for those ideas which correspond to Platonic hypothesis.

Since all ideas are subsumed by the notion of metaphorical communication of ideas of hypothesis, and, since language as a whole is bounded thus by those same principles, the notion of metaphorical provocation of hypothesis is the crucial case for all communication.

In the instance of every new Platonic hypothesis, language appears primarily as a mode of posing paradoxes to such effect that a speaker's new idea, which cannot be identified literally in existing language, can be replicated nonetheless in the mind of the hearer.⁵⁶ This leads us to the broader proposition, that ideas are not primarily sensual imageries, but are, primarily, elementarily, those valid, intelligible conceptions which cannot be named at first communication by a recognizable term of established usage. That is to say, that all valid ideas first appeared to existing language in no other form of communication but metaphor. Among such new ideas, the highest class, subsuming all other classes, is that of axiomatic-revolutionary ideas. Ideas of this class refer to a quality of sovereign mental activity within the speaker, an idea whose form is that of, variously, Platonic hypothesis, higher hypothesis, or hypothesizing the higher hypothesis. For reasons outlined above, all ideas were introduced to language first in the guise of metaphor. Then, and, even after many generations of use, those ideas were, and are still subject to those same functional notions of idea demonstrated by the case for Platonic hypothesis.

Perhaps the best illustration of metaphor, is the paradoxical quality of Plato's Parmenides. The same principle so shown by the Parmenides, is employed as the central feature of Nicolaus of Cusa's original solution to the ontological paradox of Archimedean quadrature.⁵⁷ The metaphor is the ontologically required, indivisible concept which unifies a paradoxically juxtaposed set of predicates for the case the latter reflect the same function. For Plato's Parmenides, the indivisible one is always existent in the ontological form of change, Heraclitus's ontologically unique quality of universally elementary change. The form of this change may be compared to Cantor's principle of transfinite equivalence; for Cantor's mathematics, Heraclitus's change is the highest type in Plato's universal Becoming. In Cusa's title De Docta Ignorantia⁵⁸ and De Circuli Quadratura,⁵⁹ the passage from the “Parmenides paradox,” of an endless series of regular polygons, to the circular perimeter as an ontologically higher form of an axiomatic existence, is characterized by a shift from Euclidean space, to the higher, non-algebraic domain of space-time; the axiomatic least-action, or isoperimetric definition of the circle is closed action expressing a constant change, and equivalence, a higher type than formal Euclidean geometry, or algebra.

In both cases, Plato's Parmenides and Cusa's axiomatic-revolutionary treatment of quadrature, we are presented with examples of a true metaphor in approximately the barest-bones form of representation. Cusa's non-algebraic generation of the circle, as constant change, is the metaphor represented by Archimedean quadrature. That circle's existence cannot be competently defined in the axiomatic framework of ordinary Euclidean geometry; to construct a circle, we must employ a ruse of construction excluded from the underlying set of axioms and postulates of Euclidean theorem-lattice. We must employ rotation, as one does by drawing the circle with a compass. Rotation is the ordering of action in non-algebraic space-time, not Euclidean space.

This cannot be brushed aside with the argument that I am stretching a point here. There is a four-hundred-fifty year, connected historical development, from the origin of Cusa's discovery, through Leonardo da Vinci, Kepler, Fermat, Huygens, Leibniz, Bernoulli, and then to Hermite, et al. at the close of the nineteenth century, to define rigorously the transcendental distinction of &Mac185;.⁶⁰ It is often, that proverbial, smug hand-waving at the blackboard is employed to evade even the most devastatingly crucial issues. Such has been the long, stubborn refusal to acknowledge that rotation is, axiomatically, ontologically external to a formal Euclidean theorem-lattice, or, as Augustin Cauchy's calculus has often been read to evade, the truth is that asymptotic limits are not theorems of the theorem-lattice employed to describe the relevant function.

All formal language, such as a grammatically literate spoken language, is laden with equivalent axiomatically ontological limits. Thus, contrary to the nominalists, all important ideas are introduced to a subsequent state of communicable recognition by means of initially metaphorical identification.

Those were the considerations, although more crudely formulated at the time, which obliged me to include in my 1948-1952 work on negentropy a corresponding treatment of the principal characteristics of metaphor in communication. For the purpose of this study, I chose then musical settings of poetry which had been composed during the 1780-1900 interval. The composers selected were chiefly Mozart, Beethoven, Schubert, Franz, Schumann, Loewe, Brahms, and Hugo Wolf. The central sub-topic of this study was two or more alternative musical settings of the same poem. The poets upon whom I concentrated were Goethe and Heine. The focus was upon the use of musical forms of metaphor in relationship to the natural musical vocalization in hearing and the poetic enunciation of the spoken line.

Later, beginning 1982, at my urging, aspects of my 1952 results were reconstructed with improvements by some of my musician associates. The latter study, of the 1982-1991 interval, is reported in the recently published Book I of A Manual on the Rudiments of Tuning and Registration.⁶¹ The object of both this latter and the original study was to show the connection between creativity per se's expressions in both the domain of natural philosophy and Classical art-forms. To treat the implications of negentropy for communications in general, thus to refute “information theory” adequately, it was necessary to demonstrate a relevant degree of equivalence of creativity per se in one medium to that in the other.

