For related articles, scroll down or click here.

Because of its length, this article is posted on 2 pages. Click here for Part 2. Footnotes are at the bottom of each page.

The central feature of my original contribution to the Leibniz science of physical economy, is the provision of a method for addressing the causal relationship between, on the one side, individuals' contributions to axiomatically revolutionary advances in scientific and analogous forms of knowledge, and, on the other side, consequent increases in the potential population-density of corresponding societies. In its application to political economy, my method focuses analysis upon the central role of the following, three-step sequence: first, axiomatically revolutionary forms of scientific and analogous discovery; second, consequent advances in machine-tool and analogous principles; finally, consequent advances in the productive powers of labor.

These discoveries were initially the outgrowth of 1948-1952 objections to the inappropriateness of Norbert Wiener's application of statistical information theory to describing both the characteristic distinctions of living processes and of communication of ideas.¹ I countered with a contrary, non-statistical definition of negentropy, as that meaning of the term might be derived from the common, physically distinguishing characteristic of an evolutionary biosphere. This non-statistical counter-definition of negentropy was then stated in terms of a successfully self-developing physical economy; the efficient impact of scientific discoveries' communication within such a negentropic physical-economic process was treated as most typical of the communication of ideas in general.

That was the initial core of my discovery, up to the year 1952. Yet, up to that point, the appropriate mathematical representation of such a form of physical-economic negentropy was still wanted. The third step, taken through an intensive 1952 study of Georg Cantor's 1897 Beiträge,² opened the doors of the transfinite domain upon a fresh insight into relevant features of Bernhard Riemann's contributions.³ Thence, the applied form of my definition of physical-economic negentropy acquired the title of “LaRouche-Riemann Method.”⁴

Initially, during 1948-1952, I made two principal arguments against Norbert Wiener's application of statistical method to living processes. The first of these two was, that, insofar as we employ the term “negative entropy” to signify the characteristic distinction of living processes in general, the phenomenon referenced cannot be described either in terms of a simple time-reversal of thermodynamical statistical entropy, or in terms of the term “energy” used as a notion reducible to a scalar measure of heat. The second of the two objections was, that, for similar, related reasons, statistical information theory has no appropriate application to the processes of generation and communication of ideas.

On the first of these two classes of objections, the kernel of the matter is, that, for the case of an indefinitely successfully self-developing biosphere, the imputable ratio of free energy to energy of the system increases at the same time that the total energy of the system increases, and, that, similarly and concurrently, the ratio of free energy to rising energy-flux density is also rising.

The second of the two objections is brought to light more conveniently, by examining the analogous case of a successfully evolving physical economy. The obviously intrinsic advantage of this choice of subject-matter is that metrical characteristics of the phenomena are predefined in the clearest way: input-output relations of physical labor and physical consumption, defined in per capita and per square-kilometer measures. The most readily accessible illustration of this argument is provided, broadly, by successful models of modern, post-fourteenth-century economies of the type addressed by Leibniz's 1672-1716 work of founding that science of physical economy also known as the science of technology.⁵ Such cases are typified by the characteristic feature of generally increasing intensity of use of heat-powered machinery. The measurement of such model cases in terms of both per capita and per square-kilometer caloric values of input and output, leads to an array of inequality relationships, by means of which the most relevant relations can be measured comparatively in terms of chronological successions of changes of state of each such economy studied as an integrated whole process.

Only the evolutionary model of such a heat- powered process of increase of the productive powers of labor brings the meaningful issues into focus. By contrast, any zero-growth, non-evolutionary model of physical economy is axiomatically entropic, and corresponds to no durably successful model of national or global economy.

For the evolutionary case, progress in scientific and analogous forms of knowledge is the driver of those changes in practice which lead toward a consequent expression of the indicated, life-like negentropic forms of economic development. It should be stressed, that this role of generation and communication of ideas is illustrated by considering Leibniz's study of the proposals for an industrial development based upon the combination of heat-powered machinery and analogous thermodynamical development of modes of production and transport generally. This Leibniz case is a bench-mark from which the history of physical economy in general may be traced backward and forward in time.

That Leibniz case, of increase of the productive powers of labor through employment of the heat-powered machine, has two readily identified, ironically juxtaposed aspects. First, immediately, there is the simpler aspect, the increase of productive powers of labor, in some functional correlation with increase of heat power supplied efficiently per capita and per square kilometer. In the complementary aspect, on account of nothing other than some improvement in employed principles of design, one machine, using no more power than a comparable second machine, yields greater increase of the productive powers of labor. The second case, the general notion of an efficient improvement in design principle, illustrates the notion of technology.

