The Essential Subjectivity of Science
Lurking among the numerous accomplishments of modern science, there is the absurd, but popular delusion, that "physical science" is both "materialist" and "objective." The worst, and most widespread forms of this delusion assume, first, that scientific method is essentially statistical, and that "mathematical science" is associated with measurement of forces acting along a straight-line pathway between two points. This popular delusion was key to the widespread "systems analysis" hoaxes, such as Professor Norbert Wiener's "information theory."
The proof, that such definitions of "objective science" are absurd, is elementary; that proof is given as a central feature of this author's introductory course in Leibniz's science of Physical Economy.45 We summarize the background considerations, point by point
1. If man were a mere animal, that is, like a baboon, a creature innately disposed to what is called "primitive hunting and gathering" modes of social reproduction, at no time could the living human population of this planet have exceeded about ten millions individuals 2. The increase in the human population, and the associated improvements in life-expectancy and standard of existence, are the cumulative benefit of what we may identify most simply and fairly as "scientific and technological progress." The measure of this function of progress is an increase in the potential population-density of the human species; this represents a higher per-capita standard of living and longevity, combined with a decrease in the total number of hectares required to sustain an average individual human life 3. These improvements are expressed functionally through a succession of successful, radical changes in human productive behavior, a succession akin to the series of discontinuities associated with A, B, C, D, E, ... referenced above. These changes in the behavior of successive levels of upward development of society are analogous in form or function, and effect, to successful, upward biological evolution of species among the lower forms of life.46
4. Thus, the problem of both discovering and choosing a Type of sequential ordering of thought-objects, corresponding to a negentropically ordered succession of revolutionary scientific modifications in known scientific principles, is a subjective matter. It is a matter of discovering which subjective Type of creative-mental generation of thought-objects corresponds to a negentropic sequence of increase in man's cultural potential for increasing potential population-density.
Thus, from this point of view, the subject of science is that higher-order of thought-objecta transfinitewhich correlates formal scientific progress with rate of increase of this science-driven rate of growth of a culture's potential population-density. In other words, man willfully increasing mankind's power to perpetuate ever-more successfully his own species' dominating existence within the universe.
This view is in contrast to the popularized materialist mythos of so-called "objective science," of man as the contemplative mathematician-observer.
"I see myself creating, as I define creation, as a common principle of that array of named thought-objects of fundamental discovery associated with such as Plato, Archimedes, Cusa, Leonardo, Kepler, and Leibniz. I locate my own creating-activity in respect to an effort to attribute a higher thought-object, a Cantorian Type, to the manifold composed of such historic names of original discoverers. This attribution of a specific choice of order for such an 'aleph-manifold,' and of attributing a Type to that choice of ordering, is the immediate subject of my inquiry.
"This Type defines a relatively fundamental scientific principle, as an hypothetical choice of such a principle; in Plato, this is referenced as 'hypothesizing the higher hypothesis.' I now correlate that hypothetical choice of Type with a manifest ordering of science-driven growth of relative potential population-density, of relatively superior and inferior modes of physical-economic culture."
This correlation is the characteristic activity of physical science; seeking to subsume all such hypothesizing of the higher hypothesis as a manifold of a yet higher Type, is physical science.47
As described in other locations,48 the details of this phenomenon are of the following form. The hypothetical inference of a new Type of ordering of crucial thought-objects of fundamental scientific discovery as a manifold (or, sub-manifold), in respect to a single Type of crucial (or, "unique") paradox, subsumes an experimental design for some crucial expression of this new hypothesis. That subsumes, in turn, the design of either an experimental apparatus, or an observational method akin to such an apparatus.
Thus, from fundamental discovery of (transfinite) ordering-principle, through the design of an experiment, through that experimental design expressed as a new principle of machine-tool (or, analogous) design, is the generation of a discovery of scientific principle transmitted and assimilated into a general increase of social productivity. In every step of that process, the essential thing is the generation of a new conceptual thought-object by, within, and in accord with the sovereign, individual creative mental processes of the mind of a sovereignly individual person.
We should emphasize by aid of such means as reiteration, that the process just outlined is Plato's "hypothesizing the higher hypothesis." The higher hypothesis is the Type of cardinality to which corresponds a manifold (or, sub-manifold) of thought-objects arranged in a certain choice of ordering. The choosing of such a particular such higher hypothesis, the hypothesizing of the selection of one or more such higher hypotheses for such an array of individual thought-objects, is itself the consideration of a manifold of such alternative Types. The latter manifold's Type is what we should signify by physical science.
In other words, physical science is essentially the process of discovering those rules of creative behavior of our individual mental processes which lead us to discoveries of a Type through which general culture may be changed to optimize the rate of increase of our species' potential population-density. In this fashion, physical science is essentially subjective.
Admittedly, that does not complete the argument. If a certain type of "hypothesizing the higher hypothesis" is physical science, then increases in potential population-density, so successively achieved, show us that the intelligible form of lawful ordering of nature is coherent with the process of perfection of our hypothesizing the higher hypothesis. Thus, it is our successful hypothesizing of the higher hypothesis, in this fashion, rather than our sensory impressions, the which is the proper basis for determining the lawful composition, and ontological characteristics, of that real physical universe which lies beyond the full reach of our mere senses.
Our creative-mental processes do not address directly sensory objects as sensory objects per se. Human thought knows only change; we know only a thinkable correspondence between a change in our behavior and a correlated change in the manifest behavior of nature. It is a correspondence of the two Types of change which constitute the entirety of real physical science. That correspondence is what is intelligible for us; we must discover everything else respecting nature from this approach to the elementary primacy of change, to the universal elementarity in space-time of nothing but change.
This point is clearer, if we look now at the historical source of the leading opposition to the picture we have presented.
The 'Materialist' Opposition
The leading opponent of our Leibnizian view of science, and the modern opponent of Plato, Cusa, Leonardo da Vinci, Kepler, and Leibniz, for example, is the so-called "materialist," or "mechanistic" standpoint of Francis Bacon, Robert Fludd, Elias Ashmole, René Descartes, John Locke, and Isaac Newton. This "materialist" dogma was introduced to seventeenth-century France and England by the then newly-established cult of the Rosicrucians. The essence of this gnostic Rosicrucian dogma is typified by René Descartes' deus ex machina49 and Isaac Newton's maxim hypotheses non fingo.50 This is also the axiomatically "hereditary" origin of such modern forms of radical positivism as von Neumann's "systems analysis," Professor Noam Chomsky's Korschite "linguistics," and Wiener's "information theory" hoax.
Consider as much of this Rosicrucian cult's dogma as is essential to locate the origins of that popular delusion we recognize most readily as the mythos of "objective science." The derivation of the Rosicrucian cult is the best vantage-point for this undertaking.