As I have identified this recently in “History as Science,”⁶² the case of the Indo-European language family shows language in general to be premised centrally upon three elements.

First, the spoken language as typified by reading Classic Vedic hymns and Sanskrit from the standpoint of philologist Panini.⁶³ This working assumption of the 1948-1952 period was referenced then chiefly to the Classical English-language poetry, from Shakespeare through Shelley and Keats. Years later, the argument was given a selected crucial test against the Italian of Dante Alighieri's Commedia.

Second, the visual space-time field of geometry. This correlates with the most essential feature of spoken action, the transitive verb. By this use of the verb, we are able to locate qualities of transformation in space-time.

Third, music. All spoken language is governed by musical principles, even in the rudest of violations of those principles.⁶⁴ The application of this to choral singing among naturally determined different species of singing voices is again bel canto polyphony. Bel canto polyphony determines faultlessly a well-tempered tuning of the temper used by Bach, Mozart, Beethoven. This is determined by the natural harmonics of the biological speaking and singing apparatus of human beings all as members of but a single species. Thus, the system of well-tempered, Classical,⁶⁵ bel canto polyphony was not an historical accident of taste preferred only by some people, in some time and place. This was the musical medium implicitly ordained by God; it is implicitly imbedded in the genotype common to all members of the human species, past and present. The same argument governs the principles of vocalization of a spoken form of language.⁶⁶ Music is derived from the natural vocalization of Classical forms of poetry, as the Vedic hymns typify this general case.

It should be interpolated here, as a relevant point to be stressed. “Text” in the sense the term is used by “Deconstructionists” such as Jacques Derrida, does not—or, certainly should not—exist.⁶⁷ As the pagan god was reminded, his invention of writing was useful, with some potentially disastrous side-effects, of which Derrida is one. Written texts should be heard by the writer and reader as it is being read, or written. The music—the vocalization of the spoken word, as shadowed on the written page—is an integral part of speech, as the geometry of space-time is also an integral part of speech, as Plato was first to show, as Leonardo da Vinci and Kepler later emphasized.

Fifty years ago the following point was not considered further than our present account has gone up to this moment. Even this much of the treatment of relevant musical matters so far, already includes some supporting material dating from times later than 1952. This, and the point now to be added respecting Plato's regular solids, are included here as they provide crucial supporting evidence for those conclusions respecting the theory of knowledge already reached, if on a narrower basis, forty years ago.

The Classical Greeks, who knew well-tempering in Plato's time,⁶⁸ recognized, more broadly, that natural beauty in art was characterized, in vision and in hearing, by harmonic orderings consonant with those of living processes. The whole design of the Classical Athens Acropolis attests to this.⁶⁹ Plato documents this.⁷⁰ Two key followers of Nicolaus of Cusa, Luca Pacioli and Leonardo da Vinci, demonstrate⁷¹ that; Johannes Kepler bases the beginnings of a comprehensive mathematical physics upon the common harmonic characteristics of vision, music, and Plato's five regular solids. In modern language, this current in mathematical physics indicates Kepler to be the initiator (guided by Pacioli and da Vinci) of what is most fairly named today “quantum field theory.”⁷²

We are speaking of a theory of knowledge. We are gauging these queries against Riemann's referenced warning, on the subject of the metrical features of a continuous manifold. Thus: how can man come to know the crucial implications of the five Platonic solids? What is the nature of the available evidence on this matter? What was available to Plato's Classical Athens?

We have referenced the Acropolis. The Greeks knew the principles as artistic, and architectural proportions according to an harmonics of circular sections. They recognized, thus, as natural visual beauty harmonic orderings consonant with that Golden Section which is characteristic of Plato's five solids. This Golden Section-pivoted harmonics was recognized, as by da Vinci⁷³ and Kepler⁷⁴ later, as that characteristic which distinguished living from non-living processes. It is the metrical characteristic of actions governed by negentropy, as I defined negentropy, earlier here, and forty-odd years ago. The Golden Section was also recognized by Plato, for example,⁷⁵ as the characteristic of musical training. We have just considered the natural basis for that well-tempered system of bel canto polyphony, congruent with the Golden Section, which is implicitly determined by the human genotype. In short, vision and hearing are the imbedded metrical guides to our communicable forms of representation of our universe, in terms of the Golden Section's implications. Nonetheless, it is in the implicitly well-tempered underlay of the determination of a least-action mode of vocalized speech and singing, where lies the aspect of language in which this metrical principle of thinking is imbedded. The well-tempered, bel canto polyphonic domain is the model for a quantum field, the model for a quantum-field conception of the metrical qualities of our physical space-time universe.

That leads directly to the principal point respecting a theory of knowledge.

Knowledge is accessible to mankind only in the forms corresponding to a theory of Cantorian types, in terms of hypothesis, higher hypothesis, and hypothesizing the higher hypothesis. We can know only change, the notion of universal elementarity of change which is associated with the writings of Heraclitus and Plato. That change is known to us in terms of hypothesis, or, in Cantor's terms, types.