For purposes of analysis, the term technology must denote a set of all those machine-tool and analogous principles of design which may be derived commonly from, implicitly subsumed by a specific, axiomatically unique quality of scientific or analogous discovery. Reference the refined design of a crucial experiment employed to demonstrate the proof of principle of a crucial scientific hypothesis. Each type of such refined experimental design for that same crucial hypothesis subsumes a set of machine-tool principles, or a technology; all of the sets subsumed by crucial proof-of-principle design for that same hypothesis constitute a family of such sets, or a family of technologies derived from that proof of principle. Thus, does scientific discovery lead, typically, through subsumed technologies, toward consequent increases in the productive powers of labor. The relevant task of analysis in physical economy is to show that such generation and transmission of valid creative discoveries, as ideas, is the source of the realized negentropy of physical economies, and, hence of negentropic increases of the potential population-density of mankind in our universe. My argument, in opposition to statistical information theory, was, that the generation and transmission of such noetic (negentropic) ideas exhibits fundamentally the principle underlying, bounding externally, the transmission of ideas in general.

This discovery posed two paradoxes. The first of these paradoxes is the formal difficulties posed by stating that the characteristic of all physical-economic processes which meet persistently the standard of increasing potential population-density, is negentropy. The apparent paradox lies in the fact that I defined negentropy as corresponding to an increase of the ratio of free energy to energy, and to energy-density of the system, under the condition that the energy of the system is continually increasing both per capita and per square kilometer. The second of these two paradoxes is the notion of the functional role of technology's mathematical discontinuities in the theory of heat-powered machinery.

Perhaps it may be said, that, as treasures of pagan mythology are guarded by dragons, forbidding paradoxes often deter the timid from reaching out to the crucial discoveries otherwise within their reach. These apparent paradoxes of my argument proved not the weakness, but rather precisely the strength of my case against positivists such as Wiener, John von Neumann⁶ et al.

To define my post-1951 attack upon the metrical problem, consider the following.

The two paradoxes identified above should be recognized as echoing the issue of Isaac Newton's confession as to the source of his so-called “Clockwinder” paradox. Newton warned, thus, that the false-to-nature image of an entropic universe had infected his Principia through defects inhering in what he regarded as his only available choice of mathematics.⁷ But for my adolescent grounding in such relevant works as the Clark-Leibniz Correspondence⁸ and Monadology,⁹ I, too, would probably have been frightened off the track of my discovery by the appearance of the indicated paradoxes.

The influence of Leibniz upon my view of these two paradoxes is situated historically, summarily, as follows.

In synopsis, the relevant background of Newton's “Clock-winder” problem” is this. Although the solar-astronomy roots of modern mathematical science reach back far beyond 6,000 B.C. in Vedic Central Asia¹⁰ and in the culture of China,¹¹ a comprehensive, mathematical basis for a unified body of science (“natural philosophy”) was first founded by Nicolaus of Cusa, et al. during the early middle decades of Europe's fifteenth-century Golden Renaissance of Cusa, Piccolomini, Toscanelli, Leonardo da Vinci, Raphael, et al.¹² The complication, leading to Newton's “Clockwinder” problem, was the spread of a Venice-directed opposition to the Council of Florence, an attack which featured the neo-Aristotelian empiricism of such Gasparo Contarini associates as Pomponazzi¹³ and the Franciscan cabalist Francesco Zorzi.¹⁴ Through this continuing influence upon England of such Venetian potencies as the notorious Paolo Sarpi, we have Baconian empiricism and British philosophical liberalism generally.

Respecting the two paradoxes originally posed to me by my theses against statistical information theory, the relevant problems in mathematics are a tangle of two respectively distinct, but interlocked sets of problems. Once this tangle is understood from an historical vantage-point, my solution to the cited paradoxes is more readily intelligible. The founding work of modern science is Nicolaus of Cusa's De Docta Ignorantia,¹⁵ in which the pivotal mathematical discovery referenced is Nicolaus' revolutionary treatment of Archimedes' theorems on quadrature of the circle.¹⁶ Nicolaus' new solution for these theorems¹⁷ is also a form of demonstration of the general solution for the ontological paradox depicted within Plato's Parmenides dialogue.¹⁸ Nicolaus's discovery is, in fact, an illustration of Plato's principle of human knowledge: hypothesizing the higher hypothesis.¹⁹

To this, the anti-Renaissance associates of Gasparo Contarini counterposed, violently, the dogma of neo-Aristotelian empiricism, the deductive treatment of sense-certainty, which is otherwise recognizable as the philosophical “materialism” of the Renaissance's seventeenth and eighteenth centuries' principal adversary, the Enlightenment. Thus the spread of the Enlightenment's cabalistic empiricism is typified by the influence of such notables as Francis Bacon, Robert Fludd, Elias Ashmole, René Descartes, Isaac Newton, John Locke, and Immanuel Kant.

The view of the problem of quadrature from the standpoint of Plato's Parmenides shows, perhaps most efficiently, the root of Newton's “Clockwinder” failure, and exposes also the more general form of practical differences in scientific results between the two opposed, Renaissance and Enlightenment, methods of work. This shows explicitly, in this way, the implication of my initial treatment of my own two scientific paradoxes. The gist of the matter is as follows.

