Riemann for Anti-Dummies: Part 69 : Change is All Ye Know On Earth — And in Heaven

Change is All Ye Know On Earth–And in Heaven

by Bruce Director

It is one thing to re-utter Heraclites’ fragment, “On those who enter the same river, ever different waters flow”, as the aphorism,“Nothing is constant but change,” as it has now become known, but it is entirely different, and significantly more important, to know the meaning of what Heraclites spoke. Even those who regard themselves as cognoscenti fail to understand this, as more often than not, they mistakenly consider change to be ephemeral. Competence in science, and more generally, sanity, depends upon being free from this error, and the mental obsessions that arise from it. However, such liberation comes not through formal abstract contemplations, which are intrinsically vulnerable to Eleatic-type sophistries, but, as Heraclites’ original formulation implies, through a pedagogical confrontation with the real, physically changing universe itself–a confrontation that all humans are, fortunately, blessed to experience. Continue reading Riemann for Anti-Dummies: Part 69 : Change is All Ye Know On Earth — And in Heaven

Riemann for Anti-Dummies: Part 68 : An Insider

August 28, 2006

An Insider’s Guide to the Universe

by Bruce Director

Though all humans are blessed to spend eternity inside the universe, many squander the mortal portion, deluded they are somewhere else. These assumed “outsiders” acquire an obsessive belief in a fantasy world whose nature is determined by {a priori} axiomatic assumptions of the deluded’s choosing, and an insistence that any experimental evidence contradicting these axioms must be either disregarded, or, if grudgingly acknowledged, determined to be from “outside” their world. Typical of such beliefs are the notions of Euclidean geometry, empiricism, positivism, existentialism, or that most pernicious of pathologies afflicting our culture today: Baby-Boomerism.

Continue reading Riemann for Anti-Dummies: Part 68 : An Insider

Riemann for Anti-Dummies: Part 67 : A View From The Top

August 28, 2006 (6:25pm)

The View From the Top

by Bruce Director

For more than three millennia the motion of a spinning top has been a source of great amusement for children, scientists, and philosophers. A careful examination of its motion provides insight into the underlying dynamics of the universe and exposes the fraud of absolute space. Plato took great delight in the embarrassment the top’s motion caused his Eleatic and Sophist adversaries, who argued that motion and change did not really exist. Nicholas of Cusa enjoyed the image of a spinning top as a beautiful expression of the universe’s self-boundedness. From a Riemannian standpoint, that same simple spinning top is still a great source of fun, unfolding Cusa’s concept into the domain of hypergeometries, and twisting Plato’s modern adversaries into gnarled knots. Continue reading Riemann for Anti-Dummies: Part 67 : A View From The Top

Riemann for Anti-Dummies: Part 66 : Gauss’s Arithmetic-Geometric Mean: A Matter of Precise Ambiguity

PREFACE

On the Subject of Metaphysics

By Lyndon H. LaRouche, Jr.

March 18, 2006

As I was reminded by a discussion-partner during the course of this past week, I am among the relatively few, surviving exceptions to an epidemic loss of actual knowledge of even the rudiments of metaphysics among presently living generations. Some hours after that conversation of the past week, Bruce Director’s long-promised draft introduction to the subject of the Arithmetic-Geometric Mean reached me on my current brief tour in Europe. This latest pedagogical by Bruce comes near to the core of a competent illustration of the practical meaning of metaphysics as such. I thought it useful to add the following bit of prefatory spice to Bruce’s work which follows immediately here. Continue reading Riemann for Anti-Dummies: Part 66 : Gauss’s Arithmetic-Geometric Mean: A Matter of Precise Ambiguity

Riemann for Anti-Dummies: Part 65 : On the 375th Anniversary of Kepler

November 16, 2005

On the 375th Anniversary of Kepler’s Passing

by Bruce Director

“In anxious and uncertain times like ours, when it is difficult to find pleasure in humanity and the course of human affairs, it is particularly consoling to think of the serene greatness of a Kepler. Kepler lived in an age in which the reign of law in nature was by no means an accepted certainty. How great must his faith in a uniform law have been, to have given him the strength to devote ten years of hard and patient work to the empirical investigation of the movement of the planets and the mathematical laws of that movement, entirely on his own, supported by no one and understood by very few! If we would honor his memory worthily, we must get as clear a picture as we can of his problem and the stages of its solution.” Continue reading Riemann for Anti-Dummies: Part 65 : On the 375th Anniversary of Kepler

Riemann for Anti-Dummies: Part 64 : Hypergeometric Harmonics

Hypergeometric Harmonics

by Bruce Director

In the same year that Riemann published his {Theory of Abelian Functions}, he also produced a companion piece of equal significance titled {Contributions to the Theory of Functions representable by Gauss’s Series F(a,b,c,x). The content of these works was the polished product of material Riemann developed in a series of lectures delivered at Goettingen University during the 1855-56 interval and through earlier discussions with Gauss and Dirichlet. Continue reading Riemann for Anti-Dummies: Part 64 : Hypergeometric Harmonics

Riemann for Anti-Dummies: Part 63 : Dynamics not Mechanics

Riemann For Anti-Dummies Part 63: Dynamics not Mechanics

by Bruce Director

Despite the prevailing popular opinion to the contrary, human beings are not mechanical systems. So, if you wish to begin to understand the science of physical economy, you must know the science of dynamics, as distinct from, and superior to, the Aristotelean sophistry called mechanics.

The distinction between dynamics and mechanics is not semantic. It is fundamental. Dynamics concerns causes. Mechanics concerns effects. Dynamics treats processes as a whole. Mechanics treats the interaction among some of its parts. They are as different as truth from rhetoric, ideas from words or, love from sex. Though this distinction arises directly through an investigation in the domain of physical science, its implications, as Leibniz himself emphasized, are universal. For without an understanding of the dynamics of situation, it is impossible to know anything about politics, history, science or art. Continue reading Riemann for Anti-Dummies: Part 63 : Dynamics not Mechanics

Riemann for Anti-Dummies: Part 62 : On the Continuum of the Discontinuum

On the Continuum of the Discontinuum

by Bruce Director

In the Laws, Plato’s Athenian stranger laments that the Greek people’s pervasive ignorance of the incommensurability of a line with a surface and a surface with a solid, had rendered them more like “guzzling swine” than true human beings. While the same lament can be sounded with respect to today’s society, modern citizens must also include an understanding of those higher transcendentals which are the subject of Riemann’s “Theory of Abelian Functions”, if they are to avoid the lamentable fate of Plato’s fellow Greeks. To rectify this condition, Plato recommended that these subjects should be learned in childhood, through play. Continue reading Riemann for Anti-Dummies: Part 62 : On the Continuum of the Discontinuum

Riemann for Anti-Dummies: Part 61 : To What End Do We Study Riemann’s Investigation of Abelian Functions?

March 5, 2005 (8:08pm)

To What End Do We Study Riemann’s Investigation of Abelian Functions?

by Bruce Director

On June 10, 1854, Bernhard Riemann shocked the world by stating the obvious:
For more than two thousand years scientists had accepted, as dogma, that the axioms of Euclidean geometry were the only foundation for science, despite the fact that science had been left in the dark as to whether these axioms had any physical reality. “From Euclid to Legendre, to name the most renowned writers on geometry, this darkness has been lifted neither by the mathematicians nor by the philosophers who have labored upon it.” The axioms of Euclidean geometry are, Riemann said, “like all facts, not necessary but of a merely empirical certainty; they are hypotheses; one may therefore inquire into their probability, which is truly very great within the bounds of observation, and thereafter decide concerning the admissibility of protracting them outside the limits of observation, not only toward the immeasurably large, but also toward the immeasurably small.” Continue reading Riemann for Anti-Dummies: Part 61 : To What End Do We Study Riemann’s Investigation of Abelian Functions?

Riemann for Anti-Dummies: Part 60 : The Power To Change, Change

The Power to Change, Change

New ideas, like people, come into this world naked. To effectively perform their mission, they must be provided with clothes. But unlike children who can speak for themselves, ideas must be dressed in words and images which, upon careful reflection, indicate how they were conceived. And though it would be a grave error to mistake a person’s substance for his or her outward appearance, the spirit of an idea (which also cannot be captured by a superficial account of its form) can, nevertheless, be evoked by that form’s animation. Yet there are those for whom the generation of such thoughts had seemed impossible, and to whom the very existence of such creations signifies a power they had denied could be. They focus only on the form, gossiping about its appearance, chiding its unconventionality or smothering it with an effusive description of its external features.

The history of science is replete with examples of such new creatures, which have been defended from such sophistries by their authors’ careful constructions: Archytas’s solution for the doubling of the cube, or Gauss’s complex surfaces, to name but two of those reviewed in previous installments of this series. Though such constructions arise as the unexpected solutions to specific problems, they embody the powerful new thoughts which made such solutions possible. Thus, their recreation evokes that more general principle, which, though forever connected to its origins, emerges as a new universal idea, and gains for all time, a universal increase in the cognitive power of man.

Now we turn our attention to another such creation, previously mentioned, but not yet adequately developed in this forum: the hypothesis which lies at the foundations of Riemann’s surface.

Riemann first presented to the world his new idea in his doctoral dissertation of 1851, and elaborated its implications in his 1854 habilitation lecture, his 1857 treatises on Abelian and hypergeometric functions, and his posthumously published philosophical fragments. From all these sources, and the historical context in which they were produced, we can reconstruct Riemann’s new idea as the solution to the unresolved physical paradoxes brought to the fore by C.F. Gauss’s extension of Leibniz’s calculus into the complex domain.

But from this exercise we acquire much more. As we form Riemann’s surfaces in our mind as images, we begin to recognize the quality of mind which produced this solution, and a more universal thought takes shape as well. From this point forward these images evoke in our minds that universal thought. Thus, we can bring Riemann’s creation to life, not simply as a solution to a formal mathematical problem, as its outward appearance is most frequently portrayed, but as an expression of an epistemological concept that has revolutionized human thinking.

What is a Surface?

When Riemann and Gauss speak of surfaces, they do not mean visible objects embedded in a linearly-extended Euclidean-type space. Rather, they speak of what Riemann identified in his 1854 habilitation paper as “multiply-extended continuous manifolds.” Such manifolds are not defined by a set of a priori axiomatic assumptions, but are concepts arising from an investigation of physical action determined by universal physical principles.

“In a concept whose various modes of determination form a continuous manifold, if one passes in a definite way from one mode of determination to another, the modes of determination which are traversed constitute a simply extended manifold and its essential mark is this, that in it, a continuous progress is possible from any point only in two directions, forward and backward. If now one forms the thought of this manifold again passing over into another entirely different, here again in a definite way, that is, in such a way that every point goes over into a definite point of the other, then will all the modes of determination thus obtained form a doubly extended manifold. In similar procedure one obtains a triply extended manifold when one represents to oneself that a double extension passes over in a definite way into one entirely different, and it is easy to see how one can prolong this construction indefinitely. If one considers his object of thought as variable instead of regarding the concept as determinable, then this construction can be characterized as a synthesis of a variability of “n + 1” dimensions out of a variability of “n” dimensions and a variability of one dimension.” (Riemann’s Habilitation Lecture)

Before continuing with this more general investigation of multiply extended manifolds, let us take as an example, the specific cases of the catenary and the catenoid.

As was illustrated in the last installment of this series, Leibniz, through his infinitesimal differential calculus, created a means to express physical action as the intended effect of a universal physical principle that is acting, universally, but differently, at every infinitesimal interval of time and space. In the case of the catenary, this is expressed by the fact that the visible shape of the catenary is determined by the changing effect of the physical principle of least-action on a chain supporting its own weight. Though this effect is different at each point along the catenary curve, these differences are determined by that general principle which integrates them so as to produce a physically stable chain.

Now, compare this case with that of the surface of a catenoid, formed, for example, by a soap film suspended between two circular hoops. (See Figure 1.)

Figure 1

Here the soap film is suspended between the hoops along catenary curves, but this family of catenaries are themselves integrated, along circular pathways, into a surface. Thus, like the catenary, the visible shape of the catenoid is determined by the changing effect of the physical principle of least action, but instead of those changes varying only along one curve, as in the case of the catenary, those differences occur within a rotational manifold of catenary curves. Therefore, the general principle that determines the stable shape of the soap film is expressed as the integrated effect that unites these two distinct but connected types of differences under one, higher principle.

From the standpoint of Riemann’s habilitation lecture cited above, the catenary comes under his concept of a simply extended manifold, whereas, the catenoid is a type of doubly extended manifold. Using Leibniz’s calculus, the changing effect of least-action can be expressed geometrically, by a type of animation called a differential equation. In the case of a simply extended manifold, the changing effect of least action is expressed as the changing curvature of a curve, and for a doubly extended manifold, the changing curvature of a surface. Thus, in both cases, the visible form of the action is expressed as a function of the changing effect of the universal principle of least action.

It must be underscored that in both cases, the visible shape of the curve or the surface is the effect of the characteristic {physical} “modes of determination” of the manifold. These modes of determination are not visible, but they are faithfully reflected in the visible effects. The challenge for science is to be able to measure these effects, and from their variations in the infinitesimally small, determine the principles that integrate them into a unified extended manifold of action.

As Riemann stated in his lecture notes On Partial Differential Equations and their Applications to Physical Problems:

“What has been shown to be a fact by induction, that differential equations form the actual foundation of mathematical physics, can also be shown a priori. True elementary laws can operate only in the infinitely small, and apply to points in space and time. But such laws are in general partial differential equations, and the derivation of laws for extended bodies and time periods requires their integration. Thus we need methods for deriving from laws in the infinitely small such laws in the finite, and, more precisely, derived with complete rigor, without permitting oneself omissions. Because it is only then that we can test them by way of experience.”

Riemann was basing his work in this regard on the previous achievements of Gauss, specifically, Gauss’s treatises on curved surfaces, conformal mapping and potential theory, some of which has been discussed in previous installments of this series and some of which will be discussed in future pedagogicals. For purposes of illustrating the concepts essential for this discussion, we return to the case of the catenoid.

Like the catenary, the catenoid is able to maintain its stable shape, because the tension exerted at every point is equalized by the changing effect of the principle of least action. Or, inversely, the shape of the catenary is the unique form which equalizes the tension exerted along the chain, by the effort of the chain to support its weight under the effect of gravity. But, unlike in the catenary where the tension is exerted back and forth along a curve, in the catenoid, the tension is exerted in an infinite number of directions radiating out from every point.

This changing effect that produces equal tension can be expressed, geometrically, by the changing curvature of the catenary curve, that same effect can be expressed, for the catenoid, by the changing curvature of the surface. But here the problem becomes more difficult, because the curvature of the curves within the surface is always different, depending on the curve’s direction.

Gauss solved this problem by defining several ways to measure the curvature of a surface. He recognized that at every point on any surface, though there be an infinite number of curves radiating along the surface from that point, one of them was the most curved and one of them was the least, and that these curves were always perpendicular to each other. (See Figure 2.)

Figure 2

Gauss measured the curvature of these curves, after Leibniz, by the size of the osculating circles to the curves at that point. (See Figure 3.)

Figure 3

Since the larger the osculating circle, the smaller the curvature, and vice versa, Gauss defined the measure of curvature at each point, as the inverse of the product of the radii of the extreme osculating circles. The characteristic curvature of a surface could thus be expressed by how this measure of curvature changed over the extent of the surface.

Riemann’s Surfaces

Whereas the catenoid is exemplary of a type of surface determined by a physical principle, such surfaces, in most cases, do not necessarily manifest a visible form. Nevertheless, a non- visible, but physical form can be clearly defined, as Riemann indicates, by generalizing the idea of a surface, to a concept of a multiply-extended manifold of physical action. From this concept of a physical manifold, a visible, geometrical form can be constructed, that faithfully reflects the characteristics of the physical manifold. For example, the surface of the Earth, as Gauss understood it, is the physical manifold that is everywhere perpendicular to the pull of gravity. This doubly extended manifold is measured by the changing effect that gravity has on the direction of a plumb bob. This surface, while not directly visible, is nevertheless physically determined, and thus, defines a physical geometric thought object, whose characteristics reflect the principles of physical action of gravity. Another example investigated by Gauss, is the physical manifold of the Earth’s magnetic effect, which, as a triply-extended manifold, is measured by the changing directions of a compass needle at different positions on the Earth. Here, too, the surface is not visible, but is, nevertheless, a physically determined surface.

Gauss called these types of non-visible physical surfaces “potential” surfaces, because they express the potential for action of a physical principle. These manifolds do not express what is {visible}. Rather they express what is {possible}, with respect to the physical constraints imposed by a universal physical principle on the manifold of action, just as the physical principle of least action imposes a certain characteristic curvature on a hanging chain or soap film. Thus, when investigating any physical process, it is necessary to form a concept of the manifold of physical principles in which that process is occurring. These principles impose a characteristic potential for action, which, Leibniz, Gauss and Riemann showed, can be expressed, geometrically as a characteristic curvature of the manifold. (Riemann’s Dirichlet Principle Riemann for Anti- Dummies Part 58.)

Following Gauss, Riemann recognized that in the type of least action physical manifold exemplified by the catenoid or Gauss’s potential surfaces, the curves of maximum and minimum curvature are harmonically related, which means that their mutual curvatures change at the same rate, in perpendicular directions. For example, in the catenoid, the size of the osculating circles of maximum curvature and minimum curvature increase or decrease at the same rate when moving from place to place on the surface.

It should again be emphasized, that this harmonic relationship is an effect, not a cause. The principle of least action is the cause, which, in the visible domain is reflected in this harmonic relationship.

Gauss had recognized in his Copenhagen Prize Essay, that this harmonic relationship, between the curves of maximum and minimum curvatures of a surface, is a characteristic of functions in the complex domain. Gauss applied this discovery by showing that the problem of mapping one surface onto another, so that all angular relationships are preserved, could be solved by finding a complex function that transformed one set of harmonically related curves into another.

Riemann extended Gauss’s idea by showing that from such complex functions, an entire class of physical manifolds can be expressed, whose “harmony would otherwise have remained hidden.” For Riemann, complex functions were a means to express a transformation from one physical manifold into another, as an effect of changing the curvature of the manifold’s determining characteristics. Further, as we will soon see from the implications of the essential features of Riemann’s surfaces, Riemann’s generalization of Gauss enabled him to express a new type of physical transformation, in which the potential for action is not simply transformed, but is increased in power.

Or, in other words, Riemann’s complex functions could express the power to change, change.

To pedagogically illustrate Riemann’s idea we will first introduce it, clothed in the words and images of geometry. From there, its more universal implications can more easily be brought to light.

This is a method similar to that used in investigating Archytas’s solution for doubling the cube or Gauss’s solutions to the fundamental theorem of algebra. In both cases the requirements of a specific geometric solution reveal that the original paradox was ontological, not mathematical, as it had first been presented. For example, the fact that in order to double the cube, Archytas had to produce the multiply connected manifold of physical action that generates, in a single act, the torus, cylinder and cone, demonstrates that the principle that generates the cube is outside the apparently spherical boundary of visible space.

Similarly, as Gauss showed in his proof of the fundamental theorem of algebra, the apparent mathematical paradox of the ?-1, was, in reality, not a problem of algebra, but a problem of algebraists, who were fixated on a false conception of the physical universe. As Gauss noted, the paradox of the ?-1, is not a formal mathematical one, even though it may have appeared in that form. Rather these paradoxes actually reflect “the deepest questions of the metaphysics of space”, as Gauss himself said.

Riemann exploited one such mathematical paradox to force to the surface his new conception concerning manifolds of physical action.

That paradox arises , in its geometrical context, when investigating complex functions represented, as Gauss did in his Copenhagen Prize Essay, as conformal transformations of one surface onto another. In some cases, these mappings are quite straightforward, such as the stereographic projection of a sphere onto a plane, or the conformal mapping of the spheroidal shape of the Earth onto a sphere. In these cases, there is a one to one correspondence between every point of one surface and every point of the other.

However, in some complex functions (most emphatically, all those transcendental functions that correspond to physical action), this one to one relationship does not occur. For example, the complex squaring function, maps the points of one surface onto another twice. Or, in the case of a complex cubic function, the points of one surface are mapped onto another three times.

This can be illustrated with an animation (See Figure 4a.) representing a complex cubing function.

Figure 4a

Since the complex cubing function triples the angle of rotation, a 120 degree rotation of the surface of the circular disk on the left maps to a full rotation of the surface of the circular disk on the right. Likewise, the second 120 degree rotation of the surface of the circular disk on the left also maps to a full rotation of the circular disk on the right. So to with the third 120 degree rotation. Thus, each point of the disk on the right, corresponds to three points on the disk on the left. This inversion can be seen more clearly in Figure 4b which maps the disk on the left as a function of the action of the disk on the right.

Figure 4b

Thus, there appears to be a mathematical ambiguity because one clearly defined action produces three distinct effects, and there is no formal algebraic way to distinguish one of these effects from the other. However, as Gauss did in the fundamental theorem of algebra, Riemann, applying Leibniz’s method of analysis situs found a physical geometrical form, which uniquely characterized the quality of physical action, that underlies such multi-valued complex functions.

He rejected the method of Cauchy, who sought to define an algebraic calculation to distinguish one effect from the other. Instead, Riemann focused on the unique points of the mapping that were not multi-valued the singularities. As Gauss did in his solution to the fundamental theorem of algebra, Riemann created a construction whose geometric characteristics expressed the organizing principle of the manifold by the relationship between what the singularity and everything else in the manifold.

To visualize Riemann’s idea, we can use the above example of a cubic function. For such a function, Riemann conceived of each of the three different effects produced by the cubic rotation, to be different branches of one action. He represented each as a different layer on a surface of three sheets. These layers were connected to each other at the point of singularity, where their values all coincided. He called this point a branch point. (See Figure 5.)

Figure 5

Thus, in our example, the effect of the first 120 degree rotations are spread over the first layer, the effect of the second 120 degree rotation is spread over the second layer, and the effect of the third 120 degree rotation is spread over the third layer. But since the action is continuous so must be the effects. To express this, Riemann connected these layers to each other by making a cut emanating outward from the branch point, which he called a branch cut. Along this cut, he connected the first layer to the second, the second to the third and the third back to the first. (See Figure 6.)

Figure 6

In this way, Riemann expresses a physical relationship in which a single continuous action that produces a multiplicity of effects are integrated into one continuous manifold.

It is important to underscore, that though we can create a vivid image in our mind of Riemann’s surface, it is impossible to physically construct it in visible space. Nevertheless, Riemann’s imagined surface, because it reflects an ontological principle of action, produces a very real effect. From it we can gain a greater insight into the relationship between a manifold of universal physical principles and their physical effects.

The ontological implications of Riemann’s surfaces begin to shine through if we investigate a slightly more complicated example. The next set of animations illustrate Riemann’s surface for a fourth power algebraic function, which produces a Riemann’s surface with four layers. The sets of perpendicular lines on the left and quartic curves on the right represent the effect of the transformation on the manifold’s harmonically related curves of maximum and minimum curvature.

In the first animation, these four roots are represented by four colored dots, that are shifted, in Riemann’s surface to lie on top of each other at the branch point. (See Figure 7.)

Figure 7

Now, think of this Riemann’s surface as a type of manifold of physical action, and investigate the effect of this transformation on different actions. In Figure 8a we see the effect within this manifold on a circular rotation around one of the roots.

Figure 8a

The original action is depicted in the left panel and its effect is depicted on the right. In Figure 8b, we see the same action depicted from a side view of Riemann’s surface, showing that the pathway loops around the one layer containing the root and then continuing, without looping through the rest of the layers.

Figure 8b

In Figure 8b we see top down view of 9b. Compare this with the result in 8a.

In Figures 9a, 9b, and 9c we see a similar representation of the same type of action but around a different singularity, and with a similar effect.

Figure 9a

Figure 9b

Figure 9c

In Figures 10a, 10b, 10c we see the effect on a closed pathway that encompasses no singularities.

Figure 10a

Figure 10b

Figure 10c

But, in Figures 11a 11b, and 11c we see a simple closed pathway that circles two singularities.

Figure 11a

Figure 11b

Figure 11c

Here the effect of the transformation is to produce a double loop around the branch point. Thus, within this type of manifold, an action that encompasses a greater number of singularities produces a greater effect. In other words, Riemann’s surface expresses a type of multiply extended continuous manifold in which the power of any action is a function of the number of singularities encompassed by that action.

In Figures 12a, 12b, 12c we see this power is increased once again when the pathway of action encompasses three singularities.

Figure 12a

Figure 12b

Figure 12c

And, in Figures 13a, 13b, 13c, a further increase in power is represented by a loop around 4 singularities which is transformed into a quadruple loop around the branch point.

Figure 13a

Figure 13b

Figure 13c

Here we have reached the limit of the power of this particular type of complex function. To increase the function’s power, a new principle must be introduced, which will add a new singularity to the manifold of physical action. With the incorporation of this new singularity, into the manifold, a greater potential for action is achieved.

This defines a still higher type of transformation, one that adds new singularities to the manifold of action. But in these algebraic examples, these singularities are added one by one. Riemann showed, than Abel’s extended class of higher transcendental functions, when expressed on Riemann’s surface, express a type of transformation that increases the rate and the density at which singularities can be added.

This part of the investigation will have to be continued until next time. But, at least we have begun to scratch the surface.

Riemann for Anti-Dummies: Part 59 : Think Infinitesimal

Think Infinitesimal

by Bruce Director

“It is well known that scientific physics has existed only since the invention of the differential calculus,” stated Bernhard Riemann in his introduction to his late 1854 lecture series posthumously published under the title, “Partial Differential Equations and their Applications to Physical Questions”. For most of his listeners, Riemann’s statement would have been fairly straight forward, for they understood the physical significance of Leibniz’s calculus as it had percolated over the preceding sesquicentury through the work of Kaestner and Gauss. A far different condition exists, however, for most of today’s readers, whose education has been dominated by the empiricism of the Leibniz-hating Euler, Cauchy and Russell. While such victims might find the formal content of Riemann’s statement agreeable, its true intention would be as obscure to them as the Gospel of John and Epistles of Paul are to Karl Rove and his legions of true believers.

The empiricist will not understand Riemann’s statement, for the simple reason that what he associates with the words “differential calculus” is a completely different idea than what Riemann and Leibniz had in mind. To the victim of today’s empiricist-dominated educational system, the infinitesimal calculus concerns only a set of rules for mathematical formalism. But to the scientist, the infinitesimal calculus is a kind of Socratic dialogue, through which man transcends the limitations of sense-perception and discovers those universal principles that govern all physical action.

The empiricist rejects Leibniz’s notion, because he accepts Aristotle’s doctrine that “physics concerns only objects of sense”, whereas Plato, Cusa, Leibniz and Riemann emphasized, physics concerns objects of {thought}. These thought-objects, or “Geistesmassen” as Riemann called them, refer to the universal principles which {cause} the objects of sense to behave the way they are perceived to behave. Not being directly accessible to the senses, such principles appear to come from “outside” the visible world. However, a great mistake is made if one concludes from this, as the sophists do, that these principles come from outside the universe itself. In fact, these principles, being universal, are acting everywhere, at all times, and in every “infinitesimal” interval of action, osculating the objects of sense as if tangent to the visible domain.

It is this relationship between the observed motions of the objects of sense, and the universal principles acting everywhere on them, that Leibniz’s differential calculus is designed to express. Through it, a universal principle, as it is seen and unseen, is enfolded into a single thought, showing us what is known, and indicating to us what is yet to be discovered. A scientist who turns away, under Aristotle’s, Sarpi’s, or Russell’s, influence, from these objects of thought, to objects of mere sense, is acting as if his own mind has ceased to exist, which, in fact, it has.

Just as Riemann correctly asserts, that scientific physics began with the invention of the differential calculus, it can be justly stated that the differential calculus began with Cusa’s excommunication of Aristotle from science. While it is true that some of the methods of Leibniz’s calculus were beginning to develop in the work of Archytas and Archimedes, this development was arrested when Aristotle’s doctrines became hegemonic in European culture, following the murder of Archimedes by the Romans. Cusa reversed this disaster and reoriented European science away from its obsession with objects of sense, and back to the Pythagorean/Socratic focus on the idea.

Cusa insisted that perception is not caused by sensible things, but that things are sensible because the mind has the power to sense. In turn, the mind is able to sense, because it possesses a still higher faculty of rationality; and it is able to rationalize because it possesses a still higher faculty of intellect; and it is able to intellectualize because man is created in the infinitesimal image of God.

From this standpoint, Cusa rejected Aristotle’s sophism that less change equals greater perfection, which made God a tyrannical force who keeps the world perfect by opposing change. Instead, Cusa recognized that the capacity for change in the physical universe, and in the human mind, indicated the perfectability of both, and that it was God’s intention to perfect his Creation through the cognitive powers of Man. Thus, it is the power of the mind to perceive change, not objects, through which Man relates to the physical world and increases his knowledge of, and power in, the universe.

Having freed science to recognize change as primary, Cusa concluded that all physical action must be non-uniform, which Kepler experimentally validated with his discovery that the principle of universal gravitation produces harmonically-related elliptical orbits. As Kepler insisted, the observed changing motion of the planet in its orbit, rather than an apparent deviation from Aristotle’s illusory idea of fixed perfection, was, in reality, the intended effect of the principle of universal gravitation, which is acting, universally, but whose effect differs, in every infinitesimally small interval of action. In this respect, Kepler likened the principle of universal gravitation to an idea, (using the Latin word {species} to describe it), but distinguishing it from a human idea, because it lacked the quality of willfulness unique to human cognition. Man could grasp this idea, Kepler understood, by forming a concept (thought-object), which expressed the physical action as a consequence of a universal intention, analogous to, as Cusa emphasized, the way human action is the consequence of human intention. While Kepler made significant progress in creating geometrical expressions for this relationship, he recognized the need for a new form of metaphor, and demanded that future generations make further progress to this end.

It was Leibniz who defined the required concept, to which Riemann refers as the beginning of scientific physics. Leibniz recognized that what was required was a new form of mathematical expression, one that expressed the relationship between the universal principle and its changing effect on the observed motion. Most importantly, this new expression must work in reverse, because that is the way it is encountered in scientific investigation. That is, though the effect of the principle is perceived through the motion, merely describing what is observed states nothing about the principle. To scientifically know the cause of the motion, it is necessary to express the motion as an effect of the principle.

To grasp this thought, Leibniz utilized a form of investigation that had been previously developed by Cusa: the relationship of maximum and minimum. As Cusa specified, the maximum and minimum coincide in God, but in the created world the maximum and minimum appear as opposites. Thus, to know any physical process, it is necessary to have a concept through which the opposite extremes of that physical process are recognized as the maximum and minimum effects of a single, unified, intention. For example, in every interval of an elliptical orbit, the planet’s motion is different at the two extremes of that interval, no matter how small an interval is taken. There are but two exceptions. One is the entire orbit, the other is the moment of change itself, which comprise, respectively, the maximum and minimum effect of the principle of universal gravitation on the planet. In the minimum, the universal principle’s effect is always different, but it is differing according to a well-defined principle. The mathematical expression of this differentiation, Leibniz called “the differential” which always exists integrated into the whole action. In the latter form, its mathematical expression, was called by Leibniz, “the integral”.

From this relationship, Leibniz invented a type of animation, which he called “differential equations”, in which the maximum effect is expressed as a function of the minimum. As Riemann noted, this put science on a completely new footing, for in experimental investigations it is the minimum expression that is measured, from which the maximum must be determined, as, for example, in the case of Leibniz’s and Bernoulli’s determination of the catenary, Gauss’s determination of the orbit of Ceres, or, Gauss’s or Riemann’s investigations in geodesy, geomagnetism, electromagnetism and shock waves. With Leibniz’s differential calculus, such investigations could be undertaken, for the first time, with the necessary epistemological rigor.

Of course, Leibniz’s differential equations do not express the principle directly. But, they can express the changing effect of the principle at every moment. On this basis, the principle, can be known by inversion, as that idea which produces the effect expressed by the differential equation. To emphasize the point: the differential equation is not the principle, but it expresses the ever changing footprints that the principle leaves on the visible domain. While this description, clothed in either words or geometry, is necessarily ironic, the thought-object to which it refers, is recognized, in the mind, with absolute precision.

It is crucial to emphasize that Leibniz’s calculus is a mathematical expression of a physical idea. As is obvious with respect to physical action, the differential and the integral express the minimum and maximum effect of the same universal physical principle. Yet the empiricists attacked Leibniz’s calculus by abstracting it from physics and presenting it only as a mathematical formalism. They produced through sophistry, an apparent mathematical paradox, by treating both as if they existed separately from the physical principle they expressed. From this formal mathematical standpoint, the sophist argued, the differential does not exist, because in the moment of change, the time elapsed and distance traversed are both, formally, null. From this, the sophistry continues, the integral can’t be expressed, because it is the sum of infinitely many null magnitudes.

Leibniz countered this sophistry by always insisting on the physical nature of his calculus. In a 1702 letter to Varignon, he posed the following paradox to the algebraists:

Construct two similar triangles from the intersection of two lines. (See Figure 1.)

In the construction, the legs of the large triangle are in the same proportion to each other, as the legs of the smaller one. Now, move the oblique line in a motion parallel to its original position. (See animated Figure 2.)

Under this motion, the smaller triangle gets smaller while the larger triangle gets bigger, but the proportionality of their sides remains the same. At some moment, the smaller triangle passes through the point of intersection of the two original lines, emerging, in the next moment, on the other side, to begin growing again.

The algebraists insisted that, at the moment that the small triangle passed through the point, its sides both appeared to be null, and so it is impossible to express their proportion, or more absurdly, that their proportion ceased to exist at that moment. Leibniz countered that the triangle passed through that point as the result of a physical motion, intended to maintain the proportionality of the sides of the triangles. Consequently, it was the motion that produced the constant proportionality, as its intended effect. At the moment the small triangle passed through the point, the motion did not cease and, thus, neither did the proportionality of the triangles. That proportionality reflected a principle of physical action which is known, in the mind, by a thought-object associated with a certain intention. The point is but a moment in the motion. It does not exist without respect to the physical action. Only when the mathematical expression is separated from the physical action, does the algebraic contradiction appear. The appearance of such a contradiction may indicate a problem with the thinking of the empiricist, but the problem lies only there, not in the physical universe itself.

Conversely, to insist that the algebraic contradiction has an ontological significance, is to induce a state of mental disassociation in the mind of the scientist. This is precisely the intention of Euler, Lagrange, and especially Cauchy, who replaced Leibniz’s idea of the infinitesimal with Cauchy’s idea of the limit. Cauchy argued that the limit removed the algebraic contradiction of the infinitesimal. But in doing so, Cauchy was actually inducing insanity, by removing the connection of the mind to the physical universe in which it exists. This, of course, was his intended effect.

That the cognitive capacity of the mind was the real target of the oligarchy’s attack on Leibniz’s calculus, was confessed to a popular audience by Richard Courant and Herbert Robbins in their English language 1941 book, {What is Mathematics}:

“…the very foundations of the calculus were long obscured by an unwillingness to recognize the exclusive right of the limit concept as the source of the new methods. Neither Newton nor Leibniz coudl bring himself to such a clear-cut attitude, simple as it appears to us now that the limit concept has been completely clarified. Their example dominated more than a century of mathematical development during which the subject was shrouded by talk of `infinitely small quantities’, `differentials’,`ultimate ratios’ etc. {the reluctance with which these concepts were finally abandoned was deeply rooted in the philosophical attitude of the time and in the very nature of the human mind}” (emphasis added, poor punctuation in the original bmd.).

The empiricist sees objects in motion and imagines them to move in a space that is as empty as he believes his own mind to be. A scientist envisions a manifold of universal physical principles, embodied as animated objects of thought that enliven the objects before his eyes. To the former, change is an annoying inconvenience that disrupts his ultimately futile attempts to maintain his accepted axiomatic-formal structure. To the latter, change is the happy indicator of the moving effect of universal principles acting, universally, yet differently, at all infinitesimal intervals of time and space.

The Differential Calculus Animated

The most effective pedagogical means to illuminate Leibniz’s concept of the differential calculus, is through a series of animations that illustrate its application from Kepler, to Huygens, to Leibniz, to Gauss, to Riemann. In what follows, we rely on the animations to do most of the talking, with these written words providing only the barest of stage directions:

Animated Figure 3: Kepler’s principle of equal areas. Kepler conceived of the orbital path as the changing effect of the principle of universal gravitation, which varied inversely to the distance between the sun and the planet. Kepler understood the motion in any interval to be the sum of the infinitely many changing radial distances within that interval, which reflected the planet’s motion at every moment. He could not calculate this sum precisely, but he recognized the result corresponded to the area swept out (See animated figure 3a.),

which he measured through his famous method of the three anomalies. (see animated figure 3b.)

Kepler’s method of calculation led to the paradox which prompted Leibniz to develop his concept of the differential and the integral.

Animated Figure 4: Huygens attempted to tackle the problem of non-uniform motion by expressing one non-uniform motion as a function of another, by the method of involute and evolute. In animated figure 4a the yellow curve is formed by the motion of unwrapping the white string from blue curve.

The yellow curve is called the involute. The blue curve is called the evolute. Thus, the changing curvature of the involute is a function of the changing curvature of the evolute. The white string is always perpendicular to the involute and always tangent to the evolute. Because of this, the involute is the envelope of circles whose centers all lie on the evolute. (See animated figure 4b, 4c.)