However, the distinction between truthfulness and falsehood, respecting principles of nature, requires an experiment, an experiment which can be of but one type, physical economy as the practice of maintaining progress in increasing the potential population-density of mankind. This is uniquely the form of experiment which tests the relative validity of those choices of higher hypothesis (types) which govern the generation of those axiomatic-revolutionary discoveries which foster increase of potential population-density.

Thus, the popularized notion of “objective science” is so dangerously misleading that we must regard it as absurd, or even worse. Knowledge is subjective, in the sense that we must act upon principles of discovery which can be known to us only by proving their validity in practice in terms of the benefit to mankind as a whole, a benefit which is crucially centered upon the requirement of the continuing increase in the potential population- density of our species as a whole.

The source of our personal knowledge to this effect, is the reliving of history from this standpoint. The idea of a Christian Classical humanist education, such as that of Groote's Brothers of the Common Life, or the Schiller-Humboldt reforms, the reliving of moments of great, axiomatic-revolutionary discovery, as if to replicate that moment from within the mind of the original discoverer in one's own mind, is a typification of the relevant way in which the child and youth must be developed morally and formally at the same time. By means of such an education, emphasizing the principles stated here, the mind of the child and youth, repeatedly experiencing the replication of valid axiomatic-revolutionary hypotheses in this way, is enabled to apply the same mental capacity, of hypothesizing, to the ordering (“necessary predecessor,”/“necessary successor”), the Cantorian equivalence of a series of valid hypotheses. Thus, this latter equivalence, or higher hypothesis, is the proper referent for the term scientific method. Since conflicting scientific methods may be compared by the same method of hypotheses, the student's mind is equipped, and thus impelled to enter into consciously hypothesizing the higher hypothesis. This activity within the individual defines a self-critical capability in respect to all aspects of his or her individual practice, and to observing the manifest mental processes and characteristic practice of others, including entire nations and cultures, past, present, and prospective future. Thus, by this developed subjective mental discipline, which is the proper notion of the scientific faculty, the individual judges relative truth, relative falsehood, right and wrong, superior and inferior qualities, and kindred judgment of those qualities for which mere “matters of taste” are not to be tolerated by a people which prizes its own continued moral fitness to survive. From this relative knowledge, we are assured of a few things of an essential practical importance respecting absolute matters.

For example, Cantor references this domain by equating his own transfinite to Plato's Becoming, and his absolute to Plato's Good. Becoming is physical space-time, in which development occurs through change. Absolute, or Good, is reflected in the process of Becoming, as a process of perfecting, conceived as a perfected instant, a One, everywhere more than co-extensive with the Becoming. That said, return to the Becoming, and to those notions which have a relatively changeless quality, relative to the marginal uncertainty of approximations. Once we grasp the idea, that man is distinguished absolutely above all other living creatures, solely by our willful capacity for effecting voluntarily axiomatic-revolutionary improvements, increases in mankind's command over nature, that voluntary creative activity, the activity of Platonic hypothesis, that axiomatic-revolutionary activity, compared with the resulting change in man's per-capita power over nature, is the phenomenon to which all rational employment of the term “knowledge” is referenced.

It is not the observed relations among sense phenomena, which is the subject of knowledge. The proximate subject of knowledge is the changes in sensory phenomena's patterns of behavior which have been, are being effected cumultatively, historically, through the creative faculty of hypothesis generation. It is the relationship of such changes to increases in potential population-density, and to man's breaking through barriers of technology, to make richly habitable the deserts, or barren planets beyond our own, which test, historically to present date, those adducible principles of higher hypothesis which are thus shown to be the most reliable known choices of guides to truth respecting man's relationship to nature. All along, there are certain virtually absolute social truths, with the moral force of natural law,⁷⁶ embedded in the cumulative evidence of historically successful, Platonic higher hypothesis.

First, the sacredness and lawful sovereignty of the individual person's life, by reason of that creative faculty expressed as Platonic hypothesis.

Second, the subsumed sacredness of the parental household, for its interdependent loving (agapic⁷⁷) functions of procreation and nurture of new, individual personalities through the ages of infancy, childhood and youth, to blossoming as a young adult with developed creative powers.

Third, the derived sacredness and functions of those institutions we know as republics under natural law, those more powerful, less mortal agencies whose function is to defend the sacredness of individual creative life, to defend the institution of the parental household, and to foster and protect the benefits of creative individual work to the advantage of all present and future generations of mankind.

We now come to certain concluding points of summation so crucially important, that I must set them somewhat apart from the immediately preceding pages of this concluding section. The first of these is my fresh proof of the monad.