The Archimedean quadrature of the circle relies upon the so-called method of exhaustion famously employed by Plato's collaborator, Eudoxus. By simultaneously inscribing and circumscribing regular polygons, of the same species, and by increasing the number of sides of these polygons, equally and concurrently, we may estimate the value of pi accurately to any desired decimal place. Slovenly thinking would argue, mistakenly, from this, that the perimeters of the two polygons must ultimately coincide with a circular perimeter.²⁰ The same species of philosophical problem arises in deriving the uniqueness of the five Platonic solids. In the case of quadrature, what exhaustion proves, is that, never, even at conjectural infinity, could the number of sides be increased sufficiently to produce coincidence of the polygonal and circular perimeters. Thus is illustrated the fact that a circle, as a species, is not constructible by a geometry premised hereditarily upon the axiomatic assumption of self-evident point and straight line; another, axiomatically different geometry must be adopted, one in which circular action supplants axiomatic definition of point and straight line.

Two points representing the case are relevant for understanding my solution to the negentropy paradoxes.

First, very briefly, the fact that point and straight line are theorem-existences in a geometry premised upon circular action, but not the reverse, shows that the non-circular forms externally (epistemologically) bounded by circular action (in this sense of external bounding) have only that inferior, dependent existence, dependent upon the necessary existence of the higher. This, notably, is an argument congruent with the ontological proof of existence of God. Thus, the mind must, so to speak, leap from the falsely imagined elementarity of the simpler, to recognize that the elementarity lies actually in the superior. Thus, does human reason free man from subjugation to the bestiality of neo-Aristotelian sense-certainty. This appearance of an ontological leap typifies the phenomenal guise of creative thought.

This is the same species of problem posed by Plato's Parmenides, that problem, which, as paradox, blocks the pathway to that true knowledge, which is opposite to mere sense-certainty, derived uniquely, not from simple deductive sense-certainty; this true knowledge is typified by the recognition that a necessary existent, which bounds externally a set of phenomena of mere sense-certainty, is the relative ontological reality, the relative One, which adumbrates the mere shadow-existence of sensory appearances. Thus, Cusa's treatment of quadrature implicitly defined (“hereditarily”) the non-algebraic higher mathematics which Leibniz and Jean Bernoulli proved physically by the case of light refraction, a quarter-millennium later.²¹ This gave modern science two levels of mathematics, the lower, the algebraic, and the higher, the non-algebraic, the latter later called transcendental.

Second, still later, by the same method of discovery employed in Plato's Parmenides, and used by Cusa in his treatment of Archimedean quadrature, Georg Cantor, 200 years after Jean Bernoulli's announcement,²² announced the discovery of a third, still higher domain of mathematics, the transfinite, superseding the transcendental.²³ It is only a view of the relatively subsumed, transcendental, space-time continuum, a view obtained from the standpoint of the transfinite, which permits an adequate comprehension of cognitive problems underlying the deductively apparent paradoxes of negentropy. By 1951, the specific, narrowly defined difficulty which confronted me was, that any function defined in terms of those successive, axiomatic transformations which correspond to generalized, continuing scientific-technological progress, cannot be represented functionally by any generally accepted form of classroom mathematics. I view that as a more general form of the difficulty which trapped a misled Newton into an entropic, “Clock-winder” morass.

I expressed my own notion of negentropy in such paradoxical terms which posed that conception most simply. To this purpose, I adopted conditionally the implicit assumption of customary, classroom algebraic physics, that any body of algebraically formal scientific knowledge, up to the moment of an axiomatic-revolutionary advancement of principle, is being perfected formally as a consistent, deductive theorem-lattice. In that case, the arrival of the axiomatic-revolutionary discovery represents, deductively, an absolute mathematical discontinuity separating axiomatically knowledge preceding the discovery from that which follows. So, the formal representation of a function corresponding to a succession of such axiomatic discoveries is depicted essentially as a function in terms of what appeared to deductive formalism as absolute mathematical discontinuities.

It follows, that if the discoveries of that succession each represent implicitly an increase of the productive powers of labor, the historically cumulative density of the formal discontinuities so portrayed represents an increasing power of knowledge. This notion of power of a so-selected succession of formal discontinuities, describes the needed alternative to ordinary classroom notions of function. Such is the functional form of this alternative definition of both biological and physical-economic negentropy.

My 1952 study of Cantor's Beiträge provided the key to developing this conception further. Following that study, later the same year, I was electrified by re-reading the relevant, most crucial passage of Riemann's habilitation dissertation.²⁴ Applying the Cantorian implications of my own notion of negentropy to Riemann's stated crucial problem of a continuous manifold “sent sparks flying in all directions.” Cantor's transfinite was key to bringing the two elements together in this way, my own and Riemann's.

