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The Truth About Temporal Eternity


Appendix A: The Ontological Superiority of Cusa's Solution Over Archimedes' Notion of Quadrature 

The Truth About Temporal Eternity

Raphael Sanzio's "School of Athens" 

On that account, the present circumstances of the late twentieth century are comparable to, and probably more ominous than Europe's situation during the early fourteenth. Since no later than 1905, despite some elements of progress, even some admirable ones, the overall pattern of this century has been one of global decay of civilization through two ruinous world wars. Over the period since the assassination of U.S. President John F. Kennedy, there came a worse, presently ongoing collapse, into a neomalthusian "New Dark Age," into a "New Ager's postindustrial utopia." Despite the notable accomplishments which also have been contributed during these decades, the twentieth century has been, in the large, not "modern history," but rather "modernist history."
World War I was horrible, but the aftermath was worse. The moral decay dominated the 1920's everywhere, notably including postwar Weimar Germany.
Like Friedrich Nietzsche, these followers of Comintern cultural commissar Georg Lukacs were all existentialists: Adorno, Hannah Arendt and her lover Martin Heidegger, Horkheimer, and the rest of the Weimar Republic's "Frankfurt School." The difference among these positivist synthesizers of Sigmund Freud and Karl Marx, if only temporarily, was that Heidegger became Hitler's chief custodian of the Nazis' Nietzschean philosophical purity, while others, being Jewish, soon found their 1930's careers outside of Germany.^{1} At the end of that war, while the postwar Heidegger was being excused (rather hastily, some thought) for his propounding of Nazi dogma, the doctrine of the Frankfurt School's Adorno and Arendt was applied to certain among Hitler's opponents. Thus, some German Catholic theologians, in particular, were instructed by the AngloAmerican occupation to teach the democratic principles of Arendt's former lover, the thenrecent Nazi celebrity, Martin Heidegger.^{2}
Heidegger thus became a leading postwar influence among the theologians at Germany's Tübingen University. Karl Rahner, and the famous "liberation theologist," Hans Kung, among many others, reflect this. If it were "not politically correct" these days to mention the rope in the house of the hanged, similarly, even the bare word "truth" might be deemed offensive in the existentialist precincts of the Frankfurt School, or of its admirers.
Meanwhile, from France, existentialist Heidegger's cousins, socalled "Deconstructionists" such as Jacques Derrida, have spread their campaign against even the mere name of truth through the U.S.A.'s Modern Language Association; they have established their nihilist views as the reigning dogma of "multiculturalism" at most universities in the U.S.A. today.^{3} There, especially over the past two decades, truthfulness has come to be virtually banned, outlawed not only in the classrooms, but even from many Federal courtrooms. The most extreme version of the law of the racist Confederate States of America now reigns at some of the highest levels of those courts. As a result, more and more, Federal decisions embody a worse than Nazilike,^{4} "new McCarthyite" radical positivism derived from John Locke,^{5} a positivist hostility to truth which has now virtually replaced those principles of Leibnizian natural law originally embedded within the U.S. Declaration of Independence and Federal Constitution.^{6}
More broadly, although Queen Victoria's worldwide empire of gunboats and musketry is ostensibly a thing of the past, London had used its position as the most witting of the victors in two World Wars of this century to impose the empiricist, "Third Rome" ideology of Shelburne's and Palmerston's imperialism^{7} as a more or less globally hegemonic way of thinking. That empiricism rules imperially, still, the opinionshaping of most leading circles not only in Britain's former colonies, but also within the majority of most influential public opinion throughout most of the world, in politics, in the news media, in the classroom, and in the simpleminded whinings of the populists.
So, when the time came that Pope John Paul II issued his Veritatis Splendor to the Roman Catholic bishops throughout the world,^{8} that world had come into an apocalyptic time, like that of Biblical Sodom and Gomorrah, a time when official and private lying had become the hegemonic policy of public and personal practice worldwide, more pervasive in both official and private daily life than at any time in modern recollection. In the year 1993, as among Christian communities, the general condition of mankind was far worse overall than at that time, decades earlier, when the dupes of satanic Theodor Adorno first instructed the German theologians to adopt the the dogma of Hannah Arendt's former lover, Heidegger's neoRousseauvian "liberation" dogma, that one should unleash upon the world one's inner, infantile swine.
In the preceding paragraphs we have glimpsed a significant segment from a continuing current of European conflict between opposing forces for and against the cause of truth. We have defined thus a period extending through approximately sevenhundredfifty years of European history, from the death of Frederick II to the release of Veritatis Splendor. Therefore, now consider the proposition: After having once fallen into an apocalyptic, fourteenthcentury collapse of a formerly bright civilization, and later escaped from that "New Dark Age" into the brightest moment of rebirth in a millennium and a half of worldhistory, the fifteenth century's Golden Renaissance, how is it that Europe would permit itself, ever again, to be lured into yet another "New Dark Age"?
I.
The Golden Renaissance
Let us view the cause of truth, as the essence of an agesold conflict is shown most clearly by the most recent five and a half centuries of European history, since the 1440 sessions of the ecumenical Council of Florence.^{9}
Today's plausible reading of the available empirical evidence is that the human species, as we might define it for today, has existed upon this planet for not less than some two millions years. Yet, speaking from the vantagepoint of Leibniz's science of physical economy, we can report with certainty, that the increase in the potential populationdensity of mankind during the recent fivehundredfiftyodd years, since that Council, exceeds the sumtotal of all such human development over the millions of years preceding that.
The search for the secret of the unprecedented success of the revolution launched in the setting of that Council directs our inquiries into two interrelated, but distinct lines of inquiry. For most, it will be relatively less difficult to appreciate what they will consider the socalled "objective side" of this historical phenomenon. They will ask: What is the efficient connection between the quality of practical measures taken by the Renaissance and its heirs, and the practical results? Those socalled "objective" results can be expressed in the improved quality of personal life made possible for the many, and may be expressed also in other ways which correspond to a sustainable pathway of successive increases of mankind's potential populationdensity. The other side of this history, which is to receive the more intense consideration here, is the "subjective side": the study of those forms of mental life through which such efficient means of progress were rendered intelligible subjects both of conscious reflection and of willful practice of desired change.