In other words, these circles are everywhere tangent (osculating) to the involute, and their radii are everywhere tangent to the evolute. Thus, the curvature of the involute expresses the effect of the principles acting tangentially on the evolute and vice versa.

Now, instead of thinking about these osculating circles being formed by the curves, think of the curves being formed by the osculating action of a circle whose size and position are changing according to a principle of motion. (see animated figure 4d.)

In this way, the curves are more truthfully understood, as the intended effect of a principle of change that is acting, everywhere tangent, to their visible expression.

Huygens used this relationship to build his famous pendulum clock, on the principle that the cycloid had both the property that its involute was another cycloid, and that it was the curve of equal-time for a body falling according to gravity. (See animations 4e,4f,4g,4h.)

Animated Figure 5: While Huygens’s method of the involute and evolute expresses non- uniform motion, it relies on a purely mechanical procedure, instead of expressing a principle of change directly. Leibniz solved this problem by expressing this principle of change through differential equations. To measure the differential, Leibniz projected the changing action in the infinitesimally small into the visible domain, in a manner similar to Plato’s cave metaphor. To do this, Leibniz generalized the investigations of Fermat by defining a series of functions that depended on the changing curvature that resulted from the physical action. (See figure 5a.)

In particular, Leibniz investigated the motion of the subtangent, whose length is a function of the direction of the tangent, which in turn is function of the changing curvature. Leibniz considered the triangle formed between the point of tangency , the intersection of the ordinate with the axis, and the intersection of the tangent with the axis, as a projection into the visible domain, of the changing action in the infinitesimally small.

To get an intuitive grasp of this method, take the example of the parabola (See animated figure 5b )

which illustrates the changing motion of the parabola’s subtangent. Fermat had shown that the subtangent of the parabola is always twice the distance of the abscissa to the vertex. From this standpoint, the parabola can be entirely defined as the effect of a principle of change. Instead of thinking of the subtangent as a function of the parabola, think of the effect of the parabola as a function of a subtangent which is always bisected by its vertex, which, in turn, is defined as the point at which the subtangent is at its minimum. This way of thinking is a very elementary, pedagogical description of a “differential equation”.

From this standpoint, Leibniz was able to discover the existence of physical principles that were not expressed by the visible form of the motion, but {were} expressed through the characteristics of differential equations. For example, the visible shape of the exponential curve can be defined as the curve produced by a continuous motion that is arithmetic in one direction, and geometric in a perpendicular direction. (See animated figure 5c.)

Yet, there is a unique property to this action, which Leibniz found through his infinitesimal calculus. The exponential curve is the curve whose subtangent is always constant. (See animated figure 5d.)

In other words, the exponential curve is the curve whose characteristic of change is the same as itself!

This discovery highlights a crucial distinction between Leibniz’s method and Huygens’. With respect to the involute and evolute, the cycloid is the curve whose change is the same as itself. But from the more general method of Leibniz, it is the exponential curve that exhibits a characteristic of self-similarity. The importance of this distinction is underscored by Leibniz’s discovery of the relationship between the exponential curve and the catenary, which highlights the fact that the catenary expresses, more universally, the principle of least-action, not the cycloid.

As a result of this investigation, Leibniz discovered an entirely new type of transcendental function. He realized that even though every exponential had constant subtangents, the absolute length varied with the constant of proportionality. Leibniz found the existence of a new number, which he called, “b”, which forms the exponential curve whose subtangent is equal to unity. (See figure 5e.)

(Euler later changed the name of this number to “e”, whose historically misleading and quasi-blasphemous moniker it still wears today.)

With this new power to investigate physical action as the effect of a principle of change, new characteristics are brought to the surface that otherwise had remained hidden. For example, when the apparently uniform circular action is investigated from the standpoint of change of its subtangent, the existence of two discontinuities emerge, that otherwise were not visible. This is where the subtangent becomes infinite, which correspond, in Gauss’s idea, to the \/-1 and -\/-1. (See animated figure 5f.)

Riemann for Anti-Dummies: Part 58 : Bernhard Riemann

Bernhard Riemann’s “Dirichlet’s Priniciple”

by Bruce Director

In his revolutionary essay of 1857, {Theory of Abelian Functions}, Bernhard Riemann brought to light the deeper epistemological significance of the complex domain, through a new and bold application of a principle of physical action which he called, “Dirichlet’s Principle”. Riemann’s approach, combined with what he enunciated in his habilitation dissertation of 1854, not only ushered in a revolution in scientific thinking: it ignited a counter-reaction as fierce as the one launched, for the same reasons, against Cusa, Kepler, Fermat and Leibniz by the Venetian-British controlled empiricist school of Gallileo, Newton, Euler and Lagrange; a counter-reaction that continues to rage to this day, and with implications that reach far beyond the specific setting of Riemann’s 1857 paper. Despite the volumes that have been written on this subject, from Riemann’s time to ours, an honest examination of the history of the matter reveals that, just as Gauss demonstrated the fraud of Euler, Lagrange and D’Alembert in his 1799 proof of the fundamental theorem of algebra, Riemann was right, and his critics, like today’s Straussian controllers of Bush and Cheney, were malevolent frauds.
We cannot know for sure whether, when Riemann chose to call this method an application of “Dirichlet’s Principle”, he expected to provoke the reaction he received, or if he was merely stating what would have been obvious for anyone within the extended network of Abraham Kaestner’s students. Nevertheless, it is fortunate for us that he used that name, as it enables us to fairly accurately reconstruct, not only the scientific origins of Riemann’s thought, but the historical-political process from which it arose.

Lejeune-Dirichlet

Johann Peter Gustav Lejeune-Dirichlet was a pivotal figure in early 19th Century science. Born in 1805 to a family of Belgian origin living near Aachen, his early education took place in Bonn. At the age of 16, with a copy of Gauss’s {Disquisitiones Arithmeticae} under his arm, he went to Paris to audit the lectures at the College de France and the Faculte des Sciences. After a year, Dirichlet became employed as a tutor by General Maximilien Sebastien Foy, a republican member of the Chamber of Deputies, who introduced him to Alexander von Humboldt. After Foy’s death in 1825, von Humboldt recruited Dirichlet back to Germany, arranged for him to get a degree (even though Dirichlet refused to speak Latin), and eventually succeeded in obtaining for him a professorship at the University of Berlin. There, in addition to meeting, and marrying, Moses Mendelssohn’s granddaughter, Rebecca, (the sister of the composer Felix Mendelssohn), Dirichlet developed a fruitful collaboration with Jacobi and Jakob Steiner , including a tour with both to Italy in 1843 under Alexander von Humboldt’s sponsorship.
In 1847 Riemann arrived in Berlin to study with Dirichlet, Jacobi and Steiner, having spent the previous two years studying with Gauss. In 1849 he returned to Goettingen to complete his studies and in 1851, under Gauss’s direction, published his doctoral dissertation, {The Foundations for a general Theory of Functions of a Complex Variable Magnitude}, in which he first applied his principle, without mention of Dirichlet. When Gauss died in 1855, Dirichlet was appointed his successor, bringing himself back into contact with Riemann, who, just seven months earlier, had received permission to teach, after delivering his habilitation lecture, {On the Hypotheses which Lie at the Foundations of Geometry}. In 1857, Riemann published the {Theory of Abelian Functions} in which, for the first time, he identified the principle on which his new theories were based, as “Dirichlet’s Principle”. Two years later, Dirichlet died, and Riemann, now 33 years old, was appointed to his chair, a position he held until his own, unfortunately too early death, only seven years later.

The Potential

What Riemann called “Dirichlet’s Principle”, arose out of Gauss’s application of the complex domain to his investigations in geodesy and terrestrial magnetism; the former organized in collaboration with Schumacher beginning in 1818, and the latter initiated by Alexander von Humboldt in 1832. Both projects had enormous practical benefits. Each produced detailed maps of their respective physical effects which were vital for infrastructure development, and Humboldt’s project organized, for the first time, an international collaborative network of scientists who would impact the development of the physical economy from the Americas to Eurasia for generations. But Gauss recognized that both projects posed deeper epistemological questions for science. Writing in his {General Theory of Earthmagnetism} in 1839, Gauss said a complete and accurate map of the observations is not, in itself, a proper goal for science. “One has only the cornerstone, not the building, as long as one has not subjugated the appearances to an underlying principle.” Citing the case of astronomy as an example, Gauss said that mapping the observations of the apparent motion of the heavenly bodies onto the celestial sphere is just a beginning. Only once the underlying principle of gravitation is discovered, can the actual orbits of the planets be determined.
Gauss recognized that the first step in geodesy and geomagnetism is the measurement of changes in the effect of both phenomena on the measuring instruments. In the case of geodesy, that meant changes in the direction of a plumb bob, or plane level, as those changes are mapped onto the celestial sphere. The case of geomagnetism is more complicated. Here he was measuring changes in the direction of a compass needle, with respect to three directions and time. The general question was: what is the characteristic nature of the principle of gravitation or geomagnetism that would produce these apparent effects? The specific task was: how, from these infinitesimally small measured changes in the apparent effects, can that general characteristic be determined?
It is the second question that brings us more directly into contact with what Riemann called “Dirichlet’s Principle”. However, the task of understanding Dirichlet’s Principle will be made much easier if we first look at the elementary, but congruent case of the catenary.
The relevant focus for this discussion is the devastating rebuke which Leibniz and Bernoulli delivered to Gallileo and Newton over the case of the catenary. Gallileo had insisted that all that need, or could, be known about the catenary was a description of its visible shape. On the other hand, Leibniz and Bernoulli insisted that the shape of the catenary was merely the visible effect of an underlying physical principle, and the correct shape could not be determined until the underlying principle was known. As was developed in previous installments of this series, Leibniz and Bernoulli determined the characteristic nature of that principle, by determining first, the changing physical effect of that principle in the infinitesimally small, and then, by inversion, the overall characteristic of the principle. The result was Leibniz’s discovery that the shape of the hanging chain reflected the least-action effect of the principle of universal gravitation, and that this effect could be expressed geometrically as the arithmetic mean between two contrariwise exponential functions.
It is of extreme importance to emphasize that we are speaking here of the physical hanging chain, not a formal mathematical expression. In a formal mathematical expression the exponential curves have no boundary. The physical hanging chain does–the positions of the hanging points. Consequently, the specific shape of the chain is determined by the position of the hanging points relative to the weight and length of the chain. If the positions of the hanging points change, the position of every link in the chain also changes, but always in accordance with the relationship cited above. In other words, as the boundary conditions of the physical chain change, so does the chain’s specific path, but that path’s general form, required by the principle of least-action, is always a catenary. It will never become a parabola or any other curve. (See Figure 1.)


This example illustrates an aspect of the method that Leibniz originally called “analysis situs”, or what Gauss and Carnot later called “geometry of position”, that is relevant to an understanding of Riemann’s “Dirichlet’s Principle”. The positions of the individual links in the chain are a function of the relationship of the boundary conditions (position of the hanging points relative to the length of the chain) to the characteristic curvature of the principle of gravitation, and not by a pair-wise relationship among the links themselves. In other words, the position of any individual link is not determined by a distance to the right or left and a distance up and down from its neighbors, as the Cartesians and Newtonians would insist. Rather, the position of each link is a function of the characteristic of change of the physical action as a whole. Any change in the boundary conditions changes the position of every link, { as a whole} in conformity with the least-action principle of the catenary. Thus, the unseen physical principle’s effect in the visible domain is expressed by the characteristic of change the principle of least-action demands. This is what determines the specific positions of the links. That is, position is a function of change.
Gauss recognized that the principles underlying geodesy and geomagnetism could be understood by an extension of Leibniz’s method. He rejected the popularly accepted, but provably false method of Newton, that attempted to explain these phenomena as the pair-wise interaction of material bodies, according to the algebraic formula of the inverse square. (See {Riemann for Anti-Dummies–Part 53 “Look to the Potential”}. Instead, Gauss insisted that these phenomena, like the catenary, must be understood as one process, and that the local variations in the position of the plumb bob or the compass needle are a function of the characteristic of the principle governing the phenomenon as a whole. That whole, Gauss called “the potential”, which is the Latin equivalent of the Greek, “dynamis” or Leibniz’s “kraft” or “vis viva”. Gauss invented the idea of a “potential function” to express the least-action effect of the physical principle over an area or volume, in a similar, but extended, manner to that used by Leibniz to express the effect of gravity in producing the curvature of the hanging chain. To accomplish this, Gauss extended Leibniz’s idea of a function into the complex domain.
This transformed Leibniz’s functions, which characterized a single minimal pathway, into Gauss’s “potential function,” which characterized a whole class of minimal pathways: in effect, a function of functions. In other words, if Leibniz’s catenary is understood to be a minimal pathway determined by one set of two functions, Gauss’s potential function takes the next step: to a function that unifies two (or more) {sets} of functions. Riemann would later show that these sets of minimal pathways implicitly defined minimal surfaces, as, for example, the catenoid formed by a soap film suspended between two circular rings.
These sets of functions are not arbitrary. They are related by a special type of relationship, called by the descriptive names, “spherical functions”, or “harmonic functions.” An harmonic or spherical function is a set of orthogonal functions all of whose curvatures are changing at the same rate.
This can be most easily illustrated pedagogically with some geometric examples. A set of concentric circles and radial lines comprises an harmonic function because both the circles and the radial lines intersect orthogonally and both have constant curvature. (See Figure 2)

A more illustrative example is a set of orthogonal ellipses and hyperbolas. (See Figure 3.)

To get an intuitive grasp of their harmonic relationship, carry out the following thought. Each ellipse is associated with a confocal orthogonal hyperbola. Beginning at the point where both curves meet the axis, create in your mind a connected action that moves simultaneously on both curves. (See Figure 4.)

Note that as the curvature on the hyperbola becomes less curved, so does the curvature on the corresponding ellipse, and at the same rate.
Thus, harmonic functions relate two sets of different curves such that the rate of change of their respective curvatures is always equal. (Using Leibniz’s calculus, we could calculate this relationship precisely, but an intuitive understanding is sufficient for present purposes.)
Furthermore, a set of harmonic functions need not be familiar curves such as circles, lines, ellipses, or hyperbolas. In fact, very complicated sets of functions can be harmonic. (See Figure 5.)

By contrast, a set of circles and hyperbolas is not harmonic, because the curvature of the circle is constant, while the curvature of the hyperbola is changing. Consequently, the two sets of curves are not orthogonal. (See Figure 6.)

Gauss recognized that Leibniz’s principle of least-action with respect to the surfaces and volumes encountered in phenomena like terrestrial gravitation and magnetism, could be expressed by harmonic functions. One set of curves of the harmonic function expressed the pathways of minimal change in the potential for action, while the other, orthogonal curves expressed the pathways of maximum change in the potential for action. For example, if the Earth were perfectly spherical, its minimum and maximum of potential action could be expressed by a series of concentric spherical shells and orthogonal planes. A cross-section of such a configuration would be harmonically related circles and radial lines. If the Earth were perfectly ellipsoidal, its potential would be expressed by a set of triply orthogonal ellipsoids and hyperboloids whose cross sections would be the harmonically related set of ellipses and hyperbolas illustrated above.
But, as Gauss emphasized, the shape of the Earth is much more complicated than a sphere or an ellipsoid, with respect to both gravity and magnetism and the pathways of minimal and maximal potential for action were not such simple and well known curves as circles, lines, ellipses or hyperbolas. Thus, a more complex harmonic function must be found to express these principles. Such a function could not be determined a priori, but only from the measured changes in the effect of the Earth’s gravity or magnetism.
The question for Gauss was: how to determine the true physical shape of the Earth, or the characteristic of the Earth’s magnetism, from the measured infinitesimally small changes in its potential obtained by his geodetic and magnetic measurements?
This begins to get us closer to a first approximation of what Riemann called “Dirichlet’s Principle”.
To make a precise determination of the Earth’s surface, or magnetic effect, as Gauss did, is quite complicated, but the principle on which his method was based is within the scope of this pedagogy. If one recognizes, as Gauss did, that the changes in the direction of the plumb bob are measuring the changes in direction of the potential function, then the physical shape of the Earth has the same relationship to this potential as the hanging points have to the catenary. In other words, the surface of the Earth must be understood as merely the boundary of the potential, or, as Gauss put it: “The physical surface of the Earth is, in a geometric sense, the surface that is everywhere perpendicular to the pull of gravity.”
A reference to the ancient Pythagorean problem of doubling the line, square and cube can shed some light on this idea. The line is bounded by points, the square by lines and the cube by squares. The size and position of these boundaries is determined by the length, area or volume they enclose. For example, it is the square that determines the size and position of its sides, even though it is the latter that you see and the former that you don’t. The sides of the square are lines, but they are produced by a different power, (potential), than the lines produced from other lines. Similarly, the size and position of the squares that form the boundaries of a cube are produced by a different power (potential), than the squares formed by the diagonal of another square. Thus, even though the power can not be seen, it can be measured by its unique, characteristic effect on the boundaries of its action.
Now apply this same method of thought to the physical principles discussed above. The catenary is a curve whose boundaries are points. A catenoid is a surface whose boundaries are curves. The surface of the Earth is the boundary of a gravitational volume. The magnetic effect of the Earth is still more complicated, and will be taken up in more detail in a future pedagogical.
This connected relationship between the boundary conditions of a physical process and the expression of the principle of least-action with respect to that physical process, is the relationship to which Riemann is referring when he speaks of “Dirichlet’s Principle.”

From Gauss to Dirichlet to Riemann

After succeeding Gauss in 1855, Dirichlet began lecturing on Gauss’s potential theory at Goettingen while Riemann was preparing his {Theory of Abelian Functions}. What Gauss, Dirichlet and Riemann all recognized, was that complex functions, as the extension of Leibniz’s concept of the catenary and natural logarithms, were uniquely suited to express the least-action pathways of potential functions.
Gauss had already demonstrated this in his 1799 proof or the fundamental theorem of algebra, where he showed that a complex algebraic expression produces two surfaces whose curvatures are harmonically related. What Riemann attributes to Dirichlet, is the principle that given a certain boundary condition, the function that minimizes the action within it is a complex harmonic function.
Warm up to this idea on the familiar territory of the catenary. The boundary conditions here are the positions of the hanging points. The “interior” of this boundary is the curve itself. Within the curve is a singular point–the lowest point. If the boundary conditions change, by changing the position of the hanging points, so does the position of the lowest point. To state Dirichlet’s principle in this simplified context, the catenary is the least-action pathway of a hanging chain with these specified boundary conditions and singularity. If the boundary conditions change, the shape of the curve changes correspondingly, in accordance with the preservation of the principle of least-action.
Riemann inverted Dirichlet’s principle: {since the physical principle of least-action is primary, the position of the hanging points and the lowest point completely determine the shape of the chain!}
Now, make this same investigation with respect to a catenoid formed by a soap film between two circular rings. This catenoid is a physical least-action, or minimal surface. Embedded in this surface is an orthogonal set of curves of minimal and maximal action. (Riemann later showed that these curves are harmonically related. This will be illustrated in a future installment of this series.) Experiment by changing the shape of these boundaries from circles, to ellipses, to irregular smooth shapes, to polygons. When you change the position or shape of the boundaries of this surface, the shape of the surface and the embedded curves change accordingly, but the least-action principle is preserved.
Now, generalize this idea with some other pedagogical examples, illustrated in the following animations. In Figure 7 we see a set of harmonically related circles and radial lines that intersect at the center of the circles, being transformed while maintaining their harmonic relationship.

If the position of that intersection point changes, the radial lines must be transformed into circular arcs, and their endpoints move along the boundary in order to maintain their harmonic relationship. In the animation, we see this effect as the point of intersection moves, first away from the center, and then in a circular path around the center. This motion causes all positions inside the boundary to change {as a whole}. What doesn’t change is the harmonic, i.e. least-action, relationship.
This could also be thought of inversely: that the changes in position of the intersection of the radial lines at the boundary, cause their point of intersection to move in a circular arc, and their form to change from lines to circular arcs.
Or, infinitesimally small changes in the curvature of the pathways are determined by the conditions at the boundary with respect to the position of the singularity.
Compare this action with the change in the position of the lowest point of the catenary as the positions of the hanging points change, as illustrated in the animation Figure 1. There, a change in the boundary points produced a change along a single curve. Here, a change in the boundary curve produces a change in a set of harmonically related curves within a surface.
Compare this with the problem Gauss confronted in, for example, determining the location of the Earth’s magnetic poles from infinitesimally small changes in the Earth’s magnetic effect. Gauss understood that those small changes were connected to the position of the singularities, i.e. magnetic poles, of the Earth’s magnetic effect. However, the exact location, or even the number, of those poles was still unknown in Gauss’s time. On the basis of the measurements obtained by von Humboldt’s network, Gauss determined where those poles must be located. The famous American Wilkes expedition of 1837 was launched, in part, to confirm Gauss’s findings, which it did.
In Figure 8, this same effect is illustrated by moving the point of intersection of the radial lines along the path of a lemniscate.

Notice again how this change in the position of the singularity, changes the condition at the boundary, so that all the resulting relationships remain harmonic.
Figure 9 animates the same process in which the shape of the boundary has been changed to an ellipse, which correspondingly changes the shape of the orthogonal curves into hyperbolas, and the intersection point into two foci.

Of course, it could also be said that the radial lines are changed into hyperbolas, which changes the circles into ellipses, and the intersection point in to two foci. Or, that the intersection point is changed into two foci, which changes the the boundary into an ellipse, and the radial lines into hyperbolas.
In short: {a physical process of least action is a connected action. Changing any aspect of the process, changes everything else in the process correspondingly, so as to preserve the least-action characteristic of the process. That is, it is the physical principle of least-action that is primary.}
It was Riemann’s genius to recognize, through this application of Dirichlet’s Principle, that the principle of least-action of a physical process could be understood completely by the relationship between the boundary conditions and the singularities, and that this relationship could be expressed uniquely by Riemann’s geometric concept of complex functions. Moreover, Riemann showed that the characteristic of least-action of a physical process could be changed, in a fundamental way, only by the addition of a new principle. That change in principle is expressed in a complex function, as a corresponding increase in the number of singularities. In his {Theory of Abelian Functions} Riemann demonstrated this by applying Dirichlet’s Principle to the higher transcendental functions of Abel.
The deeper significance of this discovery can only be hinted at in this installment, and will be taken up in more depth later, but it can be illustrated by the animation of Figure 10, which expresses the principle of least-action with respect to an elliptical function.

Riemann demonstrated that all elliptical functions, being functions formed by the interaction of two connected principles, are expressed in the complex domain as surfaces with two boundaries. (These boundaries are marked in green in the animation.) In this animation you can see each boundary changing differently, but connectedly, with the other, causing corresponding changes in the minimal pathways, while at all times maintaining the overall harmonic relationship of the function. In other words, the characteristic curvature of these least-action pathways is determined, in this case, by the connected interaction of two distinct principles.
A comparison between this and the previous examples indicates what Riemann emphasized, that the only way to fundamentally change the characteristic of action of a physical process is by the addition of the action of a new principle. This more advanced question will be investigated more thoroughly in future pedagogicals.
A suggestive example from economics can also help illustrate this principle. What is the relationship between all physical economic relationships and the economic boundary conditions of physical infrastructure and cultural development? What is the relationship between these boundary conditions and the singularities represented by the introduction of new technologies? What is the effect on all economic relationships, of a change, positive or negative, in these physical economic boundary conditions?
Four years after Riemann’s death Karl Weierstrass criticized Riemann’s application of Dirichlet’s Principle on formal mathematical grounds. Weierstrass contended that it was inappropriate to speak mathematically of least-action, unless a formal mathematical proof could be presented proving that a mathematical minimum, or maximum, existed. While it is possible to produce a formal mathematical example which has no minimum, all {physical} process are characterized by bounded least-action. For example, as Cusa showed, there is no absolute maximum or absolute minimum polygon because the polygon is bounded maximally by a circle (which is not a polygon) and minimally by a line (which is also not a polygon). Or, while a mathematical catenary can be extended into infinity, the physical one is always bounded by the hanging points. For Riemann, as for Gauss and Dirichlet, Weierstrass’s demand for a formal mathematical proof of a minimum was less than unnecessary, it was a sophistry. The universal physical principle of least-action was sufficient to supply the proof.
Weierstrass’s critique was seized upon by the formalists who were desperate to roll back the achievements of Kaestner, Gauss, Dirichlet, Jacobi, Abel, Riemann et al. and return science to the slavish days of Euler, Lagrange and D’Alembert. Consequently, while the form of Riemann’s discoveries has been widely discussed, the substance of his thinking has, by and large, been suppressed, until, it found new life in the more advanced discoveries LaRouche.

Riemann for Anti-Dummies: Part 57 : Pythagoras As Riemann Knew Him

Pythagoras as Riemann Knew him

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.

But 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.

Figure 1

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.

Figure 2

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}.

Power From the Standpoint of the Complex Domain

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. (See pedagogical discussion {Dance With the Planets.})

–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.

Figure 3

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.

Figure 4

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. (See The Dramatic Power of Abelian Functions, Riemann for Anti-Dummies Part 54.)

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 ).

Figure 5

Figure 6

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.).

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.

Connectivity and Curvature

But, there is another significant characteristic of these higher transcendental functions which Riemann emphasized, but which only comes to light when Gauss’s general principles of curvature are taken into account. This can be introduced pedagogically by taking note of the change in the characteristic curvature of the surface associated with different transcendental functions. For example, a sphere, which is simply-connected, is everywhere positively curved, but a torus, which is doubly-connected, is positively curved only on the “outside”, but negatively curved on the “inside”. (Ironically, and interestingly, this combination of positive and negative curvature gives the torus a total curvature of zero!) Thus, a higher transcendental power is associated not only with a change in connectivity, corresponding to a change in the density of singularities, but also with a change in the characteristic curvature. Thus, a change in the power of a transcendental function , which occurs through the revolutionary discovery of an existing, but previously undiscovered universal principle, changes the characteristic curvature of the manifold of physical action.

To illustrate this, we must again turn back to the work of Gauss. In his {General Investigations of Curved Surfaces}, Gauss showed that on a positively curved surface the sum of the angles of a triangle is always greater than two right angles (180 degrees), whereas on a surface that is negatively curved, the sum of the angles of a triangle is always less than two right angles. Inversely, the characteristic curvature of a surface can be determined by the characteristics of the triangles that exist on it.

Furthermore, this characteristic curvature of a surface determines what Kepler called the types of congruences (harmonics) possible on that surface. For example, on a surface of zero curvature, six equilateral triangles can form a perfect congruence, because these triangles will all have angles of 60 degrees, and six such angles form one complete rotation. On the other hand, on a sphere, since any equilateral triangle will always have angles that are greater than 60 degrees, three, four or five triangles, but never six, will form a perfect congruence. Thus, from Gauss’s standpoint, the uniqueness of the five regular solida can be demonstrated to be a consequence of the characteristic curvature of spherical action.

But something very different happens on surfaces of negative curvature. Since here the angles of an equilateral triangle are always less than 60 degrees, perfect congruences can be formed by any number of triangles greater than six.

The problem Gauss understood, was that while surfaces of positive curvature could be represented as objects in visible space, such as a sphere, negative curvature acted on the visible domain from outside. Consequently, no negatively curved surface could be faithfully represented directly as a visible object! Gauss discovered, however, that the relationships of negatively curved surfaces could be represented visibly, but only as projections in the complex domain. Although Gauss never published his results, his notebooks document the direction of his thinking. Figure 8 shows one of Gauss’s drawings depicting the projection of a congruence formed by eight triangles, each with three 45 degree angles. Such triangles could only exist outside the visible domain, on a negatively curved surface.

Figure 8

To understand this projection, think of it as an analogy to the stereographic projection of the sphere onto the plane. In that case, the circles of longitude are projected onto radial lines, and the circles of latitude are projected onto concentric circles. (See Figure 9.)

Figure 9

The circles of longitude are orthogonal to all circles of latitude, as are the radial lines to the concentric circles in the plane. But, whereas the circles of longitude all converge on the north pole, the radial lines spread out, approaching Cusa’s infinite circle. Note, that these radial lines will, therefore, be orthogonal to the “infinite”. Spherical triangles on the sphere are projected onto the plane as triangles whose sides are circular arcs, and whose angles are the same as on the sphere. (See Figure 10.)

Figure 10

But, though the angles are preserved by the stereographic projection, distance is not. Consequently, as the distances measured approach the north pole of the sphere, the distances in the image on the plane increase exponentially.

Now look at Gauss’s projection of a negatively curved surface. Instead of an infinitely extended plane, the negatively curved surface projects onto a bounded disc. Here the sides of the triangles are formed by circular arcs, which, like the radial lines of the stereographic projection, are orthogonal to the boundary of the surface. Also, as in the stereographic projection, angles are preserved, but distances are not. But unlike in the projection of a sphere, where the distances become exponentially large as the boundary (“infinite”) is approached, the distances in the projected image of a negatively curved surface, become exponentially shorter. (See Figure 11.)

Figure 11

With this work of Gauss in mind, we can now begin to illustrate the relationship Riemann showed, between the increasing density of singularities associated with higher transcendental functions, and a change in the characteristic curvature of the manifold.

This can be illustrated pedagogically by comparing the difference between the elliptical transcendental and the hyper-elliptical. As developed earlier, the elliptical transcendental, which generates four singularities, is expressed as a Riemann function on a torus, on which there are two distinctly different types of curves that go around the torus and the curves that go “through the hole”. (See Figure 12.)

Figure 12

This doubly-connected action maps into a network of rectangles. (See Figure 13 & Riemann for Anti-Dummies 56). As we just discovered through Gauss, such a congruence of rectangles can only be formed on a surface of zero or positive curvature.

Figure 13

But the next highest transcendental, the hyper-elliptical, generates six singularities, and as Riemann showed, must be expressed on a triply-connected surface, such as a torus with two holes. On such a surface there are four distinct closed curves, instead of the two for the torus. (See Figure 14.) A mapping of these four pathways yields an octagonal congruence. (See Figure 15.)

Figure 14

Figure 15

As Gauss showed, such a congruence can only exist on a surface of negative curvature, and so its appearance in the case of the hyper-elliptical transcendental is the image of a physical action, characterized by negative curvature, acting from outside the visible domain.

Thus, as we now think of the hierarchy of the so-called “inexpressibles”, from the algebraic, to the circular transcendentals, to the elliptical transcendentals, to the hyper-elliptic and higher, we can understand a successive transformation in curvature from zero (rectilinear/algebraic), positive (spherical/exponential), to positive/negative (elliptical/toroidal), to negative (hyper-elliptical/Abelian).

Riemann emphasized that it is the relationship among these three characteristic curvatures, positive, zero and negative, that characterizes physical action. We cannot think of physical action as being characterized by any one type of curvature, but must consider the change in curvature that corresponds to the “power” governing the action. In the Habilitation lecture, Riemann posed a pedagogical construction of three such surfaces, represented by a sphere, cylinder, and the inside of a torus, all intersecting at one circle. (See Figure 16.) The circle is the unique pathway that at all times exists on all three types of curvature at once. Think of this circle as a new type of “infinitesimal”, a moment of change from one manifold to another of greater transcendental “power”.

Figure 16

This relationship between curvature and the higher transcendentals is of extreme importance for the future development of modern physical science. As Riemann stated in his Habilitation lecture, the characteristics of physical action change when extended from the observable range, into the astronomically large, such as the Crab Nebula and the microscopically small, such as the sub-atomic domain. Such changes correspond to an increasing density of universal principles, i.e., singularities, which in turn is reflected as changes in the characteristic curvature, and connectivity, of the manifold of physical action.

As science extends its investigations into these domains, an ever increasing number of universal physical principles will be discovered and incorporated into our knowledge of the universe. Such increases are associated with transcendental functions of increasingly higher power, of the type suggested by Riemann a type whose power is akin to that which connects us, through the mind of Keats, to those ancient people depicted on that Grecian urn.

Riemann for Anti-Dummies: Part 56 : Riemannian Spherics


RIEMANNIAN SPHERICS

When Carl Friedrich Gauss repeatedly stated his conviction that Euclidean geometry was not true, his thoughts were connected to the pre-Euclidean science of the Pythagoreans and Plato. However Gauss’s “{anti}-Euclideanism” was not a mere restatement of its antecedent. Rather, Gauss, and later Riemann, sublimated the ancient Egyptian-Pythagorean science of “spherics” with a new spherics, that had been demanded by Cusa’s discovery of transcendental physical action, the development of Kepler’s discovery of the harmonics of elliptical planetary orbits, and Leibniz’s generalization of both, as the universal principle of least-action.

This is the vantage point from which to gain a firmer grasp of Riemann’s treatment of Abelian and hypergeometric functions.

The relevant characteristics of the Pythagorean concept of spherics are summarized in the Plato dialogue, named for the astronomer Timaeus, who, attributing the origin of his discourse to the testimony of wise men from more ancient times, recounts the nature and creation of the universe. Timaeus begins by noting that investigations of the created world must focus on the eternal universal principles from which it is patterned. Though “apprehensible by reason and thought,”, those principles are, of necessity, reflected in the world’s physical form. Accordingly, when speaking about the physical universe, Timaeus emphasizes, it is most important to distinguish between the original principles and their created copy, and one must be careful to recognize that a simple description of the latter cannot suffice for an explanation of the former.

“Accordingly, in dealing with a copy and its model, we must affirm that the accounts given will themselves be akin to the diverse objects which they serve to explain; those which deal with what is abiding and firm and discernible by the aid of thought will be abiding and unshakable; and in so far as it is possible and fitting for statements to be irrefutable and invincible, they must in no wise fall short thereof; whereas the accounts of that which is copied after the likeness of that model, and is itself a likeness, will be analogous thereof and possess likelihood; for as Being is to Becoming, so is Truth to Belief. Wherefore, Socrates, if in our treatment of a great host of matters regarding the Gods and the generation of the Universe we prove unable to give accounts that are always in all respects self-consistent and perfectly exact, be not thou surprised; rather we should be content if we can furnish accounts that are inferior to none in likelihood, remembering that both I who speak and you who judge are but human creatures, so that it becomes us to accept the likely account of these matters and forbear to search beyond it.” (29c-d)

Timaeus goes on to say that the Creator, being good, intended for that goodness to be reflected in the creation and so he endowed it with intelligence and soul, and brought it into existence as a self moving, self-subsisting living creature.

“He fashioned it to be One single Whole, compounded of all wholes, perfect and ageless and unailing. And he bestowed on it the shape which was befitting and akin. Now for that living creature which is designed to embrace within itself all living creatures, the fitting shape will be that which comprises within itself all the shapes there are; wherefore he wrought it into a round, in the shape of a sphere, equidistant in all directions from the center to extremities, which of all shapes is the most perfect and most self-similar, since he deemed that the similar is infinitely fairer than the dissimilar….

“Such then was the reasoning of the ever-existing God concerning the god which was one day to become existent, whereby He made it smooth and even and equal on all sides from the center, a whole and perfect body compounded of perfect bodies, and in the midst thereof He set soul, which He stretched throughout the whole of it; therewith He enveloped also the exterior of its body; and as a circle revolving in a circle He established one sole and solitary Heaven, able of itself because of its excellence to company with itself and needing none other beside, sufficing unto itself an acquaintance and friend. And because of all this He generated it to be a blessed God..” (34-b.)

This spherical form, Timaeus explains, expresses not only the goodness and perfection of a universe patterned from reason and intelligence, but it also expresses the motions of the visible objects in the heavens as well. These visible objects, whose individual motions trace circular arcs onto the celestial sphere, form the “moving image of eternity” we call time. Timaeus proceeds to recount, in detail, the diverse motions of these celestial objects, the totality of which are visible expressions of that which is eternal and universal.

But Plato emphasized that these characteristic spherical motions, while a means to express the characteristics of an idea, cannot, by themselves, be taken as knowledge of the idea. But, being patterned after the eternal, their study can aid us in grasping what can only be apprehended by reason and thought:

“Thus, said I. These sparks that paint the sky, since they are decorations on a visible surface, we must regard, to be sure, as the fairest and most exact of material things, but we must recognize that they fall far short of the truth, of the movements, namely of real speed and real slowness in true number and in all true figures both in relation to one another and as vehicles of the things they carry and contain. These can be apprehended only by reason and thought, but not by sight, or do you think otherwise?

By no means, he said.

Then, said I, we must use the blazonry of the heavens as patterns to aid in the study of those realities, just as one would do who chanced upon diagrams drawn with special care and elaboration by Daedalus or some other craftsman or painter. For anyone acquainted with geometry who saw such designs would admit the beauty of the workmanship, but, would think it absurd to examine them seriously in the expectation of finding in them the absolute truth with regard to equals or doubles or any other ratio.” (Rep. 529e-530b)

The Boundaries of the Sphere

Because these visible expressions could also mislead reason, Plato cautioned against the methods of the Sophists and Eleatics, (forerunners of today’s mathematical formalists and information theorists), whose practice, like their modern counterparts, was to induce insanity in their victims by restricting their attention to the visible form of mathematical objects that have been disconnected from the universal principles from which they arose, and, from the ideas whose shadows they were intended to represent. As he warned in the {Republic}:

“This at least, said I, will not be disputed by those who have even a slight acquaintance with geometry (earth-measure), that this science is in direct contradiction with the language employed in it by its adepts.

How so? he said.