This combined view of the universe of physical economy's experience, seen as a functional continuum, guided me to construct revisions in the applicable theory of knowledge: to exclude all residues of sense-certainty's notion of linear ontological elementarity, and to replace these entirely by the elementarity of universal, negentropically evolutionary change, in Heraclitus' and Plato's sense of the ontological elementarity of nothing but change. This required that the popular idea of a mathematical certainty must be put aside, to be superseded by a corrected view of the theory of knowledge. No system of deductive contemplation of our sense-experience can be human knowledge; we know the universe only to the degree we surpass sense-certainty by reflection upon the willful means through which we increase man's power over our universe.

This aspect of mankind's relationship to nature is the central feature of the Leibniz science of physical economy. All matters are subject to crucial tests in terms of choices of pathway of scientific changes in axioms, pathways which generate successive increases in mankind's potential population-density, as the latter relationship to our universe is measured relative to our planet Earth.²⁵

I argued that this physical-economic definition of knowledge implicitly defines a superior scientific method, and, therefore, a fresh overview of the term “mathematics” from a higher standpoint.

In recent decades, I have underscored the following, subsidiary form of that latter argument. I argue that what these reflections pose for mathematics is typified by the ontological paradox of method central to Plato's Parmenides. That dialogue is to be recognized, taken together with Cusa's treatment of quadrature for this purpose, as a forerunner of Cantor's conception of the transfinite, and also as a precedent for Kurt Goedel's derived, comprehensive refutation of the radical positivist fallacies permeating axiomatically the central mathematical theses of Betrand Russell, John von Neumann, and other beliefs of that positivist genre, including Wiener's information theory.²⁶

Typical of this ontological implication of the Parmenides is Cusa's discovery, that the circle does not come into existence, “even at infinity,” by means of any merely formal geometry of the axiomatically rectilinear theorem-lattice kind. As an outcome of that discovery by Cusa, circular action, also known (later) as Leibnizian least action, is recognized ontologically as an independently higher form of existence, an existence which bounds externally all merely algebraic space-time.

From this argument, it follows, that the term “reason” must not be used as Kant does, must not be degraded to a mere synonym of mechanistic, linear “logic.” Reason must signify, typically, valid modes of those kinds of axiomatically-revolutionary discovery, modes by means of which ontologically higher forms of existence, such as Cusa's circular action, are shown to be the necessary existence bounding externally an array of inferior, predicated phenomena. Hence, the recommended use of the descriptive term “creative reason,” to place the needed emphasis upon this intelligible use of the terms “creative” and “reason.”

Such is the principle of creative reason demonstrated by Cusa's treatment of quadrature. One should return to this application of Plato's Parmenides by Cusa, to illustrate the proper, constructive-geometrical standpoint from which to comprehend the ontological implications of Cantor's superseding of transcendental, merely mathematical, merely symbolic space-time, by the higher ontological standpoint of transfinite physical space-time.

It must be recognized, in this way, that the successive levels of mathematics—algebraic, Leibnizian non-algebraic (transcendental), transfinite—define a transfinite array of predicates of a shared common type.²⁷ All three of these are each traceable directly from Cusa's treatment of Archimedean quadrature.²⁸ Each is separated formally from its predecessor by an axiomatic-revolutionary change, a true mathematical discontinuity (singularity). Each change is affected in an equivalent way, referenced to a common point of origin; and, thus, the array qualifies as a type. Each change illustrates the Platonic principle of hypothesis; the array as a type illustrates the Platonic principle of higher hypothesis. That array of successively higher types which is physical scientific (as distinct from merely mathematical) progress, is a higher type of a transfinitely ordered array of higher hypotheses: in other words, a higher type, corresponding to Plato's notion of hypothesizing the higher hypothesis.

Thus, Cantor's discovery of that transfinite which bounds externally the mathematically transcendental, might appear to be the solution for the mathematical appearance of a paradox in my definition of negentropy. Certainly, this was an indispensable step, but did not represent a complete solution of that paradox. Negentropy is essentially a notion of causality; mathematics, even a merely mathematical notion of the transfinite, is not a true physics, but only a higher form of symbolism; such mathematics cannot represent causality as such. Another step was required. A turn to Riemann's work, later during 1952, pointed the direction to the needed next step.

Situate Riemann's significance for my work, by restating briefly the context for the 1952 reading of, especially, Riemann's Hypothesen.

From 1948 on, through 1951, my anti-reductionist notion of negentropy was developed into approximately the form it may be broadly described today. Yet, until my “electrified” reactions to successive, 1952 studies in the work of Cantor and Riemann, it remained unclear to me how to situate this seemingly paradoxical conception with respect to generally accepted forms of classroom mathematical physics.

The geometrical solution to this paradox was supplied, in large part, by aid of Cantor's Beiträge, but only with respect to mathematical formalities. As already stated, mathematics as such cannot represent causality, and the central feature of my notion of negentropy is causality as the elementarity of physical space-time. An ensuing study of relevant features of Riemann's arguments respecting the metrical qualities of a continuous manifold, prompted a conceptual insight into this remaining difficulty.