The study of the interrelationship between those two sides of our topic, but with emphasis upon the subjective side, is the route by which we shall explore here a rigorous proof of a principle of existent truth. To this purpose, we shall emphasize those aspects of this proposition which can be addressed competently only from the included standpoint of the author's fundamental discoveries in the domain of physical economy.^{10} Once a few indispensable preliminaries are satisfied, we shall focus directly upon the indicated two sides of the matter. First, we must summarize those clinical features of the Golden Renaissance which define the scope of the key historical evidence required as the most critical zone for our investigations.
The central feature of the growth unleashed so uniquely by the Golden Renaissance's influence, has been the establishment of a new kind of political institutions, the institutions of a system of sovereign nationstate republics, each based upon a literate form of a popular language, and all dedicated, in their internal affairs and relations with other states, to a form of natural law which is traced historically^{11} through St. Augustine's writings,^{12} and reaffirmed by Gottfried Leibniz. The Renaissance's rich comprehension of such natural law also defined the notion of science in a new way.
This new form of political institution, wherever it emerged, was committed, inclusively, to fostering those beneficial changes in individual and national practice which are made available to mankind through fundamental scientific progress.^{13} It was this coincidence of natural law with both the new notion of a sovereign nationstate republic, and a consistent notion of physical science, which has caused the increase of the total human population from the several hundred millions maximum of times prior to 1400, to over five billions today, (see Figure 1)^{14} and potentially to a technologicallydetermined, and rising level of more than twentyfive billions.
The natural principle which was responsible for this sudden upward turn was not new. That ancient principle, called into play to produce this Renaissance effect, is that characteristic of the individual person which has always set the human species absolutely apart from, and above all other known creatures existing within Temporal Eternity. Through creative potential inherent in each human individual, but by no different means, the human species is enabled to increase its potential populationdensity willfully in a manner and degree which is impossible for any other species. As we shall stress here, this definition of the term creative is most easily recognized as the quality of mind typically embodied in the valid axiomaticrevolutionary discoveries of physical science.
This principle of creative potential within the individual person is the same quality of man's likeness to God already known to Mosaic Judaism in Genesis 2628.^{15} In Latin, Genesis 1:27 is referenced by the words "imago Dei (in the image of God)." We shall demonstrate, in the most rigorous way, that, as we have just stated, the two meanings, the power of valid "fundamental," or "axiomaticrevolutionary" discovery in physical science, and the creativity of "imago Dei" differ no more than as but different facets of one and the same quality. If human individuals were not endowed with this distinctive quality of imago Dei, science were impossible.
Presently, the earliest known trace of mankind's development of an actually scientific form of knowledge, is the surviving elements of the demonstrably prehistoric solarsidereal astronomical calendars of Vedic Central Asia, China, and Egypt. The already advanced Vedic solar astronomical calendars date explicitly from no later than 6,0004,000 b.c. , the Chinese perhaps earlier, like the preVedic IndoEuropean, and the prepyramid Egyptian solar astronomy probably as early as the Vedic, or approximately so. It is possible that calendars and navigation based upon scientific knowledge of equinoctial and longer sidereal and solar cycles date from a much earlier time; we have grounds to infer this, but corroborating material evidence of this is unreported to us presently. Nonetheless, once we become familiar with the distinctive characteristics of creativethought patterns—as opposed to deductive ones—conclusive evidence of a creativity coherent with imago Dei is reflected to us as its faded, fragile shadows cast tenderly upon mere shards of even the most primitive ancient artifacts.
There are many precursors of modern science, including those works of Plato which are the nearest approximation of its principle from ancient history. We neither exaggerate, nor do we dishonor the contributions from the distant past if we insist upon the demonstrable truth that these were but precursors of the science first established by the Renaissance.
Indeed, the practical difference between that Renaissance and earlier forms of Christian civilization, is epitomized by that founding of modern science. The key conceptions on which this development was premised are included topics of Nicolaus of Cusa's On Learned Ignorance (De Docta Ignorantia).^{16} From the standpoint of mathematics, among the many topics which that book addresses, the crucial feature is a demonstration of the proper application of the socratic method to overturn ultimately even the most widely and deeply believed professionals' axiomatic assumptions of all known formal mathematics existing up to that time. Hence, this use of socratic method is named de docta ignorantia. The key illustration employed to this latter effect in that book, is his successfully axiomaticrevolutionary application of the principle of Plato's Parmenides to solve the ontological paradox in Archimedes' theorems on quadrature of the circle.^{17}
As the relevant considerations of that time are articulated in the most concentrated and rigorous way by Cusa, this Renaissance revolution in political and scientific institutions proceeded from the evidence that all things which are knowable to mankind are accessible to intelligibility, and, therefore, that all mankind, through its leading institutions, is implicitly accountable to God for knowing natural law and acting accordingly.
The environment of the scientific revolution erupting in this Renaissance Italy is identified by such contemporaries of Cusa's as Filippo Brunelleschi and Paolo del Pozzo Toscanelli, and, later, by not only such avowed students of Cusa's works as Luca Pacioli, Leonardo da Vinci, and Johannes Kepler,^{18} but also Pascal, Huygens, and Leibniz.^{19} An enhanced view of Cusa's influence on fifteenth through nineteenthcentury scientific progress is afforded by reference to Cantor's writings on relevant highlights of the history of science, at the close of the nineteenth century.^{20}
Unfortunately, European history since 1440 has not been so onesidedly good as the foregoing might suggest at first reading. Unfortunately, there was an extremely powerful opposition, which has been working ruthlessly from the fifteenth century to the present day in the attempt to exterminate even modern memory of those policy measures which characterize both the Renaissance Council of Florence and the science which that Council contributed crucially to setting into motion. That hatefilled opposition to the Renaissance, which was typified early on in the neoaverroist Aristotelianism of Padua's Pietro Pomponazzi, represented the interests of that Venicecentered, international financial oligarchy whose usurious practices had been central in the earlier collapse of Europe into the "Dark Age" of the fourteenth century.
