Spring, 2002

Dear Mr. Lyndon LaRouche,

... I am a student at a university at Frankfurt, Germany. After a half year period of studiying your works, reading your publications and verifying them I found your view of history very much resembling my own.

The reason I'm writing you today is very much discussed here in Germany.It is the PISA assessment study. After tracing its history back from OECD to its real origins and theryby identifying it as an subversive body, I got first doubts on the possible danger of it. Now my question: 1. Have you yet conducted any research on PISA, if so, where could I find it? 2. I agree with you on your theory on "classical education". According to PISA, countries performing "robotic" logical or mathematical education such as Japan or Korea, score best on PISA,,whereas Germany with its remnants of "classical education" bewilderingly low. But reading your 1999 book "Der Weg zum Aufschwung" I became convinced that this "robotic teaching" is absolutely wrong.

Now, how could this dreadful development possibly stopped? I'm talking of practical and efficient solutions with much personal involvement such as lectures at universities or scientific studies. Have you done anything in this direction?

Mr LaRouche, I thank you for your kind reply and best wishes for your and your wife's personal health.

Lyndon LaRouche's Email REPLY

Sometimes, even often, perhaps, the best way to attack an apparently nebulous subject-matter, such as today's animal- training of students to appear to pass standardized designs of tests, is to flank the apparent issue, in order to get to the deeper, underlying issues which the apparent subject-matter merely symptomizes. I respond accordingly.

There is a growing number of persons, chiefly university students, who have become active in our work here, and who represent special educational needs and concerns. These concerns include the insult of being subjected to virtually information-packed, but knowledge-free, and very high-priced education. More significant, is being deprived of access to the kind of knowledge to which they ought to have access as a matter of right. In various sessions in which they have tackled me in concentrations of one to several score individuals each, many of the topics posed add up to a challenge to me: "What are you going to do to give us a real education?" There is nothing unjust in that demand; I welcome it. However, delivering the product in a relatively short time, is a bit of a challenge.

I have supplied some extensive answers to that sort of question, but let me reply to your question by focussing upon what I have chosen as the cutting-edge of the package I have presented.

In the same period he was completing his Disquisitiones Arithmeticae, young Carl Gauss presented the first of his several presentations of his discovery of the fundamental theorem of algebra. In the first of these he detailed the fact that his discovery of the definition and deeper meaning of the complex domain provided a comprehensive refutation of the anti-Leibniz doctrine of "imaginary numbers" which had been circulated by Euler and Lagrange. Gauss, working from the standpoint of the most creative of his Goettingen professors, Kaestner, successfully attacked the problem of showing the folly of Euler's and Lagrange's work, and gave us both the modern notion of the complex domain, as well as laying the basis for the integration of the contributions of both Gauss and Dirichlet under the umbrella of Riemann's original development of a true anti- Euclidean (rather than merely non-Euclidean) geometry.

In his later writings on the subject of the fundamental theorem, Gauss was usually far more cautious about attacking the reductionist school of Euler, Lagrange, and Cauchy, until near the end of his life, when he elected to make reference to his youthful discoveries of anti-Euclidean geometry. Therefore, it is indispensable to read his later writings on the subject of te fundamental theorem in light of the first. From that point of view, the consistency of his underlying argument in all cases, is clear, and also the connection which Riemann cites in his own habilitation dissertation is also clarified.

Now, on background. Over the past decades of arguing, teaching, and writing on the subject of scientific method, I have struggled to devise the optimal pedagogy for providing students and others with a more concise set of cognitive exercises by means of which they might come to grips with the central issue of method more quickly. I have included the work of Plato and his followers in his Academy, through Eratosthenes, and moderns such as Brunelleschi, Cusa, Pacioli, Leonardo, Kepler, Fermat, Huyghens, Leibniz, Bernoulli, and Leibniz, among others of that same anti-reductionist current in science. All that I can see in retrospect as sound pedagogy, but not yet adequate for the needs of the broad range of specialist interest of the young people to whom I have referred. I needed something still more concise, which would establish the crucial working-point at issue in the most efficient way, an approach which would meet the needs of such a wide range of students and the like. My recent decision, developed in concert with a team of my collaborators on this specific matter, has been to pivot an approach to a general policy for secondary and university undergraduate education in physical science, on the case of Gauss's first presentation of his fundamental theorem.