The seventeenth-century Rosicrucian cult of Fludd, Ashmole, et al. was a resurfacing of a notorious, usury-practicing, medieval sect known variously as the Cathars, Bogomils, or, more commonly, "The Buggers."51 This sect, which infested the market centers of northern Italy and southern France ("Languedoc"), was one of many varieties of kindred gnostic cults sprung up over the centuries from such very ancient pagan origins as the Phrygian cult of Cybele-Dionysus, the Delphic cult of Apollo-Dionysus, the Hellenic cult of Osiris, and the sundry Babylonian and Canaanite mystery religions.
The relevant feature of these gnostic forerunners of Ashmolean Rosicrucianism is the doctrine of utter depravity of the "flesh" which is the direct source of the materialist dogmas of Bacon, Descartes, Locke, Newton, et al. The sexual perversions of the Cathars are a direct, doctrinaire correlative of this materialist dogma of theirs. Briefly, one of the cult's Elect was forbidden to place his semen in the vagina of a woman, lest he cause the procreation of newborn human flesh! The spirit inhabiting the Elect must be kept apart from the utter depravity of the fleshly process of human procreation.52
That said, consider the case of science-driven increase of society's potential population-density. The origin of a new, valid, fundamental discovery, is a mental act of creation, a spiritual act, the generation of such a thought-object. The derivation of a design of experimental apparatus, and then a machine-tool principle, from the new thought-object, is the source of a powerful material effect. This is the connection which the Rosicrucian Descartes insisted must be broken: deus ex machina, and which Newton forbade: hypotheses non fingo.
What kind of society do these Manichean, or Bugger Elect represent? The Elect are forbidden to interfere with nature; they cannot till the soil, nor perform other productive labor. They are permitted to subsist by begging for alms, or to loan their accumulation of monetary savings from alms-gathering in usury. The Elect form, thus, a parasitical class subsisting by tribute and usury.
The strength of such a usury-practicing gnostic conspiracy, is that the Elect of the "Bugger" sect could sell a note for twelve or more ducats in Lyons, which could be redeemed by the bearer at discount for ten ducats, or less, in Padua. Thus, spider-web networks of Elect "Buggers" spread across northern Italy and southern France of the Garonne-Tarn and Rhône regions, in symbiosis with the other principal usury-practicing "Elects" among Lombard bankers and Jewish money-lenders.
The following summary is fair. As the oligarchical, usury-practicing I Nuovi faction of the Venetian merchant-bankers spread their parasitical, oligarchical power, by such vehicles as the Levant Company, into England, the Netherlands, and the old Hanse regions of Northern Europe generally, the Netherlands and England became the target for the launching of such Levant Company spin-offs as the Bank of England, the City of London financial center, and the Dutch and British "India" companies. London became thus the "new Venice," a union of the usurious Levant Company "Lombards" with the Rosicrucian cult of Bacon, Ashmole, et al. These seventeenth-century developments were the roots of the combined work of the Liberal Party and (later) Fabians of the eighteenth and nineteenth centuries, in seeking to establish London as the capital of a "Third Roman Empire," a worldwide form of pax universalis, a British Empire which would be a revival of the pagan Roman Empire of Augustus, Tiberius, Caligula, Nero, and Diocletian.
Originally, science was solely a creation of the Platonists of the Golden Renaissance, chiefly the work of those fifteenth-century moral and intellectual giants who are best typified by Cardinal Nicolaus of Cusa and Leonardo da Vinci. This tradition was continued by the work of such as Kepler, Gilbert, Fermat, Desargues, Pascal, Huygens, Leibniz, and the Bernoullis. That seventeenth-century Leibnizian tradition was carried into the nineteenth century by such figures as France's Gaspard Monge, and Germany's Gauss and Riemann. This tradition is sometimes called "continental science," to distinguish it from the Cartesian, empiricist, and positivist outgrowths of the Rosicrucian influence.
The cases of Bacon, Fludd, Descartes, and Newton established the counter-science variously expressed as Cartesianism, empiricism, and positivism. The hegemony of this cult's "Enlightenment" materialism in most science classrooms today, is the result of British participation in victories in most of the wars of the past three hundred years. The supremacy of the Rosicrucian's materialist dogma in today's scientific establishment is not a scientific, but a purely political phenomenon.
The practical issue of this political division in the science establishment, is the overarching conflict between the two principal, conflicting social systems which have, almost entirely, dominated European history since Solon's defeat of the oligarchical usurers of ancient Athens, more than two-and-a-half thousand years ago. This is the point made by Friedrich Schiller's contrasting the humanist, republican constitution of Solon to the American-Confederacy-like law of Lycurgus' Spartan slave society.53
To sustain scientific and technological progress requires appropriate education of virtually all participants in the society's productive processes. A population so educated will not tolerate indefinitely that division of society's population into oligarchs and helots which was characteristic of Lycurgus' Sparta, the pagan Roman Empire, and the American Southern Scottish Rite Jurisdiction's Confederate States of America. The brutish ignorance to which the slaveholders' oligarchical system degraded not only the Confederates' "poor whites," but also most of the so-called "planter aristocracy,"54 illustrates the point at issue. The so-called "socialist" zero-technological growth decrees of the Roman Emperor Diocletian are a notable, consistent precedent for the brutish degeneracy pervading the old Confederacy.55
On the other side of the same issue of policy, an ignorant people is not capable of self-government. To govern oneself requires the capacity for efficient comprehension of qualities of processes which are, by their nature, intrinsically beyond the developmental capacity of the scientifically illiterate strata. As several founders of the U.S. federal republic warned, the survival of such a democratic republic as theirs under natural law required a certain minimal quality of compulsory education.56 Friedrich Schiller presented the conceptual basis for the most successful model of Christian classical humanist education, the reforms of Wilhelm von Humboldt.57
Under the influence of such a quality of universal compulsory secondary education, that educated citizenry will conspire to free itself from any oligarchical rule. Yet, without such an intrinsically anti-oligarchical form of education, a society could not generate, transmit, or assimilate efficiently scientific and technological progress in a general way. The self-interest of the oligarchy, as a social formation, is to destroy nations practicing generalized scientific and technological progress, and then seek to outlaw, throughout the world, both classical education and the practice of scientific progress. That is the entropic Type of cultural policy represented by the "(guild) socialism" of Diocletian, wherever the like appears, down through the ages of history since not later than the Phrygian Cybeline cult of Dionysus.
Like Kant's pro-irrationalist Critiques later, Descartes' gnostic deus ex machina dogma sought to paint a picture of the material world independent of that indispensable subjective agency, the creative mental processes upon which the discovery of all scientific knowledge depends absolutely. Kant did not deny the efficient existence of creative powers of scientific discovery, but pronounced deliberative creative acts to be impossible.58
That is the kernel of what passes for sophisticated philosophical materialism. To the credulous simpleton, the materialist demagogue exhibits himself as a solid, down-to-earth good fellow, one, perhaps, with all four feet firmly planted on the ground. "We materialists believe in nothing we cannot experience first-hand, with our own good five senses." To thoughtful, literate audiences, such cheap rhetoric is not persuasive; the argument of the Kantian unknowable thing-in-itself and Descartes' deus ex machina is offered, instead.
