Text: Pythagoras as Riemann Knew Him by Bruce Director There is a widely circulated report that when Pythagoras discovered the incommensurability of the side of a square to its diagonal, he sought to conceal its discovery on pain of death to whomever would disclose it. But such an account is of dubious veracity, as it attributes to Pythagoras an attitude more appropriate to his enemies than to his collaborators. For it was the Eleatics, Sophists and Aristotle, who insisted that what was inexpressible could not be known; and it was Aristotle's Satanic disciples, as Bertrand Russell would come to exemplify, who demanded physical death for those who posed the potential for discovering new ideas; and it was Aristotle's method itself, when practiced as directed, that caused so much mental disease from his day to ours. For Aristotle: control what can be expressed, and you control what can be known. On the other hand, those who considered themselves Pythagoreans realized that the inexpressible was the frontier, not the barrier, of human thought. As Plato expressed it in the {Laws}, those who don't know the significance of the incommensurability of the line with the square, and the square with the cube, were closer to "guzzling swine" than human beings. The issue for the Pythagoreans was not that the inexpressible could not be known, but simply that it could not be expressed, in terms consistent with an {a priori} set of axioms, postulates and definitions, as Aristotle insisted. Thus, for the Pythagoreans, the discovery of something inexpressible was not a cause for alarm, but a joyful occasion to demonstrate, that man was not constrained by mere Aristotelean logic, but was, unlike a swine, free and unbounded. Therefore, as Plato insisted, it is of great benefit, and to be highly recommended, that political leaders discover for themselves the significance of incommensurability, in the terms that that discovery was known to Pythagoras and Plato. However, the true profundity of that discovery becomes much more fully illuminated when viewed from the standpoint of its more advanced development--the complex domain of Gauss and Riemann as that concept is expressed by Gauss's 1799 {New Proof of the Fundamental Theorem of Algebra}, and Riemann's crucial 1854 Habilitation lecture, and his 1855-57 lectures and writings on elliptical, Abelian and hypergeometric functions. These breakthroughs show that the principles discovered by the Pythagoreans were simply the first of an extended, and virtually unbounded, succession of transcendental functions, that express the increasing power of the human mind to discover, and communicate, ideas concerning universal physical principles. - Knowing Is Not Calculating - Much to the disdain of the Leibniz-hating followers of Euler, Kant, Lagrange and Cauchy, Riemann insisted that physical principles could be known, and given a mathematical expression, "virtually without calculation." In taking this approach, Riemann was directly in the Pythagorean tradition of Plato, Cusa, Kepler and Leibniz, who all recognized, that to know a physical principle, meant to have an {idea} concerning that principle's generative power, the which could never be discovered, nor expressed, by merely calculating that power's visible effects. As Gauss noted in comparing Euler's attempt to determine the orbit of a comet by calculation (an effort that left poor E. blind in one eye), with his own uniquely successful determination of the orbit of Ceres, "I too would have gone blind had I calculated like Euler!" Gauss's comment was consistent with, and inspired by, Kepler's earlier attack on the Aristotelean Petrus Ramus's diabolical demand that the tenth book of Euclid, (which concerns the incommensurables) be banned. Ramus insisted, as did Aristotle, that since only ratios of whole numbers were susceptible to finite calculation, no physical action was knowable, that could not be calculated thus. (Ironically, Gauss's, {Disquisitiones Arithmeticae}, {Treatises on Biquadratic Residues I & II} and the subsequent work of Lejeune Dirichlet and Riemann on the subject of prime numbers show, that even the principles governing whole numbers cannot be expressed by the linear arithmetic advocated by Aristotle and Ramus.) In his {Hamonices Mundi}, Kepler demonstrated that the physical principles that govern planetary motion cannot be expressed by the ratios of whole numbers, but only by those magnitudes which the Aristoteleans considered "inexpressible", specifically the magnitudes associated with the regular divisions of the circle, the five regular spherical solids, and the harmonic relations of the musical tones. This posed an ontological paradox for the Aristotelean. The principles governing physical action were inexpressible in terms acceptable to the Aristotelean. Therefore, as Aristotle's syllogism went, the physical universe was unknowable. For Kepler, the principle governing physical action could be {discovered}, by physical hypothesis, and {known} as a simple, i.e. unified, idea ({Geistesmasse}) . The effect of that principle could be expressed mathematically only by the appropriate, "inexpressible", magnitudes. An inexpressible magnitude was thus known, not in itself, but as that which was produced by the effect of a discovered physical principle. {In other words, the principle is not known by a magnitude. Magnitude is known by the principle whose effect it expresses.