Goettingen's Leipzig-rooted Abraham Kaestner, was a universal genius, the leading defender of the work of Leibniz and J.S. Bach, and a key figure in that all-sided development of the German Classic typified by Kaestner's own Lessing, Lessing's collaborator against Euler et al., Moses Mendelssohn, and such followers of theirs as Goethe, Schiller, and of Wolfgang Mozart, Beethoven, Schubert, the Humboldt Brothers, and Gerhard Scharnhorst. On account of his genius, Kaestner was defamed by the reductionist circles of Euler, Lagrange, Laplace, Cauchy, Poisson, et al., to such a degree that plainly fraudulent libels against him became almost an article of religious faith among reductionists even in his lifetime, down to modern scholars who pass on those frauds as eternal verities to the present time. Among the crucial contributions of Kaestner to all subsequent physical science, was his originating the notion of an explicitly anti-Euclidean conception of mathematics to such followers as his student the young Carl Gauss. Gauss's first publication of his own discovery of the fundamental theorem of algebra, makes all of these connections and their presently continued leading relevance for science clear.

This shift in my tactics has the following crucial features.

The crucial issue of science and science education in European civilization, from the time of Pythagoras and Plato, until the present, has been the division between the platonic and reductionist traditions. The former as typified for modern science by Cusa's original definition of modern experimental principles, and such followers of Cusa as Pacioli, Leonardo, Gilbert, Kepler, Fermat, et al. The reductionists typified by the aristoteleans (such as Ptolemy, Copernicus, and Brahe), the empiricists (Sarpi, Galileo, et al., through Euler and Lagrange, and beyond), the "critical school" of neo-aristotelean empiricists (Kant, Hegel), the positivists, and the existentialists. This division is otherwise expressed as the conflict between reductionism in the guise of the effort to derive physics from "ivory tower" mathematics, as opposed to the methods of (for example) Kepler, Leibniz, Gauss, and Riemann, to derive mathematics, as a tool of physical science, from experimental physics.

The pedagogical challenge which the students' demands presented to me and to such collaborators in this as Dr. Jonathan Tennenbaum and Mr. Bruce Director, has been to express these issues in the most concise, experimentally grounded way. All of Gauss's principal work points in the needed direction. The cornerstone of all Gauss's greatest contributions to physical science and mathematics is expressed by the science-historical issues embedded in Gauss's first presentation of his discovery of the fundamental theorem of algebra.

All reductionist methods in consistent mathematical practice depend upon the assumption of the existence of certain kinds of definitions, axioms, and postulates, which are taught as "self- evident," a claim chiefly premised on the assumption that they are derived from the essential nature of blind faith in sense- certainty itself. For as far back in the history of this matter as we know it today, the only coherent form of contrary method is that associated with the term "the method of hypothesis," as that method is best typified in the most general way by the collection of Plato's Socratic dialogues. The cases of the Meno, the Theatetus, and the Timaeus, most neatly typify those issues of method as they pertain immediately to matters of the relationship between mathematics and physical science. The setting forth of the principles of an experimental scientific method based upon that method of hypothesis, was introduced by Nicholas of Cusa, in a series of writings beginning with his De Docta Ignorantia. The modern platonic current in physical science and mathematics, is derived axiomatically from the reading of platonic method introduced by Cusa. The first successful attempt at a comprehensive mathematical physics based upon these principles of a method of physical science, is the work of Kepler.

From the beginning, as since the dialogues of Plato, scientific method has been premised upon the demonstration that the formalist interpretation of reality breaks down, fatally, when the use of that interpretation is confronted by certain empirically well-defined ontological paradoxes, as typified by the case of the original discovery of universal gravitation by Kepler, as reported in his 1609 New Astronomy. The only true solution to such paradoxes occurs in the form of the generation of an hypothesis, an hypothesis of the quality which overturns some existing definitions, axioms, and postulates, and also introduces hypothetical new universal principles. The validation of such hypotheses, by appropriately exhaustive experimental methods, establishes such an hypothesis as what is to be recognized as either a universal physical principle, or the equivalent (as in the case of J.S. Bach's discovery and development of principles of composition of well-tempered counterpoint).

Gauss's devastating refutation of Euler's and Lagrange's misconception of "imaginary numbers," and the introduction of the notion of the physical efficiency of the geometry of the complex domain, is the foundation of all defensible conceptions in modern mathematical physics. Here lies the pivot of my proposed general use of this case of Gauss's refutation of Euler and Lagrange, as a cornerstone of a new curriculum for secondary and university undergraduate students.

