Author: Bruce Director

THE BEGINNINGS OF DIFFERENTIAL GEOMETRY

Fifty-two years after Gauss’ 1799 doctoral dissertation on the fundamental theorem of algebra, his student, Bernhard Riemann, submitted, to Gauss, an equally revolutionary doctoral dissertation that took Gauss’ initial discovery into a new, higher, domain. Riemann’s thesis, “Foundations for a general theory of functions of a single variable complex magnitude”, built on the foundations of Gauss’ own work, established a complete generalization of the principles of physical differential geometry that was set into motion by Kepler nearly 250 years earlier.

It is beneficial, and perhaps essential, as a preliminary to a more detailed discussion of Riemann’s work itself, to review three exemplary discoveries of physical principles, that taken together, trace the historical development of the ideas leading into Riemann’s work: Kepler’s principles of planetary motion; the Leibniz-Bernoulli discovery of the principle of the catenary; and Gauss’ own work in geodesy. All three, while seemingly diverse, are in fact intimately connected. They all deal, in one way or another, with investigations into the nature of universal gravitation, and, taken together, they comprise a succession of concepts of increasing generality and power.

Begin first with Kepler. Taken in its entirety, from the Mysterium Cosmographicum to the Harmonice Mundi, Kepler’s work demonstrates that the action governing any planet at any moment is a function of the principle that organizes the solar system as a whole; the principle of universal gravitation. Kepler discovered that this principle has an harmonic characteristic, which determines that the planetary orbits are elliptical, not circular. The unique shape of each individual elliptical orbit is determined, not by each planet alone, nor by the pair-wise interaction of that planet with the Sun, but by the harmonic relationship among the maximum and minimum speeds of all the planets. In other words, the action of the planet at any moment is determined by these extremes, between which, the planet’s orbit “hangs”. The magnitudes of these “hanging points”, are not arbitrary, but when taken all together, conform, approximately, to the harmonic ordering of the musical scale.

The eccentricity of the planetary orbits posed a challenge to Kepler because he had no mathematical means to determine the exact position, direction and velocity of each planet at every moment, so he demanded the invention of a new mathematics. Kepler prescribed that such a mathematics must be able to determine how the harmonic principle that determines the planet’s extremes, is expressed, throughout the entire orbit, and he took the first steps toward developing that mathematics. (See Riemann for Anti-Dummies Parts 1-6)

Responding to Kepler’s demand, Leibniz and his collaborator, Johann Bernoulli developed the calculus, the most general expression of which is demonstrated by their joint effort on the catenary. At first glance, the catenary appears similar, in principle, to a planetary orbit, in that the shape of the curve seems to be determined by the position of the points from which chain hangs. As the position of these “hanging points” changes, the chain re-orients itself, so that its overall shape is maintained. In this respect, the relationship of these hanging points to all the other points on the catenary, initially seems analogous to the relationship between the extreme speeds of a planet to the entire orbit. But, as Bernoulli showed in his book on the integral calculus, all points on the catenary, except the lowest point, are, at all times, hanging points. (The reader should review Riemann for Anti-Dummies Parts 10 “Justice for the Catenary”, and chapter 4 of “How Gauss Determined the Orbit of Ceres”, to perform the experiments indicated therein.(fn. 1.)) This is, in fact, an inversion of the principle expressed in Keplerian orbits. In the case of the planet, the orbit, “hangs” between its two extremes. For the catenary, the extreme, that is the lowest point, is the one point that does no hanging. (In Cusa’s terms it is the point that is simultaneously motion and no-motion.) Applying Leibniz’ calculus, Bernoulli demonstrated how the catenary is “unfolded” from this lowest point. (fn. 2.)

Leibniz, in turn, demonstrated that this physical principle also reflected the characteristic exhibited by the logarithmic (exponential) function. (See Leibniz paper on catenary.) Thus, the hanging chain is characterized by the same transcendental principle that subsumes the generation of the so-called algebraic powers, and which is exhibited in other physical processes such as biological growth and the musical scale, as well. Consequently, the characteristics of the logarithmic (exponential) function, is an expression of a physical principle, not a mathematical one.

Now, compare the above described examples with Gauss’ discovery of the Geoid. From 1818 to 1832 Gauss carried out a geodetic survey of the Kingdom of Hannover. This involved determining the physical distances along the surface of the Earth by laying out triangles and measuring the angles formed by the “line of sight” sides. The paradox Gauss confronted was that the relationship between the lengths of the sides of the triangles and the angles, is a function of the shape of the Earth. (fn.3.) However, the shape of the Earth could not be known in advance of the measurements. The problem was further complicated by the fact that all the measurements were taken with respect to the direction of the pull of gravity, as determined by the direction of a hanging plumb bob. Like the relationship between the angles of a triangle and the lengths of the sides, the direction of the pull of gravity depends on the shape of the Earth. For example, if the Earth were spherical, the plumb bob would always point toward the center of the Earth. If the Earth were ellipsoidal, the plumb bob would point to different places, depending on where on the ellipsoid the measurement was being taken. Gauss showed that the problem was even more complicated, because the Earth’s shape was very irregular. (See Riemann for Anti-Dummies Part 17.)

Here Gauss was confronted with exactly the same type of problem as Kepler and Leibniz before him. Existing mathematics could not measure such an irregular shape. All previous approaches began with an a priori assumption of the shape of the Earth, one that conformed to existing mathematical knowledge. (This brings to mind Gallileo’s foolish insistence that the catenary was a parabola because that was the shape in the mathematical textbooks which looked most like a catenary. The chain, however, did not read Gallileo’s preferred texts.) Gauss abandoned all such attempts to fit the Earth into an assumed shape, declaring that the geometrical shape of the Earth is that shape that is everywhere perpendicular to the pull of gravity. In other words, instead of assuming an imaginary shape, and measuring the real Earth as a deviation from the imaginary one, Gauss rejected the fantasy world altogether. (Something more and more people should want to do these days as the global monetary systems disintegrates.) The physically determined shape that Gauss measured has since become known as the Geoid.

While the Geoid is an irregular surface, its irregularity is “tuned” so to speak by the motion of the Earth on its axis. Like the planetary orbit, or the hanging chain, that motion determines the positions of two, “hanging points”, specifically the north and south pole, from which the Geoid hangs.

However, since the Geoid is a surface, it has a different relationship to its poles, than the planetary orbit to its extremes, or the catenary to the lowest point. The latter two cases express the relationship between singularities and action on a curve. The former expresses the relationship between singularities and action on a surface, from which the action along the curves is derived.

The problem Gauss confronted was that since the physical triangles he measured on the surface of the Geoid were irregular, how could the lengths of the sides be determined from the angles, without first knowing the relationship between the lengths and the angles, i.e., the shape of the surface? To solve the problem, Gauss recognized that since all his measurements were angles, he could free himself from having to assume the Earth’s shape before he could determine his measurements, if he could project these angles from one surface to another, for example, from the geoid, to an ellipsoid, to a sphere and back again. Like Kepler and Leibniz, Gauss could not do this within the existing mathematics. So he invented a new one.

Gauss described the beginnings of this new mathematics in several locations, most notably his 1822 memoir on the subject of conformal mapping, that was awarded a prize from the Royal Society of Sciences of Copenhagen. Riemann relied heavily on this paper for the foundations of his own doctoral dissertation.

Conformal mapping is a term, invented by Gauss, to refer to transformations from one surface to another in which the angles between any curves on that surfaces are preserved. In his memoir, Gauss described conformal mapping as a transformation where, “the lengths of all indefinitely short lines extending from a point in the second surface and contained therein shall be proportional to the lengths of the corresponding lines in the first surface, and secondly, that every angle made between these intersecting lines in the first surface shall be equal to the angle between the corresponding lines in the second surface.”

To get an idea of what this means, perform the following experiment. Take a clear plastic hemisphere and draw a spherical triangle on it with heavy black lines. Go into a dark room and, using a flashlight, project the triangle onto the wall. If you hold the flashlight at the center of the hemisphere, the curved lines of the spherical triangle will be transformed into straight lines. If you then move the flashlight from the center of the hemisphere to a pole, the projected straight lines will become curved again, and the angles between them will be equal to the angles between the sides of the original triangle on the hemisphere.

To discover experimentally the difference between these two projections, tape cardboard circles of differing sizes onto the plastic hemisphere. (The circles should vary from quite large to quite small.) Now perform the same projection with the flashlight as before. When the flashlight is at the center of the hemisphere, these circles project to ellipses. When the flashlight is at the pole of the hemisphere, the circles become more circular, with the smaller circles become more circular than the larger ones. In the first case, the transformation of the circles into ellipses indicates that the proportion by which figures are transformed changes depending on the direction of the transformation with respect to the poles. The second case shows that the transformations are proportional in all directions.

Thus, the conformal mapping of one surface to another involves a change in rotation and direction. Having done the work on Gauss’ fundamental theorem of algebra, you should be able to recognize, as Gauss did, that this type of change could only be represented in the complex domain, which is where we will begin next time.

FOOTNOTES

1. Any two points on opposite sides of the lowest point hold up the weight of the chain hanging between them. The force required to hold up this weight is proportional to the sines of the angles made by the tangents to the catenary at this point, and a vertical line rising from the point at which the tangents intersect.

2. The reader is urged to preform the experiment described in the indicated NF article. Take a string and tie a weight in the middle of it. Hold the ends of the string in each hand and let the weight hang between them. As you move your hands apart, the tension you feel on your hands will increase. If you begin with your hands close together, the tension is relatively small. As you pull your hands apart, the tension increases, slowly at first, but the rate of increase in the tension grows, the farther apart your hands are to one another. Now try to move your hands apart, while the string slides between your fingers, so that the string on one side remains horizontal. The other hand will move in the shape of the catenary.

3. The reader can grasp this by comparing triangles drawn on a piece of paper, a sphere and an irregular shaped surface, such as a watermelon.

If we look at the known cases of constructable polygons, the triangle, square and pentagon, each is constructable by a series of nested steps, in which a “knowable” magnitude is constructed, and then from that magnitude, another “knowable” magnitude is constructed, until the side of the polygon is found. For example, the triangle is constructed by first constructing the hexagon from the radius of the circle. Then the side of the triangle is constructed from the side of the hexagon. The square is constructed from one diameter and a second diameter is constructed perpendicular to it. The pentagon is constructed by first constructing the golden mean, and then the side of the pentagon is derived from the golden mean.

In each of the above examples, each magnitude in the chain is constructed from the its predecessor by simple circular action. Consequently, such magnitudes are commensurate with the type of magnitudes associated with doubling of the square, i.e. second degree magnitudes, which are generated by simple circular action. As distinguished from third degree magnitudes that are associated with doubling the cube, which as was seen in the construction of Archytus, require the complex action of rotation and extension.

