Riemann For Anti-Dummies Part 35


Rene Descartes (1596-1630) was, for all intents and purposes, a Bogomil. The geometry that bears his name, is brainwashing. Anyone exposed to it, unless cured, will suffer from cognitive deficiency. Symptoms include impotence and an inability to distinguish fantasy from reality. 

Gottfried Leibniz, writing to Molanus, circa 1679, recognized the deleterious effects of Cartesianism, "Cartesians are not capable of discovery; they merely undertake the job of interpreting or commenting upon their master, as the Scholastics did with Aristotle. There have been many beautiful discoveries since Descartes, but, as far as I know, not one of them has come from a true Cartesian.... Descartes himself had a rather limited mind." 

Descartes' method is impotent. It lacks power. Go back to the investigations of the Pythagoreans, Archytas, Menaechmus and Plato, on the matter of doubling the line, square and cube. These discoveries demonstrated, the relationship between objects and the principles from which they are generated. Each principle possess a characteristic power. The succession of objects-- line, square and cube-- are produced by a succession of higher powers (dunamis). These powers are not defined by the objects. The objects are produced by the powers. The powers cannot be known through the senses. The characteristics of the physical powers are, nevertheless, made sensible through their harmony, which only the mind has the power to grasp. 

As can be seen from the solutions to doubling the cube by Archytas and Menaechmus, the harmonic relationship among these powers reflects a characteristic curvature, that, when projected onto straight lines, produces the relationships the Pythagoreans recognized as the arithmetic, geometric and sub-contrary, (or harmonic) means. The arithmetic mean is three numbers related by a common difference: c-a=b-c, or, c=1/2(a+b). Geometrically, it is represented by the half-way point along a line; musically it corresponds to the interval of the fifth. The geometric mean is three numbers in constant proportion: a:b::b:c. Geometrically it is represented by the middle square between two squares; musically it corresponds to the Lydian interval. The harmonic mean is the inverse of the arithmetic mean: 1/c=1/2(1/a+1/b). It is expressed geometrically in the hyperbola and musically by the interval of the fourth. These harmonic relationships are number shadows cast by the curved onto the straight. (See Riemann for Anti-Dummies 33. EIR website.) 

Riemann generalized these Greek discoveries by his notion of multiply extended magnitude. The line is an artifact of a simply-extended manifold, the square an artifact of a doubly-extended manifold, and the cube an artifact of triply-extended manifold. For Riemann, as for Pythagoras, Archytas, Menaechmus, Plato, et al., each increase of degree of extension, from "n" to "n+1", occurs by the addition of a new principle, not a new independent "dimension". Consequently, a square cannot be produced from a line, nor a cube from a square, because the square is generated by a different principle than the line, as the cube is generated from a different principle than the square. But, Riemann also made clear, that extension alone is insufficient to determine physical geometry. Another principle is necessary: physical curvature. (See Riemman for Anti-Dummies, Parts 28, 29, 33, 34). 

In Descartes' make-believe world, the concept of power is excised. "Any problem in geometry can easily be reduced to such terms that a knowledge of lengths of certain straight lines is sufficient for its construction," is the opening of his treatise on analytical geometry. 

As a true Bogomil, Descartes is perverse. He begins ass backward, starting with numerical relationships, stripped of their power, and pretending to generate curves, from only these numberical relationships which he wrote down in the form of an algebraic equation. This is pure fakery, as Descartes never derived any curve from these equations. All the numerical relationships had already been discovered by Apollonius, through the investigations of the relationship between curvature and power. Descartes never generated a single curve whose harmonic relationships had not already been discovered by the Greeks. Descartes' intention was to strip the power from ideas and the idea of powers from geometry. 

To illustrate this point concretely, look at Menaechmus' solution for the problem of doubling the cube, presented in Riemann for Anti-Dummies 33. Menaechmus demonstrated that the magnitude that doubles the cube is formed by the intersection of a parabola and an hyperbola. Each curve embodies a different set of proportions that emerge when the curved is combined with the straight. For example, the hyperbola is formed by the corner of a rectangle whose sides change such that the area remains the same. The parabola is formed by the corner of a rectangle in which one side is always the square of the other. These rectangles are made up of straight lines, whose proportionality is determined by the curves. The curves posses the power to produce that proportionality, and that power is expressed in the relationship between the curve and the straight lines produced by it. In other words, only a faker or a fool would separate the curve, the straight-lines and the proportionality that produces this complex of action. As Menaechmus demonstrates, when the hyperbola and parabola are combined, a power is expressed by the resulting proportionality, which is higher than exists in either curve independently. 

For Descartes, the straight lines are independent entities, created without reason. The curve and the associated powers are deviations from these straight lines. "Here it must be observed that by a^2, b^3, and similar expressions, I ordinarily mean only simple lines, which, however, I name squares, cubes, etc., so that I may make use of the terms employed in algebra," he confessed. Thus, the fantasy make believe world of independent straight lines is taken as primary and the real world of physical action, is only a deviation from the fantasy world. Since, as Leibniz stated, this way of thinking is incapable of producing discoveries, the only intention of those teaching it, is to condition the students into believing the fantasy world has more power than reality. (The baby-boomer populist's obsession that money equals economic security is a typical result of this type of education.) 