The explicit solution to the remaining margin of paradox is not to be found within those writings of Riemann which were published during his lifetime.²⁹ The relevant, electrifying, crucial passage from the habilitation dissertation had produced its needed effect only because two leading notions from the history of science were brought to bear upon that 1952 re-reading. The first of those two was the Heraclitus-Plato concept of the unique, universal, physical elementarity of change.³⁰ Re-read Riemann's crucial passage to the effect that the continuity of negentropy, as elementary change, is the ontological type, or characteristic, which defines a continuous manifold as continuous. The second of these two is Leibniz's 1714 Monadology. For emphasis, read that Monadology as it was incompetently attacked by Leonhard Euler.³¹ On this latter account, regard Cantor's transfinite in its aspect as a devastating refutation of Euler's blunder, and, thus, a definitive, formal rehabilitation of Leibniz's Monadology. Viewing my 1952 reading of the Riemann Hypothesen more broadly, five crucial conceptions were thus conjoined by this treatment of Riemann's uniquely relevant argument. First, the Heraclitus-Plato notion of the unique physical (i.e., causal) elementarity of nothing but change. Second, Leibniz's monads. Third, the Cantor mathematical transfinite. Fourth, my notion of negentropy. Finally, Riemann's treatment of the metrical paradoxes of a continuous manifold. If one substitutes for the materialist's fantastic discrete elementarities of sense-perception-like objects, the Leibnizian sovereignty of existence of the individual monad, and if one were to show necessary and sufficient reason that a continuum, premised uniquely upon an elementary ontological quality of negentropic change, must necessarily develop such efficient monads, the paradox, as paradox, were implicitly resolved.

That proof of the existence of monads which will be shown here, as I developed it, is provided from the combined standpoint of both the theory of knowledge and physical economy. An intervening, preparatory report must be provided at this point: assuming that negentropy of the relevant form does exist, what are the elementary mathematical implications of the existence of such a phenomenon?

From the standpoint of a discrete manifold, the discontinuity which is typical of a negentropic “power” function occupies a space-time location within the transcendental manifold analogous to the transinfinitesimal difference between an indefinitely extended algebraic quadrature and never-obtainable congruence with the relevant circular perimeter. It represents thus a Dedekind-like “cut,” an interruption in the continuity of any otherwise apparently continuous line of the maximum of transcendental density of denumerable locations. It appears in merely mathematical space-time as an otherwise empty location of virtually-zero, virtually null-dimensional scale.

This is analogous to proposing for physics, that the discreteness of any sub-atomic, ostensibly elementary particle consists only of the virtually null-dimensional, mathematically circumscribed singularity embedded within a functional notion of that volume of merely mathematical space-time which the particle, as a phenomenon, is estimated to occupy.

The portent of this, is that the non-algebraic (transcendental) mathematical domain defines the location of phenomena in space-time. It cannot represent causality as such. It can pin-point the space-time “location of matter” with virtually inexhaustible refinement, but it does not define physical existence in any other sense than that of space-time location. As useful, even indispensable as this may be, it does not define a physical space-time, the latter the higher domain within which causality is expressed.

It is thus indicated, that we must not confuse the two mutually distinct ontological states, mere space-time and physical space-time. We must think of the transcendental as a certain image of space-time, a subsumed phase-space of the higher, externally bounding, transfinite domain of physical space-time.

Such reflections should prompt a reflection upon the character of those Cantor writings, notably his Grundlagen and Mitteilungen, which preceded his Beiträge. The Beiträge unveils the formal discovery of the transfinite; the preceding writings, especially those cited two predecessors, enable us to recognize the process of Cantor's thinking, grounded, from the outset, in Karl Weierstrass's treatment of some of the demonstrable boundaries of Fourier analysis.³² Cantor's extensive review of both ancient and modern philosophy³³ is an integral part of his preparations for developing the concept of the transfinite. As Cantor stresses the implications of his proof, that a higher- order mathematics, the transfinite, bounds externally the transcendental, space-time domain, requires us to adopt afresh Plato's theory of knowledge. Specifically, Cantor's transfinite domain corresponds precisely to the intent of Becoming in Plato's theory of knowledge, as Cantor himself insists; similarly, the Absolute, which bounds demonstrably the transfinite, corresponds ontologically to that Good which bounds externally Plato's Becoming.³⁴

This view of the Cantor to Plato parallels is not an optional topic in mathematics today. The central structural feature of the organization of the transfinite domain as a whole is Plato's theory of knowledge: hypothesis, higher hypothesis, and hypothesizing the higher hypothesis.³⁵ Cantor's notion of type and equivalence are cognate with that threefold structure of Plato's theory of knowledge.³⁶

