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RIGOR: METHOD OR MORTIS

ICLC Communications 15 May, 1983

You know me, kids. No pussy-footing down the old path of least resistance. I tell it like it is, and I like to say it just once. So, how come these lousy report-cards from school all the time. Tell me, kids : how many times I got to tell you : To get anywhere in this world, you got to tell them what they want to hear.
Collected Maxims of Al Gebra.

By Lyndon H. LaRouche, Jr .

On Saturday morning, Johnathan Tennenbaum and I met for about two hours, partly to discuss amplifications and supplements to his draft dissertation of April 30, 1983, Development of Conical Functions, Particularly Elliptic Functions, as a Language for Relativistic Physics. We also discussed the principal difficulty of pedagogy : that after all these years, most among our own members, as well as nearly all of today' s professional mathematicians and physicists, seem unable to grasp the most elementary notion of what "rigorous scientific method" signified to Cusa, da Vinci, Leibniz, Euler, Gauss, or Riemann : a lack of a developed sense for the difference between a rigorously defined reality and a clever, plausible explanation.
A quick glimpse at one side of an exemplary bit of dialogue displays some typical spoor of the problematic beast to which I refer:

"How would you answer the objection that ... ?" ... "But then, he's going to argue that ... " ... "That could be the right answer, but he's not ready to listen to anything like that. We have to show hom how it works in a way he' s willing to understand." ... "Even if you' re absolutely right, I think it would hurt our influence with a lot of good guys, if we didn't put it into the kind of physics they' re ready to accept."

Learning to win something better than merely passing grades, year after school year, whether in mathematics, physics, or anything else studied, tends to turn certain of the higher faculties of the mind into a mass of oozing, gelatinous schlimihl. Mere sophistry, the sophistry which is the active principle of pedagogical rhetoric -- "accepted forms of argument, pro and con," becomes embedded in the student's, and former student's mind, as a consequence of being conditioned for so long a period of childhood, adolescent and young adulthood, into substituting one' s successful adaption to "authoritative opinion" for the kinds of proofs of relative rightness or wrongness which lie at the basis of a span than five hundred years to date in each and all advances in mathematical physics. A very elementary illustration of the difference between sophistry and rigorous method. After successfully passing a class-session or two concerning proofs that the circle is the only self-evident form of existence in geometry, the rhetorician would twist what he had merely learned into a delphic sophistry of the general type :

"In other words, the Circle is the only self-evident existence in Euclidean space, and the straight line and point exist as its principal predicates."

The kernel of the sophistry in that statement is the use of the intransitive form of the verb "to be" = "->is".

"...the circle is ... straight line and point exist as"

The correct statement would be of the form:

"First, by that topological theorem, we have shown that the circle of closer rotation is unique in visible space. We have also shown that the straight line and point can be defined completely by folding the same circle defined by that theorem twice against itself. We have also shown that the entirety of geometry can be constructed using nothing but the circle and the point and the line as defined by the circle."

The latter statement uses the transitive form of statement of action:

" ... have shown ... be defined completely ... folding ... be constructed ... using ... defined by ... ."

In this latter form of statement, we examine the lawful composition of the universe in terms of our ability to transform that universe by mans of willful human activity, by consciously directing our wills to select such modes of activity as a method for transforming the indicated effect. If we employ the "is" of "to be" thereafter to describe the transformation effected by means of our willful action, the "is" is not a relationship between ostensible subject and predicate of the formal statement; the "is" is intransitive in respect to the unstated transitive verbal expression for which it is substituted in such restatements. Rigor is located for the speaker conscious of this point, as his action of locating the which the "is" is being implicitly employed as surrogate.

The most characteristic fallacy of the delphic statement is the manner rhetoric employs the verb "to be :" the action of pointing and saying "that is..." as a statement of what is purported to be a self-evident fact. As we have discussed this matter on numerous occasions, orally in and writings and lectures delivered by various members, the "is"-form of the "middle term" characteristic of aristotelean logic, is employed to the implied effect of eliminating all participation of causality from statements of "fact," statements of perception, as aristotelean or empiricist method defines "perception."