For us, the relevant experience on which physical science must be premised, is not fixedness, but change: the correlation of a change in our scientific thinking for practice, with the resulting change in the responsive behavior of nature. Unlike that theology as such which references the Absolute of Plato's the Good,59 mere physical science does not know the Absolute, but only Cantor's Transfinite. The domain of the transfinite is, at its highest level, Plato's hypothesizing the higher hypothesis, the domain of physical space-time, the domain of change, of perfecting that which remains unperfected. Thus, for physical science, the science of physical space-time, experience is change, and change is the elementary substantial feature of all scientific experience.
As the illustrative case of the experiment shows, change begins as an ostensibly non-material, subjective act of valid creative discovery of new, un-utterable Geistesmassenthought-objects. This first step in the causal sequence of human action is spiritual, not "material." Under the "foremanship" of the relevant thought-object, a crucial experimental design is fashioned, a material medium for the spiritual cause, which latter is the thought-object. So, we had next, the derivation of the new machine-tool principle, and the medium through which man's per-capita power over the universe, per square-kilometer, is increased. The latter is the relevant material effect.
It is this sequence, this spiritual change causing the material change, which every successful experiment demonstrates. The materialist insists that the results of the experiment must be described only in such ways as leave the generation of the relevant new thought-object out of account. Since the universe responds to the experiment as it is actually developed, as prompted by an initially spiritual cause, materialism, with its materialist's fanatic adherence to formal deductive consistency, falsifies the universe by such reductionist fallacy of composition.
Metaphor As Classical Tragedy
During 1948-1952, the period this author first completed the theses presented afresh here, he thought that to prove his case, that Wiener's "information theory" is a dangerous hoax, one had to direct against radical positivist Wiener the same form of refutation which this author had then earlier composed against the elementary fallacies of Immanuel Kant's virulently anti-Leibniz Critiques;60 the last of those, Kant's Critique of Judgment61 may be taken as a point for our purposes here.
This meant, then as now, that one must first attack Kant's neo-Aristotelian formalism, Kant's formal, reductionist pseudo-proof, that creative processes of original scientific discovery of principle are unknowable a priori. Additionally, it was clear then, as now, that just as Kant goes from this, in his Critique of Judgment, to deny any rational principle of knowledge of aesthetics, so we must show, that the same intelligible principle underlying creative, valid, original scientific discovery of principle, must be the governing principle of creativity in classical fine arts.
Then, in 1948-1952, as now, our central focus was that step-wise relationship between crucial scientific discovery and employment of derived machine-tool-design principles we have identified in this present location earlier. To show this same principle at work in classical fine arts, we focused upon classical poetry, emphasizing the chosen cases of Schiller and Goethe, and, then, using Goethe as the vehicle for treating the German Lied as represented by Mozart, Beethoven, Schubert, Brahms, and Wolf settings of Goethe.62 Here, we emphasize, instead of the Lied, the model of classical tragedy.
The disadvantage of employing classical tragedy as an illustration, is that there are so few truly notable tragedians, as distinct from great classical composers (from Praetorius through Brahms). Only Aeschylos, Cervantes,63 Marlowe, Shakespeare, and Schiller, chiefly, are exemplary of truly successful tragedians. Only the historian Schiller, among these few, mastered explicitly a statement and demonstration of the principles of composing classical tragedy. Nevertheless, the compelling advantage of using the case of tragedy here, is that, implicitly, it most perfectly situates in art-form the Cantor notions of cardinality and power (the German Mächtigkeit), as Cantor defines these to include the problems of ordering the aleph-manifold.64
Consider as classical drama the array of exemplary crucial scientific discoverers we listed here earlier: Pythagoras, Plato, Archimedes, Cusa, Leonardo da Vinci, Kepler, Gilbert, Desargues, Fermat, Pascal, Huygens, Leibniz, the Bernoullis, Gaspard Monge, Karl Gauss, Bernhard Riemann, Eugenio Beltrami, and Georg Cantor. Arrange the crucial discoveries associated with these personalities, to imply an ordering-principle (higher equivalence, Type) which we may equate metaphorically to the name of science. Then, construct a contrasting, entropic array, typified by such followers of Rosicrucian "Buggery's" materialist principle as Bacon, Fludd, Hobbes, Descartes, Locke, Newton, Cauchy, Clausius, Klein, Kronecker, Helmholz, Maxwell, Rayleigh, Boltzmann, Russell, von Neumann, Wiener, et al. This is an entropic Type which we may rightly equate metaphorically to the name of anti-science. There, we have the principal historical background elements of dramatis personae from which to conduct a truly classical tragedy according to Schiller's principle.
The basis for constructing a drama inclusive of these two, mutually exclusive Types, is that the formal elements of each of the manifolds might each reference the same phenomena of scientific history as the other, although the ordering principle by means of which the opposing type knows the element metaphorically may be totally irreconcilable with the opposing one.
The tragedy based upon such a conjunction might be built up in the following way.
Given, a society whose prevailing custom in science is the "post-modernist" version of the entropic Type, but a society in which a few potential heroes know that the crucial elements of the society's scientific-economic practice might be ordered according to the negentropic Type, as readily as to the presently hegemonic entropic choice. Define a situation in which the failure of a potential hero to act with pungency and force upon that latter option, means a devastating military or other kind of great suffering for his or her nation. Let this unhappy consequence occur, ostensibly because the potential hero fails to seize his last available opportunity, at the punctum saliens, to bring about the required shift of emphasis in the society's policy practice. The potential hero's seizure by what is fairly termed "psychosexual impotence," like Hamlet's, takes the form perhaps, of fearing to lose his academic security or pension, should he confront directly the entropic faction in this matter.
This failure of the potential hero defines the tragedy. This failure is implicitly of an intelligible Cantor Type; but, that is not a fully adequate representation of the notion of this tragedy.
The tragedy is performed before an audience. The performance of that drama, presented to that audience, begins to succeed if the audience is made conscious of the opposition of the two Types, and of the potential hero's situation. Thus, the audience, by taking the dramatic character's express consciousness as the object of the spectator's conscious attention, is seeing the drama, and the characters depicted, as if from above. If the audience also recognized something of itself in each of these characters, the drama has reached a second milestone in the direction of success.
Next, the negentropic alternative must ultimately uplift the spirits of the spectators; that is the spark of true life, evoked so within the audience, and imparted thus, by fusion, to the audience's consciousness of the succession of dramatic montage sensed of events on stage.