Cantor's emphasis upon the Classical philosophical theory of knowledge was in no sense gratuitous or even dispensable. Like the Cantor of my 1952 studies, I faced the requirement for a kind of proof which cannot be supplied merely by any localized sort of laboratory experiment. The appropriate experiment can be conducted only in the domain of physical economy in general. One must re-pose the Classical theory of knowledge as a study of the science of physical economy from the vantage-point of the study of the internal history of fundamental (“axiomatic”) discoveries of higher principle within physical science in general. One must then prove whatever is adduced from the study in respect to progress in principles of composition in the Classical forms of plastic and non-plastic arts.³⁷ This proof, or its reflections, therefore occupies a leading place in my writings on political-economy or policy-shaping in general.³⁸

The characteristic, absolute superiority of our human race over all lower species, is expressed implicitly by mankind's rise from a bestial, baboon-like, rock-artist-like potential population density of circa ten millions living individuals, to a technologically-determined potential of more than 25 billions today. This change is owed entirely to a quality which the Christian's Latin terms imago Dei and capax Dei, the Mosaic tradition of Genesis 1, that man, male and female alike, is cast in the image of God. This likeness is by virtue of that power of creative reason which is most simply illustrated by a revolutionary-axiomatic superseding of inferior by superior principle of scientific practice.³⁹ Thus, in effect, mankind is the only super-species, the only species which can willfully self-develop itself to the physical-economic equivalent of a succession of successively higher species.

To state this pivotal point very briefly, this quality of being such a “superspecies” of creative reason is the image of negentropy as far as the human mind is capable of defining that notion. As such a “superspecies,” insofar as our physical-economic practice is premised upon such a continuing process of science-driven increase of our power of physical-economic practice, per capita and per square kilometer of our earth's habitable surface,⁴⁰ our conscious reflection upon our revolutionary practice is this idea of negentropy, this notion of the ontologically transfinite. This identifies a Platonic conceptualization of that ontological reality which adumbrates the mathematical imagery of Cantor's Beiträge. That is what is fairly described as my updated presentation of Leibniz's principles of a general theory of knowledge.

My argument on this point is summarily as follows.

Continue to PART II

top of page

1. Cf. Norbert Wiener, Cybernetics, or Control and Communication in the Animal and the Machine (New York: John Wiley, 1948); 2nd ed., (Cambridge, Mass: M.I.T. Press, 1961).

2. Georg Cantor, "Beitrage zur Begrundung der transfiniten Mengenlehre," in Georg Cantors Gesammelte Abhandlungen, ed. by Ernst Zermelow (Hildesheim, 1962), pp. 282-356; English translation: Contributions to the Founding of the Theory of Transfinite Numbers, trans. by Philip E.B. Jourdain (1915) (New York: Dover Publications, 1941).

3. Bernhard Riemann, "Uber die Hyporhesen welche der Geometrie zu Grunde liegen," in Mathematische Werke, 2nd ed. (1892), ed. by Heinrich Weber in collaboration with R. Dedekind. English translation: "On the Hypotheses Which Lie at the Foundations of Geometry," in David Eugene Smith, A Source Book in Mathematics (New York: Dover Publications, 1959), pp. 411-425.

4. From late 1979 to the close of 1983, the international newsweekly Executive Intelligence Review produced a quarterly economic forecast based upon the LaRouche-Riemann method. This report was constructed quarterly from, primarily, a GNP-defined data-base, using a set of constraints supplied by this author. During this peri- od, that was the only consistently reliable published forecast available from any U.S. source. This forecasting was discontinued during early 1988, at this author's recommendation. The margin of fakery in U.S. government and Federal Reserve System data rendered any report using such data worthless. See"Riemannian analysis predicts industrial top shutdown," Executive Intelligence Review, Vol. VI, No. 41, Oct. 23-29, 1979; and "'Spectral Analysis' of Collapse," New Solidarity, Vol. X, No. 71, Nov. 9, 1979, p. 8.

5. See G.W. Leibniz, "On the Establishment of a Society in Germany for the Promotion of the Arts and Sciences" (1671) and "Society and Economy" (1671), Fidelio, Vol. I, No. 2, Spring 1992 and Vol. I, No. 3, Fall 1992.

6. For John Von Neumann's initial proposal to simulate economics and other "social phenomena" by sets of linear inequalities, see "Zur Theorie der Gesellschaftsspiele," Math. Ann. 100, 1928, pp. 295-320, reprinted in John Von Neumann: Collected Works (New York: Pergamon Press, 1963), Vol. V, pp. 1-26. See also, John Von Neumann and Oscar Morgenstern, The Theory of Games and Economic Behavior (Princeton, N.J.: Princeton University Press, 1944); and Von Neumann's posthumously published The Computer and the Brain (Silliman Lectures) (New Haven: Yale University Press, 1958).