Therefore, those polygons, whose constructions could be reduced to a nested chain of second degree magnitudes are, in principle, constructable. All others are not.

The crucial insight of Gauss was to recognize that each polygon (“planetary system”) could be constructed as a chain of “orbital periods” and “sub-periods”. The character of the magnitudes associated with these periods and sub-periods, is determined by the number-theoretic characteristics of the prime number, or more specifically, the prime number minus 1.

Herein lies the “profound connection” between the generation of transcendental magnitudes and higher Arithmetic. The arithmetical characteristics determine the geometry, while the geometry, in turn determines the arithmetical characteristics. Unlike formalists such as Euler, Lagrange and D’Alembert, Gauss saw no distinction between the geometrical and the arithmetical characteristics. The same physical principle that governed the circle, ruled number. What the circle concealed, number revealed. One need only be able, as Plato said, “to see the nature of number with the mind only.” (Remember that the Greek word from which “arithmetic” is derived has the same root as the Greek word, “harmonia”.)

For Gauss, the circle is not simply an object in visible space, but rather an artifact of an action in the complex domain. Successive divisions of the circle reflect a succession of different types of actions corresponding to the hierarchy of powers. The vertices of an “n” sided regular polygon are the “n” roots of 1. Inversely, as was shown last week, these vertices can be generated as a succession of powers.

Ironically, the principles of this so-called “imaginary” domain determine what is possible in the visible domain. Gauss showed that the deeper principle of their generation becomes known under examination of, what he called the “residues of powers” in his “Disquisitiones Arithmeticae”.

Each prime number modulus has a characteristic period of residues with respect to a series of powers. For example, the modulus 5 produces the period of residues {1, 2, 4, 3, 1, 2, 4, 3, 1, 2, 4, 3,etc.}, with respect to the powers of 2, and the period of residues {1, 3, 4, 2, 1, 3, 4, 2, 1, 3, 4, 2, etc, }, with respect to the powers of 3. (See Riemann for Anti-Dummies Parts 20-23.)

(Since the powers of 2 and 3 yield complete, albeit different, periods, they are called “primitive roots” of 5. Compare this result to the periods generated from the residues of the powers of 2 and 3 relative to modulus 7. In the case of 7, 3 is a primitive root, whereas 2 is not.)

These periods are completed periods and are not altered when all the elements are multiplied by any number. For example, multiply {1, 2, 4, 3} by any number, and take the residues relative to modulus 5. The resulting period will be the same as the one you started with. Similarly, for the period {1, 3, 4, 2}. (The reader is strongly encouraged to perform these experiments.)

Each complete period also has the two sub-periods. For the case of modulus 5, those sub- periods are {1, 4} and {2, 3}, which “orbit” each other. When either sub-period is multiplied by 2 or 3, they are transformed into the other. When multiplied by 1 or 4, they remain unchanged.

Similarly, the modulus 7 produces the period of residues, {1, 3, 2, 6, 4, 5, 1, 3, 2, 6, 4, 5} with respect to the powers of 3. It contains 2 sub-periods of 3 elements each, {1, 2, 4} and {3, 6, 5 } and 3 sub-periods of 2 elements each, {1,6}, {3, 4}, and {2, 5}. (Much will be gained if the reader tries multiplying the elements of each sub-period to see what transformations occur.)

Modulus 17 produces the period of residues, {1, 3, 9, 10, 13, 5, 15, 11, 16, 14, 8, 7, 4, 12, 2, 6} with respect to the powers of 3. It contains 2 sub-periods of 8, {1, 9, 13, 15, 16, 8, 4, 2} and {3, 10, 5, 11, 14, 7, 12, 6}; 4 sub-sub-periods of 4, {1, 13, 16, 4}; {9, 15, 8, 2}; {3, 5, 14, 12}; and {10, 11, 7, 6}. And, finally, 8 sub-sub-sub-periods of 2, {1, 16}, {3, 14}, {9, 8}, {10, 7}, {13, 4}, {5, 12}, {15, 2}, {11, 6}.

Notice that in all cases, the sum of the numbers of a period or sub-period is always congruent to 0 relative to the modulus, and that the lengths of all periods are always the modulus minus 1 or a factor of the modulus minus 1.

The Determination of the Polygon’s Orbits

Being an artifact of an action in the complex domain, these individual vertices each corresponds to a complex number. The “n” complex numbers, corresponding to the “n” vertices, of an “n” sided polygon, comprise a complete period of “n” roots. The problem Gauss confronted was how to determine the positions of the individual vertices (“orbits”) of a polygon ?

Gauss’ discovery was to show that each of these “orbits” was completely determined by the harmonic nature of the whole. That harmonic principle is reflected in the nested chain of periods and sub-periods of the residues of powers. Gauss worked by inversion. Like Kepler with planetary orbits, Gauss understood that the harmonic principle determined the individual positions, so he developed a method to work from the top down, that is, from the harmonies to the notes, so to speak, showing how to “read” this chain of periods and sub-periods to determine the positions of the vertices, (“orbits”) of the polygon.

For pedagogical purposes it is most efficient to illustrate by continuing with the example of the pentagon.

The first step in determining the vertices of the pentagon is organize the vertices into a “harmonic” period. As was shown last week, all the vertices can be generated as a series of powers from any one of them. Therefore, Gauss began with one of the vertices and generated all the others as a series of powers. But, to bring out the “harmonic” characteristic, they had to be ordered according to the principle exhibited by the residues of the primitive root. Continuing from the example of the pentagon from last week, that would mean generating the period from the powers of a^{2^0}, a^{2^1}, a^{2^2} and a^{2^3}. Taking the residues of these periods relative to modulus 5, these vertices will now be in the order, {1, 2, 4, 3}.

This period of can be divided into the two sub-periods {1, 4} and {2, 3}, that define the first set of magnitudes required to construct the pentagon. To determine the value of these magnitudes, Gauss considered them as “roots”, and since there are two of them, they must be “roots” of a quadratic equation. Call the value of {1, 4}= r1 and the value of {2, 3} = r2.

Here again Gauss worked by inversion. Even without knowing what the values for r¹ and r² are, except that they are “roots” of some quadratic equation, Gauss could work backwards from the harmonic relationship between them, to determine what must produce them.

To solve this problem, Gauss relied on the relationship between the roots and coefficients of algebraic equations (introduced without demonstration). That relationship is that if a quadratic equation is in the form x² + Ax + B = 0, the sum of the roots equals -A and the product of the roots equals B.

Back to our example. Even without knowing the values of the individual vertices, we can know the sum and the products of them. The sum of the sub-periods {1, 4} and {2, 3} is {1 + 2 + 4 + 3}. This means adding together the complex numbers that correspond to the vertices 1, 2, 4, 3. Each complex number denotes a complex quantity of combined rotation and extension. To add complex numbers, you carry out the rotation and extension in series. In this example, you first carry out the rotation and extension that produces vertex 1. Then from the endpoint of vertex 1, carry out the rotation and extension that corresponds to vertex 2, and so forth. Geometrically, this turns the “inside-out” pentagon, “inside in”. (See figure.) From this it can be seen that the sum of 1 + 2 + 3 + 4 = -1.

Similarly, we can also determine the product of the sub-periods, even without knowing the values of the individual vertices. The product of the sub-periods {1, 4} and {2, 3} is {(1 + 2) + (1 + 3) + (4 + 2) + (4 + 3)}. Taking the residues relative to modulus 5 this equals {3 + 4 + 1 + 2} which also equals -1. (See figure.) ( This is also evident from the fact that 1 x 4 x 2 x 3 = 24 which is congruent to -1 mod 5.)

Therefore {1, 4} and {2, 3} are the “roots” of the quadratic equation where A = 1 and B = -1, or, x² + x – 1 = 0. That means the {1, 4} = r1 = (-1+?5)/2 and {2, 3} = r2 = (-1-?5)/2.

The final step for the construction of the pentagon is to find the two vertices from the just discovered values of each sub-period. For example, the vertices 1 and 4are the “roots” of the sub- period {1, 4}, and the vertices 2 and 3, are the “roots” of the sub-period {2, 3}.

In sum, the action that generates the pentagon is a nested chain of second degree actions, and therefore, “knowable” geometrically.

What Gauss has demonstrated in general, is that any polygon is generated by a nested series of actions determined by the periods and sub-periods formed by the residues of powers. Since the number and length of these periods and sub-periods is determined by the factors of the modulus minus 1, the degree (or power) of each action will be determined by these factors.

For example, the construction of the heptagon will be determined by one cubic and one quadratic action. The 11-gon will be determined by one fifth power and one quadratic action; the 13-gon by one cubic and two quadratic actions; the 19-gon by two cubics and one quadratic action.

On the other hand, the 17-gon, 257-gon, the 65, 537-gon are all generated by a chain of quadratic powers, and are therefore geometrically “knowable”

Anyone who makes the effort to re-live this discovery of the 18 year old Gauss, will discover a corresponding increase in their own cognitive power.

THE POWERS OF ONE

On the morning of March 30, 1796, Carl Friedrich Gauss discovered that the way people had been thinking for more than 2000 years was wrong. That was the day, when, after an intensive period of concentration, he saw on a deeper level than anyone before, the “profound connection” between transcendental magnitudes and higher Arithmetic.

The first public announcement of his discovery was at the initiative of E.A.W. Zimmerman, a collaborator of Abraham Kaestner, who headed the Collegium Carolineum, the school for classical studies, where Gauss had received his preparatory education. The notice was carried in the April 1796 issue of Allgemeine Literaturzeitung:

“It is known to every beginner in geometry that various regular polygons, namely the triangle; tetragon; pentagon; 15-gon, and those which arise by the continued doubling of the number of sides of one of them, are geometrically constructable.

“One was already that far in the time of Euclid, and, it seems, it has generally been said since then that the field of elementary geometry extends no farther; at least I know of no successful attempt to extend its limits on this side.

“So much the more, methinks, does the discovery deserve attention, that in addition to those ordinary polygons there is still another group, for example the 17-gon, that are capable of geometric construction. This discovery is really only a special corollary to a theory of greater scope, not yet completed, and is to be presented to the public as soon as it has received its completion.”

Carl Friedrich Gauss
Student of Mathematics at Goettingen

“It deserves mentioning, that Mr. Gauss is now in his 18th year, and devoted himself here in Brunswick with equal success to philosophy and classical literature as well as higher mathematics.”

E.A.W. Zimmerman, Prof.