Consider, from the standpoint of language as I have defined language: How do we know with the authority of necessary and sufficient reason, that man possesses an individual soul? It is most appropriate to state the case of the monad in that form, because for Gasparo Contarini's Aristotelian cronies, such as the exemplary Pomponazzi, for all consistent Aristotelians, the individual soul could not exist. Thus, for all empiricists, and other neo-Aristotelians, the individual soul does not exist, but rather a “bolshevik,” e.g., a “collective soul.” For whomever rejects the notion of Platonic hypothesis, the individual soul cannot exist; that is the functional connection I am stressing here. -

Turn to our earlier treatment of the subject of metaphor.⁷⁸ Any idea, in its guise either as an original discovery, or in its transmission de novo as it might have been an original discovery, cannot be transmitted as a literal intent of the language-medium employed, but only as the intent which reposes in the individual user of that language. The idea cannot be addressed by any formal analysis of the language-medium employed. This predicament is a consequence of the fact that any true discovery corresponds to a formally absolute discontinuity in any system of deductive representation previously employed. Relative to language as such, true ideas lie only in the individual, creative mental processes of each person participating in the communication.

This illustrates, and also demonstrates implicitly the relationship between a true, i.e., negentropic continuous manifold and individual existence of the form shown as the originally metaphorical character of all communicated ideas. The truth on this point has been right under everyone's nose for millennia past. Here lies the kernel of Leibniz's Monadology, and my own. Here lies the key to exposure of a politically corrupted Leonhard Euler's perversely falsified attack upon Leibniz's Monadology.⁷⁹ The crucial point here is this; no idea corresponding to a Platonic hypothesis may be communicated to another person except as metaphor; no language can explicitly, literally transmit a true idea. Ideas are transmitted by aid of use of language, but this in a manner comparable to the common features of Plato's Parmenides and Cusa's solution for the paradox of quadrature. Ideas do not exist among individuals, but only within individuals. They exist with individuals only by being generated de novo within each person. They may be communicated only by use of paradox, i.e., metaphor, to provoke the replication of the original generation of the idea within, and by means of the sovereignly individual creative mental, hypothesis-generating processes of that individual person.

That shows us the following. By virtue of the creative-mental, hypothesis-generating processes of the person, each and all persons are singularities within, of the physical space-time domain. They are higher monads. That point is crucial. This next is also crucial.

The form of both higher hypothesis, and hypothesizing the higher hypothesis, is the form of negentropy as I have defined negentropy in opposition to Wiener et al. Thus, to take higher hypothesis as a subject of conscious reflection is to be conscious thought-object.⁸⁰

This next is also crucial, similarly.

Also, that which defines the individual person as having intelligibly a personal soul, is the principle of Platonic hypothesis. To wit: the reason Aristotelians could never solve, or even comprehend the Parmenides paradox is not only that the joke against the Eleatics is equally applicable to Aristotle and to Sophists generally. The reason no language could communicate ideas literally is that ideas are generated by functions of discontinuities, that ideas are characteristically of the domain of higher transfinite types. This is the characteristic of negentropy; this is also the proof of the uniqueness of the individuality of the monad, of the person.

This next, then, is also crucial.

The idea of a true continuum must be nothing other than a continuous function of hypothesis-generation, an higher hypothesis. That higher hypothesis must be of the characteristic form of negentropy, a form equivalent to the verb “to create.”

This next crucial argument follows.

All true human knowledge is of the form of hypothesizing the higher hypothesis. Thus the forms of this process of generating knowledge are the forms equivalent to knowledge of the real world, that real world which is mankind increasing its per-capita power over physical space-time. That increasing is the equivalence of the higher hypothesis as itself a process. That process, taken as a subject of willful consciousness, is human knowledge, is science in the most comprehensive meaning of the term science since the work of Cusa and Leibniz.

Next, the crucial issue here: that which is elementary within the process of conscious knowledge, defined in this way, is the idea which corresponds to what is elementary in that transfinite universe of Becoming which lies outside our skins.

From the side of language which corresponds to geometry, metaphor addresses a universe which is elementarily negentropic change. This view of elementarity, opposite to that of the neo-Aristotelian materialists Bacon, Galileo, Newton, et al., is the sure-footed advantage gained by shifting consciousness from obsessive fixation upon sense-certainties, to a consciously critical examination of those internal mental processes by means of which supposed, and real knowledge is generated. That is the shift from the blind, mystical materialist faith in the elementary particles of Democritus and Lucretius, to the elementary reality of change as such. This is a formal solution for the continuum paradox. Summarize that solution as follows. In place of simply a Platonic view of Heraclitus' “nothing is permanent but change,” say “Nothing is permanent but change subsumed by continuing negentropic action,” defining negentropy as I have defined it in opposition to the statistical vulgarization employed by modern, post-Mach positivists⁸¹ such as Wiener and von Neumann.

To restate the underlying, applicable argument from the domain of the theory of knowledge, knowledge is a term properly restricted in use to identify our own minds' conscious image of those of its own cognitive processes which, as a Cantorian type, account for the increase historically of man's increased power over nature, per-capita and per-square kilometer of our planet's surface.