This is usefully compared with the Jesuit' s sophistry used by Rene Descartes, in his efforts to nullify the influence of Cusa, Kepler, Desargues, Fermat, Pascal, Descartes is clever. He believes in the existence of some principle outside the physical universe, but that principle ceased to have any control over the universe once the universe had been created. "Facts" are reduced to self-evident particles roaming through empty, Euclidean space, continuing their present course until they must combine that action through intersection with like other bodies. He is much cleverer, and more dangerous that an obviously agnostic as Bacon, Hobbes, Locke, or Hume. There is a universal principle of causality in his system as a whole, a precisely located principle : located at the instant of the "big bang." "If you can run fast enough to reach infinity, you can actually see it."

"Facts" are reduced to "little, round, hard balls," sense-perceptions defined as syllogistically self-evident existence. The properties of self-evident particles may be examined, but not the axiomatic presumption of their self-evident existence. This act of arbitrary disconnection of the universe into hermetically defined particlular existence, is thus defined presumptively as the only cause for intellectual activity of science: the activity of seeking to rediscover the interconnectedness of the universality the syllogist has previously presumed, axiomatically, not to exist.

Starting with that rudimentary, characteristic methodological principle of aristotelean sophistry, the empiricist corpus of "inductive science" unfolds.

The sophist degrades the universe to a logician lattice-work of brutish subject-predicate forms of syllogistic statements respecting isolatable perceptions. These include sensory perceptions of the events persumably external to the perceptual apparatus, and also psychological "facts" defined by turning this same dogma of empiricist perception inward toward the internal features of the process of perception itself.

The syllogistic statements woven into a latticework through supposed, formal identities among "middle terms" define an infinitely-extended "material universe" of "hard facts."

The primary activity associated with mathematics is therefore assumed to be an effort to sort those syllogistic statements of perception into three classes. (1) Syllogisms perfectly connected by agreement of middle term to all syllogisms sharing the same middle term, plus all statements using the same subject as subject, or that same subject as the predicate of another syllogistic statement. (2) Syllogistic statements which have no such consistent connection within the latticework; formal fallacies, (3) Statements which, in their present form, are both partly interconnected and partly fallacious.

Formal mathematics, and equivalent versions of formal-logical topology, assign themselves the characteristic activity of repairing defects (formal fallacies) within the net of the infinite, syllogistic latticework as an entirety. (Specialists are persons who inhabit only a smaller region of the network.) This moral equivalent of an eternity condemned to playing the video-game of Pac-Man, is the poor knit-wit's spider-like obsession with perfecting each knit and purl of the latticework as a whole. He performs this activity by means of crotchets. Each particular syllogism presumed to contain a fallacy must be reworked, using crotchets. The mathematician' s spider-like mind races him through infinitely endless toil, an obsession with the life' s work of bringing each syllogistic statement in the endless, Cartesian lattice, into an infinitely connected consistency with the lattice as a whole.

Once to infinity and back, the poor knit-wit must begin the entire work all over again. He has successfully correlated the coordination of the centipedes' first, second and third legs with respect to each other, and the coordination of the third with the fourth and fifth, but the coordination of the fifth with the fourth and third does not extend to coordination of the fifth with the second. A mother-spider's house-work never ends. The mind of the true mathematician runs like a squirrel in a rotating cage, in the Mobius-band of an infinitely recursive loop.

Pure mathematics based on an adducible lattice-theory passes over from the simple appearance of an Egyptian scarab's housework, to become a recognizable higher mathematics, through the practical implications of the infinitely-recursive looping-behavior of the community of such mathematicians as a whole. As the scarabs pile their wealth higher and higher, a wonderful, mystifying phenomenon emerges. The pile of scarab' s quarks impresses the observer not only with its consistency of shapes and assortments of colors, but a higher principle seems to exude from the fruit of the entire, cumulative exertion, an undeniable aroma. Thus, we have the essence of higher formal mathematics. In Phoenician cults and Gnosticism, this aromatic essence of higher mathematics is identified as the hermetic principle; the translation of the worship of Ishtar-Astarte-Isis-Baal-0siris into Hebrew is "kabbalism." In the clinical study of epidemics of schizophrenic psychosis among assistant professors of universities' mathematics-departments, the same mystical aromas are known as "numerology."