Gauss did not construct the 17-gon. As the announcement indicates, the constructability of the 17-gon is merely a corollary of a much deeper principle–the generation of magnitudes of higher powers, as that principle was understood by Plato, Cusa, Kepler, Fermat, Leibniz and the Bernoulli’s. As with his contemporaneous work on the fundamental theorem of algebra, Gauss’ approach was explicitly anti-deductive, discovering a common physical principle that underlay both geometry and number. It was also a direct confrontation with the failed Aristotelean methods of the likes of Euler and Lagrange who understood the circle as an object in visible space and numbers as abstract formalisms.

Today’s pedagogical exercise is the first of two, intended to guide the reader through the relevant concepts of Gauss’ method. It will require some “heavy lifting” and the reader is advised to work it through all the way to the end, no matter how arduous it seems along the way, and then look back, surveying what has been gained from the vantage point of the summit. The reader is also advised to review the preliminary work on Gauss’ theory of the division of the circle that was the subject of the several past pedagogicals, as it was summarized in the Winter 2001-2002 edition of 21st Century Science and Techonlogy, and the pedagogical exercises on the residues of powers (Riemann for Anti-Dummies Parts 20-25.) (Reference will also be made to several figures)

Polygons As Powers

As Gauss’ announcement indicates, by Euclid’s time, geometers had succeeded in finding the magnitudes that divided a circle in certain ways. What was not so evident, was why those ways and not others? From the standpoint of sense certainty, the circle, like the line, appears uniform and everywhere the same. Why then, is it not, like the line, divisible into whatever number of parts one desires? What unseen principle is determining which divisions are possible, and which are not?

Yet, when the circle is considered as a unit of action in the complex domain, it becomes evident that the division of the circle is based on the principle that generates magnitudes of successively higher powers. Those who have worked through the pedagogical exercises on Gauss’ 1799 doctoral dissertation are familiar with how this works. There we saw that algebraic powers are generated by a non-algebraic, physical principle, as expressed, for example, by the catenary. This principle belongs to the domain of functions that Leibniz called transcendentala, and is expressed mathematically by the equiangular spiral, or alternatively, the exponential (logarithmic) functions. Gauss showed that these transcendental functions were themselves part of a higher class of functions that could only be adequately known through images in the complex domain.

From this standpoint, the generation of magnitudes of any algebraic power correspond to an angular change within an equiangular spiral. “Squaring” is the action associated with doubling the angle within an equiangular spiral, “cubing” by tripling, fourth power by quadrupling, and so forth. These angular changes are, consequently, what generates magnitudes of succesively higher algebraic powers. When the circle is correctly understood as merely a special case of an equiangular spiral, the generation of algebraic powers is reflected as a mapping of one circle onto another. Squaring, for example, maps one circle onto another twice, cubing maps three times, and so on for the higher powers. (The reader is referred to the figures from the pedagogical discussions on the fundamental theorem of algebra.)

The regular divisions of the circle are simply the inversion of this action. Each rotation around the “squared” circle divides the original circle in half. Each rotation around the “cubed” circle divides the original circle into thirds, each rotation around the fourth power circle, divides the original circle into fourths, and so on. Consequently, the vertices of a regular polygon, are the points on the original circle, that correspond to the complete rotations around the “powered” circle and the number of vertices corresponds to the degree of the power. For example, the fifth power will produce, by inversion, the five vertices of the pentagon; the inversion of the seventh power, will produce the seven vertices of the heptagon, etc. All the vertices of a given polygon are generated, “all at once”, so to speak, by one function, which is the inversion of the function that generates the corresponding power. (By Gauss’ time, such inversions had come to be called “roots”, not to be confused with the misapplication of that term by ignorant translators of Plato’s word, “dunamis”.) Herein lies the paradox. If the triangle, square and pentagon are inversions of the generation of third, fourth, and fifth powers respectively, how come they are constructable and other polygons are not? (Constructable is used here in the same sense as Kepler uses the term “knowable” in the first book of the Harmonies of the World. By “knowable”, Kepler meant those magnitudes that were commensurate with the diameter of the circle, part of the diameter, or the square of the diameter or its part. These magnitudes are the only magnitudes, “constructable” from the circle and its diameter, or by straight-edge and compass. All such magnitudes correspond to “square roots” or magnitudes of the second power. Magnitudes of higher powers, are not “knowable” from the circle alone, as is evident from the problem of doubling the cube, or trisecting the angle.)

Prime Numbers are Ones

It was Gauss’ insight to recognize that the solution to this paradox lay, not in the visible circle, but in the nature of prime numbers. To begin with, throw out the common formal definition of prime numbers, and consider a physical principle in which prime numbers arise. This can be most efficiently illustrated by example. Perform the following experiment: draw 10 dots, in a roughly circular configuration, and number them 0 to 9. Connect the 10 dots sequentially (0, 1, 2,…) and call that sequence 1. Now connect every other dot, (0, 2, 4, 6…) and call that action sequence 2. Then every third dot, (0, 3, 6, 9, …, for sequence 3) then every fourth dot, (0, 4, 8, …, sequence 4) and so on.

Notice, that some sequences succeeded in connecting all 10 dots, namely, sequences 1, 3, 7 and 9, while sequences 2, 4, 5, and 8 connected only some of the dots. In the case of the latter, sequences, 2 and 5 became completed actions within one rotation, whereas 4 and 8 did not become completed actions until after more than one rotation.

Numbers are not formal symbols (or objects), to be manipulated according to a set of formal rules, but are relationships arising from physical action. In the above example, the number 10 becomes a One, or, as Gauss called it, a modulus. The numbers 1 through 9 are types of actions, not collections of things. With respect to modulus 10, the numbers (actions) 1, 3, 7, and 9 are called relatively prime, because those actions do not divide the modulus. The numbers 2, and 5, are called factors of 10, because those actions do divide the modulus within one rotation. (The numbers 4 and 8, divide the modulus but not within one rotation because they are not factors themselves but they share a common factor (namely 2) with 10.)

These relationships, of factors and relative primeness, are determined only by the nature of the modulus. If you begin sequence 2 on dot #1 instead of dot #0, it still connects only 5 dots. Similarly, if you begin sequence 3 on dot #1, it will still be relatively prime to 10. Additionally, if you continue the experiment with sequences 11, 12, 13, etc., the results will be identical to the sequences 1, 2, 3, etc. except that one rotation will be added. Gauss called these numbers congruent relative to modulus 10.

Thus, the modulus defines certain relationships, relative to the entire universe of whole numbers, in which some numbers are factors, some numbers are relatively prime, and some numbers are not factors themselves, but contain factors of the modulus.

However, when one dot is added, and the same experiment is performed with respect to 11 dots, all the sequences connect all the dots. Thus, 11 has no factors and all numbers are relatively prime to it. The relationship of modulus 11 to the entire universe of whole numbers is quite different than the modulus 10.

The modulus is the One. Some moduli, such as 10, define some numbers as factors,and some numbers as relatively prime and are called “composite”. Those moduli under which all numbers are prime, are known as prime numbers.

There is nothing absolute about the quality of primeness. Relatively prime numbers gain this characteristic relative to a one (modulus). Those numbers that are prime relative to the One, are absolutely prime. (Gauss, in his treatises on bi-quadratic residues, would later show that even this characteristic of absolute primeness is not really absolute but relative to a still higher principle.)

Polygons as Planetary Systems

This leads us back to the original paradox. If the prime numbers are irreducible Ones, how come some prime number divisions of the circle are constructable and others not?

Take another look at the image of a circle in the complex domain. The vertices of a regular polygon are the roots (inversions) of a corresponding power. This relationship of “roots” and “powers” produces a type of harmonic “planetary system” for each polygon in which only those “planetary orbits” that correspond to the “roots” of that “power” are possible, and, these “roots” have a unique harmonic relationship to each other, whose characteristics are determined by the number-theoretic characteristics of the prime number.

Illustrate this pedagogically by an example. The vertices of a regular pentagon are the five “roots” of 1 and each of these “roots” is a complex number that has the power to produce a fifth degree magnitude. Such complex numbers represented the combined action of rotation and extension. Since in a circle the extension is constant, the complex numbers are at the endpoints of equally spaced radii. To construct the polygon it is necessary to determine the positions of these radii. To do this Gauss used the method of inversion and determined the positions of the radii from the harmonic relations among them. Even without knowing the positions of the radii, the harmonic relations can be known because the radii are inversions (roots) of powers. In other words, the vertices of the polygon are the endpoints of equally spaced radii.

But don’t look at the endpoints (visible objects). Look for why the radii are equally spaced. They are equally spaced because they are the roots of an algebraic power. To illustrate this use the pentagon as an example, draw a circle with five approximately equally spaced radii. This should look like an “inside out” pentagon. (Since we are investigating only the relationships among the radii at this point it is not necessary that the radii be exactly equally spaced.)

Label the endpoints of the radii 1, a, b, c, d, with “a” representing 1/5 of a rotation, “b”, 2/5, “c”, 3/5, “d”, 4/5 and “1” being 1 full rotation. If any of these individual angular actions is repeated (multiplied) five times, the resulting action will end up at 1. In other words, a⁵, b⁵, c⁵ and d⁵ are all equal to 1. Furthermore, a⁰=1, a¹=a, a²=b, a³=c, a⁴=d; b⁰=1, b¹=b, b²=d, b³=a, b⁴=c; c⁰=1, c¹=c, c²=a,c³=d, c⁴=b; d⁰=1, d¹=d, d²=c, d³=b, d⁴=a. Thus, any vertex can generate all the others. (For the general case, each of the vertices corresponds to a complex number of the form a + b ?-1, such that (a + b?-1)ⁿ =1 for all “n’s” of an “n” sided polygon.)

In the example of the pentagon, five is the modulus, the One, which establishes a certain harmonic ordering under which there are five and only five “orbits”. A different modulus would produce a different number of “orbits”, but the relationship just illustrated will remain; only the number of “orbits” will have changed, and consequently, the nature of the harmonies. Notice the congruence of these actions with our earlier experiment with dots illustrating the physical principle from which primeness, relative primeness and factors arise. Notice the similarity between the power sequences generated from each complex root, and the different number sequences used to connect the dots. This congruence is not discovered by looking at the visible objects, but by a method Leibniz called, “Geometry of Position”, or “analysis situs”, or what Gauss called, “geometrica situs”. It reflects a higher principle, independent of any particular number and begins to shed light on that “profound connection” Gauss discovered between the geometry of transcendental functions and higher Arithmetic.

Next week we’ll look further into that connection.