This assembles a manifold of not less than the following conjoined arrays of thought-objects: (1) the negentropic Type of science, as portrayed; (2) the entropic Type of anti-science; (3) the manifold of the phenomena which both of the foregoing two, opposing Types reference as correlation of their respective thought-objects; (4) the role of the potential hero, as a thought-object; (5) the audience's reflection of its own projected consciousness of itself, returned to it as fused with the drama on stage.
This manifold, so composed, must be represented by a thought-object corresponding to the tragedy as a whole. That Type is the indivisible substance, the sovereign elementarity of that tragedy as a creative work of classical fine art. Here, metaphor is plainly the indispensably ironic character of both every moment of development of the drama, and of the drama in its entirety.
The idea of metaphor as mere "symbolism," is plainly the absurd conceit of an illiterate. Symbolism returns to a mere sense-object's image. It is the experience of generating the thought-object corresponding uniquely to that tragedy as a composed, indivisible entity, not a sensuous symbol, which is the referent for the idea of that drama taken as a whole.
That idea, that Type is the essential experience of the author, as composer, and of the audience in experiencing the discovery of this new thought-object, as one might regenerate an original scientific discovery, as a thought-object, in one's own, sovereign creative-mental processes.
The tragedy addresses so, implicitly, the central feature of all individual creative-mental activity; that central feature is the act of efficient participation in humanity as an historical entirety. Nicolaus of Cusa's elaboration of the principle of capax Dei references this impulse in its highest form of expression.65 The Types associated with this creative impulse, include, most prominently, the following:
1. Man the individual as imago viva Dei, in the living image of God the Creator.66 Man is thus set apart from, and above the beasts, by virtue of the fact that the successful existence of our human species is effected by creative activity of a Type centered upon the generation, transmission, and efficient assimilation of scientific and technological progress. Without this creative activity, mankind could not continue to exist as a human species. God's quality as Creator, and man's unique affinity to that God the Creator, is knowledge which depends upon a thought-object corresponding to this creative self-image of man 2. Man as the sovereign creative individual. Although we are mortal, we exist efficiently in the present and whole future of all mankind by means of our employment of our creative-mental processes for the generation, transmission, and efficient assimilation of thought-objects equivalent to crucial features of scientific and technological progress. In this, every instance of generation of a thought-object, (whether an original discovery, or not) is a sovereign act of an individual person, rather than a "collective" effect 3. The issue of creative discovery, is not resolvable in terms of case-by-case assessment of individual isolated such discoveries. The issue is the discovery and enhancement of an ordering-principle which directs us along a negentropic pathway of valid, successive discoveries. We require a process of valid discoveries. We seek a higher rate of this Type of growth of the rate of progress. In and of itself, the abstractly isolable, particular discovery by an individual person is of a transfinite order of lesser importance, than that person's contribution to improving the negentropy of that ordering-principle of successive changes, the which defines successive increases of potential population-density as a unified manifold. This latter consideration is the form of the most readily intelligible aspect of individual participation, not only in the classical tragedy, but also the universe as an entirety
As is elaborated in other locations,67 the individual affects efficiently, so, not only present and future generations, but also past. In the domain of space-time, in which the transfinite process of successive, negentropic change is ontologically the primary reality, this change is not merely the simple outcome of an individual act, but the outcome of participation in changing the universality of the determining process, the significantly efficient result of a person's mortal existence. Thus, we, by altering, through participation, the relevant feature of outcome of participation by even remote ancestors of the presently living generations, alter the pastby altering the outcome of the past's participation in the present and future.
Thus, in tragedy, Shakespeare causes the mind of Hamlet to be obsessed by what Hamlet believes to have been the ghost of his father. Thus, as by historical subjects of classical tragedy, do great poets seek to prompt their audiences to improve significantly the way in which we arrange the participation of the past in our present and future.
In this location, so far, we have emphasized those personalities whose very names are metaphors for the crucial thought-objects of scientific discovery. The pedagogical advantage of limiting our attention to such a selection of personalities, is that the work of discovery of these selected historical persons is readily susceptible to at least a negative form of mathematical treatment; on this account, the notion of a transfinite ordering of such discoveries through the issues of Cantor's aleph-manifold, is accessible.
Once the case is understood for such scientific metaphorizing, approximately at least, the concept is more readily extended to metaphor specific to classical art and statecraft. The favor is returned; from the extension to art and statecraft, we return the conception, much enriched, to scientific matters. The principal such enrichment is a keener sense, not only that valid science is essentially subjectivecontrary to the popularized Cathar-Rosicrucian influences upon Descartes and British Empiricism; the meaning of science is not only Leibnizian physical economy, but, more broadly, contemporary man's efficient participation in the past, present, and future of the universe. We understand the essential role of classical art in making science possible, and understand the meaning of the metaphor; the highest form, the most rigorous form of mathematical physics is, thus, a body of classical poetry and music, a larger scope of classical philology derived from the polyphonic vocalization of the poetry.68 Classical tragedy is part of this philology.
As these sources identify the related points, the human singingand speakingvoice has a natural set of characteristics and values which are shown clearly by the most efficient training and use of the vocal apparatus. That "most efficient" training and use, is the "Florentine bel canto" already in use no later than the middle decades of the fifteenth century, probably in the time of the great Cosimo de Medici's leadership there. "Most efficient" signifies the ratio of projected tone to air expelled from the singer's mouth and nose. The speaking of a language, notably the enunciation of the vowels (vocalization) is thus naturally tuned (see Figure 20).69 Each natural species of adult singing (and speaking) voice has its own specific division among registers of mutually distinct "color," divisions located specifically (see Figure 21) within a specific between-note interval on the C-256-pivoted-well-tempered musical scale.
A relatively elementary illustration of the implications, is effected by attempting to compose a vocal quartet (soprano, mezzo-soprano, tenor, bass) in the medium of well-tempered polyphony, using an opening line from a classical poem such as, for English speakers, one by John Keats. First,70 use the simplistic, but rigorous scheme of Goethe's favorite song-setter, J. F. Reichardt. Begin with the soprano part, setting the first utterance of the line within the soprano's second register. Then, examine the difficulties of writing a simple, four-part canon, copying the soprano part into each of the other three voices. A novice should try to copy with equal distance below the second to third register shift for each species of voice.
Observe two of the most obvious features of these attempts. First, note in passing, the chords defined by the polyphony. Second, linger over the implications of the cross-voice sequencing. For an example of cross-voice sequencing, select a note from the bass line; read the note immediately following that in the tenor line; similarly, successively, from the mezzo-soprano and soprano lines. Repeat this cycle for each of the following tones in the bass line. Now, consider other cross-voice sequences, treating first all of the possible permutations which begin with the bass line. Note a similarity to some possible orderings within an aleph-manifold, as referenced earlier.