Typical of this opposition is the case of Britain's Sir Francis Bacon and his empiricist faction. Baconian empiricism was chiefly the work of a faction of Venetian financier oligarchs headed by one Paolo Sarpi. In Britain, from the close of the seventeenth through the midnineteenth centuries, the followers of Sarpi's faction were known as "the Venetian party," or "the British Liberals." This "Venetian party" of Marlborough, Walpole, Shelburne, et al. was also known as the Illuminati, or "Enlightenment" faction. This conflict between the two opposing forces, Renaissance versus Enlightenment, has become the characteristic, defining internal conflict of European, and, more recently, world history, down to the present date.^{21} This continuing conflict between the traditions of Cusa and Leibniz, on the one side, and our enemies Locke and the existentialists, on the other, is to be recognized in today's life as our heritage of resistance to today's "Distant Mirror" of the fourteenthcentury "New Dark Age."^{22}
A competent critical reading of every proposition crucial to what we have to report from this point onward hangs upon the reader's ability to recognize the constructive^{23} definition of the term "creativity" as that term is employed here. For that reason, we now summarize that same definition which we have employed on other locations.
For our purpose here, it were sufficient to say that Plato's Parmenides dialogue is, without exaggerating, the most important scientific pedagogical exercise composed during no less than the recent two and a half thousand years. The same conceptions are present within other dialogues of Plato; the Parmenides not only makes the most crucial point respecting all formal mathematics or mathematical physics, but accomplishes this with a stunningly rigorous compactness which the greatest thinkers since might have but dreamed of matching.^{24} The most crucial issue of all formal scientific utterance is embedded in the single ontological paradox which that dialogue defines. As in other locations where this present writer addresses that topic, he hinges the definition of scientific creativity upon the demonstration of Plato's Parmenides principle which is typified by Nicolaus of Cusa's De Circuli Quadratura.^{25} The construction of the Parmenides ontological paradox is most simply illustrated in a way which is also the most useful pedagogically, by taking up Archimedes' quadrature of the circle as a topic to which Plato's principle is most aptly applied.
One might begin the classroom blackboard exercise with a circle and a pair of respectively inscribed and circumscribed squares. Next, double repeatedly, at an equal speed, the number of sides of each of these respectively inscribed and circumscribed polygons. At that point in the lesson, our attention must be turned to the famous "method of exhaustion" associated with a mathematician of Plato's Academy of Athens, Eudoxus.^{26}
Let the class ask itself: What is the relationship between the circular perimeter and the perimeters of the polygons when the n of 2^{n} becomes extremely large? Focus upon two adjacent sides of the inscribed polygon at that instant of the ongoing process, as if in a suitably powerful microscopic enlargement. Examine the relationship between the two polygonal perimeters in that vicinity, and the segment of circular perimeter lying between them. Extend the process to a value of 2^{(n+n)}. Repeat the microscopic scrutiny. Extend the process to the degree that a polygonal side the length of one micron would require a circle larger than the currently imagined largest size of our universe. It changes, but it remains the same: the polygonal species and the species responsible for the existence of the circle can never become congruent.^{27}
At this point, the Classical scholar must recognize that this problem of quadrature has affinities with Plato's Parmenides. It appears that the circular action, which both generates the circle and is crucial for constructing the polygonal series, defines and bounds externally^{28} all the polygons of this series, but can never be a member of the series which it defines in a subsuming way.
At this juncture in the experiment, the student might pick up his drawing compass, studying it very thoughtfully: This compass has no place to exist within the set of axioms and postulates of what we term Euclidean geometry! This Archimedean construction which we followed so faithfully has a terrible error of assumption built into it, at least as that theorem has been ordinarily presented in schools. The act of circular rotation, which defines and bounds the polygonal series, is not allowed within the set of Euclid's ontologically axiomatic notions of point, and straight line as a "shortest distance between two points." The latter set belongs to the domain of mere space; circular action belongs to the domain of spacetime—as Johann Bernoulli and Gottfried Leibniz proved the latter in 1697, when they established nonalgebraic mathematical physics, and did so on the basis of the physicalgeometrical principles of refraction of radiated light.^{29} Some of the deeper implications of this for mathematical physics awaited those fundamental discoveries which Georg Cantor presented two centuries later, in 1897.^{30}
The "handwaving," Brotgelehrter professor^{31} before the blackboard ends his treatment of that topic with the sophistry of presuming that the possibility of increasing the mathematical approximation of the curve by the polygonal perimeter indefinitely signifies that "ultimately" the two must coincide. Cusa's refusal to accept that sophist's fraud was the basis for the later, 1697 establishment of the nonalgebraic higher mathematics of spacetime by Bernoulli and Leibniz.
The construction actually proves directly the opposite to what the "handwaving" professor asserted so blithely. To a scientific mind, that construction proves that never can the two coincide, because they represent different species of existence. In the domain of mathematical physical science, that quality of socratic negation is the onset of a creative mental act of axiomaticrevolutionary discovery.
This leads to a further step. If we avoid the trap of reading the word "halving" in an empty, arithmetic way, we are obliged to examine the construction by means of which the series 2^{n} might be generated in visual and furtherextended spacetime. The construction itself is bounded by circular action. The proposition must be restated accordingly: The possibility of generating indefinitely the series 2^{n} depends upon circular action; circular action is thus the crucial feature of the generatingprinciple of construction of the transfinite^{32} series of polygons, both the respectively inscribed and the circumscribed series treated as a single series. Thus, the same quality of circular action which bounds the inscribed series externally and the circumscribed series internally also determines the generating principle of both series, and, in that sense, bounds the combined series externally, from outside and above the set of axioms and postulates upon which a Euclidean geometry of simple space depends for all its consistent theorems.
Thus, creative mentation concludes, the difference between the species of polygons in Euclidean space and circular action is an ontological difference; therefore, the use of Archimedean construction to approximate a circular perimeter by averaging the difference between the two polygonal 2^{n} series, prompts the eruption to view of an underlying ontological paradox. The species of circular perimeter can not be generated honestly as a theorem from the set of axioms and postulates of formalist Euclidean space. Thus, the two species are distinct.
