Riemann for Anti-Dummies: Part 13 : The Finer Art of Science

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.

Riemann for Anti-Dummies: Part 12 : Gauss’s Division of the Circle

Gauss’ Division of the Circle

The pursuit of a discovery of a universal principle always requires the pursuer to follow the Socratic method of negation, or, as Cusa called it, “Learned Ignorance”. This is the method by which Kepler ascended from the tangle of observed motions of the planets on the inside of an imaginary sphere, to the “hypergeometric” function, whose characteristics we’ve been exploring throughout this series. While the cause of these observations is determined from the top down by that function, like the shadows on Plato’s Cave, we cannot know that function directly. Rather, we must look between the gaps in the observations, and discern the hypergeometry of which those observations are a function.

It is the nature of this method, that, as our knowledge of the underlying hypergeometry increases, new gaps appear, through which new characteristics of the hypergeometry become discernable. These new characteristics, in turn, recast the previous discoveries, in a new light. For example, the principle that light travels the shortest path, discovered by the Greeks as a characteristic of reflection, becomes a special case of the principle that light travels the path of least time, discovered by Fermat as a characteristic of refraction. In both cases, light acts according to a minimizing principle. But, in the former, that principle is with respect to a manifold of space, while the latter is with respect to a manifold of space-time. It is the manifold from which the characteristic of the path is determined, but it is the characteristics of the path, by which the manifold is discovered.

Another example may help illustrate this point. What is the meaning of the question, “When did human life begin?”? If your conception of the universe is a reductionist/Darwinian one, in which non-living processes give rise to living ones, which, in turn, give rise to cognitive processes, the answer to that question, is, a search for a mystical point in time, in which something, becomes what it’s not. (i.e., non-living processes become living, or living processes becomes cognitive.) However, from the standpoint of a multiply-connected universe, which, as LaRouche most recently wrote, is organized from the top down, cognitive, to living, to non- living, the answer to the question, as Plato, Philo, and Augustine have said, is “from the beginning.” (cognition created man) Under a conception of the universe organized from the bottom up, the trajectory of evolution is a line. (Whether that line is curved or straight, it is still a one-dimensional magnitude.) Whereas, under the top-down conception, the trajectory is of a quite different nature.

To grasp the nature of that trajectory requires the metaphors developed by Riemann, toward which this series is heading. To get there, we must first traverse the intervening steps. This is what brings us to Gauss’ division of the circle. What follows may be at times dense, as it brings together discoveries that span more than 2500 years of cognition. But roll up your sleeves and work it through.You will be well rewarded with learned ignorance.

Kepler, in the first book of the “Harmonies of the World”, showed that the divisions of the circle generate a hierarchy of types of magnitudes. Following the Greek tradition, these magnitudes are ranked according to “knowability”. Those magnitudes that are measurable directly by the diameter of the circle, have the first degree of knowability, those measurable by a part of the diameter, have the second degree, and those whose squares are measurable by the diameter, have the third. All others are incommensurable. It is these incommensurable magnitudes, on which non-uniform motion depends, as Kepler had already shown in his Mysterium Cosmographicum and The New Astronomy. Therefore, to grasp the principles of non-uniform motion, one had to grasp the principles of generation of these incommensurables.

Magnitudes of the first three degrees of knowability, are also called constructable, because they can be constructed from the circle itself, or to put it colloquially, by straight-edge and compass. (The straight-edge being the diameter and compass being the circumference of the circle.) It is in the investigation of what is constructable, that we discover a gap, an anomaly, through which we ascend to the idea, that, the seemingly uniform circle, is, in fact, not uniform at all, but a special case of non-uniform action!

Ancient Greek philosophers had fully investigated, that, while the circle is uniform in all its parts, it does not divide uniformly. The circle can be divided into two parts by folding in half. By folding in half again, and again, the circle can be divided into 2,4,8,16,etc. parts. But to divide it into three parts, that is, to inscribe a triangle in it, requires the generation of a magnitude of the second degree of knowability. (Half the diameter, is the side of a hexagon.). Once that magnitude is created, it too can be doubled, to produce a division by 6, 12, 24, etc. parts. But, to divide the circle into five parts, requires the generation of a magnitude of the third degree of knowability, specifically, the golden section. The golden section is not commensurable with the diameter or a part of the diameter, but its square is. Hence, it is constructable.

However, it had been believed for more than 2000 years, that all other divisions of the circle were non-constructable. In other words, those divisions of the circle based on prime numbers greater than five, depended on magnitudes that were beyond Kepler’s third degree of knowability. This boundary condition, suggests that something outside the circle, a higher principle, is governing. It is that principle, that Gauss sought.

To discover this principle, Gauss, in effect, inverted the problem all together. Instead of thinking of two different types of magnitudes, constructable and non-constructable, he investigated a general principle governing the generation of magnitudes, of which the constructable ones were a special case. (A suggestive example of this concept was supplied by LaRouche several years ago, when he posed that the number 5 associated with the hypotenuse of a 3-4-5 right triangle, should be considered a special type of irrational number.)

Additionally, Gauss, using a unique application of geometry of position, which, at first may seem obscure. He inverted the conception of the circle. Instead of beginning with the circle and trying to find those positions that divided it, he sought the functions that created a divided circle. Thus, the positions were produced by the division, not the division by the positions.

Gauss showed that both the above principles relied on two, seemingly unrelated, conceptions that were at the heart of Greek science, and, that had been extended by the work of Kepler, Leibniz, Bernoulli, and Fermat: the geometric generation of incommensurables, and the more shrouded principles governing the generation of prime numbers. And, spurred by Kaestner’s prodding, Gauss showed that these principles extended beyond the circle.

As Gauss described it in the opening of the final chapter of “Disquistiones Arithmeticae”, “Among the most splendid developments contributed by modern mathematicians, the theory of circular functions without doubt holds a most important place. We shall have occasion in a variety of contexts to refer to this remarkable type of quantity, and there is no part of general mathematics that does not depend on it in some fashion…. I will speak of the theory of trigonometric functions as related to arcs that are commensurable with the circumference, or of the theory of regular polygons….The reader might be surprised to find a discussion of this subject in the present work which deals with a discipline apparently so unrelated; but the treatment itself will make abundantly clear that there is an intimate connection between this subject and higher Arithmetic.

“The principles of the theory which we are going to explain actually extend much farther than we will indicate. For they can be applied not only to circular functions but just as well to other transcendental functions….”

The first principle of Greek geometry that Gauss re-worked, is described by Plato in the Meno, Theatetus and Timaeus dialogues, and concerns the generation of incommensurable magnitudes as a consequence of a change in dimension. The reader can construct a geometrical representation of this by drawing a square, then drawing the diagonal, then, drawing a new square using the diagonal for its side. As Plato demonstrated in the Meno, the second square will have twice the area of the first, but the side of the second square, will be incommensurable to the side of the first. (In Kepler’s terms, the diagonal will be the third degree of “knowability”.) If you continue this drawing, you will produce a spiral sequence of squares, whose sides are the diagonals of the previous squares, and whose areas are double, the previous squares. The Greeks called the diagonals, the “geometric mean” between the two squares.

However, something new develops if you try and replicate this process in 3 dimensions, as in the case of doubling a cube. The diagonal of the cube does not correspond to the side of a cube with double the volume. This is the famous problem, the Delian priests brought to Plato. Eratosthenes reports Plato’s famous rebuke, that the Gods had posed this problem to the Greeks, because they wanted to chide the Greeks into studying geometry in order to improve their thinking. Hippocrates of Chios had shown that the incommensurable associated with the doubling of the cube, was of a different species, than the incommensurable associated with doubling the square. In other words, a change in dimensionality, produced a different species of incommensurable. In the Timeaus, Plato reports Hippocrates discovery:

“But it is not possible that two things alone be joined without a third; for in between there must needs be some bond joining the two…. Now if the body of the All had had to come into being as a plane surface, having no depth, one mean would have sufficed to bind together both itself, and its fellow-terms; but now it is otherwise for it behoved it to be solid in shape, and what brings solids into harmony is never one mean, but always two.”

By inversion, if two or more geometric means are required to double a magnitude, the doubling of that magnitude, is an action, that originates in a dimensionality greater than two.

Gauss’ insight rested on these Platonic principles, with a crucial extension supplied by Bernoulli’s discovery of the equiangular spiral, which he called, “spiral mirabilus”. Bernoulli showed that this spiral was an exemplar of geometric growth. For example, a line extending from the center of the spiral outward, will be cut at different intervals by each spiral arm. These intervals will be the same proportion to one another, as the areas and diagonals of Plato’s squares. Similarly, lines emanating from the center of the spiral at equal angles from each other, will cut the spiral arms in geometric proportion. Thus, equal divisions of the spiral, cut the spiral into parts that are in geometric proportion.

Now, if we think of the circle as a special case of the spiral, then the division of the circle by lines (radii) emanating from the center at equal angles, cuts the circumference in arcs that are in geometric proportion. The intersection of these lines with the circumference correspond to the vertices of an inscribed polygon. Thus, to divide the circle into “n” parts, corresponds to finding “n-1” geometric means. Those divisions, that can be accomplished by finding one mean between two others, are constructable; and those requiring two or more means are not. Thus, the different divisions of the circle are actually projections, of action originating in manifolds, of higher dimensionality, than the seemingly two dimensions of the circle. Again, like the shadows on Plato’s Cave, or Learned Ignorance, we can only ascend to knowledge of those manifolds, from the anomalies embedded in their reflection.

Gauss showed that these anomalies can be discovered, if we think of the circle, not in a Euclidean/Cartesian plane, but in the complex domain. A simple example is, perhaps, the easiest way to illustrate the point. To divide a circle into 4 parts, first think of the circle in the complex domain. Pick a point on the circumference for the first vertex, and call it 1. To divide the circle into four parts we would mark off three other points that are 90 degrees from each other. According to what we said above from Bernoulli and Plato, these points are all in geometric proportion to each other. Expressed in numbers, these points form a cycle of geometric means from 1 to 1. Using the letter i to denote the square root of -1, that series is, 1, i, -1, –i. These four numbers produce a cycle, such that if you multiply each one by itself 4 times, you get 1.

Gauss’ insight was based on the following: Dividing the circle into “n” parts requires finding “n-1” geometric means between 1 and 1. Each “n” divisions requires a function with specific characteristics. Gauss discovered a general principle that governed the nature of those characteristics.

Again, it is easiest to demonstrate this by example. Put five dots on a page, roughly in a circle and number the dots counter-clockwise, 1, 2, 3, 4, 5. Now, connect dot 1-3, then 3-5, then 5- 2, then 2-4, then 4-1. In this action, you went around the circle twice, connecting 2 dots in each turn. You also drew a pentagonal star. Perform the same action, connecting 1-4; 4-2; 2-5; 5-3; 3- 1. This produces the same result as above. Try the same with 7 dots. Connect 1-3; 3-5; 5-7; 7-2; 2- 4; 4-6; 6-1. This action required 2 cycles, connecting 3 dots in each cycle (3, 5, 7) and (2, 4, 6). Notice the shape of the heptagonal star. Make another configuration of 7 dots, and connect them in the following sequence: 1-4; 4-7; 7-3; 3-6; 6-2; 2-5; 5-1. This action required 3 cycles, connecting 2 dots in each cycle. Notice that the shape of this heptagonal star is different than the previous one. If you experiment around, using different cycles, (e.g. 1-5; 5-2;, etc. ) the result will be identical to one of the two produced above. Now, continue these types of experiments with 7, 11, 13 and 17 dots. In each case, you will find that the combinations of cycles and dots, is constrained by the prime number factors of 10, 12, and 16 respectively, ( n-1).

Gauss showed that these combinations of cycles corresponds to Plato’s principle of means. Those divisions that can be resolved completely into cycles of 2, correspond to inserting 1 geometric mean between two others, as in doubling of the square, and are therefore constructable. Those prime number divisions that cannot be so resolved, correspond to inserting two or more geometric means, and, like the doubling of the cube, are not constructable. Thus, it is possible to construct figures of 2, 3, 5, 17, 257, 65,537 and any other prime number divisions of the form (22)n + 1. All other divisions cannot be constructed because they are reflections of actions of a dimensionality higher than two.

It is the thinking underlying the above discovery of Gauss, that is at the heart of Gauss’ and Riemann’s development of the theory of functions.