Not only the doctrine of mathematics of Leopold Kronecker, but also the entire physics of Newton, Cauchy, and Maxwell, is a lattice-theory of the variety we have described. In the case of Newton et al., the simplest form of such constructs, the same characteristic behavior of lattice-building outlined here has been extensively described and defended as policy by the accredited advocates of the "method of the inductive sciences." The relatively more recent descriptions provided by Harvard University' s Professor Garret Birkhoff and, earlier, by the school of radical empiricism of Russell's and Whitehead's Principia, are fully consistent with this.

To define pure mathematics according to the same axiomatic principles this mathematics itself demands for its latticework, how must we understand the substance, the ontological quality, of this quality of aroma which sets aristotelean mathematical physics apart from science based on the real universe?

What, ontologically, is the aroma arising from the ripening heap of quark-facts of Newtonian physics? What smellable essence causes us to regard the Newtonian physicist as a profesional, of a profession which rises above the unprofessional? The simple, unprofessional man pursues experience, taking as knowledge the raw droppings bequeathed to him from the hind-quarters of that experience. The simple man amuses himself to believe that the bagful he has accumulated by his labor-intensive stooping is an accumulation of truth. Our Don Quixote of a pure mathematician is a man of ineffable tastes. As much as that Quixote can discipline the Sancho Panza of a Newtonian physicist who follows him in their scientific explorations together, neither of them would wish to be known as the sort of ignorant peasant who profanes every fact picked up from the street by simply putting this substance in his mouth. Quixote does not profane his profession by dirtying his hands with such facts; he sniffs them. What distinguishes the professional character of Quixote and Panza is their obsession with sniffing between the quarks, so to speak.

What, ontologically, is that universality, that aroma, arising from the ripening heap of quarks? What is the aroma the pure mathematician pursues, in his infinitely recursive looping back and forth across the fabric of the syllogistic latticework. The Newtonians pretend to deal with "facts," but this is only a trick, employed to hide from profane eyes the hermetic mysteries permitted to be known only among the ranks of professional knit-wits. Yet, the secrets they hide from their speech, they reveal with their hands. The subject of mathematics is not consistency; the only fact of importance to mathematicians is a well-established inconsistency. The content of pure mathematics, its subject, is formal-logical inconsistency, fallacy. If there were no discernible pervasive fallacy in mathematics, pure mathematicians would promptly cease to exist.

The ontological content of pure mathematics is, by definition, that which is presumedly proven not to exist.

The pure mathematican and his Newtonian accomplice are gentlemen collaborating in magicians' enterprise of duping credulous peasants into paying entrance-fees for an exhibition of the practice of seeming to turn absolutely nothing ( the non-existent subject-matter of mathematics ) into the illusion of something: Newtonian physics. David Hume, for one, insists on this point at great length. His admirers among the modern radical-empiricists, such as Bertrand Russell and the positivists, insist upon the same point more radically. How is the trickery performed?

Construct in the imagination a vast matrix.

Start with any syllogism in the "infinitely-extended, interconnected" latticework. Choose any pathway of middle terms, intersecting any one term only once : continue until you have either returned to the starting-point or have demonstrated that this can not be accomplished in the latticework as it presently exists. In the latter alternative case, introduce one arbitrary (empty) syllogism which formally makes the otherwise non-available connection to the syllogism used as the arbitrary starting-point. Using the same starting-point again, trace a pathway from the first point [unreadable, note] middle-term which branches to a different connection than was traced in the first sequence. Proceed as in the first case, reconnecting the second entire sequence, at the end of possible branching-points to the specific term from which the second entire sequence initially branched. Repeat the operation until all of the terms of the entire latticework have been intersected by sequences once and only once by the sum-total of the sequences generated. Let us explore this exhaustion-process no further in that way. Rather, we stand back from the matrix constructed so in the imagination; what are the principled features of the formal- topologist' s logical oeprations on the latticework from this vantage-point?