Yet, by multiplyconnected circular actions, we can generate all of the valid spatial existences and theorems of a formal Euclidean geometry of simple space, without resort to Euclidean ontological axioms. Thus, the circular perimeter's existence can not be comprehended from the standpoint of the formal Euclidean geometry, but the Euclidean geometry, minus its failed ontological axioms, can be fully comprehended from the standpoint of substituting the axiomatic quality of circular action for the ontological axioms of Euclidean formalism. The spacetime of axiomatic circular action, is ontologically the superior, relatively higher species of existence.
Furthermore, that which is thus shown to determine the existence of that transfinite series, the which fully comprehends that series, is not a member of the formal theoremlattice for which the members of the series are each ostensibly theoremmembers. That is precisely an illustration of the ontological paradox which Plato used, in his Parmenides, to demolish the "hereditarily" Eleatic method of such sophists as the immoral rhetorician Aristotle.^{33} Formally, this is Plato's root for the 1897 work of Georg Cantor, in his Beiträge.^{34} On this point, Cantor is echoed famously by the original work which established Kurt Gödel as one of the firstrank scientific minds of our century, Gödel's beautifully elementary and devastating, axiomatic obliteration of the scientific pretensions of Bertrand Russell.^{35}
The generationprinciple which is a higher species than any member of the theoremset of a transfinite ordering, stands ontologically outside and above each and all members of the set. It is the One which subsumes, thus, the Many. Plato's principle precisely. The One is distinguished from the Many by the quality of change. So, in the instance of Cusa's discovery of what became known later as nonalgebraic or transcendental functions, circular action is the principle of change which bounds and defines the double polygonal series. The circular perimeter, whose ontological content is change, is a singularity, relatively an absolute, virtually zerodimensional mathematical discontinuity, which both unites and separates absolutely the two series, the inscribed and circumscribed, as avowed student of Cusa's work Johannes Kepler explores the astrophysical and other implications of this around the beginning of the seventeenth century.^{36}
Over the years, this writer has adopted the following pedagogical device to assist students in conceptualizing what we have just described here, but in a more general way, as a matter of a general principle.^{37}
Let us consider any case of a creative discovery formally analogous to what we have adduced just now from the case of Cusa's Platonic solution to the quadrature paradox. Take into account the comparison made among Plato's Parmenides, Cusa's De Circuli Quadratura, Cantor's Beiträge and related discoveries, and Gödel's devastating exposure of the axiomatic blunders of Bertrand Russell.
Let us, in the manner of a socratic dialogue, consider the proposition that all scientific propositions can be reduced ultimately to the terms of a perfected update of today's principles of generally accepted classroom mathematics. Then, let us take into account the proofs given refuting that proposition, successively, in various forms, of the principle of Plato's Parmenides: those of Plato, Cusa, Cantor, and Gödel, notably. Let us represent this treatment of the proposition in the following way.
Let us therefore propose to represent all axiomaticrevolutionary discoveries in physical science by a series of the form
A, B, C, ... n
(for which "n" is the number of the ith term of this series).
Let "A" signify a formal Euclidean geometry of simple space, and "B" signify a nonalgebraic geometry of the CusaKeplerLeibniz speciestype. Formally, we may proceed from the axiomatics of "B" to generate all valid theorems of "A," although none of these will be consistent any longer with the set of axioms of "A"; we may not reach any of the consistent theorems of "B" from the axiomatic basis of "A." From the standpoint of formalism, to reach "B" from "A" we must make an intellectual leap of the sort which Cusa effected in solving the ontological paradox of Archimedean quadrature. To the formalist, this "leap" appears an "un"rational act of blind intuition; as we shall indicate in the next topical section, it is that "intuitionist" view which is blindly irrational.
Let "C" signify the higher transfinite types discovered by Cantor. As a matter of informing the reader who may not have been aware of these relevant historical facts of earlier, we report the following additional considerations respecting Cantor's discovery.
The first statement of the mathematical problem solved formally by Cantor (1897) is Leibniz's Monadology,^{38} as that Monadology was attacked falsely by Leonhard Euler in the latter's "Letters to a German Princess" (1761).^{39} Leibniz's notion of a monadology had its formal mathematical basis for intelligibility in his general notion of an analysis situs.^{40} This issue came freshly to the surface among the collaborators and other students of the work of Carl Gauss, notably Lejeune Dirichlet, Bernhard Riemann, and Karl Weierstrass. As Riemann put the point, the issue among those leading mathematicians is that in continuous spacetime no naive denumerability of the kind attributed to an ideal purely arithmetic domain is possible.^{41}
As the White translation of Riemann's paper puts the point, "[t]his path" (a continuous manifold in the domain of mathematical formalism) "leads out into the domain of another science, into the realm of physics."^{42} Such were the ontological implications of Georg Cantor's discoveries in mathematics, which provided formal intelligibility of this continuum problem within the domain of the transfinite. This is also the related implication of Gödel's referenced work, as systemsanalysis founder John Von Neumann failed to comprehend this significance of Gödel's proof. Cantor's discovery supplied the mathematical conceptions appropriate for the domain of the nondenumerable in physical spacetime: the domain of those virtually nulldimensional, but curiously efficient singularities, the which are the hallmarks of the modern physics of the quantum field, and which are the cornerstone for a notion of "notentropic" function in the science of physical economy.^{43}
What we said of the noncommutative formal relationship between A, the algebraic domain, and B, the nonalgebraic or transcendental, is also applicable to the relationship of C, the higher transfinite domain, to B. From A to B, and from B to C, we can proceed upward only by what must appear as "arbitrary leaps" to an observer selfblinded by his own obsessive adherence to radical formalism.