Riemann for Anti-Dummies: Part 11 : Transcending Euclid

Transcending Euclid

It is crucial for anti-dummies to always bear in mind the groundwork for all modern science, that Nicholas of Cusa teaches us in “On Learned Ignorance”:

“Wherefore it follows that, except for God, all positable things differ. Therefore, one motion cannot be equal to another; nor can one motion be the measure of another, since, necessarily, the measure and the thing measured differ. Although these points will be of use to you regarding an infinite number of things, nevertheless if you transfer them to astronomy, you will recognize that the art of calculating lacks precision, since it presupposes that the motion of all the planets can be measured by reference to the motion of the Sun. Even the ordering of the heavens with respect to whatever kind of place or with respect to the risings and settings of the constellations or to the elevation of a pole and to things having to do with these is not precisely knowable. And since no two places agree precisely in time and setting, it is evident that judgments about the stars are, in their specificity, far from precise. If you subsequently adapt this rule to mathematics, you will see that equality is {actually} impossible with regard to geometrical figures and that no thing can precisely agree with another either in shape or in size. And although there are true rules for describing the equal of a given figure as it exists in its definition, nonetheless equality between different things is {actually} impossible. Wherefore, ascend to the recognition that truth, freed from material conditions, sees, as in a definition, the equality which we cannot at all experience in things, since in things equality is present only defectively.”

Such was the approach of Kepler, Fermat, and Leibniz, who rejected the method of imposing mathematical definitions on the physical universe, but, in doing so, discovered a new higher principle, that led to the creation of a new mathematics. Be it the motion of the planets, the path of refracted light, or the shape of a hanging chain, physical action does not conform to perfect mathematical rules. For example, the planets don’t move in perfect ellipses, nor does a ray of refracted light move in a perfect cycloid, nor does a hanging chain form a perfect catenary. It is in this deviation from the perfect, that the higher, harmonic, principles governing the action can be discovered. Furthermore, these deviations are not expressible in perfect mathematical ratios, such as, for example, the dependence of the planet’s or light’s path on the sine of the angle. These higher principles define the characteristics of what Riemann called “multiply-connected functions”. In other words, it is not the geometry that defines the physical action, it is the physical action that defines the geometry. That geometry, so defined, does not conform to perfect mathematical rules, but is knowable, yet, one must first grasp what is meant by knowable. Cusa’s cited imperfection of the physical universe, thus provoked the discovery of the universe’s perfectability.

The irony is, that if the universe conformed to perfect mathematical rules, it would not be discoverable. As Kepler describes this process in the “Harmonies of the World”:

“… [U]nless the Earth, our domicile, measured out the annual circle, midway between the other spheres changing from place to place, from station never would human cognition have worked its way to the true intervals of the planets, and to the other things dependent from them, and never would it have constituted astronomy.”

Cusa’s method had opened the door, not only to Kepler’s revolutionary discoveries, but, to the revolutionary discoveries of Fermat, who abandoned the path of shortest distance for the path of least time; J.S. Bach, who rejected abstract mathematical notions of musical intervals, creating the well-tempered system through his compositions; and Leibniz, who created the mathematics of non-constant curvature, that he called the infinitesimal calculus, to name but a few.

It was through this doorway, that the 18-year old Gauss walked on March 30, 1796, when he made his first entry into his scientific notebook — his discovery of the constructability of the 17-sided polygon. For more than 2000 years, it was believed that it were impossible to construct a 17-sided regular polygon with straight-edge and compass. On this day, Gauss recalled, after much hard work, the possibility of such a construction appeared to him all at once.

Gauss always considered this one of his most important discoveries, which, accompanied by a provocation from Kaestner, determined the trajectory of his entire creative life. So much so, that he asked that a 17-gon be engraved on his tombstone. (Because an engraved 17-gon would look so much like a circle, a 17-sided star was engraved instead.)

In the next installment, we will work through Gauss’ construction in detail. However, some historical background will be useful to set the stage. From 1792-1795, Gauss received his early education at the classically-oriented Collegium Carolineum in Brunswick-Wolfenbuettel. The curriculum consisted of ancient and modern languages, classical sciences, aesthetics, poetry, music, and art. On Oct. 11, 1795, he left for Goettingen, where he was attracted to the study of philology, under Christian Gottlob Heyne. The philology seminar at Goettingen had been founded by Johann Matthais Gesner, a former colleague of J.S. Bach at the St. Thomas School in Leipzig. Heyne was Gessner’s successor. Also teaching at Goettingen was Leibniz’ defender and Benjamin Franklin’s host, Abraham Gotthelf Kaestner, who had come from Leipzig in 1750, and under whom Gauss studied mathematics.

In a later letter to his Collegium professor, E.A.W. Zimmerman, Gauss said he was particularly attracted to the philology lectures of Heyne and wished to concentrate on the subject. Initially, he thought Kaestner was a dull old man, but, “I have since realized I was in error and that he is quite an extraordinary man.”

In May 1796, Gauss told Zimmerman in a letter, that he had decided to devote himself to mathematics after discovering the constructability of the 17-gon. When he first showed his discovery to Kaestner, the teacher was not at all impressed, and Gauss interpreted this as hostility to something new. He persevered, and when he discussed the concept further, Kaestner was astounded, but responded that the discovery would be of no use, and he (Kaestner) had already developed the basis for the discovery in his, “Beginning foundations of the analysis of finite magnitudes.” Gauss persisted further, obtaining an agreement from Kaestner, to have him review Gauss’ paper. After further contemplation, Gauss realized that he had to separate Kaestner’s criticism concerning the discovery’s practical use, from the rest of Kaestner’s reaction, “but, if I (Gauss) were to be able to give a more general treatment of the subject, it would be of pleasing curiosity and perhaps produce a brighter insight into this area of mathematics.” Gauss told Zimmerman that he took Kaestner’s judgment entirely to heart.

The more general treatment to which Kaestner was guiding Gauss, was toward the solution of the “Kepler challenge”, which concerned the divisibility of the ellipse, a curve of non- constant curvature. In fact, Gauss’ method for the division of the circle was based on the discovery, that the constantly-curved circle, is, actually, a special case of non-constant curvature!

Gauss’ investigation into the division of the circle, was an extension of the study done by Kepler in the first book of the “Harmonies of the World”: “On the regular figures, the harmonic proportions they create, their source, their classes, their order, and their distinction into knowability and representability.”

It is here that Kepler started his elaboration of the harmonic ordering principle that governs the physical universe. “We must seek the causes of the harmonic proportions in the geometrical and knowable divisions of a circle into equal number of parts,” he began.

To divide the circle by geometrical means, according to Kepler, is to determine the ratio of the side of the figure to the diameter. It is in the efforts to divide the circle, that the human mind discovers the different types of harmonic proportions, which Kepler ranked according to degrees of knowability.

The first degree of knowability pertains to those quantities which can be proven equal to the diameter. The second degree of knowability pertains to those quantities that can be proven to be equal to parts of the diameter. The third degree of knowability pertains to those quantities that are inexpressible in length but expressible in square. From this follows those quantities that are inexpressible, or as the Greeks called them, incommensurable. However, the ranking continues with the incommensurables. The fourth degree of knowability pertains to those lengths that are not expressible by squares, but are expressible as rectangles. Kepler continued, following the tenth book of Euclid, the further degrees of knowability of the incommensurable.

What is significant for us, is that Kepler is re-asserting the knowabiblity of incommensurable magnitudes, not only in the context of geometry, but as the very magnitudes by which the physical universe is characterized. Here Kepler carried out a strenuous polemic against Petrus Ramus, a leading Aristotelean of the day, who had sought to ban incommensurable magnitudes, not only from the physical universe, but from geometry as well!

As we will see in the next installment, it is in the division of the circle, that we encounter these different degrees of knowability. This raises the question, if the circle is, at it appears to the eye, a perfectly uniform, constantly curved figure, why, when one attempts to divide it, does one encounter magnitudes of different degrees of knowability? The reader should think of the different types of magnitudes necessary to construct a triangle, square, pentagon, hexagon, and septagon, for example. Each different division of the circle gives rise to different degrees of knowability, and some divisions, such as seven, seem to be unknowable altogether.

This is the question that Kepler investigates in the Harmonies. The question the young Gauss investigated was, “What is the principle that governs the principle of knowability?”

Riemann for Anti-Dummies: Part 10 : Justice for the Catenary

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 Gauss-Riemann theory of functions, and also more generally.

The crucial issue is the distinction between defining a principle from the standpoint of abstract geometry, versus real physics. From the standpoint of abstract geometry, least time and equal-time are represented by a cycloid, but from the standpoint of real physics, the catenary function reflects these principles. The investigation of the gap between what abstract geometry leads us to believe, and what we come to know by real physics, is at the center of the method of Cusa, Kepler, Leibniz, Gauss, Riemann, et al.

“But, wait a minute,” some might protest, “If I make a pendulum wrap around a cycloid, or, if I make a ball roll along a cycloidal path, it’s motion conforms to equal-time and least-time. Doesn’t that show that the cycloidal path corresponds to a physical principle?”

The difficulty, or even downright hostility, with which some people might react to this paradox, is paradigmatic of the mediocrity associated with relying on secondary sources, and popularly accepted gossip, instead of becoming to know, by re-living an original discovery. While LaRouche draws new, revolutionary, implications from this paradox, the distinction he makes between abstract geometry versus real physics, contrary to academically accepted chatter, is identical to the standpoint of the original discoverers; Huygens, Johann Bernoulli and Leibniz.

During the 1680’s and 1690’s these thinkers engaged in a dialogue concerning the development of the new mathematics, demanded by Kepler’s confirmation of Cusa’s hypothesis, that action in the physical universe is non-uniform. Because physical action of this type is always changing non-uniformly, it is impossible to determine the position of, for example, a planet based simply on its past positions, or as LaRouche has put it, “by connect-the dots, statistical methods”. Rather the position of the planet is determined by an underlying characteristic of change that governs the whole orbit. This problem is exemplified by the question, “How does the planet know how to move?”. To answer that question, we must first ask, and answer, “What intention is this action of the planet fulfilling?”, and, “How is that intention manifest at each moment?”

Cusa expresses this in his dialogue De Ludo Globi (The Bowling Game). The dialogue concerns a game played with a non-uniform ball that is rolled on a surface on which 9 concentric circles are drawn. The object of the game is roll the ball as close to the center as possible. But, since the ball is non-uniform, it follows a spiral, rather than a straight path. The player intends to roll the ball on a path that ultimately winds up in the center of the circle, but to do that, he must start the ball with a speed and trajectory, that after changing non-uniformly, ends up at the center. Nicholas of Cusa draws an analogy from this game to the relationship between God, Man and Nature:

“Analogously, the rational soul intends to produce its own operation; with its steadfast intention persisting, the soul moves the hands and instruments when a sculptor chisels on a stone. Intention is seen to persist immutably in the soul and is seen to move the body and the instruments. In a similar way, nature (to which certain men give the name “world-soul”), moves all things while there persists ts unchanging and permanent intention to execute the command of the Creator. And the Creator, with His eternal, unchanging, and immutable intention persisting, creates all things. “Now, what is an intention except a conception, or a rational word, in which all the respective exemplars of things are present?…”

Leibniz and his collaborators, Johann and Jakob Bernoulli, developed the calculus to increase the mind’s capacity to grasp the nature of the intentions governing non-uniform action in the universe. The effectiveness of the calculus is illustrated by Bernoulli’s determination of the brachistrone, discussed in the last installment. In that example, Bernoulli derived the cycloid as the least-time path that results, if at each moment, the speed of the body is proportional to the square root of the distance dropped. This week, we look at another example of the application of the calculus, in the determination of the geometry of the hanging chain, by Bernoulli and Leibniz.

In 1691 Bernoulli published his “Lectures on the Integral Calculus”, which remains the best elementary textbook on the integral calculus to this day. (Anyone comparing this work to the post-Cauchy calculus textbooks widely used today, will be immediately struck by how fraudulent all such treatments of the calculus are. It is a testament to the bankruptcy of modern science education, that Bernoulli’s book, rather than being the standard for all introductory courses in calculus, can be found only in obscure places in some university libraries.) Contrary to the Cauchy fraud, Bernoulli defines the calculus from the standpoint of the integral as the solution of a differential equation. In other words, the integral, for Leibniz and Bernoulli, expresses the underlying nature (intention) of a physical process which at each moment has a certain characteristic of action. The characteristic which expresses the change at each moment is what Leibniz called the differential. A whole physical processes (integral) can, thus, be expressed as a function of its characteristic change at each moment, by what Leibniz called a differential equation. The example of the problem of the hanging chain will illustrate this relationship.

Bernoulli justly claims that Leibniz’ method, “Which to a certain extent, stretch into the deepest regions of geometry,” is capable of solutions, “that until now the power of ordinary geometry had ridiculed and were unable to produce.”

Bernoulli shows how the calculus was developed to solve certain physical-mechanical problems, such as determining the path of least-time and equal time, or the shape of the hanging chain. However, Bernoulli also issued the same caveat, that had previously been sounded by Kepler with respect to the methods of Ptolemy, Brahe and Copernicus. For example, he says that investigations of the cycloid as the path of least-time and equal-time, start with certain physical assumptions, and then, as Bernoulli said, “dress[es] them up so as to transform the mechanical principle into a purely geometrical one.” The physical mechanical principles that result, such as the least-time and equal-time properties of the cycloid, are, thus, products of abstract geometry, and not true physical principles.