In each of these cross-voice sequences, observe the dissonances generated; but, do not end the matter there. Study the rules for classical canons from this standpoint. Generalize the notion of a resolution for each such dissonance. All of this is a process of the formation of a thought-object from the single polyphonic germ of one line of classical poetry. The fact that each dissonance implies a range of possible resolutions, defines a manifold of all of each. Thus, does a polyphonic setting of even a single line of classical poetry define implicitly a Cantor Type.71
The example just given is premised upon the mere rudiments of classical song-writing; yet it suffices to illustrate the notion, that music is the domain of metaphor, not of symbolism. Since this music originates in the naturally determined forms for polyphonic vocalization of classical poetry, the transfinite essence of musical composition must be recognized as an "hereditary" implication of classical poetry, and, thus, also of language in general, and drama.
In language, we have primary reference to the senses of vision and hearing; insofar as language references the senses, it refers chiefly to these two. Vision is geometry; hearing and speech are the language of music. So, language equips us to provide sensory metaphors, by means of which to reference those thought-objects pertaining to creative reason's enabling mankind's labor to master the universe, and to participate thus in assisting the work of the Creator. In tragedy, we reference the social essence of that labor, directly; in classical music, we celebrate, and strengthen so, the process by means of which we foster that creative labor.
In Summation: Negentropy
Since the writings of Nicolaus of Cusa to this effect, the paradigm for the idea of growth has been, not a mere Fibonacci Series, but, instead, Cusa's image of the ascending evolution of species; each species participates in the generation of its own, superseding, higher species.72 The Mendeleev Periodic Table of chemical elements and isotopes, rigorously examined, also implies integrally, such a negentropic ordering function. This idea of negentropic growth can be understood only from no less a standpoint than has been identified in this present report; the sweep of growth of the most valid current of modern science, from Plato, through Nicolaus of Cusa, Leibniz, Gauss, Riemann, and Cantor, is indispensable.
This form of growth must be understood to signify qualitatively more than mere linear increase of magnitude. Nor can it be confined to a mere inversion of Clausius-Kelvin statistical entropy, as Wiener foolishly misuses the work of Boltzmann to such banally inappropriate effect. True growth, to be consistent with the integral function of the Periodic Table, or Cusa's succession of ascending species, must be defined essentially as an increase not of simple magnitude alone, but, rather, an increase of quality. The simplest mathematical reflection of such quality is an increase in the density of singularities (mathematical discontinuities) per interval of action, or, better, an increase in the rate of growth of density of singularities per interval of action.
Such a negentropic series is depicted, in first approximation, by our functional series A, B, C, D, E, ..., for the case that the separation of each term from each and all of the others is equivalent (mathematicallyCantor) to the Types of a higher-order aleph-manifold.
Those aleph-manifold Types of discontinuities are apparently absolute separations, and each thus of a magnitude as near to the notion of a definite number-value of "0" as the human mind, so far, has succeeded in defining such a value as a positive one. Yet, each such singularity is not merely a separation, not a mere mathematical discontinuity, but rather, an efficiently functional singularity, whose content is equivalent to that of a thought-objecta Monad. That which Leibniz identifies as a Monad,73 that toward which Riemann points with his Geistesmassen,74 and that which is termed here a thought-object, has that functional significance.
Thus, we have situated the indispensable role of metaphor, as the essential poetic characteristic of any scientific or similarly rigorous communication. Metaphor is the key, the only possible means by which the unutterable is rendered perfectibly intelligible in communication among two or more persons.
As a matter of contrast, symbolism merely combines by reference, one sensory experience with another, or, in a worst case, the mere name for one thing with the name for another object or mere name. Symbolism is to intelligent communication as cabalistic numerology stands in opposition to both science and even mere sanity itself. Symbolism is merely combinatorial construction within the virtually empty domain of names.
An intelligent notion of metaphor hangs upon Plato's Socratic dialectic of negation. The referent is the experience of generating a true thought-object, not a sense-impression, through the processes of creative reason. Metaphor, so comprehended, is therefore the tactic by means of which two minds may coordinate an ordering among respectively similar thought-objects, in a problem-solving mode of creative thinking. This is the only means available to mortal persons, by which the unutterable thought-object is rendered, more or less adequately, perfectibly intelligible. Metaphor, so comprehended, is therefore the required essence of the secondary school classroom.
In fine art, the principle of metaphor is indicated, perhaps most sufficiently, by our rejection of "romanticism," "naturalism," and "modernism" in such art-forms as Classical music. Consider some selected highlights from the two-hundred-year history of Classical polyphony from the work of J. S. Bach through the 1890's Johannes Brahms.75 Take two particular points of reference from within that domain: that revolutionary breakthrough in Classical polyphony effected by J. S. Bach's composition of his Musical Offering,76 and what is identified as Joseph Haydn's discovery of the Motivführung principle of thoroughly integrated composition.76 Examine these two as Wolfgang Amadeus Mozart combined their effect in his celebrated "Haydn" string quartets of 1782-1785.78 This case, as continued by Beethoven, Schubert, and Chopin, illustrates the way in which all serious Classical musical composition is subsumed by the principle of metaphor.
The famous Ricercar of J. S. Bach's A Musical Offering solves a problem in counterpoint by a tactic which Leibniz would recognize as analysis situs. Mozart's intensive, regular encounter with the work of Handel and the Bachs, at the regular Sunday, Vienna salon of Baron Gottfried von Swieten occurred during the time-frame Mozart was inspired by the celebrated "Russian" string quartets which Joseph Haydn had then just recently presented. The impact of Bach's "Musical Offering" is most striking in the sixth of Mozart's "Haydn" Quartets, the C-minor, "Dissonant," K. 465. The same connection is characteristic of Mozart's famous keyboard fantasy-sonata K. 475-457, which is quoted directly by Beethoven as his own keyboard sonatas Opus 13 and Opus 111, and also the C-minor violin sonata, Opus 30, No. 2. Beethoven's Opus 13 is quoted by Franz Schubert's posthumously published C-minor keyboard sonata; the opening movement of Chopin's "Funeral March" keyboard sonata quotes Beethoven's Opus 111. All of Mozart's major compositions of the 1782-1791 interval reflect his revolutionary insight into the combined importance of the two predecessors' cited discoveries.
The greatest representation of Haydn's Motivführung principle, is the "Credo" of Beethoven's Missa Solemnis. A beautiful, and masterful presentation of the same principle, is found in the opening movement of Brahms' Fourth Symphony. With the work of the 1782-1791 Mozart, the key to comprehension, and performance of each composition, is to locate the manner in which the Motivführung principle is elaborated to define the composition as a single, indivisible, unifying conception of the development of a single germ.79
The point of these brief references to Classical music, is to show how it is that all good Classical composition, especially since Haydn's referenced discovery, defines each thoroughly composed work as representing a single, integral, indivisible thought-object, a thought-object corresponding to a specific notion of ordered development.