Such radical formalists, such as the Aristotelian or quasiAristotelian formalists Pietro Pomponazzi, René Descartes or Immanuel Kant, can interpret such "leaps" only as mysteries, as blind, irrational mysticism. Those formalist professors and their credulous admirers delude themselves as a man who denies the existence of that of which he has deprived himself. On no higher authority than their own refusal to comprehend the reality lying outside the domain of their formalism, for them, what they have not succeeded in attaining has no intelligible, has no more than a mystical existence. As Gasparo Contarini showed himself to have understood his teacher, Pietro Pomponazzi's own soul could exist for poor Pietro only once that Paduan had proven, by rigorous Aristotelian logic, that he had no soul; his God existed for him only in a similar way, a Kantian unintelligible thinginitself. Pomponazzi's soul was for him, as an Aristotelian, an imaginary object which existed only in that domain of paganist theologians' irrational mysticism. It existed only within that domain of irrationalist fictions where dwell William James' "varieties of religious experience,"^{44} within the ancient heathen domain of delphic faiths adored by consistent Aristotelian sophists.^{45} This is the tendency of weakness in today's commonplace forms of attempts to assert a principle of truth: that commonplace which has been exploited with such frequent, gloating success by the existentialists Friedrich Nietzsche, Bertrand Russell, Carl Jung, and Martin Heidegger.
Fortunately, it is not absolutely necessary to be as foolish as these formalists. What appear to the professional ignorance of the formalist as "arbitrary leaps," are fully intelligible actions, fully susceptible of unassailable proof. On that basis, an intelligible principle of creative acts of axiomaticrevolutionary discovery is accessed similarly, an intelligible principle of natural law, of universal truth, most usefully described otherwise as "The Truth About Temporal Eternity."
Thus far, we have situated the "leap" which we have designated as the formal representation of the occurrence of an axiomaticrevolutionary, or creative act of scientific progress. To render human creativity intelligible, we must define it next as also a mental object of conscious thought.
The Golden Renaissance and its continuation through some nineteenthcentury expressions of it, is typified by the mode of Christian humanist education traced from such a fourteenth through midsixteenthcentury model as Groote's and Thomas à Kempis' teachingorder, the Brothers of the Common Life. It may be traced thereafter through the Prussian educational reforms, according to the prescriptions of Friedrich Schiller, as developed and introduced by Wilhelm von Humboldt.^{46} This Christian humanist tradition is the only model policy yet developed which explicitly addresses the task of fostering the development of the powers for creative discovery in the student—in direct opposition to popularized forms of "textbookbased" education. We include in this Christian humanist tradition, much of the work of the French Oratorians, for example, as echoed in France's 17941814 Ecole Polytechnique under the direction of founder Gaspard Monge.^{47}
Return to the leading point introduced earlier, under the rubric of "Golden Renaissance."^{48} The development of the potential populationdensity of mankind, first in western Europe, and then throughout this planet, which occurred since the beginning of the fifteenth century, exceeds the accumulated net development of society throughout all man's existence on this planet before that. This is the case despite the evil, typified by Britain's "Venetian" empire, and by empiricist immorality, which has been the powerful adversary of the Renaissance, and of mankind, through all of these recent six centuries. Acknowledging the great indebtedness which that Renaissance has to the contributions of many branches of humanity earlier, the active principle of this Renaissance is the highest form of society, morally, intellectually, and materially, which has existed on this planet up through the present time.
It was born in Europe, as the Christian humanism epitomized by the writings and related work of Nicolaus of Cusa; but, as a glance toward the educational grounding of Cusa himself attests, the power of Christian humanism lies in its unmatched capacity for treasuring the greatest known true contributions of all mankind before it. Christian humanism was rooted in the rise of European civilization, as the early IndoEuropean (Classical Greek) contributions were reflected in the Platonic tradition known to the Hellenic world of the Christian apostles inside and outside of Palestine. The principles of Christian humanist education, typified as we indicate here, are the source of the extraordinary, unprecedented power of this European Renaissance.
Today, whatever parent wishes to afford his child, or his nation the fullest possibility for equality of achievement, must turn to the heritage of these Christian humanist, Renaissance principles of education. It is this Platonic tradition, as reflected in Classical humanist education, which affords us, uniquely, the means for rendering intelligible "the truth about Temporal Eternity." Once the implications of a science of physical economy are situated with respect to an intelligible principle of scientific creativity, known in these Renaissance terms, the certainty of that truth becomes for us a fully intelligible object of conscious thought.
Gather up a selection of the brightest youth of secondaryschool age, with no distinction made among their putative social rank. Rally them under a program of Classical studies, emphasizing early the greatest productions in Classical Greek and Latin, but, above all else, teaching the students, in succession, as they are prepared for each next step, to relive the known, original great, axiomaticrevolutionary discoveries of all human history to date. The case we have outlined for replicating Cusa's discovery of an axiomaticrevolutionary solution of the Archimedean quadrature's ontological paradox, is a model for what we signify as "reliving the original discoverer's state of mind during the act of effecting the original discovery."
For the case of mathematics, and so forth, the student gains not a scorable classroom, textbook mastery of approved observations, experiments, and standard formulas, but, instead, an actual reliving, within himself or herself, of the intellectual experience of the original discovery. Thus, a part of the living tissue of the student's mind is occupied by a living quality of replication of the mind of the greatest, longdead discoverers. That student learns thus to command the living minds of the greatest discoverers of the past, revived within himself or herself, to master the original problems of the present.
What transpires, during the reliving so of a succession of original discoveries, is the mustering of the student's inborn and partially developed capacity for the kind of creative thinking which appears in the mistaken opinion of an Aristotelian formalist as an "irrational leap of intuition." The occurrence of that reenactment of the mind of the original discoverer within the mind of the student, defines that experience as an object of the student's consciousness. This mental object appearing to the student's consciousness in that way, is constructed of the student's own, mustered capacity for an act of creative discovery congruent with that of the original discoverer. The content of that constructed image is nothing other than a process of creative thinking, a species of thinking absolutely distinct from logical formalism.
Indeed, it is our ability to contrast the creative "thoughtobject"^{49} with those of our own states of mind we recognize as merely the inferior level of logical formalism, which is the basis for most scientific creative work.
As the chosen example of Cusa's discovery illustrates this point, most of the fundamental and closely related mathematicalphysical discoveries in all known history correspond to this particular model of what is termed Platonic higher hypothesis. One drives a logical construction to beyond its limits, in the most rigorous way possible, searching for a devastating, axiomatic quality of ontological paradox in those extremes of vastness or smallness. Once such a paradox is provoked into appearing, the Eudoxian "method of exhaustion" by means of which the paradox is evoked, is examined from the standpoint of the solutionprinciple of Plato's Parmenides. That tactic, or method of generating a succession of revolutionary hypotheses, represents thus an higher hypothesis.