Now, look at the problem associated with the catenary, to which Bernoulli and Leibniz also applied the methods of the calculus. A chain or rope hanging under its own weight assumes a unique geometrical shape. That shape, however does not conform to any curve found in any textbook on geometry. Here the calculus is employed to determine, “What is the geometry that characterizes this physical process.” It is important to re-state this inversion. The cycloid is the path that geometry produces, on the assumption that the universe acts in a certain way. The catenary is the path the universe produces to enable the hanging chain to assume a stable, “orbit”. In the former, geometry produces the principles, in the latter, the principles produce the geometry. The Catenary

To grasp this distinction, look at the catenary, as Leibniz and Bernoulli did.1 Bernoulli’s treatment is found in a German translation of his 1691 “Lectures on the Integral calculus”.2 The physical properties of the hanging chain are described in Chapter 4 of “How Gauss Determined the Orbit of Ceres” Fidelio, which the reader should review.) As emphasized there, the catenary shape, formed by the hanging chain, is akin to a planetary orbit, in that every position along the curve, is a function of the physical principles that produce the curve. If any part of the curve is changed, the entire curve re-orients itself, so as to maintain the non-uniform curvature of the catenary (See Figure 1.) Galileo attempted to apply his “ivory tower” methods to investigations of this phenomenon by trying to fit the hanging chain into his pre-existing assumptions of geometry. The closest shape he could find, was that of a parabola. However, reality didn’t want to be girdled, no matter how hard Gallileo tried. Joachim Jungius, by experiment, definitively proved that Gallileo was wrong, but he could not determine what the curve of the hanging chain was. So the question remained, what was the geometry of the hanging chain? Or, more generally, what does the curvature of the hanging chain show us about the geometry of the physical universe?

Figure 1

Since none of the curves of pure geometry fit this physical process, the physical process required the development of a new geometry. Begin then with the physical properties of the hanging chain. Hang a chain and it assumes a characteristic shape. (See Figure 2, Figure 3 , and Figure 4.)

Figure 2

Figure 3

Figure 4

Contrary to naive intuition, that shape is the same, regardless of the material of which the chain is made, or the position of the suspension points, or other factors.

The shape, thus, reflects a universal physical principle. But the chain is not just sitting there doing nothing. It is always in motion, so to speak. Each point along the chain is feeling a tension. The link on one side of the point is pulling it in one direction along the curve, and the link on the other side of the point is pulling it equally in the opposite direction along the curve. These countervailing tensions are the same for every point along the chain, regardless of how much chain is hanging between them. This is also contrary to naive intuition, which would assume that the points closer to the suspension points, for example, would have more tension on them, and thus have to be made of stronger material, since they have more of the weight of the chain to support. If the length of the chain is increased or decreased between any two points, the amount of weight supported by those points changes, but the equality of tension at each point doesn’t change. The hanging chain assumes a shape, such that as the length (weight) of chain changes, the principle of equal tension remains. Thus, each position of this non-uniform curve, is a function of a physical principle. It is to this physical property, that Bernoulli applied Leibniz’ calculus.

Taking the above described property of equality of tension as the “differential”, Bernoulli sought to determine what is the nature (integral) of the curve that would produce this characteristic at each point. He began with an experimental corollary. He demonstrated that the force (Kraft) the chain exerted between any two points on opposite sides of the catenary would be the same as if the entire weight of the chain between those points, was concentrated in a body, that hung from strings that were tangent to the catenary at those points (See Figure 5.)

 

Figure 5

The relationship of the forces at these points is dependent on the sine of the angles the tangents make with a vertical line drawn through the weight. The reader can conduct a simple experiment to discover this for himself. (See Figure 6 and Figure 7.)

Figure 6

Figure 7

The lowest point on the chain is a singularity, as it is the one place where the force doesn’t change, regardless of whether the length of the chain is increased or decreased on either side of it. Paradoxically, this point supports no chain, while supporting all the chain.(See Figure 8 and, Figure 9). Bernoulli shows that the shape of the hanging chain, which Huygens called the catenary curve, is that path that must be followed, so as to maintain an equal force on this lowest point. In order to satisfy this intention, the chain must manifest a unique geometrical configuration (See Figure 10.)

Figure 8

Figure 9

Figure 10

This contradicts any assumption that space conforms to a uniform geometry, that is infinitely extended in three dimensions, such as is suggested by the axioms, definitions and postulates of Euclidean geometry. Rather, the physical properties of the chain interacting with the Earth produces a unique type of curvature to which the chain must conform in order to be stable. It is not the geometry that determines the shape of the chain, but the physics that determine the geometry. Coincident with Bernoulli’s discovery, Leibniz discovered another principle underlying the geometry of the hangting chain. In the next installment, we will present Leibniz’s side of the story.

Riemann for Anti-Dummies Part 10a

JUSTICE FOR THE CATENARY (CONTINUED)

The last installment presented Bernoulli’s discovery of the unique geometry exhibited by a hanging chain. While Bernoulli discovered the characteristics of the catenary, it was Leibniz who asked, and answered, “Why does the chain assume this shape and not some other?”.

To summarize Bernoulli’s discovery: a chain hanging under its own weight, in order to form a stable “orbit”, assumes a unique shape, that does not correspond to any geometrical configuration that was known to mathematicians in Bernoulli’s time. Huygens called this shape the catenary curve. Bernoulli derived the geometrical properties of the catenary from the physical properties of the chain; specifically that in order for the chain to be stable, it must distribute the tension equally throughout its length. The catenary is: that which produces this physically determined characteristic of change, or, what Leibniz called, the integral. That physically determined characteristic is manifest at all positions along the chain. The nature by which that characteristic changes from position to position, Leibniz called the differential. Thus, the geometrical shape the chain assumes, is, that shape which expresses this unique physical property.

Such thinking enrages mathematicians of the Newton, Euler and Cauchy variety. “Mathematics first, physical reality second”, might as well be their motto, which is just another version of the same psychosis exhibited today by those who attribute some magical economic value to money, particularly, “my money”. But it is quite natural for thinkers in the tradition of Plato, Kepler and Cusa, who comprehend that mathematics is only a metaphor to express a certain level of knowledge about the intention that a physical process is carrying out.

Cusa expresses this way of thinking in his dialogue De Ludo Globi (The Bowling Game). The dialogue concerns a game played with a non-uniform ball that is rolled on a surface on which 9 concentric circles are drawn. The object of the game is roll the ball as close to the center as possible. But, since the ball is non-uniform, it follows a spiral, rather than a straight path. The player intends to roll the ball on a path that ultimately winds up in the center of the circle, but to do that, he must start the ball with a speed and trajectory, that after changing non-uniformly, ends up at the center.

Cusa draws an analogy from this game to the relationship between God, Man and Nature:

“Analogously, the rational soul intends to produce its own operation; with its steadfast intention persisting, the soul moves the hands and instruments when a sculptor chisels on a stone. Intention is seen to persist immutably in the soul and is seen to move the body and the instruments. In a similar way, nature (to which certain men give the name “world-soul”), moves all things while there persists its unchanging and permanent intention to execute the command of the Creator. And the Creator, with His eternal, unchanging, and immutable intention persisting, creates all things.

“Now, what is an intention except a conception, or a rational word, in which all the respective exemplars of things are present?…”

In the case of the hanging chain, the universe was presenting a paradox not unlike the one Kepler confronted in his determination of the geometry of the non-uniform motion of a planet, or like the case of Pierre de Fermat’s discovery that light travels according to the path of least-time. In these examples, the physical action measured did not conform to a geometry that could be deduced from the axioms, postulates and definitions of Euclidean geometry. In fact, in each case, the physical action contradicted the conclusion, deduced from those axioms, postulates and definitions, that space was a uniform continuum, infinitely extended in three orthogonal directions.

Like Kepler, Bernoulli rejected the “curve fitting” methods typified by Ptolemy, Copernicus, Brahe, Gallileo and Newton. These Aristoteleans assumed that space was a sort of infinite empty box, in which physical objects interacted with one another along straight lines or perfect circles. For them, man’s knowledge of such physical action was limited to mapping whatever observations were made onto perfect circles, straight lines, or some combination of same. On the other hand, Kepler, Bernoulli, and Leibniz made no such a priori assumption about the nature of space. Rather they sought to determine what is governing the physical process “in between”, so to speak, what is seen. The irony is that what is actually governing the physical process is not directly observable, but it must be discovered from paradoxes that are produced, when ,what is seen, contradicts our assumptions.

A further comparison with Kepler’s astronomical discoveries and Fermat’s work on light, is useful. Kepler showed that the unique path of a planet in the solar system is governed, not by a pair-wise interaction between the planet and the sun, but, by what Gauss and Riemann would later call a “hypergeometry”. The characteristics of that hypergeometry were expressed by Kepler’s principles of planetary motion, which have been discussed at length in earlier installments of this series. Similarly, Fermat showed that the path the light took was governed, not by a Euclidean notion that the shortest path is the shortest distance, but by a “hypergeometry” in which the shortest path is the path of least-time.

A further review of Fermat’s discovery will prove relevant. When reflected in a mirror, light assumes the geometry such that its angle of incidence and angle of reflection are equal. But, the question remains, “Why does the light assume this geometry, and not some other?” While Aristoteleans bristled at the mere posing of this question, Plato’s followers were compelled to ask and then answer it, leading to a discovery of a characteristic of the “hypergeometry” governing the phenomenon. When confronted with this observation, Plato’s followers demonstrated that the equal angles were a consequence of the hypergeometric requirement that light must follow the path of shortest distance.

However, under refraction, the light does not travel the shortest distance, nor are the angles of incidence and refraction equal. Kepler and others, particularly Willlibroad Snell, determined that the geometry of light under refraction, was such, that the sine of the angles of incidence and refraction were proportional. But again, this is the geometry of the observed phenomenon, not the characteristic of the hypergeometry governing it. In other words, the question, “Why are the sines of the angles of incidence and refraction proportional?” remained unasked, and unanswered. It was Fermat’s great discovery to show, that this geometrical relationship was itself a consequence of the universal principle, that light travels in the path of least-time. Upon reflection, the shortest distance exhibited by reflecting light, is simply a special case of the principle of least-time, expressed by refraction.

So, why does the chain assume the shape that it does? Or, in other words, “what are the characteristics of the hypergeometry governing the chain’s action?”

Leibniz’ discovery was based on his and Bernoulli’s re-working of the discoveries of Pythagoras, Theodorus, Theatetus, and Plato, as recounted in part in Plato’s dialogue, “The Theatetus”. These investigations concern the first level of paradoxes that arise, when considering the difference between linear action and rotational action.

To grasp these paradoxes, conduct the following exercise:

First, draw a line segment, then double it, then double it again, and so forth. Then, begin with the a similar segment, and triple, once, twice, three times, etc.

Now, do the same thing again, except instead of beginning with a line segment, begin with a square.

Notice that when doubling, or tripling the line, the result is always a line. However, when doubling or tripling a square, the result is an alternating series of squares and rectangles. In the Theatetus dialogue, Plato presents the paradox that the rectangles are incommensurable with the squares. (Re-draw the alternating series of squares and rectangles as all squares. Begin with a square; draw its diagonal. Using that diagonal as a side draw another square. Now draw the diagonal of that square, and so on. This should produce a spiral of squares.)

Thus, doing the same thing in two different geometries, produces two different results. To use the terminology of Gauss and Riemann, the dimensionality of the manifold determines the nature of the action in it.

The Greeks expressed these two different manifolds, in numbers, as arithmetic (linear) and geometric (rotational), and measured the relationship of action in each manifold, by the characteristic intervals, or “means” that each process defined. The “arithmetic mean” is the characteristic interval between two linear magnitudes, specifically the midpoint of a line. The “geometric mean”, is the characteristic interval of rotation, specifically, half a rotation.

Bernoulli, Huygens, and Leibniz investigated this paradox in a new light. Bernoulli discovered that both the arithmetic and geometric could be combined in one representation, by an equiangular spiral.

Leibniz represented both the arithmetic and geometric by what he called the “logarithmic” curve. (Leibniz’ logarithmic curve is constructed such that the horizontal change is arithmetic, while the vertical is geometric. See http://www.schillerinstitute.com/fid_97-01/011_catenary.html)

It was Leibniz’ surprising discovery that the catenary curve can be constructed from the logarithmic curve. Thus, the catenary is the arithmetic mean between two logarithmic curves, and inversely, the logarithmic curve is the geometric mean of the catenary!