Their language is most ludicrous, though they cannot help it, for they speak as if they were doing something and as if all their words were directed toward action. For all their talk is of squaring and applying the adding and the like, whereas in fact the real object of the entire study is pure knowledge.

This is absolutely true, he said.

And must we not agree on a further point?

What?

That it is the knowledge of that which always is, and not of a something which at some time comes into being and passes away.

That is readily admitted, he said, for geometry is the knowledge of the eternally existent.” (Rep. 527a.)

The Sophists insisted that geometry could state nothing truthful concerning “knowledge about that which always is” because geometrical objects, being objects of sense, lie in the domain of that which is always changing. Consequently for the sophist, geometrical objects, like all other objects of sense, could be connected to each other by a set of formal rules, described and manipulated, but such actions were {by definition} divorced from the eternal physical principles governing the universe itself. By limiting scientific inquiry to such objects of sense perception, as Aristotle also maintained, the Sophists could awe their audiences with dazzling demonstrations, such as the simultaneous existence and non-existence of the one and the many, while convincing them that knowledge of universal principles was, in the end, beyond their grasp.

By contrast, the Pythagorean-Platonic scientists admitted no objects into geometry that did not arise in connection with an effort to grasp an unseen, eternal, universal principle. Such objects, such as the Pythagorean concept of the sphere, were brought into existence by an action of a mind intending to gain greater mastery over the principles that govern the universe itself. Consequently for these Socratics, all such geometrical objects are representations of ideas that are apprehensible only by reason and thought. “Thought-objects” so generated contain within them, the essential characteristics of the universal principles they are intended to grasp. Being thus connected to universal principles, paradoxes generated by the investigation of such objects, point to the existence of unknown principles, that, once discovered, lead to the generation of new ideas, that demand new types of geometrical expression.

In the science of Classical Greece, this is exemplified by the uniqueness of the five regular solids. Nothing in the smooth, everywhere same, form of a sphere gives any indication of the existence of a bounding principle that determines that there be five, and only five, unique solids that equally divide that everywhere constant surface. But, the fact that only five such solids can be constructed, reveals the existence, and reflects the characteristics, of that bounding principle. Thus, even though the sphere is the shape “that comprises all shapes”, the five regular solids indicate the existence of a still higher principle from which the sphere’s power is derived. This higher power, though outside the visible boundary of the sphere, is, nevertheless, expressed on the sphere’s surface, by the uniqueness of the five regular solids.

That that higher principle can be found only by surpassing the boundaries of the sphere, was already evident to the Pythagoreans, as indicated by Archytas’s construction for the doubling of the cube by the intersection of a torus, cylinder and cone. This stands in contrast to the doubling of the square, whose generating principle can be expressed by circular action existing within the same manifold (plane) in which the square itself is generated. On the other hand, the cube, which is formed in the sphere, cannot be doubled by means of spherical action. As Archytas demonstrated, a more complex unfolding of the sphere is required. Thus, the characteristic of the cube, as well as the other four regular solids, indicated to the Pythagoreans and Plato, the existence of a still higher principle that bounds the apparently self-bounded spherical form.

Because of paradoxes like these, Plato insisted that the study of these solids should be emphasized, in addition to the Pythagorean quadrivium of arithmetic, geometry, astronomy and music, in the education of political leaders, so that those leaders could develop the cognitive powers to recognize universal principles and act only on the basis of truth, not on the basis of arbitrary opinion.

For opposing reasons, Aristotle introduced the arbitrary fiction that the form of the universe is an unbounded, empty, linearly extended void, as codified in the Elements of Euclid. Under this scheme, spherical action is not the reflection of the universal principles from which it is generated. It is but only one, of infinitely many undifferentiated possible types of action. For Euclid, the sphere’s existence, like the rest of the objects of Euclidean geometry, rests only on the authority of an {a priori} set of definitions, axioms and postulates. For Euclid, and those indoctrinated by sophism, empiricism, etc. this false, fantasy world of linear action becomes the only true one from which all physical processes deviate.

Ironically, the Pythagorean-Platonic method expressed as self-bounded spherical action reflects the unlimited potential for the discovery of new physical principles, because the existence of these new principles is indicated through the emergence of paradoxes, such as the uniqueness of the five regular solids. On the other hand, the Aristotelean-Euclidean fantasy of an apparently unbounded, linearly extended void, that, {by definition}, permits no physical principles within its midst, acts as a barrier to the introduction of new ideas, restricting human knowledge to a definite and finite domain: a barrier that held back human progress for nearly 1500 years, from the murder of Archimedes in 231 B.C. until the dawn of the European Renaissance in the 14th century.

In sum: the self-bounded is unlimited potential; the unbounded is forever constrained.

Surpassing the Boundaries of the Self-Bounded Sphere

The Aristotelean barrier was cracked when Nicholas of Cusa re-introduced the Pythagorean-Platonic concept of spherics, in the new, more advanced form of his method of “On Learned Ignorance”. Like his forerunners, Cusa understood that mathematical objects, generated as metaphors, were a means for expressing new discoveries:

“Therefore, in mathematicals the wise wisely sought illustrations of things that were to be searched out by the intellect. And none of the ancients who are esteemed as great approached difficult matters by any other likeness than mathematics….

“Proceeding on this pathway of the ancients, I concur with them and say that since the pathway for approaching divine matters is opened to us only through metaphors, we can make quite suitable use of mathematical signs because of their incorruptible certainty.”

Cusa’s approach was emphatically anti-Aristotelean. Instead of the unlimited, linearly extended infinite void, in which geometrical objects rattled around, devoid of intention, Cusa saw the apparent infinite merely as the domain of yet to be discovered universal principles, that bounded the finite and determined the characteristic of action within it. The characteristics of that set of principles can be discovered by ascending upward from their finite manifestation towards the higher expression:

“…when we set out to investigate the Maximum metaphorically, we must leap beyond simple likeness. For since all mathematicals are finite and otherwise could not even be imagined; if we want to use finite things as a way for ascending to the unqualifiedly Maximum, we must first consider finite mathematical figures together with their characteristics and relations. Next, we must apply these relations in a transformed way, to corresponding infinite mathematical figures. Thirdly, we must thereafter in a still more highly transformed way, apply the relations of these infinite figures to the simple Infinite, which is altogether independent even of all figure. At this point our ignorance will be taught incomprehensibly how we are to think more correctly and truly about the Most High as we grope by means of a symbolism.”

Cusa’s approach redefined the self-bounding characteristic of the Pythagorean sphere. For Cusa, the visible sphere, though apparently self-bounded, was actually bounded by a higher domain of universal principles that lay outside the sphere itself. Because the domain of these principles exists outside the domain of finite physical objects, it appears, from the finite perspective of the objects, to be infinitely far away.

Thus, the characteristics of the visible sphere, most importantly for Cusa, the incommensurability between the spherically curved and the linearly straight, reflected the higher principle from which the curved and the straight were both generated. Thus, for Cusa, in opposition to Aristotle, the curved was not a deviation from straightness. Rather, the incommensurability between them, reflected one single principle, in unfolded form. Cusa expressed this relationship by the ironical juxtaposition of the idea of a finite sphere and line with an hypothesized infinite one:

“I maintain, therefore, that {if} there were an infinite line, it would be a straight line, a triangle, a circle and a sphere. And likewise {if} there were an infinite sphere, it would be a circle, a triangle, and a line.” (emphasis added).

From this standpoint, Cusa redefined the physical expression of Pythagorean spherics. There are no perfectly circular or spherical motions in the created world, Cusa correctly insisted. Nor is there perfect equality in harmonics, statics, or other physical phenomena. There is no absolute center to the universe, nor is there a single set of poles. However, this imprecise nature of the physical world does not indicate its deviation from the false-perfection of the {a priori} linearly-extended fantasy world of Aristotle’s Euclidean-type geometry. Rather, this imprecision indicated the characteristics of the true physical-geometry of the universe, which cannot be described by simple abstract geometrical shapes, but must be discovered by physical investigations:

“Thereupon you will see through the intellect, to which only learned ignorance is of help that the world and its motion and shape cannot be apprehended. For the world will appear as a wheel in a wheel and a sphere in a sphere having its center and circumference nowhere, as was stated.”

But, the fact that the physical universe cannot be described by simple, static shapes, does not make it unknowable. It is through this very imprecision, as projected into the visible domain, (as for example, onto the Pythagorean sphere) that the true higher ordering principle can be discovered, as knowledge of the Creator’s intention:

“Who would not admire this Artisan, who with regard to the spheres, the stars, and the regions of the stars used such skill that there is though without complete precision both a harmony of all things and a diversity of all things? This Artisan considered in advance the sizes, the placing, and the motion of the stars in the one world; and He ordained the distances of the stars in such a way that unless each region were as it is, it could neither exist nor exist in such a place and with such an order nor could the universe exist….”

“But all things reply to him, who in learned ignorance asks them what they are, or in what manner they exist, or for what purpose they exist…”

On this basis, Kepler, who called Nicholas of Cusa “divine”, broke the centuries-old barrier that had been established by Aristotelean dogma, and established a new, modern physics not tied to any {a priori} considerations, but only to physical ones. As Kepler noted in the opening of his {New Astronomy}, it may be the first presumption of reason that action in the physical universe is circular, but physical experiment determined this to be untrue. Consequently, all {a priori} geometrical descriptions of the motions of the planets, such as those adopted by Ptolemy, Copernicus, and Brahe must be abandoned in favor of a physical-geometry based only on physical causes. For Kepler, the paradoxes among the tracings of the motions of the planetary orbits onto the celestial sphere provided the clues that led him to discover the principles that were determining them. Thus, Kepler investigated not the objects of the sky, nor simply their motions as projected onto the celestial sphere. Rather, he investigated the relationships between these projected motions, from which he apprehended the characteristics of the unseen principles from which those motions were determined.

It is from this standpoint that Kepler was able to elaborate the harmonics of the non-uniform, essentially elliptical motion of the planetary orbits, giving an entirely new meaning to the physical significance of the ancient Pythagorean discoveries concerning the boundaries of spherical action, such as the five regular solids, conic sections, and musical harmonics. The question posed by Kepler to all future generations, was to develop a new type of mathematics that could express these higher principles. Standing on Leibniz’s shoulders, this is what Gauss and Riemann developed as the complex domain.

The Physical “Shape” of the Universe

As Riemann emphasized in his 1854 habilitation lecture, his philosophical fragments, and his lectures on Abelian and hypergeometric functions, and as Gauss had spoken earlier, the common, intuitive notions concerning geometrical objects presupposes a set of assumptions concerning the fundamental nature of the “space” in which those objects arise. Like Cusa, Kepler, and Leibniz before them, both Gauss and Riemann insisted it was the action of physical principles that determined the characteristics of what is called “space”, not a set of arbitrary {a priori} assumptions. While these physical principles are unseen, the effect of their action is measurable.

To express the connection between these measurable effects and the unseen principles themselves, Gauss and Riemann developed a new form of spherics, by extending Leibniz’s infinitesimal calculus into the complex domain.

For Gauss this meant rejecting all arbitrary notions of the sphere as a three-dimensional object bouncing around in an empty, linearly-extended void. Instead, Gauss considered the sphere to be what Riemann would later call, a doubly-extended manifold. In this way, Gauss eliminated from the sphere (and all other doubly-extended manifolds), all arbitrary ideas of linear extension. Linear distance between any set of points on the surface is determined solely as a function of angular displacement. The relationship between angular displacement and linear extension reflected the effect of an invariant characteristic of the surface. That characteristic Gauss called “curvature”.

By “curvature” Gauss did not mean “not-straight”. Gauss’s idea of “curvature” is an expression of the characteristic principles that determine action on that surface. But, since the characteristic curvature is not visible, it must be discovered by its effect on all action on that surface.

For example, the elliptical arc of a planetary orbit is determined by the harmonic characteristics of the principle of universal gravitation. To the planet, such an arc is a “straight-line”. To our senses the planet’s orbit is a variable arc of a great circle on a sphere. But to our minds, as Kepler showed, we can discover that those arcs are merely the projection onto a sphere of an action that is occurring in a different surface a surface we cannot see. However we can imagine this other surface, not as a visible image, but as that set of physical principles which has the power to produce the physical effects whose visible images are projected onto the celestial sphere. Furthermore, since we are moving in that surface ourselves, we must apprehend its characteristic curvature, from within it, by measuring the effect of that curvature in the relatively infinitesimally small–a method that Leibniz called “analysis situs”.

Gauss’s idea of a surface, therefore, is not a simple visible shape but a manifold of physical principles. To better determine the characteristics of these manifolds, Gauss developed the means to map these manifolds onto a sphere. Under these mappings the characteristic curvature of the unseen surface is expressed as anomalies within the domain of spherical functions. Working backwards, we can then “unfold” these projections, to determine the characteristics of the manifold from which they are projected. (See Riemann for Anti-Dummies Parts 44-46.)

The Riemann Sphere

Armed with Gauss’s work, Riemann generalized the method. As indicated in his fragment on “n-dimensional” manifolds, Riemann considered the geometrical ideas of line, surface and solid as only special intuitive examples of a singly, doubly and triply extended manifold, whose modes of determination are physical principles, not linear extensions. However, Riemann emphasized as did Gauss, Leibniz, Kepler, Cusa, and Plato before him, that the construction of the appropriate metaphorical representation was essential to achieve a greater understanding of these principles, and to communicate these discoveries to others, so as to bring them under the willful control of mankind.

Riemann developed these ideas in his lectures on Abelian and hypergeometric functions, whose deeper implications were not realized until LaRouche’s more advanced, revolutionary breakthroughs in the science of physical economy.

Riemann’s starting point was Gauss’s original idea that the physical characteristics of doubly-extended manifolds, (i.e. surfaces) could be represented in the complex domain as functions of complex numbers. This is based on Gauss’s idea of representing complex numbers as a function of position on a surface.

This is {not} the idea of representing complex numbers as points on an abstract Cartesian grid as is universally taught in reductionist mathematics education today. We emphasize, as Gauss did, that any function of position must be in reference to some physically determined starting point and a physically determined direction. For example, the determination of position on the celestial sphere, must be in reference to some starting point (the pole), and some direction (the direction of the position of sunrise on the vernal equinox). Once these two parameters (origin and direction) are determined, all other positions on the surface can be expressed with respect to them. This is distinct from the Aristotelean, Euclidean, Cartesian, Kantian idea of an undifferentiated, linearly-extended space. As Gauss pointed out with reference to Kant, it is impossible to determine anything about the nature of space except by reference to physical objects. In other words, Gauss did not begin with a Cartesian-Kantian idea of an empty surface and then assign some arbitrary origin and direction to it. Rather, physics establishes the origin and reference direction, which in turn established the surface.

Riemann showed that once such a physically determined doubly-extended surface has been established, the characteristics of action within that surface can be expressed as a complex function. Here again, Riemann adopted Gauss’s approach of the “Copenhagen Prize Essay” on conformal mappings. For example, the stereographic projection of the sphere onto the plane is an expression of the complex exponential. (See Figure 1.)

Figure 1

Thus, complex functions considered as conformal mappings, express the transformation of one surface, with a certain characteristic of action, into another, with a different characteristic of action.

For pedagogical purposes let us investigate this matter in more detail with respect to the ancient Pythagorean problem of doubling the square, as seen from the higher perspective of the complex domain. As Gauss demonstrated in his new proof of the fundamental theorem of algebra, the most general expression of an algebraic power is represented geometrically as a specified change in angle and length. That is, squaring is expressed by doubling the angle and squaring the length with respect to that physically determined origin and direction. Cubing is expressed by tripling the angle and cubing the length, etc. (See Figure 2.)

Figure 2

So, for example, in the simplest case of squaring as a complex function, a semi-circular pathway will be mapped onto a full circle. (See Figure 3.)

Figure 3

Thus, a complete circle will be mapped onto a complete circle twice. (See Figure 4.)

Figure 4

Transforming an entire surface in this way, thus maps half the original surface on top of the other. (See Figure 5.)

Figure 5

Since it is impossible, in a Cartesian empty space, to distinguish the first mapping from the second, Riemann invented the idea of a surface that has two layers with the second mapping laying on top of the first. Thus, by squaring, every point on the original surface is transformed into two different points, one laying directly over the other, on the “squared” surface. Those points that have only one value (in this example, the origin), Riemann called a “branch point”. (See Figure 6.) These branch points correspond to physical singularities on the surface as the characteristic of action with respect to them is different than the rest of the surface. A surface with no branch points, Riemann called, “simply-connected”; a surface with one branch point he called “doubly-connected”; a surface with two branch points he called “triply-connected”, etc.

Figure 6

Riemann emphasized that this type of mapping can not occur in a linear, infinitely-extended surface, but only in a bounded one. The reader is encouraged to discover this for yourself. You will discover that it is impossible to think of transforming one doubly-extended manifold (i.e. surface) into another without a boundary. Even if the nature and characteristics of that boundary are not yet determined, its necessary existence is.

This is another indication of why physical action can only be expressed in the complex domain, as there is no physical process that is not bounded. For this reason, Riemann insisted that one could not express complex functions on a Cartesian plane, but only on a bounded surface, as represented by what today is called, “The Riemann Sphere”. (See Figure 7.)

Figure 7

Now, back to Riemann’s surfaces. Riemann confronted the problem that when a mapping produces a multi-sheeted surface, a paradox results, because a continuous curve on one surface becomes two separate curves on two (or more) different layers of the second surface. For this reason he specified the need for a cut, which he called a “branch-cut” that extended from the branch points to the boundary. The various layers of the surface are then connected to one another along these branch cuts, so that all pathways on one of the layers can be continued onto all the other layers continuously. (See Figures 8a, 8b, 8c, 8d, and 8e. These figures include some computer generated representations as well as a hand drawn representation by Riemann himself.)

Figure 8a

Figure 8b

Figure 8c

Figure 8d

Figure 8e

On the Riemann sphere, the entire function can be represented because unlike the Cartesian plane which has no boundary, the Riemann sphere is bounded. The accompanying animation, (See Figure 9.) illustrates the squaring function projected onto Riemann’s sphere. As the boundary on the plane extends outward from the origin, its projection approaches the “north pole” of the sphere. In this example, the squaring function forms a two-sheeted surface on the sphere, whose branch cut runs from one pole to the other.

Figure 9

These animations illustrate the action of the singularity on the Riemann sphere. Here two closed pathways, one encircling the branch point at the origin (10a.) and the other not, (10b.) are illustrated.

Figure 10a

Figure 10b

Under squaring, the pathway encircling the branch point at the origin folds over itself and is mapped onto two sheets. The pathway that does not contain the branch point remains entirely on one layer. (See Figure 10c.)

Figure 10c

However, something is revealed on the Riemann sphere that would otherwise remain hidden. The sphere reveals the existence of another branch-point at the north pole, which in the Cartesian plane can only be considered “the infinite”. This can be seen by performing the same squaring action, but under the inversion that turns the function “inside out”. What was the origin is now the “infinite” and vice versa. (See Figure 11.) This has the same effect on closed pathways. Inversion turns the pathways inside out, which now encircle the branch point at the “infinite”, or the north pole on the sphere. (See Figure 12.) Just as the branch point at the origin acted as a singularity, so now does the other branch point, that had been hidden in the “infinite” in the Cartesian plane, but on the Riemann sphere it is plainly brought into view.

Figure 11

Figure 12

Beyond the Sphere

On this basis, Riemann was able to demonstrate, using Leibniz’s principle of “analysis situs”, that all the essential characteristics of a complex function, are determined by the relationship of the boundary to the branching points, or the other singularities.

This discovery was crucial to Riemann’s treatment of elliptical and Abelian functions. As we discussed in previous installments (See Riemann for Anti-Dummies Parts 49, 52 and 54) these functions, that initially arose from the investigations of Kepler’s elliptical planetary orbits, express a succession of higher transcendental powers. Riemann demonstrated, using a method similar to the one used by Gauss in his proof of the fundamental theorem of algebra, that those complex functions associated with these higher transcendentals generate surfaces with two sheets, but with more than the two branch points expressed by the algebraic, circular, hyperbolic or exponential functions. (See Figure 13.) As Riemann showed, each successively higher power, generates a new set of branch points.

Figure 13

For example, as illustrated in the previous installments, Gauss showed that the elliptical transcendentals were distinguished from the circular, hyperbolic, and elliptical, by their double periodicity. (See Figure 14.)

Figure 14

Riemann showed that this characteristic, by necessity, produces four branch points, which require two branch cuts. (See Figure 15.)

Figure 15

A surface with two branch cuts must be doubly connected, as for example a torus (See Figure 16.) Thus, each new species of transcendental function is characterized by a change in the characteristic topology of the surface, from an “n” connected manifold to an “n+1” connected manifold.

Figure 16

This is the type of transformation associated with the discovery and application of a new physical principle.

Now looking back from the standpoint of this new type of Riemannian spherics, we can recognize that the Pythagorean spherics, is only the simplest type of a succession of self-bounded surfaces of increasingly higher “connectivity”. The provocative question now posed is: What bounds this succession of self-boundedness?

This same question was posed by Gauss in a different way concerning the determination of what is the true geometry. In a letter to Olbers on April 28, 1817, Gauss asserted, like Cusa, that the true geometry could never be seen directly but must be discovered:

“Perhaps in another life we will come to another insight into the essence of space, which is now unreachable for us. Until then, one must not put geometry with arithmetic, purely a priori, but closer to in rank with mechanics…”

Riemann for Anti-Dummies: Part 55 : What Are the Real Objects of Physical Science?

The Dramatic Power of Abelian Functions

To understand Riemann’s treatment of Abelian functions, situate that discovery within the context of the history in which it arose, reaching back to the pre-Euclidean Pythagoreans of ancient Greece, and forward to LaRouche’s unique and revolutionary discoveries in the science of physical economy. Imagine that entire sequence, all at once, as a dramatic history, leap over time, project the past into the future, the future into the past, and both into the present, so that centuries of accomplishment are telescoped into a single, instantaneous thought.

Dynamis

In the opening scene of Plato’s {Theatetus} dialogue, Euclides of Megara informs Terpsion that he has just seen Theatetus being carried to Athens, near death, after being wounded in the battle at Corinth. This experience prompts Euclides to recall that Socrates, when near his own death many years before, had told him that “Theatetus will be a remarkable man if he lives.” Now, contemplating Theatetus’s life at its end, Euclides reports that Socrates had also recounted his first conversation with Theatetus when the latter was merely a boy. Having written down Socrates’s report, Euclides now reads the history to Terpsion, the which forms the bulk of Plato’s dialogue. As Euclides tells it, the conversation began with the young Theatetus being praised by his teacher Theodorus, because the former had surpassed the insights of the latter. As Theatetus explained, Theodorus had taught him about the incommensurablity of lines that double, triple, quintuple, etc. a square, by demonstrating each as a separate and distinct power, beginning with tripling and ending with seventeen, where, “for some reason he stopped.” But, Theatetus continued, “Since the number of powers are innumerable, the notion occurred to us of including them all under one name or species.”

Then by reference to a simple geometrical construction, Theatetus indicates the existence of three distinct {species} of powers: those that generate lines, those that generate squares and those that generate solids. Each species comprises an entire manifold of separate and distinct individual powers. But, each manifold, Theatetus explains, can be thought of under a single principle: linear, quadratic, or cubic.

Not mentioned directly in the dialogue, but prominent in the background, is the related discovery of Archytas, who showed that each species is associated with a different type of physical action: linear, circular and toroidal, respectively.

These discoveries of Theatetus and Archytas demonstrate a power of the mind that the Eleatics, Sophists and later, Aristotle, denied existed, and one that is essential to Riemann’s treatment of Abelian functions: {the capacity of the mind to rise above the finite determinations of sensible objects, and to recognize the higher universal principles that determine them}.

As Theatetus demonstrated, it is possible to {know} an entire species of powers, without having to construct each one individually, by knowing the principle that determines what each individual power can do e.g., double a square– and what they can’t– double a cube. To know that all these individual powers are of one species, {and} that the principle that generates square powers can never generate cubic ones, requires the mind to rise above the characteristics unique to each individual magnitude and recognize the nature of the entire species, {and} the nature of its boundaries.

Aristotle insisted that the human mind did not have this power because, being limited and mortal, it could not rise from finite determinations to universal ones:

“Some, as the Pythagoreans and Plato, make the infinite a principle in the sense of a self- subsistent substance, and not as a mere attribute of some other thing. Only the Pythagoreans place the infinite among the objects of sense (they do not regard number as separable from these), and assert that what is outside the heaven is infinite. Plato, on the other hand, holds that there is no body outside (the Forms are not outside because they are nowhere), yet that the infinite is present not only in the objects of sense but in the Forms also….”

“It is plain, too that the infinite cannot be an actual thing and a substance and a principle….

“Thus the view of those who speak after the manner of the Pythagoreans is absurd….

“This discussion, however, involves the more general question, whether the infinite can be present in mathematical objects and things which are intelligible and do not have extension, as well as among sensible objects. Our inquiry (as physicists) is limited to its special subject matter, the objects of sense, and we have to ask whether there is, or is not, among them a body which is infinite in the direction of increase.” (Physics, Book 5)

Aristotle’s argument is pure sophistry. By {defining} physics to concern only sensible objects, he excludes all consideration of the universal principles that generate those objects. Once excluded, he asserts that such principles play no role in the physical world, because being infinite, they cannot actually exist.

Contrary to Aristotle, as Theatetus demonstrates, in and through Plato’s dialogue, or as Socrates himself makes clear in numerous other locations, most notably the {Phaedo}, immortality, not temporal boundaries, characterizes the human spirit. It can freely transcend the finite limits of the sensual domain through its power to comprehend, not only things, such as lines, squares and cubes, but the powers that generate them. It can transform itself, as Theatetus had shown, from comprehending different powers individually, as Theodorus had taught, to comprehending the concept of an entire species of innumerable individual powers, as he had discovered.

And, as the above account of Plato’s account, of Euclides’s account, of Socrates’ account, of Theatetus’s account of his own discovery illustrates, the human spirit is not constrained by the mortal life in which it originates.

Powers that Generate Powers

The method described in the {Theatetus }dialogue is an elementary, but historically and pedagogically important example of the type of thinking that underlies Riemann’s method, elaborated in his {Theory of Abelian Functions}. There Riemann is dealing with the principles that generate higher and higher species of functions, beyond those contemplated by the Pythagoreans, Theodorus, Theatetus, Archytas and Plato. While Riemann is looking back on what was anticipated by these great ancient minds, he does so from the vantage point of the discoveries of Cusa, Kepler, Leibniz, Gauss and Abel. What these later discoverers demonstrated, was that the species of powers indicated by Plato et al., were superceded by a succession of higher species of successively greater power. Riemann, extending Gauss’s concept of the complex domain, created the means to comprehend the higher power that generates this extended class of higher powers. His treatment of Abelian functions gives expression to the quality of mind that can reach beyond the domain of sense perception, transcend what {appears} to be infinite, recognize the existence of new species of transcendental powers, and {generate} them as higher forms of cognition.

To understand Riemann’s elaboration of Abelian functions it is crucial to emphasize the epistemology underlying Cusa’s earlier discovery of a new species of power, by the method that Cusa called {“Learned Ignorance “}.

Cusa discovered that the incommensurability of the curved to the straight, as exemplified by the incommensurability between a circle and its inscribing and circumscribing polygons, indicated the existence of a new species of power, that transcended those named by Plato, et al. Cusa proved, and Leibniz later elaborated, that this new species, which Leibniz called {transcendental}, could generate all lower, {algebraic} powers, whereas the inverse–to generate the transcendental from the algebraic–is impossible.

Like Plato, Cusa recognized that to comprehend these higher species of powers, the mind had to look beyond the boundaries of the domain of sense-perception, to discover the higher powers that govern it from the outside:

“I know that what I see perceptibly does not exist from itself. For just as the sense of sight does not discriminate anything by itself but has its discriminating from a higher power, so too what is perceptible does not exist from itself but exists from a higher power. The Apostle said, “from the creation of the world” because from the visible world as creature we are elevated to the Creator. Therefore, when in seeing what is perceptible I understand that it exists from a higher power (since it is finite, and a finite thing cannot exist from itself; for how could what is finite have set its own limit?), then I can only regard as invisible and eternal this Power from which it exists….” (Actualized-Potential)

Cusa gave his method a mathematical representation, but {not by the formal, logical deductive procedures of Aristotle, or his later followers, Euler, Lagrange, and the bigoted Cauchy}. In Cusa’s mathematical representation, what appears to be mathematically infinite, that is, what is over, or beyond, the finite, (what Cantor would later call the transfinite) indicates the existence of a new power or principle acting on the domain of sense-perception from outside. Thus, pushing finite mathematical expressions to the boundary, and beyond, reveals the characteristics of those sought for, but yet undiscovered, higher powers:

“…when we set out to investigate the Maximum metaphorically, we must leap beyond simple likeness. For since all mathematicals are finite and otherwise could not even be imagined: if we want to use finite things as a way for ascending to the unqualifiedly Maximum, we must first consider finite mathematical figures together with their characteristics and relations. Next, we must apply these relations, in a transformed way, to corresponding infinite mathematical figures. Thirdly, we must thereafter in a still more highly transformed way, apply the relations of these infinite figures to the simple Infinite, which is altogether independent even of all figure. At this point our ignorance will be taught incomprehensibly how we are to think more correctly and truly about the Most High as we grope by means of a metaphor.” (On Learned Ignorance, Book I.)

Cusa elaborated this metaphor extensively. He examined carefully the difference between the finite relations of the point, line, polygon and circle, and their transfinite relations. In the finite, these objects are each distinct, but transformed into their infinite forms, they become congruent, indicating the existence of the higher principle from which their finite relations are unfolded. Thus, the lower, finite forms are bounded (and governed) by the higher, which, from the standpoint of the lower, {appears} to be infinitely far away. Therefore, to know anything about the relationships in the finite, one must view them from the standpoint of the conditions at that seemingly infinite boundary. As we will soon illustrate, this relationship reaches a more developed form in Riemann’s treatment of Abelian functions.

Like Plato, Cusa associated this capacity of the human mind, to reach beyond the boundary of the apparent infinite, and identify new species of numbers as powers, with the immortality of the soul:

“Likewise, the exhibiting of the mind’s immortality can suitably be pursued from a consideration of number. For since mind is a living number, i.e., a number that numbers, and since every number is, in itself, incorruptible (even though number seems variable when it is considered in matter, which is variable), our mind’s number cannot be conceived to be corruptible. How, then, could the author of number [viz., mind] seem to be corruptible? Moreover, no number can deplete the mind’s power of numbering. Hence, since the motion of the heavens is numbered by the mind, and since time is the measure of motion, time will not exhaust the mind’s power. Rather, the mind’s power will continue on as the limit, measure, and determination of all things measurable. The instruments for the motions of the heavens instruments produced by the human mind attest to its not being the case that motion measures mind rather than mind’s measuring motion. Hence, mind seems to enfold by its intellective operation all movement of succession, and mind brings forth from itself rational operations, or rational movement. Thus, mind is the form of moving. Hence, if when anything is dissolved, the dissolution occurs by means of motion, then how could the form of moving be dissolved through motion? Since mind is an intellectual life that moves itself i.e., is a life that gives rise to the life which is its understanding how would it fail to be always alive? How could self-moving motion cease? For mind has life conjoined to it; through this life it is always alive. (By way of illustration: a sphere is always round, by virtue of the circle that is conjoined to it.) If mind’s composition is as the composition of number, which is composed of itself, how would mind be dissolvable into not-mind?…

…thus, he who numbers unfolds and enfolds. Mind is an {image} of Eternity, but time is an {unfolding} of Eternity…”(The Laymen on Mind)

Unlike the heathen Aristotle, for Cusa, the infinite is not outside the universe. What appears to be infinitely far away from the standpoint of sense-perception, is only the boundary marking the transition to the unseen, but efficient, domain of universal physical principles. Those principles reach through that boundary, into the visible domain, determining the characteristics of action within it. The mind, in turn, reaches back, transcending the apparently infinite, and discovering the characteristics of those principles. {Thus, the appearance of the infinite in mathematical expressions, simply indicates the necessity for a transformation of the mind’s idea a transformation that makes a universal principle that had appeared to be inaccessible, known.}

The Physical Generation of Powers

From Cusa, the road to the discovery of those higher species of transcendental powers that are the subject of Riemann’s work, goes through the investigations of Gauss and Abel who were provoked by:

–Kepler’s discovery that the principle of universal gravitation generated planetary orbits that were essentially elliptical, and;

–Leibniz’s determination that these orbits reflected a universal principle of physical least- action expressed by the catenary curve.

In rejecting the Aristotelean formalism of Ptolemy, Copernicus, and Brahe, and basing his investigation only on physical considerations, Kepler was compelled to reject the mathematically simpler circular orbits for the physically determined elliptical ones. This produced a crisis, because, whereas in a circular orbit, there is an incommensurability between the angle and the sine, in an elliptical orbit, there is an incommensurability between the arc and the angle as well. Kepler’s attempts to measure elliptical motion from the standpoint of the circular functions (i.e., as the connected action of a rectilinear triangle and a circular arc), produced the paradox known as “The Kepler Problem”. Kepler developed a method to approximate this measurement for practical purposes but, for him, this was insufficient. Kepler was not a pragmatist. Though practical solutions were vital to his work, Kepler was insistent on knowing the principle as well. When knowledge of that principle eluded him, Kepler demanded that future geometers solve the problem, insuring that even his own death would not end his quest.

While Kepler did not discover the higher principle of elliptical motion, he indicated where to look. In his related work on optics, Kepler proposed to look at the ellipse as merely one phase in a single function that generated all the conic sections. He expressed this geometrically through his famous projection that carried the circle into a hyperbola. (See Figure 1.) The emergence of an apparently infinite boundary between the ellipse and the hyperbola, within an otherwise continuous function, indicated the direction in which this undiscovered principle could be found.

Figure 1

The appearance of the infinite boundary between the ellipse and the hyperbola reflects, of course, the double cone construction of the conic section. On one side of the “infinite”, the circle, ellipse and parabola, only the lower cone is cut by an intersecting plane. It is when that plane touches the upper cone, that the hyperbola is formed. (See Figure 2.)

Figure 2

Leibniz, through an application of his infinitesimal calculus, showed that the curvature of the hanging chain can be expressed as the arithmetic mean between two exponential curves. (See Figure 3.) These exponential curves do not exist in the perceptible domain of the hanging chain. Nevertheless, they express a characteristic of the {physical manifold} that is acting, from outside the domain of sense-perception, on the chain, in every infinitesimal interval.

Figure 3

As Leibniz emphasized, these exponential curves express that transcendental species of power that generates all the lower algebraic ones. (See Figure 4.)

Figure 4

This same relationship is also expressed, in different forms, by the equiangular spiral and the hyperbola. (See Figure 5a, and Figure 5b.)

Figure 5a

Figure 5b

In connection with this investigation of the catenary, Leibniz coined the term {“function”} to denote a specified transformation. So, for example, the exponential function transforms arithmetic relations into the geometric ones. Its inverse, the logarithmic function, transforms geometric relations into arithmetic ones. As the arithmetic mean between two exponential functions, the catenary, therefore, is a function of two functions.

When Leibniz pushed this investigation to its boundary, by attempting to determine the function that generates the logarithms of negative numbers, he met the appearance of the square root of -1, which he identified as existing in a real, but “imaginary” domain.

That Undiscovered Country

Thus, the young Gauss, confronted with the appearance of what Leibniz called, “a fine and wonderful recourse to the divine spirit, almost an amphibian somewhere between being and non-being”, granted the square roots of negative numbers their “full civil rights”. From this standpoint he reexamined the “Kepler Problem”, and discovered that elliptical motion was governed by a higher species of transcendental, that had been anticipated by, but not known to Kepler or Leibniz.

The physical significance of complex numbers emerged quite naturally from Leibniz’s catenary principle and the paradox of negative logarithms. (See Figure 6.)

Figure 6

Since the catenary is formed as the arithmetic mean between {two} exponential curves, Leibniz sought the function that generated both, which could be expressed as that which generated their geometric mean. But, there is no continuous transformation, within the plane of the catenary, and the exponentials hanging behind it, that can transform one exponential into the other. The only physical action that has the power to produce both curves, is a rotation orthogonal to the plane of the catenary. (See Figure 7.)

Figure 7

If the direction of one exponential is designated as positive and the other direction negative, the geometric mean between them is expressed as the square root of a negative number. (See Figure 8.)

Figure 8

This gives rise to the following paradox: In the plane of the two exponentials, the geometric mean between them is a straight-line. But, when looked at from the higher standpoint of physical action, that straight-line is actually the axis of rotation from which is generated an entire surface orthogonal to the plane of the chain. Thus, to represent the physical action of the chain, the unseen exponentials that hang behind it, and the unseen principle that generates the exponentials themselves, requires both the plane of the chain, and the plane of the action orthogonal to it.

Gauss conceived the idea of representing this complex domain on one surface. That surface acts as a stage for the imagination on which {both} the physical action and the universal principles that govern it can be represented.

The two exponentials in the plane of the catenary are generated by an action that is completely outside their visible domain. The point at which the axis of rotation touches the surface generated by that rotation, i.e. the center of the circle formed by that rotation, is the only intersection between the visible domain and the surface on which the principle of generation is represented.

Just as on the Classical stage, a single line, (“Its Greek to me.”) in the context of the drama, can indicate an historical transformation of an entire culture, on the surface representing the complex domain, an entire principle, in the context of a complex function, can be represented by the action at a single point.