The formalist state of mind is obsessed with method of formal proof, formal consistency with a set of underlying, axiomatic assumptions. Creative discovery signifies overthrowing some of those axiomatic assumptions; for such a case a formal proof is not possible.^{50} The person who does not immediately recognize the empirical distinction between the two distinct species of thinking, is neither a scientist nor a competent policyshaper or other professional in the field of education.
The student advantaged to enjoy such a Christian humanist mode of secondary education, thus locates knowledge not in mere "facts," but in the process of generating knowledge within those creative processes which are empirically defined for that student by the repeated reliving of the moments of valid discovery by original discoverers. That student, by the time he or she is graduating from such an institution, can recognize readily the significance of Plato's term hypothesis. He or she can recognize those kinds of discovery achieved through overturning previously held axiomatic assumptions: valid such discoveries are Platonic hypotheses. Similarly, once the student comprehends individual hypothesis in this mode, the student is able to employ the method of hypothesis to define the higher One subsuming a large array of individual valid, axiomaticrevolutionary discoveries (hypotheses). All of the discoveries which, as a (e.g., transfinite) series are generated by a common (higher) hypothesis respecting the method of generating such discoveries, are a Platonic Many commonly subsumed by a Platonic One. That higher hypothesis, the One, is a higher hypothesis. We have already indicated the use of the solutionprinciple of Plato's Parmenides, to solve a paradox generated by "method of exhaustion," as a model example under the definition of higher hypothesis supplied here.
Similarly, the existence of alternative forms of higher hypothesis obliges the student trained in consciousness of hypothesis to hypothesize higher hypothesis, in the sense that higher hypothesis is defined by hypothesizing hypothesis.
Admittedly, we are employing the term "hypothesis" here in a manner different from that in generally accepted classroom use, or, in the formalization of plane and solid geometry. In rebuttal to any objections along those lines, three points can and should be made. One: Plato was there first; two: his definition of hypothesis conforms to an adequate definition of mathematics and physical science. As the emergence of, first, nonalgebraic, and, later, transfinite mathematics demonstrates, mathematics as a whole becomes incomprehensible unless we approach the matter historically from the standpoint which Plato represents by his definition of hypothesis. Three: Today's commonly accepted classroom definition of "hypothesis" came into being because Aristotelian and Hellenistic formalists sought to castrate geometry, by degrading it from a constructive (e.g., synthetic)^{51} geometry, to a sterilely fixed, formalist theoremlattice.
Wherever modern science occupied itself with fostering revolutionary progress in mankind's power to survive as a species, rather than rote teaching of dead algebraic dogma, the practical revolutionary implications of Plato's notion of hypothesis came back into play.
Hypothesis, considered formally (i.e., statically) signifies what modern theoremlattice doctrine would recognize as an "hereditary principle." Given, any set of axioms and postulates, treated as interdependent, the expandable array of theorems which may be derived as consistent with each and all of those axioms and postulates is transfinitely defined as a Cantorian type. Thus, formal proof belongs only to the inferior domain of showing consistency with such a fixed hypothesis, as representable formally by a fixed set of axioms and postulates. The theorems of that fixed lattice are a Platonic Many, and the corresponding hypothesis a Platonic One.
However, hypothesis is not located fundamentally in terms of the fixed theoremlattice with which the results of a particular hypothesis may be associated. As the Parmenides indicates, the ontological content of hypothesis is change, the Cantorian type of change which it incorporates as the process of creativemental action which brought it into being. It is in this aspect, as change, that a succession of hypotheses, as a Many, corresponds to its appropriate One, an higher hypothesis.
Consider now the implications of the following series of conditions.
From the historical vantagepoint identified thus far, it is implicit that no generally accepted mathematics has the qualifications for proving anything but consistency; in the search for scientific truth, we must rely upon entirely different means. The appropriateness of any particular choice of mathematics is located in the adducible Platonic form of hypothesis to which that mathematics, representable as a theorem lattice, corresponds transfinitely. Yet, neither consistency, nor appropriateness are synonyms for scientific truth. The quality of relative truth of an hypothesis, if it, in particular, satisfies the conditions of relative truth, is derived from the principle of generating hypotheses.
That principle also may be termed a method of scientific discovery which subsumes that hypothesis. This principle is an higher hypothesis in the same sense that the application of the solutionprinciple of Plato's Parmenides to an Eudoxian ontological or related paradox has been used here as illustration of a relatively common choice of higher hypothesis. Even relative truth is to be found in no place inferior to the domain of higher hypothesis.
Consider another notion of mathematical form of higher hypothesis, one not included in the terms of that higher hypothesis premised upon a Platonic treatment of Eudoxian ontological paradoxes: Consider harmonic orderings which are either coherent, or not coherent with the Golden Section as an externally bounding, asymptotic limit: the higher hypothesis upon which Johannes Kepler premised his construction of the Solar System according to a quantumfield principle.^{52} The history of this harmonic principle for generating hypotheses, from Plato, through Kepler, and beyond, is also an higher hypothesis.
Those two higher hypotheses may be combined, to form a third. The first, Eudoxian form of hypotheses corresponds to the sense of vision = spacetime. The second, quantumfield, corresponds to the sense of hearing, and of natural vocalization by a full spectrum of the six characteristic adult voicespecies of spoken/sung languages.^{53}
The consideration (hypothesizing) of these three, each welldefined notions of higher hypothesis, illustrates the significance of the term hypothesizing the higher hypothesis. This mental activity locates us ontologically within a domain which Plato terms "The Becoming." This definition of "Becoming" Georg Cantor equates to his generalized Transfinite.^{54}
This poses, as Cantor emphasizes, the equivalence of what Plato identifies as the "Good" to what Cantor designates as his "Absolute." This Becoming, or generalized transfinite corresponds to the highest possible ontological significance of physical spacetime, as does Cantor's generalized transfinite. This, generalized, corresponds to what this writer chooses to identify, descriptively, as "temporal eternity." That descriptive term, temporal eternity, is required to distinguish a transfinite notion of "eternity" from the "timeless absolute" of the Good.^{55}
That Good, or Absolute, is defined by hypothesizing the generalized "hypothesis of the higher hypothesis." The resulting conception can be nothing but the bounding of temporal eternity by an intelligent, timeless Absolute which is efficiently coincident at each moment, in each place, with all moments and places of all temporal eternity: The Absolute One, the Good.