Now compare Bernoulli’s discovery with Leibniz’s. Bernoulli discovered the geometry of the catenary as a consequence of the physical characteristics of the hanging chain. Leibniz showed, that, that geometry, is itself a consequence, of a characteristic of the hypergeometry governing the chain’s physical action. In other words, the chain is being “guided”, so to speak, by an unseen (logarithmic) curvature. It is a demonstration of Leibniz’ method of “analysis situs” that he discovered the nature of that unseen curvature, from the seen. The guide, from what is being guided.

(Think of the little experiment described above. [See Figure 6, and Figure 7] The curve one must follow in order to maintain the equal force is being “guided” by the curvature of the logarithmic curve. This is a physical demonstration that space is not Euclidean.)

Figure 6

Figure 7

Just as the planet’s action is an expression of the principles underlying the solar system, and light’s path an expression of the principle of least-time, so to the hanging chain’s path is an expression of the principle expressed by the logarithmic curve. But, is there a multiplicity of “hypergeoemtries”, or is there some unifying principle that unites these three seemingly disparate phenomena?

That is the discovery of Gauss and Riemann.

Riemann for Anti-Dummies: Part 9 : Bernouilli’s Brachistochrone

Bernouilli’s Brachistichrone:

An Exemplary Case of the “Science of the Moments of Becoming”

In response to Kepler’s call for the development of a mathematics appropriate to non-uniform motion, Leibniz invented a new form of geometry of position, that he called, the “infinitesimal calculus”. While a horror may well up in the minds of some at these words, such terrors can be calmed, were one to realize, that the source of this consternation, is due entirely to the Aristotelean assault on Leibniz, by Newton, Euler, Lagrange, Cauchy and their mindless adherents, who imposed on Leibniz’ beautiful invention, the scowling, constipated formalism of his enemies.

What Kepler’s discovery required, was a geometry that measured the position of the planet, with respect to the principle of change governing the planet’s motion. The achievements of the Greeks proved insufficient, as those investigations sought to determine positions with respect to other positions. What Leibniz supplied was a geometry of position that determined position with respect to a principle of change.

To attune our minds to Leibniz’ invention, turn to another investigation of the geometry of position, developed during the same time, under the leadership of J.S. Bach. As Bach’s compositions demonstrate, musical notes are not positions, that in turn determine intervals, which in turn determine scales and then keys, and then the whole well-tempered system. As any listener to a Bach composition can easily recognize, the position of any note, is an ambiguity, that becomes less ambiguous, as the composition unfolds, and the intervals so generated, and their inversions, are heard with respect to the well-tempered system of bel canto polyphony as a whole. It is the change, with respect to the whole well-tempered system, that determines the notes, not the notes that determine the change.

So too, with a planet in a Keplerian orbit. The position of the planet at any given moment, is a function of the harmonic ordering of the solar system. Two positions of a planet mark off an interval of an orbit, but that interval is not defined by the positions, rather, the positions are defined by the change that occurs in that interval of action. Since in a non-uniform orbit, the speed and trajectory of the planet is always changing, Kepler demanded a means to measure that change at each moment. Leibniz delivered, developing his new geometry of position, i.e., the infinitesimal calculus.

As mentioned in the last installment of this series, a good pedagogical example of Leibniz’ discovery, is its application in John Bernoulli’s discovery of the brachistichrone curve. (What follows is a summary of the concepts of Bernoulli’s construction. It will require some work on the part of the reader, and is intended to be read in conjunction with Bernoulli’s original essay, an English translation of which can be found in D.E. Smith’s, “Source Book of Mathematics”.

In 1697, Bernoulli put out a challenge in Leibniz’ Acta Eruditorum, to all mathematicians in the world. The problem was stated:

“Mechanical Geometrical Problem on the Curve of Quickest Descent.”

“To determine the curve joining two given points, at different distances from the horizontal and not on the same vertical line, along which a mobile particle acted upon by its own weight and starting its motion from the upper point, descends most rapidly to the lower point.”

The prize promised was not gold or silver, “for these appeal only to base and venal souls, for which we may hope for nothing laudable, nothing useful for science. Rather, since virtue itself is its own most desirable reward and fame is a powerful incentive, we offer the prize, fitting for the man of noble blood, compounded of honor, praise, and approbation; thus we shall crown, honor and extol, publicly and privately, in letter and by word of mouth the perspicacity of our great Apollo.”

As Bernoulli pointed out, the problem posed could not be solved, even by the method of maxima and minima of Fermat. For in those cases, Fermat sought the maximum and minimum from among a given set of quantities or loci, such as the point of maximum curvature of a conic section. Instead, Bernoulli’s problem was to find a minimum curve, among an infinity of possible paths. Every position on this sought after curve, was determined by a principle of change. So, what had to be discovered was, from a given a principle of change, i.e., least-time, how are the positions of the body determined. This is the equivalent of finding the correct orbit of a planet, not merely a possible one. Or to put it in metaphysical terms, “How can we know, how a falling body knows, to find the path of least descent?”

As you will see, Bernouilli was not posing an abstract mathematical puzzle, for the mere sake of befuddling others, the solution to this problem led to important discoveries in mechanics, as well as metaphysics.

Bernoulli’s attack on this problem began with what he called “Fermat’s metaphysical principle”, that light always seeks out the path of least time. It was a discovery of the ancient Greeks, that when light was reflected from a mirror, the path it took was the shortest length. However, when light was refracted by traveling through different media, such as water and air, the path of the light was not the shortest length. Fermat, discovering that the velocity of light was slower in denser media, demonstrated that the light changed its direction at the boundary between the two media, so as to follow the path of least time. This, of course, was consistent with the Greek discovery. In reflection, since the light travels through only one medium and therefore doesn’t change velocity, the shortest path, is also the path of least time. But, when there’s a change in medium, the light travels the shortest path in space-time, or the path of least-time.

Bernoulli’s approach was to follow the light, so to speak, to the path of least time. If the path of a ray of light traveling through a medium, whose density is continuously changing, according the same principle as that of a body falling under gravity, the the least time path of the light, will be the same as the least time path of the body.

But, how to discover the path, when we only know the principle of change, and have no positions to which to orient? At each moment along the light’s path, the light would be changing its speed and direction, such that its overall travel took the least time. Thus, similar to the motion of a planet, at each such moment, the light was ceasing to be what it was, and becoming what it will be. At each moment, the position of the light was a function of the principle of maintaining the least-time path.

Fermat had shown, that as light moved from a rarer to a denser medium, it slowed down, and its path became more vertical. For example, if light were traveling through air to water, the angle its path made with a vertical line, changed at the boundary between the air and water. If the angle its path made with the vertical in the air changed, the angle it made with the vertical under the water changed accordingly. However, the two angles did not change proportionally. Rather, they changed such that the sines of the angles were always in the same proportion.

So, at each “moment of becoming” along the light’s path, the light’s velocity and trajectory were changing, such the sine of the angle the light’s path made at that moment, was always proportional to the sines of the angles at all such “moment’s of becoming.”

To find the brachistichrone, Bernoulli thought of the medium in the following way:

“If we now consider a medium which is not uniformly dense but is as if separated by an infinite number of sheets lying horizontally one beneath another, whose interstices are filled with transparent material of rarity increasing or decreasing according to a certain law; then it is clear that a ray which may be considered as a tiny sphere travels not in a straight but instead in a certain curved path. This path is such that a particle traversing it with velocity continuously increasing or diminishing in proportion to the rarity, passes from point to point in the shortest time.”

Under this idea, at each horizontal sheet, the speed and direction of the light changes. The principle under which its speed and direction changes at each horizontal sheet, Leibniz called, the differential. The totality of all such differentials, (what Leibniz called the integral), is the sought after brachistichrone curve.

From one “moment of becoming” to the next, the position of the light changes, as it passes vertically from one density to the next. Each such vertical change in position is accompanied by a horizontal change in position, that corresponds to the sine of the angle of inclination at each “moment of becoming”. (Bernoulli’s geometrical construction of the above can be found on p. 652 of Smith.) Bernoulli adopted Leibniz’ notation for these ideas, calling the vertical change, dy, the horizontal change, dx, and the resulting change in the path of the light, dz. The proportion between the vertical and the horizontal, dy:dx, and the resulting change in the path, dz, is a function of the rate at which the density of the medium is changing.

Bernoulli shows, that if the density of the medium is changing according to the rate at which a body falls under its own weight, (specifically, that the velocity changes according to the square root of the vertical distance) then the resulting curve is a cycloid. “…you will be petrified with astonishment when I say that this cycloid, the tautochrone of Huygens is our required brachistochrone…” he declared.

But, Bernoulli noted that this was not a discovery of a particular physical phenomenon, but a metaphysical discovery of a universal principle:

“Before I conclude, I cannot refrain from again expressing the amazement which I experienced over the unexpected identity of Huygen’s tautochrone and our brachistochrone. Furthermore, I think it is noteworthy that this identity is found only under the hypothesis of Galileo so that even from this we may conjecture that nature wanted it to be thus. For, as nature is accustomed to proceed always in the simplest fashion, so here she accomplishes two different services through one and the same curve, while under every other hypothesis two curves would be necessary the one for oscillations of equal duration the other for quickest descent. If, for example, the velocity of a falling body varied not as the square root but as the cube root of the height falalen through, then the brachistochrone would be algebraic, then tautochrone on the other hand transcendental; but if the velocity varied as the height fallen through then the curves would be algebraic, the one a circle, the other a straight line.”

Riemann for Anti-Dummies: Part 8 : The Significance of Precise Ambiguity in Science

The Significance of Precise ambiguity in Science

It is evident from all great examples of classical art, that the only way to precisely communicate an idea, is through ambiguities, in between which, one mind can say to another, “I know exactly what you mean.” It is also evidence, that cognition is an embedded principle of the universe, that, when man communicates with it, the universe speaks in similarly precise ambiguities.

When Cusa and Kepler led mankind out of the long dark age of Aristotelean slavery, the world of non-uniform motion they discovered, did not obey the precise mathematical calculations the “ivory tower” academics had deluded themselves into believing. Nevertheless, this new world was susceptible to precise measurement. All that was required was the invention of a new type of mathematics–a metaphor that reflected, rather than excluded, the principle of cognition.

This new mathematics arose from a three-way dialogue across the centuries. The participants included: Renaissance thinkers, such as Cusa, Leonardo, Kepler, Fermat, Leibniz, Kaestner, Gauss, Riemann et al.; ancient Greeks, such as, Pythagoras, Theatetus, Theodorus, Plato, Eratosthenes, Archimedes, et al.; and the universe itself. This “Great Deliberation” can still be heard today, if one works to discover its language.

It is historically accurate, and pedagogically useful, to think of the Kepler challenge as the “motivfuehrung” of this dialogue. The paradox Kepler identified is a fulcrum, whose solution requires the assimilation of the entire Socratic tradition of Greek science, while, at the same time, demanding an as yet undiscovered principle.

Subsequent discoveries of non-uniform physical action, such as Fermat’s least-time path of light; Huygens’ cycloid; Leibniz’ catenary; or Bernoulli’s brachistichrone, showed that Kepler’s challenge was exemplary of a whole class of physical problems that were characterized by non-uniform action. These discoveries posed, indisputably, the inseparability of mind and matter, as expressed by Kepler’s concept of the mind of the planet. Non-uniform physical action implies the question, “How does the planet, light, falling body, or hanging chain, or rolling wheel, know how to move, and, even more importantly, how can the human mind know what the physical universe knows?”

A paradox arises because in non-uniform action the position of, for example, the planet, is a function of a principle of change. Such positions cannot be thought of as points in space, but rather, as the effects of “moments of becoming” transformations from what the planet was, to what it will be. These transformations are themselves a function, not of the planet alone, but of the multiply-connected harmonic orderings that underlie the solar system as a whole, expressed, for example, by Kepler’s three principles, the function of the Platonic solids, and the function of the musical intervals formed by the planet’s motion. Thus, the planet’s position from moment to moment is ambiguous with respect to its previous position, or to any other position inside or outside the orbit. But, the position is less ambiguous with respect to the underlying physical characteristics that govern the “moments of becoming”. The ambiguity decreases with the discovery of new physical principles. Since the human mind must discover these principles from the inside, so to speak, these physical characteristics, not positions, are what must be measured at each “moment of becoming.”

To answer this question, Cusa, Kepler, Fermat, Leibniz, et al., turned to the Socratic tradition in Greek science, in which this hylozoic principle was a central feature. As indicated in earlier installments in this series, Kepler, and Fermat attempted to apply the Greek investigations of loci, or what today is called geometry of position, to this new class of physical problems. But, the paradoxes posed by these newly discovered physical principles required a further advance over the Greeks. The Greek investigations concerned loci that were determined by a set of conditions, with respect to a set of positions. For example, the locus of positions that are equally distant from a single position, is a circle. The locus of positions, the sum of whose distance from two points remains constant, is an ellipse. The locus of positions of the centers of circles that go through two points is a line. As these examples illustrate, the resulting locus is a function of the conditions applied to some set of positions. (See Riemann for Anti-Dummies Part 6.)