By adopting only one, arbitrarily chosen point of departure, the sequence of sequences constructed presents the logician with two most significance features. First, there is a pathway from the initially chosen point of reference to the first terms of each of the sequences. This both imposes and provides the logician a chosen main-pathway for moving exhaustively through the entire lattice. The second feature is the set of imaginary terms concocted to close each sequence not otherwise closable with its specific term used as starting-point, including the particular main-sequence developed as the sequence of first terms of all of the sequences in sequence.

The purpose of the logician' s abstraction of such a matrix is his commitment to reducing the entire lattice to consistency with a single syllogism, such as "All matter is originally green." or "All elementary particles are composed of non-existent elements called quarks." That reduction would represent the approximate precondition for the mass-suicide of all logicians: the logical inconsistency of logic, on which the existence of logicians depends, would have disappeared. The universe of formal fallacy, in which the mind of the logician exists, would have evaporated.

The invention of the celebrated non-existence called the "quark" was precisely the creation of an empty syllogism -- by two mathematicians meeting in a California saloon -- to provide imaginary closure for a sequence of first-terms of a latticework matrix of the type we have specified.

How is it possible that professionally educated people of not inconsiderable scientific experience, could belief such an obvious hoax as the quark-hoax? The treatment of latticework-doctrine we have summarily described here is well documented in literature that is very well known among professionals. Any mathematician or mathematical physicist who attempts to rework the steps of the quark-dogma, can not deny that the entire logical procedure is nothing but an exercize of the sort of higgledepiggly we have described, the most banal abacadabra. Ask yourself what occurs to the mind of a victim, if that human individual is subjected to a very intensive equivalent of putting a laboratory-rat daily through a very elaborate maze. Imagine that the rat must start each day along a new track, cross none of the points of intersection of the maze he crossed on any preceding day, and must, each day, return through the maze to the point from which he started that day. On those days the poor, tortured rat's choice of sequence does not permit him to return to the starting-point of the day' s excursion, "generously" allow the poor, tormented rat-man to regain his starting-point by inventing one imaginary syllogism which connects the blind-end of his sequence to the starting-point. At the end, let the rat-man run through a sequence based on those imaginary syllogisms' presumed consistency with the first days' starting-point; force poor rat-man to construct one additional imaginary set of syllogisms which create the logical appearance of infinite connectivity of all points of the imaginaries' sequences together with the real starting-point of the first day of the maze-torture exercize. Will the poor, brainwashed rat-man believe, or will he not believe, that the last of the imaginary set of syllogisms represents his discovery of the lawful principle of formal logic vhich provides total consistency to the entirety of the latticework? Subject an ordinary human being to a formal education in which he or she is steeped in the equivalent of Newtonian physics, under psychological pressures of a rat-maze setting of "rewards and punishments," and that poor victim will end up believing that the satanic experimenter is God himself, and that his own final set of imaginary syllogism is the fundamental law of the universe.

Such a process of aristotelean brainwashing subsumes two alternate varieties of approach to the matter of experimental scientific inquiry. By "varieties," we signify that the two distinguishable varieties of are a common "species."

One view, the experimental view, emphasizes the desirability of discovering an experience which corresponds to the empty syllogism. If the observer experience is itself logically consistent with the latticework, then the empty syllogism is considered "proven." The deductive theorem, the mathematical analysis of experimental data, has precisely this significance in empricist-physics practice. It is not the existence of the data, as "facts," which proves the theorem; the proof islocated in demonstrating logical consistency with an entire syllogistic lattice-work by means of using as premises a set of theorems which are, themselves, established as logically consistent with the results of the present status of continuing, infinitely recursive looping through the lattice-work by logicians.

