Riemann for Anti-Dummies: Part 50 : The Geometry of Change

Riemann For Anti-Dummies Part 50


In his famous letter to Hugyens concerning his discovery of the significance of the square roots of negative numbers, G.W. Leibniz stated clearly his recognition that this investigation originated with the scientists of ancient Greece: “There is almost nothing more to be desired for the use which algebra can or will be able to have in mechanics and in practice. It is believable that this was the aim of the geometry of the ancients (at least that of Apollonius) and the purpose of loci that he had introduced….”

Understanding the implication of Leibniz’ statement is crucial to grasping the deeper significance of Gauss’ 1799 treatment of the fundamental theorem of algebra.

Leibniz’ statement will either baffle, or enrage, a modern academic, but such reactions only typify a broader social disease the inability, as LaRouche has repeatedly emphasized, to recognize the essential difference between human and beast. Like any disease, this one spreads through infectious agents that attack the defenses of the victim, causing the victim’s own system to act as an agent for the aggressor. The cure for such conditions is to strengthen the targeted population’s natural immunities, enabling them, not only to fight the disease, but to become permanently resistant to its effects. In this case, those natural immunities are the cognitive powers of the human mind. Hence, the therapeutic effects of pedagogical exercises and classical art.

What Leibniz, Gauss, and their ancient predecessors understood, is that the essential distinction between man and animal is the capacity of the human mind to reach behind the domain of the senses and discover those unseen principles that govern the changes perceived in the physical universe. However, being unseen, those principles can only be discovered through changes (motions) within the domain of the senses, which in turn give rise to paradoxes concerning the relationship of the seen to the unseen. Consequently, it is the coupled interaction between the seen and the unseen that must be comprehended. Physical motion gives rise to the willful motion (passion) of the mind from one state to a higher one. As Leibniz indicates, no formal system, such as algebra or Euclidean geometry, is capable of representing this characteristic of change that emerges from the interaction between the seen and the unseen. Only a geometry of change, such as the pre-Euclidean “spherics” of Thales and the Pythagorean school, the geometry of motion associated with Archimedes, Eratosthenes, and Apollonius, Leibniz’ infinitesimal calculus, or Gauss’ concept of the complex domain, has such power.

Just as the origins of the discovery of the complex domain begin in the ancient Mediterranean cultures of Egypt and Greece, so do the roots of its adversary. The mode of attack has been to induce the false belief that the physical world which is seen, and the immaterial world which is unseen, do not interact, but are hermetically separated. This belief is typified by the mystery cults of ancient Babylonian and Persian cultures. The Eleatics, (such as Parmenides and Zeno) sought to introduce this corruption into Greek culture, against Heraclites and the Pythagoreans, by insisting that change is merely an illusion and does not exist. (fn. 1)

Socrates made mincemeat of Parmenides’ Eleatic argument, so those who would today be called satanic, switched tactics, expressing the same evil intent through forms of Sophistry, such as admitting that change exists, but then arbitrarily defining change as the opposite of the Good and defining the Good as that which does not change and is not corrupted by change.

After Plato discredited the trickery of Sophistry, Aristotle, while distancing himself formally from the Sophists, nevertheless propounded the same evil in a new guise. For example, writing in his “Nichomachean Ethics”, Aristotle said :

“This is why God always enjoys a single and simple pleasure; for there is not only an activity of movement but an activity of immobility, and pleasure is found more in rest than in movement. But change in all things is sweet, as the poet says, because of some vice; for as it is the vicious man that is changeable, so the nature that needs change is vicious; for it is not simple nor good.”

Aristotle adopted this same view towards physical motion, stating in his “Physics” that motion originates only from within a body, and that irregular motion, because it contains more change, is of a lesser degree than regular motion, which is of a lesser degree than rest.

Like the Sophists and the Eleatics, Aristotle was not developing an original argument, but reacting against Plato’s repeated demonstration that the material and the immaterial are coupled:

” for this creation is mixed being made up of necessity and mind. Mind, the ruling power, persuaded necessity to bring the greater part of created things to perfection, and thus and after this manner in the beginning, when the influence of reason got the better of necessity, the universe was created.” (Timaeus).

And it is the power to gain knowledge of the universe through the interaction of the seen with the unseen, the temporal with the eternal, that is human nature. Change is a characteristic, not of viciousness and vice, but of perfection:

“But, now the sight of day and night, and the months and revolutions of the years, have created number, and have given us a conception of time and the power of enquiring about the nature of the universe; and from this source we have derived philosophy, than which no greater good ever was or will be given by the gods to mortal man…God invented and gave us sight to the end that we might behold the courses of intelligence in the heaven, and apply them to the courses of our own intelligence which are akin to them, the unperturbed to the perturbed; and that we, learning them and partaking of the natural truth of reason, might imitate the absolutely unerring courses of God and regulate our own vagaries. The same may be affirmed of speech and hearing;…Moreover, so much of music as is adapted to the sound of the voice and to the sense of hearing is granted to us for the sake of harmony; and harmony, which has motions akin to the revolutions of our souls, is not regarded by the intelligent votary of the Muses, as given by them with a view to irrational pleasure, which is deemed to be the purpose of it in our day, but as meant to correct any discord which may have arisen in the courses of the soul, and to be our ally in bringing her into harmony and agreement with herself; and rhythm too was given by them for the same reason, on account of the irregular and graceless ways which prevail among mankind generally, and to help us against them.” (Timaeus.)

The tension of this Socratic irony, of the unchanging principles of change, is the means by which man, and the universe as a whole, perfects itself. As Kepler notes in the “New Astronomy”, it is the tension from the discovery that the planetary orbits are not circular, “that gives rise to a powerful sense of wonder which at length drives men to look into causes.”

Remove that tension, as Aristotle, Euler, Lagrange, et al., do, and you excise from Man his human nature, rendering him defenseless against those oligarchical forces who seek to enslave him.

The Square Root of -1 and Motion