But, non-uniform motion required that the locus be thought of as determined by a set of conditions relative to a characteristic of action. For example, the positions of a planet in its orbit are not a function of the relationship of the planet to some other point or body. Rather, the position of the planet in its orbit, is a function of the harmonic ordering of the solar system as a whole. It is these principles which determine the planet’s position at each moment, not the planet’s relationship to a fixed point, such as the geometrical center of the orbit, or the equant, or even a fixed body, such as the Sun.

Kepler stated this point explicitly in his “Epilogue Concerning the Sun, By Way of Conjecture,” from the “Harmonies of the World” (see Riemann for Anti-Dummies Part 3.) Kepler adopted the conception of Pythagoras and Plato:

“…The relation of the six spheres to their common center, thereby the center of the whole world, is also the same as that of unfolded Mind (dianoia) to Mind (noos)….”

But, he warned against thinking that this ordering principle of Mind, is located in a fixed place. That it has, “a royal throne in the solar body”, as the Pythagoreans believed. Echoing Cusa, Kepler said, “But, we Christians, who have been taught to make better distinctions, we know, that the eternal and uncreated `Logos’, which was `with God’, is contained by no abode, although He is within all things, excluded by none….”

This distinction is crucial for solving the Kepler challenge. While Kepler demonstrated the Sun as the mover of the planets, the Sun itself was subordinate to a higher principle, reflected in the harmonic ordering principles, which were present at every moment of the planet’s action, and discoverable by the human mind. Thus, the need for a new mathematical metaphor that can describe position as a function of a principle of change.

That, is ultimately what Leibniz set out to create with his infinitesimal calculus, building on Fermat’s investigations. (Riemann for Anti-Dummies Parts 6 & 7.) Some examples will help construct the concept. What follows requires some work, and the introduction of concepts that might seem difficult. This is unavoidable, however, to lay the groundwork for Gauss and Riemann’s later work. The serious reader is encouraged to work through the examples, or otherwise be condemned to the dumbed down domain of Riemann for Dummies. (Due to the inability to present diagrams in this form of communication, I will refer the reader to illustrations in readily available resources.)

Look at the wax paper constructions of the conic sections on page 11 of the “Gauss” Fidelio. (If you haven’t made these constructions in a while, it would be worth your while to make a set.) In these constructions, each conic section can be thought of in two ways. One, is as a locus with respect to some fixed positions, such as a curve generated by the intersection of a plane with a cone. By inversion, each conic section can be thought of as the envelope of a set of tangents. In the first, it is the curve that determines the position of the tangents, while in the inversion, the position of the tangents determines the curve. If you begin with the curve, the positions of the tangents are precisely determined. But, if you begin without the curve, how are the tangents determined, so that their positions envelop the resulting curve?

To solve this problem, Leibniz sought not to determine the positions of the tangents, but to determine the function that governed the way the positions of the tangents changed. To do this Leibniz thought of each tangent as a “moment of becoming”, and the change between the tangents as a “differential”. The sought after function he called a “differential equation”. (Don’t get scared off by the word “equation”. In Leibniz’ sense, like Fermat’s and Kepler’s, equation is not a formula, but a set of relationships.)

To illustrate this, take the example of the parabola, beginning with Fermat’s geometry of loci. (See Riemann for Anti-Dummies Part 7.) In that example, Fermat showed that while the length and angle of each tangent to the parabola changed non-uniformly, when projected onto an axis cutting the parabola in half, a 2:1 ratio was always maintained. (This example is detailed in Part 7.) Leibniz recognized that this 2:1 ratio was the characteristic of change, differential, between each “moment of becoming”. Thus, Leibniz thought of the tangents to the parabola, as those lines, produced by a sequence of differentials, that changed according to this 2:1 ratio. The parabola, in turn, was the “integral” of this differential equation.

To put this in less technical sounding terms, the “sense certainty” approach generates the parabola as a set of positions, which in turn determines the positions of a set of tangents, which in turn determines a characteristic change from one tangent to the next. From the standpoint of Leibniz’ calculus, the characteristic change (differential equation) determines the positions of the tangents, which in turn determine the parabola.

Leibniz, of course, did not limit the application of his calculus to conic sections, or even to known curves. Instead, he and Jean Bernoulli, showed that even hitherto unknown curves could be discovered from their corresponding differential equations. The example of Bernoulli’s discovery of the brachistichrone curve illustrates this point. (The reader is referred to Bernoulli’s essay on the subject. An English translation is available in Smith’s Source Book of Mathematics p. 644.) In future installments, we will work through this example in more detail, but, for pedagogical reasons, we summarize the relevant questions here.

Bernoulli had thrown out a challenge to determine the curve on which a body falling under the influence of gravity, travelled in the least time. No one but Leibniz presented a solution. As will become evident later, Bernoulli’s challenge concerned not simply this particular problem, but was directed towards illustrating his and Leibniz’ method for discovering universal principles, and exposing the bankruptcy of Newton and Gallileo.

In summary, Bernoulli’s ingenious solution is as follows: Fermat had demonstrated that light, refracted by a change in the density of the medium, travelled the least time path. Were light travelling through a medium, whose density was continuously changing, its path would be a curve of least time. If the medium were changing according to the same proportion, by which a body fell according to gravity, then the resulting path of the light, would correspond to the least-time curve of the falling body.

To restate this problem in the language we’ve developed in this series. The locus of positions of a body falling in least time is unknown to us. However, we know that light travelling through a medium of continuously changing density will find the least-time path. We want to know what the light knows. The position of the light at each “moment of becoming” is ambiguous, but the change from moment to moment (the differential) is known precisely, and defines a differential equation. The sought for least-time curve is the integral of that differential equation.

The Significance of Precise ambiguity in Science

Riemann for Anti-Dummies: Part 7 : Towards a Hylozoic Calculus

Towards a Hylozoic Calculus

It always comes as a shock to the mathematically schooled, that Leibniz’ infinitesimal calculus is not, essentially, a mathematical procedure. It is just as shocking to the unschooled, but mathematically intimidated, that it were impossible to grasp the deeper implications of Leibniz’ discovery, without digging into its mathematical expression. These two, seemingly opposite, subjective reactions, arise from a common misunderstanding about the nature of man. They are the twin pathologies of the same disease–the Aristotelean/Gnostic separation of mind from matter.

Leibniz, of course, did not suffer from the disease that afflicted his enemies. In a 1678 letter to Countess Elizabeth, he recounts how he was drawn to the study of mathematics in pursuit of greater knowledge, but not as an end in itself:

“I cherished mathematics only because I found in it the traces of the art of invention in general; and it seems to me that I discovered, in the end, that Descartes himself had not yet penetrated the mystery of this great science….

“…I can state that it is for the love of metaphysics that I have passed through all these stages. For I have recognized that metaphysics is scarcely different from the true logic, that is, from the art of invention in general; for, in fact, metaphysics is natural theology, and the same God who is the source of all goods is also the principle of all knowledge. That is because the idea of God contains within it absolute being, that is, what is simple in our thoughts, from which everything that we think draws its origin.”

And in a 1716 letter to Samuel Mason, Leibniz added, “The ancients considered mathematics as the passage from physics to metaphysics or to natural theology, and they were right.”

Leibniz’ calculus is exemplary of such a passage. It was provoked by the paradoxes brought to light by Kepler’s discovery of the elliptical orbit of Mars. Kepler had demonstrated that action in the physical universe was characterized by non- uniform motion. But, as discussed in previous installments in this series, the universe presented Kepler with a new challenge. Kepler could measure the planet’s non-uniform motion, from the standpoint of the universal principles that governed the planet’s orbit, but the inverse, to measure the universal principles that governed the orbit, from the planet’s motion, required a new discovery.

Kepler had asked the universe a question and discovered the universe’s answer. The universe, in turn, posed a question back, that revealed a hitherto unthought of paradox in Kepler’s cognitive process. This irony, that the discovery of a physical principle, in turn provokes a new discovery of mind, comes as no surprise to those healthy minds not afflicted with Aristotle’s disease, and it demonstrates, as Plato and Cusa taught us, that mind and matter are inseparable.

This irony should also come as no surprise to someone who has come to know what LaRouche has shown, i.e. that the higher principle of cognition, governs the principles of living processes, which governs the principles of non-living processes. Thus, the principle of cognition is expressed in living and non-living processes, albeit in paradoxical form. As in the method of Plato and Cusa, it is through such paradoxes (such as the Kepler challenge) that we are led to the discovery of these higher principles.

Our grasp of this method of discovery by inversion, is aided by reference to the principles of musical polyphony. Think of how the underlying characteristics of the bel-canto, well- tempered system of polyphonic musical composition, are discovered with respect to the principle of inversion. For example, the true characteristic singularity of the Lydian interval only clearly emerges in the context of the principle of inversion within a musical composition.

To get a handle on the problem, think of the planet’s motion from the standpoint of the mind of the planet. The planet’s path around the Sun is governed by the characteristics of the elliptical orbit, which, in turn, are governed by the harmonic ordering of the solar system. These characteristics are expressed at each moment of the planet’s motion, as a non-uniformly changing speed and trajectory of the planet. At each such moment, the planet’s action is ceasing to be what it was, and becoming what it will be. At each such “moment of becoming”, the planet has a definite speed and trajectory, which is changing such that the resulting interval maintains a constant proportion to the whole orbit, i.e. equal areas. The change to the planet’s speed and trajectory at each moment, is also changing from moment to moment. How does the planet know, at each “moment of becoming”, how its action must change?

The subsequent discoveries of non-uniform physical action by Fermat, Leibniz, Huygens and Bernoulli, posed a similar challenge. How does the light know how to change its trajectory in order to find the path of least-time? How does the hanging chain know how to find the curve of least-tension, i.e. the catenary curve? More importantly for Fermat and Leibniz, how can the human mind know, what the planet, light, or chain, knows.

These questions perplexed and infuriated the Aristotelean/Gnostics like Descartes. Either they rejected the idea of universal principles, or they sought some mechanistic explanation, that separated the principle of mind from matter. For Aristotle, Gallileo, Newton and Descartes, action in the universe occurs along straight-lines or perfect circles. Such uniform action has no need of cognition, as bodies moving according to straight-lines or circles, don’t change except by an “outside” force. For Aristotle, that “outside” force–cognition is not only outside the physical universe, but outside the human mind as well.

Fermat and Leibniz had no such problem. They understood that principles such as least-time and Kepler’s orbits, reflected a universal characteristic, which was present at each “moment of becoming”. The solution to the Kepler problem was, thus, to determine the universal characteristics in the “moments of becoming”.

As Leibniz expressed this with respect to light:

“Indeed, neither can the ray coming from C make a decision [1] about how to arrive, by the easiest way possible, at points E, D, or G, nor is this ray self-moving towards them [2]; on the contrary, the Architect of all things created light in such a way that this most beautiful result is born from its very nature. That is the reason why those who, like Descartes, reject the existence of Final Causes in Physics, commit a very big mistake, to say the least; because aside from revealing the wonders of divine wisdom, such final causes make us discover a very beautiful principle, along with the properties of such things whose intimate nature is not yet that clearly perceived by us, that we can have the power to explain them, and make use of their efficient causes, along with their artifacts, such as the Creator employed them in order to produce their results, and to determine their ends. It must be further understood from this that the meditations of the ancients on such matters are not to be taken lightly, as certain people think nowadays.”

This is the purpose for which Leibniz’ calculus was developed as a form of geometry of position. As we saw in the last installment of this series, Fermat began this work by generalizing the Greek concept of geometry of position (topos or locus). Now, he laid the foundations for the calculus, by inverting this concept. In effect, Fermat, and later Leibniz, were defining the completed paths from the standpoint of the “moments of becoming.” Fermat called this his “Method for Determining Maximum and Minimum and on Tangents to Curved Lines”. Leibniz extended this concept, by seeing in these “moments of becoming”, what he called, differentials.

To illustrate this method, we will begin, as Fermat and Leibniz did, with a simpler example than the Kepler challenge. (The Kepler problem proved to present paradoxes whose solution required the subsequent discoveries of Gauss and Riemann.)

Make a parabola using the wax paper method illustrated in Figure 1.9(c) on page 11 of the “Gauss” Fidelio. In this construction the parabola is formed as an envelope of the tangents, which is an inversion of forming the tangents from the parabola. Through the focus of the parabola draw an axis about which the parabola is symmetric. (Call that line A.) Now, pick one of the tangents and draw a line from the point of tangency that is perpendicular to line A. (Call the point of intersection with A, x.) Now, extend the tangent line until it intersects line A. (Call that point y.) Repeat this with several tangent lines. You should notice that as the point of tangency gets farther from the vertex, point y also gets farther from the vertex and vice versa. And, at the vertex, points x and y coincide.