Hidden Harmonies Become Heard

By adopting Gauss’s approach, Riemann was able to create a stage, on which an innumerable class of transcendentals of successively higher power could be brought within the scope of the imagination. Each new species of transcendental, beginning with the simplest species of transcendentals discovered by Cusa, to the elliptical functions of Gauss, to those higher species discovered by Abel, is distinguished from its predecessor by the addition of a new principle. Under Riemann’s idea, each new principle is represented by a change in the geometrical characteristics, {the topology} of the surface.

Here Riemann adopted Leibniz’s method of {analysis situs} as applied by Gauss in his 1799 dissertation on the fundamental theorem of algebra. Gauss showed that formal algebra could not distinguish the implications of a change from one power to another. This is reflected, as Gauss ironically emphasized, in the inability, from the standpoint of formal algebra, to answer algebra’s most important question: “How many solutions are there for a given algebraic expression?” It was only when these expressions were thought of geometrically, as surfaces, that their essential characteristics could be made known. Then, it was clear, that the highest power of the function determined a topological characteristic (the number of “humps”) that was independent of any variations in the lower powers or the coefficients of that function. (See Figure 9a, and Figure 9b.)

Figure 9a

Figure 9b

Thus, Gauss’s employment of Leibniz’s geometry of position, {analysis situs}, expressed the epistemological relationship that the principle associated with the highest power dominated all lower ones. A change in the degree of power is, therefore, a change in the governing principle, and is expressed by a transformation of the entire geometry of the surface. Thus, the essential characteristics of an entire species of algebraic functions could be known completely, without the useless formal calculations of Euler, Lagrange and D’Alembert.

To illustrate Riemann’s approach, we begin with a look at the simplest species of transcendental, those associated with the circular, hyperbolic and exponential functions. As Leibniz and Bernoulli demonstrated the catenary, expressing the universal physical principle of least-action, embodied all three of these functions in one single physical action.

However, from the standpoint of the “simple” domain, these three functions seemed to have entirely different characteristics. For example, the circle and the hyperbola are conic sections, the exponential is not. In the visible domain, the circle is periodic, the hyperbola and exponential appear infinitely extended.

Leibniz imagined that a higher principle united all three transcendentals, but he never elaborated his idea. The sophisticated, but Leibniz-hating Euler, tried to obscure Leibniz’s insight with a series of algebraic formalisms, (such as his infamous equations: ei*Pi-1=0; and ei*x=cos[x]+i*sin[x]) which have bedeviled generations of students and scientists to this day.

Yet through Gauss’s and Riemann’s physical conception of the complex domain it can be easily demonstrated that these three functions express one unified species of transcendentals, and are all derived from the complex exponential. Gauss recognized this as early as August 14, 1796, writing in his notebook: “By the way, (a+b*?-1)m+n*?-1 has been explained”.

Here Gauss indicates he has extended the idea of the exponential function to the complex domain. As noted above, the exponential function is the transformation of arithmetic relationships into geometric ones. In the “simple” domain, this appears geometrically as the characteristic curvature of a single curve, as, for example, the equiangular spiral, hyperbola or exponential curve. In Gauss’s complex domain, the complex exponential function transforms the arithmetic relations of an entire surface into geometric ones, transforming all curves into new ones.

This can be seen most easily from the geometrical example in Gauss’s Copenhagen Prize Essay on conformal mapping (on which Riemann relied.). There Gauss shows that the complex exponential function corresponds to the stereographic projection of a sphere onto a plane a projection first developed by the ancient Greeks in connection with the mapping of the celestial sphere. (See Figure 10.)

Figure 10

Under this projection, the circles of “latitude” on the sphere are projected onto concentric circles on the plane, whose radii are expanding exponentially. Circles of “longitude” on the sphere are transformed into radial lines on the plane.

Don’t think of this transformation as a static image. Think of it from the standpoint of physical action. Think of the motion along a circle of latitude on the sphere. What is the corresponding motion on the plane? Think of the motion along a circle of longitude on the sphere. What is the corresponding motion in the plane? Think of motion along a path of equal heading on the sphere (loxodrome). What is the corresponding motion along in the plane? When you think of it in this way, you can begin to see how all relationships on the sphere are transformed, lawfully, into different relationships on the plane.

Here a crucial new idea emerges that cannot be seen in the “simple” domain. What appeared to be “infinitely” long radial lines of the plane, become “finite” circles of longitude on the sphere. Action on the plane that appears to go off into the “infinite” approaches a single point on the sphere. From the standpoint of Cusa, if the terminus of all the radial lines in the plane is thought of as the “infinite” circle, its image is a single point on the sphere. In other words, the principle bounding the action on the plane, that appears to be infinite and unknowable, is brought into the imagination, by the action associated with a single point on the sphere. Thus, in the complex domain, we can faithfully represent Cusa’s notion that in the infinite, the center and circumference, minimum and maximum, coincide.

It is important to emphasize, however, that physical action, such as the hanging chain, is never “infinite”. The catenary is the curve of a chain hanging between two positions. Thus, “this side of the infinite”, the physical exponential is always bounded. It is the universal principle that it reflects that is {transfinite}.

Another characteristic now comes into view that otherwise remained hidden with the “simple” view of the exponential. In the simple domain, the exponential curve appears aperiodic But, this is only an illusion caused by viewing the exponential too simply. As can be seen in the image of the stereographic projection, the radial lines are unbounded, but the circles of latitude are periodic. (As you can imagine, this period is real. It is Euler’s fraud to insist that the period of the complex exponential is “imaginary”.)

The hidden periodicity of the complex exponential can be seen more dramatically when viewed as a transformation of an orthogonal grid of curves on a plane, or, as Riemann suggested, on the surface of a very large sphere. (See Figure 11.)

Figure 11

Here all “straight” lines are transformed into a network of equiangular spirals, bounded by spirals with maximum rotation and minimum extension (i.e., circles), and spirals with maximum extension and minimum rotation (i.e., radial lines).

Think of this image from the standpoint of action. Under the transformation of the complex exponential, motion along a “straight”line is transformed into spiral motion, including the extreme cases of the circles and the radial lines. The non-periodic motion along the vertical lines is transformed into periodic circular action. (See Figure 12.)

Figure 12

The aperiodic motion along horizontal lines is transformed into motion along the exponentially extending radii, which remains aperiodic. Motion along a diagonal line is transformed into ever expanding spiral action. (See Figure 13.)

Figure 13

Now compare this result with what was noted above concerning the two exponential curves associated with the hanging chain. In the visible domain, this action is aperiodic. However, the rotation orthogonal to the plane of the hanging chain has a periodicity. Here we can recognize the physical reality of the exponential function’s “imaginary” period.

As Leibniz and Bernoulli demonstrated in their work on the catenary, the hyperbolic functions are functions of the exponential: the hyperbolic cosine is one-half the sum of two exponentials and the hyperbolic sine is one-half the difference of two exponentials. (See Figure 14.)

Figure 14

As Leibniz also insisted, the circular functions are also functions of the exponential, but only in a different, “imaginary”, domain. (See Figure 15.) The circular cosine is one-half the sum of two complex exponentials, while the circular sine is one-half the difference of two complex exponentials.

Figure 15

When viewed from the standpoint of Gauss’s and Riemann’s complex domain, all these functions can be expressed by the same type of action: the arithmetic mean between two complex exponentials moving in opposite directions, as illustrated by the circles in the accompanying animation. (See Figure 16a, Figure 16b, Figure16c, Figure 16d, and Figure 16e.) Together, the hyperbolic and circular functions form the four different variations of the same type of action. Thus, generated by the same type of action, they are all of the same species of transcendental.

Figure 16a

Figure 16b

Figure 16c

Figure 16d

Figure 16e

The Elliptical Transcendentals

The young Gauss, contemplating the origin of the “Kepler Problem”, was puzzled by the inability to express elliptical motion by the circular, hyperbolic or exponential functions. But, when he looked at a similar problem, that of the lemniscate, from the standpoint of the complex domain, he discovered a different species of transcendental that was expressed by elliptical motion. When the characteristics of that species are viewed on Riemann’s stage, the difference between the elliptical and the lower transcendentals becomes obvious.

While Gauss did not state it this way directly, his January 1797 decision to begin serious investigation of the lemnsicate alludes to the clue suggested earlier by Kepler’s concept of the conic sections.

As noted above, Kepler thought of all the conic sections as being generated from one continuous function. The emergence of an infinite boundary, between the ellipse and the hyperbola, reflects the transition to the inclusion of the upper cone in the double conical function. Thus, the entire function involves action in two different directions (parallel and perpendicular to the base of the cones). When thought of from the standpoint of Cusa, the appearance of the infinite signifies that a higher, undiscovered principle exists that encompasses both the perpendicular and parallel action as one. It is from this higher type of function that Kepler’s conic section function is unfolded.

This function must have the same relationship to the circular and hyperbolic functions as those functions have to the algebraic. That is, the higher species must have the capacity to generate the lower, but not the inverse. This is akin to the relationship between the quadratic and cubic species of magnitudes as viewed from the standpoint of Archytas. The principle of toroidal motion that generates cubic magnitudes can generate the quadratics as well, but not the inverse.

Now what is the nature of these elliptical transcendentals and how can we discover it? Let’s reconstruct Gauss’s investigation of the lemniscate from the standpoint of the clue supplied by Kepler, and the later solution supplied by Riemann.

That the lemniscate would become the focus of Gauss’s attention, flows quite naturally from the extension of Kepler’s concept of the conic sections into the complex domain. As presented in previous installments of this series, the lemniscate can be generated as the inversion of the hyperbola in a circle, or the stereographic projection of a hyperbola onto a sphere. (See Figure 17 and Figure 18.) From the standpoint of these two projections, it can be seen that the circle is the geometric mean between the hyperbola and the lemniscate. Thus, we seek a higher function that transforms the hyperbola, through the circle, into the lemnsicate.

Figure 17

Figure 18

It is important to take note of the method of inversion employed here. In the case of the two exponential curves and the catenary, the geometric mean between the two exponential curves could not be found within the visible domain of the curves and the hanging chain. The search for the principle that would generate both exponentials led us into the complex domain. In the case of the conic sections, Kepler’s function generates one from the other. But the appearance of the infinite in the middle of the function coaxes us to consider a higher function. That function generates, not another conic section, but a lemnsicate.

Now look at Kepler’s function, as he envisioned it projected onto a plane, and, from Riemann’s standpoint, projected onto a sphere. (See Figure 19.) In the former case, the circle is transformed into an ellipse, which is transformed into an hyperbola, and then into a line. In the latter, the circle is transformed into a projected ellipse, which is transformed into a lemnsicate, and then into two coincidental hemispheric circles.

Figure 19

Thus, to grasp the higher principle we have to think of both functions happening simultaneously: the one on the stage of the plane that is generating the conic sections, and the one on the stage of the sphere, that is generating a lemniscate. But from our vantage point we see both on {one} stage the stage of the complex domain.

Note that the infinite boundary that appears in the plane, appears in the spherical projection as the crossing point of the lemniscate. This presence of the {transfinite}, represented as a single point in the lemniscate, signifies the existence of an additional principle, that is outside the domain of the conic sections, but, {inside} the domain in which the lemnsicate resides. As noted in a previous installment, Gauss showed that the crossing point on the lemniscate also reflects the boundary between the regions of positive and negative curvature on the torus. (See Figure 20.)

Figure 20

Does the presence of this new principle indicate that the lemniscate is associated with a different species of transcendental than the circular or hyperbolic functions? Gauss said yes, and Riemann provided the basis to recognize this geometrically.

Gauss recognized the existence of this added principle in the lemniscate by the double periodicity of the lemnsicatic functions. Where, for example, the circular functions are periodic with respect to the interval 0 to 2Pi, the lemnsicatic functions are periodic with respect to the interval 1 to -1 {and} the interval \/-1 and -\/-1. (See Riemann for Anti-Dummies 52.)

Gauss expressed the geometrical characteristic of double periodicity in his work on bi-quadratic residues, which he published in 1832. However, its discovery was much earlier, as indicated by his notebook entry, “I have discovered a remarkable connection between the lemniscate and bi-quadratic residues”.

That connection is illustrated by the geometrical representation Gauss gave, with respect to simple and complex moduli, in his {Second Treatise on Bi-Quadratic Residues}. There Gauss shows that for real numbers, the modulus can be expressed geometrically by a simple line segment or curve. (See Figure 21a and, Figure 21b.)

Figure 21a

Figure 21b

However, in the complex domain, a complex modulus is expressed by a parallelogram on a surface. (See Figure 22.)

Figure 22

This same geometrical distinction, as Gauss noted, also appears in comparing the lemniscate with the circular functions in the complex domain. In the case of the circular, hyperbolic or exponential functions, motion in one direction was periodic, while motion perpendicular to it was not. As illustrated above, this type of motion could be mapped onto a surface such as a sphere.

However, for the lemnsicate, the motion is periodic in two directions simultaneously. (See Figure23a, Figure 23b, and Figure23c.)

Figure 23a

Figure 23b

Figure 23c

Such motion can not be represented on Riemann’s sphere, where one direction, such as the circles of latitude, are periodic, but, in the orthogonal direction, the circles of longitude, are not periodic without crossing the infinite.

Riemann showed that the elliptical functions could only be expressed on a surface that allowed for two distinct periodic curves. Such a surface can be constructed, Riemann showed, by taking the parallelogram of Gauss’s complex modulus and connecting the opposite sides to each other, forming a surface with the topology of a torus. (See Figure 24.)

Figure 24

On a torus, there are two distinct curves, one that goes around the outside, and one that goes through the hole. (See Figure 25.)

Figure 25

Riemann emphasized that the torus is an entirely different type of surface than a sphere. {And, there is no continuous function that can transform a sphere into a torus, without the introduction of a new principle. Once introduced, that new principle effects a complete transformation of all relationships on that surface.}

Through this investigation into the distinction between the higher species of elliptical transcendentals and the lower species, we are able to recognize that the difference between these two species of transcendentals reflects a fundamental change in the number of principles acting through each species. As Abel showed, an entire class of transcendentals can be constructed of successively higher power . In Riemann’s elaboration, the change from one species to the higher, conforms to a transformation from a domain of “n” principles to a domain of “n+1” principles. As in Archytas’s construction for the doubling of the cube, or Gauss’s method of his 1799 dissertation on the fundamental theorem of algebra, such transformations can be made intelligible on the stage of the complex domain, as a discontinuous transformation in the {topology} of that domain.

More importantly, however, armed with this new power to make intelligible the generation of higher species of transcendentals, we can now envision the transformation that must occur, in any domain when a new principle is added. Thus, through the willful employment of the mind’s power of discovery, we can reach past the boundary of what seems to be infinite, and bring new principles into our conscious possession.

And this brings us back to Theatetus, who, as a young boy, shows us today, that an entire species can be known, completely without calculation.

Riemann for Anti-Dummies: Part 54 : The Dramatic Power of Abelian Functions


The Dramatic Power of Abelian Functions

To understand Riemann’s treatment of Abelian functions, situate that discovery within the context of the history in which it arose, reaching back to the pre-Euclidean Pythagoreans of ancient Greece, and forward to LaRouche’s unique and revolutionary discoveries in the science of physical economy. Imagine that entire sequence, all at once, as a dramatic history, leap over time, project the past into the future, the future into the past, and both into the present, so that centuries of accomplishment are telescoped into a single, instantaneous thought.

Dynamis

In the opening scene of Plato’s {Theatetus} dialogue, Euclides of Megara informs Terpsion that he has just seen Theatetus being carried to Athens, near death, after being wounded in the battle at Corinth. This experience prompts Euclides to recall that Socrates, when near his own death many years before, had told him that “Theatetus will be a remarkable man if he lives.” Now, contemplating Theatetus’s life at its end, Euclides reports that Socrates had also recounted his first conversation with Theatetus when the latter was merely a boy. Having written down Socrates’s report, Euclides now reads the history to Terpsion, the which forms the bulk of Plato’s dialogue. As Euclides tells it, the conversation began with the young Theatetus being praised by his teacher Theodorus, because the former had surpassed the insights of the latter. As Theatetus explained, Theodorus had taught him about the incommensurablity of lines that double, triple, quintuple, etc. a square, by demonstrating each as a separate and distinct power, beginning with tripling and ending with seventeen, where, “for some reason he stopped.” But, Theatetus continued, “Since the number of powers are innumerable, the notion occurred to us of including them all under one name or species.”

Then by reference to a simple geometrical construction, Theatetus indicates the existence of three distinct {species} of powers: those that generate lines, those that generate squares and those that generate solids. Each species comprises an entire manifold of separate and distinct individual powers. But, each manifold, Theatetus explains, can be thought of under a single principle: linear, quadratic, or cubic.

Not mentioned directly in the dialogue, but prominent in the background, is the related discovery of Archytas, who showed that each species is associated with a different type of physical action: linear, circular and toroidal, respectively.

These discoveries of Theatetus and Archytas demonstrate a power of the mind that the Eleatics, Sophists and later, Aristotle, denied existed, and one that is essential to Riemann’s treatment of Abelian functions: {the capacity of the mind to rise above the finite determinations of sensible objects, and to recognize the higher universal principles that determine them}.

As Theatetus demonstrated, it is possible to {know} an entire species of powers, without having to construct each one individually, by knowing the principle that determines what each individual power can do e.g., double a square– and what they can’t– double a cube. To know that all these individual powers are of one species, {and} that the principle that generates square powers can never generate cubic ones, requires the mind to rise above the characteristics unique to each individual magnitude and recognize the nature of the entire species, {and} the nature of its boundaries.

Aristotle insisted that the human mind did not have this power because, being limited and mortal, it could not rise from finite determinations to universal ones:

“Some, as the Pythagoreans and Plato, make the infinite a principle in the sense of a self- subsistent substance, and not as a mere attribute of some other thing. Only the Pythagoreans place the infinite among the objects of sense (they do not regard number as separable from these), and assert that what is outside the heaven is infinite. Plato, on the other hand, holds that there is no body outside (the Forms are not outside because they are nowhere), yet that the infinite is present not only in the objects of sense but in the Forms also….”

“It is plain, too that the infinite cannot be an actual thing and a substance and a principle….

“Thus the view of those who speak after the manner of the Pythagoreans is absurd….

“This discussion, however, involves the more general question, whether the infinite can be present in mathematical objects and things which are intelligible and do not have extension, as well as among sensible objects. Our inquiry (as physicists) is limited to its special subject matter, the objects of sense, and we have to ask whether there is, or is not, among them a body which is infinite in the direction of increase.” (Physics, Book 5)

Aristotle’s argument is pure sophistry. By {defining} physics to concern only sensible objects, he excludes all consideration of the universal principles that generate those objects. Once excluded, he asserts that such principles play no role in the physical world, because being infinite, they cannot actually exist.

Contrary to Aristotle, as Theatetus demonstrates, in and through Plato’s dialogue, or as Socrates himself makes clear in numerous other locations, most notably the {Phaedo}, immortality, not temporal boundaries, characterizes the human spirit. It can freely transcend the finite limits of the sensual domain through its power to comprehend, not only things, such as lines, squares and cubes, but the powers that generate them. It can transform itself, as Theatetus had shown, from comprehending different powers individually, as Theodorus had taught, to comprehending the concept of an entire species of innumerable individual powers, as he had discovered.

And, as the above account of Plato’s account, of Euclides’s account, of Socrates’ account, of Theatetus’s account of his own discovery illustrates, the human spirit is not constrained by the mortal life in which it originates.

Powers that Generate Powers

The method described in the {Theatetus }dialogue is an elementary, but historically and pedagogically important example of the type of thinking that underlies Riemann’s method, elaborated in his {Theory of Abelian Functions}. There Riemann is dealing with the principles that generate higher and higher species of functions, beyond those contemplated by the Pythagoreans, Theodorus, Theatetus, Archytas and Plato. While Riemann is looking back on what was anticipated by these great ancient minds, he does so from the vantage point of the discoveries of Cusa, Kepler, Leibniz, Gauss and Abel. What these later discoverers demonstrated, was that the species of powers indicated by Plato et al., were superceded by a succession of higher species of successively greater power. Riemann, extending Gauss’s concept of the complex domain, created the means to comprehend the higher power that generates this extended class of higher powers. His treatment of Abelian functions gives expression to the quality of mind that can reach beyond the domain of sense perception, transcend what {appears} to be infinite, recognize the existence of new species of transcendental powers, and {generate} them as higher forms of cognition.

To understand Riemann’s elaboration of Abelian functions it is crucial to emphasize the epistemology underlying Cusa’s earlier discovery of a new species of power, by the method that Cusa called {“Learned Ignorance “}.

Cusa discovered that the incommensurability of the curved to the straight, as exemplified by the incommensurability between a circle and its inscribing and circumscribing polygons, indicated the existence of a new species of power, that transcended those named by Plato, et al. Cusa proved, and Leibniz later elaborated, that this new species, which Leibniz called {transcendental}, could generate all lower, {algebraic} powers, whereas the inverse–to generate the transcendental from the algebraic–is impossible.

Like Plato, Cusa recognized that to comprehend these higher species of powers, the mind had to look beyond the boundaries of the domain of sense-perception, to discover the higher powers that govern it from the outside:

“I know that what I see perceptibly does not exist from itself. For just as the sense of sight does not discriminate anything by itself but has its discriminating from a higher power, so too what is perceptible does not exist from itself but exists from a higher power. The Apostle said, “from the creation of the world” because from the visible world as creature we are elevated to the Creator. Therefore, when in seeing what is perceptible I understand that it exists from a higher power (since it is finite, and a finite thing cannot exist from itself; for how could what is finite have set its own limit?), then I can only regard as invisible and eternal this Power from which it exists….” (Actualized-Potential)

Cusa gave his method a mathematical representation, but {not by the formal, logical deductive procedures of Aristotle, or his later followers, Euler, Lagrange, and the bigoted Cauchy}. In Cusa’s mathematical representation, what appears to be mathematically infinite, that is, what is over, or beyond, the finite, (what Cantor would later call the transfinite) indicates the existence of a new power or principle acting on the domain of sense-perception from outside. Thus, pushing finite mathematical expressions to the boundary, and beyond, reveals the characteristics of those sought for, but yet undiscovered, higher powers:

“…when we set out to investigate the Maximum metaphorically, we must leap beyond simple likeness. For since all mathematicals are finite and otherwise could not even be imagined: if we want to use finite things as a way for ascending to the unqualifiedly Maximum, we must first consider finite mathematical figures together with their characteristics and relations. Next, we must apply these relations, in a transformed way, to corresponding infinite mathematical figures. Thirdly, we must thereafter in a still more highly transformed way, apply the relations of these infinite figures to the simple Infinite, which is altogether independent even of all figure. At this point our ignorance will be taught incomprehensibly how we are to think more correctly and truly about the Most High as we grope by means of a metaphor.” (On Learned Ignorance, Book I.)

Cusa elaborated this metaphor extensively. He examined carefully the difference between the finite relations of the point, line, polygon and circle, and their transfinite relations. In the finite, these objects are each distinct, but transformed into their infinite forms, they become congruent, indicating the existence of the higher principle from which their finite relations are unfolded. Thus, the lower, finite forms are bounded (and governed) by the higher, which, from the standpoint of the lower, {appears} to be infinitely far away. Therefore, to know anything about the relationships in the finite, one must view them from the standpoint of the conditions at that seemingly infinite boundary. As we will soon illustrate, this relationship reaches a more developed form in Riemann’s treatment of Abelian functions.

Like Plato, Cusa associated this capacity of the human mind, to reach beyond the boundary of the apparent infinite, and identify new species of numbers as powers, with the immortality of the soul:

“Likewise, the exhibiting of the mind’s immortality can suitably be pursued from a consideration of number. For since mind is a living number, i.e., a number that numbers, and since every number is, in itself, incorruptible (even though number seems variable when it is considered in matter, which is variable), our mind’s number cannot be conceived to be corruptible. How, then, could the author of number [viz., mind] seem to be corruptible? Moreover, no number can deplete the mind’s power of numbering. Hence, since the motion of the heavens is numbered by the mind, and since time is the measure of motion, time will not exhaust the mind’s power. Rather, the mind’s power will continue on as the limit, measure, and determination of all things measurable. The instruments for the motions of the heavens instruments produced by the human mind attest to its not being the case that motion measures mind rather than mind’s measuring motion. Hence, mind seems to enfold by its intellective operation all movement of succession, and mind brings forth from itself rational operations, or rational movement. Thus, mind is the form of moving. Hence, if when anything is dissolved, the dissolution occurs by means of motion, then how could the form of moving be dissolved through motion? Since mind is an intellectual life that moves itself i.e., is a life that gives rise to the life which is its understanding how would it fail to be always alive? How could self-moving motion cease? For mind has life conjoined to it; through this life it is always alive. (By way of illustration: a sphere is always round, by virtue of the circle that is conjoined to it.) If mind’s composition is as the composition of number, which is composed of itself, how would mind be dissolvable into not-mind?…

…thus, he who numbers unfolds and enfolds. Mind is an {image} of Eternity, but time is an {unfolding} of Eternity…”(The Laymen on Mind)

Unlike the heathen Aristotle, for Cusa, the infinite is not outside the universe. What appears to be infinitely far away from the standpoint of sense-perception, is only the boundary marking the transition to the unseen, but efficient, domain of universal physical principles. Those principles reach through that boundary, into the visible domain, determining the characteristics of action within it. The mind, in turn, reaches back, transcending the apparently infinite, and discovering the characteristics of those principles. {Thus, the appearance of the infinite in mathematical expressions, simply indicates the necessity for a transformation of the mind’s idea a transformation that makes a universal principle that had appeared to be inaccessible, known.}

The Physical Generation of Powers

From Cusa, the road to the discovery of those higher species of transcendental powers that are the subject of Riemann’s work, goes through the investigations of Gauss and Abel who were provoked by:

–Kepler’s discovery that the principle of universal gravitation generated planetary orbits that were essentially elliptical, and;

–Leibniz’s determination that these orbits reflected a universal principle of physical least- action expressed by the catenary curve.

In rejecting the Aristotelean formalism of Ptolemy, Copernicus, and Brahe, and basing his investigation only on physical considerations, Kepler was compelled to reject the mathematically simpler circular orbits for the physically determined elliptical ones. This produced a crisis, because, whereas in a circular orbit, there is an incommensurability between the angle and the sine, in an elliptical orbit, there is an incommensurability between the arc and the angle as well. Kepler’s attempts to measure elliptical motion from the standpoint of the circular functions (i.e., as the connected action of a rectilinear triangle and a circular arc), produced the paradox known as “The Kepler Problem”. Kepler developed a method to approximate this measurement for practical purposes but, for him, this was insufficient. Kepler was not a pragmatist. Though practical solutions were vital to his work, Kepler was insistent on knowing the principle as well. When knowledge of that principle eluded him, Kepler demanded that future geometers solve the problem, insuring that even his own death would not end his quest.

While Kepler did not discover the higher principle of elliptical motion, he indicated where to look. In his related work on optics, Kepler proposed to look at the ellipse as merely one phase in a single function that generated all the conic sections. He expressed this geometrically through his famous projection that carried the circle into a hyperbola. (See Figure 1.) The emergence of an apparently infinite boundary between the ellipse and the hyperbola, within an otherwise continuous function, indicated the direction in which this undiscovered principle could be found.

Figure 1

The appearance of the infinite boundary between the ellipse and the hyperbola reflects, of course, the double cone construction of the conic section. On one side of the “infinite”, the circle, ellipse and parabola, only the lower cone is cut by an intersecting plane. It is when that plane touches the upper cone, that the hyperbola is formed. (See Figure 2.)

Figure 2

Leibniz, through an application of his infinitesimal calculus, showed that the curvature of the hanging chain can be expressed as the arithmetic mean between two exponential curves. (See Figure 3.) These exponential curves do not exist in the perceptible domain of the hanging chain. Nevertheless, they express a characteristic of the {physical manifold} that is acting, from outside the domain of sense-perception, on the chain, in every infinitesimal interval.

Figure 3

As Leibniz emphasized, these exponential curves express that transcendental species of power that generates all the lower algebraic ones. (See Figure 4.)

Figure 4

This same relationship is also expressed, in different forms, by the equiangular spiral and the hyperbola. (See Figure 5a, and Figure 5b.)

Figure 5a

Figure 5b

In connection with this investigation of the catenary, Leibniz coined the term {“function”} to denote a specified transformation. So, for example, the exponential function transforms arithmetic relations into the geometric ones. Its inverse, the logarithmic function, transforms geometric relations into arithmetic ones. As the arithmetic mean between two exponential functions, the catenary, therefore, is a function of two functions.

When Leibniz pushed this investigation to its boundary, by attempting to determine the function that generates the logarithms of negative numbers, he met the appearance of the square root of -1, which he identified as existing in a real, but “imaginary” domain.

That Undiscovered Country

Thus, the young Gauss, confronted with the appearance of what Leibniz called, “a fine and wonderful recourse to the divine spirit, almost an amphibian somewhere between being and non-being”, granted the square roots of negative numbers their “full civil rights”. From this standpoint he reexamined the “Kepler Problem”, and discovered that elliptical motion was governed by a higher species of transcendental, that had been anticipated by, but not known to Kepler or Leibniz.

The physical significance of complex numbers emerged quite naturally from Leibniz’s catenary principle and the paradox of negative logarithms. (See Figure 6.)

Figure 6

Since the catenary is formed as the arithmetic mean between {two} exponential curves, Leibniz sought the function that generated both, which could be expressed as that which generated their geometric mean. But, there is no continuous transformation, within the plane of the catenary, and the exponentials hanging behind it, that can transform one exponential into the other. The only physical action that has the power to produce both curves, is a rotation orthogonal to the plane of the catenary. (See Figure 7.)

Figure 7

If the direction of one exponential is designated as positive and the other direction negative, the geometric mean between them is expressed as the square root of a negative number. (See Figure 8.)

Figure 8

This gives rise to the following paradox: In the plane of the two exponentials, the geometric mean between them is a straight-line. But, when looked at from the higher standpoint of physical action, that straight-line is actually the axis of rotation from which is generated an entire surface orthogonal to the plane of the chain. Thus, to represent the physical action of the chain, the unseen exponentials that hang behind it, and the unseen principle that generates the exponentials themselves, requires both the plane of the chain, and the plane of the action orthogonal to it.

Gauss conceived the idea of representing this complex domain on one surface. That surface acts as a stage for the imagination on which {both} the physical action and the universal principles that govern it can be represented.

The two exponentials in the plane of the catenary are generated by an action that is completely outside their visible domain. The point at which the axis of rotation touches the surface generated by that rotation, i.e. the center of the circle formed by that rotation, is the only intersection between the visible domain and the surface on which the principle of generation is represented.

Just as on the Classical stage, a single line, (“Its Greek to me.”) in the context of the drama, can indicate an historical transformation of an entire culture, on the surface representing the complex domain, an entire principle, in the context of a complex function, can be represented by the action at a single point.

Hidden Harmonies Become Heard

By adopting Gauss’s approach, Riemann was able to create a stage, on which an innumerable class of transcendentals of successively higher power could be brought within the scope of the imagination. Each new species of transcendental, beginning with the simplest species of transcendentals discovered by Cusa, to the elliptical functions of Gauss, to those higher species discovered by Abel, is distinguished from its predecessor by the addition of a new principle. Under Riemann’s idea, each new principle is represented by a change in the geometrical characteristics, {the topology} of the surface.

Here Riemann adopted Leibniz’s method of {analysis situs} as applied by Gauss in his 1799 dissertation on the fundamental theorem of algebra. Gauss showed that formal algebra could not distinguish the implications of a change from one power to another. This is reflected, as Gauss ironically emphasized, in the inability, from the standpoint of formal algebra, to answer algebra’s most important question: “How many solutions are there for a given algebraic expression?” It was only when these expressions were thought of geometrically, as surfaces, that their essential characteristics could be made known. Then, it was clear, that the highest power of the function determined a topological characteristic (the number of “humps”) that was independent of any variations in the lower powers or the coefficients of that function. (See Figure 9a, and Figure 9b.)

Figure 9a

Figure 9b

Thus, Gauss’s employment of Leibniz’s geometry of position, {analysis situs}, expressed the epistemological relationship that the principle associated with the highest power dominated all lower ones. A change in the degree of power is, therefore, a change in the governing principle, and is expressed by a transformation of the entire geometry of the surface. Thus, the essential characteristics of an entire species of algebraic functions could be known completely, without the useless formal calculations of Euler, Lagrange and D’Alembert.

To illustrate Riemann’s approach, we begin with a look at the simplest species of transcendental, those associated with the circular, hyperbolic and exponential functions. As Leibniz and Bernoulli demonstrated the catenary, expressing the universal physical principle of least-action, embodied all three of these functions in one single physical action.

However, from the standpoint of the “simple” domain, these three functions seemed to have entirely different characteristics. For example, the circle and the hyperbola are conic sections, the exponential is not. In the visible domain, the circle is periodic, the hyperbola and exponential appear infinitely extended.

Leibniz imagined that a higher principle united all three transcendentals, but he never elaborated his idea. The sophisticated, but Leibniz-hating Euler, tried to obscure Leibniz’s insight with a series of algebraic formalisms, (such as his infamous equations: ei*Pi-1=0; and ei*x=cos[x]+i*sin[x]) which have bedeviled generations of students and scientists to this day.

Yet through Gauss’s and Riemann’s physical conception of the complex domain it can be easily demonstrated that these three functions express one unified species of transcendentals, and are all derived from the complex exponential. Gauss recognized this as early as August 14, 1796, writing in his notebook: “By the way, (a+b*?-1)m+n*?-1 has been explained”.

Here Gauss indicates he has extended the idea of the exponential function to the complex domain. As noted above, the exponential function is the transformation of arithmetic relationships into geometric ones. In the “simple” domain, this appears geometrically as the characteristic curvature of a single curve, as, for example, the equiangular spiral, hyperbola or exponential curve. In Gauss’s complex domain, the complex exponential function transforms the arithmetic relations of an entire surface into geometric ones, transforming all curves into new ones.

This can be seen most easily from the geometrical example in Gauss’s Copenhagen Prize Essay on conformal mapping (on which Riemann relied.). There Gauss shows that the complex exponential function corresponds to the stereographic projection of a sphere onto a plane a projection first developed by the ancient Greeks in connection with the mapping of the celestial sphere. (See Figure 10.)

Figure 10

Under this projection, the circles of “latitude” on the sphere are projected onto concentric circles on the plane, whose radii are expanding exponentially. Circles of “longitude” on the sphere are transformed into radial lines on the plane.

Don’t think of this transformation as a static image. Think of it from the standpoint of physical action. Think of the motion along a circle of latitude on the sphere. What is the corresponding motion on the plane? Think of the motion along a circle of longitude on the sphere. What is the corresponding motion in the plane? Think of motion along a path of equal heading on the sphere (loxodrome). What is the corresponding motion along in the plane? When you think of it in this way, you can begin to see how all relationships on the sphere are transformed, lawfully, into different relationships on the plane.

Here a crucial new idea emerges that cannot be seen in the “simple” domain. What appeared to be “infinitely” long radial lines of the plane, become “finite” circles of longitude on the sphere. Action on the plane that appears to go off into the “infinite” approaches a single point on the sphere. From the standpoint of Cusa, if the terminus of all the radial lines in the plane is thought of as the “infinite” circle, its image is a single point on the sphere. In other words, the principle bounding the action on the plane, that appears to be infinite and unknowable, is brought into the imagination, by the action associated with a single point on the sphere. Thus, in the complex domain, we can faithfully represent Cusa’s notion that in the infinite, the center and circumference, minimum and maximum, coincide.

It is important to emphasize, however, that physical action, such as the hanging chain, is never “infinite”. The catenary is the curve of a chain hanging between two positions. Thus, “this side of the infinite”, the physical exponential is always bounded. It is the universal principle that it reflects that is {transfinite}.

Another characteristic now comes into view that otherwise remained hidden with the “simple” view of the exponential. In the simple domain, the exponential curve appears aperiodic But, this is only an illusion caused by viewing the exponential too simply. As can be seen in the image of the stereographic projection, the radial lines are unbounded, but the circles of latitude are periodic. (As you can imagine, this period is real. It is Euler’s fraud to insist that the period of the complex exponential is “imaginary”.)

The hidden periodicity of the complex exponential can be seen more dramatically when viewed as a transformation of an orthogonal grid of curves on a plane, or, as Riemann suggested, on the surface of a very large sphere. (See Figure 11.)

Figure 11

Here all “straight” lines are transformed into a network of equiangular spirals, bounded by spirals with maximum rotation and minimum extension (i.e., circles), and spirals with maximum extension and minimum rotation (i.e., radial lines).

Think of this image from the standpoint of action. Under the transformation of the complex exponential, motion along a “straight”line is transformed into spiral motion, including the extreme cases of the circles and the radial lines. The non-periodic motion along the vertical lines is transformed into periodic circular action. (See Figure 12.)

Figure 12

The aperiodic motion along horizontal lines is transformed into motion along the exponentially extending radii, which remains aperiodic. Motion along a diagonal line is transformed into ever expanding spiral action. (See Figure 13.)

