Riemann for Anti-Dummies: Part 1

Indicative of the cognitive deficiency of the Baby Boomers and subsequent generations, is the proliferation of “How To” books, under the appellation, “`X’ for Dummies.”

Such tomes originated, as did many of the more destructive trends in the late 20th Century, in the computer-based information/entertainment business. Desiring ever-increasing values in stock prices, the high priests of information ensnared a growing number of lost souls, by explaining the purported intricate workings of computer programs, using baby talk and pictures, so that “Dummies” could understand them. This appeal to ignorance spread like fungus in August, as growing numbers of willing fools, eagerly identified themselves as “Dummies.” A recent survey of Amazon.com turned up 714 such books, ranging from computer subjects such as “Windows ’98 for Dummies” and “Internet for Dummies,” to “Investing for Dummies”, “Buying a Car for Dummies,” “Classical Music for Dummies,” “Parenting for Dummies,” “Gourmet Cooking for Dummies,” “Dating for Dummies,” “Sex for Dummies,” “Fundraising for Dummies,” and “Leadership for Dummies, abridged version.” (No joke. Check it out for yourself!) Soon, under pressure from HMOs, medical schools will soon adopt this training method. Instead of educating highly skilled physicians, “health care providers” will be tutored in such subjects as “Brain Surgery Made Simple” and, “Cardiology in Three Easy Steps.”

Of course, there is nothing new here. 500 years ago Paolo Sarpi and his protege Galileo, pushed a similar scam, against the Renaissance, with a, “Science for Dummies” series, which taught that the universe could be made simple, if the human mind were removed. Since then, Sarpi’s minions, from Newton, to Euler, to Kronecker, to von Neumann, have induced countless numbers of otherwise intelligent individuals, to turn from the truthful, but challenging, science of Plato, Cusa, Kepler, Fermat, Leibniz, Kaestenr, Gauss, and Riemann, to a simplified world without Mind “The Universe for Dummies.”

Now, given that the future of civilization depends on diminishing the attractiveness of being a dummy, we will introduce, from time to time over the coming period, a series of pedagogical discussions, on the above-titled subject, “Riemann for Anti-Dummies.” This investigation will take us through the development of what has been generally called Gauss’ and Riemann’s theory of functions. While formal mathematical presentations of this subject abound, both in a form suited for high priests of mathematical and computer sciences, and also in a form more suited “for dummies,” virtually all these surveys lack any real understanding of the thinking underlying Gauss’ and Riemann’s discoveries. From our standpoint it were more appropriate to characterize the Gauss/Riemann theory of functions, as a metaphorical science, rather than a formal mathematical one. That being the case, we will avoid beginning with a formal definition of a function, preferring to leave such matters unstated, so that the reader can construct a concept, rather than learn a definition.

While it may occasionally be tacitly acknowledged, it has become obscured to the point of being unrecognizable, that the Gauss/Riemann theory of functions is rooted in the revolutionary science of Kepler. For that reason, it is necessary to restate, albeit in an abbreviated form, certain aspects of Kepler’s discoveries, that have been dealt with at more length in previous pedagogical discussions.

For Kepler, like all humans before and since, the motions of the planets of our solar system, and the stars beyond, are observed, only indirectly and measured as positions on the inside of a celestial sphere. These observations, when mapped simultaneously, onto one sphere, produce a tangled network of circular arcs. From this jumble, it becomes known upon reflection, that the motions so represented, appear to be irregular. How then, can the human mind, untangle these curves, and construct a function, of which this jumble is a projection?

Contained in these projections, are a set of unique positions (singularities), such as the points representing the place where the motion of the planets change direction; or the points representing the positions in which the planet’s motion changes from becoming increasingly fast, to increasingly slow; or the points representing where the planet’s reach a maximum or minimum deviation from the ecliptic; or the points where the projected orbits of different planets appear to intersect. Working backwards, the relationship among these singularities, affords an investigator the ability to construct a function, that would produce the observed result.

As Kepler puts it in the “New Astronomy”:

“The testimony of the ages confirms that the motions of the planets are orbicular. It is an immediate presumption of reason, reflected in experience, that their gyrations are perfect circles. For among figures it is circles, and among bodies the heavens, that are considered the most perfect. However, when experience is seen to teach something different to those who pay careful attention, namely, that the planets deviate from a simple circular path, it gives rise to a powerful sense of wonder, which at length drives men to look into causes.