Fermat showed, in the case of the parabola, that, the distance from the point x to the vertex of the parabola was always ? the distance from x to y. In this way, he could describe action along a parabola, by inversion, as that action which maintained a 1:2 ratio between points x and y. This proportion, like the equal area principle of a Keplerian orbit, reflects the change at each “moment of becoming”.

(The case of the parabola is merely an illustration for pedagogical purposes. Fermat’s method was a general one, that enabled him to solve a myriad of transcendental problems, that, by the way, made Descartes furious.)

In future installments we will work through some other examples to help solidify this method. But, to keep your mind active, look way ahead and consider where this goes.

Any problem posed from the standpoint of geometry of position, implies an assumption about the domain in which the action occurs. In the above mathematical example of the parabola, it is tacitly assumed that the parabola lies in a Euclidean plane. Yet, in a physical problem, a grave mistake would be made, if one assumed that the action, such as the refracting ray of light, or the motion of a planet, were taking place against the backdrop of a Euclidean space. But, it will still be a mistake, to merely replace the Euclidean space with some other non-Euclidean manifold. And, we would still be in error, if we adopted what we thought was an anti-Euclidean concept, but limited our investigation to non-living physical processes. In other words, in considering the geometry of position of a planet’s motion about the Sun, we cannot limit the investigation to the planet’s position with respect to non-living matter in the solar system. As stressed above, we must investigate the planet’s changing position as a function of non-living, living and cognitive processes.

Now look again at the revolutionary experimental discoveries reported in JBT’s last two pedagogical discussions giving fresh evidence of the universal principles ordering living and non- living processes. Viewed from the standpoint of LaRouche, that the principle of cognition orders living and non-living processes, the following question is provoked to a future Leibniz: How are the universal principles, of cognitive, living, and non-living processes, expressed in each “moment of becoming”? The answer requires the development of a new type of geometry of position, which might be called a, “hylozoic calculus.”

Riemann for Anti-Dummies: Part 6

Happiness as a physical principle

In the Dedication to the “Fourth Book of the Heroic Deeds and Sayings of the Noble Pantegruel”, Francois Rabelais refers to a discovery of Greek father of medicine, Hippocrates.

“The question over which we sweat, dispute, and rack our brains, is not whether the physician’s visage depresses the patient, if he is frowning, sour, morose, severe, ill-humored, discontented, cross, and glum; nor whether he cheers the patient if his expression is joyful, serene, gracious, frank and pleasant. There is no doubt on that score. The real question is whether the patient’s depression or cheerfulness arises from his apprehensions on reading these signs in his physician’s face, and from his consequent deductions of the probable course and issue of his disease; or whether it is caused by the transmission of the serene or gloomy, aerial or terrestrial, joyous or melancholy, spirits from the doctor to the person of the sick man as is the opinion of Plato and Averroes.”

So too is the relationship between investigator and the physical universe. Approach the universe with the joyful, serene, gracious, frank and pleasant expression of a Kepler, Fermat, Leibniz, Kaestner, Gauss, Riemann, or LaRouche, and its secrets are displayed in its reflected smile. Put on the frowning, sour, morose, severe, ill-humored, discontented, cross, and glum, visage of Gallileo, Newton, Descartes, Euler, and Russell, and all you will see is a grid of scowling straight lines.

In point of fact, the history of science, from Kepler on, can be divided into two camps, the smilers and the scowlers. Kepler had demonstrated that Cusa’s principle, of non-uniform physical action, was, in truth, the characteristic of action in the solar system. A flood of new discoveries followed, demonstrating the validity of this principle with respect to other physical phenomena: Fermat’s discovery of the principle of the least-time path for light; Huygen’s discovery of the isochrone, Bernoulli’s discovery of the brachistochrone; Leibniz’ discovery of the catenary; to name but a few.

In each case, as the “Kepler Challenge” showed, the physical principle under investigation, was not susceptible to precise calculation, but, had to be grasped from the standpoint of the higher principles underlying the generation of the phenomena. The example of the “Kepler Challenge” illustrates this problem most clearly. If the entire orbit is known, the time elapsed between any two planetary positions can be determined to an arbitrary degree of precision, by Kepler’s equal area principle. But, if the time elapsed was known, Kepler had no direct means to determine the planetary positions. His “imperfect” solution was to divide the planetary orbit into 360 intervals, calculate the time elapsed for each interval, compile a table of these results, and refer all calculations to the table. The imperfection arose because these small intervals are themselves as non-uniform as larger ones. Only if the planetary motion can be known at each “moment of becoming”, could such a method provide any degree of precision. Yet, there are an infinite number of such moments in any orbit, implying the need for infinite knowledge, and so Kepler made his famous call for future geometers to solve for him this paradox.

This paradox is not merely formal. The individual positions of the planet, are themselves a function of the whole orbit, which, itself, is a function of the whole solar system. Yet, like the shadows in Plato’s cave, the orbit and the solar system do not present themselves to our minds directly, but only through the changing observable positions of the planets. Kepler discovered the nature of the function by which the solar system determined the orbit, and the orbit determined the planetary positions, but he nevertheless required the inversion, to measure the effect of both these functions, at each moment of becoming, i.e., to discern the how the shadows were made, from only the shadows.

For Kepler, this state of learned ignorance was a happy one, albeit not without a certain amount of angst: “And we, good reader can fairly indulge in so splendid a triumph for a little while(for the following five chapters, that is), repressing the rumors of renewed rebellion, lest its splendor die before we shall go through it in the proper time and order. You are merry indeed now, but I was straining and gnashing my teeth,” Kepler tells us in the “New Astronomy”

But, it provoked nothing but rage from Aristoteleans, who desperately sought to limit the world to that which is susceptible to precise calculation. “Wipe that smile off your face,” the Newtonians would say, “We don’t need these complicated, imprecise, convoluted hypotheses, of Kepler. We can get a more precise measure of planetary motion, using the inverse square law.” (“Don’t pay attention to the fact that we have to eliminate everything in the universe except two bodies, for our calculations to work,” they scowl. “We get results.”)

In response to Kepler’s challenge, Pierre de Fermat, set about to overcome those limitations brought to the `fore when one tries to measure action along a path of non-constant curvature, such as an elliptical orbit, or the least-time path of light traveling through different media. He took as his starting point, the earlier Greek investigations into these types of problems. Appolonius, Archytus, Aristaeus, Eratosthenese, among other Platonic thinkers, called these types of investigations theorems of position. (The Greek word used was “topos”, meaning “place”. Today these problems are identified by the Latin word, “locus”.)

Under the Greek concept of locus, geometrical curves are not defined as self-evident entities, but as the concept characterizing those positions resulting from a certain type of action. Proclus re-states this ancient Greek conception as, “those (results) in which the same property obtains over some entire locus, and as `locus’ (topos) the placement (thesis) of a line or a surface which makes one and the same property.”

A couple of examples would be:

1) the locus of all positions equidistant from a given position is a circle.

2) the locus of all positions whose distance from two given positions have a constant sum is an ellipse.

3) the locus of all positions that form an equal angle between two fixed points is a circle.

4) the locus of positions of intersection between a plane and a cone are a circle, ellipse, parabola, or hyperbola, depending on the angle at which the plane intersects the cone.

Or:

5) the locus of all positions of a planet, moving about the Sun non-uniformly, such that it sweeps out equal areas in equal times, is an elliptical orbit.

(It should not be overlooked, that examples 1-4 refer to purely geometrical loci, while example 5 refers to a locus in physical space-time.)

Significantly, the Greeks separated loci into several types according to their method of generation, such as planar loci, solid loci, or loci according to means. The limitation of the Greek investigations, according to Fermat, was that such concepts were investigated one by one, whereas Fermat sought a generalized approach.

This is where Fermat began:

“None can doubt that the ancients wrote on loci. We know this from Pappus, who, at the beginning of Book VII, affirms that Apollonius had written on plane loci and Aristaeus on solid loci. But, if we do not deceive ourselves, the treatment of loci was not an easy matter for them. We can conclude this from the fact that, despite the great number of loci, they hardly formulated a single generalization, as will be seen later on. We therefore submit this theory to an apt and particular analysis which opens the general field for the study of loci.”

In essence, Fermat’s proposed generalization was to consider all loci in the following way. Consider two straight line segments intersecting at right angles. While holding one of the segments fixed, allow the other to move along it, both vertically and horizontally, while keeping the intersecting angle right. The locus of positions of the free end of the moving line, will then trace a curve. This curve will be a function of the motion of the moving line.

To visualize this, take two sticks (bamboo skewers work well) and hold them at right angles to each other. Experiment by keeping the vertical stick fixed and slide the horizontal stick over it. The free end of the horizontal stick will trace a line parallel to the vertical stick. Try this again, this time moving the horizontal stick vertically, while at the same time moving it horizontally. If the rate of vertical motion equals the rate of horizontal motion, the free end of the moving stick will trace a diagonal line. Now try the same thing, but this time make the horizontal motion, the square of the vertical. The free end will trace a parabola. If the vertical motion is arithmetic, while the horizontal motion is geometric, the curve will be exponential. (Additional exercises and examples will be forthcoming in the next installment.)

Fermat expressed the relationship of the vertical to horizontal motion by an equation. Rather than deal with each individual locus, he could now consider types of loci, which were described a specific types of equations. Bernoulli would later introduce another, simpler, type of equation, in which the point of intersection remained fixed, while the moving stick rotated about that point, while still able to slide. He could describe these loci by a relationship between an angle and a distance. Gauss, would later generalize this concept further, by replacing the straight lines with any curve whatsoever, and he and Riemann would take this one step further, determining multi-dimensional loci. (These matters will be developed in future installments.)

“Grrrrrr”, snarled the Cartesians, in a fit of inquisitional constipation. (Their Aristotlean minds always got headaches from such thoughts.) “All this relative motion is confusing. Let’s make this simple stupid.” Instead of loci, Descartes and his progeny would create the, unfortunately all too familiar, grid of infinitely many intersecting right lines. Descartes’ equations described, not a locus of motion, but the fixed positions in this grid, where a given geometrical figure intersected the straight-lines.

Nothing exemplifies a frowning visage, and has been responsible for implanting a glum expression on the faces of so many students, than the abominable cage, known today as the “Cartesian co-ordinate system”. It is high time, that this fraud be debunked, once and for all.

Descartes equations were nothing more than a set of instructions about how to connect the dots in the fixed grids of absolute space. For Fermat, like for Kepler, there was no fixed grid in absolute space. These curves were generated by a type of motion, which types would then be susceptible to further investigation.

Fermat’s theory of loci was not sufficient to overcome the Kepler paradox, but it was a necessary intervening step for Leibniz’ happy development of the next phase in the science of the moment of becoming.

Riemann for Anti-Dummies: Part 5

No person is qualified for political leadership today, or for that matter, even competent to vote, unless they know the significance of what has been called, “The Kepler Problem”. So important is this paradox, and so widespread is its fame, that every significant discovery in physical science since, can trace its origins to it, and every major scientific thinker upon whose shoulders we now stand, can find their first sparks of creative insight in its contemplation. And yet, today, except as a name for an obscure mathematical formula, most know nothing of it, which is a fitting measure of this stage in civilization’s descent.

The problem itself embodies the essential characteristics of any crucial discovery, a paradox of physics and mind. It arises from Kepler’s discovery that non-uniform motion was not merely an appearance, but a physical characteristic of action in the universe, as demonstrated by the eccentric orbits of the planets around the Sun. This discovery immediately presents an ontological problem, that was stated earlier by Cusa, in Book 2 of “On Learned Ignorance”:

“Wherefore it follows that, except for God, all positable things differ. Therefore, one motion cannot be equal to another; nor can one motion be the measure of another, since, necessarily, the measure and the thing measured differ. Although these points will be of use to you regarding an infinite number of things, nevertheless, if you transfer them to astronomy, you will recognize that the art of calculating lacks precision, since it presupposes that the motion of all the other planets can be measured by reference to the motion of the sun. Even the ordering of the heavens with respect to whatever kind of place or with respect to the risings and settings of the constellations or to the elevation of a pole and to things having to do with these is not precisely knowable. And since no two places agree precisely in time and setting, it is evident that judgments about the stars are, in their specificity, far from precise.”

This paradox manifests itself in an eccentric planetary orbit, because no interval of that orbit can measure any other interval of the same orbit. For in every interval of the orbit, no matter how small, the distance from the Sun to the planet, the speed of the planet, the distance travelled along the orbit, and the curvature of the arc, are always changing. (See Figure 5.3 p.27, Summer 1998 Fidelio, for an illustration. The interval between all segments, denoted as “Q” in the figure, exemplifies this characteristic of non-constant curvature.)