The serious person can take heart that such symptoms need not indicate an incurable condition, but it is only the recurring effects of the malicious teaching methods most people today have suffered through. It may be helpful to those suffering from these effects, to take a clinical look at how this “deductivizing” was introduced into modern educational practices by G.W.F. Hegel’s grandson-in-law, Felix Klein. As a talented mathematician, Klein was not as radical a reductionist or as openly fascistic as Russell, Kronecker or Helmholtz. Yet his method was pure Bogomilism, nevertheless. Rather than try and obliterate the creative discoveries of Leibniz, Gauss and Riemann, Klein adopted a seemingly “middle ground” so to speak, in which the discoveries were stripped of their creative insight, and re-cast in deductive, i.e. impotent, form.

While Klein had an extensive influence over the teaching methods of a wide domain of scientific subjects, it is sufficient, for our purposes at this moment, to look at his treatment of Gauss’ early discoveries, to obtain the clinical benefit of freeing those individuals, who, knowingly or not, have been victimized by Klein’s crime.

As discussed in the recent pedagogicals of this series, Gauss’ early discoveries have their origin in the paradoxes arising from the investigations of “powers” as that concept is defined by Plato, and how these paradoxes arise in the classical problems of doubling the cube and trisecting an angle. For Plato, Cusa, Kepler, Leibniz, Kaestner, Gauss, Riemann, these investigations led into the deepest questions concerning the relationship of man to the universe. However, in his 1895 lectures, “Famous Problems of Elementary Geometry”, Klein reduces these problems to the following, which will seem uneasily familiar to most students today:

“In all these problems the ancients sought in vain for a solution with straight edge and compasses, and the celebrity of these problems is due chiefly to the fact that their solution seemed to demand the use of appliances of a higher order…”

This already is complete fraud. Plato’s circle did not consider the straight edge and compass as “appliances”, but as Kepler summarizes the question in the first book of the “Harmonies of the World”, the question under investigation was the “knowability” of magnitudes. That is, which magnitudes were “knowable” from the circumference and diameter of a circle, and which were “unknowable”.

Klein continues, “At the outset we must insist upon the difference between practical and theoretical constructions. For example, if we need a divided circle as a measuring instrument, we construct it simply on trial. Theoretically, in earlier times, it was possible (i.e. by the use of straight edge and compasses) only to divide the circle into a number of parts represented by 2ⁿ, 3 and 5 and their products. Gauss added other cases by showing the possibilty of the division into parts where p is a prime number of the form p = (2^2p) + 1, and the impossibility for all other numbers. No practical advantage is derived from these results; the significance of Gauss’ developments is purely theoretical.”

Klein’s separation of the theoretical and practical is pure evil Bogomilism, in addition to being a fraud. One need look no further, than Erathosthenes’ account of the history of the duplication of the cube, as reported by Theon of Smyrna:

“Eratosthenes in his work entitled “Plotinicus” relates that, when the god proclaimed to the Delians through the oracle that, in order to get rid of a plague, they should construct an altar double that of the existing one, their craftsmen fell into great perplexity in their efforts to discover how a solid could be made the double of a similar solid; they therefore went to ask Plato about it, and he replied that the oracle meant, not that the god wanted an altar of double the size, but that he wished, in setting them the task, to shame the Greeks for their neglect of mathematics and their contempt of geometry.”

Where is the separation of the theoretical from the practical in Eratosthenes account? Was it purely a theoretical matter, that the Delians had become so morally corrupt by their neglect of the cognitive powers of the mind, that they had become victims of a deadly plague?

As the Thirty Years War began to unfold in full horror, Kepler, on the occasion of the twenty-fifth anniversary of the publication of his “Mysterium Cosmographicum”, invoked the “practical” benefits of the power of cognition, “would that even now indeed there may still, after the reversal of Austrian affairs which followed, be a place for Plato’s oracular saying. For when Greece was on fire on all sides with a long civil war, and was troubled with all the evils which usually accompany civil war, he was consulted about a Delian Riddle, and was seeking a pretext for suggesting salutary advice to the peoples. At length he replied that, according to Apollo’s opinion Greece would be peaceful if the Greeks turned to geometry and other philosophical studies, as these studies would lead their spirits from ambition and other forms of greed, out of which wars and other evils arise, to the love of peace and to moderation in all things.”

And Gauss, himself, when installed as head of the Goettingen University Observatory, pronounced that the political troubles that had befallen Europe at that time, arose from a contempt for purely cognitive discoveries.

Klein is deadly wrong. Gauss’ discoveries were not purely theoretical. Recognizing that is crucial to being able to grasp elementary mathematics from a truly advanced, (LaRouchian) standpoint.

Riemann for Anti-Dummies, Part 26

IDEAS CAST SHADOWS, TOO

It can be a source of confusion for the naive, and a means of deception of the wicked, to restrict the meaning of Plato’s metaphor of the cave, to those objects that originate outside of one’s skin. As all great scientists have come to know, ideas cast shadows, too. A true scientist never mistakes the shadows for the idea, seeking instead to discover the idea from between the shadows. Those who merely manipulate shadows are called sophists.

This defines the clear distinction between the concept of the complex domain of Gauss and Riemann, and the sophistry of Euler, Lagrange and D’Alembert. The former understood complex numbers as a simple case of a hierarchy of multiply extended magnitudes, or as Gauss called them, “shadows of shadows.” The latter considered complex numbers, “impossible,” but susceptible to complicated, but ultimately meaningless, symbolic manipulation, whose very complexity is intended to obscure its trickery.

A passion for sophistry pervades modern academia, as exemplified by J. E. Hofmann, who penned the forward to the 1970 republication of Abraham Kaestner’s “Geschichte der Mathematik.” As LaRouche indicated in footnote 42 of his new piece, “At the End of a Delusion,” Hofmann complains that Kaestner did not show sufficient respect for the achievements of the great mathematicians of his time Euler, Lagrange, and D’Alembert. It is precisely Kaestner’s disrespect for these sophists for which he deserves our great admiration and respect today. As the history of the discovery of the complex domain demonstrates, Hofmann’s blunder is not only a matter of a lack of comprehension of the subject, it is also indicative of the illiteracy of modern academia.

Hofmann’s error is immediately exposed by examining the 1799 doctoral dissertation of Kaestner’s student Carl F. Gauss, on “A Proof of the Fundamental Theorem of Algebra.” There, the 22 year old Gauss, matriculating for his doctorate under Kaestner, openly and explicitly castigates, Euler, Lagrange, and D’Alembert as sophists on the matter of the existence of complex numbers, showing the same disregard for Euler, Lagrange, D’Alembert, for which Hofmann cricizes Kaestner.

It is revealing that all modern biographers of Gauss have gone out of their way to dismiss Gauss’ relationship to Kaestner, who Gauss called, “A poet among mathematicians and a mathematician among poets..” It was Kaestner, the passionate defender of Leibniz and Kepler, the host of America’s Benjamin Franklin, who first raised the questions leading to the development of anti-Euclidean geometry, and, who provoked the young Gauss into deciding to pursue a life of scientific investigation. Kaestner’s biting wit and sharp-tongued polemics against the sophistry of Euler, Lagrange, and D’Alembert, the fools who would fall for their methods, sticks in the craw of the his Romantic enemies to this day. While Gauss never adopted the polemical style of his teacher, he shared Kaestner’s contempt for “ivory tower” sophistry, and expressed it in his life’s work, as a plain reading of 1799 doctoral thesis shows. After Kaestner’s death in 1800 and the ensuing rise of the fascist Napoleon, Gauss became more circumspect in his public pronouncements, but his distaste for what he called, “the screeching of the Beothians” never waned.

While a fuller account of this history must still be elaborated, it can already be stated without equivocation, that those who demean Kaestner, and hold Euler, Lagrange, and D’Alembert in high esteem, do so in defense of the degraded conception of man that produced modern fascism.

The next installment will provide a pedagogical presentation of Gauss’ doctoral thesis. This week focuses on the essential pre-history of the development of complex numbers.

As discussed in “Riemann for Anti-Dummies Part 18; ‘Doing the Impossible,'” the possible existence of complex numbers was posed in a paradox by Cardan in 1545. In his Ars Magna, Cardan pointed to the existence of what he called a “subtile” magnitude through a specific problem, to wit: “Find two numbers that add up to 10 and when multiplied together equal 40.”

Cardan recognized that this problem contained the paradox that arises from the difference between a line and a surface, because addition implies linear magnitudes, while multiplication implies a surface.

Begin with a line AB which has a length of 10. Divide the line into two parts, that produce the maximum area when multiplied together, which will be two segments of 5, which when multiplied together produce an area of 25. The sought after area is 40. Subtract 40 from 25 which yields an area of -15, which is produced by (?-15)(?-15). Thus, if you add ?-15 to one of the segments of 5 and subtract it from the other, the problem is solved, since (5 + ?-15) + (5 – ?-15) = 10; and (5 + ?-15)(5 – ?-15) = 40!

“This subtility results from arithmetic of which this final point is as I have said as subtile as it is useless,” Cardan proclaimed perplexed.

The paradox arises when one limits the conception of magnitudes to the sense perception characteristics of lines and areas, resulting (in Cardan’s example) in a magnitude of negative area.

A similar paradox arises in an even similar example. Think of a line segment of length x. Now think of a different line segment of length y. Now think of adding x to y to produce the line segment z. No matter what length you choose for x and y, you will always be able to think of a line segment whose length is z. In other words, one extensible magnitude added to another extensible magnitude, produces a third extensible magnitude.

But, what happens when you try and subtract one extensible magnitude from another? No problem if you try and subtract a smaller magnitude from a larger. But, if you try and subtract a larger extended magnitude from a smaller, you get a negative length! (For this reason negative numbers were often referred to as “false” numbers.)

It’s as if subtracting a larger line from a smaller, or a larger area from a smaller, pokes us into a world, that includes objects other than lengths and areas. Or, we must recognize that lengths and areas are only shadows and should not be mistaken for the {idea} of extended magnitude. To comprehend the {idea}, we have to go behind the shadows, by “seeing” between them. Subtracting a longer line from a shorter one, shows us a world of extensible magnitudes that exist behind the visible sense perceptions of magnitudes associated with lengths, and reveals that a more general idea of magnitude must include not only length, but also direction.

The paradox arising from subtracting a larger area from a smaller one, areas, proves more subtile. As we reviewed in Part 18, Leibniz and Huygens corresponded on the implications of the existence of the square roots of negative numbers, of which Huygens would say, “there is something hidden there which is incomprehensible to us.”

To which Leibniz would reply, “The imaginary number is a fine and wonderful recourse of the divine spirit, almost an amphibian between being and non-being.”

By contrast Euler, Lagrange and D’Alembert would prove adept at complicated manipulations of algebraic equations that included the square roots of negative numbers, while insisting at all times that such magnitudes were “impossible.”

