Riemann for Anti-Dummies Part 13

THE FINER ART OF SCIENCE

How often have you heard, after briefing someone on the strategic situation and LaRouche's unique role in leading mankind out of this crisis, the retort, "I just don't think one man can have the answer." Such a response, not only indicates a narrow, petty, and small minded way of thinking, but, it actually displays a gross illiteracy concerning the history of ideas. In fact, all scientific (and artistic) discoveries were made by one person, who, when that discovery was made, was the only person in the universe who, "had the answer". In truth, the human race has existed to date, because, one person, "had the answer", when no one else did.

Such genius is characterized by the ability, and the willingness, to find principles, in small deviations from the expected, where everyone else finds either no deviations, or excuses such deviations as mere errors. For example, Kepler's discovery that the planetary orbits were elliptical, was provoked by a deviation of 8 minutes of an arc, between the observations, and the results he expected from his hypothesis that the planets moved on eccentric circles. Or, Gauss' determination that the shape of the Earth was both non-uniform and irregular, was provoked by the 16 seconds of an arc deviation, between the measurements of the height of the pole star and the measurements of his geodetic triangulation from Goettingen to Altona. Or, LaRouche's determination of the trajectory of the economy, from the small changes in the mental activity of the population.

In each example, as in all scientific discoveries, it was these small deviations from the "normal" from which new revolutionary concepts were derived.

This process was described by B. Riemann in some philosophical fragments, published for the first time in English in the Winter 1995-1996 issue of Twenty First Century Science and Technology:

"Natural science is the attempt to understand nature by means of exact concepts.

"According to the concepts through which we comprehend nature our perceptions are supplemented and filled in, not simply at each moment, but also future perceptions are seen as necessary. Or, to the degree that the conceptual system is not fully sufficient, future perceptions are determined beforehand as probable; according to the concepts, what is "possible" is determined ( thus what is "necessary" and conversely, impossible). And the degree of possibility (of "probability") of each individual even which is seen as possible, in light of these concepts, can be mathematically determined, if the concepts are precise enough.

"To the extent that what is necessary or probable, according to these concepts, takes place, then this confirms the concepts, and the trust that we place in these concepts rests on this confirmation through experience. But, if something takes place that is unexpected according our existing assumptions, i.e. that is impossible or improbable according to them, then the task arises of completing them or, if necessary reworking the axioms, so that what is perceived ceases to be impossible or, improbable. The completion or improvement of the conceptual system forms the "explanation" of the unexpected perception. Our comprehension of nature gradually becomes more and more complete and correct through this process, simultaneously penetrating more and more behind the surface of appearances.

"The history of causal natural science, in so far as we can trace it back, show that this is, in fact, the way our knowledge of nature advances. The conceptual systems that are now the basis for the natural sciences, arose through a gradual transformation of older conceptual systems, and the reasons that drove us to new modes of explanation can always be traced back to contradictions and improbabilities that emerged from the older modes of explanation.

"The formation of new concepts, in so far as this process is accessible to observation, therefore takes place in this way."

In that same fragment, Riemann goes on to say:

"1. When is our comprehension of the world true?

"`When the relations among our conceptions correspond to the relations of things.'

"The elements of our picture of the world are completely distinct from the corresponding elements of the reality which they picture. They are something within us; the elements of reality are something outside of ourselves. But, the connections among the elements in the picture, and among the elements of reality which they depict, must agree, if the picture is to be true.'

"The truth of the picture is independent of its degree of fineness; it does not depend upon whether the elements of the picture represent larger or smaller aggregates of reality. But, the connections must correspond to one another; a direct action of two elements upon each other may not be assumed in the picture, where only an indirect one occurs in reality. Otherwise the picture would be false and would need correction. If, however, an element of the picture is replaced by a group of finer elements, so that its properties emerge, partly from the simpler properties of the finer elements, but partly from their connections, and thus become in part comprehensible, then this increases our insight into the connection of things, but without the earlier understanding having to be declared false."

