Text: POWER FROM THE STANDPOINT OF THE COMPLEX DOMAIN Posted By: Rosalinda Date: Saturday, 26 June 2004, 10:47 p.m. In Response To: PYTHAGORAS AS RIEMANN KNEW HIM (Rosalinda) The above review is pedagogically helpful as a starting point for approaching the work of Gauss and Riemann. As these simple examples illustrate, physical processes are the effects of a connected action of physical powers (principles). Each power is expressed by a distinct species of magnitude. But, when a physical action is generated as the effect of a connected action among a group of powers, it generates a manifold, the which expresses a new, and completely different, characteristic species of magnitude. Riemann called such manifolds, "multiply-connected". A strong word of caution is in order. As will become more clear as we work through Riemann's ideas, by "multiply-connected", Riemann did not mean the Aristotelean idea of a set of theorems connected to one another through a lattice of logical formalism. Rather, Riemann's multiply-connected manifold is a unity of demonstrable physical principles, which, like Leibniz's {monads}, are distinct, but connected, not directly to each other, as if point-wise, but only through the higher organizing principle of the manifold itself. A few physical examples, with which readers of this series will be familiar, will help illustrate this point: --As Kepler's principles of planetary motion illustrate, the planet's motion, at every infinitesimal moment, is being determined by the connected action of all those principles that govern action in the solar system. This action is expressed mathematically by the combined effect of Kepler's treatment of the five regular solids, the principles of elliptical motion, and the harmonic relationship among the musical pitches. As Gauss later showed through his determination of the orbit of Ceres, and his later work on the secular perturbations of the planets and asteroids, there are an even larger number of physical principles affecting the motion of the planet at each moment, than those expressed by Kepler. Gauss showed that the manifold of these connected principles can only be expressed in the complex domain. --The case of the intersection of a beam of light with a boundary between two different media, such as air and water, in which some of the beam is reflected and some of the beam is refracted. On the macroscopic level, we can see that this action must be thought of as occurring in a manifold that connects the two principles, reflection and refraction. But as we take this investigation into the microscopic domain, many more principles, those governing action in the atomic and sub-atomic domain, come into play, requiring a re-conceptualization of the manifold, into one with the power to connect a greater number of principles. --The catenary's expression of the universal principle of least-action as the arithmetic mean between two, oppositely directed exponentials. Each exponential itself denotes a manifold that transcends all algebraic powers. The catenary, therefore, must exist in a manifold that connects two such transcendental manifolds. In this higher manifold, both exponentials are acting, not only arithmetically, as indicated by their visible relationship, but also geometrically, the latter acting in the direction perpendicular to the visible plane of the hanging chain. (See Figure 3.) As Gauss showed, a manifold with the power to act on both exponentials arithmetically and geometrically, must be expressed as a surface in the complex domain. In all of the above examples, the powers determining the physical action, are acting, from outside the visible domain, but their effects are present everywhere. Therefore, as Riemann made clear in his 1854 Habilitation lecture, to understand physical action, we must ban from science all considerations of geometry formed from a set of {a priori} axioms, postulates and definitions, and consider only {ideas} concerning physical manifolds, whose "modes of determination" are physical principles. With axiomatic assumptions now eliminated from geometry, the characteristic of action associated with Euclidean geometry, i.e., infinitely extended linearity, in three directions, disappears as the phantasm it always was. Instead, the characteristics of such a physical manifold are determined only by the physical principles which form the "modes of determination" of the physical action under consideration. In his work, Riemann established the elementary principles to construct an image that faithfully reflects the means by which such physical "modes of determination" determine the characteristic of action in such a multiply-connected manifold, by showing how the effect of these principles determines the topology and characteristic curvature of the image. Most importantly, what is gained by Riemann's method, is a means to determine and express the type of change that occurs, by the discovery of a new physical principle. Riemann based his discovery on the previous work of Gauss, most notably, Gauss's 1799 treatise on the fundamental theorem of algebra, and Gauss's work on the general characteristics of curvature. Thus, it is most efficient pedagogically, to begin with a quick review of these features of Gauss's work. In rejecting the methods of Euler, Lagrange, and D'Alembert, Gauss showed that any formalist treatment of algebraic expressions, according to the logical rules of algebra, lead to a contradiction, (i.e. the square root of -1), within the domain of the formal system of algebra itself. This was not the result, Gauss insisted, of some hidden flaw within the logical system. It was a flaw of the system itself, arising from the fact that the algebra of Euler, Lagrange and D'Alembert was merely a logical system. As Gauss emphasized, the system could not be reformed, it had to be abandoned all together. In other words, Gauss did not come to save the system of algebra. He came to free science from its mind-killing constraints. As Gauss showed, the inherent flaw in the formalist's algebra, was the treatment of an algebraic power by simple rules of arithmetic. Gauss, in referring back to the Pythagorean principles of the doubling of the line, square and cube, insisted that the "power" in an algebraic expression must be understood to reflect a physical principle. For example, an algebraic expression of the second degree, must concern what Riemann would later call a "doubly- extended" relationship such as areas; an expression of the third degree, must concern a "triply- extended" relationship such as among volumes. A change from one power to another, therefore, denoted a change in the number of principles under investigation, not the number of times