To hammer this home and to prepare the ground for taking on Riemann's physical differential geometry, look at two physical examples: the conic section orbit of a heavenly body around the sun; the catenary; and Gauss' Geoid. 

In the first case, the heavenly body is conforming to a unique curved pathway around the sun, which Kepler and Gauss demonstrated was a conic section with the sun at a common focus for all orbits. Thus, the orbits define a physical pathway, and the sun a physical origin. The straight-lines that have physical significance are the ones related to the physical action. For example, the major axis of an elliptical orbit is the line that connects the points of minimum and maximum speed, which are also the points of maximum curvature. The parameter of the orbit is the line going through the sun that is perpendicular to the major axis of the conic section. The minor axis of the elliptical orbit is the line connecting the points of minimum curvature of the orbit. These lines express the harmonic relationships of the arithmetic, geometric and harmonic means, which in turn reflect the higher powers, the "reason" why the planet's orbit takes the shape it does. (See Appendix to "How Gauss Determined the Orbit of Ceres", Summer 1998 Fidelio.) 

Now look at the catenary. Despite Descartes' boast that his method could solve any problem in geometry, the hanging chain proved him wrong. The catenary presents a different problem than the conic section orbits. It did not conform to any known geometrical figure, so its nature had to be discovered only from its physical characteristics. This presented a problem for Descartes because unless the nature of the curve was known, he could not determine where to put his straight lines. 

Leibniz and Bernoulli demonstrated, that physical nature of the catenary is expressed by the relationship between any point on the chain, and the lowest point. That relationship is measured by the tangents to the curve at these two points. (See "Justice for the Catenary", Schiller Institute website.) The tangent to the lowest point is always perpendicular to the pull of gravity, i.e. horizontal. The relationship of the force between any point on the catenary and this lowest point, is measured by the sines of the angles formed by the tangents to these two points, and a vertical line drawn from the lowest point. In other words, the physical action at any point on the catenary, is expressed by a "differential" relationship between the angles formed by these three lines. The horizontal tangent to the lowest point, which is perpendicular to the pull of gravity, a vertical line drawn from that point, which is along the direction of the pull of gravity, and the tangent to the point on the curve. 

Leibniz and Bernoulli showed that this "differential" change does not conform to any previous known algebraic curve. It does not exist in Descartes' world. Descartes could not determine how to construct this curve from straight lines. (Anyone indoctrinated in Descartes method will be getting very uncomfortable now.) But, obviously the chain exists in the real world. As we just observed, the only lines that are relevant are those determined, physically, by the changing relationship of the catenary to the pull of gravity and the perpendicular to the pull of gravity. This changing relationship is not determined by Cartesian geometry. It is determined by the physical curvature of the pull of gravity. Leibniz and Bernoulli demonstrated, that this relationship is expressed by the exponential and hyperbolic functions, both of which are expressions of a succession of higher powers, and as such, undiscoverable by the Cartesian method. (See Riemann for Anti-Dummies 33. EIR website.) 

Gauss' Geoid presents a still different problem. In the previous two examples, the "differential" of action was along a pathway determined by the principle of universal gravitation. In these cases, the "differential" could be determined with respect to a doubly-extended magnitude. (The major axis and parameter for the orbit and the pull of gravity and its perpendicular for the catenary.) In determining the shape of the Earth, Gauss confronted the addition of a new principle. Instead of measuring along a pathway in a doubly-extended surface, he was measuring changes of the surface itself. For pedagogical purposes, think of measuring a triangle on a perfect sphere. How does the shape of that triangle change as the area of the triangle increases? Compare this with measuring a triangle on an irregular surface, such as a watermelon. One the sphere, the sides of the triangles change because they are circles in all directions. However, on a watermelon, the sides of the triangle change according to a different principle depending on the direction. To measure this type of change, Gauss invented a new type of complex differential, which will be developed more fully in future pedagogicals. 

To summarize the epistemological issues raised in this pedagogical, we quote Leibniz disputing Descartes theory of motion: 

"There was a time when I believed that all phenomena of motion could be explained on purely geometrical principles, assuming no metaphysical propositions...But, through a more profound meditation, I discovered that this is impossible, and I learned a truth higher than all mechanics, namely that everything in nature can indeed be explained mechanically, but that th e principles of mechanics themselves depend on metaphysical and, in a sense moral principles, that is, on the contemplation of the most perfectly effectual efficient and final cause, namely, God... 

"...I discovered that this, so to speak, inertia of bodies cannot be deduced from the initially assumed notion of matter and motion, where matter is understood as that which is extended or fills space, and motion is understood as change of space or place. But rather, over and above that which is deduced from extension and its variation or modification alone, we must add and recognize in bodies certain notions or forms that are immaterial, so to speak, or independent of extension, which you can call powers, by means of which speed is adjusted to magnitude. These powers consist not in motion, indeed, not in conatus or the beginning of motion, but in the cause or in that intrinsic reason for motion, which is the law required for continuing. And investigators have erred insofar as they considered motion, but not motive power or the reason for motion, which even if derived from God, author and governor of things, must not be understood as being n God himself, but must be understood as having been produced and conserved by him in things. From this we shall also show that it is not the same quantity of motion (which misleads many), but the same powers that are conserved in the world."  

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