This is precisely the issue that the young Gauss attacked in his proof of the fundamental theorem of algebra. These were not “impossible” magnitudes, Gauss insisted, but “shadows of shadows.” One can think of an image of such shadows by thinking of a unit circle in the complex domain divided by two perpendicular diameters, which intersect the circumference of the circle at 1, -1, ?-1, -?-1. Think of a point rotating counter-clockwise around this circle. Now think of the image of that point, as if it were observed by looking at the circle edge on. One would only see a point moving back and forth along a line from 1 to -1 and back again. In other words, the so-called “imaginary” part is always there, but you have to look behind the shadows to “see” it.

As Gauss told his friend Hansen in 1811:

“These investigations lead deeply into many others, I would even say, into the Metaphysics of the theory of space, and it is only with great difficulty can I tear myself away from the results that spring from it, as, for example, the true metaphysics of negative and complex numbers. The true sense of the square root of -1 stands before my mind (Seele) fully alive, but it becomes very difficult to put it in words; I am always only able to give a vague image that floats in the air.”

Riemann for Anti-Dummies Part 25

SCHILLER AND GAUSS

In his “Aesthetic Estimation of Magnitude”, Friedrich Schiller discusses a crucial ontological paradox that confronts science when it tries to exceed existing axiomatic assumptions:

“The power of imagination, as the spontaneity of emotion, accomplishes a twofold business in conceptualizing magnitude. It first gathers every part of the given quantum into an empirical consciousness, which is {apprehension}; secondly, it assembles the {successively collected} parts into a pure self-consciousness, in which latter business, that of {comprehension}, it acts entirely as pure understanding. That concept of “I” (empirical consciousness), in other words, combines with each part of the quantum: and through reflection upon these successively performed syntheses, I recognize the identity of my “I” (pure self- consciousness) in this series as a whole; in this way, the quantum first becomes an object for me. I think A to B to C, and so forth, and while I watch my activity; as it were, I say to myself: in A, as well as in B, and in C, I am the acting subject.

“Apprehension takes place {successively}, and I grasp each partial conception after the other….The synthesis, however, takes place {simultaneously}, and through the concept of the self-identity of my “I” in all preceding syntheses, I transcend anew the temporal conditions under which they had occurred. All those different empirical conceptions held by my “I” lose themselves in a single pure self-consciousness; the subject, which had acted in A, and B, and C, and so forth, is I, the eternally identical self…

“If the power of reflection transgresses this limit, and seeks to bring together mental images, which already lie beyond the limit, into unity of self-consciousness, it will lose as much in clarity as it gains in scope. Between the circumference of the entirety of a mental image and the distinctness of its parts, is an ever insuperable, specific relationship, wherefore in each addition of a large quantum we lose as much backward as we gain forwards and when we have reached the end-point, we see the starting point vanish.”

Schiller is not referring to quanta, which have magnitude, simply with respect to quantity, but as Leibniz, Gauss and Riemann did, as {universal principles}:

“Everything which has parts, is a quantum. Every perception, every idea formed by comprehension, has a magnitude, just as the latter has a domain and the former a content. Quantity in general, therefore, cannot be meant, if one speaks about a difference of magnitude among objects. Here we speak about such a quantity as characteristically belongs to an object, that is to say, that which is not simply a {quantum}, but is at the same time a {magnum},”

Think of Schiller’s concept with respect to the successive discoveries of Kepler and Gauss concerning planetary motion. If we think of the position and speed of the planet at any given moment, as a quantum, it is indeterminable, except as that quantum is a characteristic of the whole orbit. In that sense, the indeterminable position and speed at the moment, becomes determinable, only as an interval, a part, of the whole orbit. The magnitude associated with that interval, is the area swept out. This, magnitude cannot be measured by the successive addition of the speeds and positions of the planet, which, owing to the non-uniformity of the orbit are indeterminate, but, only as these are grasped as an interval of the whole.

But, the orbits, in turn, are not self-defined, and their magnitudes are indeterminable as individual orbits. Rather, the magnitudes of the individual orbits can only be determined as intervals with respect to the harmonic ordering among all the orbits at once. Inversely, that harmonic ordering cannot be determined by successive addition of each individual orbit, but only as intervals of the whole.

Further, as Gauss’ investigation of the asteroids demonstrated, these harmonic orderings are themselves changing, according to a still higher harmonic ordering.

In other words, if we seek to determine the position and speed of the planet at any moment, we are stymied until we are led to the orbit as a whole. And, if we seek to determine the nature of an individual orbit, we are stymied anew, until we are led to all the orbits. And, further, if we try to determine the harmonic ordering of all the orbits, we will be once again stymied, until we are led to the ordering of the harmonic ordering. From this vantage point, the individual position and speed of the planet, which was our first object of investigation, recedes, as the deeper underlying principles come to the fore.

In the terms of Leibniz’ calculus, the differential can be known only as a function of the integral. Or, under Schiller’s idea, if each principle is thought of as a quantum, it can only be measured with respect to a magnum, which in turn, is a quantum, to a, higher, yet to be discovered magnum. In terms of Riemannian differential geometry, it is the highest principle, which determines all lower ones.

Seen in this way, the principle of Mind, of which Kepler speaks as governing the motion of the planet, is not a simple conception of a mind interacting one on one between the planet and the Sun, but a principle of Mind, as Schiller speaks of above, that comprehends its actions from a higher and higher standpoint, which determine, the seemingly indeterminable action in the small.

Gauss’ investigation of bi-quadratic residues, and his and Riemann’s further development of differential geometry, provide the pedagogical/epistemological capacity for our minds to grasp this concept.

For Gauss, as for Plato, Fermat, and Leibniz, individual numbers are not self-defined, but are rather defined by a higher principle, which Gauss called congruence. Each modulus, thus, defines a certain indivisible “orbit” in which all the numbers from 1 to the modulus minus 1, are ordered. The ordering within any individual “orbit” is itself a function of the characteristic of the modulus. For example, if the modulus is an odd-even prime number, such as 5, 13, 17, etc. -1 is a residue of the modulus minus 1 power, and ?-1 is the residue of the 1/4 the modulus minus 1 power. If the modulus is an odd-odd prime number, ?-1 never emerges. However, this characteristic of prime numbers, is not determined by the individual prime numbers, but is rather a function of the, still as yet undiscovered, “orbit”, that determines prime numbers.

This characteristic of number led Gauss to search for a higher principle, which he discovered by extending the concept of number from simply-extended magnitudes, to doubly- extended magnitudes, which he called complex numbers.

The significance of this is best grasped pedagogically, by way of an example directly out of Gauss’ second treatise on bi-quadratic residues.

Gauss thought of the complex domain as mapped onto a plane that is covered by a grid of equally spaced squares, the vertices of each square signify what Gauss called complex whole numbers. Each complex whole number is of the form a +bi, where i stands for ?-1, and a and b are whole numbers. Gauss called a² + b² the “norm” of the complex number. Gaussian prime numbers, are those complex whole numbers, whose norms are prime numbers.

Gauss’ example uses the complex prime number 5+4i. Taking this as the modulus, the entire complex domain is “partitioned” into diamonds, whose sides are the hypothenuses of right triangles whose legs are 5 and 4. (See last week’s pedagogical.) Each diamond encloses 41 (5² + 4²) individual complex whole numbers, which are all incongruent to each other, relative to modulus 5+4i.

(You can illustrate this, if you take the diamond whose vertices are the complex numbers 0, 5+4i, 1+9i, -4+5i, as no two numbers within this diamond will be separated by doubly- extended interval greater than 5+4i. Now, construct another diamond whose vertices are 5+4i, 10+8i, 6+13i, 1+9i. Each complex number within this new diamond will all be incongruent to every other within the diamond, but, each complex number of the second diamond will be congruent to that complex number that is in the same relative position in the first diamond, specifically, the number whose difference with it is 5+4i.)

Gauss then takes the complex number 1+2i as a primitive root of 5+4i. To grasp the meaning of this concept, see what happens, geometrically, when 1+2i is raised successively to the powers, in a new type of geometric progression. First you have (1+2i)⁰ = 1; Next is (1+2i)¹ = 1+2i; These two numbers define a triangle whose vertices are 0, 1, 1+2i. This will form a right triangle, whose legs are 1 and 2 with hypotenuse ?5. The angle at the vertex 0 will be 63.4349 degrees, the angle at 1 will be 90 degrees and the angle at 1+2i will be 26.5651 degrees. Now construct a similar triangle to this, using the hypotenuse of this first triangle, as the shorter leg, placing the 90 degree angle at the vertex 1+2i. This will define a new vertex at -3+4i, which is (1+2i)². Repeat this process, constructing another similar triangle, with right angle at -3+4i, and the side 0, -3+4i as the short leg. This defines a new complex number, -11-2i, which is (1+2i)⁴.

This chain of similar right triangles, is but a general case of the famous chain of right triangles constructed by Theodorus, as reported by Theatetus in Plato’s dialogue.

Each new vertex of this chain of similar right triangles, is thus a new, higher, power of 1+2i, and all lie on a unique logarithmic spiral. In other words, as this particular logarithmic spiral winds its way around the complex domain, the complex whole numbers it intersects are the powers of 1+2i. Thus, the powers of 1+2i are determined by a higher principle, of logarithmic spiral action. They are as moments in a orbit, or orbits in a planetary system.

Gauss continued this process, by continuing this spiral, so as to define 41 (5² +4²) powers of 1+2i, and investigated these spiral points in a complex domain, “partitioned” into diamonds by modulus 5+4i, with the beginning diamond having 0 at its center. (This is the diamond whose vertices are (-1/2 – 4i), (4 – i), (+4i), (-4 + 1/2 i).) Now think of these diamonds spreading out, partitioning the complex domain, as the spiral winds its way around. Each time the spiral intersects a complex whole number, that number will be a power of 1+2i, and that number will be inside a particular diamond. Gauss showed that the first 40 complex whole numbers the spiral intersects, will each be in a different relative place within their respective diamonds, than any other previous or succeeding one. In other words, each complex whole number the spiral intersects, will be congruent to only one of the complex whole numbers in the beginning diamond. Most importantly, the 10th power of (1+2i) would be congruent to i, the 20th power to -1, the 30th power to –i, and the 40th power to 1. Then the cycle would repeat!

And so, if we begin with individual numbers we soon see these numbers can not be self determined, and we are led to the generating principle of congruence. But, these congruences produce “orbits” which can not be self-determined and we are led to a still higher principle of extended magnitudes. With each successive step, the individual numbers recede and as the higher principles come more to the fore in our minds.

And, yes, there is a still higher principle at work which Gauss discovered was connected directly to the Kepler problem. This was indicated by one of the earliest entries in his diaries that read, “I have discovered an amazing connection between bi-quadratic residues and the lemniscate”.