That is the roadmap to guide us through the work now to follow.
In significant part, the implications for classroom mathematics of what has been presented here thus far, is fairly straightforward. Let us go directly, therefore, to a point which may not seem to be so straightforward. Next, let us construct the relevant anomaly; then, examine that anomaly's import for the determination of truth. We begin so, next, with the most crucial feature of a science of physical economy: the issue of "notentropy."
Leibniz, Hamilton,^{56} and others have defined the general form of the physicaleconomic transformation which corresponds to successful growth of any economy. It is implicit in that statistical "model," that there exists a level of growth—of net increase of the percapita, perhousehold, and persquarekilometer "productive powers of labor"^{57}—which is just barely above the level at which entropy ("dying") takes over. In order to construct a system of linear inequalities to describe the form of the phenomenon, it is not necessary to know in advance the precise value at which that transition from entropic to "notentropic" occurs. Initially, we are designing the experiment, so to speak; that experiment will indicate to us the relevant values for scaling.
So far, so good. Then, comes the excitement. The mathematical function so described is formally nondeterministic, no matter what the scaling values prove to be. One of the early results of this experience, is to look at all of generally accepted classroom mathematics, and mathematical physics in a disturbingly fresh way.
Let us now build up a mathematical description of the conditions which must be satisfied to maintain the current human population of this planet above the level of entropy in mankind's potential populationdensity.^{58} Note, that a zeroentropy, "equilibrium" state, between entropy and notentropy, is, in this function, a mathematical discontinuity corresponding to a condition which does not, and could not exist in a reallife physicaleconomic process (and not in a respectable conjectural model, either).
The description begins with a simple requirement that the rate of increase of potential populationdensity be greater than zero. This requires some improvement: in effect, technological progress; this is a modification of social behavior which enables man to overcome some boundary condition ostensibly barring the way to maintaining an abovezero level of increase of potential populationdensity. This is expressed as a transmission of a selfimproving culture, to the effect of improved skills being added to the heritage of earlier generations' contributions.
This already defines three constraints: increases per capita, per square kilometer, and of physical productivity per capita and per household.
This function is delimited not only by technological progress, but by the conditions required to realize that progress. Those conditions are expressed chiefly as improvements in the appropriateness of the area used, per square kilometer and per capita, and improvements in the tools and materials of production. These require expression in terms of structural changes in the division of physicallyproductive labor.
Look at this general model under conditions emerging millions of years later, especially the changes required to sustain the progress (in potential populationdensity)—where they have occurred, in fact—during the recent six hundred years of European and North American development.^{59} The significance of focussing upon this segment of the evidence is that the vastly more rapid rate of increase of mankind's potential populationdensity, beginning in the Renaissance, more than fivehundredfifty years ago, affords us a more concentrated expression of the determining quality of change.
The characteristic of this recent six hundredodd years of European culture and its influence, is the increase in the rate of urbanization. The reasons for that increase are implicit in the set of constraints already listed here: the requirement of increasing emphasis upon improvements in suitability of landarea and in tools, and also the implicit cultural requirement of an increase in the physical standard of household consumption and in lifeexpectancies. Such changes imply already an increase in urbanization relative to the percentile of the total laborforce required for physically essential rural occupations. These changes are much slower and marginally more modest in earlier periods of history (and, of course, prehistory), but, nonetheless, are efficiently present always, positively or in their neglect.
Urbanization signifies more than a rising intensity of these changes. New categories of change emerge lawfully from out of the belly of the old. Not only does the percapita, and persquarekilometer requirement of general infrastructural development (water, transportation, power, sanitation, etc.) become much more significant, but the effects of an indispensable rise in capitalintensity and powerintensity, per capita and per square kilometer, produce sideeffects of great significance. These required qualitative structural changes in the social division of (principally) physicalproductive labor, confront us with the required set of descriptive constraints in their most anomalous form.
It is sufficient for our purposes to consider only a few of the outstanding features.
Make a cut in time, through an interval in that physical spacetime process which is the role of production in effecting the social reproduction of the human species. The combination of skills of productive labor and preconditions for productive employment of that labor, represent a social cost. Designate the rate of flow of this total social cost, seen as the productive process in flux, at the brief moment immediately before the cut, as "energy of the system." See the rate of useful physical output of the productive process, at the brief moment after the cut as "output of the system." Compare these two values in terms of an implied function corresponding to changes in the values of a ratio of the two: of "output of the system," thus defined, to "energy of the system," thus defined.
Consider this ratio in terms of the percapita, perhousehold, and persquarekilometer values of each of these respective terms of the ratio, and of the ratio itself. Effect this comparison, in these listed terms of reference, in terms of "marketbaskets." There are two broad classifications of marketbaskets: households' consumption marketbaskets, expressed per capita and per household; producers' marketbaskets, per capita and per square kilometer. Both are expressed in terms of projectible potential populationdensity (e.g., the Netherlands or Belgium as a comparative standard of reference for humanity today, at today's level of technology available).
There are two magnitudes chiefly to be measured: time (in available workingyears of adult life of members of the laborforce), and comparative quantity and quality of physical goods contained within each of households' and producers' marketbaskets. To these physical goods must be added several required types of services: education, medical, and science. These latter three are included in both the households' and producers' marketbaskets.
The result of applying such categories of measurement to the actual modern history of physical economy is chiefly the following. The increase of the potential populationdensity of society as a whole is dependent chiefly upon the following constraints, applied to the function of the ratio as we have just described it.
These four constraints, so situated, describe a process which satisfies the definition of "notentropic." The "history" of the evolutionary development of the Earth's biosphere, is also such a "notentropic" process, as, not irrelevantly, Cardinal Nicolaus of Cusa defined the correct notion of evolutionary development in his "Vision of God."^{61}1. The percapita and perhousehold consumption must increase in terms of comparative quality and quantity of contents of the total marketbasket. Yet, the time required to produce that enhanced percapita and perhousehold marketbasket must be less than that required to produce the earlier, poorer quality and quantity of percapita and perhousehold marketbaskets.