The other, more sedentary, contemplative view, is satisfied to have created the equivalent of the discovery of a set of empty syllogisms equivalent to the syllogisms proving the existence of the experimentally non-existent quark.

The former is empiricism- The latter is radical empiricism. The one drinks gin; the other prefers cocaine.

Neo-aristoteleanism and its empricist offshoots are admittedly not the form of the block against the elementary fundamentals of scientific rigor commonplace within our ranks. The problem which our members have yet fully to overcome, a problem naturally most clearly reflected from among those with mathematical or physics training, is not empiricism as such, but is most closely approximated by the lowering of the level of quality of Germany' s science during the late nineteenth and early twentieth centuries, a collapse symptomatized by the emergence of "intuitionism."

Whatever the methodological tendency more or less commonplace in our own circles, the neo-aristoteleans dominate and almost control the science-departments of all leading universities, and do control the dominant scientific associations. Just as late nineteenth-century science' s vitality was greatly undermined by the growing influence of the Kroneckers, Dedekinds, Helmholtzes, and increasing British and Viennese radical-empiricism' s influence in Germany, so the far more degraded professional-scientific ruling climate of our own times is the social pressure threatening sterility in our own scientific work. Like a dagger, or deadly poison, the influence of aristoteleanism is not human, but a dagger can kill a man nonetheless, and neo-aristoteleanism can destroy entirely the creative potential of an otherwise gifted professional mind.

To analyze the range of compromises between science and neo-aristoteleanism's social hegemony in the scientific profession, we must first identify our second benchmark : What are the most relevant distinctions of the real mathematical physics, the deadly adversary of neo-aristoteleanism?

The Internal History of Science

As we have proven in published locations, the literary record of all modern European mathematical physics begins with the dialogues of Plato, most emphatically the Timaeus; In this and other, correlative locations, Plato develops three leading, interdependent conceptions, on whose combined effect the entirety of all successful fundamental discoveries during more than five hundred years of modern European physics entirely depends.

There three root-discoveries are: (1) The rigorous elaboration of the principle of the hypothesis of the higher hypothesis; (2) The principle employed by Plato, that the circle is the only self-evident existence in geometry, and that no other axioms or postulates can be tolerated within the extent of mathematics; (3) The discovery elaborated by a collaborator at the Cyrenaic temple of Ammon, that only five kinds of regular polyhedra can be constructed in visible space.

The problem for later European scientific work was chiefly that, until Cardinal Nicholas of Cusa rediscovered the principle of the circle during the middle of the fifteenth century, it was not possible for students of the Timaeus to conduct a rigorous reworking of Plato's discovery. It was Cusa's work on mathematical physics, including his preliminary statement of the solar hypothesis later perfected and proven by Kepler, which was mediated through the post-1480 work of da Vinci, Pacioli, the School of Raphael, et al., to lead into the establishment of mathematical physics proper by the work of Kepler, Gilbert, and Desargues.

Leibniz successfully completed Kepler's specifications for a differential calculus by 1676, through aid of the work on geometrical determination of differential series of rational numbers of Desargues' student, Pascal. It was Karl Gauss who solved the task of developing elliptical functions, as required by specifications of Kepler. It was the Leibnizian Ecole Polytechnique of Monge-Carnot, plus the Leibnizian Gauss, whose combined contributions culminated in the completion of mathematical physics as to methodological fundamentals with the work of Riemann, Weierstrass, and the 1871-1883 work on the transfinite by Cantor.

That internal history is more adequately elaborated to this point in other available locations, whose content it is not our purpose or obligation to replciate (sic) here. For our present purposes, it is sufficient to identify the broad outlines of that setting for our immediate discussion of several subsumed points.