Figure 13

Now compare this result with what was noted above concerning the two exponential curves associated with the hanging chain. In the visible domain, this action is aperiodic. However, the rotation orthogonal to the plane of the hanging chain has a periodicity. Here we can recognize the physical reality of the exponential function’s “imaginary” period.

As Leibniz and Bernoulli demonstrated in their work on the catenary, the hyperbolic functions are functions of the exponential: the hyperbolic cosine is one-half the sum of two exponentials and the hyperbolic sine is one-half the difference of two exponentials. (See Figure 14.)

Figure 14

As Leibniz also insisted, the circular functions are also functions of the exponential, but only in a different, “imaginary”, domain. (See Figure 15.) The circular cosine is one-half the sum of two complex exponentials, while the circular sine is one-half the difference of two complex exponentials.

Figure 15

When viewed from the standpoint of Gauss’s and Riemann’s complex domain, all these functions can be expressed by the same type of action: the arithmetic mean between two complex exponentials moving in opposite directions, as illustrated by the circles in the accompanying animation. (See Figure 16a, Figure 16b, Figure16c, Figure 16d, and Figure 16e.) Together, the hyperbolic and circular functions form the four different variations of the same type of action. Thus, generated by the same type of action, they are all of the same species of transcendental.

Figure 16a

Figure 16b

Figure 16c

Figure 16d

Figure 16e

The Elliptical Transcendentals

The young Gauss, contemplating the origin of the “Kepler Problem”, was puzzled by the inability to express elliptical motion by the circular, hyperbolic or exponential functions. But, when he looked at a similar problem, that of the lemniscate, from the standpoint of the complex domain, he discovered a different species of transcendental that was expressed by elliptical motion. When the characteristics of that species are viewed on Riemann’s stage, the difference between the elliptical and the lower transcendentals becomes obvious.

While Gauss did not state it this way directly, his January 1797 decision to begin serious investigation of the lemnsicate alludes to the clue suggested earlier by Kepler’s concept of the conic sections.

As noted above, Kepler thought of all the conic sections as being generated from one continuous function. The emergence of an infinite boundary, between the ellipse and the hyperbola, reflects the transition to the inclusion of the upper cone in the double conical function. Thus, the entire function involves action in two different directions (parallel and perpendicular to the base of the cones). When thought of from the standpoint of Cusa, the appearance of the infinite signifies that a higher, undiscovered principle exists that encompasses both the perpendicular and parallel action as one. It is from this higher type of function that Kepler’s conic section function is unfolded.

This function must have the same relationship to the circular and hyperbolic functions as those functions have to the algebraic. That is, the higher species must have the capacity to generate the lower, but not the inverse. This is akin to the relationship between the quadratic and cubic species of magnitudes as viewed from the standpoint of Archytas. The principle of toroidal motion that generates cubic magnitudes can generate the quadratics as well, but not the inverse.

Now what is the nature of these elliptical transcendentals and how can we discover it? Let’s reconstruct Gauss’s investigation of the lemniscate from the standpoint of the clue supplied by Kepler, and the later solution supplied by Riemann.

That the lemniscate would become the focus of Gauss’s attention, flows quite naturally from the extension of Kepler’s concept of the conic sections into the complex domain. As presented in previous installments of this series, the lemniscate can be generated as the inversion of the hyperbola in a circle, or the stereographic projection of a hyperbola onto a sphere. (See Figure 17 and Figure 18.) From the standpoint of these two projections, it can be seen that the circle is the geometric mean between the hyperbola and the lemniscate. Thus, we seek a higher function that transforms the hyperbola, through the circle, into the lemnsicate.

Figure 17

Figure 18

It is important to take note of the method of inversion employed here. In the case of the two exponential curves and the catenary, the geometric mean between the two exponential curves could not be found within the visible domain of the curves and the hanging chain. The search for the principle that would generate both exponentials led us into the complex domain. In the case of the conic sections, Kepler’s function generates one from the other. But the appearance of the infinite in the middle of the function coaxes us to consider a higher function. That function generates, not another conic section, but a lemnsicate.

Now look at Kepler’s function, as he envisioned it projected onto a plane, and, from Riemann’s standpoint, projected onto a sphere. (See Figure 19.) In the former case, the circle is transformed into an ellipse, which is transformed into an hyperbola, and then into a line. In the latter, the circle is transformed into a projected ellipse, which is transformed into a lemnsicate, and then into two coincidental hemispheric circles.

Figure 19

Thus, to grasp the higher principle we have to think of both functions happening simultaneously: the one on the stage of the plane that is generating the conic sections, and the one on the stage of the sphere, that is generating a lemniscate. But from our vantage point we see both on {one} stage the stage of the complex domain.

Note that the infinite boundary that appears in the plane, appears in the spherical projection as the crossing point of the lemniscate. This presence of the {transfinite}, represented as a single point in the lemniscate, signifies the existence of an additional principle, that is outside the domain of the conic sections, but, {inside} the domain in which the lemnsicate resides. As noted in a previous installment, Gauss showed that the crossing point on the lemniscate also reflects the boundary between the regions of positive and negative curvature on the torus. (See Figure 20.)

Figure 20

Does the presence of this new principle indicate that the lemniscate is associated with a different species of transcendental than the circular or hyperbolic functions? Gauss said yes, and Riemann provided the basis to recognize this geometrically.

Gauss recognized the existence of this added principle in the lemniscate by the double periodicity of the lemnsicatic functions. Where, for example, the circular functions are periodic with respect to the interval 0 to 2Pi, the lemnsicatic functions are periodic with respect to the interval 1 to -1 {and} the interval \/-1 and -\/-1. (See Riemann for Anti-Dummies 52.)

Gauss expressed the geometrical characteristic of double periodicity in his work on bi-quadratic residues, which he published in 1832. However, its discovery was much earlier, as indicated by his notebook entry, “I have discovered a remarkable connection between the lemniscate and bi-quadratic residues”.

That connection is illustrated by the geometrical representation Gauss gave, with respect to simple and complex moduli, in his {Second Treatise on Bi-Quadratic Residues}. There Gauss shows that for real numbers, the modulus can be expressed geometrically by a simple line segment or curve. (See Figure 21a and, Figure 21b.)

Figure 21a

Figure 21b

However, in the complex domain, a complex modulus is expressed by a parallelogram on a surface. (See Figure 22.)

Figure 22

This same geometrical distinction, as Gauss noted, also appears in comparing the lemniscate with the circular functions in the complex domain. In the case of the circular, hyperbolic or exponential functions, motion in one direction was periodic, while motion perpendicular to it was not. As illustrated above, this type of motion could be mapped onto a surface such as a sphere.

However, for the lemnsicate, the motion is periodic in two directions simultaneously. (See Figure23a, Figure 23b, and Figure23c.)

Figure 23a

Figure 23b

Figure 23c

Such motion can not be represented on Riemann’s sphere, where one direction, such as the circles of latitude, are periodic, but, in the orthogonal direction, the circles of longitude, are not periodic without crossing the infinite.

Riemann showed that the elliptical functions could only be expressed on a surface that allowed for two distinct periodic curves. Such a surface can be constructed, Riemann showed, by taking the parallelogram of Gauss’s complex modulus and connecting the opposite sides to each other, forming a surface with the topology of a torus. (See Figure 24.)

Figure 24

On a torus, there are two distinct curves, one that goes around the outside, and one that goes through the hole. (See Figure 25.)

Figure 25

Riemann emphasized that the torus is an entirely different type of surface than a sphere. {And, there is no continuous function that can transform a sphere into a torus, without the introduction of a new principle. Once introduced, that new principle effects a complete transformation of all relationships on that surface.}

Through this investigation into the distinction between the higher species of elliptical transcendentals and the lower species, we are able to recognize that the difference between these two species of transcendentals reflects a fundamental change in the number of principles acting through each species. As Abel showed, an entire class of transcendentals can be constructed of successively higher power . In Riemann’s elaboration, the change from one species to the higher, conforms to a transformation from a domain of “n” principles to a domain of “n+1” principles. As in Archytas’s construction for the doubling of the cube, or Gauss’s method of his 1799 dissertation on the fundamental theorem of algebra, such transformations can be made intelligible on the stage of the complex domain, as a discontinuous transformation in the {topology} of that domain.

More importantly, however, armed with this new power to make intelligible the generation of higher species of transcendentals, we can now envision the transformation that must occur, in any domain when a new principle is added. Thus, through the willful employment of the mind’s power of discovery, we can reach past the boundary of what seems to be infinite, and bring new principles into our conscious possession.

And this brings us back to Theatetus, who, as a young boy, shows us today, that an entire species can be known, completely without calculation.

Riemann for Anti-Dummies: Part 53 : Look to the Potential

Look to the Potential

In his 1857 {Theory of Abelian Functions}, Bernhard Riemann stated that the foundation of his theory of higher transcendental functions depended on what he called “Dirichlet’s Principle” and the method of thinking discussed by Gauss in his lectures on forces that act in proportion to the inverse square. These references help situate the historical specificity of Riemann’s discovery, the understanding of which, is essential to understanding the discovery itself.

The Gauss lectures to which Riemann referred were summarized by Gauss in an 1840 memoir titled, “General Propositions relating to Attractive and Repulsive Forces acting in the inverse ratio of the square of the distance”. There Gauss implicitly cast the fight between the physics of Kepler and Leibniz vs. the mystical pseudo-physics of Newton, Euler, Lagrange, and LaPlace, within the 2500 year ongoing drama between the two diametrically opposed views of man, as typified by the conflict between Plato’s concept of {power}, and Aristotle’s concept of {energy}. However, Gauss’s memoir is only one scene in the history. It was written under the political pressures of the post-1789 attack against the American Revolution that asserted, by force, the Aristotelean concept of man as animal associated with such frauds as Newton, Euler, and Lagrange. As such, it does not explicitly reveal all the elements of the play. But, as in any Classical drama (as well as physical processes), the organizing principles of the entire composition are always present and active in every part of every scene. Consequently, from a knowledge of the history of ideas, and the historical context in which Gauss was educated, lived and worked, it is possible to recreate, in the imagination, that great play to whose scene our attention is now drawn.

The Fraud of Newton’s Inverse Square “Law”

Given that Isaac Newton, by his own admission, was a fraud and a man convinced he had no soul (“Hypothesis non fingo”), it is a cause of some amazement that he was ever held in high esteem. This, of course, can be more easily understood, once one realizes that the high regard with which Newton has been held by some, was never for his physics, which, on its face, doesn’t work, but rather, for his degraded view of man from which his physics was derived.

A great mythology has been built up around Newton’s so-called theory of gravity, that asserts as a “law”, that every material body in the universe possesses an innate force that attracts every other material body in the universe according to the product of the masses divided by the square of the distances between them. Most modern textbooks and professors of physics demand blind obedience to this “imperial” law, from all who wish to be indoctrinated into their freemasonry. Such obeisance is demanded, not because this law is held to be true, but because it is so obviously false. In the tradition of Mephistopheles, these high priests seek to bind their supplicants to their cause, by forcing them to willingly embrace a doctrine that is contrary to human reason. And in the tradition of Marlowe’s Faust, too often these supplicants pay the price.

In fact, as Riemann enjoyed pointing out, Newton himself confessed his own immorality to Reverend Richard Bentley in 1693:

“Newton says: `That gravity should be innate, inherent, and essential to matter, so that one body can act upon another at a distance through a vacuum, without the mediation of anything else, by and through which their action and force may be conveyed from one to another, is to me so great an absurdity, that I believe no man who has in philosophical matters a competent faculty of thinking can ever fall into it.’ See third letter to Bentley.” (Riemann’s Gesammelte Werke).

To accept Newton’s fraud is to accept the Aristotlean concept on which it is based: that man, like animals, is limited to interacting with the physical universe (and by implication other human beings) only through the objects of sense perception, i.e. material bodies, the which exist in an empty, infinitely-extended Euclidean type space. Any change, or motion, that occurs among these bodies (or humans), originates in something innate to the bodies themselves, and not in any higher principle.

As Aristotle stated it:

“This then is one account of `nature’, namely, that it is the immediate material substratum of things which have in themselves a principle of motion or change.” (Aristotle’s Physics, Book II.)

The person adopting such a view is eliminating from the universe the active principles, or “powers”, as Plato called them, that exist outside the domain of sense-perception, but which make possible that which the senses perceive. For Aristotle, all that can be known about the interaction of physical bodies, is contained in the visible attributes of the bodies themselves (extension), and not in any higher principle. Similarly, all human interaction, for Aristotle (or Newton), is limited to the pair-wise interaction of human bodies, and not on such overarching governing influences as the individual human being’s willful interaction with history, language- culture and the simultaneity of eternity.

By contrast, Plato, and later Cusa, insisted that it is those higher powers, not the objects of sense, on which the mind of the scientist must focus. As Cusa expressed this Platonic principle in his dialogue “On Actualized-Potential”:

“CARDINAL: Temporal things are images of eternal things. Thus, if created things are understood, the invisible things of God are seen clearly for example, His eternity, power, and divinity. Hence, the manifestation of God occurs from the creation of the world.

BERNARD: The Abbot and I find it strange that invisible things are seen.

CARDINAL: They are seen invisibly just as when the intellect understands what it reads, it invisibly sees the invisible truth which is hidden behind the writing. I say “invisibly” (i.e. mentally) because the invisible truth, which is the object of the intellect, cannot be seen in any other way.

BERNARD: But how is this seeing elicited from the visible mundane creation?

CARDINAL: I know that what I see perceptibly does not exist from itself. For just as the sense of sight does not discriminate anything by itself but has its discriminating from a higher power, so too what is perceptible does not exist from itself but exists from a higher power. The Apostle said, “from the creation of the world” because from the visible world as creature we are elevated to the Creator. Therefore, when in seeing what is perceptible I understand that it exists from a higher power (since it is finite, and a finite thing cannot exist from itself; for how could what is finite have set its own limit?), then I can only regard as invisible and eternal this Power from which it exists….”

As a follower of Cusa, Kepler looked to this Platonic concept of higher powers to develop his notion of universal gravitation. Kepler understood the principle of universal gravitation to be that immaterial “species” (which is the Latin equivalent for the Greek word “idea”) that had the “power” to move the material planets. This species could be “seen” mentally, as Cusa indicates, as an object of the intellect (Geistesmassen in the language of Riemann and Herbart). This thought-object itself is recognized in the visible domain, again as Cusa indicates, by its harmonic characteristics, expressed, as Kepler showed, by such relationships as the five regular solids, the equal area principle of planetary motion, the musical relationships among the planets, etc. A characteristic of this species was, (as Leonardo da Vinci had shown for light), that its effect diminished with distance according to the inverse of the square of the distance.

{Thus, for Kepler, as for Plato and Cusa, the action of material bodies was not the result of some nature innate to the material itself, but was the visible, measurable effect of an invisible, but efficient and cognizable, physical “power”.}

This power to cognize such physical powers, and to act directly on those powers to control the physical processes themselves, expresses the efficient power of human cognition. This itself is a demonstration, contrary to Aristotle, Descartes and Newton, that the universe is not indifferent to human thought.

As Leibniz later wrote in his {Discourse on Metaphysics}:

“I do not accuse our new philosophers (Descartes), who claim to banish final causes from physics. But I am nevertheless obliged to confess that the consequences of this opinion appear dangerous to me, especially if I combine it with the one I refuted at the beginning of this discourse, which seems to go so far as to eliminate final causes altogether, as if God proposed no end or good in acting or as if the good were not the object of his will. As for myself, I hold, on the contrary, that it is here we must seek the principle of all existences and laws of nature, because God always intends the best and most perfect.”

Led by the Venetian monk Paolo Sarpi, the enemies of Cusa and Kepler sought to extricate from human practice the cognitive powers of the mind by eliminating consideration of physical powers from science. The lackeys Galileo, Descartes, and Newton, elaborated Sarpi’s intention, by developing a new pagan religion, called empiricism, based on an imperial decree that science should concern itself solely with the description, by precise mathematical laws, of the objects of sense-perception.

Exemplary, is Newton’s “law” of gravity. Where Kepler’s mind recognized universal gravitation as an idea that governed the motion of the celestial bodies, Newton saw only the bodies. Where Kepler recognized that the motions of the bodies were the visible effect of an invisible physical power, Newton saw bodies copulating, instantaneously at a distance, through an, ultimately mysterious, force, whose measure is proportional to the inverse of the square of the distance between them.

Leibniz’s Notion of Vis Viva

The exemplary fraud of empiricism immediately relevant to this discussion, was emphasized by Leibniz in his refutation of Descartes’ theories of motion. Adhering to Sarpi’s empiricism, Descartes insisted on Aristotle’s idea that material bodies were totally separate from immaterial principles, ideas or powers. Consequently, what could be known about the universe, according to Descartes, was limited to the perceptible characteristics of material bodies, such as size, shape, mass, and speed. Nothing, at least nothing knowable, existed outside these visible characteristics, and so, the motion of material bodies was assumed to occur in an empty Euclidean-type space.

Thus, disregarding the existence of physical principles, Descartes insisted that motion of a material body could only be measured by its visible effects. From this he deduced that a moving body contained a certain “quantity of motion” which, as Aristotle had indicated, was innate to the body itself. Descartes measured this “quantity of motion” as the mass of the body times its speed. When two bodies collided, the quantity of motion of one body was transferred to the other, which Descartes elevated to a universal principle that he called “conservation of quantity of motion.” When a falling body hit the ground, its effect was measured by the mass times its speed on impact.

Leibniz rejected Descartes’ empiricism on epistemological grounds. Writing, for example, in his 1690, {On The Nature of Body and the Laws of Motion}:

“There was a time when I believed that all the phenomena of motion could be explained on purely geometrical principles, assuming no metaphysical propositions, and that the laws of impact depend only on the composition of motions. But, through 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 the 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, and cannot in any way be deduced from the blind composition of motions. And thus, I learned that it is impossible for there to be nothing in the world except matter and its variations, as the Epicureans held.”

For Leibniz, the motion of a material body was the result, not of an innate nature of the body, but of a higher power, a capacity for motion that he called vis viva or “living force”. Thus, Leibniz did not seek to measure the visible effect of the motion. He sought to measure the “metaphysical” principle on which this physical effect depended.

In numerous locations, most notably, {Specimen Dynamicum}, Leibniz demonstrated that the higher principle, “living force”, was measured by the mass times the {square} of the speed. (fn 1.) As Leibniz demonstrated, the same force was necessary to raise a 1 pound body 4 feet as to raise a 4 pound body 1 foot, but the “quantities of motion” would be different. For the 1 pound body will hit the ground with a “quantity of motion” of 2 while the 4 pound body will hit the ground with a “quantity of motion” of 4. Thus, to produce the same quantity of motion requires a force proportional, not to the mass times the speed, but to the mass times the {square} of the speed.

Leibniz wrote, “There is thus a big difference between motive force and quantity of motion, and the one cannot be calculated by the other, as we undertook to show. It seems from that that {force} is rather to be estimated from the quantity of the {effect} which it can produce; for example, from the height to which it can elevate a heavy body of a given magnitude and kind, but not from the velocity which it can impress upon the body.”

The reader is encouraged to work through Leibniz’s demonstration for themselves. What is crucial to this discussion is that Leibniz’s physics is based on measuring the physical powers, not the visible effects. Descartes’ “quantity of motion”, measured by mass times speed, is an effect. Leibniz’s ” living force”, mass times the {square} of the speed, is the measurement of the principle that determines the effect.

Gauss’s Concept of Potential

While Leibniz’s discrediting of Descartes established the Leibnizian concept of quantity of force as the necessary measure of physical action, the fraud that Newton’s inverse square law was an adequate measure of universal gravitation persisted–enforced by the enemies of the American Revolution associated with Voltaire, Euler, Lagrange and the oligarchical controllers of Napoleon.

Kaestner, most notably, insisted, repeatedly and polemically, that Kepler and Leibniz were correct: that Descartes and Newton were only measuring the observable effects of physical action, while Kepler and Leibniz measured the physical principles themselves. Gauss’s 1799 new proof of the fundamental theorem of algebra demonstrated that even deeper physical principles could be measured, when expressed by the physical relationships of the complex domain.

After Kaestner’s death and Napoleon’s rise to power, Gauss was forced to be more cautious in his public discussions of these deeper issues, but he always insisted on focusing on physical principles. For example, in his great treatise on celestial mechanics, {The Theory of the Heavenly Bodies Moving About the Sun in Conic Sections}, Gauss treats Kepler’s principles as primary, and the inverse square law as a derived effect:

“The laws above stated differ from those discovered by {our own} Kepler in no other respect than this, that they are given in a form applicable to all kinds of conic sections…If we regard these laws as phenomena derived from innumerable and indubitable observations, geometry shows what action ought in consequence to be exerted upon bodies moving about the sun, in order that these phenomena may be continually produced. In this way it is found that the action of the sun upon the bodies moving about it is exerted just as if an attractive force, the intensity of which is reciprocally proportional to the square of the distance, should urge the bodies towards the center of the sun…” (fn. 2)

That there had even been any debate over the fraud of Newton’s inverse square law indicates only the viciousness of the political power that was used to enforce the adherence to such a foolish doctrine. Even on its own terms, the efforts to explain physical phenomena by taking the inverse square law as primary, leads to an even greater absurdity than that pointed out by Leibniz about Descartes “quantity of motion”.

For example, to apply the inverse square law, Newton treated all physical bodies as if their entire mass were concentrated in a simple Euclidean point. Furthermore, even with this contrived assumption, it is only possible to calculate the pair-wise relationship of two bodies. Once a third body is introduced, the calculations become, in principle, unsolvable.

But obviously, the universe is not limited to point-masses interacting pair-wise. As with his fundamental theorem of algebra, Gauss used an obvious fallacy–the one everyone danced around with sophistry, making fools of themselves by introducing into science, wild-eyed pagan rituals, in propitiation of an arbitrary authority– to establish, rigorously, the higher principles of science.

This is the standpoint of Gauss’s above mentioned lectures on forces acting according to the inverse square, where he treats the general problem, as it manifests itself in gravity, magnetism and electricity.

“Nature presents to us many phenomena which we explain by the assumption of forces exerted by the ultimate particles of substances upon each other, acting in the inverse proportion to the squares of their distance apart,” Gauss begins his treatise, indicating that the inverse square law is only an assumption, not a physical principle.

Gauss then demonstrates that the effort to explain these natural phenomena by the assumption of the inverse square leads to an inherent contradiction. First in the case of the difficulty of establishing the physical relationships among the interactions of many material bodies, and then as that problem applied to the obvious an extended body. That is, since no point-masses exist, how is it possible to calculate the infinite number of pair-wise interactions between all the parts of one body on all the parts of another?

Gauss’s solution is to throw out the inverse square law completely, and return to the method of Leibniz’s idea of measuring the underlying principle that produces the visible effect. In this case, however, Gauss goes beyond Leibniz, establishing a higher function that determines the physical geometry in which Leibniz’s living force exists.

Gauss called this function {potential}. His choice of the term{ potential }was deliberate, as potential is one of the Latin translations of the Greek idea of “power”. From Gauss’s standpoint, potential denotes a physical geometry, a phase-space, whose characteristic curvature, determines the nature of what action is possible in that phase-space. Thus, the motion of a material body is not occurring in an assumed Euclidean-type empty space, it occurs in a physical phase space whose characteristic curvature expresses the {potential} for the action.

The potential is not seen physically. It is seen only as an object of the mind. Yet, it is this mental object that governs the relationships among the objects of sense. Not being visible, it must be created, in the imagination, using the method of Plato and Cusa, from the paradoxes arising experimentally from physical action.

For purposes of pedagogical efficiency, this idea is best illustrated by examples.

First, take the example of Gauss’s measurements of the surface of the Earth. As Gauss emphasized, these are not measurements of an abstract mathematical geometric shape. These are physical measurements. A physical plumb bob is held on a string, a plane leveller is used to determine its perpendicular, and an angle is turned relative to these two directions. What is being measured, the material shape of the Earth, or that which is causing the plumb bob and bubble to move? If the latter, then what are the characteristics of that cause? Is it the combined interaction of every particle of the plumb-bob acting pair-wise with every particle of the Earth, as Newton’s method would demand? Or, does the motions of the plumb bob indicate the characteristic curvature of the {potential} for action of universal gravitation with respect to the Earth, as Gauss indicated?

Or similarly, when Gauss measured the Earth’s magnetism by the motion of a magnetic needle. Is that motion the result of pair-wise interactions of every magnetic particle of the needle with every magnetic particle of the Earth? Or, do the motions of the magnetic needle indicate the characteristic curvature of the {potential} for action with respect to the Earth’s magnetic properties?

Or, the case of two statically charged electrical objects? Is the repulsive or attractive force between these objects the result of the pair-wise interaction of every electrical particle of one object on every electrical particle on the other?

Gauss rejected such obviously foolish attempts as attempting to add up all the pair-wise interactions. Instead, he investigated these physical effects only as a consequence of the characteristics of the potential function: “[t]he investigation of the properties of this function will be itself the key to the theory of the attracting or repelling forces,” Gauss wrote.

As Gauss showed in his memoir, to investigate the properties of the potential function means to discover the characteristics of its curvature from the physical action.

Riemann Functions

To get an intuitive idea of Gauss’s concept of the potential, it will be pedagogically easier to first illustrate a simple example of Riemann’s more general concept of complex functions. With Riemann’s geometrical concepts in mind, we can then look back on the physical geometry of Gauss’s potential function.

Take the case of the simple “squaring” of a complex number. As Gauss showed in his fundamental theorem of algebra, complex numbers can be represented as a quantity of spiral action. “Squaring” that action is applying that spiral action twice. (See Figure 1.)

Figure 1

Riemann developed a geometrical concept for the general form of a function of a complex variable. Representing complex numbers on a surface, as Gauss did, Riemann considered a function of a complex variable as a Gauss mapping of that surface onto another surface. For example, the complex “squaring” function defines for every point on surface “a”, an image on surface “b”, which is the square of the original point on “a”. (See Figure 2.)

Figure 2

Riemann, following Gauss’s investigation of curved surfaces, showed that such a function transforms all relationships of one surface into a different set of relationships on the other surface. So, for example, an orthogonal grid of straight-lines on surface “a” is transformed into an orthogonal grid of parabolas on surface “b”. (See Figure 3.)

Figure 3

An orthogonal grid of circles and straight-lines on “a” is transformed into a similar grid on “b”. (See Figure 4.)

Figure 4

An orthogonal set of ellipses and hyperbolas on “a” transforms into a more complex set of relationships on “b”. (See Figure 5.)

Figure 5

Riemann then generalized two principles from Gauss. The first from Gauss’s study of conformal mappings and curved surfaces (See Riemann for Anti-Dummies Parts 44 through 48.), the second from Gauss’s investigation of the potential function.

In the first principle, Riemann recognized that if the geometrical conditions Gauss identified in his paper on conformal mapping were met, than images on surface “b” would all be conformal to their pre-images on surface “a”. (fn 3.) In other words, all angles would be preserved. (See Figure 6.)

Figure 6

In the second case, Riemann generalized a crucial concept from Gauss’s potential function. In a complex function there arise unique singularities that determine the general characteristic curvature, but around which that characteristic curvature changes. In this example of the squaring function, that singularity is the origin. Any action that includes this singularity will be different than a similar action that does not include the singularity. In animation 1 you see the effect of a square moving around a phase-space free of singularities. While it changes its shape, the angles between the sides remain orthogonal. But also notice the changes in the shape of the figure that includes the singularity within its boundaries. The presence of the singularity inside that shape, changes the whole shape’s relationship to the curvature and its shape changes dramatically differently than the shape of the singularity-free square.

Animation 1

In animation 2 you can see that even if the singularity free square is inside the boundary of the shape that contains the singularity, it still is unaffected, as long as the singularity remains outside of it. Its as if the singularity and the boundary are acting on each other, not at a distance, but as an effect of the organizing principle of the function.

Animation 2

Riemann also identified another geometrical characteristic that arises in this example of the squaring function. A point on surface “b” will correspond to two different points on surface “a”. From this, Riemann invented a new type of geometrical thought-object, now known as a “Riemann surface.” In this example, surface “b” will consist of two sheets connected both at the origin and the boundary. (See Figure 7.) In this way, the two different points of “a” will now correspond to two different points of “b”, each being on a different sheet.

Figure 7

It must be emphasized that the Riemann surface is a mental object, a metaphor, arising from the interaction of the mind with the physical universe. It is not an object of sense. However, it is more real than an object of sense, because it expresses the principles that govern an object of sense. But, it is not merely an abstract mental object divorced from the physical world, as its existence is completely connected to the physical processes whose investigation give rise to its formation. (A modern mathematician, degraded by his or her acceptance of the method of Descartes and Cauchy, will snarl at this paragraph.)

Gauss’s Potential from the Standpoint of Riemann Functions

Much more will be developed in future pedagogicals concerning these Riemann functions, but with these geometric ideas in mind, look again at Gauss’s concept of a potential function.

From this standpoint, Gauss’s potential function can be understood to express the curvature of a physical phase-space with respect to those universal principles acting in that phase-space, that are being investigated. The investigation of a different set of universal principles will define a different physical phase-space with a different curvature.

To keep it simple, consider the phase-space (potential function), of a material body with respect to universal gravitation. As expressed by a Riemann surface function, this phase-space has a characteristic curvature defined by the relationship between the boundary of the phase- space and the relationship of that boundary to the singularity.

For example, think of a physical action, from the standpoint of Leibniz’s concept of “living force”, that traverses a course in this phase space outside the surface of the body. Such as, the motion of a pendulum on the Earth or some other planet. Gauss’s potential function expresses the changes in the relationship of the pendulum’s “living force” to the phase space, as that pendulum swings within that potential phase-space. Note that the path of the pendulum does not encircle the Earth or the planet.

Now, think about the path of a satellite orbiting the Earth or the planet. A certain amount of force would have to be applied to move the satellite into its orbit, but the potential function indicates the existence of certain physical pathways, which, once achieved, require no changes in the relationship between Leibniz’s “living force” and the continued motion of the satellite. Note that such orbits complete encircle the planet.

(We omit, for now, the very curious question of the potential function inside the planet, or the characteristic of the potential function at the surface. These will be investigated in a future pedagogical.)

Note the relationship between this physical geometry expressed by Gauss’s potential function and the more general case of Riemann functions. Note the relationship of the material body in Gauss’s potential function to the singularity in Riemann’s surface functions.

It is in this more general form that the significance of Riemann’s theory of Abelian functions, for the frontiers of science, politics and art, is found.

FOOTNOTES

1. The fact that modern physicists refer to this quantity “mv2” as “kinetic energy” is itself a type of fraud whose purpose is to obscure the epistemological significance of Leibniz’s idea by clothing it with the name from Aristotle.

2. Emphases added by BMD. The use of the designation, “our own Kepler” is a direct reference to Kaestner who repeatedly criticized German science for turning its back on “our Kepler” and embracing the inferior British science of Newton.

3. It is one of the great frauds of the history of science, that these Gaussian conditions are referred to as the Cauchy-Riemann equations. The significance of these conditions was first identified by Gauss and then generalized by Riemann. Cauchy’s fraudulent formal treatment of these equations was an attack on the geometrical ideas of Gauss. To attach Cauchy’s name to Riemann’s in this regard is equivalent to the fraud of claiming the philosophy of John Locke instead of Leibniz’s as underlying the American Declaration of Independence and the American Revolution.

Riemann for Anti-Dummies: Part 52 : Abelian Functions and the Difference Between Man and Beast

Riemann For Anti-Dummies Part 52

ABELIAN FUNCTIONS AND THE DIFFERENCE BETWEEN MAN AND BEAST

All Aristoteleans are liars. In fact they must lie. For Aristoteleans believe that their minds are empty vessels, indifferent to what is put in them. They project this view of themselves onto the Universe, which, they insist, must conform to their degraded view of man: an empty box devoid of principles, and subject to no cognizable lawfulness. Devoid of principles both within and without, all statements, in the view of such Aristoteleans, can not be true, but only consistent within a logical deductive framework that rests not on universal principles, but on some arbitrary authority that determines a set of axioms, postulates and definitions.

Leibniz posed this issue in the preface to his {New Essays On Human Understanding}, which was written in response to the pro-beast-man John Locke’s {Essays On Human Understanding}:

“Our differences are about subjects of some importance. There is the question about whether the soul in itself is completely empty like tablets upon which nothing has been written (tabula rasa), as Aristotle and the author of the {Essay} maintain, and whether everything inscribed on it comes solely from the senses and from experience, or whether the soul contains from the beginning the source [principe] of several notions and doctrines, which external objects awaken only on certain occasions, as I believe with Plato and even with the Schoolmen, and with all those who find this meaning in the passage of St. Paul (Romans 2:15) where he states that the law of God is written in our hearts.”

Under the doctrine of Aristotle and Locke, man is no different than a beast– a point also posited by Liebniz, in the {New Essays on Human Understanding}:

“Also, it is in this respect that human knowledge differs from that of beasts. Beasts are purely empirical and are guided solely by instances, for, as far as we are able to judge, they never manage to form necessary propositions, whereas man is capable of demonstrative knowledge [sciences demonstratives]. In this, the faculty beasts have for drawing consequences is inferior to the reason humans have. The consequences beasts draw are just like those of simple empirics, who claim that what has happened will happen again in a case where what strikes them is similar, without being able to determine whether the same reasons are at work. This is what makes it so easy for men to capture beasts, and so easy for simple empirics to make mistakes. Not even people made skillful by age and experience are exempt from this when they rely too much on their past experiences. This has happened to several people in civil and military affairs, since they do not take sufficiently into consideration the fact that the world changes and that men have become more skillful in finding thousands of new tricks, unlike the stags and hares of today, who have not become any more clever than those of yesterday.”

Recently this author, along with several members of the LaRouche Youth Movement in the United States and Mexico, witnessed, on several occasions, a pedagogical demonstration of the above described conflict. In response to the distribution of LaRouche’s {Visualizing the Complex Domain} and {The Pagan Worship of Isaac Newton} in the context of a presentation on the epistemological and historical significance of Gauss’s attack on Euler, Lagrange and D’Alembert in his 1799 proof of the Fundamental Theorem of Algebra, various professors and students of mathematics were observed reacting with an hysterical defense of Newton and Euler and insisting that all knowledge must be stated in the form of logical, deductive mathematics. These individuals were observed insisting that obvious historical falsehoods were in fact true, solely because they could utter them with great energy. Such objections, coming from Aristoteleans, were, of course, disingenuous lies. It was not their beloved idols, Newton and Euler, alone that they were defending. More fundamentally, they were defending their right to lie through the Aristotelean methods which Newton and Euler exemplify. The insistence on knowable truth “was against the rules” and warranted the observed outbursts, much as an enraged animal defends his perceived boundaries of his territory.

With this in mind, now take a look at the history of the development of what have become known as “Abelian Functions”.

Niels Henrik Abel

In 1826, the Norwegian Niels Abel, then 24 years old, arrived in Paris as part of a tour of continental Europe. Abel had been sent to the continent by his teachers, Bernt Holmboe and Christopher Hansteen. Holmboe was Abel’s first teacher and he introduced the young Abel to Gauss’s {Disquisitiones Arithmeiticae} which Abel mastered quickly. Abel was particularly intrigued by Gauss’s remark in Section VII of that work, where Gauss states that his theory of the divisions of the circular functions could be applied to the lemniscate and elliptical functions as well.

Hansteen was a direct collaborator with Gauss, Alexander von Humboldt, and Dallas Bache, in the Magnetic Union and he was responsible for taking magnetic measurements throughout northern Europe. Recognizing Abel’s potential, Holmboe and Hansteen arranged to finance a tour of the continent so that Abel could interact with the leading thinkers of his day.

After a stay in Berlin, Abel proceeded to Paris, where he met the Jesuit-controlled Augustin-Louis Cauchy, whom he called, “a bigoted Catholic a strange thing for a man of science.” Cauchy had made his reputation writing excessively long dissertations on the formal manipulations of algebraic equations and complex functions (which were directed mainly as an attack on Leibniz’s idea of the infinitesimal) and he had established himself as a type of inquisitor within the French scientific community.

Abel, who had already published numerous ground-breaking discoveries, submitted a treatise titled, “Memoire on a General Property of a Very Extensive Class of Transcendental Functions” to the French Academy of Sciences. As the leading mathematical figure in the Academy, Cauchy was entrusted with the manuscript.

Disgusted with the reactionary climate of Paris, Abel left for Vienna, and then went back to Norway where, devoid of any income, he lived in poverty, contracted tuberculosis and died at the age of 26.

Meanwhile, the bigoted Cauchy took Abel’s manuscript home and kept it from being published. Only in 1829, after Abel’s death, when C. G. J. Jacobi heard of Abel’s memoire from Legendre, did its existence come to light. On reading a copy Jacobi wrote:

“What a discovery is this of Mr. Abel’s… Did anyone ever see the like? But how comes it that this discovery, perhaps the most important mathematical discovery that has been made in our century, having been communicated to your Academy two years ago, has escaped the attention of your colleagues?”

Despite Jacobi’s insistence, Cauchy still sat on the manuscript, allowing it to be published only after great pressure in 1841, 15 years after it was submitted, and 12 years after Abel’s death.

The subject of Abel’s discovery is indicated in the opening of the memoir:

“The transcendental functions hitherto considered by mathematicians are very few in number. Practically the entire theory of transcendental functions is reduced to that of logarithmic functions, circular and exponential functions, functions which, at bottom, form but a single species. It is only recently that some other functions have begun to be considered. Among the latter, the elliptic transcendentals, several of whose remarkable and elegant properties have been developed by Mr. Legendre, hold the first place. The author has considered, in the memoir which he has the honor to present to the Academy, a very extended class of functions…..”