2. Urban physicalproductive employment and marketbaskets output must increase relatively over rural, up to an asymptotic limit of feasible reduction in percentile of rural.
3. Producers' goods marketbaskets must increase relative to households' goods marketbaskets, both in time of production and in quality and quantity of percapita and persquarekilometer composition.
4. Thus, the designated "energy of the system," per capita, per household, and per square kilometer, must increase absolutely. However, the following must also apply. Let the difference between the numerator and denominator of the ratio, after deducting for "overhead" factors, be designated as relative "free energy" of the process; the ratio of "free energy" to "energy of the system" must increase.^{60}
We have thus defined a powerful anomaly, the most important and most ancient in known science since the time of Plato. This "notentropic" image of both processes, the physicaleconomic and the evolutionary development of the biosphere, can be measured in the manner we have indicated here, and in analogous, more or less refined ways. It is always measurable so; in that sense, it satisfies broadly our general notion of a succession of terms of a mathematical function, a function which may indeed be contrasted with any modern statistical model for any of various sorts of entropic functions. The effect of this comparison upsets people, especially semiliterate sciencesports fans cast in the roles of cheering spectators in the grandstand of the mathematicalphysics professionals' derbies. We are confronted thus with an anomaly: for numerous among the relevant professionals, an extremely disconcerting sort of sharp formal discontinuity in the domain of generally accepted classroom mathematics.
From some professionals' quarters, in recent decades, the popularized response to the appearance of this disturbing anomaly has been what we might fairly describe by the term "reaction formation," the radical positivist's dogma of "negentropy": the low probability assigned to a virtual timereversal of the Boltzmann Htheorem function for statistical entropy in a stereotypical mechanical gas, or analogous system.^{62} We suggest the term "reaction formation," since there is plainly no conformity between the constraints of the "notentropic" form of the process described, and a simple timereversal of the Htheorem determination of statistical entropy. The popularized response is the wildly desperate "handwaving" of the professor hoping to escape from the lecturehall unscalped. Rather than resort to such desperate, and ultimately futile handwaving gestures, the professional need but examine some fascinating, very revealing characteristics of this anomaly.
Put most simply, although we can describe the process mathematically, either in the terms given here, or more refined terms to the same net effect, no extant form of generally accepted classroom mathematics can represent this process as a deterministic mathematical model. Rather than collapsing to mewl in muted hesychastic hysteria over the mortal injury to his beloved textbook formalism, the professional ought to experience joy, to discover here a phenomenon in the physical world which every competently trained twentiethcentury mathematician knows from the domain of higher mathematical formalities: the principle of the ontologically transfinite implicit in Georg Cantor's 1897 Beiträge.
Focus upon the physicaleconomic process, as represented in the modern industrialsociety phase outlined. The source of the increases in physical productivity which define the determination of the function described, is a process of continuing scientifictechnological progress subsumed (as a Platonic "Many") by a higher process of valid axiomaticrevolutionary forms of scientific (and analogous^{63}) discovery. Those axiomaticrevolutionary discoveries have a form of absolute mathematical discontinuities, relative to any formal theoremlattice, such as a formal logic or mathematics. Consequently, in relationship to any generally accepted classroom mathematics of today, any valid mathematical description of the effects of a notentropic physicaleconomic process is axiomatically nondeterministic.
There are two other cases immediately to be considered, to address the matter of notentropic processes more generally. First, obviously, the case of the evolutionary biosphere, over the most recent billions of years. Second, the relevant, analogous conceptual overview of the Mendeleyev Periodic Law, as the evidence stands today. The advantage of choosing the physicaleconomy form of notentropy as the subject, is that this shows us that some analogous form of discontinuity, analogous to axiomaticrevolutionary forms of mental creativity, necessarily distinguishes a merely chemical process of the relevant sort from a living one. Reciprocally, this urges us to consider a view which is admittedly conjectural, but a compelling one, that mental creativity is a qualitatively higher species of the same notentropic principle which distinguishes living from nonliving processes. Is this principle also reflective of processes whose ostensibly elementary location appears in the subnuclear domain, perhaps more deeply ensconced than 10^{@ms18} centimeters? A quantumfield view of the Periodic Law suggests this is a case to be investigated, employing what we know of notentropic processes in physical economies.^{64}
Our views on approaches to questions of notentropy in living processes and the Periodic Law so indicated, we can dispense for the moment with further consideration of such other topics; it is the determination of notentropic economic processes by creative forms of mental activity which is our immediate subject here, from which we shall derive what is to be said on the subject of certainty of truth.
We, speaking of ourselves collectively as Leibniz's and U.S. Treasury Secretary Alexander Hamilton's modern industrial society,^{65} have in our hands the readily comprehensible evidence of the way in which valid axiomaticrevolutionary discoveries in physical science cause directly increases in the productive powers of labor.
The translation of the discovery of an (Platonic form of) hypothesis into its formal mathematical or related expression, requires a revised set of axioms and postulates for all relevant topical areas of scientific thought. This revision defines corresponding differences between the old and new theorems for every subtopic of application of the respectively new and old theoremlattice. Each such case of a difference implies a corresponding form of crucial experiment, for which the salient points of axiomatically determined differences serve as the critical features of design of such experiment. The refinement of such a valid experimental design is implicitly the model for design of corresponding, new machinetool or analogous principles. The transmission to the "point of production" of both the knowledge provided by the discovery, and improved design of workplace, etc., yields the relevant increase in physical productivity per capita, per household, and per square kilometer.
All of the effects of this transformation are implicitly measurable, and intelligible in that form. However, the very nature of the motiveforce of the increase in physical productivity signifies that the notentropic function apparently represented by these measurements, unlike statistical "negentropy" socalled, is not a deterministic one. Such are the relevant limits of authority of generally accepted classroom mathematics.
Appendix A: The Ontological Superiority of Cusa's Solution Over Archimedes' Notion of Quadrature
FOOTNOTES
Appendix A: The Ontological Superiority of Cusa's Solution Over Archimedes' Notion of Quadrature
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