Consequently, the healthy mind seeks the ever finer elements that reveal those, yet undiscovered, cycles governing action in the universe. Such cycles had been there all along, but, once discovered, the universe, in which they were acting, changes, by virtue of their now becoming an object of human cognition. This, in turn, enables man to begin a new quest for even finer elements, a search whose possibility depends on the just discovered new cycles. It is the intention, at this point in this series, "Riemann for Anti-Dummies", to investigate those finer discoveries, on which the new concepts of Gauss and Riemann are based. Significantly, the deeper implications of those concepts were not fully recognized, even by Gauss and Riemann, until LaRouche's discoveries in physical economy.

The finer elements that gave rise to these new concepts of Gauss and Riemann were centered in the investigations of the inter-related areas of astrophysics, geodesy, electromagnetism and life. The most efficient path to grasp the Gauss/Riemann theory of functions is through a pedagogical presentation of them, which defines the intended trajectory of this series.

1. Astrophysical-- Kepler had left open for discovery, a planetary orbit between Mars and Jupiter based on a dissonance between the angular speeds between those two planets, which dissonance, Kepler notes, was evidenced by the smallest deviation perceptible. Gauss' determination of the orbit of Ceres, and the subsequent discoveries of other asteroids, confirmed Kepler's hypothesis. The orbits of these finer elements were consistent with the principles Kepler had discovered for the six visible planets. The eccentricities, inclinations, and interweavings, of these orbits made hitherto unobserved, but suspected, orbital irregularities, measurable. Gauss' investigation showed that these irregularities, were, in fact, not irregularities, but evidence of finer cycles that permeated the whole solar system. This extended colligation of cycles gave rise to a new concept of manifold in the minds of Gauss and Riemann.

2. Geodesic-- Earlier measurements of the Earth had shown that its shape was non- uniform (ellipsoidal) rather than uniform (spherical). Gauss spent nearly 20 years making and supervising careful physical measurements of the Earth's gravitational and magnetic characteristic, and relating those measurements to astronomical ones. Gauss' meticulous effort revealed that these characteristics deviated slightly from the concept of simple non-uniformity, as, for example, in an ellipsoid, and he showed the error of assuming any shape a priori. Instead, Gauss developed the idea of the shape of the Earth as a non-uniform and irregular manifold of measurement, today called the "Geoid". To measure this concept, Gauss, and later Riemann, extended Leibniz' calculus from concerning the characteristic of action along pathways, to the characteristic of action in the surfaces on which those pathways exist.

3. Electromagnetism and Light-- The work of Ampere and Fresnel posed the paradox that the assumed characteristics of action in space in the macroscopic realm became discontinuous in the microphysical realm. Such paradoxes led to Gauss' and Riemann's development of retarded potential, and Riemann's concepts of complex functions.

4. Life-- The functional relationship between living and non-living processes were investigated by Riemann, notably in his researches into the characteristics of a sound wave, in the human ear, and in the air.

But, it would be wrong to leave the suggestion that a thinking mind would be content with existing concepts, until knocked on the head by some physical deviation, instead of actively seeking out such paradoxes. Think of this process as a type of a well-composed fugue, in which the theme and counter-theme become indistinguishable as to cause, and only the whole composition remains in the mind as a One. For example, the discoveries of Kepler, Fermat and Leibniz had already provoked Abraham Kaestner to knock down the remaining pillars of ivory tower mathematics with his attacks on the a priori acceptance of Euclidean geometry. Once that was initiated, Kaestner shifted life's trajectory of the young Gauss, by (as discussed in the last two installments) provoking that young man to draw out the deeper implications of his first discovery, the constructability of the 17-gon. That small shift, which Kaestner induced into Gauss' mind, contained the insights that emerged, years later, in the investigations of the physical paradoxes just described.

That's where this series is headed.