What Johnathan (sic) et al. have done, starting with a program of developing conical functions for which I have at several point introduced fruitful specifications, has been, thus far, to develop a new proof for Gauss's arithmetic-geometric mean, and to uncover the direct conenction (sic) between that proof and a more generalized approach to a general theory of elliptic functions. My own crucial contributions to this have been chiefly two: (1) Beginning my 1981 specifications for the method of generating proof of the 24-key well-tempered system by means of a self-similar conical (complex) function, to state and repeatedly reassert that such a conical function is the minimal primitive precondition for mathematical, negentropic functions consistent with the program for a revolution in mathematical physics embedded within Riemann's 1854 habitational dissertation. (2) To specify that a pair of conical functions was the minimal, primitive root of mathematical, negentropic functions adequate for generating a Riemann potential surface subsuming negentropic transformations. Such a pair of conical functions must be treated as the hyperspatial coordinates for determination of the "world-line"-values of the generated potential surface.

My specifications for the program based on elaboration of the second specification have been to determine at what "level" of further elaboration the "family" of functional spaces based on this specified, primitive root, corresponded in some provably unique fashion to certain classes of important, known mathematical functions, and to elaborate the specifications of functional space at higher levels in such a manner as to reach the level of elaboration corresponding to determined *** of required results in terms of quantum electrodynamics.

Except for my part in stating such and related primitive specifications, the results achieved have been entirely the work of Johnathan and those others from whose work he has drawn various forms of assistance. However, in this present communication it is most useful that I concentrate on my own part in this project. It is my purpose in choosing this course of argument, to underscore the fact that all fundamental discoveries in mathematical physics (for example) are mathematically trivial, at least in the conventional sense of what mathematical work represents. All fundamental discoveries of the past internal history of science have been mathematically trivial in that same sense; the form of the discovery was to sweep away some underlying assumptions on which the complicated edifice or existing worked-out science was constructed, and to establish new conceptual foundations for elaborated scientific work, changing the character of elaborated mathematical physics (for example) as a whole by implication. Cusa's rediscovery of the principle of the circle is exemplary of this fact. My purpose in stressing the autobiographical aspects of this matter is to expose the basis upon which I chose the indicated and related specifications, and upon what authority I had foreknowledge that the results which be fruitful in the directions proposed.

As the earlier review of a syllogistic latticework took our attention inside the maze of the neo-aristotelean mind, so, in contrast, it is most advantageous now that we share a view of scientific method from the inside, and make that inside-view as personalized, as sensuously immediate, as possible.

A few observations on the development of the mathematical physics of platonic physics by Cusa, da Vinci, and Kepler, sets the stage.

The view that the existence of a circle is determined by ruler-and-compass methods falls victim to the fact that it is impossible to show that a circle or its equivalent area can be derived from the starting-point of existence of points and lines. Even the use of the compass to assist in constructions from such starting-points fails, for the reason that the use of the compass in such cases actually determines straight lines between points of intersections. The methodological root of this latter failure glares out from mere inspection of the manner the compass is employed to generate the circle: , the act of rotation to generate a closed curve..*.., To rotate to a point, and so, implicitly, to halt at the point, contributes not an act of rotation to the process of construction, but only a radius between the center of the arc and the point at which the rotation halts.

However, the reverse construction is readily accomplished. The straight line (diameter, in this case) is rigorously determined by precisely folding the circle against itself once. The point is rigorously determined by folding that semi-circle against itself. A line is a "degenerated" circle (the first-order geometrical derivative, singularity). A point is a generated line (second derivative, singularity), which is also, lii^e the line, a singularity defined by the existence of the circle. We can not go from "self-evident" points and lines to circles,( but we go directly from a "self-evident" circle to lines and points. To create a line or point, we can proceed by generating a circle, but we cannot generate a circle by deriving it from points and lines. We must introduce the act of rotation, which is not a "property" of axiomatically defined lines, points, or combinations of the two.

The basic construction-"theorem" for showing that the circle is uniquely the closed curve enclosing the greatest relative area — by folding half-circumferences against one another, and so forth, shows that the circle is primarily determined in a different way than by compass-construction. What this "other way" really signifies about our universe more deeply, is not fully clear to us until we have worked our way through Kepler's proofs, and have so arrived at the notion of the principle of least action adduced by Leibniz. Rotation is not a static condition, but an action. The closed, continuous action which defines the circle, identifies the least amount of action corresponding to the work done in enclosing an area equivalent to that of the circle.