Our investigation of this remark, will have to wait for a future installment.

Riemann for Anti-Dummies Part 24

LET THERE BE LIGHT

As you heard Riemann proclaim in the opening remarks of his Habilitation lecture, without a “general concept of multiply-extended magnitudes in which spatial magnitudes are comprehended,” you are left in the dark. You can not know the nature of the physical universe, the validity of an idea, the economic value of human activity, the strategic significance of a current, or historical, event, or your personal identity in the simultaneity of eternity, to name but a few of the more important matters on which one would wish to shed light. Yet, the principles to which Riemann refers are far too little understood by those who must urgently be able to make such judgements.

Referencing Gauss, Riemann cites two characteristics necessary for the determination of multiply-extended magnitudes, dimensionality and curvature, neither of which can be determined a priori, but only by physical measurement. Such magnitudes are not mathematical quantities, but are universal physical principles, produced by a manifold of physical action, and, are relative to the manifold, not absolute.

Take some examples from the arsenal of ideas built up over the course of this series to illustrate the point.

1. As Kepler demonstrated, the non-uniform elliptical planetary orbit defines the magnitude of action within an orbit, as equal areas, rather than the arbitrary mathematical magnitudes of equal arcs or equal angles. The solar system as a whole, in turn defines a magnitude of action for individual orbits, consistent with the five Platonic solids, and the principles of musical polyphony. Thus, the action of a planet at any moment can only be measured as a function of the whole orbit, which orbit in turn is measured as a function of the whole solar system. While the orbit defines one species of magnitude (equal areas) the solar system as a whole defines a distinct and different species of magnitude (harmonics), which “reach down” into all parts of the individual orbit, even though the latter cannot be derived simply from the former.

2. The shortest path of reflected light defines a magnitude of action measured by equal angles. The least time path of refracted light defines a magnitude of action measured by the proportionality of the sines of the angles of incidence and refraction. In other words, under reflection the angles measure the change in the direction of the light, while under refraction, the angles are determined by the sines. In the manifold of physical action of reflected light, there is no change in medium, consequently no change in velocity of light, and so the effect of the sines “disappears” into the equality of angles. But in the higher dimensional manifold of refraction, the truth comes out, that it is not the angles that measure the action, but the inverse, the transcendental magnitudes of the sines.

It is important to keep in mind, that in both these examples, “dimension” is not a mathematical construct, but is associated with a distinct physical principle, which is then associated with a distinct species of magnitude, and, as Riemann emphasizes, the number of dimensions is increased by the discovery of each new physical principles.

This concept of magnitude is consistent with Schiller’s expression in “On the Aesthetic Estimation of Magnitude”:

“All comparative estimation of magnitude, however, be it abstract or physical, be it wholly or only partly determined, leads only to relative, and never to absolute magnitude; for if an object actually exceeds the measure which we assume to be a maximum, it can still always be asked, by how many times the measure is exceeded. It is certainly a large thing in relation to its species, but yet not the largest possible, and once the constraint is exceeded, it can be exceeded again and again, into infinity. Now, however, we are seeking absolute magnitude, for this alone can contain in itself the basis of a higher order, since all relative magnitudes, as such, are like to one another. Since nothing can compel our mind to halt its business, it must be the mind’s power of imagination which sets a limit for that activity. In other words, the estimation of magnitude must cease to be logical, it must be achieved aesthetically.

“If I estimate a magnitude in a logical fashion, I always relate it to my cognitive faculty; if I estimate it aesthetically, I relate it to my faculty of sensibility. In the first case, I experience something about the object, in the second case, on the contrary. I only experience something within me, caused by the imagined magnitude of the object. In the first case I behold something outside myself, in the second, something within me. Thus, in reality, I am no longer measuring, I am no longer estimating magnitude, rather I myself become for the moment a magnitude to myself, and indeed an infinite one. That object which causes me to be an infinite magnitude to myself, is called sublime.”

Think in these terms about Gauss’ development of the complex domain in the context of his work on biquadratic residues, where Gauss demonstrates that it is actually impossible to construct a concept of magnitude devoid of dimensionality. As the discoveries that Plato made famous in his Meno, Theatetus, and Timaeus dialogues, action along a line, a surface, or a solid, is associated, in each case, with distinct species of magnitude. The species of magnitude, associated with the manifolds of lower dimensions, are found in the manifolds of higher dimensions but not vice versa. Consequently, a paradox arises, if one attempts to measure action in manifolds of higher dimensions, by magnitudes that are produced in a manifold of lower dimensionality.

Look at this from the standpoint of the simple operations with numbers, addition, subtraction, multiplication, division. (Riemann added Leibniz’ integration and differentiation, to the domain of simple operations, and this will be taken up in future installments.) As the Theatetus reports, addition of doubly-extended magnitudes, (i.e. areas) cannot be measured by simply-extended magnitudes (i.e. lines), and yet, until Gauss, all operations of Arithmetic were constrained by the underlying assumption that each manifold could be measured by the same species of magnitudes. This paradox reemerged from the Renaissance on, as the paradox associated with the ?-1. Cardan, Leibniz, Huygens, and Kaestner, all understood that this paradox required the need for a higher conception of magnitude, while Newton, Euler and others, dismissed this magnitude as “impossible” .

For Gauss, action in a doubly-extended manifold, could only be measured by doubly- extended magnitudes, which he called “complex numbers”. These numbers are determined by two actions, rotation and extension, or alternatively, simultaneous horizontal and vertical action, such as in the bubble of a carpenters level. (It is about time to replace the commonly used term “Cartesian coordinates” when referring to horizontal and vertical action, with the more historically and conceptually accurate, term, “Fermat coordinates”.)

From this standpoint look at the basic concepts of Arithmetic with respect to both simply- extended and doubly-extended magnitudes. Under Gauss’ concept of congruence, all numbers are ordered with respect to the interval between them or the modulus. With respect to simply- extended manifolds, that interval corresponds to a line segment. But, with respect to a doubly- extended manifold, that interval has two parts, up-down and back forth. Illustrate this with an example from Gauss’ second treatise on bi-quadratic residues. The modulus 5+2i “partitions” the entire complex domain, by a series of squares whose sides are the hypothenuses of right triangles whose legs are 5 and 2. For example, the square whose vertices are 0, 5+2i, 3+7i, – 2+5i. All complex numbers inside this square are not congruent to each other. Now draw adjacent squares, such as the squares whose vertices are 5+2i, 10+4i, 8+9i, 3+7i; or, 3+7i, 8+9i, 6+14i, 1+12i. All the numbers inside these squares are also not congruent to each other, but each number is congruent to the one number in each of the other squares, which is in the same relative position within the square. For example, 2+4i and 7+6i, occupy the same relative position within their respective squares, consequently, the difference (interval) between them is the modulus 5+2i.

Thus, the simple periodicity generated by congruences with respect to real numbers, is transformed into a double periodicity with respect to complex moduli. In a simply-extended manifold, therefore, subtraction determines the linear interval between two numbers, while in a doubly-extended manifold, subtraction determines the area interval between two numbers.

Gauss next developed a concept of doubly-extended “complex” multiplication, which will require you to re-think what you were taught about multiplying numbers in elementary school. Simply-extended multiplication was defined by Euclid as:

“15. A number is said to multiply a number when that which is multiplied is added to itself as many times as there are units in the other, and thus some number is produced.”

But, even Euclid admits an inadequacy of this concept in the next definition:

“16. And, when two numbers having multiplied one another make some number, the number so produced is called plane, and its sides are the numbers which have multiplied one another.”

But, we already have discovered from Theatetus, that adding in a simply-extended manifold, (lines) and adding in a doubly-extended manifold (areas) are not the same, so how can adding one number to itself “as many times as there are units in the other” (definition 15.) produce the areas described in definition 16? This darkness arises from the lack of a concept of multiply-extended magnitude.

The matter is resolved by a higher concept. If you look again at Theatetus’ alternating series of squares and rectangles, or the expanding series of squares from the Meno, you can see that adding areas produces a rotation and an extension. For example, the square whose area is 1 is transformed into the rectangle whose area is two, by rotating a line whose length is 1 90 degrees and multiplying its length by 2. The next transformation, to a square whose area is 4, is produced by the action of rotating the longer side of the rectangle and additional 90 degrees, and multiplying its length again by 2.

As Gauss’ follower Neils Henrick Abel said, “To know the truth, you must always invert.” An inversion, therefore, will show us the general principle that is, the action of adding rotation and multiplying lengths, produces the geometric progression.

So in the complex domain, multiplication is the action of adding rotations and multiplying lengths.

Illustrate this first with respect to prime numbers, by the example of multiplying (1+2i)(1-2i). 1+2i denotes a rotation of 45 degrees and a linear extension of ?5. 1-2i denotes a rotation of 45 degrees, in the opposite direction and a linear extension of ?5. To multiply the two magnitudes, add the rotations, (which together equal 0) and multiply the extensions (?5)(?5) = 5. Hence, 5, a prime number in a simply-extended manifold, is a composite number in a doubly- extended manifold. However, no such geometric action will produce odd-odd prime numbers such as 7, 11, 19, etc.

Gauss saw this paradox as an excellent pedagogical demonstration of the principle that the nature of the manifold determines nature of the magnitudes. Since prime numbers produce all numbers by multiplication, but cannot be produced themselves by multiplication, Gauss has shown that these magnitudes, (prime numbers) that produce other magnitudes (composite numbers) are themselves produced by the manifold in which the action (multiplication) takes place . Some numbers are prime (undeniable facts) but when a new principle (dimension) is added even those undeniable facts, are changed!

Now, construct a geometric progression from a complex number, by multiplying that number by itself repeatedly. For example, start with 1+i which denotes a rotation of 45 degrees and an extension of ?2. Then multiply 1+i times 1+i. This produces a rotation of 90 degrees and an extension of 2. Repeating this again produces a rotation of 135 degrees and an extension of 4. If you continue this action you will see unfolding points on a logarithmic spiral.

From this Gauss demonstrated that the periodicity produced by the residues of a geometric progression, actually reflected magnitudes of a higher manifold. In the next installment, we will illustrate this discovery.

Riemann for Anti-Dummies Part 23

THE CIVIL RIGHTS OF COMPLEX NUMBERS

As the unfolding of current history demonstrates, it is reality that determines policy, not the other way around. This should come as no surprise to a scientific thinker knowledgeable in the method of Plato, Cusa, Kepler, Leibniz, Fermat, Gauss, Riemann and LaRouche. It is, however, shocking for anyone unfortunate enough to have accepted, wittingly or unwittingly, the delusion of Aristotle, Kant and Newton, that extensible magnitude exists outside the domain of universal physical principles.