one number is multiplied by another. By constructing his surfaces as images that reflect this physical idea of power, the addition of a new power is reflected in the image, as a change in what he called the geometry of position, or topology, of the surface. (See Figure 4.) Thus, what is counted in algebra is not numbers, but powers. For Gauss, it was mind deadening brainwashing to consider an algebraic expression as a set of formal rules. Instead, he insisted, such expressions are, at best, only a short-hand description of a physical action, whose real characteristics could only be truthfully expressed through his geometric constructions. Riemann insisted that only a method similar to Gauss's could be applied when investigating the transcendental, elliptical and Abelian functions. As Leibniz had already indicated, such functions, by their very nature, could never be expressed by any formal algebraic- type means. For example, assigning a set of rules for calculating the expression "sine of x" does not give us any knowledge of the transcendental relationship between circular and rectilinear motion, let alone the profound connection that Leibniz discovered between circular, hyperbolic and exponential functions. Yet, as Leibniz emphasized, following Kepler and Cusa, universal {physical }action could only be expressed by such non-algebraic, "inexpressible" magnitudes. Thus, for Riemann, to "know" a transcendental function, meant to know its geometrical characteristics, because all attempts at formal expression, as typified by the work of Euler, Lagrange, and the bigoted Cauchy, were always impotent. In Riemann's geometrical expressions, as in Gauss's, the change from one transcendental power to another, is reflected as a change in the topology of the Riemann surface. For example, the circular/hyperbolic transcendental, which is associated with the catenary, is simply periodic, has two branch-points, and thus can be characterized by the topology of the sphere. (See Figure 5.) Whereas the elliptical transcendental associated with the elliptical orbit of a planet, or the motion of a pendulum, is doubly periodic, with four branch-points, and is characterized by the topology of the torus. (See Figure 6. See Riemann for Anti-Dummies 49, 52, 54, and 56 ). Just as in the case of Gauss's treatment of algebraic powers, each transcendental power is distinct. Consequently, the transition from one transcendental to another, because it involves the addition of a new principle, is not continuous. Like a discovery of a revolutionary new idea, the shift to a new transcendental, suddenly and completely, transforms all pre-existing relationships, that had been considered, until then, fundamental. For example, think of how Riemann expressed the effect of a simply periodic transcendental function, through the image of a stereographic projection of a sphere onto a plane. In this image, the circles of latitude on the sphere are images of concentric circles in the plane, and, as such, are orbicular. But, the circles of longitude are images of radial lines which converge at the image of the "infinite", i.e., north pole. Consequently, motion along these longitudinal circles can never be periodic, as a complete rotation must always "cross over the infinite". In this way, Riemann's image fixes in our mind the idea of a physical process in which simple periodicity is a physical characteristic, not simply a mathematical formalism. On the other hand, a doubly periodic action is a connected action with two distinct periods. Such an action could never be represented on a sphere with an infinite boundary. As Riemann showed in his treatment of the elliptical transcendentals, the type of surface on which these elliptical transcendentals "live", must correspond topologically to a torus, whose "hole" allows for these two distinct, but connected, periods. However, as Riemann emphasized, the transformation from a sphere to a torus is discontinuous, because an entirely new possibility of action is added. In this way Riemann showed, that the essential characteristics of a transcendental function, {and} the characteristic of a change in transcendental power, could be made intelligible, even though such characteristics were utterly "inexpressible" in formal algebraic terms. Riemann called the type of transformation just illustrated, a change in the "connectivity" of the manifold. For Riemann, the sphere is "simply-connected", because it has no hole and requires only one closed curve to cut it into two distinct parts. The torus, on the other hand, is a surface that Riemann called "doubly-connected", because it has one hole and requires two closed curves to cut the surface into two distinct parts. A "triply-connected" surface is one that has two holes, etc. (See Figure 7.). Riemann emphasized that connectivity is a characteristic, like the number of "humps" in Gauss's surfaces, that is independent of all measure relations of that surface, or calculations within the formal expression. For example, in the case of an algebraic expression, it doesn't matter how wildly the coefficients of the expression vary, the physical characteristics of the action that expression describes are determined solely by the number of principles involved, as denoted by the expression's highest "power". This is what is reflected by the topology (number of "humps) of the corresponding Gaussian surface. In the case of Riemann's investigation of the higher transcendentals, the "power" of the transcendental is expressed by a similar type of invariant characteristic, the surface's connectivity. It is important to note here, but reserve for the future its more complete development, that Riemann showed that this characteristic change in the topology of the image, is a function {solely} of the "power" of the transcendental function, which, in turn, is determined by the number of characteristic singularities generated by that transcendental function. Thus, the "holes" in a Riemann surface do not signify "nothingness", or that something is missing or left out. Rather the number of holes signifies the density of singularities associated with the power of the transcendental function. In this way, Riemann showed, in his lectures on Abelian and hypergeometric functions, that Abel's "extended class" of transcendentals could be expressed by surfaces of increasing degrees of connectivity, or what Riemann called "multiply-connected" surfaces. A change in the number of singularities associated with a transcendental function, is expressed as a change in the connectivity of the surface that expresses that function. http://www.rumormillnews.com/cgi-bin/forum.cgi?read=51182

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