This leaves one correlated topic of language to be considered at this juncture, the notion of the quantum field, as that notion is to be traced from Plato's treatment of the five Platonic solids, through the modern work of Pacioli,⁸² da Vinci⁸³, and Kepler.⁸⁴ The special connections to language now to be stressed here, is the fact that the principles of well-tempered polyphony were already well-tempered quantum field is already a natural characteristic of the mental image of our speaking and hearing any spoken (or, sung) language. This heard characteristic of those language images correlates to such expressions as the Golden Section with the visual, i.e., geometric facet of language. In the field of vision, this notion of quantum field is also associated with the notion of qualities of color attributed uniquely to respectively partitioned sectors of an ostensibly continuously defined frequency-domain of the visible field. We may thus speak, in this sense, of innate ideas, ideas which appear to us as comprehensible, intelligible ideas only from that higher consciousness of our own conscious processes which is Plato's hypothesizing the higher hypothesis.

Thus, the notions of monad, negentropy, and quantum field are innate ideas whose existence and nature are susceptible of being rendered intelligible to us, if we look at the use of language as a medium for generating those forms of metaphor needed to communicate valid, genuinely creative discoveries of principle by individual persons. If we employ the contributions of such figures as Plato, Cusa, Leibniz, and Cantor to assist us in making ourselves conscious of our own conscious processes, in terms of hypothesis, higher hypothesis, and hypothesizing the higher hypothesis these innate and related ideas are made intelligible to us. To the degree the human creative processes have been educated, through aid of reliving original acts of creative discovery over a long span of history, to define higher hypothesis governing new discoveries of principle for human practice, that individual mind, seeing its own relevant conscious activity of hypothesis-generation in that way, in that context, is seeing there a mirror of the lawful universality of our universe in its aspect as Platonic Becoming. It is in that view of matters that proper notions of knowledge in general, and scientific principles more narrowly, are to be adduced.

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41. At the time of the French Revolution, Jeremy Bentham (1748-1832) was employed by British East India Company executive and British Prime Minister Shelburne to run a "radical writers shop" at Shelburne's Bowood estate. Bentham and another East India Company operative, Samuel Romilly, penned many of the speeches that were delivered by Jacobins Marat and Danton during the height of the Paris revolt. It was essential for Shelburne and Bentham that the French republican, pro-American forces be crushed, and France be prevented from adopting a constitutional form of government modeled on the U.S. Constitution. Thus, while supporting the ultra-monarchist forces around Count Mirabeau, the British East India Company simultaneously provided covert financial aid to the Jacobins. Records of payments to Marat, Danton, and other Jacobin leaders are still on file at the British Museum.

42. The systematic destruction of France's Ecole Polytechnique is a leading example of how the Congress of Vienna's cultural policies were imposed. The Ecole had been the world's leading and most vigorous center of advancement of the physical sciences during the 1794-1814 period, under the leadership of its founder, the great Gaspard Monge. Through political intervention, Pierre Simon, Marquis de LaPlace and LaPlace's protégé Augustin Cauchy were assigned to destroy the Ecole's instructional program, exemplified in the notorious cases of Niels Heinrik Abel and Evariste Galois, both of whose work was first suppressed and then plagiarized, following the victims' early deaths. LaPlace's first act in this démarché was to organize the expulsion of Monge. Despite the continued, if much reduced, influence of the collaborators of Monge and Lazare Carnot in France, French science slipped rapidly from its preeminent position worldwide, to a poor second, as Germany's scientific ascendency emerged under the tutelage of the Humboldt brothers and leadership of circles associated with Carl Gauss during the 1820's. See Felix Klein, Development of Mathematics in the Nineteenth Century, trans. by M. Ackerman (Brookline, Mass.: Math Science Press, 1979); see also E.T. Bell, Men of Mathematics (New York: Simon & Schuster, 1937).

43. Alexander Dallas Bache (1806-1867), a brilliant graduate of West Point, carried the prestigious name and tradition of his greatgrandfather Benjamin Franklin. During the 1820's and 1830's, nationalist strategists in Franklin's old Philadelphia political machine (led by Nicholas Biddle, the president of the Bank of the United States, publisher Mathew Carey, and German emigré economist Friedrich List) successfully organized the initial industrialization of the U.S. In 1837, Biddle sent Bache to Europe to work with scientists and educational leaders, including Carl Gauss, Wilhelm Weber, and Alexander von Humboldt. Back in the U.S., Bache formed a patriotic group of the best American scientists, known as the "Lazzaroni" (Italian for "beggars"), in close cooperation with the German and allied French scientists. Bache's group designed and organized the U.S. Naval Academy. As head of the U.S. Coast and Geodetic Survey, Bache was chief strategist for the emergence of an advanced U.S. military-industrial capability, and was a leading advisor on intelligence to President Abraham Lincoln.

44. Louis Agassiz (1807-1873), leading zoologist and geologist of the nineteenth century, and one of the greatest naturalists of all time. He was born in Switzerland, trained in Germany at the University of Erlangen, and later worked with the leading French naturalist, Cuvier. In 1846, Agassiz moved to the United States and, as chief professor of the Harvard Lawrence Scientific School, he become a leading member of Alexander Dallas Bache's "Lazzaroni." Together with Admiral Charles Henry Davis, Bache, and Joseph Henry, Agassiz helped found the U.S. National Academy of Sciences in 1863. See his Contributions to the Natural History of the United States (Boston: Little, Brown & Co., 1857-62; reprint New York: Arno Press, 1978).