This is the standpoint from which Gauss introduced his concept of the complex domain, beginning with his doctoral dissertation on the fundamental theorem of algebra, his Disquisitiones Arithmeticae, his treatises on geodesy and curvature, and his second treatise on biquadratic residues. From his earliest work, Gauss adopted the standpoint of his teacher Kaestner, and Leibniz before him, that the characteristic of extensible magnitude is a function of the manifold out of which those magnitudes were created.

It is in this light that one must view the discussions in the previous week’s installments. Gauss has rejected any {a priori} conception of magnitude, and instead derived the characteristic of numbers from a set of generating principles. First, by generating numbers from the juxtaposition of simple cycles, and then from the standpoint of a geometric cycle of cycles. As such, the relationships among numbers can not be found in the numbers themselves, but only in the relationship of those numbers to the manifold in which they exist. Like Leibniz’ monads, numbers don’t relate to each other directly, but only through the manifold from which they are created.

A quick review from last week illustrates the point. Take the “orbit” generated by the residues of the powers of the primitive root of 11 and 13.

Modulus 11:
     Index:  0,      1,     2,     3,     4,     5 
   Residue: {1,-10},{2,-9},{4,-7},{8,-3},{5,-6},{10,-1}
     Index:  6,      7,     8,     9,    10 
   Residue: {9,-2}, {7,-4},{3,-8},{6,-5},{1,-10} 

Modulus 13: 
     Index:  0,      1,     2,      3,      4,      5,    6 
   Residue: {1,-12},{2,-11},{4,-9}, 8,-5}, {3,-10},{6,-7},{12,-1} 
     Index:  7,      8,     9,      10,     11,     12 
   Residue: {11,-2},{9,-4}, {5,-8},{10,-3},{7,-6}, {1,-12}

In both cases the orbit begins with 1 and ends with 1, ordering all the numbers between 1 and the modulus minus 1, according to a principle. That principle, at first does not appear obvious, but on further investigation, it reveals itself to be highly ordered. At the halfway point of the “orbit,” (the 5th power for 11 and the 6th power for 13), the residue is -1, which when squared equals 1.

In the case of 13, a further division by half is possible. This gets us to the 3rd power, whose residue is 8, which, when squared is congruent to -1 modulus 13. In other words, -1 is at half the orbit; the square root of -1 is at half of the half.

This phenomenon hints at a paradox that reveals the underlying geometry of the ordering principle that generates the numbers. In the above example we were “experimenting” with positive and negative whole numbers. Naive sense-certainty indicates that these numbers can be represented completely as equally spaced intervals along an infinitely extended straight line, with positive numbers lining up in one direction and the negative numbers in the other. However, under such a conception, the square root of -1 does not exist as a magnitude, yet, its existence was just discovered as the biquadratic root of 1, modulus 13. (8 = ?-1 mod 13; 8² = -1 mod 13; 8⁴ = 1 mod 13.)

In other words, a species of magnitude exists, that can not be logically deduced from a manifold of one dimension. Euler concluded that such magnitudes were, therefore, “impossible.” Gauss, on the other hand, would not be restricted to a one-dimensional manifold, when an anomaly required an extension into two dimensions, in which such “impossible” magnitudes become “possible.” Not only were such magnitudes possible, but Gauss proclaimed, they deserved “complete civil rights.” As he stated in his announcement to the second treatise on biquadratic residues:

“It is this and nothing other, that for the true establishment of a theory of bi-quadratic residues, the field of higher arithmetic, that otherwise extends only to the real numbers, will be enlarged also to the imaginary, and these must be granted complete and equal civil rights, with the real. As soon as one considers this, these theories appear in an entirely new light, and the results attain a highly surprising simplicity.”

In a manifold of two dimensions, the relationship among objects is not restricted to the back and forth relationship of objects along a line, but also includes a relationship of up and down, so to speak. Be careful, this is not two separate relationships, back-forth and up-down. Rather it is one, doubly-extended relationship. As Gauss stated:

“Suppose, however, the objects are of such a nature that they can not be ordered in a single series, even if unbounded in both directions, but can only be ordered in a series of series, or in other words form a manifold of two dimensions….”

The root of this conception lies not in mathematics, but in physical geometry. In a fragmentary note, “On the Metaphysics of Mathematics,” Gauss described a doubly-extended relationship using the metaphor of a carpenter’s level. The bubble in the level can only move back and forth, if the ends of the level move up and down. Furthermore, Gauss repeatedly noted, such concepts as back and forth, up and down, left and right, can not be known, as Kant claimed, mathematically. Instead, such concepts are only known with respect to real physical objects.

This type of action is represented geometrically by two-dimensional magnitudes which Gauss called complex numbers. Gauss represented these numbers as the vertices of a grid of equally spaced squares on a plane. Be mindful. It is not the grid that generates the numbers. It is the {idea} of a doubly-extended manifold, that generates doubly-extended magnitudes, that form the grid. As in the case of the bubble in the carpenter’s level, any relationship between two complex numbers is a combination of horizontal and vertical action along the grid.

This geometrical representation of complex numbers flows easily from the geometry of the “orbits” generated by the residues of powers. For example, take the case of 13, (or any odd-even prime number modulus) as illustrated above. Think of the cycle of residues as a closed orbit, beginning with 1 and returning to 1. Halfway around the orbit is -1. One quarter the way around the orbit is the square root of -1. Three quarters around is minus the square root of -1.

This is the geometrical relationship that is reflected in the characteristics of the residues, and is nothing more than a generalization of the principle that Plato presents in the {Meno} and {Theatetus}, for the special case of squares. In that case, the diagonal of the square, which forms the side of a square whose area is double the original square, is called the geometric mean. The diagonal has the same relationship to the two squares, as -1 does to 1, and the square root of -1 does to -1.

Gauss described the manifold of complex numbers this way:

“We must add some general remarks. To locate the theory of biquadratic residues in the domain of the complex numbers might seem objectionable and unnatural to those unfamiliar with the nature of imaginary numbers and caught in false conceptions of the same; such people might be led to the opinion that our investigations are built on mere air, become doubtful, and distance themselves from our views. Nothing could be so groundless as such an opinion. Quite the opposite, the arithmetic of the complex numbers is most perfectly capable of visual representation, even though the author, in his presentation has followed a purely arithmetic treatment; nevertheless he has provided sufficient indications for the independently thinking reader to elaborate such a representation, which enlivens the insight and is therefore highly to be recommended.

“Just as the absolute whole numbers can be represented as a series of equally spaced points on a line, in which the initial point stands for 0, the next in line for 1, and so forth; and just as the representation of the negative whole numbers requires only an unlimited extension of that series on the opposite side of the initial point; so we require for a representation of the complex whole numbers only one addition: namely, that the said series should be thought of as lying in an unbounded plane, and parallel with it on both sides an unlimited number of similar series spaced at equal intervals from each other should be imagined, so that we have before us a system of points rather than only a series, a system which can be ordered in two ways as series of series and which serves to divide the entire plane into identical squares.

“The neighboring point to 0 in the first row to the one side of the original series corresponds to the number {i,} and the neighboring point to 0 on the other side to -i and so forth. Using this mapping, it becomes possible to represent in visual terms the arithmetic operations on complex magnitudes, congruences, construction of a complete system of incongruent numbers for a given modulus, and so forth, in a completely satisfactory manner.

“In this way, also, the true metaphysics of the imaginary magnitudes is shown in a new, clear light….”

Consequently, the domain of whole numbers has been extended beyond simply positive and negative numbers, to numbers of the form “a+bi“, where “i” stands for the square root of -1. These numbers are represented as points on a plane, in which “a” expresses the horizontal action while “b” the vertical action. For example, 2+3i would be represented by a point 2 to the right of 0 and 3 up from 2; 5+4i would be represented by 5 to the right of 0 and 4 up from 5. The difference (interval) between 2+3i and 5+4i would be 3+i, which is the combined amount of horizontal and vertical action required to move from 2+3i to 5+4i.

In Gauss’ complex domain, the fundamental characteristics of numbers are re-defined. Of particular importance is prime numbers. Here, numbers that are prime in a simply-extended manifold, are no longer prime in the complex domain. For example, 5, can be factored into the complex numbers (1+2i)(1-2i); or 13 into (3+2i)(3-2i). Gauss showed that all odd-even prime numbers are no longer prime in the complex domain, while all odd-odd prime numbers are still prime. Gauss went on to discover a new type of prime number that he called complex primes, which are now called “Gaussian primes.” These are complex numbers of the form a+bi, where a² + b ² is a prime number. (The geometrical demonstration of this principle will be developed in a subsequent installment.)

Thus, prime numbers, the “stuff” from which all numbers are made, are themselves not primary. Instead, they are defined by the nature of the manifold in which they exist. A one-dimensional manifold produces a certain set of prime numbers, whose “primeness” is absolute within a one-dimensional manifold, but relative with respect to a two-dimensional manifold. In turn, a two-dimensional manifold produces prime numbers whose characteristic “primeness” is different from what constitutes “primeness” in a one-dimensional manifold. The characteristic “primeness” of one-dimensional prime numbers can be derived from the characteristic of “primeness” of two dimensions, but not vice versa. Implicit in this, is a hierarchy of dimensionality, in which the singularities of n-dimensions are subsumed and transformed by manifolds of higher dimensions. Gauss himself anticipated such an idea stating:

“The author reserves the possibility of treating these matters, only barely touched upon in this paper, more fully at a later date, at which time we shall also answer the question, why such relations between things as form manifolds of more than two dimensions might not provide additional species of magnitudes to be admitted in general Arithmetic.”

This is only a taste of the manifold of ideas manifest in the minds of the hearers of Riemann’s habilitation lecture. The more this manifold begins to order your mind’s thoughts, the more lively Riemann’s ideas will become.

Your Education Was Not Merely Incompetent

If you felt a little disconcerted to sit in the same lecture hall with C.F. Gauss, listening to B. Riemann deliver his habilitation address, do not despair. Be happy. You are being afforded the opportunity to discover that your education was not merely incompetent, it was also malicious. Incompetent, in that your teachers were most likely totally ignorant of the most significant original discoveries upon which the human race has depended for survival; malicious, in that the system to which the teachers acquiesced, had no intention of producing individuals capable of making such discoveries. As we now see from the events unfolding around us, a system which does not intend to produce creative individuals, has no intention of surviving. Therefore, rejoice at the occasion to clear your head of the restrictive fixation on facts, laws, opinions, and popularly held beliefs, and set about the task of producing creative thinkers.