The real music of such a Classical composition is a thought-object, for which the sensed aspect of the music is an indispensable metaphor. The thought-object appears "between the notes," so to speak, as the apparent, absolute mathematical discontinuities of the functional, non-linear series, A, B, C, D, E, ..., taken as a whole, defines implicitly (negatively) the thought-object corresponding metaphorically to that series.
The most obvious of the discontinuities of a musical score, are the simple intervals defined by the time-intervals between tones, and by (negatively) duration of tones. The simplest notion of the ordering of intervals is a scale or mode. The changes from one to another scale or mode, are a higher ordering; and, so on.
These values are not relative values, but are situated with respect to an absolute, well-tempered scale of C = 256 cycles, and are also situated with respect to vocalized poetic forms of speech, and, so forth and so on.
Therefore, the representation of that metaphorically situated thought-object, the which is the intent of the composition, requires rigorously clean polyphonic transparency. Differences must not arise except as differences are necessary to metaphorical representation of the relevant thought-object.
Thus, the performers must not simply perform the notes. They must, first, experience the relevant thought-object, and then read the notes to the purpose of causing the experienced dissonances and other differences in the performance to correspond to nothing but the metaphorical development of the unifying thought-object.
If we compare this overview of Classical music with Classical tragedy, seeking to grasp the common developmental characteristic of both media, we have a correct view of fine art, as Kant did not, a conception of art which corresponds to science as we have portrayed science here. If we comprehend the unity of a composition, one of any species of the fine arts, as being that composition's existence as a truly metaphorical work of art, and, if we seek out that conception of unity, as a precondition for our representation of that work of art, we are on the proper track.
A recent edition of selected Cantor correspondence contains a citation which is typical of Cantor's view of a certain important matter, and is directly relevant to the disgusting, and destructive incompetence of "information theory's" pretense to the name of "science": The majority of modern mathematicians, through the brilliant success of their self-perfecting formal character, which admits of more and more applications to the mechanical side of nature, have become flushed with a victory, which causes them to degenerate into materialistic one-sidedness and makes them blind to any objective-metaphysical knowledge and thus also to the foundations of their science.80
The root of that against which Cantor complains here, as he did frequently to the same effect in other locations, is the materialist tradition of "Buggery" as imposed upon seventeenth-century empiricism and Cartesianism by the Rosicrucian/Theosophist cult. This neo-paganist, materialist, "Enlightenment" cult, directed its energies toward uprooting and crushing the Christian Platonic tradition of Cusa, Leonardo da Vinci, Kepler, Leibniz, et al. Thus, it sought to uproot and destroy such specific, crucial thought-objects as Cusa's negative definition of the elementarity of circular action as universal least action, of Leonardo da Vinci's treatment of Golden Section harmonics, of Kepler's partition of elementary spherical space-time into negative and positive curvatures, of the seventeenth-century development of the interdependent notions of non-linear and least-action function.
As Cantor demonstrates, especially by aid of his richly historical treatment of his subject, driving non-algebraic function to the remotest boundaries of both macrocosm and microcosm, works to such effect that the Platonic principle of negation enables us to discover the necessary, intelligible existence of causal agency far beyond the furthest reach of non-algebraic function. Before this discovery could be made, it were necessary, not only to discover non-algebraic functions, but to show, from this vantage-point, that all ontological assumptions premised axiomatically upon an arithmetic or an algebraic standpoint are intrinsically false. Only by establishing such unique relative authority of non-algebraic function and its intrinsic, Leibnizian principle of universal least action, could the basis be found for discovery of the higher manifold.
If we today look back to Kepler's distinction, respecting harmonic implications, between positive and negative spherical curvatures, and note the derivation of modern "non-algebraic" function theory from such roots, we should recognize in this way why ignorance of the elementary discoveries of Cusa, Leonardo, Kepler, Leibniz et al., would blind modern victims of an empiricist education into seeing nothing but the mechanistic, entropic implications of positive curvature, being thus blind to the interrelated, dominant principles of negative curvature and least action. Hence, they cannot understand the nature of those limits of non-algebraic function upon which Cantor's most crucial discoveries rest.
What Cantor shows in this way cannot be compared, or contrasted to formal notions of function, in any ordinary sense. What Cantor demonstrates in fact, by the argument elaborated in his 1895-1897 Beiträge,)81 is that the formal aspect of the ordinary notion of mathematical function, even non-algebraic function, is but a metaphorical reflection of an entirely different ordering, an ordering of thought-objects, which order itself is, ontologically, also such a thought-object.
We enter thus into a world of such conscious objects, that their origins, their nature, their place, and their implicit effect, can be communicated to other minds; but, in this case, the conscious objectthe thought-object itselfis unutterable in any mode of communication as such. In these cases, the communication of the object itself, from one mind to the other, occurs either by causing, dialectically, the creation of that other object in the mind of the hearer, or by prompting the hearer to recall such an earlier experiencing of the generation of that thought-object.
That is also to say, that such communication cannot be effected as the transmission of "information"; but, rather, only by subordinating the process of communication to the most intensive and strict methods of Platonic, dialectical reasoning. Hence, all "information theory," insofar as it pertains to human thought, is not merely a fraud, but a monstrously destructive attack upon an entire crippled generation of victims. Unfortunately, under the evil influence of John Dewey and his like, and the more evil influence of the Frankfurt School and, now, the current, "New Age" and related reforms generally in effect in our classrooms today, that destructive fraud has become the hoax, which today passes for a more or less accepted standard for education.
Now, we conclude with closing words on the matter of the problem of intelligibility in the communication of thought-objects.
Refer, once more, to the pedagogical series of formal theorem-lattices, A, B, C, D, E, .... The intelligibility of both A and B, for example, as member-terms of such a non-linear function series, is found in the change of "hereditary principle"of the axiomatic basiswhich distinguishes B from A. The ontological quality of this function of change is located formally "within" the aleph-manifold Type of discontinuity between each pair of terms. That change, so formally located, is the causal feature of the process as a unified whole. The equivalent aspect shared among all such changes in that series, defines a Type, and also defines a thought-object corresponding, as a One, to the generation of the Many terms of this series.
This aspect of the matter is expressed in the communication-process by the Platonic form of negation of the "hereditary," axiomatic principle separating one set of underlying assumptionsas for theorem-lattice Afrom all other sets of a seriessuch as theorem-lattice B, or C, or D, .... Communication in this Platonic mode, as employed, for example, by Cusa, is the only possible Type of communication of those conceptionsthought-objectswhich are not susceptible of explicit representation within the linear "band-pass" of any medium of communication itself.