“It is just this from which astronomy arose among men. Astronomy’s aim is considered to be to show why the stars’ motions appear to be irregular on earth, despite their being exceedingly well ordered in heaven, and to investigate the circles wherein the stars may be moved, that their positions and appearances at any given time may thereby be predicted.”

Ptolemy, [Tycho] Brahe, and Copernicus all sought to construct a function, that when projected onto a sphere, produced a tangle of curves, approximate to the observed motions of the planets. Each produced one, and each was radically different than the others. All three approximations, were equal, “to within a hair’s breadth” of each other, yet deviated from the historical observations, and in their ability to predict future observations, by a small, but measurable amount. Yet, there was no way to distinguish which one was true, or for that matter, why it were not possible to construct an infinite number of other functions, that provided an equally good approximation of the observations.

In seeking to construct the desired function, Kepler rejected the mathematical methods of his predecessors. “But where Copernicus did so through mathematical arguments, mine were physical, or rather metaphysical.”

The required function, therefore, must not simply account for the appearances, but it must account for the physical motion of the planet, which required an inquiry into the reason for the planet’s motion:

“Now the planet must execute a perfectly circular path in the pure aether by its inherent force, epicyclic in the first model and eccentric in the second. It is therefore clear that the mover is going to have two jobs: it must have a faculty strong enough to move its body about, and second, it must have sufficient knowledge to find a circular boundary in the pure aether, which is itself not divided into such regions. This is the function of Mind.”

What is the nature of this Mind, and how can it be discovered? “God, like one of our own architects, approached the task of constructing the universe with order and pattern, and laid out the individual parts accordingly, as if it were not art which imitated Nature, but God himself had looked to the mode of building of Man who was to be.”

The function Kepler constructed, was not a formal logical construction, but a metaphor, using geometry, words and number, to reflect the characteristics of Mind, to the Mind. The sought after function cannot be stated in simple formal mathematical terms, but only through its elaboration, which is contained in the entire corpus of Kepler’s work. Some of the principles that characterize the function can be stated summarily as: the motive force of the solar system is located in the Sun, which is at rest; the effect of the Sun’s motive force diminishes with distance; the orbits of the planets are ordered according to the five Platonic solids; the planet’s distance from the Sun varied during its orbit, thus producing a regular but non-constant motion, reflected in the constant ratio between the area swept out by the planet, to the time elapsed during that interval; the shape of these orbits so produced are ellipses which all intersect at the Sun; all planetary orbits are further constrained such that the square of the total time for one orbit is equal to the cube of planet’s mean distance to the Sun; and the ratio of the angular velocities, between the planets, correspond to unique musical intervals.

The above conditions, are only some of the characteristics, determined by the sought after function that characterizes the motions of the solar system. It is the overall nature of the function, that determines each characteristic. Ironically, we know the nature of the function, from the characteristics, yet we cannot discover the characteristics, without a concept of the function.

Next week we will review their development more closely, but to complete this week’s polemic, here’s the version the oligarchy created for dummies. First, get rid of the Platonic solids and the harmonic intervals. Reduce the rest to three simple mathematical expressions, and call them, “Kepler’s laws.” Then further reduce these three expressions, to one simple mathematical expression, called the “inverse square law.” Then hire a psycho-pathological occultist named Isaac Newton, to write a “Universe for Dummies.” Then hire a new generation of even more degraded academics, (preferably from the Island of Laputa) who produce, another version — “Universe for real Dummies,” and sell it as textbooks for secondary schools and universities, and put it on the Internet.

A healthy commentary on such scientific methods was recounted by Rabelais.

“Why is it,” asked Gargantua, “that Friar John has such a handsome nose?”

“Because,” replied Grandgousier, “God wished it so, and he makes us in such shape and to such end as pleases his divine will, even as a potter fashions his pots.”

“Because,” said Ponocrates, “he was one of the first at Nose-fair. He chose one of the finest and biggest.”

“Stuff and nonesense,” said the monk. “According to true monastic reasoning, it was because my nurse had soft breasts: when she suckled me, my nose sank in, as if into butter, and there it swelled and grew like dough in the kneading-trough. Hard breasts in nurses make children snub-nosed. But come, come! Ad formam nasi cognoscitur, ad te levavi. (By the shape of his nose he is known, I have lifted up mine eyes to thee.)