The principle which Cusa re-discovered, by seeing the error in Archimedes' method for exploring the quadrature of the circle, was not a principle of geometry as we usually understand Euclid's Elements to exemplify geometry. With Cusa, as with the geometry of the Academy of Athens, before the aristotelean concoction produced as the Elements in Egypt, geometry was nothing but physics, and physics was nothing but geometry. The principle of the circle places the principle of least action of physics as the only self-evident assumption from which the entirety of synthetic geometry is properly derived.

Cusa's discoveries unleashed the process leading into Kepler, Gilbert, Desargues, et al., and Leibniz. The leading feature of this connection was that Cusa's rediscovery of the principle of the circle made possible a rigorous comprehension of the entirety of Plato's Timaeus (most emphatically). The features of this leading toward my own primitive specifications must be summarily identified at this juncture.

The boundedness of visible space, implicit in the fact that only five regular species of polyhedra could be constructed within visible space, led to Plato's hypotheses:

(1) That the harmonic self-similarity congruent with intervals of fifth, third, etc., and their complements was a law of distribution of events in the visible manifold.
(2) That the visible manifold was a kind of distorted reflection of real processes occurring with a continuous manifold.
(3) That the lawful principles of causality in the universe are immediately expressed as empirics - of statements in terms of the notion of an hypothesis of the higher hypothesis (not the higher hypothesis, nor of simple experimental hypothesis).

The most crucial feature of the work of daVinci and Pacioli, as it bears on the subject immediately under development, is the focus upon the observation, that all living processes have the morphological forms of growth and function of self-similar proportions congruent with the Golden Section, whereas nonliving processes apparently do not. The fact that the Fibonacci series, the a prioristic population-rgowth (sic)series, converges upon the self-similar proportionings of the Golden Section, implicitly exposes the peculiar significance of the Golden Section on this account. The Golden Section, as the characteristic harmonic principle of a physical process, is the minimal, primitive expression of negentropic processes' harmonics.

Kepler proved the points labelled above as 1,2, and 3 of the Timaeus for the solar system (and implicitly much more). By proving 1, it proved also 2 and 3. It also demonstrated, by the participation of the harmonic characteristic of the fifth in the composition of the solar system, that the universe of astronomy is negentropic. Gauss's demonstration that the harmonic values of the orbit of Pallas coincided with Kepler's given orbital values for a destroyed planet his solar hypothesis required, established that Kepler's proof, insofar as it proved the cited points identified as 1,2, and 3 of the Timaeus, was a conclusive proof of those points. Any physics which does not proceed thereafter from the points of rigor concerning geometry we have identified here, was necessarily an absurd physics as to method.

Kepler's proof was conclusive empirical proof that only the geometrical method of Plato, Cusa.da Vinci, et al. was a rigorous method for physics.

As I have shown for economic processes, such processes are characteristically negentropic, and the definition of "energy" required to correlate the fact of increasing energy-flux density with advances in technology overthrows the notion that energy can be measured in scalar units such as kilowatt-hours or calories for any purpose bearing upon the underlying lawful ordering of the universe. Hence, the arbitrarily concocted Law of Conservation of Energy is an absurdity, and Einstein's E= me2 is to that degree fundamentally absurd, although of restricted usefulness. Einsetin (sic) clearly did not understand the ABC of Riemannian physics, or he could not have perpetrated that and other reductionist blunders. The empty construct, called the Cauchy-Riemann function, is also illustrative of rabid incompetence toelrated widely in contemporary mathematical physics.

The normalization of an economy at value "1" over successive "reinvestment cycles," which is integral to my method of economic analysis, correlates "1" with increasing values (modally) of potential relative population-density per-capita. This corresponds to a series of concentric circles (moving outward). All of the LaRouche-Riemann functions for an economy are based upon this.