The essential feature of all such Platonic communication is predominantly twofold. First, the essential thought-object, to which all other thought-objects should be referenced, is the notion of negentropy as that has been implicitly, metaphorically defined here (that takes into account "anti"-negentropy.) Second, that the reality to which our thought-object manifold's Type must correspond, is mankind's successfully negentropic social reproduction of our speciesimago viva Deiin the universe, by our negentropically-ordered changes in mankind's practice upon that universe.
It must be in art as in science. Truthfulness is Socratic irony, and Truth is a metaphor. We cannot say what we mean; but, we can render our conceptions of, and intent to change intelligible to other minds, by aid of a rigorous regard for the fact that information so-called is never more than metaphor.
45. LaRouche, "In Defense of Common Sense," chaps. II-IV; and "The Science of Christian Economy," chaps. II-IV, VI; in Christian Economy, op. cit.
46. Cf. Nicolaus of Cusa. Cusa's view in an early work, On Learned Ignorance, is that "God has implanted in all things a natural desire to exist with the fullest measure of existence that is compatible with their particular nature .... There is in them a discernment that is natural and in keeping with the purpose of their knowledge, which ensures their natural inclination serving its purpose and being able to reach its fulfillment."
Later, in "The Vision of God," (1464), Cusa develops the conception that each species, with its natural faculties as they develop, "yearns" for the existence of a higher species, as man does for the knowledge of the Absolute, of God. Here, Cusa's idea of negentropic species-evolution as the characteristic of Creation, is expressed by the poetic conception terminus specie. The universe consists of negentropic growth of higher orderings, whose microcosm is human reason. The species recognizes this divine order of creation, in its own way, and becomes a singularity in the transition from one ordering to the next. Thus the species has a terminus specie, the actualization of infinity in one point, which enables further development. "This power, which I have from Thee, and in which I possess a living image of the power of Thy almightiness, is the free will through which I am capable of either increasing or reducing the capacity to recive Thy beneficence."
47 See LaRouche, U.S. Science Policy, chap. III, section "The Geometry of Creative Reason," op. cit.
48 See LaRouche, "In Defense of Common Sense," chaps. IV, XII; "Project A," chaps. XVII, XVIII; and "The Science of Christian Economy," chaps. IV, VII; in Christian Economy, op. cit.
49 On Descartes' deus ex machina, see LaRouche, U.S. Science Policy, chap. IV
50 Sir Isaac Newton, in his The Mathematical Principles of Natural Philosophy (New York: The New York Philosophical Society, 1964), states "hypothesis non fingo" (I don't make hypotheses), and explains his reasons for this on grounds of induction versus hypothesis
51 See LaRouche, U.S. Science Policy, chap. IV, op. cit.
We first hear of the Bogomils in the tenth century a.d.i n Bulgaria. (In Bulgarian, Bogomil means "beloved of God.") Among their beliefs is the characteristically gnostic one, that the Father of Jesus Christ was not the Creator of the world. For the Bogomils and later the Cathars, the power of the devil worked through the nature and constraints of the material world; matter and spirit were never meant to co-abit. This division and its corresponding principles of good and evil, light and darkness, is broadly called dualism. For the origins of the Bogomil or Cathar cults in Manicheanism, and the Albigensian Crusade against them, see LaRouche, Christian Economy, pp. 485-486, op. cit.
52 The Cathar cult was known in France as the Bulgarian cult, or "Les Bougres," which translated into English as "the Buggers." Because of the cult's peculiar sexual perversion, which flowed from their gnostic doctrine of separation of matter and spirit, it resorted to various other kinds of sexual activity, and thus the name "Bugger" became associated in English with homosexuality.
Overt gnostic cultism continues to this day, including the sexual perversions. In Colombia, for example, the head of the Universal Christian Gnostic Church, Samael Aun Weor, is the author of a book entitled Perfect Marriage, which asserts: "The age of sex is coming, the New Age of Aquarius .... Sexual magic will be officially admitted in the universities of the new Aquarian Age." The book continues: "To create a child, you do not need to spill semen. The spermatozoid which escapes without spilling semen is a choice spermatozoid of a superior nature, totally mature. The result of such impregnation is a new creation of exremely high order. That is how we can form a race of Supermen. In the mysteries of Eleusis, the sacred dances, the naked dances, the burning kiss and sexual connection, they make men unto Gods ... the Sufi dances and the whirling dervishes are tremendiously marvelous." Aun Weor is also the author of The Social Transformation of Society, which sketches the Gnostics' political program for Latin America. The Gnostic Church has been the political controller of the M-19 narcoterrorists who today share power with the government of Colombia
53 See Friedrich Schiller, "The Legislation of Lycurgus and Solon," in Friedrich Schiller, Poet of Freedom, Vol. II, ed. by William F. Wertz, Jr. (Washington, D.C.: Schiller Institute, 1988)
54 See Fred Henderson, "Free Trade, The Confederacy, and Slavery," The New Federalist, Vol. V, No. 36, Nov. 11, 1991, pp. 5-6; "The Lee myth is debunked but not the more dangerous mythmakers," Executive Intelligence Review, Vol. 18, No. 38, Oct. 4, 1991, p. 62ff
55 The decrees of the Roman Emperor Diocletian (284-305 a.d. attempted to freeze the economic crumbling of the Roman Empire by fixing prices and wages by law. This led in the fourth century to the reforms of the Emperor Theodosius, which established legal enforcement of the occupation which each Roman citizen was forced to follow for his entire life. These Malthusian reforms were the earliest attempt to impose socialist decrees by totalitarian government. See Global Showdown, §2.3 (Washington, D.C.: Executive Intelligence Review, 1985), on the edicts of Diocletian and his successors
56 See, for example: Benjamin Franklin, "Proposals Relating to the Education of the Youth in Pennsylvania," Philadelphia (1749). Thomas Jefferson, "A Bill for the More General Diffusion of Knowledge" (1779), in Thomas Jefferson: Writings, ed. by Merrill D. Peterson (New York: Library of America, 1984): "[T]he most effectual means of preventing [tyranny] would be, to illuminate, as far as practicable, the minds of the people at large .... [Therefore] it becomes expedient for promoting the public happiness that those persons, whom nature hath endowed with genius and virtue, should be rendered by liberal education worth to receive, and able to guard, the sacred deposit of the rights and liberties of their fellow citizens, and that they should be called to that charge without regard to wealth, birth, or other accidental condition or circumstance." John Adams, "Thoughts on Government" (1776), in American Political Writing During the Founding Era: 1760-1805, Vol. I, ed. by Charles S. Hyneman and Donald S. Lutz (Indianaplis: Liberty Press, 1983). Benjamin Rush, "A Plan for the Establishment of Public Schools and the Diffusion of Knowledge in Pennsylvania; To Which Are Added, Thoughts upon the Mode of Education, Proper in a Republic" (1786), in American Political Writing, op. cit.