} Here, Kepler took his approach directly from Nicholas of Cusa, who, citing the Pythagoreans in {The Laymen on Mind}, insisted that such inexpressible magnitudes, such as the proportion of the side of a square to its diagonal, or the relationships among the musical tones, lead to an understanding of "a number that is simpler than our mind's reason can grasp": "By comparison then, see how it is that the infinite oneness of the Exemplar can shine forth only in a suitable proportion a proportion that is present in terms of number. For the Eternal Mind acts as does a musician, who desires to make his conception, visible to the senses. The musician takes a plurality of tones and brings them into a congruent proportion of harmony, so that in that proportion the harmony shines forth pleasingly and perfectly. For there the harmony is present as in its own place, and the shining forth of the harmony is made to vary as a result of the varying of the harmony's congruent proportion. And the harmony ceases when the aptitude-for-proportion ceases." John Keats makes clear in his great poem, {Ode on a Grecian Urn}, that all human knowledge is gained in this way. Looking at the urn, Keats sees the images of an ancient Greek society-- images of real people who lived and died, with passions much like ours. Yet all the questions he poses, which attempt to determine what the formalist would consider precise knowledge of those people and their culture, go unanswered. However, what is completely known, with absolute precision, is that {principle} of whose effect this urn is an image--the eternal power of human thought: When old age shall this generation waste, Though shalt remain, in midst of other woe Than ours, a friend to man, to whom thou say'st, "Beauty is truth, truth Beauty" that is all Ye know on earth, and all ye need to know. - Toward an Extended Class of Higher Transcendentals - To understand Riemann's essential discovery, we must take a quick look back, at the early development of the knowledge of inexpressibles, from the higher standpoint of Riemann's work. Begin with the magnitude which doubles the line. It can double the line but not a square. Yet, the magnitude that doubles the square is inexpressible, in terms of the magnitude that doubles the line. Inexpressible, but known--as that magnitude, that expresses the effect, of the physical principle, that has the {power}, (i.e., {dynamis}), to double a square. Thus, this simple, yet inexpressible magnitude, is known. The magnitude that doubles the square, however, cannot triple, nor quadruple, nor quintuple, etc., a square. These magnitudes are associated with different physical actions. Though each is distinct, they are nevertheless mutually related, and expressed by the general relationship, which the Pythagoreans called one geometric mean between two extremes. Thus, each particular square power is generated by a still higher species of power--the power that generates all individual square powers. This higher power can be given a clear mathematical expression as the geometrical relationships among the sides of the connected right triangles formed by a certain motion in a semi-circle. (See Figure 1.) While this construction expresses the effect of this power, as one unified action, it is not the power itself. The power is in the {idea} of that which has the power to generate all individual square powers. By giving the effect of this idea such an expression, our {mind's} power to control, and act on this {physical} power, is increased. But to know more of this idea, we must know not only what it can do, but what it cannot. This square power, while unlimited with respect to squares, is impotent to double a cube. The doubling, tripling, etc. of the volume of a cube, is the effect of a different species of power, which the Pythagoreans understood could be expressed as two geometric means between two extremes. As Archytas's construction demonstrates, the generation of this cubic power, can be given a mathematical expression by the proportions generated by a series of connected right triangles formed by the relative motion of two orthogonal semi-circles. (See Figure 2.) The relationships among the right triangles so produced, though changing, always express two geometric means between two extremes. This construction expresses not only the effect of the cubic power, but also the connection between the cubic and the square power, because here, the effect that generates the square powers, is itself generated as an effect, of the motion that generates the cubic. Even more importantly, the Archytas construction provides an insight, if seen from the standpoint of Cusa, into that still higher power, from which the square and cubic powers are themselves generated. While the specific magnitudes that correspond to the edges of squares and cubes are generated in the above construction as specific relationships among the lines forming the sides of right triangles, those relationships are determined not solely by lines, but by the connected effect of circular and rectilinear action. This can be seen clearly in the above cited figures. In figure 1, the relationships among the sides of the triangle are formed as an effect of the connection between the uniform motion of "P" along the circular arc which generates the non-uniform motion of "Q" along a line. But in figure 2, "Q" now moves both along a straight-line, {and} around a circular arc, while the motion of "P" is along both a circular arc {and} along the curve formed by the intersection of a torus and cylinder. Thus, it is a type of doubly-connected circular action that generates the rectilinear relationships that determine the effective changes in squares and cubes. Cusa, in {On the Quadrature of the Circle}, became the first to identify, and prove, that this circular action was an effect of an entirely different species of power, than the cubic and square powers. Leibniz identified this species of power as {transcendental}, as distinct from the lower species of powers (such as the cubic and square), which he called {algebraic}. http://www.rumormillnews.com/cgi-bin/forum.cgi?read=51180

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