The History of Abelian Functions

It was still not until Riemann’s 1857 “Theory of Abelian Functions” that the full significance of Abel’s discovery was brought to light, and it was not until LaRouche’s discoveries in the science of physical economy that the true significance of Riemann’s insights are made clear. In future pedagogicals we will go into more detail concerning the actual constructions of Abel and Riemann, from the standpoint of the higher development of these ideas by LaRouche. However, before embarking on those investigations, it is necessary to set the stage from the historical standpoint.

While the development of the higher transcendentals of Abel and Riemann properly begins with Kepler, it is essential to recognize Kepler’s discoveries from the standpoint of the Pythagorean, Platonic concept of power, as distinct from the Aristotelean concept of energy.

As Plato demonstrates in the Meno, and Theatetus, objects in the visible domain, such as lines, squares and cubes, are generated by powers that are not knowable through the senses. Nevertheless, such powers are perfectly cognizable, through the power of number. The Pythagorean/Platonic idea of number as a proportion that signifies a power, is distinct from the Aristotelean idea of a number that counts objects of the visible domain.

Speaking on this same subject in {The Laymen On Mind}, Nicholas of Cusa distinguishes these two concepts of number:

“I deem the Pythagoreans who, as you state, philosophize about all things by means of number to be serious and keen philosophers. It is not the case that I think they meant to be speaking of number qua mathematical number and qua number proceeding from our mind. (For it is self-evident that that sort of number is not the beginning of anything.) Rather, they were speaking symbolically and plausibly about the number that proceeds from the Divine Mind of which number, a mathematical number is an image. For just as our mind is to the Infinite, Eternal Mind, so number that proceeds from our mind is to number that proceeds from the Divine Mind. And we give our name “number” to number from the Divine Mind, even as to the Divine Mind itself we give the name for our mind…”

From this standpoint, we recognize the existence and characteristics of the powers that generate the actions we observe, from the characteristics of the numbers associated with the proportions those actions produce. For example, the number associated with the doubling of a square, is a special case of one geometric mean between two extremes. In the particular case of the square, the generating power is expressed by the incommensurability between the side of the square and its diagonal. The Aristotelean sees this type of number as “irrational” because it is more complicated than the simple whole number ratios that express simple linear proportions. But for Cusa, this incommensurability is actually simpler, because it indicates the existence of a higher power:

“Moreover, from the relation of the half-tone to a full tone and from the relation of a half a double proportion, this relation being that of the side of a square to its diagonal I behold a number that is simpler than our mind’s reason can grasp…”

In more general terms, the number associated with the diagonal of a square to its side is a special case of a whole class of magnitudes one geometric mean between two extremes which is a function of a type of curvature, i.e. circular rotation. As Archytas demonstrated for the case of the doubling of the cube, there exists a still higher class of magnitudes two geometric means between two extremes which are generated by a different curvature, i.e. conical action, as illustrated by his construction of the torus, cylinder and cone.

Cusa later demonstrated that these classes of numbers, which Leibniz would later call “algebraic”, are all subsumed by a higher class of numbers, that Leibniz called transcendental. Further, Cusa indicated that the generation of these classes of numbers is governed by the succession of discoveries of new physical principles:

“Likewise, the exhibiting of the mind’s immortality can suitably be pursued from a consideration of number. For since mind is a living number, i.e., a number that numbers, and since every number is, in itself, incorruptible (even though number seems variable when it is considered in matter, which is variable), our mind’s number cannot be conceived to be corruptible. How, then, could the author of number [viz., mind] seem to be corruptible?”

Starting from Cusa’s epistemological standpoint, Kepler demonstrated that the motion of the individual elliptical orbits of the planets are governed by a universal principle that cannot not be expressed by the numbers associated with simple circular action. To resolve this problem, Kepler demanded the development of a new mathematics.

That mathematics was supplied by Leibniz’s infinitesimal calculus, which, when applied to the problem of the catenary, demonstrated that the circular functions and the logarithmic/exponential functions, were united by the principle of least-action expressed by the catenary. That unified relationship pointed to the discovery of what Gauss would later call the complex domain. (See Riemann for Anti-Dummies Part 50. ).

But, when Leibniz’s calculus was applied to the elliptical orbits directly, a paradox resulted. This paradox was not a mathematical one, rather it indicated the existence of a new physical principle that could only be characterized by, a hitherto undiscovered, new type of number. (See Riemann for Anti-Dummies Parts 49 & Part 51.)

Kepler had already anticipated the existence of this higher type of transcendental in his investigation of the implications of conic sections for optics. Here Kepler recognized that all the conic sections could be generated by one continuous function. He expressed that function by the motion of the focus of the conic section. Thinking of a circle as an ellipse in which both foci are coincident, the other conic sections are generated by the motion of one focus. (See Animation 1.) Kepler noted the existence of a discontinuity, between the ellipse and the hyperbola, a discontinuity straddled by the parabola, which Kepler said had “one side toward the curved and the other side toward the straight.”

Animation 1

The discovery of this higher type of elliptical function drew the attention of the young Gauss through his investigation of the lemniscate. As discussed in previous installments, the lemniscate expresses the higher unifying principle among all the conic sections, as exemplified by its relationship as the inversion of a hyperbola in a circle (fn.1). (See Figure 1.)

Figure 1

And, from this relationship, Gauss understood the lemniscate to be the expression of a new type of transcendental that was higher than the circular and logarithmic transcendentals. This type of transcendental, like the algebraic magnitudes, or the circular and logarithmic functions, could not be expressed directly, but only by inversion. In other words, it could only be known as “that which expresses the power that generates the characteristic of this species of action.”

To get an intuitive grasp of Gauss’s insight, think about the generation of the circle from the circular functions. This requires the mind to get out of the domain of sense perception and into the domain of principles. For the circle is characterized by uniform motion. Yet, the circular functions, i.e., the sine and cosine, are non-uniform. From the domain of sense perception, it is more “comfortable” to generate the non-uniform from the uniform. But, from the domain of principles, it is the other way around. The non-uniform motion of the sine and cosine express the higher generating principle that produces what appears to be uniform motion. As Cusa insisted, and Kepler demonstrated, uniform motion does not exist in the physical universe. It is only an artifact of non-uniform, transcendental action.

Gauss’s method of the division of the circle proved, from the standpoint of the complex domain, this dependence of uniform on non-uniform motion. It was this investigation of the circle, which Gauss saw as a special case of ellipse, that led him to investigate the lemnsicate, the which later inspired the young Abel.

One way to illustrate this relationship was presented in the last installment of this series. (See Animation 2.) Another way is the following.

Animation 2

A circle can be generated by the uniform motion of one end of a line of fixed length (radius) which rotates while the other end is stationary. From the Pythagorean theorem, the relationship of the cosine and sine to the radius is proportional to the square root of 1minus a square. (See Figure 2.)

Figure 2

A circle can also be generated by varying the length of the moving line according to the cosine (or sine) of the angle it makes with a fixed line. (See Animation 3.) Expressing the length of the moving line (cosine) in terms of the sine, makes the arc of the circle vary according to the square root of 1-sine2.

Animation 3

However, when we allow the length of the moving line to vary by the cosine of double the angle, we produce two perpendicular lemniscates. (See Animation 4.) If we vary the length by the square root of double the angle, we generate one lemnsicate.

Animation 4

It can be remembered from the pedagogicals on the fundamental theorem of algebra, that doubling the angle squares the sine. (See Bringing the Invisible to the Surface: Gauss’s Declaration of Independence, Summer/Fall 2002 Fidelio.) Thus, if we express the length of the moving line in terms of the sine, the arc of the lemnsicate varies according to the square root of 1minus the square of a square, or the square root of 1-sine4.

From this relationship, Gauss recognized that in the complex domain the principle that generated the lemniscate expressed a fundamentally different type of relationship than the principle that generated the circle. First of all, the circular function, albeit non-uniform, generates uniform motion. But, the lemniscate function generates non-uniform motion. In the case of the circle, the sine (or cosine) is periodic. For example, the sine varies from 0,1,0,-1,0 for each rotation around the circle. (See Figure 3.)

Figure 3

But, since the functions that generated the lemniscate vary according to the fourth power, these functions have two periods. 0,1,0,-1,0 and 0,i,0,-i,0. (where i =square root of -1.) (See Figure 4.)

Figure 4

Thus, the power that generates all conic sections, as expressed by the lemniscatic functions is a higher type of transcendental, which generates the non-uniform action of the ellipse by two distinct, but connected, relationships.

These early investigations of Gauss were never published and they didn’t become known until Gauss’s notebooks were discovered in the 1890’s. But, from the intriguing remark in the {Disquisitiones Arithmeticae} the young Abel reconstructed Gauss’s discovery for himself and then went still further. Abel recognized that the lemniscate, and the related elliptical functions, were only the first step of an “extensive class of higher transcendental functions.” Thus, the circular and logarithmic functions were but a special case of the elliptical, which in turn were a special case of what have since become known as “Abelian” functions.

But, such functions were not supposed to exist in the bigoted animal world of the Aristotelean Cauchy, so he tried to cover them up with a lie.

The truth won out. And there, in part, begins Riemann’s theory of Abelian Functions. Inversion expresses the proportion that the distance from the center of the circle is to a point on the hyperbola is to the radius of the circle, as the radius of the circle is to the distance from the center of the circle to the corresponding point on the lemniscate.

Riemann for Anti-Dummies: Part 51 : The Power of Number

Riemann for Anti-Dummies Part 51

THE POWER OF NUMBER

Nicholas of Cusa begins “On Learned Ignorance”, by reaching back to the method of Pythagoras:

“Therefore, every inquiry proceeds through proportion, whether an easy or difficult one. Hence, the infinite qua infinite, is unknown; for it escapes all proportion. But since proportion indicates an agreement in some one respect and, at the same time, indicates an otherness, it cannot be understood independently of number. Accordingly, number encompasses all things related proportionally. Therefore, number, which is a necessary condition of proportion, is present not only in quantity but also in all things which in any manner whatsoever can agree or differ either substantially or accidentally. Perhaps for this reason Pythagoras deemed all things to be constituted and understood through the power of numbers.”

Here, Cusa adopts the view of Plato, that numbers arise from the inseparable interaction between the human mind and the physical universe. The mind expresses the concepts it creates, about the principles it discovers, through the power of numbers. These concepts themselves become objects of the mind’s investigation, and the relationships among these “thought-objects”, also give rise to concepts, which are themselves expressible through the power of number.

Numbers in and of themselves have no power. They are like ironies that point, by inversion, to the principles that govern the physical universe. Those principles don’t exist in the numbers. They exist “behind” the numbers. Thus, the significance of Cusa’s reference that the Pythagoreans deemed all things to be {constituted and understood} through the power of numbers.

Against Cusa’s understanding of Pythagoras is the far different fraud perpetrated by the Sophists who, then and now, insist on separating the conjoined idea {constituted and understood}. For them, as for Aristotle, how things are {constituted}, and how things are {understood}, are two separate and mutually exclusive actions. In his “Metaphysics”, Aristotle attacks both Plato and the Pythagoreans for their insistence that ideas are an active principle in the Universe:

“But the lauded characteristics of numbers, and the contraries of these, and generally the mathematical relations, as some describe them, making them causes of nature, seem, when we inspect them in this way, to vanish; for none of them is a cause in any of the senses that have been distinguished in reference to the first principles. In a sense, however, they make it plain that goodness belongs to numbers, and that the odd, the straight, the square, the potencies of certain numbers, are in the column of the beautiful. For the seasons and a particular kind of number go together; and the other agreements that they collect from the theorems of mathematics all have this meaning. Hence they are like coincidences. For they are accidents…”

Aristotle’s method is pure sophistry. As Cusa and Plato both indicate, number arises in the {human} mind through the effort to discover the unseen universal principles that govern the world behind the senses. Thus, the power of numbers is {deliberate}, not accidental. However, Aristotle plays the trick of separating the world of mind and the world of matter, and so, for him, any connection between number and the physical world is purely accidental.

But Aristotle does not have clean hands. He is the hired-gun who provides the oligarchy with the method it needs, to create the cultural basis it requires, to assume its arbitrary authority over humanity. By introducing this false separation between the universe of sensible things, the unseen principles that govern them, and the thoughts by which we understand those principles, Aristotle excises human cognition as an active principle from the Universe. Once cleansed of cognition, he creates a false universe, indifferent to the power of human thought, unknowable, and governed by mysterious forces accessible only to those with access to special incantations, such as the moans of the oracles of ancient Delphi, or the formulas of formal mathematics.

That is the form of sophistry from which Cusa rescued civilization.

This is not an arcane technical argument, but one that goes to the heart of the difference between man and beast.

As Cusa stated this explicitly in “On Conjectures”:

“The natural sprouting origin of the rational art is number; indeed, beings which possess no intellect, such as animals, do not count. Number is nothing other than unfolded rationality…

“…we conjecture symbolically from the rational numbers of our mind in respect to the real ineffable numbers of the divine Mind, we indeed say that number is the prime exemplar of things in the mind of the Composer, just as the number arising from our rationality is the exemplar of the imaginal world.”

Cusa’s view was later adopted by Leibniz, who wrote in “Reflections on the Souls of Beasts”:

“However, lest we seem to equate man and beast too closely, it should be known that there is an enormous difference between the perception of humans and beasts. For besides the lowest degree of perception that is found even in insensible creatures, and (as been explained) a middle degree which we call sensation and acknowledge in beasts, there is a certain higher degree which we call thought. But thought is perception joined with reason, which beasts so far as we can observe do not have.

“…However, a human being, insofar as he does not act empirically but rationally, does not rely solely on experience, or a posteriori inductions from particular cases, but proceeds a priori on the basis of reasons. And this is the difference between a geometer, or one trained in analysis, and an ordinary user of arithmetic, teaching children, who learn arithmetical rules by rote, but do not know the reason for them, and consequently cannot decide questions that depart from what they are used to: such is the difference between the empirical and the rational, between the inferences of beasts and the reasoning of human beings….Thus, brutes (as far as we can observe) do not acquire knowledge of the universality of propositions, because they do not understand the ground of necessity. And even if empirics are sometimes led by inductions to universally true propositions, this nonetheless happens only accidentally, not by force of entailment.”

Creating Number

This concept of number is what Gauss had in mind when, provoked by the paradoxes associated with the “Kepler Problem”, he sought the discovery of those hitherto unknown higher transcendentals indicated by Kepler’s discovery. Gauss recognized, in the tradition of the Pythagoreans, Plato, Cusa and Leibniz, that these new numbers, like all numbers, cannot be defined by any set of formal, deductive rules, such as in the methods of Aristotle, or his philosophical protege, the Leibniz-hating Euler. Rather, Gauss understood that these new transcendentals, like all numbers, could only by defined by {inversion} with respect to a physical process.

To illustrate this point, take a case you think you are familiar with, and look at it, as Gauss did, from an entirely new standpoint: the number associated with the relationship of two squares whose areas are in the proportion 1:2, which is sometimes called, “the square root of two”.

Construct a square from a circle. Now construct a square whose area is double. How do you know the areas of these two squares are in the proportion 1:2? Not by looking at them. There is nothing in the visible appearance of the two squares from which you can {know} the proportion of their areas. You can know this only from the method of construction, as Plato discusses that method in the Meno dialogue.

Now, look at the diagonal and the side of the square. These look the same. Both are lines. There is nothing in the visible appearance of these lines from which you can {know} their proportion, other than to say one is a little longer than the other. Yet, the Pythagoreans {proved} that the two lines are {incommensurable}, or in other words, had no common measure. This incommensurability cannot be demonstrated from the visible appearance of the lines, but, only, as the Pythagoreans did, by investigating the relationship itself between magnitudes that are commensurable in length and those commensurable as squares. The Pythagoreans showed that no linear proportion could possibly exist, whose square is in the proportion 1:2. (See Jason Ross article Fall 2003 21st Century.).

Most importantly, this characteristic of incommensurability is independent of the actual size of the squares. It depends only on the proportion between their areas, which is determined only by construction. As such, this incommensurability expresses a characteristic of the physical process of the construction of the squares. Consequently, the number called “the square root of 2”, cannot be defined by any set of rules, definitions or procedures such as a logical deductive method, but only by an inversion, as {that which expresses the principle with the power to generate two squares whose areas are in the proportion 1:2}.

In the visible domain, we proceeded in the opposite manner. We constructed the squares and produced the magnitude called, “the square root of two”. But from the standpoint of physical principles, it is the incommensurability of the magnitude called the square root of two which expresses the {power} (possibility) to produce two squares whose areas are in the proportion 1:2.

As Thales, Pythagoras, Theodorus, and Theatetus, further demonstrated, the square root of two is merely a special case of a more general species of relationships generated from circular action, which they called one geometric mean between two extremes. (See Figure 1.) As point P moves from A to O, the length of the line QP will always be the geometric mean between the lengths of line OQ and QA. When OQ is one half of QA, then QP is the magnitude called “the square root of 2”.

Figure 1

The particular proportion is independent of the size of the circle or the actual lengths of the lines. It depends solely on the position of P with respect to A and O, the which is determined by the circular “orbit” on which P travels. Thus, the circular “orbit” on which P travels produces, as a whole, a complete “type” or “species” of proportions, with respect to the magnitudes OQ, QP and QA.

This is one of the simplest examples of the Greek method of the geometry of position (“topos”) known today by the Latin word, “loci”, in which proportions are understood as generated from some physical action.

Another famous example of this method of loci is Archytas’s construction of cubic magnitudes. Here, an entirely new species of magnitudes is generated by combining two degrees of circular rotation orthogonal to each other, producing the torus and cylinder whose intersection defines a different “orbit” on which P travels. Again, these relationships are dependant only on the characteristic of the “orbit”, not on the size of the circles, or the surfaces generated from them.

The solid loci, or conics, of Menaechmus and Apollonius, follows the same principle, of generating both square and cubic magnitudes from {action acting on action}.

But here we seem to have run into a boundary. The change in the motion that generates one mean between two extremes (squares) to the motion that generates two means (cubes), was effected by introducing a second degree of circular action acting orthogonally to the first. Visible space, however, does not permit the addition of a third degree of circular action acting orthogonally to these two. Does that mean that no higher powers are possible? And, if they are possible, how are they manifest physically, as distinct from some formal mathematical algebraic definition?

Ultimately, it was Leibniz’s discovery of the catenary principle that demonstrated the physical existence of these higher powers. But that discovery rested on a previous one by Cusa, which is necessary to review for the sake of the subject matter to follow.

Look again at the generation of geometric means from circular action, but this time from the standpoint of inversion. As was demonstrated above, the circular “orbit” of P generates the entire species of geometric means. But what about the inverse? Can the entire species of geometric means produce a circle?

Cusa demonstrated that the answer was no, that, in fact, there exists a higher principle that generates the circle, which Leibniz later called, “transcendental”. These “circular transcendentals” are identified with those interrelated magnitudes known as trigonometric functions, and the relationships among them express the incommensurability between the curved and the straight. (See Figure 2.)

Figure 2

As Cusa showed, this incommensurability between the curved and the straight, is a different type than the incommensurability expressed by squares or cubes, and, the higher “algebraic” powers. Later, Leibniz, through his discovery of the catenary principle and its relationship to natural logarithms, demonstrated that all species of algebraic powers are generated by these transcendental functions, and it is these transcendental magnitudes, not the algebraic, that express the relationships that arise in the physical universe.

It is crucial to restate this point in this form:{ The transcendental functions, not the algebraic, are those functions that are inverse to circular rotation}. The significance of this statement will become more clear from the standpoint of the discovery of the higher, elliptical functions, by Gauss and Riemann.

Elliptical Functions From Kepler to Leibniz

Through his education by E.A.W. Zimmerman and A.G. Kaestner, both leading defenders of Leibniz and Kepler, Gauss was focused, from his early adolescence, to investigate the implications of the paradoxes arising from the “Kepler Problem” that were left unresolved by Leibniz’s invention of the infinitesimal calculus. (See Riemann for Anti-Dummies, Part 49, Aug. 16, 2003.)

As Kepler demonstrated, the ellipticity of a planetary orbit demanded a new type of geometry of position, one that expressed the position of the planet as a function of the characteristic of change of the orbit as a whole. Kepler’s effort to develop this concept resulted in his famous principle of equal areas. He recognized that in every interval of an elliptical orbit, no matter how small, the motion of the planet at the beginning of that interval is different than at the end. The only interval excepted is one entire orbital period. In that case, the planet is doing the same thing at the beginning and end of the interval. Thus, Kepler made the entire orbit the primary interval of action, and measured the planet’s intermediate motion as portions of the whole orbit.

In The New Astronomy, Kepler, citing Archimedes, measures this relationship by the proportion of the total area of the planet’s orbit, to the area swept out in a given interval. He defines the area swept out by the planet as the “sum” of the infinite number of radial distances:

{“For the whole sum of the radial distances is, to the whole periodic time, as any partial sum of the distances is to its corresponding time.”}

For Kepler, the planet’s motion was thus measured by the changing proportion between the part of the orbit and the whole. This proportion, Kepler understood to be the “time-elapsed” within any given interval.

It is extremely important to recognize the difference between Kepler’s actual principle and the Newtonian-algebraic formulation, misidentified as “Kepler’s second-law”, and stated in text-book gossip circles as: “the planet sweeps out equal areas in equal times”. This historically and epistemologically false characterization is a clinical example of an Aristotlean-type sophistry in two important ways. First, it falsely characterizes Kepler’s discovered principle as a mathematical “law”, thereby excising out the cognitive action. Second, by stating this law in the form of a proportion between area and time, it transforms time and space, by fiat definition, into independent absolute magnitudes. In this way, the Universe is turned on its head. Instead of recognizing the characteristics of space and time as functions of the physical motion of the planet, the fantasy pseudo-world of absolute time and space is held to define the planet’s motion.

Responding to Kepler’s demand for a generalization of his principle, Leibniz developed the infinitesimal calculus, by extending Kepler’s proportionality between the part and the whole, into infinitesimal intervals of action.

For Leibniz, the infinitesimal is physically determined as a proportionality, as Cusa understood proportionality, between the inseparable part and whole of a physical process. His critics reacted by reaching back to the ancient sophistries of the Eleatics, and, like Zeno, posed the paradoxes of physical motion from the standpoint of an arbitrary formal mathematical definition of a curve. Leibniz defended himself from these attacks during his lifetime, but after his death, the oligarchy recruited Euler to give a more “academic” imprimatur to these attacks, as expressed most blatantly in his Letters to a German Princess.

In response to Euler, Leibniz was posthumously defended, on precisely this point, by those who were fighting to establish the American Republic against the oligarchy that employed Euler, most notably by A.G. Kaestner. (See Appended essay by Kaestner, Moral from the History of the Infinitesimal Calculus.)

Elliptical Functions of Gauss

Ironically, while Leibniz’s discovery of the catenary principle demonstrated the power of his infinitesimal calculus, its application to the elliptical orbit, the problem that had provoked its invention, led to the paradox known as the “Kepler Problem”. The essential characteristics of this paradox can be understood by investigating the difference between a circular orbit and an elliptical one from the standpoint of Kepler and Leibniz.

In a circular orbit the area swept out can be measured directly by the angle. In the elliptical orbit, as Kepler showed, the area swept out is measured by a circular sector and a rectilinear triangle. (See Figures 3).

Figure 3

The incommensurability between the triangle and the circular sector leads to the “Kepler Problem”. (See Figure 4.)

Figure 4

Gauss recognized that this paradox arose from the effort to measure the elliptical action of the planet by circular functions. The problem here was that while the circular functions reflect the incommensurability between the arc and the line, in elliptical action there exists an incommensurability between the arc and the angle as well. (See Figure 5.) Gauss realized that since these two types of incommensurability were connected, both arising from a unified elliptical action, there must exist a new, more general type of elliptical transcendental, of which the circular functions were only a special case. The complex domain was required to make intelligible the relationships associated with these new transcendentals.

Figure 5

This investigation forms the core of Gauss’s youthful work, as a review of his early notebooks and correspondence reveals. Gauss presented some aspects of these discoveries more formally in his 1799 new proof of the Fundamental Theorem of Algebra and his Disquisitiones Arithmeticae, as well as his later works in astronomy, geodesy, and curvature. But because of the tyranny imposed on Europe in the post-1789 reaction to the American Revolution, (especially after the 1799 consolidation of Napoleon’s rise to power and its aftermath) some of these discoveries were never published, and only found their expression in the work of the next generation of scientists, most notably, Dirichlet, Riemann, Abel, and Jacobi, who were themselves influenced by Gauss, and the Leibnizian networks with which Kaestner was associated, such as the Humboldts, Schiller, Herbart et al.

Because of this fragmentary nature of Gauss’s early writings, much of Gauss’s thinking is expressed in abbreviated form. It becomes pedagogically much easier, therefore, to present the nature of his discovery, from the standpoint of his later work on curvature and Riemann’s elaboration of those ideas, most notably, in his famous treatise on Abelian functions. As in Gauss’s 1799 proof of the fundamental theorem of algebra, both Gauss and Riemann indicated the superiority of geometrical construction to algebraic formulas for conveying ideas.

The crux of Gauss’s method was that of the ancient Greeks, Cusa, Kepler and Leibniz: that nothing could be known from the visible manifestation of the circle or the ellipse. Rather, these visible characteristics, such as the uniformity or non-uniformity of the arcs, were a function of some underlying principle. That principle, however, was not visible, and could only be discovered by inversion. In other words, those principles could not be seen, but could be known, as, that which produces the visible characteristics of curvature.

To illustrate Gauss’ inverse method geometrically, take a new look at the trigonometric relationships. In the visible domain, the trigonometric relationships are generated as an effect of circular motion. But, as Cusa indicated (for which Kepler called him “divine”), the harmonic characteristics are found not in circular action alone, but in the incommensurability between the curved and the straight. Thus, Gauss thought of uniform circular motion as being merely the visible artifact of the more complex motion associated with circular functions.

This is illustrated in the accompanying animations. In animation 1, the circular rotation generates the trigonometric relationships. Animation 2 illustrates the inverse, where the visible effect of the circle is created as an artifact of the movement of the cosine and sine. In the visible domain, this seems impossible. How can you know the relationship of the cosine to the sine without first drawing the circle? But in the domain of reason, the essential nature of the sine and cosine can be {known} by inversion, as those functional relationships that produce circular areas.

Animation 1

Animation 2

In the case of the circle, this method may seem a bit arcane and clumsy, and so it was resisted by virtually every scientific thinker of Gauss’s time. But Gauss recognized that this method of inversion, rooted in the method of the ancient Greeks, was required to discover the nature of the elliptical transcendentals. For because of the incommensurability of the arc to both the line and angle, there was no way to generate, from the visible characteristics of an ellipse, a characteristic elliptical function, and all efforts to measure elliptical motion from circular transcendentals failed.

Gauss posed the elliptical problem in exactly these terms. He {knew} the higher elliptical functions must exist, and that they could not be defined directly. “What characteristics must such functions have to produce the ever changing elliptical motion?” Gauss can be imagined to have asked. “How can such characteristics be made intelligible?” This is the only way these functions can become known. Not directly, but only as that which is inverse to elliptical motion.

This thought will undoubtedly provoke psychological resistance in the modern reader, steeped in the culture of Aristotle and empiricism, who so strongly desires logical proofs presented in the visible domain. But it is the method of all Classical science and Classical art. It requires the mind to move; to willfully create new concepts. Hence, the benefit of reliving Gauss’s true discovery.

When the algebraists, such as Euler, Lagrange, et al. had tried to express the elliptical motion using their formalized, non-physical and, therefore, false version of the calculus they produced a formal representation of the elliptical motion that defied all their efforts at calculation. Gauss flanked them all by focusing on the simplest such case, the lemniscate of Bernoulli, which had been shown to be a special case of an elliptical type non-uniform curve.

Gauss’s choice of flank was rooted in Kepler’s insight into conic sections. Kepler had generalized Apollonius’s conics by recognizing the projective relationship among the conic sections as a whole. (See Hyperbolic Functions: A Fugue Across 25 Centuries.). Kepler had shown, by inversion, that all conic sections were generated by a single principle. However, he made particular notice of the significance of the discontinuity in the visible manifestation of this principle, expressed as an infinite boundary between the circle/ellipse and the hyperbola.

It was Gauss’s insight that the lemniscate expressed the higher, unified, principle that generated the conic sections. In its projected, visible form, the lemniscate is the locus of positions in which the product of the distances from a point on the curve to two foci, is equal to the square of ? the distance between those foci. (See Figure 6.). Or, in other words, the distance from one focus to the center of the lemniscate is the geometric mean between the two distances from the curve to each of the foci respectively.

Figure 6

However, the lemniscate has a more general relationship to the generating principle of the conic sections, and to the elliptical functions in particular, that can be grasped intuitively from the higher standpoint of Gauss and Riemann. On the one hand, the lemniscate can be generated as the inversion of the hyperbola in a circle. (See Figure 7.)

Figure 7

From this more advanced standpoint, the hyperbola can be seen as the stereographic projection from a sphere onto a plane of a lemniscate. (See Figure 8)

Figure 8

Here we can begin to see emerge the essential characteristics of the elliptical transcendentals, from the standpoint of Gauss’s principles of curvature. (See On Principles and Powers, Fidelio, Summer 2003.) The lemniscate on a sphere is generated as a mapping of the transition between the sections of positive and negative curvature of a torus. (See Figure 9.) And as Riemann would later demonstrate, the torus expresses a different topology (geometry of position) than the sphere or the ellipsoid. On the sphere or ellipsoid any closed curve separates the surface into two parts. But this is not the case on the torus where there are two distinct types of pathways, one around the torus, and the other through the “hole”. This characteristic, Riemann showed, expressed the double periodicity of the elliptical functions. (See Figure 10.)

Figure 9

Figure 10

Thus, enfolded in the lemniscate, and also in the ellipse, is the characteristic of double periodicity which is unfolded in the form of the torus. Additionally, hidden in the torus, if cut in the right way, one finds, the lemnsicate.

Here is the geometry of position that establishes the unseen, but nevertheless real, “orbit” which exists behind the elliptical functions.

We will come back to this discussion in future installments of this series. But for now, from this high perch, look back to Archytas with a justified sense of happiness.

Appendix

Moral from the History of the Infinitesimal Calculus

by A.G. Kaestner

When the question of calculating the infinite first arose, the most famous mathematical wise men had an aversion to it. Their habitual methods of discovering mathematical truths appeared to them to be clear and secure; whereas with the new one, they found dark secrets, much that was uncertain, and in the main, a degree of subtlety which they would rather forgo.

To convert these scorners, a cure was supplied by the camp of Leibniz and his friends, roughly as follows:

It was demonstrated that the calculation of the infinite was in agreement with all prevailing customary theories, in that it easily and comfortably led to truths which previously could only be attained by tiresome cogitation, and, finally, because it enlarged hitherto existing knowledge, such that the summit of Archimedes’ discovery was its lowest boundary; with it, one could answer, in total completeness, questions which could only be answered incompletely, or not at all, by the previously known feats of mathematical skill. And thus, the calculation of the infinite won the respect of an eye which, even without it, had made so many, so great discoveries.

How much would the Christian faith not gain, if its followers were to show, through their own acts, that with regard to the exercise of virtue, it has the same superiority over every other religion, as makes a Christian deserving of being admired by Socrates?

But many of these followers, and even their teachers, strike me today like someone who would go around constantly spouting higher mathematics and dropping Euler’s name, and who would declare anyone who could not integrate to be a dunce, but who would personally make errors as frequently as he was asked to calculate a Rule of Three!

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

Riemann For Anti-Dummies Part 50

THE GEOMETRY OF CHANGE

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

Riemann for Anti-Dummies: Part 49 : The Hidden History of the Complex Domain

Riemann for Anti-Dummies Part 49

THE HIDDEN HISTORY OF THE COMPLEX DOMAIN

When Kepler discovered the elliptical nature of the planetary orbits, he uncovered a paradox whose solution would require the development of an entirely new way of thinking, and he called on future generations to develop it. This “Kepler Problem”, as it has since become known, was not merely a mathematical lacuna, but reflected the ontological paradox indicated by Nicholas of Cusa in “On Learned Ignorance” and other locations. Kepler’s demand provoked Leibniz to develop the infinitesimal calculus, which revealed a new manifestation of that same paradox. This led Leibniz to indicate that the solution existed in a higher, yet to be discovered, domain of the imagination. Reflecting on these developments, the young Carl F. Gauss discovered that what both Kepler and Leibniz had sought. He called it the complex domain.

The above sketch is the true history of the origin of the discovery of the complex domain. It was known to Gauss’s immediate collaborators and followers, but today it lies hidden, even to the relatively best scientific thinkers. What has been substituted is the myth that complex numbers arise as “impossible” solutions to formal algebraic equations a myth whose malignancy has infected today’s popular thinking far beyond the domain of pure mathematics. The source of the myth is not new, but begins with Venice’s Paolo Sarpi’s launching of modern empiricism, and continues as a progression of degeneration through Hobbes, Descartes, Newton, Euler, Kant, Lagrange, Hegel, Cauchy, Klein, and down into the modern forms of existentialism and information theory associated with Synarchism. Today, as always, its target is the intellectual and emotional powers of mind associated with the development of the nation state, as embodied in the hard won Declaration of Independence and Constitution of the United States.

Consequently, anyone wishing to know the principles of science, must understand the connection between the history of the development of the complex domain, and the fight for the founding and preservation of the United States. Conversely, anyone wishing to know the latter, must understand its connection to the discovery of the complex domain.

For this reason, it is crucial that the record be set straight.

What follows is a summary overview. An opening statement, so to speak, designed to lay out what the investigation will show. It is best worked through slowly in small discussion groups. Over the coming installments of this series, we will explore these matters more deeply. This will take us, albeit not without a certain amount of hard work, directly to Riemann’s investigation of the higher transcendentals which he called Abelian functions.

The “Kepler Problem”

The ultimate source of Kepler’s discovery of the planetary orbits lies not in some particular astronomical theorem, but, as Kepler repeatedly emphasized, in his conception of Man. Unlike Ptolemy, Copernicus, and Tycho Brahe, Kepler understood that Man was distinguished from lower forms of life, by the capacity of mind that enabled him to rise above the limitations of sense-perception and discover those unsensed principles that govern the universe. As such, Kepler rejected the Aristotelean methods of Ptolemy, Copernicus and Brahe, all of which sought only to model the appearances of the motion of the heavenly bodies. Instead, as Kepler emphasized, his approach was to derive the apparent motion of the planets as a function of their physical causes. Consequently, where Ptolemy, Copernicus, and Brahe looked only to the domain of sense-perception, Kepler looked to the interaction between what is perceived and the principles that efficiently control what is perceived.

This latter point is crucial. Kepler did not ignore the world of sense-perception. Rather, he understood that what appeared to the senses was caused by principles that were unseen. But since those principles could not be sensed directly, they could only be known through contradictions and paradoxes that emerge in the domain of the senses. From these paradoxes, the mind has the power to form a synthetic visualization, so to speak, through which it can grasp for itself and communicate to other minds, those unseen principles indicated through the paradox. This synthetic visualization is emphatically not a symbolism. Rather it is a metaphor that preserves the paradox so as to guide the mind to the indicated principle that lies behind it.

It is in this light that the “Kepler Problem” must be seen. This problem arises when trying to determine the non-uniform motion of a planet. Cusa had already indicated that motion in the physical universe did not conform, as Aristotle had insisted, to fixed perfect circles. Instead, Cusa indicated that the perfection of the universe were better expressed by the principle of change, and, as such, action in the physical universe must be non-uniform, i.e., changing.

By deriving motion of the planets from physical causes, Kepler was led to the discovery that Cusa was right. Kepler demonstrated that the non-uniform motion of the planet through its orbit, as measured by the speeding up or slowing down of the observed movement of the planet against the background of stars on the inside of the celestial sphere, was not just an appearance, as Ptolemy, Copernicus and Brahe all held, but actually reflected the true physical motion of the planet. Thus, the planet didn’t just appear to be always changing, it {is} always changing. Consequently, the planet’s motion was being governed, not by a fixed principle such as would be characterized by perfect circles, but by a principle of change.

Kepler showed that that principle of change was embodied in the Sun, which had the power to move the planets by an immaterial “species” (idea), just as an idea in the mind moves the body. The measured speeding up and slowing down of the planet indicated the existence of an elliptical orbit in which the planet’s distance to the Sun is always increasing or decreasing. The always changing speed and direction of the planet, thus reflects the principle of change inherent in the principle of universal gravitation, as that principle was understood by Kepler, not the dumbed down bastardization associated with Newton.

However, the elliptical orbit presented a new type of paradox. As Kepler stated it, the relationship between the planet’s position and the time elapsed could be measured by his famous principle of equal areas. But these areas were measured by the combination of a circular sector and a rectilinear triangle. (See Figure 1.) Because of the “heterogeneity” of these two areas, this relationship could only be determined retrospectively: that is, if one knew where the planet had been, it was possible to measure the portion of the planet’s total period that had elapsed during that interval. However, Kepler said, there was no elegant way to determine, prospectively, where the planet would be, in a specified interval of time.