This paradox was enough to entice most people to stick with the Aristotelean custom and tradition that banished non-uniform motion from the universe. Why would God, the Aristotelean would argue, create action in the universe that evaded precise calculation?

But, Kepler, on the other hand, understood that this is precisely what God preferred, as the principle of change must be manifest in every part of the universe. It was only when man realized the motion of the planets were non-uniform, that man would inquire into its causes. Kepler overcame the apparent obstacle, not by measuring the motion directly, but by measuring the effect of the motion, as expressed in his principle of equal areas.

However, he could only overcome this obstacle in one direction, so to speak. He could calculate the effect of the planet’s motion in an orbit, if he knew the orbit as a whole, that is, if he knew the position of the Sun relative to the circumference, the eccentricity, the greatest and shortest distances between the Sun and the planet, etc. This was done, as the {Fidelio} article summarizes, by measuring the area swept out during any interval of the planet’s orbit. Intervals that swept out equal areas, corresponded to equal portions of the planet’s total periodic time. [See figure.] But, if he tried the inverse, that is, if he knew one position of the planet, and he needed to calculate the planet’s next position, after a specified area was swept out, he was blocked by what Cusa showed to be the transcendental incommensurability between the arc and the straight line. (See Fidelio p. 28.) In other words, he could determine the time elapsed from the orbit, but he couldn’t determine the orbit from the time elapsed. This is what’s called the “Kepler Problem”.

But, is this merely a problem of calculation, or is it an indication of something more fundamental?

Look again at the paradoxical nature of a non-uniform orbit. On the one hand, the relationship of the Sun to the planet individually, and the relationship of the Sun to all the planets, define a characteristic of the planet’s complete orbit. The planet’s distance from the Sun, it’s speed, and direction, are thus defined for each “moment” of action. It is this global characteristic that Kepler’s equal area principle measures.

The “Kepler Problem” arises, when we try to measure these global characteristics from the standpoint of each of “moment” of action.

“Why even bother?” the Dummy might ask.

“Because, it is the only way human cognition can know anything,” is the short answer. The completed orbit is never seen, nor is it ever known, as a completed orbit. Rather, it is known by its influence on the planet at every “moment” of action. In each of these “moments”, or rather, “moments of becoming”, the planet is ceasing to be what it was, and becoming what it will be next. What it was and what it will be, is the completed orbit. So, at each such “moment of becoming”, the planet is simultaneously in one moment and all moments of the completed orbit.

An analogous paradox can be grasped subjectively, if one thinks back, from present to past, on one’s life, in the context of history. In this direction, it is possible to determine the influence of one’s life’s orbit on each past moment. But, in the other direction, how can one determine the historic curvature of life’s orbit, in this present “moment of becoming”?

The “Kepler Problem” can thus be re-stated as, “How can we measure the influence of the completed orbit at each “moment of becoming”?

Kepler relied on what he admitted was an “imperfect” compromise. He divided the entire area of the orbit into 360 segments, and calculated the areas of each segment, using the equal area principle. The imperfection of this method is obvious. Even these small segments are comprised of an infinite number of “moments of becoming”, so by their very nature they are imprecise.

He knew all too well the imperfection, saying in his “New Astronomy”, “… [I]nsofar as it lacks geometrical beauty, I exhort the geometers to solve me this problem…. It is enough for me to believe that I could not solve this, a priori, owing to the heterogeneity of the arc and sine. Anyone who shows me my error and points the way will be for me the great Apolonius.”

Perhaps “Kepler’s Problem” is better named, “Kepler’s Challenge”. He freed man from the unchanging custom and tradition of perfect circles and straight lines, and left him in a world with principles, but without precise calculation. But, this, “gives rise to a powerful sense of wonder, which at length drives men to look into causes.” Such wonder found form in the minds of Fermat, Leibniz, Kaestner, Gauss, Riemann, and Cantor, where calculation was supplanted by the “science of the moments of becoming.”

Riemann for Anti-Dummies: Part 4

when Kepler demonstrated the non-linear characteristic of the solar system, and consequentially, the entire physical universe, he set in motion a revolution in thinking, that to this day, is either hated or misunderstood, by scientists and laymen alike. Witness the discussion with a Baby Boomer mathematician who works for NASA, that took place at a recent chapter meeting. After a presentation on the congruence between LaRouche’s successful forecasting of the current systemic financial breakdown, and Gauss’ determination of the orbit of Ceres, the well-educated specialist asked, “Did Gauss know of the elliptical orbits? If so, then he must have had the inverse-square law.” When the historical illiteracy of his assertion was pointed out, the specialist replied, “Oh. I always thought the elliptical orbits were a consequence of the inverse square law. I never knew otherwise.”

Or, take the case of the slave of today’s popular opinion who confidently gauges his or her economic well-being from the standpoint of their own personal financial situation. Like the brother of the protagonist of Poe’s “Descent into the Maelstrom”, such fools are doomed to sink ever deeper into the abyss, hanging desperately onto the ship, when safety is easily won by leaving the temporary security of the ship’s greater bulk, for the seeming insecurity of a light barrel, which the whirlpool eagerly rejects.

Those who wish not to be counted among these legendary fools, find themselves compelled to actually comprehend Kepler’s great discovery.

It is commonly misunderstood, that Kepler’s discovery was the result of a numerical discrepancy between the observed positions of the planets, and the positions predicted by Ptolemy, Brahe, and Copernicus. While such discrepancies were certainly a marker that something was wrong in the state of astronomy, the paradox that provoked Kepler’s passion was not a numerical one. Rather, it was an epistemological one: If, God composed the solar system, so that action occurred in perfect circles, or inversely, in Gallileo’s straight lines, why then was the planets’ motion non-uniform? In other words, it was not a paradox in the domain of perception, but a paradox in the domain of Mind. The paradox arose from the irreconcilability of two ideas concerning the relationship between man and nature. On the one hand, the idea that physical action occurs according to perfect circles and straight lines, while seemingly more sensible to the naive mind, requires the universe to perform an irrational dance, in order to conform to its dictates. On the other hand, the concept that action in the physical universe was actually non-uniform, seemed to require man to embrace a less perfect geometry, but conformed more to planets’ actual motion. The former implicitly assumes that either man, nature, or both were irrational. The latter acknowledges, initially, a less precise geometrical construction , but in accepting a less simple mathematics, it hopes to gain a more perfect one.

Recall to mind again, the opening words of Kepler’s “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 star’ 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.”

The specific difficulty is this: Once, Kepler insisted that the Sun moved the planets by a force whose effect diminished with distance, the irregular speed of the planet could be known as the result of an eccentric orbit. This is because in an eccentric orbit, the distance between the planet and the Sun, is always changing, getting either longer or shorter. Thus, as the planet moves around the Sun, the effect of the Sun’s force is always diminishing or increasing, which slows the planet down, or speeds it up. (See the figures on pages 26, 27, and 33 of the Summer 1998 Fidelio.)

[See also http://csep10.phys.utk.edu/astr162/lect/binaries/visual/kepleroldframe.html]

This is distinct from the characteristic of a circular orbit, in which the distance between the planet and the Sun is always constant, and so a planet moving in such an orbit, will move at a constant rate. (This required the imposition of irrational demigods to speed the planet up or slow it down as it moved in this circular path.)

The question which struck fear in the hearts and minds of Aristoteleans to this day was: “How could the planet know to stay on this eccentric orbit?”

Since in an eccentric orbit, the planets’ distance from the Sun is always changing, the planet is have to constantly re-define its path, in every interval of action no matter how small. There is no way for the planet to define this path from the standpoint of simple straight line action between the planet and the Sun. Rather, the planet’s orbit must be defined by something outside the orbit itself. That “something” is what Kepler referred to in his “Epilogue” as the relationship of the Sun to the each planet individually and the relationship of the Sun to all the planets, which is the same as the relationship of noos to dianoia.

But, how can this “something” be measured? For this, Kepler had to settle for what he acknowledged was an imperfect method.

As Kepler recounts in the “New Astronomy”:

“My first error was to suppose that the path of the planet is a perfect circle, a supposition that was all the more noxious a thief of time the more it was endowed with the authority of all philosophers, and the more convenient it was for metaphysics in particular. Accordingly, let the path of the planet be a perfect eccentric…”

“Since, therefore, the times of a planet over equal parts of the eccentric are to one another as the distances of those parts, and since the individual points of the entire semicircle of the eccentric are all at different distances, it was no easy task I set myself when I sought to find how the sums of the individual distances may be obtained. For unless we can find the sum of all of them (and the are infinite in number) we cannot say how much time has elapsed for any one of them. Thus the equation will not be known. For the whole sum of the distances is to the whole periodic time as any partial sum of the distances is to its corresponding time.” (Diagram 5.3 on page 27 of the Fidelio, illustrates Kepler’s problem.)

In other words, while the speeds and distances are constantly changing, Kepler looks to the “sum” of the distances, or the area swept out, which remains constant for equal intervals of time. [See figure.] Thus, with respect to the relationship of the Sun to the individual planets, the “something” expresses itself in this proportionality. However, Kepler ran into a serious problem here. For while if he knew two positions of the planet, he could determine the time it took the planet to move from one position to the other, by calculating the area swept out, he was unable to do the inverse. That is, calculate the positions of the planet, if given the time elapsed. (In terms of diagram 5.4, for example, Kepler could calculate the area swept out (time elapsed) between position P1 and P2, but he was unable to determine a position P3, such that the time elapsed (area swept out) from P1 to P2 is equal to that of P2 to P3. See Fidelio for a complete review of what has since become known as the “Kepler Problem”.)

This begins to answer the question, “How could the planet know to stay on this eccentric orbit?”, but, it begs the next question, “How does the planet know to stay on this eccentric, and not some other?”

For this Kepler turned to the relationship of the Sun to all the planets. All the eccentric orbits were constituted, Kepler found, so that the square of the periodic time was equal to the cube of the average distance to the Sun. (See chapter 7 of the Fidelio.)

Now we have two characteristics of this “something” else, that determines the planetary orbits. Each orbit, though always changing, has its own constant of area to time elapsed, and all the planets have the same constant of the square of the periodic time and the cube of the distance from the Sun. These combined with the ordering according to the five Platonic solids, and the harmonic intervals of the planet’s extreme velocities, characterizes that “something” by which the planet knows to stay in its non-constant orbit.

And that “something else”, is what Gauss and Riemann would later call a function.

Riemann for Anti-Dummies: Part 3 : On Kepler

ON KEPLER

No mortal yet as climbed so high,
As Kepler climbed and died in need, unfed:
He only knew to please the Minds
And so, the bodies left him without bread.
–Abraham Gotthelf Kaestner

In “On Copernicanism and the Relativity of Motion,” G. W. Leibniz presents a proposition that might provoke you.

“To summarize my point,” Leibniz writes, “since space without matter is something imaginary, motion, in all mathematical rigor, is nothing but a change in the positions [situs] of bodies with respect to one another, and so, motion is not something absolute, but consists in a relation.”

It follows, then, that it were impossible to answer the question, “Which is moved, the Earth or the Sun?”

There are several common reactions to this question, among which are:

1. From the standpoint of naive sense perception: “The Sun is moved, because I see the Sun, stars and planets, all moving around me.”

2. From the standpoint of popular opinion: “The Earth moves around the Sun. Everyone knows that. We learned it in school.”

3. From the standpoint of naive sense perception and popular opinion, dressed up in more sophisticated evening clothes: “The Earth moves around the Sun. This can be discovered by observing the motion of all the planets, measuring the intervals between the various periods of retrograde motion, and comparing those intervals with the Sun’s apparent yearly cycle. All these observations, taken as a One, clearly indicate the Earth is moving around the Sun.”

There are, of course, numerous other variants. However, as Leibniz correctly states, try as you might, there is no way you can say, whether the Sun or the Earth is moved. The best you can hope, to truly say, is that the Earth and the Sun are in motion relative to one another.

“How then,” you might ask, “Could Kepler so surely state, that the Sun is at rest and the Earth is in motion?”

As usual, the problem is, that you asked the wrong question. Kepler never asked, “Is the Sun or the Earth moved?” Rather, he asked, “Is the Sun or the Earth, the mover?”

See the difference between the two questions. The former concerns only a change in place, which Leibniz shows is always relative. The latter concerns a physical ordering principle. In the first question, the verb “is”, refers to something other than the Sun and the Earth, i.e., the imaginary space in which the two bodies interact. In the second question, the verb “is”, refers to the physical cause of the motion, “the mover.” The second question is susceptible to being answered, while the first, is only susceptible to being debated by academics and fools.