57 See Friedrich Schiller, "Aesthetical Lectures (1792-1793)" and Wilhelm von Humboldt, "On Schiller and the Course of His Spiritual Development," both in Friedrich Schiller, Poet of Freedom, op. cit. Humboldt, who predicated his work on the influence of and education provided him by Schiller, was for a time responsible for all educational policy in Prussia
58 See Immanuel Kant, Critique of Pure Reason, trans. by Norman Kemp Smith (New York: St. Martin's Press, 1965); Critique of Practical Reason, trans. by Lewis White Beck (Indianapolis: Bobbs-Merrill Company, 1956); also, in particular, Critique of Judgment, trans. by J.H. Bernard (New York: Hafner Press, 1951), §30-54, p. 152ff.: "[Genius] cannot describe or indicate scientifically how it brings about its products .... [A] Homer ... cannot show how his ideas ... come together in his head, simply because he does not know, and therefore cannot teach others."
59 Plato's arguments connecting the idea of the Good (or the Absolute Infinite as expressed by later Christian Platonists), both to the evolution of the physical universe, and to the process of Becoming proper to human reason, are developed with more and more arduous rigor in a number of dialogues: Theaetetus, Parmenides, Sophist, Republic, Philebus, Timaeus, Critias.
60 See footnote 58
61 Immanuel Kant, Critique of Judgment, op. cit.
62 See A Manual on the Rudiments of Tuning and Registration, Vol. I (Washington, D.C.: Schiller Institute, 1992), chap. 11
63 See footnote 1
64 Georg Cantor, Theory of Transfinite Numbers, op. cit.
65 See Nicolaus of Cusa, "On Conjectures," in Philosophisch-Theologische Schriften, Vol. II (Vienna: Herder & Co., 1982), p. 158. "Man is indeed god, but not absolutely, since he is man; he is therefore a human god. Man is also the world, but not in a contracted way everything, since he is man; man is therefore a microscosm or a human world. The region of humanity therefore embraces God and the whole world in its human potentiality."
66 See Nicolaus of Cusa, "On the Filiation of God," in Philosophisch-Theologische Schriften, op. cit.,, p. 640. "Indeed, just as God is the actual essence of all things, so is the intellect, separated and united in itself vitally and reflexively, a living similitude of God. Therefore, as God Himself is the essence of all things, so the intellect, the similitude of God, is the similitude of all things. Cognition, however, is effected through similitude. However, since the intellect is an intellectual living similitude of God, it knows, when it knows itself, everything in itself as the one."
See also Philo of Alexandria, op. cit., §XXIII: "Moses tells us that man was created after the image of God and after His likeness (Gen. 1:26).... Let no one represent the likeness as one to a bodily form; for neither is God in human form, nor is the human body God-like. No, it is in respect of the Mind, the sovereign element of the soul, that the word "image" is used; for after the pattern of a single Mind, even the Mind of the universe as an archetype, the mind in each of those who successively came into being was moulded.... [The human mind] opens by arts and sciences roads branching in many directions, all of them great highways.... (W)hen on soaring wings it has contemplated the atmosphere and all its phases, it is borne yet higher to the ether and the circuit of heaven, and is whirled round with the dances of planets and fixed stars, in accordance with the laws of perfect music, following that love of wisdom which guides its steps. And so, carrying its gaze beyond the confines of all substance discrenible by sense, it comes to a point at which it reaches out after the intelligible world."
67 See LaRouche, "In Defense of Common Sense," chap. XI; "Project A," chap. II; and "The Science of Christian Economy," chap. V; in Christian Economy, op. cit.
68 Cf. A Manual on Tuning, op. cit. References are the Preface: "The Classical Idea," passim; chap. 2: "The Six Species of Singing Voice"; chap. 9: "The Principles of Bel Canto"; chap. 10: "The Synthetic Geometry of Composition"; and chap. 11: "Artistic Beauty: Schiller vs. Goethe."
See also LaRouche, "Solution to Plato's Paradox of the 'One and the Many,' " and Jonathan Tennenbaum, "The Foundations of Scientific Musical Tuning," Fidelio, Vol. 1, No. 1, Winter 1992
69 A Manual on Tuning, op. cit., chap. 10
70 Cf. A Manual on Tuning, op.cit., chap. 11
71 The attribution of musical notions to Cantor's work is ironically most appropriate. Cantor was an able amateur musician, of a musical tradition traced to his maternal grandfather Kapellmeister Ludwig Böhm, whose violinist brother, Joseph, was the teacher of the great virtuoso Joachim. (Adolf Frankel, Das Leben Georg Cantors, cited in Georg Cantors Gesammelte Abhandlung, op. cit., p. 452.) It was this Ludwig Böhm who delivered the definitive performance of Beethoven's late string quartets on Beethoven's behalf
72 See footnote 46 for Cusa's concept of species-evolution
73 See, G.W. Leibniz, Monadology, trans. by George Montgomery (LaSalle: Open Court Publishing Co., 1989)
74 See footnote 3
75 The, unfortunately, popularized myth of an "Hegelian" division of musical history, into successive "baroque," "classical," and "romantic" periods, should be simply ignored as nonsense. The work of Classical composers such as J. S. Bach, his famous sons, Haydn, Mozart, Beethoven, Schubert, Mendelssohn, Schumann, Brahms, et al., is separated by a moral principle of composition from the contrasting, irrationalist principle of ascending chromatic eroticism adopted by such nineteenth-century Romantics as Berlioz, Liszt, and Wagner, et al.
76 Johann Sebastian Bach, "Musical Offering," BWV 1079 (New York: G. Schirmer, 1944)
77 Joseph Haydn, Opus 33, "Russian" string quartets (Mineola, N.Y.: Dover Publications, 1985), ed. by Wilhelm Altman
78 Cf. Bernhard Paumgartner, Mozart (München: 1991), chap. 31, pp. 299-311; p. 548
79 See A Manual on Tuning, op. cit., chap. 12 passim, on the principled approach of Beethoven and Brahms to composing a set of variations on a theme
80 Cited in Herbert Meschkowski and Winfried Nilson, eds., Georg Cantors Briefe, (Heidelberg: Springer Verlag, 1991), pp. 9-10, 478; from J. Bendiek, "Ein Brief Georg Cantors an Pater Ignazius Jeiler O.F.M.," Franzisch Kannischer Studien 47, 1965, pp. 65-73. "Die modernen Mathematiker in ihrer Mehrheit durch den glänzenden Erfolg ihres stets sich vollkommenden Formalwesens, das immer mehr Anwendungen auf die mechanische Seite der Natur zulässt, in einen Siegesrausch hineingeraten sind, der sie zur materialistischen Einseitigkeit verkommen lässt and sie für jegliche objektiv-metaphysische Erkenntnis and daher auch für die Grundlagen ihrer Wissenschaft blind macht."
81 See footnote 28.
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