Figure 1

“But given the mean anomaly (time elapsed-bmd), there is no geometrical method of proceeding to the equated, that is, eccentric anomaly (position bmd). For the mean anomaly is composed of two areas, a sector and a triangle. And while the former is numbered by the arc of the eccentric, the latter is numbered by the sine of that arc multiplied by the value of the maximum triangle, omitting the last digits And the ratios between the arcs and their sines are infinite in number. So, when we begin with the sum of the two, we cannot say how great the arc is, and how great its sine, corresponding to this sum, unless we are previously to investigate the area resulting from a given arc; that is, unless you were to have constructed tables to have worked from them subsequently.

“This is my opinion. Insofar as it is seen to lack geometrical beauty, I exhort the geometers to solve me this problem:

Given the area of a part of a semicircle and a point on the diameter, to find the arc and the angle at that point, the sides of which angle and which arc, encloses the given area. Or, to cut the area of a semicircle in a given ratio from any given point on the diameter.

It is enough for me to believe that I could not solve this a priori, owing to the heterogeneity of the arc to the sine. Anyone who shows me my error and points the way will be for me the great Apollonius.”

Thus, having rejected Aristotle’s perfect circles, Kepler had to measure the planet’s motion not merely by the arc of a circle, but by the relationship of that arc to its sine. However, as Cusa had indicated, the ratio of the sine to the arc is “infinite”. Thus, the physical motion of the planet depended on a quantity whose characteristics could be precisely known, but could not be precisely calculated. The ellipse was not the orbit, it was that which is lawfully produced by the principle of universal gravitation.

Kepler did not directly observe this elliptical orbit, and to this day neither has anyone else. Nevertheless, the elliptical orbit was known to Kepler (and is known to anyone who relives his thoughts), more surely than if it had been directly seen. However, there is something more to the planet’s orbit. The ellipse is only the visible manifestation of the principle of universal gravitation, produced by the always changing action of gravitation on the planet at every moment of its motion.

The question posed by the paradox of the “Kepler Problem” is: what is the hidden characteristic of universal gravitation that produces the elliptical shape of the orbit?

Since that characteristic cannot be seen, it must be discovered through the paradox it presents in its visible expression. That paradox is the indeterminacy expressed by the “Kepler Problem”. The paradox does not exist for the planet. The planet “knows” at all times what it is doing. It moves seamlessly; always changing. The paradox exists in the expression of that continuously changing orbit within the visible domain of the elliptical orbit, through, as Kepler expresses it, the infinite ratio of the sine to the arc.

Leibniz’ Infinitesimal Calculus

What was required to solve this paradox was a new type of geometry that could express the relationship between the seen and the unseen. That is what Kepler demanded, and that is what Leibniz supplied.

Working from Fermat’s method of inverse tangents and maximum and minimum, Pascal’s investigations of conic sections, and Huyghen’s development of involutes and evolutes, Leibniz invented the infinitesimal calculus as a new type of geometry that expressed not only what was seen, but the relationship of what is seen to the principles that lay behind it. For reasons that will become clear below, Leibniz’ calculus, though inspired to solve the “Kepler Problem”, didn’t provide its solution. Such a solution would require Gauss’ discovery of the complex domain.

The most direct means of grasping the principles of Leibniz’ calculus is through his, and Johann Bernoulli’s application of the calculus to the solution to the catenary problem. As has been developed more thoroughly in other locations, the catenary expresses a principle of universal least-action. That principle is what produces the unique shape of the catenary, which, like the planetary orbit, acts on the hanging chain differently at each point. But unlike the planetary orbit, the catenary does not conform to a conic section.

To determine the shape of the catenary, Leibniz and Bernoulli first investigated the physical manifestation of this continuously changing least-action. This is most easily demonstrated pedagogically by the often cited experiment showing the physical determination of the catenary curve using a string and a hanging weight. (See Riemann for Anti-Dummies 46.) As Bernoulli showed in his treatise on the integral calculus, the total effect of the principle of least action is expressed by the general shape of the catenary, while its continuously changing manifestation at each (infinitesimal) point along the chain, is expressed by the relationship between the sines of the angles formed by the tangents. (See Figure 2.) In other words, the principle of least action exists outside the catenary, but “touches” it at every point as if it were acting tangent to the curve. Consequently, the continuously changing relationship of these tangents produces a visible expression of the everywhere present, but non-visible, principle of least action.

Figure 2

As this crucial example of the catenary illustrates, the infinitesimal calculus is not the geometry of the domain of sense-perception. It is the geometry of the interaction between the domain of sense-perception and the principles that lie behind it. It is a generalization of Kepler’s method of measuring the planet’s motion in any given interval, by its relationship to the whole orbit. It is in that relationship, of the whole (integral) to the part (differential) that expresses the interaction of the unseen to the seen. That which is seen is produced by the unseen acting according to a discoverable principle at each infinitesimal interval.

As mentioned above, Bernoulli’s application of the calculus to the catenary problem demonstrated that, like the planetary orbit, the physical principle governing the catenary was dependent on the sine of the angle. But, Leibniz further demonstrated that the catenary curve was also determined by another type of relationship that he called logarithmic. (See Figure 3.)

Figure 3

As has been developed more fully in earlier installments, this logarithmic relationship is the generalization of the principle of higher powers that was developed by the Pythagorean- Socratic current in Classical Greece, as typified by the investigations into the doubling of the line, square and cube. As Plato emphasizes, each type of action is associated with a magnitude of a different power, which are each mutually incommensurable. The more general form of these types of powers is expressed by the logarithmic spiral and the exponential function. Provoked by the type of discovery expressed by the catenary, Leibniz investigated this more general form of these “Platonic” powers, and discovered that they were related to a still higher power, expressed through the form of the so-called “natural logarithm”.

This type of discovery led Leibniz to distinguish between two types of magnitudes transcendental and algebraic. The algebraic are those types of magnitudes exemplified by the powers that double the line, square and cube. The transcendental are those types of magnitudes exemplified by the trigonometric and the exponential functions. As Cusa had insisted, the transcendental powers are a higher type. All algebraic magnitudes can be generated from transcendentals, but not the inverse.

As mentioned above, the planetary orbit, while dependent on a transcendental function, i.e., the sine, is expressed by a conic section, i.e., an ellipse. Apollonius had already demonstrated, as was later expanded by Fermat and Pascal, that the conic sections expressed the relationship of the second (square) algebraic power.

As such, Leibniz distinguished between those types of curves, such as conic sections, that could be expressed algebraically, and those, like the catenary, that could only be expressed by transcendental functions.

The catenary in particular, however, expressed a relationship between the two types of transcendentals. On the one side, as Bernoulli showed, it expressed the trigonometric, whereas on the other side, Leibniz showed it expressed the exponential. Thus, the physical principle of universal least-action indicated a connection between these two types of transcendentals. Yet, mathematically these two types of transcendentals appeared to be generated differently.

It is in the investigation of the connection between these two types of transcendentals that Leibniz encountered the square root of minus 1. He expressed this as the paradox of logarithms of negative numbers. (See Riemann for Anti-Dummies Part 38.) He insisted that these logarithms existed, but not in the visible domain, but in a domain still to be imagined. As he discussed the matter in a letter to Huygens, “… 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…. “

He was convinced that the square roots of negative numbers provided the paradox that would open the door to this new domain where the interaction between the seen and the unseen could be expressed. He referred to them, as “amphibians somewhere between being and non- being”.

Gauss, The Kepler Problem, and The Complex Domain

It was left to the young Gauss, when he was between the ages of seventeen and twenty, to discover that the higher transcendentals to which Leibniz had pointed, demand the development of the complex domain as that thought-object through which the interaction between the seen and the unseen could be expressed. Herein would emerge the source of the “Kepler Problem” that the motion of the planets in elliptical orbits are governed not by the circular transcendentals, but by a higher form of “elliptical” transcendental, the characteristics of which could only be discovered and expressed in the complex domain.

Gauss never published his findings, except to indicate in a famous remark in the “Disquisitiones Arithmeticae”, that his method for the division of the circle could also be applied to other transcendentals, particularly the lemniscate. Gauss promised to publish these results in a future time. But, due to the oppressive conditions brought on by the rise of Napoleon and its aftermath, Gauss never brought forth the promised text. His remark, however, prompted two brilliant youth, the Norwegian, Niels Henrik Abel, and the German, C. G. Jacobi, (both of whom were students of Gauss’s collaborators, Hansteen and Dirichlet , respectively) to independently develop these results. Upon seeing their work, Gauss remarked that he had already developed this theory when he was a youth. At the time, these comments were dismissed as arrogant boasting by those in the scientific community who were submitting to fascist terror, as exemplified by the supreme bigot, Augustin Cauchy.

Nevertheless, Jacobi believed Gauss was right, writing in Crelle’s Journal in May 1839, that it was the investigation of elliptical transcendentals which led Gauss to introduce complex numbers . Yet, it wasn’t until 1898, when Gauss’ youthful diaries became available, that the truthfulness of Gauss’s statements was fully confirmed.

From a review of these diaries, it is clear that from the beginning, Gauss was engaged in one unified investigation: the development of the complex domain as that idea necessary to express those higher transcendental relationships governing the physical universe.

While popular academia grudgingly admitts today that Gauss was the actual discoverer of elliptical functions, they still falsely present this discovery as an extension of the formal algebraic approach of Euler and Lagrange. While Gauss’ 1799 dissertation already exposes this as a lie, the diaries give an even more complete picture of Gauss’ train of thought.

The diaries begin on March 30, 1796, with the announcement of the discovery that the 17-gon is constructible by straight-edge and compass. As has been developed in other locations (Riemann for Anti Dummies Parts 30 and 31 and JBT’s lecture on the heptadecagon in L.A.) Gauss’ discovery depends on recognizing that the visible circle is an artifact of a process that can only be expressed in the complex domain. From the standpoint of the visible domain, it appears that the circle can only be divided into 2, 3, or 5 parts by straight-edge and compass, as was believed for more than two thousand years. Yet Gauss showed that the principle on which the division of the circle depends, can only be expressed in the complex domain. From this standpoint, the division of the circle is seen as the more general form of the ancient Greek problems of doubling the square and the cube.

The doubling of the square and cube are determined by those principles that generate one or two means, respectively, between two extremes. Gauss showed, that from the standpoint of the complex domain, the division of the circle into “n” parts is the problem of finding the principle that generates “n-1” means between two extremes. If “n-1” is 2 to a power of 2, then the problem can be transformed into a succession of square roots, and is susceptible of being constructed by straight edge and compass.

Work through Gauss’ method for dividing the circle for yourself, using the above cited pedagogicals as guide. For this discussion, bear in mind this crucial point. An action that is carried out in the visible domain, dividing a circle into “n” parts, is determined by a set of relationships that cannot be expressed in the visible domain. Rather, those relationships can only be expressed in the complex domain, where the interaction between the visible and the principles that lie behind it, can be made manifest.

Is the complex domain real? You can actually, with straight-edge and compass, divide a visible circle into 17 equal parts. But you cannot know how, or that you even can, accomplish such a task, unless you investigate, as Gauss did, the principle on which that act is accomplished, in the complex domain.

During this time, Gauss was also investigating the “Kepler Problem”, and began to recognize what had blocked its solution. The general characteristics of the problem can be illustrated pedagogically in the following way:

The circular trigonometric transcendentals vary periodically according to the angle a moving radius makes with a fixed diameter. (See Figure 4, Animation 1.) The circular sine has a period of 0 to 1 to 0 to -1 and the cosine from 1 to 0 to -1 to 0, as the radius rotates around the circle. For every one rotation, this cycle repeats. Thus, for example, the sine of 30 degrees, 390 degrees, 750 degrees are all the same.

Figure 4

Animation 1

However when this same relationship is extended to an ellipse, something dramatically changes. In an ellipse the length of the radius changes as it moves around the circle. (See Figure 5, Animation 2.)

Figure 5

Animation 2

But how much it changes relative to the angle depends on the eccentricity of the ellipse. (See Figure 6, Animation 3.)

Figure 6

Animation 3

If the idea of the circular sine and cosine are extended to the ellipse, then both the sine, cosine, and length of the radius each vary periodically with the angle, but, how they vary depends on the eccentricity of the ellipse. This, obviously, is a much more complicated relationship than the circular functions. Whereas the circular functions exhibit a very simple periodicity, the elliptical ones are much more complicated. For this reason, the elliptical functions were generally disregarded and all efforts to solve the “Kepler Problem” focused on measuring elliptical action by circular functions, which proved very elusive.

At Kaestner’s prompting in the Spring of 1796, Gauss sought the more general principle on which his method for dividing the circle depended, by extending that investigation into the division of non-uniform curves. While aiming for the elliptical functions, he set his first attack on the simplest case of such a function: the lemniscate.

The lemniscate is a special case of an elliptical curve. It looks like a figure 8 (See Figure 7)

Figure 7

It was originally investigated by Jacob Bernoulli, who was investigating the physics of elastic rods. As such, it was originally known as the “curva elastica” until Bernoulli gave it the Latin name, “lemniscate”, which means ribbon. The lemniscate is the locus of positions the product of whose distances from two foci is always equal to the square of the distance between the foci. (See Figure 8).

Figure 8

This property of the lemniscate has a similarity to the property of the ellipse. In the ellipse the sum of the distance from the foci to the curve is always a constant. (See Figure 9.) For the lemniscate it is the product of the distances. However, unlike the ellipse which is an algebraic curve of the square power, the lemniscate expresses the quartic power. (There is a simple geometric demonstration of the above statement which would be a digression to develop here. The reader is encouraged to discover it as a pedagogical exercise.)

Figure 9

The lemniscate has another geometrical property important for this discussion. It can be formed by the inversion of a hyperbola in a circle. (See Figure 10.)

Figure 10

Gauss’ investigation into the lemniscate began no later than August 1796, when he wrote out a series of eleven “Mathematical Exercises” on various subjects. Of significance for this discussion is the fifth exercise, in which Gauss begins to develop the characteristics of what he would later call sinus lemniscatus, (lemniscate sine).

On January 8, 1797, Gauss wrote, ” I have begun to investigate carefully curves dependent upon the leminiscate integral .”

This followed on March 19, 1797, by an entry announcing that the division of the lemniscate into “n” parts depends on “nn(t?-1) power, accompanied by the additional notation: “Imaginary quantities: The general criteria are sought according to which it is possible to distinguish complex functions of many variables from the non-complex ones.” (the cited symbolic notation reads: n times n raised to the t times square root of -1 power.)

Gauss’ method of the division of the circle and his generalization to the lemniscate and the elliptic functions, makes use of his clear geometrical understanding of the complex exponential. Protests from formalist sycophants not withstanding, Gauss’ idea is completely different than Euler’s formal algebraic mish mash peddled in virtually every text book today.

Gauss had this concept early on. On August 14, 1796, he wrote, “By the way, (a + b?-1)(m + n?-1), has been explained.” And two days later, he wrote, “The highest things are already of the mind. Let it stand firm in order that they be protected.” (Note: the cited notation reads: a plus b times the square root of -1 raised to the m plus n to the square root of -1 power.)

The Lemniscate and Elliptical Functions

We can get a preliminary taste for Gauss’ discovery of the elliptical transcendentals by investigating them using the geometrical methods established by Gauss and developed more fully by Riemann. In this way we will obviate, at least for now, the unnecessary use of formalism, which otherwise might be required for a more complete investigation. This is a method that both Gauss and Riemann emphasized, stating in numerous locations, that while certain relationships can only be expressed initially as formulae, the results can be given much better clarity through geometrical representation. It must be born in mind, as Gauss emphasized:

“The demonstration is presented using expressions borrowed from the geometry of position, for in this way, the greatest acuity and simplicity is obtained. Fundamentally, the essential content of the entire argument belongs to a higher domain, independent from space, in which abstract general concepts of magnitudes, are investigated as combinations of magnitudes connected by continuity, a domain, which, at present, is poorly developed, and in which one cannot move without the use of language borrowed from spatial images.”

Look at figure 11 showing the lemniscate as the inversion of the hyperbola in the circle. Now, let’s investigate the principles that generate this relationship among these three functions, through the method of complex mappings as developed by Gauss in his work on curved surfaces and later extended by Riemann.

Figure 11

The mappings we will investigate all depend on the complex exponential as Gauss indicated in his August, 14, 1796 notation. As was illustrated in Riemann for Dummies 48, the complex exponential can be represented geometrically as a stereographic projection. Under this mapping, the arithmetic relationships of one surface are transformed into geometric relationships on another. As, for example, the lines of latitude and longitude on a sphere are transformed into exponentially spaced concentric circles and radial lines on a plane. (See Figure 12)

Figure 12

When this same complex exponential is mapped from one plane to another, it is expressed by the transformation of a grid of equally spaced lines into the same exponentially spaced concentric circles and radial lines of the stereographic projection. (See Figure 13)

Figure 13

WARNING TO THOSE INFECTED WITH CARTESIANISM: These mappings are not graphs. They indicate the transformation of one set of least action physical relationships into another by the introduction of a new physical principle.

Figure 14

Of particular importance for this discussion, is the periodicity of the complex exponential. (See Figure 14, Animation 4.)

Animation 4

From this standpoint, let’s unfold the hidden relationships among the hyperbola, circle and lemniscate. To do this, we adopt Gauss’ method of inversion. Instead of investigating the circular, hyperbolic and lemniscatic functions as derived from the relevant curves, we investigate how the curves are derived from the relevant functions.

Begin first with the circle. Think of the circle as that which is produced by a right triangle whose hypotenuse is constant, but whose angle varies. It is obvious that the circle is generated by two periodic functions: the sine and the cosine.

Now turn to the hyperbola. The hyperbola is generated by a rectangle whose sides change but whose area remains the same. (See Figure 15). This constant area determines two functions, the hyperbolic sine and cosine. (See Figure 16). However, it is evident from the figure, that unlike the circle, the hyperbolic sine and cosine are not periodic. The are always getting bigger and bigger.

Figure 15

Figure 16

Now look at the lemniscate. Gauss defined two transcendental functions for the lemniscate, called the lemniscatic sine and lemniscatic cosine (See Figure 17)

Figure 17

Something strange happens when we examine these functions. If we apply Gauss’ method of inversion, and derive the lemniscate from these functions, we see that to define the entire lemniscate, the sine appears to have two periods for each full rotation and the cosine appears to have one. (See Animation 5).

Animation 5

Keep this curiosity in mind.

As developed in the previous installments on hyperbolic functions (See Riemann for Anti-Dummies 33) the hyperbolic cosine is expressed physically by the catenary, (See Figures 18 and 19) and as such can be expressed by the exponential.

Figure 18

Figure 19

When this principle is expressed as a Gauss- Riemann complex mapping, the hyperbolic functions are formed from the midpoint of a line joining two of the circles formed by the exponential, that are rotating in opposite directions. (See Animation 6). From this standpoint, a periodicity emerges that otherwise had remained hidden.

Animation 6

Now, look again at the circle. As Leibniz indicated, the circular functions are also expressions of the complex exponential. (See Riemann for Anti-Dummies Parts 37 and 38.) (See Figures 20 and 21). When these functions are expressed as a Gauss-Riemann complex mapping, the full expression of their periodicity is brought forth. They too are formed by the countervailing rotations of the exponential circles, but these circles are formed from the mappings that are at right angles to the hyperbolic.

Figure 20

Figure 21

In sum, the circular and hyperbolic functions form the four possible variations of the mappings of the complex exponential. This brings out a unity between the hyperbola and circle that lies behind their visible appearance as conic sections, but on which that visible appearance is based.

In these examples we can see how the complex domain provides the means to create a synthetic visualization of the interaction between the visible domain and the unseen principles that lie behind it.

Now, look at the lemniscate. To unfold the lemniscate, Gauss applied the same principle of inversion illustrated above for the circle and hyperbola. A further elaboration of Gauss’ discovery will be developed in a future installment. But the characteristic of Gauss’ discovery is evident from the geometric characteristics that emerge when the lemniscate functions are expressed in the complex domain.

What Gauss discovered was that the lemniscate functions, and more generally, the elliptical functions are all doubly periodic. (See Figures 22, 23, and 24). This is no where evident from the relationships that appear in the visible domain. Furthermore, he demonstrated that the circular and hyperbolic functions were special cases of the higher class of elliptical transcendentals.

Figure 22

Figure 23

Figure 24

From this standpoint, Gauss established that the ellipse was governed by a higher type of transcendental which subsumed the circular, hyperbolic and exponential transcendentals. But these higher transcendentals were only expressible in the complex domain. In other words, the principles on which the planetary orbits really depended, were reflected by, but not visible in, the domain of sense-perception, or its representation. To passion to know these principles demanded the development of the complex domain.

Ironically, Kepler’s approach for calculating an approximate solution for the “Kepler Problem”, is essentially still the best approach for determining the position of a planet in its orbit. But Gauss’ determination to investigate the paradox it revealed, led to his discovery of those higher transcendentals on which the underlying principles of planetary orbits are based. This paved the way for even deeper investigations.

Riemann for Anti-Dummies: Part 48 : Riemann’s Roots

Riemann For Anti-Dummies Part 48

RIEMANN’S ROOTS

In December 1822, C.F. Gauss submitted a paper to the Royal Society of Science in Copenhagen titled, “General Solution of the Problem: To Map a Part of a Given Surface on another Given Surface so that the Image and the Original are Similar in their Smallest Parts”.

Notably, the paper contained the motto: “Ab his via sternitur ad maiora” (“These results pave the way to bigger things”).

They did. Nearly 30 years later (1851), B. Riemann submitted, to Gauss, his doctoral dissertation on functions of a complex variable, which, along with his 1857 works on Abelian functions and the Hypergeometric series, developed the further implications of the method Gauss had initiated. The deeper epistemological implications of these results, however, were only brought to light in our present time, through Lyndon LaRouche’s discoveries in the science of physical economy, as in his most recent work, “On Visualizing the Complex Domain”. Therein is established the highest vantage point from which to re-live the discoveries of Gauss and Riemann.

Initially, the Royal Society had posed a more limited subject for the so-called “Copenhagen Prize Essay” than the one ultimately addressed by Gauss. The original question was directed toward solving some particular problems involved in the production of geographical maps. At the suggestion of his collaborator, the astronomer H.C. Schumacher, Gauss proposed the more general question to the Royal Society. After several years without anyone providing a serious solution to the question he posed, Gauss submitted his own solution, and, of course, was awarded the prize.

Obviously, Gauss was less interested in winning the prize than taking the opportunity to present the more general results he had been developing his whole life, beginning with his earliest work on the division of the circle and the fundamental theorem of algebra. The Royal Society’s challenge afforded Gauss the opportunity to demonstrate the extension of Leibniz’ calculus, under the concept of the complex domain that Gauss had developed in those earlier works. While this application provided the solution to the more limited practical problem of producing accurate maps, as Gauss indicated, it was really about something much more fundamental: specifically, the improvement in the capacity of the mind to grasp and communicate truths concerning the unsensed principles that govern the universe.

Mapping the Sensorium

The roots of Gauss’ method go deep into the history of Mankind’s efforts to increase its power in and over nature, beginning with the earliest attempts to map sense perceptual space- time by the development of calendars and geographical and astronomical maps. These maps expressed not merely the visible changes of the heavenly bodies. The unsensed principles were reflected as well, in the form of anomalies, paradoxes, and distortions. Thus, each map already implies another map that lies “behind’, so to speak, the visible map.

Members of the LaRouche Youth Movement currently involved in observing the motion of Mars, are confronting the types of paradoxes that arise from the development of such maps. Go out and look at Mars rising in the eastern sky. The arrangement of the visual image of Mars and the surrounding stars on the inside of the celestial sphere gives rise to a mental image, i.e., map. Over the course of the night, the motion of Mars and the stars, relative to the observer, changes, forming a succession of mental images, which gives rise to a map of the changes in the night’s succession of maps, or, in other words, a map of maps. From night to night, the image of Mars changes its relationship with respect to the images of the other stars. This change gives rise to a higher map, formed from each night’s map of maps. This map of maps of maps gives rise to an even higher type of map, a map that reflects the underlying principles governing the changes among the maps of the sense perceptual images. This higher map only becomes fully intelligible from the standpoint of Gauss’ and Riemann’s complex domain.

While these investigations are most ancient, the roots of our modern knowledge begin with Thales’ (624-547 B.C.) effort to map the celestial sphere onto a plane by the gnomonic, or central projection. Thales’ method was to define on the plane, the image of every point on the sphere, by drawing a line from the center of the sphere, through the surface, until that line intersected the image plane. (See Figure 1.) (The same result can be demonstrated physically by using a light source located at the center of a transparent hemisphere to cast shadows of figures drawn on the sphere onto a flat wall.)

Figure 1

This projection immediately presents us with a crucial paradox. Certain relationships among the images on the sphere are different than the relationships among their images on the plane. For example, the image of a spherical triangle whose vertices are three stars on the celestial sphere, is a rectilinear triangle on the plane. Consequently, the angular relationship among the three stars cannot be preserved in the image, for the sum of the angles of a spherical triangle is always greater than 180 degrees, while the sum of the angles of a plane triangle is always equal to 180 degrees.

However, the measurable relationships among the images of the stars on the celestial sphere are determined by angular measurements, which are not accurately represented by Thales’ gnomonic projection. The first solution to this problem is attributed to Hipparchus (160-125 B.C.), to whom is attributed the discovery of the stereographic projection. In this projection, the sphere is mapped onto the plane from one of its poles. (See Figure 2.) (This can be demonstrated physically by moving the light source in the previous experiment from the center of the sphere to its pole. Notice the resulting change in the relationship among the shadows.)

Figure 2

In this stereographic projection, the lines of longitude of the sphere are transformed into radial lines on the plane. The latitude lines on the sphere are transformed into concentric circles. If we think of the point touching the plane as the “south” pole, and the point of projection as the “north” pole, the circles of latitude in the “southern” hemisphere all map to circles inside the circle which is the image of the equator. On the other hand, the circles of latitude in the “northern” hemisphere all map outside this circle. The radial lines on the plane make the same angles with each other, as the longitude lines of which they are images. However, the radii of the concentric circles that are images of the circles of latitude, increase exponentially, the farther the latitude circles they represent get from the “south” pole and the closer they get to the “north” pole.

In the case of the stereographic projection, the angles among the images are preserved.

Another projection that preserves angles was developed by Gerhard Kremer (1512- 1594), otherwise known as Mercator. On the Mercator projection, the image of the equator is a straight line, and the images of the circles of longitude are perpendicular lines spaced equally along it. The images of the circles of latitude are straight lines parallel to the equator, but the distance between them increases. (See Figure 3.) This is because on a sphere the longitude lines get closer together as they approach the poles. Consequently, the ratio of the distance along the surface of a sphere for a given angle of latitude, to the distance along the surface for the same angle of longitude, changes from the equator to the poles. (See Figure 4.) This change is reflected in the Mercator projection by increasing the distance between the lines of latitude so that the proportion between the lengths of latitude and longitude is the same as on the sphere.

Figure 3

Figure 4

The Mercator projection, although entirely different than the stereographic, also preserves angles. It also has the characteristic that the so-called rhumb line, or path that makes the same angle with all lines of longitude, is a straight line. This exemplifies the types of paradoxes that emerge, for the straight-line is the shortest path on the flat plane of the projection. But on the sphere, the shortest path is a great circle, and the rhumb line is a spiral path called a loxodrome, which is longer than the great circle arc. (See Figure 5.)

Figure 5

The Mapping of Principles

Thus, at first glance, the development of these angle-preserving projections (a characteristic that Gauss would later call, “conformal”), has a very important significance for the representation of the images of the Sensorium. Nevertheless, Gauss had something much more significant in mind. The distortions and paradoxes that result from these projections are not only due to the visual representation, but reflected something “behind” the visible. By developing a general means for transforming one surface into another conformally, Gauss paved the way for Riemann’s more general investigations into the nature of these transformations themselves, and their relationship to the underlying principles behind the Sensorium.

To do this, Gauss rejected the reactionary, a priori Euclideanism of Kant. For him a “surface” is not an object embedded in empty Euclidean space that is infinitely extended in three directions. Rather, a “surface” is generated by some physical action. For example, we don’t measure the celestial sphere by two angles. The celestial sphere is a physically determined idea, generated by the physical action of rotation with respect to the direction of the pull of gravity, and the direction around the horizon from some physically determined direction, such as the position of the rising (or setting) Sun.

In Riemann’s terms, these two acts of rotation are the physical modes of determination of the celestial sphere. Were these modes of determination created by some other physical action, they would produce a different surface. This is the basis for a physically determined geometry. The transformation of one surface into another, Gauss demonstrated, is accomplished by finding a function that transforms one pair of modes of determination into another.

This is exactly what takes place in a stereographic projection. The two modes of determination, represented as circles of latitude and longitude on the sphere, are transformed into concentric circles and radial lines, respectively, on the plane. In the Mercator projection, the same two modes of determination of the sphere, are transformed differently, that is, into straight lines. Nevertheless, both projections are conformal. Thus, the characteristic of conformality reflects a more general principle, not specific to a particular projection.

Gauss recognized that for a projection to be conformal, it must transform one surface into another equally in all directions. This is expressed geometrically by the fact that the images of circles are also circles. This can be illustrated physically in two ways. Think of an image on a stretchable, say, rubber, surface. If the surface is stretched proportionally in all directions, then the shape of the image will be the same, only larger. If it is stretched by a different amount in different directions, the image will be distorted. The former, represents a conformal transformation, the latter, non-conformal.

Another physical example has been pointed out in previous pedagogicals. Take a clear plastic hemisphere and tape to it circles of differing sizes, around what would represent a circle of latitude on a sphere. Shine a light from the center of the hemisphere. Look at the shadows. The circles are transformed into ellipses. Now, move the light to the pole. The shadows become more circular, with the shadows of the smaller circles being the most circular. This illustrates the difference between the gnomonic projection and the stereographic. In the former, the circles are stretched differently in different directions, thus producing ellipses. As such, this projection is non-conformal. In the latter, the circles are stretched into circles, i.e., they are conformal.

This characteristic of circular action, Gauss had already developed as the principle of complex numbers, as early as 1796, in his discovery of the division of the circle, and his 1797 fundamental theorem of algebra (published in 1799). That is, a complex number, Gauss showed, was not arbitrarily defined as the solution to an algebraic equation. Rather, the complex number expressed that rotational action, which transcended, and thus determined, all possible algebraic magnitudes. The transformation of one complex number into another, therefore, was the transformation of one rotational action into another rotational action, exactly the condition necessary for the mapping to be conformal.

For this reason, Gauss considered not the visible surface, but its representation in the complex domain. Each point on the surface corresponded to a complex number, which in turn was determined by the physical modes of determination of the surface. To transform one surface onto another, required transforming the modes of determination of one surface into the modes of determination of the other, which in turn transformed each complex number of the first surface into a definite complex number of the second. This is what Riemann would later call, “a function of a complex variable.”

Figure 6

This was illustrated above by the examples of the stereographic and Mercator projections. In the former case, the circles of latitude and longitude that were equally spaced around the sphere were transformed into exponentially spaced concentric circles and radial lines on the plane. Gauss notes that this corresponds geometrically to the transformation of the complex exponential. (See Figure 6.) The Mercator projection corresponds to a transformation of the stereographic projection. (See Figure 7.) To map a sphere onto a plane, Gauss applied these complex transformations, to the modes of determination of a sphere, that is, the two modes of rotation.

Figure 7

In his paper, Gauss demonstrated why these types of projections would be conformal. This can be illustrated geometrically, by looking at the behavior of a small square undergoing the desired transformation. As the accompanying animation illustrates, in order for the diagonals of the square to remain perpendicular, the sides of the square must change accordingly. Gauss expressed this geometric condition by a formula in the language of Leibniz’ calculus, which was restated by Riemann in his doctoral dissertation. It is one of the continuing frauds of modern mathematics, that this formula has become known as the “Cauchy-Riemann” formula, despite the fact that Augustin Louis Cauchy added nothing to its development. For the sake of historical accuracy, and mental health, this relationship should really be known as the Gauss-Riemann relationship.

Still Lurking Behind the Scenes

Within Gauss’ discovery, something is lurking behind the scenes, a spirit from the domain of unseen principles. In both the above examples, the projection becomes increasingly distorted– as the projection approaches the north pole in the case of the stereographic, and as the projection approaches both poles in the case of the Mercator. At the poles, the projection “blows up” completely and ceases to exist. Is this just a failing in the projection, or, is this an indication of some yet unknown, hidden principle?

In this question lay the “bigger things” for which Gauss paved the way.

Riemann for Anti-Dummies: Part 47 : Defeating I. Kant

Riemann for Anti-Dummies Part 47

DEFEATING I. KANT

In the opening of his Habilitation lecture, Bernhard Riemann proposed to establish the foundations of geometry on a rigorous basis:

“Accordingly, I have proposed to myself at first the problem of constructing the concept of a multiply extended magnitude out of general notions of quantity. From this it will result that a multiply extended magnitude is susceptible of various metric relations and that space accordingly constitutes only a particular case of a triply-extended magnitude. A necessary sequel to this is that the propositions of geometry are not derivable from general concepts of quantity, but that those properties by which space is distinguished from other conceivable triply-extended magnitudes can be gathered only from experience”.

Riemann’s program poses a paradox for those habituated to the doctrine of Immanuel Kant and its more extreme, modern form–existentialism. How can the propositions of geometry be determined by experience?

Kant had insisted that:

“Space is not an empirical concept which has been derived from outer experiences. For in order that certain sensations be referred to something outside me…the representation of space must be presupposed. The representation of space cannot, therefore, be empirically obtained from the relations of outer appearance. On the contrary, this outer experience is itself possible at all only through that representation.”…”Geometry is a science which determines the properties of space synthetically; and yet a priori. It must in its origin be intuition; for from a mere concept no propositions can be obtained which go beyond the concept as happens in geometry. For geometrical propositions are one and all apodeictic, that is, are bound up with the consciousness of their necessity; for instance that space has only three dimensions. Such propositions cannot be empirical or, in other words, judgments of experience, nor can they be derived from any such judgments.”

Kant was not very original. Nearly two centuries earlier, Johannes Kepler, through his discovery of universal gravitation, had already liberated science from similar Aristotelean dogmas that, from the murder of Archimedes until the Renaissance, had enslaved European civilization. Kant was deployed to put the chains back on. Those doctrines had taught that experience, (which, for them, was limited to sense perception), can tell us nothing about the physical world. For example, our experience of phenomena such as the motion of the planets and other heavenly bodies, is limited to the perceptions of the changes of position of points of light on the inside of a great sphere of unknown radius, whose center is always the location of the observer. For the Aristoteleans, the actual motions, as well as the principles that govern them, are inherently unknowable, and so they must be referred to some a priori determined set of propositions, such as those of Ptolemy, Copernicus or Brahe. These propositions, in turn, are ultimately derived from Euclidean-type axioms, postulates and definitions, which Kant insisted, are the only possible form by which we can conceive of space:

“Space is a necessary a priori representation, which underlies all outer intuitions. We can never represent to ourselves the absence of space, though we can quite well think it as empty of objects. It must therefore be regarded as the condition of the possibility of appearances, and not as a determination dependent upon them. It is an a priori representation, which necessarily underlies outer appearances.”

According to Kant: these propositions are not decidedly truthful; they make no judgement about the actual motions; they are the form by which the appearances must be represented; and nothing can happen that is not possible under these propositions.

When you begin to think about this, you come face to face with the fundamental question of science (and also, politics, history and art): What is experience? Is it sense perception? Therein lies Kant’s trickery, for if experience is limited to sense perception, then indeed, it can tell us nothing about the propositions of geometry. As Kant’s sophistry insists: “Were this representation of space a concept acquired a posteriori, and derived from outer experience in general, the first principles of mathematical determination would be nothing but perceptions. They would therefore all share in the contingent character of perception; that there should be only one straight line between two points would not be necessary, but only what experience always teaches. What is derived from experience has only comparative universality, namely, that which is obtained through induction. We should therefore only be able to say that, so far as hitherto observed, no space has been found which has more than three dimensions.”

However, Riemann had something far different in mind when he spoke of experience:

“There arises from this the problem of searching out the simplest facts by which the metric relations of space can be determined, a problem which in the nature of things is not quite definite; for several systems of simple facts can be stated which would suffice for determining the metric relations of space; the most important for present purposes is that laid down for foundations by Euclid. These facts are, like all facts, not necessary but of a merely empirical certainty; they are hypotheses…”

For Riemann, as for all humans not self-degraded into Aristotelean/Kantian bestiality, experience is not sense-perception; it is the active interaction of the mind with the universe, of which it is a part. It is, as Plato insists, the formation of hypotheses, higher hypotheses, and hypothesizing the higher hypothesis. It is the investigator, investigating, how he is investigating what is being investigated. Or, as for Apollo, who sings in Percy Shelley’s Hymn, “I am the eye with which the Universe beholds itself and knows itself divine;”

The propositions of geometry can, and must, be derived from this type of experience and Riemann advanced the general methods for how this is done. He divides this task into two steps, both of which rested on foundations laid by Gauss. The first, as indicated above, is the determination of the general notion of multiply-extended magnitude. Here Riemann cites Gauss’s second treatise on bi-quadratic residues and his fundamental theorem of algebra. In those works, as well as other unpublished discussions, Gauss attacked Kant’s view as an “illusion,” and he advanced the concepts begun with the investigations of the Pythagoreans, Archytas and Plato, that physical action, not a priori intuition, gives rise to our concept of extension, as exemplified by the different principles, or powers, of physical action that extend a line, square, or cube.

