August 28, 2006 An Insider’s Guide to the Universe by Bruce Director Though all humans are blessed to spend eternity inside the universe, many squander the mortal portion, deluded they are somewhere else. These assumed “outsiders” acquire an obsessive belief in a fantasy world whose nature is determined by {a priori} axiomatic assumptions of the … Continue reading Riemann for Anti-Dummies: Part 68 : An Insider →

August 28, 2006 (6:25pm) The View From the Top by Bruce Director For more than three millennia the motion of a spinning top has been a source of great amusement for children, scientists, and philosophers. A careful examination of its motion provides insight into the underlying dynamics of the universe and exposes the fraud of … Continue reading Riemann for Anti-Dummies: Part 67 : A View From The Top →

November 16, 2005 On the 375th Anniversary of Kepler’s Passing by Bruce Director “In anxious and uncertain times like ours, when it is difficult to find pleasure in humanity and the course of human affairs, it is particularly consoling to think of the serene greatness of a Kepler. Kepler lived in an age in which … Continue reading Riemann for Anti-Dummies: Part 65 : On the 375th Anniversary of Kepler →

On the Continuum of the Discontinuum by Bruce Director In the Laws, Plato's Athenian stranger laments that the Greek people's pervasive ignorance of the incommensurability of a line with a surface and a surface with a solid, had rendered them more like "guzzling swine" than true human beings. While the same lament can be sounded … Continue reading Riemann for Anti-Dummies: Part 62 : On the Continuum of the Discontinuum →

March 5, 2005 (8:08pm) To What End Do We Study Riemann’s Investigation of Abelian Functions? by Bruce Director On June 10, 1854, Bernhard Riemann shocked the world by stating the obvious: For more than two thousand years scientists had accepted, as dogma, that the axioms of Euclidean geometry were the only foundation for science, despite … Continue reading Riemann for Anti-Dummies: Part 61 : To What End Do We Study Riemann’s Investigation of Abelian Functions? →

The Power to Change, Change New ideas, like people, come into this world naked. To effectively perform their mission, they must be provided with clothes. But unlike children who can speak for themselves, ideas must be dressed in words and images which, upon careful reflection, indicate how they were conceived. And though it would be … Continue reading Riemann for Anti-Dummies: Part 60 : The Power To Change, Change →

Think Infinitesimal by Bruce Director "It is well known that scientific physics has existed only since the invention of the differential calculus," stated Bernhard Riemann in his introduction to his late 1854 lecture series posthumously published under the title, "Partial Differential Equations and their Applications to Physical Questions". For most of his listeners, Riemann's statement … Continue reading Riemann for Anti-Dummies: Part 59 : Think Infinitesimal →

Bernhard Riemann’s “Dirichlet’s Priniciple” by Bruce Director In his revolutionary essay of 1857, {Theory of Abelian Functions}, Bernhard Riemann brought to light the deeper epistemological significance of the complex domain, through a new and bold application of a principle of physical action which he called, “Dirichlet’s Principle”. Riemann’s approach, combined with what he enunciated in … Continue reading Riemann for Anti-Dummies: Part 58 : Bernhard Riemann →

Pythagoras as Riemann Knew him There is a widely circulated report that when Pythagoras discovered the incommensurability of the side of a square to its diagonal, he sought to conceal its discovery on pain of death to whomever would disclose it. But such an account is of dubious veracity, as it attributes to Pythagoras an … Continue reading Riemann for Anti-Dummies: Part 57 : Pythagoras As Riemann Knew Him →

RIEMANNIAN SPHERICS When Carl Friedrich Gauss repeatedly stated his conviction that Euclidean geometry was not true, his thoughts were connected to the pre-Euclidean science of the Pythagoreans and Plato. However Gauss's "{anti}-Euclideanism" was not a mere restatement of its antecedent. Rather, Gauss, and later Riemann, sublimated the ancient Egyptian-Pythagorean science of "spherics" with a new … Continue reading Riemann for Anti-Dummies: Part 56 : Riemannian Spherics →

The Dramatic Power of Abelian Functions To understand Riemann's treatment of Abelian functions, situate that discovery within the context of the history in which it arose, reaching back to the pre-Euclidean Pythagoreans of ancient Greece, and forward to LaRouche's unique and revolutionary discoveries in the science of physical economy. Imagine that entire sequence, all at … Continue reading Riemann for Anti-Dummies: Part 55 : What Are the Real Objects of Physical Science? →