8. See G. W. Leibniz, "The Controversy between Leibniz and Clarke," in Gottfried Wilhelm Leibniz Philosophical Papers and Let- ters, ed. by Leroy E. Loemker (Chicago: University of Chicago Press, 1956), vol. II, pp. 1095-1169, for the problem of Newton's "Clockwinder."

9. See G. W. Leibniz, Monadology, trans. by George Montgomery (LaSalle: Open Court Publishing Co., 1989).

10. See Lokamanya Bal Gangadhar Tilak, The Orion; Or, Researches into the Antiquity of the Vedas (1893), 5th ed. (Poona: Shri J.S. Tilak, Tilak Bros., 1972), and The Arctic Home in the Vedas, Being Also a New Key to the Interpretation of Many Vedic Texts and Legends (1903) (Poona: Tilak Bros., 1956). Astronomical observations recorded in certain amongst the ancient Vedic hymns place their date of composition at an outside limit of approximately 6,0004,000 B.C. (The Orion); more speculative indications of earlier, Arctic astronomical observations in these sources, would push back fragments of these hymns to the period no later than the climate shift accompanying the ending of the last Ice Age (Arctic Home).

11. The British holist biologist Joseph Needham, whose encyclopedic writings on the history of science and technology in China dominate twentieth-century scholarship, went to great lengths to discredit or cover up the discoveries made in the nineteenth century concerning ancient Chinese astronomy. The French scientist Edouard Biot and the Dutch philologist Gustav Schlegel, proved from evidence in the Confucian classics that astronomical science was already highly developed in the third millennium B.C.; and Schlegel's research led him to hypothesize that significant mapping of the heavens existed at the extremely early date of the sixteenth millennium B.C. Needham, while acknowledging the authority and competence of these scientists, labeled their findings as "quite absurd" and "purely legendary," lying that they had little support and that they "served to discredit what real historical research might reveal"-this because, in keeping with British historiography, Needham insisted such knowledge had necessarily to be "derived from Babylonian sources." See Joseph Needham, Science and Civilization in China (London: Cambridge University Press, 1954), Vol. III; Edouard Biot, Le Tcheou-Li: ou, Rites des Tcheou, traduit pour le premier fois du chinois par feu Edouard Biot (Paris: 1851) (Taipei: Ch'eng Wen Publishing Co., 1969); Gustav Schlegel and Dr. Franz Künert, Shu King Finsterniss, Journal V.K.A.W.A.-L, Amsterdam, 1890; Gustav Schlegel, Uranograthie Chinoise (Leyden and The Hague: 1875).

12. The Golden Renaissance of the fifteenth century is centered around the 1439-1440 Council of Florence as the principal event. Nicolaus of Cusa is the principal figure of that period, whose work on science directly shaped the work of such figures as Leonardo da Vinci and Luca Pacioli and indirectly thus the entire school of Raphael and also the work of Kepler.

13. Pietro Pomponazzi (1462-1525); philosopher who enjoyed the patronage of the Contarini family, he studied and taught at the University of Padua. Pomponazzi took Averroës as his point of departure, and by dichotomizing discourse into the philosophical and the religious, argued that according to reason the soul must die with the body, but according to the teaching of Christianity, we know it to be immortal; this argument appears in his major work, De Immortalitate Animae (On the Immortality of Souls) (Bologna: 1516). See The Renaissance Philosophy of Man, ed. by Ernst Cassirer, Paul O. Kristeller, and J.H. Randall (Chicago: University of Chicago Press, 1948); also see Studi su Pietro Pomponazzi, ed. by B. Nardi (Florence: 1965).

14. Francesco Zorzi (or Giorgi), a Franciscan friar descended from the patrician Zorzi family of Venice. Authored De Harmonia Mundi (1525), a mystical work with elements deriving from the Cabbala. Zorzi supported the arguments of King Henry VIII of England when Henry sought the annulment of his marriage to Catherine of Aragon, and he was called to the English royal court, where he remained active between 1531 and his death in 1540. Zorzi was a proponent of a satanic and pseudo-Platonic school of mysticism called Rosicrucianism, which became an important component of English and British Freemasonry.

15. See Nicolaus of Cusa, De Docta Ignorantia (On Learned Ignorance), trans. by Jasper Hopkins as Nicholas of Cusa on Learned Ignorance (Minneapolis: Arthur M. Banning Press, 1985).

16. Ibid., Book I, chap. 3, pp. 52-53.

17. See Nicolaus of Cusa, "Dc Circuli Quadratura" ("On the Quadrature of the Circle"), German trans. by Jay Hoffman (Mainz: Felix Meiner Verlag); see this issue, English trans. by William F. Wertz, Jr., , p. 56.

18. As noted in the text below, there is a precise equivalence as to method between the Parmenides dialogue of Plato and the method employed by Nicolaus of Cusa to make his discovery in connection with his reading and reconstruction of Archimedes' treatment of quadrature.