In each type of action, the determination of the essential, distinguishing characteristic, Riemann noted, always leads back to “n” determinations of magnitude, in which “n” signifies the relative degree of the power governing the action. For example, a cube, which is determined by a triply-extended magnitude, cannot be determined by the doubly-extended square, nor a square by a simply-extended line.

However, there is still another consideration:

” …there follows as second of the problems proposed above, an investigation into the relations of measure that such a manifold is susceptible of, also into the conditions which suffice for determining these metric relations. These relations of measure can be investigated only in abstract notions of magnitude and can be exhibited connectedly only in formulae; upon certain assumptions, however, one is able to resolve them into relations which are separately capable of being represented geometrically, and by this means it becomes possible to express geometrically the results of the calculation. Therefore if one is to reach solid ground, an abstract investigation in formulae is indeed unavoidable, but its results will allow an exhibition in the clothing of geometry. For both parts the foundations are contained in the celebrated treatise of Privy Councillor Gauss upon curved surfaces.”

Thus, the question of discovering a physical geometry requires determining both the “n” determinations of magnitude, and their measure relations. Neither of these can be given a priori. How, then, can these matters be discovered from human cognitive, (as opposed to Kantian) experience?

The last three installments of this series, (Riemann for Anti-Dummies Parts 44, 45, and 46), explored the essential foundations of Gauss’ and Riemann’s approach. There we showed how Gauss, from his very earliest work under the tutelage of Kaestner and Zimmerman, recognized that only physical principles, not definitions, can lay the foundations for geometry, as, for example, the determination of what is a straight-line. From there, Gauss showed that these physical principles determine a characteristic curvature from which the measure relations of a surface are derived, and that these general principles of curvature are expressed, in the smallest parts, by the characteristics of the “shortest lines”, or geodesics, of the surface. What remains to discuss is this discovery’s inversion. How can the physical principles of the curvature of the surface be determined from the characteristics of the geodesics as measured by small changes in those geodesics?

Gauss’ work on this inverse problem is of crucial significance, as this is the form of investigation usually confronted in science, politics, history and art. We cannot know these physical principles by sense perception, but we can perceive their effects by some small measurable change, from which, by hypothesizing, we can determine the general principles that are determining that change. Kepler determined the general principles of curvature of the solar system as a whole, from small measured changes in the relationship between the orbits of Mars and Earth. Leibniz and Bernoulli determined the general principles of least- action from small measured changes in the shape of a hanging chain. Gauss determined the solar system’s harmonic dissonance that had been indicated by Kepler, from Piazzi’s very small measurements of Ceres’ arc. LaRouche determined the general direction of world history by measured changes in the cultural/mental outlook of the population following the death of FDR.

As Riemann indicated, an exploration of Gauss’ work in this direction is impossible without resort to abstract concepts expressible by formulae, but, these results are always capable of geometrical representation. For pedagogical purposes we will minimize the former and emphasize the latter, but limited reference to formulae are unavoidable, and will always be accompanied by the appropriate geometrical representation.

To begin to get a handle on the principles involved, take a simple case a line. When considered as a simply-extended magnitude, a line can be increased or decreased only by action along its length, that is, back and forth. Such changes can be measured only by increments of more or less, and expressed by rational numbers.

However, when that line is understood to be generated from a doubly-extended magnitude, such as the diagonal of a square or rectangle, its increases or decreases are measured by an entirely different set of relationships, as demonstrated by Plato in the Meno and Theatetus dialogues. In this case, the line is increased or decreased along its length, but only in a connected way, to changes in the lengths of the side of the square or rectangle. These changes cannot be measured by the simple ideas of more, or less, as expressed by rational numbers. Rather, they express the type of relationship that has come to be known by the “Pythagorean theorem”, which as Pythagoras and Plato emphasized, are incommensurable with simply-extended magnitudes.

Draw a rectangle and call the length of one side “x” and the length of the other side “y”. By the Pythagorean theorem, the length of the diagonal “s” can be measured as the square root of the sum of the squares of the two sides, or in shorthand, s=?(x2+y2). If the diagonal is extended by a small amount “ds”, the sides of the rectangle will be increased by proportional small amounts “dx” and “dy”. (“ds”, “dx”, and, “dy” are Leibniz’ notation for these infinitesimal increments, which he called differentials.) If this action is taking place on a Euclidean flat plane, then ds, dx and dy, will express the Pythagorean relationship, ds=?(dx2+dy2).

Thus, contrary to the textbook versions, the Pythagorean relationship is not an arbitrary formula; it expresses a characteristic relationship of a certain type of surface–a Euclidean flat plane. Inversely, if a physical process is measured by the Pythagorean relationship expressed above, that action is occurring in a Euclidean flat plane.

This measurable physical relationship, not Kant’s dictum of a priori certitude, is the only reality of a Euclidean flat plane. And, since real world physical measurements express a different relationship, the physical reality of a Euclidean flat plane is not only not necessary, it is illusory .

But, if our cognitive experience, i.e. physical measurement, determines that the Pythagorean relationship doesn’t hold, what relationship expresses a doubly-extended magnitude? A Kantian will fly into a fit of rage at this paradox. Kant insists that Euclidean space is the only possible way one can think about space, therefore, for the Kantian, Euclidean space must be the only space possible. And so, the Kantian will demand the world be treated as if it were Euclidean, even if physical measurements tell us otherwise. Fakers in the tradition of Gallileo’s deceitful attempt to curve-fit the catenary into a parabola, will have no problem with this. They will suggest limiting physical measurements to small enough regions, that the deviation from the Pythagorean relationship is below the errors of measurement. Such chicanery is, however, only self-deception, as the deviation from flatness, no matter how small it may seem, still exists, and, sooner or later, judgements made on that delusion will become impossible to ignore (as, for example, how the ongoing financial crisis was ignored by believers in such “New Economy” frauds as exemplified by the Winstar madness).

Gauss, of course, rejected such follies. He recognized that the Pythagorean relationship, as expressed in Euclidean geometry, was not sacrosanct. Rather, it was merely a special case of a more general principle. Rather than cling to the special case, Gauss discovered the foundations on which more the general principle was based.

To gain an understanding of Gauss’ discovery, it is pedagogically efficient to work through several examples, and then abstract from them the general principle at work.

Begin with the case of the physically determined celestial sphere. This surface is physically determined by the position of the observer and the direction of the pull of gravity. The former determines the center of the sphere and the latter determines the poles and the horizon. If we are bound by Kant’s constriction that we can only think of space as infinitely extended in three directions, then this sphere sits like a large object surrounded by empty space.

But, Gauss was free from Kantianism, and he understood the celestial sphere as a manifold of physical action, produced by two mutually inter-dependent angles: the angles around the horizon and the angles perpendicular to it. (For pedagogical simplicity, I will refer to these angles by the familiar names latitude and longitude respectively.)

All positions on the celestial sphere can be determined by these two physical parameters. (Thus, for the sphere, n=2, in Riemann’s terms of “n” determinations of magnitude.) From this relationship, Gauss constructed what is called a “parametric formula” in which the surface of the sphere is expressed entirely by these two parameters.

All the longitudinal circles have a common center, i.e. the observer. Positions along any one of these circles can be determined by a function of the angle from the horizon to the zenith. If this angle is called “p”, then any position along a circle of longitude can be determined as a function of the cosine and the sine of the angle “p”. (See Figure 1.) Each of these circles of longitude can be distinguished from one another, by another angle, as measured around the horizon. If that angle is called “q”, then positions around this circle can be determined as a function of the cosine and sine of angle “q”.

Figure 1

Thus, for all the positions along a single circle of longitude, q is constant while p varies, whereas for all positions along a single circle of latitude, p is constant while q varies. However, there is a significant difference between the two types of circles. The center of all the circles of longitude are the same as the center of the sphere, consequently, all circles of longitude are great circles. But, the circles of latitude all become smaller as they get farther from the horizon and closer to the poles. How much smaller they get, is function of angle p, that is, how close they are to the poles. From the geometry of the sphere, the radii of the circles of latitude are proportional to the cosine of angle p. (See figure 2.)

Figure 2

From this, all positions on a sphere can be reduced to determinations of the two parameters p and q, which reflect the physical curvature of the sphere. (footenote 1.) The empty box, which Kant insisted on, just disappeared, perhaps into the empty corners of his brain from whence it came.

Gauss now investigated the relationship between the length of an arbitrary geodesic of the sphere and the angles p and q, in order to determine a way of measuring the general principles of curvature of the surface from small measured changes in the geodesic.

To do this, Gauss established a more general form of the Pythagorean relationship. In the case of the sphere, a geodesic line can be thought of as a diagonal of a spherical rectangle, whose sides are circles of longitude and latitude. But, unlike the Pythagorean in the Euclidean flat plane, the relationship between the sides of the spherical rectangles are changing, depending on their position with respect to the horizon and the poles. Specifically, the latitudinal sides will get shorter, by a factor proportional to the cosine of angle p, as the longitudinal sides increase towards the poles. (See Figure 3.) Thus, the “spherical Pythagorean” must measure not only the relationship of the diagonal to the sides, but the also the change in this relationship as the relationship between the sides themselves changes. From this, Gauss showed, that the form of the “spherical Pythagorean”, is that the length of the geodesic “ds” =?((Cos[p])2dp2 + dq2) where ds is the change in the length of the diagonal and dp and dq are the changes in longitude and latitude. The coefficient Cos[p]2 expresses the shortening of the latitude lines as they get closer to the poles.

Figure 3

Consequently, if one is making physical measurements along what appears to be a “straight-line”, and the relationship measured corresponds to this “spherical Pythagorean”, then that “straight-line” is a geodesic on a sphere.

To further develop this idea, continue in this same vein to the two other examples used in Riemann for Anti-Dummies Part 46, the spheroid and the ellipsoid.

For the spheroid, a parametric formula can be constructed that expresses the geometrical relationship that the lines of latitude get shorter as they approach the poles, while the lines of longitude get longer. (See Figure 4.) In this case, the “spheroidal Pythagorean” must express the relationship between the length of the geodesic, “ds”, and the changing relationship between the lengths of the circles of latitude and the ellipses of longitude. This produces a somewhat more complicated formula for the “spheroidal Pythagorean”, but its geometrical representation can be gleaned from the accompanying figure.

Figure 4

By inversion, if physical measurements along a “straight- line” reflect the relationship expressed by the “spheroidal Pythagorean”, that “straight-line” must be a geodesic on a spheroid.

On the ellipsoid, as illustrated in the last installment, the relationship between the sides of the “ellipsoidal rectangles” changes as they move towards the poles and as they move around in the latitudinal direction as well. (See Figure 5.) Using Gauss’ method, it is possible to calculate a formula for this “ellipsoidal Pythagorean”, which enables us to make the same type of determinations about the general principles of curvature from small changes in the measurement of the “ellipsoidal geodesic”.

Figure 5

In his treatise on curved surfaces, cited by Riemann in the habilitation paper, Gauss developed a general form of the Pythagorean that expressed the relationship between the geodesic line and the two determinations of magnitude of that surface. For purely illustrative purposes, the form of Gauss’ generalized Pythagorean is, ds=?E2dp2+2Fdpdq+G2dq2, where, E, F, and G are themselves functions which express the changing relationship of the two parameters of the surface. E expresses the function by which p changes with respect to q; G expresses the function by which q changes with respect to p; and F expresses the function by which the area of the rectangle changes relative to changes in p and q. For the Euclidean flat plane, Gauss’ Pythagorean reduces to the familiar form.

As Riemann indicated, this relationship in its most general form is expressed by the above formula, but, as in the above examples, its results can always be represented geometrically.

With this foundation laid by Gauss, Riemann went still further, to determine an even more general idea of a Pythagorean, not only for surfaces, but for manifolds of “n” determinations of magnitude. This leads into even more interesting areas, such as the one confronted in investigating physical processes in the very small and very large, or, in the biotic domain, where the characteristic of the geodesic is changing non-uniformly. To reach that point we must, however, first master the foundations laid by Gauss and Riemann, with some assistance of the insights gained from classical art, such as Beethoven’s late string quartets.

FOOTNOTE

1. For those who wish to know, a parametric formula for the sphere is Cos[q] Cos[p], Sin[q]Cos[p], Sin[p]. For the spheroid and ellipsoid, each direction is multiplied by a simple factor.

Riemann for Anti-Dummies: Part 46 : Something is Rotten in the State of Geometry

Riemann for Anti-Dummies Part 46

SOMETHING IS ROTTEN IN THE STATE OF GEOMETRY

When Gauss issued his 1799 doctoral dissertation on the fundamental theorem of algebra, he had much more in mind than just proving that particular theorem. He was creating the foundation for a mathematics that rested only on physical principles. He chose the domain of algebra, because that’s where his enemy was weakest. The algebraists, Euler, Lagrange, and D’Alembert, had boasted that they had “freed” mathematics from the paradoxes of geometry and, like their modern counterparts, the information theorists, had reduced even the most complicated problems down to a finite set of rules, definitions and procedures. This was the oligarch’s dream. The feudalist elite was constantly confronted with the dilemma that to stay in power they must rule over stupid people. But, the principles of physical economy would always intervene to bring about the destruction of any society that failed to value human creativity. The method of Newton, Euler, D’Alembert and Lagrange, a variation of an old Babylonian trick, offered the prospect of maintaining the world in a state of perpetual stupidity, while a small group of magicians (algebraists) kept things orderly.

Gauss showed that what Euler, Lagrange and D’Alembert considered their strength–the ability to plug holes in their system with a new definition based solely on their authority–was its weakness. Just as his teacher Kaestner had done with respect to the parallel postulate of Euclid, Gauss demonstrated that the square root of minus 1, was not the imaginary fiction Euler had defined it to be, but was the spirit of the physical universe come to haunt the algebraic system. Algebra, like the oligarchical system, could not solve its fundamental problem. It must yield to physical geometry, and the algebraists, to the creative scientist.

By the time Gauss was writing his 1799 dissertation, he had already begun the process of constructing a physical geometry based on measurement. As soon as he arrived at Goettingen in 1795, he borrowed one of the university’s theodolites and spent many hours measuring out triangles on the Earth. Some of his early notebook entries show him constructing a physical geometry from these measurements without resort to the axioms, definitions and postulates of Euclidean geometry. A quarter century later, when Gauss undertook to survey the entire Kingdom of Hannover, many of his colleagues were shocked that someone of his stature would undertake a project they considered pedestrian. Yet, Gauss saw in this undertaking, the chance to further his youthful efforts to construct a physical geometry based on measurement. This work was brought to fruition in his published papers on curvature, mapping and geodesy, the which provided the foundation for Riemann’s development of complex functions.

The past two segments of this series have dealt with some of the ideas developed by Gauss in his “General Investigations of Curved Surfaces” ,which established the principles of what, today, is called, “differential geometry”. Before turning to the more universal applications of Gauss’ geodetic investigations, it is advisable to review a principle of Leibniz’ calculus through the example of the catenary.

The determination of the catenary, as the shape formed by a hanging chain, required the discovery of a physical principle that, like the square root of minus 1, was outside the rules, definitions and procedures of Euclidean geometry, and outside the domain of sense-perception. Leibniz and Bernoulli determined that, while that physical principle could not be known by some a priori rule, it could be discovered from the way it expressed itself in the smallest parts of the chain.

To illustrate this, the reader should perform the pedagogical experiment described by Bernoulli in his text on the integral calculus. Take a string and tie a light weight to the middle of it. Take one end of the string in each hand and let the weight hang freely between them. As you pull your hands apart, you will feel an increase of force exerted on your hands by the weight. If you lift one hand higher than the other, the force on the raised hand increases, while the force on the lower hand decreases.

To simulate the action that produces the catenary, now hold the string in each hand very close to the weight. Move one hand so that it pulls the weight away from the other, while allowing the string to slide through both hands. As you do this, keep the segment of the string that connects the weight with the stationary hand horizontal. (This simulates the lowest point on the catenary.) To do this, you will have to constantly raise the moving hand. The farther your hands are separated, the faster the moving hand must be raised in order for the opposite segment to remain horizontal. The curve traced out by the moving hand will be the catenary.

The reason for performing the above described experiment, is to realize that the catenary curve is physically determined. In order to keep one of the string segments horizontal, the moving hand is compelled, by a physical principle, to follow the catenary curve. The curve is not seen, but its effects are “felt” by the moving hand, at every small interval of action. That infinitesimal expression of the catenary principle, Leibniz called the “differential”.

These effects can be measured by the increasing length of the curve for equal amounts of horizontal motion. In other words, when the hands first start to move apart, the moving hand only has to be raised a little to counteract the force of the weight. But as the moving hand moves further out, the amount of vertical “lift” for small horizontal increments, increases, which in turn, increases the length of the curve for corresponding horizontal motion. (See Figure 1.)

Figure 1

(Warning to those indoctrinated in Cartesian geometry: The horizontal and vertical here are not Cartesian axes, but directions of motion, physically determined with respect to the direction of the pull of gravity.)

In the hanging chain this action happens all at once. The horizontal and vertical are not simply directions of motion, they are the physically determined boundaries at which the catenary ceases to exist. The curve of the catenary unfolds as the path of least action between these two extremes. Its length per unit of action increases as it nears the vertical extreme, and decreases towards the horizontal. Since the length of the curve is a direct function of the physical principle governing the chain’s action, it is an appropriate measure of that principle.

Gauss’ geodetical investigations led him to extend Leibniz’ calculus into a higher domain.

All measurements of the Heavens and the Earth are made with respect to a physically determined direction, as indicated by the direction of a free hanging weight on a string, called a plumb bob. The surface of the Earth is that surface that is everywhere perpendicular to the direction of the plumb bob. The question Gauss investigated in his geodesy, is, ” What is the nature of this surface?” Since it were impossible to know the answer from sense perception, Gauss determined the overall nature of the surface of the Earth from small (differential) changes in action measured on it.

To begin to grasp Gauss’ idea, begin with the simpler case of the celestial sphere. This sphere can be entirely determined by two angles, one around the circle of the horizon, and one from the horizon to the pole. These two angles define an orthogonal network of circles, which, for pedagogical purposes, we will call latitude and longitude. The longitudinal circles are great circles, which are what Gauss called “geodesics” or shortest lines, while the latitudinal circles are not. (See Figure 2.)

Figure 2

While these circles are always orthogonal to each other, nevertheless, their relationships change, depending on where they are on the sphere. As the longitudinal lines get closer to the poles, they get closer together. Thus, the length of an arc of a circle of latitude between two determined arcs of longitude changes, depending on its position with respect to the poles. For example, the distance along the arcs of latitude between two arcs of longitude separated by 10 degrees, will decrease, as the latitudes get closer to the poles.

Now, compare that with a spheroid. Here, the lines of longitude are elliptical, and the lines of latitude are circular. In this case, the length of the arcs of latitude between two determined arcs of longitude still get smaller as they approach the poles, but also, the length of the arcs of longitude between any two determined arcs of latitude get {longer} as they approach the poles. (See Figure 3.)

Figure 3

This characteristic can be measured in a geodetic survey, by measuring the lines of latitude as angular changes in the inclination of the north star. If the Earth were a sphere, equal angular changes will correspond to equal changes in length of the geodesic longitudinal lines. If these geodesic lines get longer, with equal angular inclinations of the north star, then the Earth is spheroidal. What the specific measurements of that spheroid are, cannot be known by a priori mathematical methods, but require more refined physical measurements, as developed by Gauss.

Now, look at an ellipsoid. Here the lengths of the lines of latitude between any two determined lines of longitude, change, both as they approach the poles, and as they move around in the “equatorial” direction as well. Additionally, the lengths of the lines of longitude between any determined lines of latitude, increase as they approach the poles. (See Figure 4.)

Figure 4

(Note: The accompanying computer generated graphics are supplied merely to illustrate the text. The reader is strongly encouraged to make physical demonstrations on real surfaces approximating these shapes. The reader is also encouraged to experiment with wildly irregular shapes as well.)

Gauss recognized this characteristic of change as a new type of “differential”, which, for pedagogical purposes, I will call “surface differentials”. Like Leibniz’ differentials, these surface differentials express how the overall principle of action of the surface is manifest in every small part. However, instead of directly characterizing the least-action pathways, i.e. geodesics, these “surface differentials” characterize the changing nature of the principles through which the least- action pathways unfold. In other words, the surface differential expresses the characteristic of change of the principles that determine the characteristics of change of all possible least-action pathways, i.e. geodesics, on that surface.

To get an intuitive sense of this idea, think of these surface differentials being approximated by small “rectangular” patches of the surface. (See Figure 5, Figure 6, and Figure 7). Notice how the shape of these patches changes, as their positions change on the different surfaces.

Figure 5

Figure 6

Figure 7

Gauss determined that the relationship between these surface differentials and the characteristic geodesic lines of the surface could be measured, because even though these geodesic lines are always the shortest distance between two points on the surface, the length of the geodesic with respect to the surface differential changes, according to the overall curvature of the surface.

Gauss established the relationship between the surface differential and the characteristic curvature of the changing geodesic line, as a generalization of the Pythagorean relationship between the diagonal and the side of the square or rectangle. In the case of a flat plane, (or surface of zero-curvature, as Gauss would see it), the relationship of the diagonal to the side of a square (or rectangle), expresses the power that generates areas, as distinct from the power that generates lines. Thus, the line that forms the diagonal of the square is a different type of line than that line which forms the side of the square, because it is generated by a higher power. This relationship can be measured by the relationship of the length of the diagonal, to the lengths of the sides of the square or rectangle. The common expression for this “Pythagorean” relationship is that the length of the diagonal is equal to the square root of the sum of the squares of the sides of the square or rectangle.

This “Pythagorean” relationship, Gauss showed, was just a special case of a more general principle. On a curved surface, the sides of the square are the constantly changing “sides” of the surface differential, and the diagonal is the geodesic. The principle that governs the constantly changing lengths of the “sides” of the surface differential is a function of the curvature of the surface, which, in turn, is reflected in the changing length of the geodesic diagonal. Consequently, the overall curvature of the surface is reflected in the smallest parts of the geodesic. From this Gauss devised a more general idea of the “Pythagorean”, in which the lengths of the “sides” of the surface differential are multiplied by a function that characterizes the physical curvature of the surface. As the surface differential changes according to the curvature of the surface, the length of the geodesic diagonal changes accordingly. (A future pedagogical will illustrate, geometrically, Gauss’ idea.)

To get an intuitive sense of this concept, look again at figures 5, 6, and 7. Imagine the diagonals of each surface differential. Imagine how the lengths of these diagonals change with the position of the surface differential. Now conduct a similar investigation on the physical surfaces you experimented with earlier. Draw on these surfaces geodesic diagonals to the orthogonal curves you previously drew. This can be done by holding a string taught between opposite corners of each “rectangular” patch, and tracing the string path with a marker. Notice the changes in length and direction of these diagonal’s geodesic as the curvature of the surface changes.

Now, look at a concrete example with respect to the physics of navigation. On a flat surface draw a grid of orthogonal lines. Draw a diagonal line that cuts all the vertical lines at the same angle. This will produce a straight line. (See Figure 8.) Now try the same thing on a sphere. That is draw a series of geodesic lines that cut the lines of longitude at a constant angle. The result is not a straight line, but a spiral like curve called a “loxodrome” (See Figure 9.) A navigator who has not mastered the principles of curvature, will find himself getting farther and farther from his destination, and closer and closer to the north pole!

Figure 8

Figure 9

(Figure 10 illustrates a similar process for the spheroid and ellipsoid. Notice the difference between them and the sphere.)

Figure 10

From this relationship, Gauss showed that it were possible to discover the surface differential, and thus the characteristic curvature of the surface, by measuring small variations in the length of geodesic lines. For example, in determining the length of the geodesic line that connected his observatory in Goettingen with Schumacher’s in Altona, Gauss measured a 16 seconds of an arc deviation from what that length should be if the Earth were a spheroid. That led Gauss to prove that the shape of the Earth could not conform to any a priori geometric shape, but was being determined by the physical characteristics of the Earth’s matter and its motion.

Riemann for Anti-Dummies: Part 45 : The Making of a Straight Line

Riemann for Anti-Dummies Part 45

THE MAKING OF A STRAIGHT LINE

Straight lines are not defined, they are made.

The above statement might seem jarring to one fed a steady diet of neo-Aristotelean dogma from their primary, secondary and university teachers, but it is the standpoint adopted by C.F. Gauss by the time he was 15 years old. In July 1797, at the age of 20, Gauss wrote in his notebook, “Plani possibiliatem demonstravi,” (The Possibility of the Plane Proven). He later elaborated on this idea in a January 1829 letter to Bessel, where he spoke of his conviction, “for nearly 40 years,” that “it were impossible to establish the foundations of geometry a priori.” Gauss gives as an example, the Euclidean definition of a plane, as a “surface that lies evenly with the straight lines on itself” (Euc. I, def.7.) “This definition,” Gauss wrote, “contains more than is necessary to determine the surface, and involves tacitly a theorem which first must be proven.”

An individual subjected to the aversive conditioning of today’s information society education, might think Gauss was making some esoteric quibble, of interest only to the arcane curiosity of certain specialists. In fact, the epistemological issue Gauss is addressing, is exactly the one that is the cause of much of today’s mass psychosis, upon whose successful treatment the future of civilization depends.

To grasp the point, consider the following illustration, along with Euclid’s concomitant definition of a line as “breadthless length” (Euc. I,def. 2.) and a “straight line” as, “a line that lies evenly with the points on itself.” (Euc I, def. 4.) Imagine an octahedron, or some other solid, and think of the line connecting two of its vertices. Under Euclid’s definition, this line would be straight, as it lies evenly with the vertices, and the face of the solid would be a plane because it lies evenly with the lines that form its edges. But, from the standpoint of construction, the solid is generated from a sphere. Those same vertices also lie evenly with great circle arcs along the surface of the sphere, which themselves lie evenly with the spherical surface. (See Figure of inscribed octahedron.) How can one distinguish which of the two surfaces, spherical or planar, and which set of lines, circular or linear, are the “straight” ones, by Euclid’s definition?

Inscribed Octahedron

Euclid’s definition applies equally to both types of lines and surfaces, as well as to an infinite number of other possible surfaces and lines that could conceivably lie even with the vertices of a solid. A definition alone is insufficient to distinguish one from the other, because, as Gauss says, the definition assumes a theorem concerning the physical characteristics of the surface and the lines contained in it. Such characteristics must be proven, or, in the domain of physical action, measured, by what Riemann called a “unique experiment.” In the tradition of Leibniz, Gauss called these characteristics, “curvature.” The difficulty today’s victims of information education have in grasping Gauss’ point, is that they’ve become accustomed to thinking of “straight” and “flat” in a certain pre-defined way, which, in the minds of the victim, carry the authority of “Roman law.” As long as this “rule of law” holds its sway over the victim’s mind, the afflicted person will cringe at the thought that the physical universe might disobey this defined law of straightness. But, whatever the authority with which this rule of law is pronounced, the universe decides what is straight as a matter of principle. This produces a psychological crisis that intensifies as long as the victim cowers under the arbitrary dictate of definitional straightness.

A baby and a drunk both walk a crooked path. The baby because it’s trying to discover the multiply-connected principles of physics, biology and cognition that determine its intended path. The drunk because, his damaged state prevents him from recognizing the principles he once knew, and he responds to whatever definition of straightness his inebriated impulses conjure up. The baby’s frustration, when it falls upon reaching the limit of each temporary hypothesis, is transformed into joy, when the discovery of the missing principle increases its power to proceed on its way. The stumbling frustration of the drunk, oligarch, lackey, or victim of Straussian brainwashing, stews into bi-polar rage at his loss of control, screaming, like Shelley’s Ozymandias, “Look upon my works ye mighty and despair….”

As long as one’s mental powers are impaired by arbitrary definitions whose only force is the arbitrary authority with which they’re uttered, one remains in a stupor, either intoxicated with the power to wield such authority, or, the depression brought on by submitting to it. To free the victim and restore those inherently human powers he or she once experienced, the sobering balm of classical art and science must be applied. Hence, the importance of pedagogical exercises.

The Determination of Curvature

While Gauss’ youthful insight was at odds with such contemporary authorities as Leonard Euler and his protege, I. Kant, who insisted that the principles of geometry could only be given by a priori definitions, its roots were quite ancient. Euclid’s “axiomization” of geometry was itself at odds with the very process by which the geometrical principles contained in it were discovered. As the solutions of Archytas and Meneachmus for the problem of doubling of the cube, and more generally, the construction and uniqueness of the five Platonic solids from spherical action indicate, the relationships of rectilinear geometry are derived from non- rectilinear physical action. The investigations into this type of physical action are further exemplified by the works of Apollonius, Archimedes and Eratosthenes, as well as the Pythagoreans’ demonstration that the relationships among musical tones are generated by a higher principle than the linear divisions of a string.

It was this “anti-Euclidean” approach that was adopted by Cusa, Kepler, Fermat, and Leibniz, who replaced the sophistry of an arbitrary definition of a straight line, with the idea of least-action pathway, or what later became known as a “geodesic.” The geodesic is the straightest and shortest line, whose nature is determined by the physical properties of the surface, or, from Riemann’s more general standpoint, the “n-dimensional manifold” or phase-space, in which it occurs.

For example, Kepler’s planetary orbit is the least-action pathway created by the harmonic principles of the solar system, the which “define” the elliptical orbit as its “straight line.” Similarly the principles of reflection of light “define” the shortest distance as its “straight line” while the principles of refraction “define” shortest time as straightness. The introduction of the principle of the changing velocity of light under refraction, “re-defines” the straight line, from the path of shortest-distance to the path of least-time. Or, conversely, the change in what is straight, indicates the presence of a new principle.

In each case, the definition of a straight-line is not given by some arbitrary authority, but by a set of measurable physical principles. The question for science, politics, economics and history, is how to determine those governing physical principles, from what amount to small pieces of the “straight” lines determined by them. This entails being able to discover the principles from the “curvature” of the line, and to discover new principles by measuring changes in that characteristic curvature.

Exemplary is Leibniz’ and Bernoulli’s determination of the curvature of the hanging chain. Unlike Galileo, both Leibniz and Bernoulli recognized that the chain’s curvature was determined by a physical principle. This principle does not exist in an empty, infinitely-extended Euclidean-type plane, but is produced in a physical manifold with a characteristic curvature. This manifold is bounded by physically-determined extremes, expressed by the relationship of the lowest point of the chain to the hanging points. If the hanging points coincide, there is no tension, and the chain has no curvature. If the hanging points are pulled apart, at some point the chain will break. At these extreme conditions of maximum and minimum tension, there is no curvature, and no stable “orbit” for the chain. The common-sense notion of straight, that is not-curved, only exists outside the physical manifold in which the chain hangs. In that manifold the catenary curve is the only possible “straight” line. Thus, “straightness” for the chain, is a curve–a curve determined by a measurable physical characteristics. At every small interval along the chain, the links steer a course which is constantly changing, but changing according to a measurable principle. As developed in previous installments, Leibniz and Bernoullli demonstrated that this characteristic changed according to a principle that Leibniz called, “logarithmic.”

A similar relevant case is Gauss’ earth-shaking determination of the orbit of the asteroid Ceres, from that infinitesimal portion of its orbit represented by Piazzi’s observations. All the established scientific authorities were stymied in their efforts to find Ceres’ orbit, hampered by their insistence that Ceres’ orbit was moving in an empty box and its orbit was a deviation from the definition of straight-line action that Galileo and Newton had re-imposed on physics, after Kepler’s liberation of science from such Aristoteleanism. Gauss, as a student of Kaestner, was guided by the knowledge that Ceres was following a least-action pathway in a solar system governed by the harmonic physical principles that Kepler described, and that those harmonic principles indicated a discontinuity in the region between Mars and Jupiter. Unlike his competitors, Gauss knew that the Galilean/Newtonian straight line did not exist in the physical manifold in which Ceres and the Earth were moving. So, while everyone else was looking for a path among an infinite number of possible pathways, in a manifold that did not exist, Gauss was looking for the unique least-action pathway that Kepler’s solar system would produce. His successful approach was focused on determining how those principles would be expressed in the small portion of the orbit that Piazzi had observed. (See “How Gauss Determined the Orbit of Ceres,” Summer 1998 Fidelio.)

A crucial distinction occurs when one compares the case of the catenary with the case of the Ceres orbit. The discovery of the catenary principle required the determination of a single pathway. The discovery of Ceres’ orbit involved the relationship between two different pathways, Ceres’ and Earth’s as these pathways were viewed as projections on the inside of the celestial sphere. These two pathways, though different, are both least-action, i.e. “straight,” paths within Kepler’s solar system. Thus, the solar system produces “straight-lines” of different curvatures in different parts.

Gauss found a similar situation in his geodetic measurements, where he measured a variation in the direction of the pull of gravity from place to place on the Earth. As Gauss moved north, the angle of inclination of the north star increased, but non-uniformly with the distance traveled along the Earth’s surface. But, Gauss also determined that the direction of the pull of gravity varied as he moved east to eest or some other intermediate direction. The question for Gauss was how to determine the shape of the Earth, from these variations along small parts of its surface? Or, in other words, how is the overall curvature of the Earth, and its local variations, reflected in every small part of its surface, in the same way that the physical principle of the catenary is reflected in every small part of the chain?

The Making of Curvature

These types of considerations gave raise to Gauss’ theory of curved surfaces. As illustrated in the previous installment, Gauss measured the “total” or, “integral” curvature of a surface by mapping the changes in direction of the normals onto an auxiliary sphere. Following the direction of Leibniz’ infinitesimal calculus, Gauss showed how this overall curvature was related to the curvature at every infinitesimal surface element:

“The comparison of the areas of two corresponding parts of the curved surface and of the sphere leads now (in the same manner as e.g. from the comparison of volume and mass springs the idea of density) to a new idea. The author designates as “measure of curvature” at a point of the curved surface the value of the fraction whose denominator is the area of the infinitely small part of the curved surface at this point and whose numerator is the area of the corresponding part of the surface of the auxiliary sphere, or, the integral curvature of that element. It is clear that, according to the idea of the author, integral curvature and measure of curvature in the case of curved surfaces are analogous to what, in the case of curved lines, are called respectively amplitude and curvature simply. He hestates to apply to curved surfaces the latter expressions, which have been accepted more from custom than on account of fitness. Moreover, less depends upon the choice of words than upon this, that their introduction shall be justified by pregnant theorems.”

Gauss goes on to develop the methods by which to measure what has now become known as “Gaussian curvature.” If, following the tradition of Euler, the surface is considered as the boundary of a three-dimensional solid object, then this curvature could be measured by cutting the surface at the point by two planes, normal to the surface and perpendicular to each other. The curves formed by the intersection of these planes with the object express the curves of minimum and maximum curvature at that point.

To illustrate this, cut an egg, apple, or some other curved solid in half. Then cut a similar shaped object in half at a 90 degree angle to the first cut. Compare the curves formed by these cuts. Cut another similarly shaped object at another angle. Compare the curvature of the three types of curves.

This method is totally useless for a real physical problem such as measuring the surface of the Earth, for it is obviously impossible to make orthogonal cuts in the Earth at every point and measure the curvature of the resulting curves.

To solve this problem, Gauss conceived of a curved surface as a two-dimensional object. Thought of in this way, the curvature could be determined by measuring the behavior of the “shortest” lines, i.e. geodesics, emanating from that point.

For example, the surface of a sphere can be entirely determined by a system of two sets of orthogonal circles, akin to “lines” of longitude and latitude, the former being “geodesics” and the latter not. In a sphereoid, the lines of latitude remain circular, while the longitudinal ones become elliptical. In an ellipsoid, both sets of curves are elliptical. Other examples are a psuedosphere, where the one set of curves are circles and the other tractrices, or, the catneoid, where the curves are circles and catnaries. For more irregular surfaces, the curves are irregular, but such an orthogonal system can always be developed.

From this standpoint, the common-sense notion of a “flat” Euclidean plane is just a special type of surface, with no particular, a priori, “legal” authority. The common sense notion of “straight” line becomes simply the “geodesic,” characteristic of this type of surface.

Gauss showed that the behavior of the shortest lines emanating from any point on a surface could be measured with respect to these systems of orthogonal curves by extending the method of Leibniz’ infinitesimal calculus. And, more importantly for physical science, the nature of these orthogonal curves, and consequently the curvature of the surface, could become known by the measured changes in these geodesic lines.

In the next installment we will delve into Gauss’ method more directly. For now, we supply an intuitive introduction through the accompanying animations. Here you can compare the behavior of geodesic lines emanating from a point on different surfaces, namely: a sphere, spheroid, ellipsoid, monkeysaddle, and torus.

Sphere

Sphereoid

Ellipsoid

Monkeysaddle

Torus

Notice the relationship between the behavior of these lines and the system of orthogonal curves drawn on the surface. Notice how the shape of these geodesics depends only on the nature of the surface and their direction. Also, compare the behavior of these geodesics with the integral curvature of these surfaces illustrated in the Gauss mappings that accompanied Riemann for Anti-Dummies Part 44.

The first step is to develop the power to measure the physical characteristics of the surface from the “straight” lines that surface produces. But humans possess a greater power–to discover and apply new principles, thus changing the curvature, and making “straight” lines.