The Dramatic Power of Abelian Functions To understand Riemann's treatment of Abelian functions, situate that discovery within the context of the history in which it arose, reaching back to the pre-Euclidean Pythagoreans of ancient Greece, and forward to LaRouche's unique and revolutionary discoveries in the science of physical economy. Imagine that entire sequence, all at … Continue reading Riemann for Anti-Dummies: Part 54 : The Dramatic Power of Abelian Functions →

Look to the Potential In his 1857 {Theory of Abelian Functions}, Bernhard Riemann stated that the foundation of his theory of higher transcendental functions depended on what he called "Dirichlet's Principle" and the method of thinking discussed by Gauss in his lectures on forces that act in proportion to the inverse square. These references help … Continue reading Riemann for Anti-Dummies: Part 53 : Look to the Potential →

Riemann For Anti-Dummies Part 52 ABELIAN FUNCTIONS AND THE DIFFERENCE BETWEEN MAN AND BEAST All Aristoteleans are liars. In fact they must lie. For Aristoteleans believe that their minds are empty vessels, indifferent to what is put in them. They project this view of themselves onto the Universe, which, they insist, must conform to their … Continue reading Riemann for Anti-Dummies: Part 52 : Abelian Functions and the Difference Between Man and Beast →

Riemann for Anti-Dummies Part 51 THE POWER OF NUMBER Nicholas of Cusa begins "On Learned Ignorance", by reaching back to the method of Pythagoras: "Therefore, every inquiry proceeds through proportion, whether an easy or difficult one. Hence, the infinite qua infinite, is unknown; for it escapes all proportion. But since proportion indicates an agreement in … Continue reading Riemann for Anti-Dummies: Part 51 : The Power of Number →

Riemann For Anti-Dummies Part 50 THE GEOMETRY OF CHANGE In his famous letter to Hugyens concerning his discovery of the significance of the square roots of negative numbers, G.W. Leibniz stated clearly his recognition that this investigation originated with the scientists of ancient Greece: "There is almost nothing more to be desired for the use … Continue reading Riemann for Anti-Dummies: Part 50 : The Geometry of Change →

Riemann for Anti-Dummies Part 49 THE HIDDEN HISTORY OF THE COMPLEX DOMAIN When Kepler discovered the elliptical nature of the planetary orbits, he uncovered a paradox whose solution would require the development of an entirely new way of thinking, and he called on future generations to develop it. This "Kepler Problem", as it has since … Continue reading Riemann for Anti-Dummies: Part 49 : The Hidden History of the Complex Domain →

Riemann For Anti-Dummies Part 48 RIEMANN'S ROOTS In December 1822, C.F. Gauss submitted a paper to the Royal Society of Science in Copenhagen titled, "General Solution of the Problem: To Map a Part of a Given Surface on another Given Surface so that the Image and the Original are Similar in their Smallest Parts". Notably, … Continue reading Riemann for Anti-Dummies: Part 48 : Riemann’s Roots →

Riemann for Anti-Dummies Part 47 DEFEATING I. KANT In the opening of his Habilitation lecture, Bernhard Riemann proposed to establish the foundations of geometry on a rigorous basis: "Accordingly, I have proposed to myself at first the problem of constructing the concept of a multiply extended magnitude out of general notions of quantity. From this … Continue reading Riemann for Anti-Dummies: Part 47 : Defeating I. Kant →

Riemann for Anti-Dummies Part 46 SOMETHING IS ROTTEN IN THE STATE OF GEOMETRY When Gauss issued his 1799 doctoral dissertation on the fundamental theorem of algebra, he had much more in mind than just proving that particular theorem. He was creating the foundation for a mathematics that rested only on physical principles. He chose the … Continue reading Riemann for Anti-Dummies: Part 46 : Something is Rotten in the State of Geometry →

Riemann for Anti-Dummies Part 45 THE MAKING OF A STRAIGHT LINE Straight lines are not defined, they are made. The above statement might seem jarring to one fed a steady diet of neo-Aristotelean dogma from their primary, secondary and university teachers, but it is the standpoint adopted by C.F. Gauss by the time he was … Continue reading Riemann for Anti-Dummies: Part 45 : The Making of a Straight Line →

Riemann for Anti-Dummies Part 44 PRINCIPLES AND POWERS Rembrandt van Rijn's masterpiece, ?Aristotle Contemplating a Bust of Homer?, conveys a principle that leads directly into the deeper implications of Gauss' and Riemann's complex domain. In the painting, the eyes of both figures are fixed directly before them, yet, Aristotle's gaze is insufficient to guide him. … Continue reading Riemann for Anti-Dummies: Part 44 : Principles and Powers →