45. The U.S. Coast and Geodetic Survey began operation in 1817 as a branch of the Treasury Department, and was the only Federal government scientific agency during the first part of the nineteenth century. It was directed by F. Hassler until his death in 1843, after which Alexander Dallas Bache assumed its direction. Hassler carried on an extensive correspondence with Carl Gauss, who provided both scientific advice and equipment, continuing to advise the Coast Survey under Bache. In fact, most of Bache's leading assistants were either students or correspondents with Gauss. For example, Benjamin Peirce, who took over after Bache died in 1867, was a leading student of Gauss; Admiral Charles Henry Davis translated Gauss' book on the determination of celestial orbits. See Carl Friedrich Gauss, Briefen und Gesprächen, ed. by Kurt-R. Biermann (Munich: C.H. Beck, 1990).

46. William Draper Harkins (1873-1951), professor of physical chemistry at the University of Chicago for almost forty years. His students and laboratory equipment, such as the Chicago Cyclotron, made the success of the World War II Manhattan Project possible. See biographical introduction by T.J. Young to Draper's The Physical Chemistry of Surface Films (New York: Reinhold, 1952). Young points out that Harkins and E.D. Wilson published the first calculation for nuclear fusion of hydrogen to form helium in 1915. And, in the early 1920's, Harkins, together with Gans and Newson, was the first to generate and detect the formation of an excited nucleus, (Nitrogen-16) in a Wilson Cloud Chamber, "which may be regarded as the first radioactive element produced artificially."

47. Leopold Kronecker (1823-1891), professor of mathematics at the University of Berlin, politically dominated German mathematics during the 1870's and l880's. A radical empiricist, he believed that integers alone had a basis in reality, and that all other numbers (e.g., irrationals) were figments of man's imagination; hence, Cantor's development of transfinite numbers was seen by Kronecker as a direct threat to his entire theory of mathematics. As early as 1874 Kronecker tried to block publication of Cantor's preliminary work on the non-denumerability of real numbers. Using his political influence, Kronecker threatened the editors of professional journals against publishing Cantor's work, which he denounced as "humbug-a slander which, coming from so prominent a figure, had a particularly pernicious influence. Kronecker used his influence to prevent Cantor's appointment to a professorship at Berlin or Gottingen, relegating Cantor to a post at Halle, where he was physically isolated and financially impoverished. The strain of intellectual isolation and Kronecker's constant hounding contributed to the nervous collapse suffered by Cantor in this period.

48. See Bertrand Russell, An Essay on the Foundations of Geometry (1897) (New York: Dover Publications, 1956); also "On Some Difficulties in the Theory of Transfinite Numbers and Order Types," Proc. London Math. Soc. 4, 29-53, 1907. Russell's collaboration with Alfred North Whitehead in the composition of their notorious Principia Mathematica was a desperate effort to refute Georg Cantor's Beiträge by limiting mathematics axiomatically to the crudest possible forms of analysis situs, those of greater than, less than.

49. See G.W. . Leibniz, "Système nouveau de la nature et de la communication des substances" (1695); English trans. "A new system of the nature and the communication of substances," in Gottfried Wil- helm Leibniz: Philosophical Papers and Letters, op. cit., vol. II, p. 739. See also in G. W. Leibniz: Mathematische Schriften, ed. by CI. Gerhardt (Berlin and Halle: 1849-1863; reprinted Hildesheim: 1962), vol. IV, p. 477.

50. Pietro Pomponazzi lectured on Aristotle at the University of Padua between 1487 and 1509, as well as at Ferrara and Bologna. One of his students was Gasparo (Cardinal) Contarini (1483-1542), a descendant of the Venetian oligarchical family, who became the most important Venetian operative during the period of the Protestant Reformation and the initial Catholic Counter-Reformation. Another influence on the young Contarini was Francesco Zorzi (Giorgi), who became his close friend. Among Contarini's close associates were Gregorio Cortese, the Abbot of the Benedictine Monastery of San Giorgio Maggiore, Reginald Cardinal Pole, a sometime-pretender to the English throne, and Gianpietro Caraffa, later Pope Paul IV. Pole and his friend Vittoria Colonna were central figures of the Italian crypto-Protestant movement called the "Spirituali." In 1537, Cardinal Contarini chaired the Holy See's Council on the Reform of the Church, which issued a decree citing Aristotle and condemning Erasmus, thus initiating the process leading to the Council of Trent.

51. LaRouche, "Science of Christian Economy," op. cit., pp. 241-256.

52. The Thirteen Books of Euclid's Elements, trans. by Thomas L. Heath (1925) (New York: Dover Publications, 1956).

53. See Wilhelm von Humboldt, "Preliminary Thoughts on the Plan for the Establishment of the Municipal School System in Lithuania" and "School Plan for Königsberg," which are summarized by Mananna Wertz, in "Wilhelm von Humboldt's Classical Education Curriculum," New Federalist, vol. VII, No. 10, March 15, 1993, p. 8; see also Wilhelm von Humboldt, Humanist Without Portfolio: An Anthology of the Writings of Wilhelm von Humboldt, trans. by Marianne Cowan (Detroit: Wayne State University Press, 1963). Humboldt's reform program was directly influenced by his long association with Friedrich Schiller. See "On Schiller and the Course of His Spiritual Development," by Wilhelm von Humboldt, and Schiller's "What Is, and To What End Do We Study, Universal History?" in Friedrich Schiller, Poet of Freedom, vol. II, ed. by William F. Wertz, Jr. (Washington, D.C.: Schiller Institute, 1988).