19. See the celebrated image of the Divided Line in Plato's Republic, in Plato: The Republic, Loeb Classical Library, trans. by Paul Shorey (Cambridge: Harvard University Press), vol. II, Steph. pp. 507a511e, esp. p. SlOa-e.

20. See Lyndon H. LaRouche, Jr., "On the Subject of Metaphor," Fidelio, Vol. I, No. 3, Fall 1992, pp. 18-20; see also, Nicolaus of Cusa, "Dc Circuli Quadratura," op. cit.

22. Ibid.

23. Georg Cantor, Beiträge, op. cit.

24. Bernhard Riemann, "On the Hypotheses Which Lie at the Foundations of Geometry," op. cit., pp. 422-425.

25. This view of potential population-density connotes a higher definition of our human species: first, as man in our solar system, and, next, as galactic man yearning toward a universal mankind.

26. See Kurt Gödel's "Richardian paradox," in Kurt Gödel, On For- mally Undecidable Propositions of Principia Mathematica and Relat- ed Systems, (New York: Dover, 1992); also "The Consistency of the Axiom of Choice and of the Generalized Continuum Hypothesis," Proceedings of the National Association of Science U.S.A., 24 (1938), pp. 556-557. See also Ernest Nagel and James R. Newman, Gödel's Proof (New York: New York University Press, 1958), pps. 60-63, 66, 85-86. Gödel directly refuted Von Neumann's "finitist approach" approach in a letter published in The Theory of Self- - Reproducing Automata, by John Von Neumann, edited and com- pleted by Arthur W. Burks (Urbana and London: University of Illinois Press, 1966), pp. 53-59. Gödel points out in this letter to Burks that Von Neumann's approach is 'tin line with the finitisric way of thinking," like that of Alan Turing. In remarks published postumously in Kurt Gödel: Collected Works (New York: Oxford University Press, 1990), vol. II ("Some remarks on the undecid- ability results 1972a" and "A philosophical error in Turing's work"), Godel States that "Turing in his 1937, p. 250 (1965, p. 136), gives an argument which is supposed to show that mental proce- dures cannot go beyond mechanical procedures. However, this argument is inconclusive. What Turing disregards completely is the fact that mind, in its use, is not static, but constantly developing . ....... See footnote 6 for relevant works of Von Neumann.

27. For type, see Georg Cantor, Beiträge, op. cit.

28. Nicolaus of Cusa, "Dc Circuli Quadratura," op. cit.

29. See Bernhard Riemann, "Zur Psychologie und Metaphysik," on Herbart's Gottingen lectures, in Mathematische Werke, posthumous papers, op. cit.

30. As cited by Plato in Cratylus, in Plato: Cratylus, Parmenides, Greater Hippias, Lesser Hippias, trans. by H.N. Fowler, Steph. p. 402a; for Heraclitus, see G.S. Kirk and J.E. Raven, The Presocratic Philoso- phers, pp. 184-187,197-198.

31. See Lyndon H. LaRouche, Jr., "Project A," Appendix XI, "Euler's Fallacies on the Subjects of Infinite Divisibility and Leibniz's Monads," in The Science of Christian Economy and Other Prison Writings (Washington, D.C.: Schiller Institute, 1991), pp. 407-425.

32. See Georg Cantor, "Uber trigonometrische in Gesam- melte Abhandlungen mathematischen und philosophischen Inhalts, ed. by E. Zermelo (Berlin: J. Springer, 1932; reprinted Hildesheim: Olms, 1966).

33. Here, "modern" signifies the period of Western European civilization beginning approximately A.D. 1400. This style emphasizes that both modern science and the modern form of nation-state republic were founded during the fifteenth century, both as leading, interdependent features of Europe's recovery from the rubble of the fourteenth-century "New Dark Age."

34. See, e.g., Plato's Republic, op., cit., Steph. pp. 505a-520e.

35. See footnote 19.

36. Ibid. It is most relevant to note that this Platonic theory of knowledge permeates the philosophy of Plato-student Leibniz, his Monadology emphatically; this monad also appears under the rubric of Geistesmassen in Bernhard Riemann's posthumously published notes on Herbart's Gottingen lectures (see footnote 29).

37. For example, in 1952 this author first described the Classical lied's interface between music and poetry as a "Rosetta Stone," in connection with a project refuting Norbert Wiener et al. on "information theory." See Lyndon H. LaRouche, Jr., "History As Science: America 2000," Fidelio, Vol. II, No. 3, Fall 1993, p. 32ff.

38. See Lyndon H. LaRouche, Jr.,"The Science of Christian Economy," in Christian Economy, op. cit., pp. 221-223.

39. See Lyndon H. LaRouche, Jr., "The Science of Christian Economy," op. cit., pp. 263-266; "On the Subject of God," Fidelio, Vol. II, No. 1, Spring 1993, pp. 24-33; and "History as Science: America 2000," op. cit., pp. 60-64.

40. Man's existence in the solar system is measured relative to the surface of the planet Earth.

top of page

go to part II