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".

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....."

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.