54. See Carol White, "The Roots of British Radicalism," in The New Dark Ages Conspiracy (New York: New Benjamin Franklin House, 1980), pp. 285-333; see also "Origins of the Counterculture," in Dope, Inc.: The Book That Drove Kissinger Crazy, by the Editors of Executive Intelligence Review (Washington, D.C.: Executive Intelligence Review, 1992), pp. 533-553.

55. See, e.g., Bertrand Russell, Introduction to Mathematical Philosophy (1917) (New York: Simon & Schuster, Touchstone Books, 1971), p. 21.

55. See, e.g., Bertrand Russell, Introduction to Mathematical Philosophy (1917) (New York: Simon & Schuster, Touchstone Books, 1971), p. 21.

56. This incidentally, is the proper standpoint from which to appreciate the non-mysterious implications of Kurt Gödel's famous treatment of formally undecidable propositions (see footnote 26).

57. Note both the treatment of the circle in Nicolaus of Cusa's De Docta Ignorantia earlier and then, later, the summation of that in "Dc Circuli Quadratura" (see footnotes 16 and 17).

58. Nicolaus of Cusa, De Docta Ignorantia, op. cit.

59.Ibid.

60. In 1766-1767, Johann Heinrich Lambert proved the irrationality of the numbers &Mac185; and e. Based upon Lambert's Theorem, in 1873 Charles Hermite proved the transcendence of the number e. In 1882, F. Lindemann demonstrated the transcendence of the number&Mac185; as an extension of Hermite's proof for the case of e. See: Johnann Heinrich Lambert, "Mémoire sur quelques propriétés remarquables des quantités transcendentes circulaires et logarithmiques," Histoire de l'Académie Berlin, 1761 (1768), pp. 265-322; partial trans. in D.J. Struik, A Source Book in Mathematics, 12001800 (Princeton, N.J.: Princeton University Press, 1968), pp. 369-374. Charles Hermite, "On the Transcendence of e" (1873), in David Eugene Smith, A Source Book in Mathematics, op. cit., trans. by L. Guggenbühl, pp. 99-106. For a summary of the arguments of Hermite and Lindemann, see Felix Klein, Famous Problems of Geometry (1897), in Famous Problems and Other Monographs (New York: Chelsea Publishing Co., 1955), pp. 61-77.

61. See A Manual on the Rudiments of Tuning and Registration, ed. by John Sigerson and Kathy Wolfe (Washington, D.C., Schiller Institute, 1992), esp. chap. 11 passim, pp. 199-228. See also, Lyndon H. LaRouche, Jr., "Mozart's 1782-1786 Revolution in Music," Fidelio, Vol. I, No. 4, Winter 1992.

62. LaRouche, "History as Science," op. cit., pp. 24-27.

63. Panini (c.400 B.C.), grammarian of Classical Sanskrit. PB. Junnarkar's An Introduction to Panini (Baroda: Shanti S. Dighe, 1977) includes the full text of Panini's Astadhyayi.

64. Cf. A Manual on Tuning, op. cit., chaps. 9 and 10, pp. 151-198. If the principle of least action is applied to voice training of singers, the result of this is a form of voice training associated with the bel canto tradition carved in stone by Luca della Robbia in the Cathedral of Santa Maria del Fiore in mid-fifteenth century Florence, Italy.

65. See A Manual on Tuning, pp. xv-xxix.

66. Ibid.

67. See Webster G. Tarpley, "The Evil Philosophy Behind Political Correctness," Fidelio, Vol. II, No. 2, Summer 1993, pp. 42-54.

68. Aristoxenus (born c.375 B.C.), a student of the Pythagoreans and Aristotle, developed a fully-conceived system of musical tuning presented in such works as the surviving Harmonic Elements, whose "tense diatonic" scale has been interpreted by modern writers as containing a system of equal temperament. See The Harmonics of Aristoxenus, trans. and ed. by H.S. Macran (London: Oxford University Press, 1902); see also R. Westphal, Aristoxenus von Tarent (Leipzig: A. Abel, 1883-93; reprinted 1965).

69. See Pierre Beaudry, "The Acropolis of Athens: The Classical Idea of Beauty," New Solidarity, Vol. II, No. 24, June 24, 1988, pp. 6-7; see also, Lyndon H. LaRouche, Jr., "The Classical Idea: Natural and Artistic Beauty," Fidelio, Vol. I, No. 2, Spring 1992, p. 8ff.

70. See Plato, Republic, op. cit., Steph. 509d-513e; Timaeus,, op. cit., Steph. 32a, 35b-36b, 54d-55c.

71. See Luca Pacioli, De Divina Proportione (1497) (Vienna: 1896), whose geometrical diagrams of the Golden Section-determined regular solids were drawn by Leonardo da Vinci. Reproductions of these drawings appear in The Unknown Leonardo, ed. by Ladislao Reti (New York: McGraw-Hill Book Company, 1974), pp. 70-71.