“Okay, now that you’ve shown me the right question, it is still answered by observing the motions of the Sun, the Earth, and the planets, discovering the anomalies, and then determining that the Sun is the mover,” you might assert.

Be careful. Here things can get slippery. A universal principle is not {discovered} by observation or experiment. A universal principle is discovered by an act of cognition, and its validity is demonstrated, by the replication of that act of cognition, through a crucial experiment.

As Plato, Cusa, and Kepler showed, this is not a matter of preference between equally valid methods, but concerns an ontological principle of the Universe itself.

At the end of the {Harmonies of the World}, Kepler appended an, “Epilogue Concerning the Sun, By Way of Conjecture,” where he elaborated this ontological principle.

“From the celestial music to the hearer, from the Muses to Apollo the leader of the Dance, from the six planets revolving and making consonances to the Sun at the center of all the circuits, immovable in place, but rotating into itself,” Kepler begins.

“… The relation of the six spheres to their common center, thereby the center of the whole world, is also the same as that of unfolded Mind (dianoia) to Mind (noos)….”

Here an understanding of the Greek words, can help us to a more precise grasp of the concept. Since words evoke concepts, it is an error to seek an “equivalent” word, when projecting a concept from one language to another. It were more appropriate to apply a principle similar to the Kepler/Gauss concept of congruence. This may require the use of several terms, in several languages, in order to communicate the idea, to which the word is intended to refer. In this case, the Greek words “dianoia” and “noos”, (usually translated as understanding and reason, respectively) refer to concepts developed by Plato in the sixth book of the Republic, to denote the two highest principles of knowledge. The highest principle, “noos”, subsumes the next highest principle, “dianoia”. While these are distinct principles, the higher begets the lower, denoted in the Greek by the prefix, “dia”. Kepler inserts the Greek words, not to provide an equivalent term for the Latin text, but to refer his readers to the concept developed by Plato. Even the expression of universal physical principles, must be congruent with those principles.

Plato states this principle as:

“This then is the class that I described as intelligible, it is true, but with the reservation first that the soul is compelled to employ assumptions in the investigation of it, not proceeding to a first principle because of its inability to extricate itself from and rise above its assumptions, and second, that it uses as images or likenesses the very objects that are themselves copied and adumbrated by the class below them, and that in comparison with these latter are esteemed as clear and held in honor … and by the other section of the intelligible I mean that which the Mind itself lays hold of by the power of dialectics, treating its assumptions not as absolute beginnings but literally as hypotheses, underpinnings, footings, and springboards so to speak, to enable it to rise to that which requires no assumption and is the starting-point of all, and after attaining to that again, taking hold of the first dependencies from it, so to proceed downward to the conclusion, making no use whatever of any object of sense, but only of pure ideas moving on through ideas to ideas and ending with ideas.”

With that in mind, now, back to Kepler’s “Epilogue”:

“But we duly subordinate the created mind of whatever excellence it may be to its Creator, and we introduce neither intelligent powers as Gods, as does Aristotle and the pagan philosophers, nor armies of innumerable planetary spirits with the Magi, nor do we propose that they are either to be adored or summoned to intercourse with us by theurgic superstitions, for we have a careful fear of that; but we freely inquire by natural reasons what sort of thing each mind is, especially if, in the heart of the world, there is any mind bound rather closely to the nature of things and performing the function of the soul of the world or if also some intelligent creatures, of a nature different from human perchance do inhabit, or will inhabit, the globe thus animated. But it is permissible, using the thread of analogy as a guide, to traverse the labyrinths of the mysteries of nature. I believe the following arguments can not be put aside. The relation of the six spheres to their common center, thereby the center of the whole world, is also the same relation, as that of unfolded Mind (dianoia) to Mind (noos). On the other hand, the relation of the single planets’ revolutions from place to place around the Sun, to the unvarying rotation of the Sun in the central space of the whole system, is also the same as the relation of unfolded Mind to the Mind, which is, that of the manifold of dialectics, to the most simple cognition of the Mind. For as the Sun rotating into itself moves all the planets by means of the form emitted from itself, so too as the philosophers teach Mind, stirs up dialectics, by which it understands itself and in itself all things, and by unfolding and unrolling its simplicity into those dialectics, it makes everything known. And the movements of the planets around the Sun at their center, and the unfolded dialectics are so interwoven and bound together, that, unless the Earth, our domicile, measured out the annual circle, midway between the other spheres changing from place to place, from station never would human cognition have worked its way to the true intervals of the planets, and to the other things dependent from them, and never would it have constituted astronomy.”

This relationship, of the Sun to all the planets, the Sun to each planet individually, of noos to dianoia, of the word to the idea, is at the heart of Gauss’ and Riemann’s theory of functions. With what has been developed here, we are now in a position to begin to unfold these functions in the next installment.

We end with another poem by Kaestner written for his student Christlob Mylius. The poem was meant to accompany a copy of Kepler’s {Harmonies of the World} which Kaestner sent to Mylius, who was on his way to America from Leipzig. Mylius, a cousin of Lessing, never made it. He died in London, survived by this short poem. In it, Kaestner expresses his great love for Kepler, and his oft-stated disdain for those German authorities and bread-scholars, who cowardly turned their back on their great countryman and embraced the inferior Newton instead.

To Mr. Christlob Mylius:
Along with sending over Kepler’s Harmonice Mundi
Friend, your tender ear perceives the graceful art of tones,
The world-form’s harmonies, your deeper thoughts explore,
In here, Newton’s teacher writes of both them,
Deustchland let him starve, and remains unworthy of him.

Riemann for Anti-Dummies: Part 2

Don’t tell me how the universe is constructed,” protested an early 17th-Century ancestor of today’s Baby Boomer, “I just need to know when Mars will be in Leo, when Saturn will be in opposition to the Sun, and when Jupiter will align with the Moon.

Armies are massing, and on good information, I’ve been told, that when the stars are such, the war begins. When armies amass, I make cash. Dice, whiskey and whores. I get a cut of them all. But once the fighting starts, it’s too late to speculate. I need to know now, what’s in store. Buy cheap and sell dear, they say. It’s best if I tell the others to sell. But, it only works, if they think the information that I’ve got, they have not.”

“My dear son,” the old man said, after a long pause, “You have to be human to understand matter.”

Perhaps Kepler never encountered an interlocutor as crude as this. Nevertheless, he must have confronted something similar:

“Yet, alas, of what great goods do miserable mortals despoil one another, by their shameful itching for quarrels. How profound an ignorance of their fate overwhelms them, as they have deserved. With what deplorable perverseness do we rush into the midst of the flames, in fleeing from the fire,” he wrote in 1621, three years after the eruption of the Thirty Years War.

“Would that even now indeed, there may still, after the reversal of Austrian affairs which followed, be a place for Plato’s oracular saying. For when Greece was on fire on all sides with a long civil war, and was troubled with all the evils which usually accompany civil war, he was consulted about a Delian Riddle, and was seeking a pretext for suggesting salutary advice to the peoples. At length he replied that, according to Apollo’s opinion Greece would be peaceful if the Greeks turned to geometry and other philosophical studies, as these studies would lead their spirits from ambition and other forms of greed, out of which wars and other evils arise, to the love of peace and to moderation in all things.”

Such was Kepler’s introduction to the second edition of his Mysterium Cosmographicum, published 25 years after its first appearance in 1595. The intervening period was marked by an escalation of the Venetian-led religious wars, that had by then been ongoing since 1513, and was accompanied by an orchestrated, popular rise in superstition and witchcraft, in which Kepler even saw his own mother victimized in a witch trial. Meanwhile, Kepler had extended the discoveries contained in his first work, elaborating the principles underlying the motions of the solar system, and, composing those principles into one unified function, of the type, characterized by Gauss nearly 200 years later, as “hypergeometric”.

“Astronomy has two ends,” Kepler wrote in the “Epitome of Copernican Astronomy”, “To save the appearances and to contemplate the true edifice of the world.” For Kepler the principles necessary for the latter would satisfy the requirements of the former. Numerous efforts to save the appearances had been attempted previously, and all had turned up wanting. All suffered from a common flaw, not in the sophistication of their geometrical constructions, but in their underlying intent, which was simply to predict, and not to know.

To know, instead of simply to predict, requires, “the discipline which discloses the causes of things, shakes off the deceptions of eyesight, and carries the mind higher and farther, outside of the boundaries of eyesight. Hence, it should not be surprising to anyone that eyesight should learn from reason, that the pupil should learn something new from his master which he did not know before…,” Kepler wrote in the “Epitome”.

The “hypergeometric” type of function that Kepler composed, is, thus, comprised of a set of physical constraints, that, taken as a One, permit only those motions that physically occur, to occur. Each constraint is itself a function, a sort of special case, of the more generalized “hypergeometric” function that characterizes the ordering of the Solar System.

Exemplary is Kepler’s discussion of the constraint concerning the number of the planets, as stated in the “Epitome”:

“This worry has been resolved, with the help of God, not badly. Geometrical reasons are co-eternal with God — and in them there is first the difference between the curved and the straight line. Above (in Book 1) it was said that the curved somehow bears a likeness to God; the straight line represents creatures. And first in the adornment of the world, the farthest region of the fixed stars has been made spherical, in that geometrical likeness of God, because as a corporeal God — worshipped by the gentiles under the name of Jupiter — it had to contain all the remaining things in itself. Accordingly, rectilinear magnitudes pertained to the inmost contents of the farthest sphere; and the first and the most beautiful magnitudes to the primary contents. But among rectilinear magnitudes the first, the most perfect, the most beautiful, and most simple are those which are called the five regular solids. More than 2,000 years ago Pythagoreans said that these five were the figures of the world, as they believed that the four elements and the heavens–the fifth essence–were conformed to the archetype for these five figures.

The Five Regular Solids

“But the true reason for these figures including one another mutually is in order that  these five figures may conform to the intervals of the spheres. Therefore, if there are five spherical intervals, it is necessary that there be six spheres; just as with four linear intervals, there must necessarily be five digits. “

Kepler's Intervals

Thus, the constraint that determines the number of planets and the general size of their orbits, is a special case of the “hypergeometric” function, and reflects the incommensurability of the straight to the curved. As we have shown in other pedagogical discussions, there is a different characteristic between action on a flat (straight) surface and a spherical one. The five Platonic solids represent the totality of all possible equal divisions of a sphere, while, a flat surface permits an entirely different set of tilings. Further, the characteristic periodicity of a flat surface is four-fold, while (as Kepler’s contemporary Napier showed in his studies on the pentagramma mirificum) the characteristic periodicity of a spherical surface is five. The correlation between this geometrical paradox, and the physical nature of the solar system, is evidence that the “hypergeometric” function ordering the solar system is curved, i.e., anti-Euclidean.

Yet, spherical action is not sufficient by itself to characterize planetary motion. While the total periodic time of each planetary orbit is constant, “It, nevertheless, is of irregular speed in its parts; and it makes the planet in one fixed part of its circuit digress rather far from the Sun, and in the opposite part come very near to the Sun; and so the farther it digresses, the slower it is; and the nearer it approaches, the faster it is…. For astronomy bears witness that, if with our mind we remove all deceptions of sight from that confused appearance of the planetary motion, the planet is left with such a circuit that in its different parts, which are really equal, the speed of the planet is irregular … astronomy, if handled with the right subtlety, bears witness in this case that the routes or single circuits of the planets are not arranged exactly in a perfect circle but are ellipses.”

These physical constraints, governing the irregular motion of the planets, are also a special case of that same generalized “hypergeometric” function, encountered peviously. As in the case of the physical constraints reflected in the Platonic solids, the difference between the straight and the curved are reflected here. While the motion of the planet within its orbit is constantly changing, the ratio of the area swept out between a line connecting the planet and the Sun, to the time elapsed, is always constant. Similarly, even though the sizes of the planetary orbits, and the periodic times are obviously different for each planet, the ratio between the square of the time and the cube of the mean distance from the Sun, is the same for all planets. These so-called second and third laws of Kepler, are thus, special cases of the “hypergeometric” function governing planetary motion. The one ratio is a constant within each orbit, the other across the orbits. (These matters were treated at some length in chapters five, six and seven of the Fidelio article on Gauss’ determination of the Ceres orbit, and will form the jumping off point for future investigations).

Similarly, the harmonic relationships between the extreme angular velocities of adjacent planets, that correspond to musical intervals, is another special case, of this generalized “hypergeometric” function.

“But, Galileo made it all so simple,” the Baby Boomer’s ancestral soul-mate complains. “I like his way better.”

Kepler certainly faced this problem: “As regards the academies, they are established in order to regulate the studies of the pupils and are concerned not to have the program of teaching change very often: in such places, because it is a question of the progress of the students, it frequently happens that the things which have to be chosen are not those which are are most true but those which are the most easy.”

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. Continue reading Riemann for Anti-Dummies: Part 1