However, concentric circles occur in physical space only as projections, not as products of a function definable within the bounds of 2-space. The most primitive function which expresses the projection of such concentric circles is a conical function, which, for reason of the fact that the LaRouche— Riemann function (Fibonacci-series-like growth) is negentropic, must be a self-similar conical function. This function is a complex function, which defines circles "parallel" to the circular base of the cone at self-similar intervals, and defines a7i"asgocisted spiral on the outer surface of the cone.

If we cut the center of the cone with a plane, such that any two of the successive circles at self-similarly determined intervals are parallel lines, draw a diagonal on this plane, from one end of the smaller circle's line, to the opposite end of the larger circle's larger line. This diagonal proceeds, in the direction of rotation of the circles in this example. This diagonal corresponds to an ellipse in the cone.

Now, imagine this ellipse as a moving figure, which begins as superimposed upon the smaller circle, and which grows as an ellipse as it pivots upward on a hinge located at one point on the smaller circle.

Pause a moment. What happens, topologically, when a circle grows into an ellipse? To answer this, fold the ellipse against itself as a circle is folded to determine its diameter! A singularity has been added.

When does this occur; at what exact instant does the cjrcle become an ellipse? The question contains an indeterminancy, absurdity, if read in the wrong way. The right answer defines the right interpretation of the question: When the function associated with the smaller circle "becomes1^jnegentropic,

This is the kernel of Riemann's 1854 habiltiation disserttation (sic).

Once Uwe P-H had elaborated the crucial part the topological principle of Dirichlet had performed in shaping Riemann's work, it became clear to me, during 1981, that this view of Dirichlet(s topological principle had provided Riemann the key to transforming a conical elliptical function, bared on the most-primitive form as a point of elaboration, into the germ of a Riemann surface. A continuous surface which reacts to the introduction of singularities (discontinuity) to restore continuity according to Dirichlet's principle •, is a higher-order surface subsuming the sequence of lower-order potential surfaces in which the discontinuites (sic) emerge. This is congruent with Plato's hypothesis of the higher hypothesis: it is the higher-order potential surface which defines the action of the lower-order domains. Karl Gauss had already struck the germ of this with his arithmetic-geometric mean, as Fermat had brilliantly, if mistakenly located the determination of prime numbers in a related way earlier.

This prompted immediately my proposal of the conical function to determine the well-tempered system, in 1981, and was the basis for specifying the generation of a potential surface by means of paired"conical-functional coordinates more recently. I was delighted but not mystified by the fact that Johnathan's initial application of this approach evoked immediately the Gauss arithmetic-geometric mean, and thac the mean so situated implied directly an indicated approach to elliptic functions generally. Riemann's use of Dirichlet(s principle had already implied precisely such a view of the work of his teacher, Gauss.

On the rest of this specific matter, turn directly to Johnathan's (sic) teatment of the project.

What is of fundamental methodological importance here, as the relative sterility of later German science, relative to Riemann and earlier, shows, is the ontological interpretation of the methodological approach we have illustrated here. I, like Plato, Cusa, et al., locate reality in the unseen, continuous manifold, and treat visible phenomena as the lawfully distorted projections of reality into a mirror we see as visible space, everywhere embedded, relative to our/perception, in the continuous manifold. It is within the continuous manifold, and "on the level" equivalent to the hypothesis of the higher hypothesis, that causality, action, substantiality, immediately exist. The mathematical functions for a continuous manifold immediately congruent with such ontological—causal specifications are the proper, underlying functions for physics.

It was the flight from that neoplatonic ontological standpoint during and after the late nineteenth century, which accounts fully for the realtively abrupt decline in rate of fundamental advances in science. "Intuitionism" is paradigmatic.

The late nineteenth century capitulated more or less entirely to the aristotelean dogma, that reality is located ontologically within the discrete manifold of visible space. So, causality and substance were removed from the continuous manifold of Plato,

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