72. This is not the place to take up the distinction between a so-called "quantum mechanics" and a "quantum field theory." It is sufficient to inform the reader that Planck's work leads as readily to a quantum field theory of quasi-Keplerian type, as to a strained quantum mechanics, and without the distressing paradoxes inhering in the latter. A point here is the theory of knowledge; only that implication is being treated in this part of the report.

73. See footnote 71. Leonardo's drawings and studies of plants and plant growth abound in the application of Golden Section harmonics.

74. See Johannes Kepler, On the Six-Cornered Snowflake, trans. by Col- in Hardie (Oxford: Clarendon Press, 1966), reprinted by 21st Cen- tury Science & Technology, 1991.

75. See Plato's Timaeus in Plato: Timaeus, Critias, Cleitophon, Menexenus, Epistles, Loeb Classical Library, trans. by R.G. Bury (Cambridge: Harvard University Press, 1929), Steph. pps. 32a, 35b-36b, 54d-55c.

76. This pertains to the intelligibility of principles of higher hypothesis by creative reason. On natural law generally, see G.W. Leibniz. Natural law signifies those universal, endurable principles of the world as Becoming which are naturally intelligible to individual creative reason. For example, as given in the text, the principle of the sacredness of the individual person, the derived sacredness of the family, and the derived relative sacredness of the republic form of government.

77. The term "agapic" signifies the agapic form of love in opposition to erotic love. The reference is, of course, to the Gospel of St. John, especially the famous verse 3:16, and to I Corinthians 13 of St. Paul, as the standard for defining agape.

78. Lyndon H. LaRouche, Jr., "On the Subject of Metaphor," op. cit., pp. 20-26.

79. G.W. Leibniz, Monadology, op. cit.

80. See LaRouche, "On the Subject of Metaphor," op. cit., pps. 22-23, 44-47.

81. Ernst Mach (1838-1916) initiated the effort to impose positivism on science in the twentieth century, and is generally credited with founding the fraud known today as modern "philosophy of science." While most of his scientific conclusions have long been proven false-for example, "that atoms [don't] exist-his general method, particularly his opposition to any notion of causality in science, have become prevalent in modern physics. Mach led a scientific vendetta against Ludwig Boltzmann-eventually leading to his suicide in 1906-because Boltzmann refused to completely abandon the concept of causality in thermodynamics. He afforded similar treatment to Louis de Brogue at the 1927 Fifth Solvay Conference on Physics, and later, to Erwin Schroedinger. Dc Broglie characterized these events as "a virtual coup d'etat in theoretical physics." See Morris Levitt, "Linearity and Entropy, Ludwig Boltzmann and The Second Law of Thermodynamics," Fusion Energy Foundation Newsletter, Vol. II, No. 2, Sept. 1976, pp. 3-18; see also Uwe Parpart,"The Theoretical Impasse In Inertial Confinement Fusion," Fusion, Vol. III, No. 2, Nov. 1979, pp. 31-40.

82. See Luca Pacioli, Dc Divina Proportione, op. cit.

83. See footnote 73.

84. For Kepler's use of "quantum field theory," see his Mysterium Cosmographicum (The Secret of the Universe), trans. by AM. Duncan (New York: Abaris Books, 1981); chap. 2 contains his explicit reference to Nicolaus of Cusa. For Kepler's discussion of the Divine Proportion (Golden Section), and of the geometric determination of harmonic relations, both in music and astronomy, see his Harmonice Mundi (The Harmony of the World), in Opera Omnia, vol. 5, (Frankfurt: 1864); English trans.: Books I-IV, trans. by Christopher White et al. (unpublished); Book V, trans. by Charles Glenn Wallis, included in Great Books of the Western World series (Chicago: Encyclopedia Britannica, 1952).

Owing to an editorial error, footnote 49 to Section 2 of Lyndon LaRouche's "History as Science: America 2000," which appeared in the previous issue of Fidelio (Vol. II, No. 4, Fall 1993) was incorrect as printed. The corrected note, which deals with Georg Cantor's use of the "power set" to generate the transfinite cardinal numbers, reads as follows:

49. The "power set" is the set of all subsets of a given set. Cantor applied this idea to his transfinite cardinal numbers, and proved that the power set of a given transfinite cardinal number would generate a new, higher-order transfinite cardinal. Cantor's first transfinite cardinal represents the countable or denumerable infinites. The power set of the countable infinites is the nondenumerable continuum, and Cantor demonstrated through his diagonal method that the number continuum is a higher order cardinality than the countable infinites. There may be other nondenumerable aggregates besides the number continuum, as Paul Cohen's proof of the non-demonstrability of Cantor's continuum hypothesis demonstrated. The power set of the number continuum gives a higher order cardinal, the set of all functions, and so on. The capability to generate higher and higher transfinite cardinal numbers is equivalent to Plato's concept of "hypothesizing the higher hypothesis."

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