Riemann for Anti-Dummies Part 42 ARCHYTAS FROM THE STANDPOINT OF CUSA, GAUSS, AND RIEMANN A citizen in 2003 A.D., wishing to muster the conceptual power necessary to comprehend today's historical, political and economic crisis, and to act to change it, will find it of great benefit to bind into one thought, Archytas' construction for finding … Continue reading Riemann for Anti-Dummies: Part 42 : Archytus from the Standpoint of Cusa, Gauss, and Riemann →

Riemann for Anti-Dummies Part 41 DIE WIDMUNG The following pedagogy is dedicated to the celebration of Lyn and Helga's silver wedding anniversary on Dec. 29, which is an occasion for joy, not only for said happy couple, but for all people around the world to whom this marriage has contributed such happiness over the past … Continue reading Riemann for Anti-Dummies: Part 41 : The Long Life of the Catenary →

Riemann for Anti-Dummies Part 40 COGNITIVE LEAST ACTION Throughout his various works on complex functions, Riemann notes that the hidden harmonies of the complex domain, not calculations, are the least action pathways for the discovery of truth. Riemann's concept is in keeping with the tradition from Plato to Gauss, as exemplified by Gauss' determination of … Continue reading Riemann for Anti-Dummies: Part 40 : Cognitive Least Action →

Riemann For Anti-Dummies Part 39 To paraphrase Nicholas of Cusa, consumers, like animals, don't count. They have no concept of number. Their mental world is made up only of the things they consume. How those things are produced, what power generates such things, is beyond their ken. Numbers, for them, are mere symbols, that, when … Continue reading Riemann for Anti-Dummies: Part 39 →

Riemann for Anti-Dummies Part 38 YOU ARE NOT IMPOSSIBLE When Gauss set about writing his 1799 dissertation on what he called, "The Fundamental Theorem of Algebra," he had already in his mind a fully developed concept of the complex domain as the idea that penetrated most deeply into the metaphysics of space, and he would … Continue reading Riemann for Anti-Dummies: Part 38 : You Are Not Impossible →

RIEMANN FOR ANTI-DUMMIES PART 37 THE DOMAIN OF POSSIBILITY Plato, speaking in the Laws through the voice of an Athenian stranger, holds it indispensable for leaders of society to possess elaborate knowledge of arithmetic, astronomy and the mensuration of lines, surfaces and solids. He also considers it a disgrace for any common man to lack … Continue reading Riemann for Anti-Dummies: Part 37 : The Domain of Possibility →

Riemann For Anti-Dummies Part 36 TRANSCENDENTAL HARMONIES Discoveries indicating the existence of what Gauss would later call the complex domain began with Pythagoras and his followers in the 6th Century B.C, These discoveries, which include the ratios of musical intervals, the doubling of the line, square and cube, the five regular solids, and many others, … Continue reading Riemann for Anti-Dummies: Part 36 : Transcendental Harmonics →

Riemann for Anti-Dummies Part 33 HYPERBOLIC FUNCTIONS - A FUGUE ACROSS 25 CENTURIES When the Delians, circa 370 B.C., suffering the ravages of a plague, were directed by an oracle to increase the size of their temple's altar, Plato admonished them to disregard all magical interpretations of the oracle's demand and concentrate on solving the … Continue reading Riemann for Anti-Dummies: Part 33 : Hyperbolic Functions – A Fugue Across 25 Centuries →

Riemann for Anti-Dummies Part 28 BRINGING THE INVISIBLE TO THE SURFACE When Carl Friedrich Gauss, writing to his former classmate Wolfgang Bolyai in 1798, criticized the state of contemporary mathematics for its "shallowness", he was speaking literally - and, not only about his time, but also of ours. Then, as now, it had become popular … Continue reading Riemann for Anti-Dummies: Part 28 : Bringing the Invisible to the Surface →

Riemann for Anti-Dummies Part 27 GAUSS' DECLARATION OF INDEPENDENCE In September 1798, after three years of self-directed study, C.F. Gauss, then 21 years old, left Goettingen University without a diploma. He returned to his native city of Brunswick to begin the composition of his "Disquisitiones Arithmeticae." lacking any prospect of employment, he hoped to continue … Continue reading Riemann for Anti-Dummies: Part 27 : Gauss’ Declaration of Independence →

Justice for the Catenary On the very eve of his unjust incarceration, Lyndon LaRouche issued a short, but substantial, memo on the catenary function, that was vigorously maligned by a few, and, unfortunately, largely ignored or not understood by many. The principles identified there, are critical at this stage of this pedagogical review of the … Continue reading Riemann for Anti-Dummies: Part 10 : Justice for the Catenary →