Summarily, Gauss demonstrated not only that arithmetic is not competently derived axiomatically from the notion of the so- called counting numbers, but that the proof of the existence of the complex domain within the number-domain, showed two things of crucial importance for all scientific method thereafter. These complex variables are not merely powers, in the sense that quadratic and cubic functions define powers distinct from simple linearity. They represent a replacement for the linear notions of dimensionality, by a general notion of extended magnitudes of physical space-time, as Riemann generalized this from, chiefly, the standpoints of both Gauss and Dirichlet, in his habilitation dissertation.

The elementary character of that theorem of Gauss, so situated, destroys the ivory-tower axioms of Euler et. al. in an elementary way, from inside arithmetic itself. It also provides a standard of reference for the use of the term "truth," as distinct from mere opinion, within mathematics and physical science, and also within the domain of social relations. Those goals are achieved only on the condition that the student works through Gauss's own cognitive experience, both in making the discovery and in refuting reductionism generically. It is the inner, cognitive sense of "I know," rather than "I have been taught to believe," which must become the clearly understood principle of a revived policy of a universalized Classical humanist education.

Once a dedicated student achieves the inner cognitive sense of "I know this," he, or she has gained a bench-mark against which to measure many other things.

- Lyndon.

Summer, 2002

Original Message
To: schiller@schillerinstitute.org

Hi. Mr. Larouche how are you today? I'm one of your fans. I Read Eir online. I'm interested in Mathematics. I wanna be able to solve problems related to enginnering and computing. Whats the best way to go about being the best in Mathematics? Is there any teacher or book that will help me become the best mathematician?

Sincerely,

Mr. LaRouche's Reply

Decades ago, I adopted as an especially beautiful comment on number theory, a remark by a prominent Russian mathematician, Khinchin, who wrote that solutions for important problems in number theory, are always elementary, but not necessarily simple. Today, during the recent several years, I have acquired an additional dimension to my personal responsibilities: the problems of education of a stratum of bright young people, most of whom are presently aged between 18-25 years, who have been cruelly cheated by the kind of education offered them, in secondary schools during their adolescence, and even what are considered leading universities now. These young people have demanded of me, that I outline a program of education to help them make up for what typical, currently leading educational institutions have cheated them. The solution for such a problem is elementary, but not necessarily simple.

For people in the 18-25 bracket, I propose that they begin with a good translation, from the original Latin, of Carl Gauss's 1799 publication of his original discovery of the fundamental theorem of algebra, with the further intention of working through Gauss's Disquisitiones Arithmeticae. The 1799 paper defined what is known as the complex domain. This 1799 paper has the additional importance of presenting his devastating proof of the fundamental errors of D'Alembert, Euler, and Lagrange, a proof which I emphasize as the crucial definition of the meaning of mathematical physics.

I find that the most effective way to present Gauss's fundamental theorem, is to compare the argument he makes with ancient Classical Greek geometry, especially the same problems of geometry which are addressed by Gauss in tha 1799 paper: problems such as doubling the square and the cube by geometric construction. Once the connection of Gauss's discoveries to ancient treatments of these problems, by Archytas and Plat through the work of Eratosthenes, is understood, there is a clear practical sense the meaninng of terms such as mathematics and physics.

Working through Kepler's New Astronomy, is a trip through the foundations of modern mathematical physics. Ask yourself, how did Kepler actually discover gravitation? Ask yourself: What is the importance of followers of Kepler such as Fermat, Huyghens, Leibniz, and Jean Bernouilli, in developing the underlying principle of modern mathematical physics, the Leibniz-Bernoulli demonstration of the relationship of the catenary to the universal principle of least action. For example, study closely the facts respecting the function of the catenary principle in Brunelleschi's solution for the "impossible" engineering task of constructing, with the materials available to him, of the cupola of the Cathedral of Florence.

Concentrate on the Classical, elementary, but never simple solutions for the truly interesting problems of science.

We attempt to keep a running series of pedagogical exercizes on our websites, and intend to publish a polished, suitably illustrated version of that series during the times ahead.

List of Pedagogical Articles

Speeches, Dialogues and Webcasts by Lyndon LaRouche

Link to Gauss 1799 paper (in English)

More on Education

Gauss's Pedagogy, Part I

Gauss' Pedagogy Part II

Translations of Schiller, Leibniz, Plato and Other Great Thinkers

Other Email Dialogues With Lyndon H. LaRouche, Jr.