Riemann sought to “lift the darkness” that had settled on science for more than 2000 years, by providing for science a general concept of multiply-extended magnitude. A concept in which it was recognized, that magnitude had no {a priori} characteristics, but was itself determined by the nature of the manifold in which it existed which nature was only determined by experiment. Riemann’s taking off point was Gauss’ work on physical geometry and arithmetic, which was itself the revolutionary result of Gauss’ early education in the work of Kepler, Leibniz, Bach, Kaestner, and the scientific achievements of classical Greece. Central to all these discoveries was the desire to discover the principles that generated the objects of investigation, be it physical objects, such as the motions of the planets, living processes, or objects of cognition, the latter being the most fundamental, upon which all other investigations depend.

In this regard, Plato recognized that the mind must be trained to investigate itself, to which end he prescribed the study of geometry, astronomy, music and arithmetic, the latter, because, “thought begins to be aroused within us, and the soul perplexed and wanting to arrive at a decision asks `Where is absolute unity?’ This is the way in which the study of the One has a power of drawing and converting the mind to the contemplation of true being … and, because this will be the easiest way for the soul herself to pass from becoming to truth and being….”

That search for the nature of unity underlies Gauss’ arithmetical investigations. Its revolutionary feature being that the nature of unity, is itself not a fixed, but developing and changing. This is what underlies Gauss’ concept of congruence, the ordering of numbers with respect to a modulus. This is based on the principle that numbers are not fixed objects that determine order, but are themselves ordered, according to the principle from which they are generated.

The first principle is the generation of numbers from the juxtaposition of cycles. These juxtapositions form two types of relationships. Either the cycles equally divide one another, such as a cycle of 8 and a cycle of 4, or no such division is generated, such as a cycle of 5 and a cycle of 8. In the latter case, that relationship is called, “relatively prime”. Those cycles, which when juxtaposed to all smaller cycles and One were called simply “prime”, and until Gauss were thought of as absolutely prime, or prime relative to One.

Thus when thinking about numbers from the bottom up, as formed by adding 1 to 1 to 1, the prime numbers are mysterious and arise from an unknown. However, when thought about from the top down, the prime numbers are that from which all numbers are made. The question that Gauss and Riemann contemplated was, “what principle generates prime numbers”. This led to the investigation, not of the numbers, but of the manifolds in which those numbers were generated.

The investigation of those manifolds leads to the second principle of generation. This is the principle which the Greeks called “geometric”, and was examined in last week’s installment. This is where today’s work begins.

Take the example from last week — the investigation of the cycle of residues generated with respect to modulus 11 and compare that to the cycle of residues with respect to modulus 13. For the sake of brevity, I indicate only the cycle with respect to one primitive root. The first row is the index, or power to which the primitive root is raised, and the second row is the corresponding residue. For reasons that will become apparent, we include both the positive and negative residues:

Modulus 11:
     Index:  0,      1,     2,     3,     4,     5 
   Residue: {1,-10},{2,-9},{4,-7},{8,-3},{5,-6},{10,-1}
     Index:  6,      7,     8,     9,    10 
   Residue: {9,-2}, {7,-4},{3,-8},{6,-5},{1,-10}

Modulus 13: 
     Index:  0,      1,     2,      3,      4,      5,    6 
   Residue: {1,-12},{2,-11},{4,-9}, 8,-5}, {3,-10},{6,-7},{12,-1} 
     Index:  7,      8,     9,      10,     11,     12 
   Residue: {11,-2},{9,-4}, {5,-8},{10,-3},{7,-6}, {1,-12}

In both cases, half the residues, that is, the residues of even powers, are residues of squares, (quadratic residues). The residues of the other half, the residues of odd powers, are residues of rectangles (quadratic non-residues). In the case of 13, the quadratic residues are the same whether negative or positive. While with 11, the positive quadratic residues are different than the non-residues.

This indicates an at first surprising connection between the ancient Pythagorean discovery of odd and even, which seems to pertain to numbers, and the geometric progression, which seems to pertain to figures in space. That odd and even reflected a deeper principle was described by Cusa in “On Conjectures”:

“It is established that every number is constituted out of unity and otherness, the unity advancing to otherness and otherness regressing to unity, so that it is limited in this reciprocal progression and subsists in actuality as it is. It can also not be that the unity of one number is completely equal to the unity of another, since a precise equality is impossible in everything finite. Unity and otherness are therefore varied in every number. The odd number appears to have more of unity than the even number, because the former cannot be divided into equal parts and the latter can be. Therefore, since every number is one out of unity and otherness, so there will be numbers in which the unity prevails over the otherness, and others in which the otherness appears to absorb the unity.”

It doesn’t stop with the division into even and odd, as both types have a deeper nature. The even numbers can be divided, into those even numbers, such as 10, that, when divided form two odd numbers (5 and 5), and those, such as 12, that form two even numbers (6 and 6). The former are called even-odd, the latter even-even. Likewise odd numbers can be divided into two types. Odd numbers, like 11, that are one more than an even-odd number and are called odd-odd, while odd numbers, like 13, that are one more than an even-even and are called odd-even. (Gauss called odd-even numbers 4n+1, and odd-odd numbers 4n+3.)

Now look at the mid-point of each of the above “orbits” of residues. As we showed at the end of last week’s installment, the midpoint of the orbit is both the arithmetic and the geometric mean. The arithmetic, because it is half the length of the cycle. The geometric, because its half the rotation from the 1 to 1, or the square root of 1. For modulus 11, that residue is either 10 or -1, both of which, when squared, are congruent to 1 modulus 11. For modulus 13, that residue is either 12 or -1, both of which, when squared, are congruent to 1 modulus 13.

Illustrate this in your mind, using Plato’s alternating series of squares and rectangles. In a cycle of 10 squares and rectangles, the 5th action is a rectangle, whose area is 32. That area is the geometric mean between a square whose area is 1 and the square whose area is 1024. Since the residues form a cycle that begins and ends with 1, the residue of the 5th power, mod 11, is the geometric mean between 1 and 1. Similarly, with a cycle of 12 squares and rectangles, the 6th action produces a square whose area is 64. That square is the geometric mean between a square whose area is 1 and a square whose area is 4096. With respect to modulus 11, the geometric mean is a rectangle, while for modulus 13, the geometric mean is a square.

But, there’s a difference between modulus 11 and modulus 13, as 11 is odd-odd, which means the half-way point is an odd number, that is 5. While 13 is odd-even, and is susceptible of further division, into quarters.

The residue of the 1/4 power relative to modulus 13 is either 8 or -5, both of which when squared twice, are congruent to 1 modulus 13. But, when both are squared once, they are congruent to -1 modulus 13. In other words, 8 and 5, -8 and -5, are congruent to the square root of minus 1 modulus 13.

Thus, the square root of -1 has clearly defined existence with respect to an odd-even modulus, while it has no existence in a manifold generated with respect to an odd-odd modulus.

From the naive standpoint, it would appear that the square root of -1 is a product of characteristic of oddness. But, as Cusa states, oddness is a quality in which unity prevails over otherness. So, rather than look for the square root of -1 in nature of oddness, look for the nature of oddness in the characteristic of unity.

This is precisely the way Gauss approached the problem. Rather than think of a manifold of a simply extended unity, he conceived of a manifold of a doubly extended unity, in which the square root of -1 is a “natural product” so to speak. He called this manifold the complex domain.

In his words:

“From this, we had already begun to ponder these objects in 1805, and we soon came to the conviction that the natural source of a general theory be sought in an extension of the field of Arithmetic.

“While higher arithmetic, has until now dealt only with questions pertaining to whole numbers, propositions concerning biquadratic residues appear in their complete simplicity and natural beauty, only if the field of arithmetic is extended to include imaginary numbers, without limitation, the numbers of the form a+bi forms its object, where the customary i denotes the square root of -1 and a and b are all whole numbers between minus infinity and plus infinity.”

Next week we’ll put flesh and bones on this new concept.

It is Principles, Not Numbers, That Count

As we continue the investigations into the “hints” from Gauss, to which Riemann referred in his 1854 habilitation lecture, it is vitally important to maintain the perspective of a member of the audience in the lecture hall that June day when Riemann delivered his revolutionary address. Don’t be a fearful, passive observer. Go in. Take the open seat next to the 77 year old Gauss and hear these living ideas, not only as they were spoken then, but as they are today, alive and transformed in the mind and work of LaRouche, for which Riemann provides brief hints.

Listen as Riemann boldly proposes “to lift the darkness” that has existed for more than 2000 years, by elaborating a “general concept of multiply-extended magnitude”. But, before you can even begin to lift that darkness, you must first realize that the lights aren’t on.

That is the basis on which the preliminary exercises into the investigations of the geometry of numbers was begun last week.

Gauss is training the mind to give up all deductive, a priori, notions of number. Instead of investigating numbers, we investigate what generates them. It is the principle of generation to which we must turn our thoughts, aided by concepts from Classical art. The numbers are simply players, guides to what’s in between.

The first principle of generation of numbers, to which Gauss points, is the generation of numbers by the juxtaposition of three cycles. While this concept was introduced in a new form in the Disquisitiones Arithmeticae, by the concept of congruence with respect to a modulus, the principle underlying it is perhaps the earliest, and most elementary concept of number. In this case, no number exists on its own. Rather, all numbers exist as players, whose parts are a function of their relationship to one another and a One, which Gauss called a modulus. Thus, all numbers are ordered according to the characteristics of the modulus. Those characteristics are themselves determined by an underlying generating principle, which will become more clear below. The so-called, “natural”, counting numbers are only the special case, of numbers ordered with respect to the modulus 1.

The second principle of generation to which Gauss turned his attention, is generating numbers from a cycle of cycles, specifically, the “geometric” cycle. Here each cycle is generated by the function described by Plato in the Theatetus dialogue, as reflected in the alternating series of squares and rectangles produced by some repeated action, such as doubling or tripling, etc. However, Gauss, as Plato, Kepler, Leibniz, Bernoulli and Fermat before him, understood that the alternating series of squares and rectangles, was itself only a shadow of a higher principle of generation, that had to be discovered.

Naive sense certainty says this geometric progression is not a cycle at all, but open ended and continuously growing. Yet, as the experiments at the end of last week’s installment illustrate, if each “stage” of the geometric progression is thought of as a cycle, and each such cycle is juxtaposed to a third cycle (modulus), an underlying periodicity is revealed, indicating the characteristics of the cycle that generated each stage.

The principle of generation of that underlying cycle, is best investigated by experiment. Hopefully, you carried out the experiment indicated last week. If so, you will have no trouble producing the necessary geometric constructions.

Construct a chart of the residues of powers with respect to modulus 11 by first making a row of numbers from 0 to 10. These denote the powers. Then make a separate row of the residues of the powers of 2 through 10, writing each residue under the corresponding power. The result should be the following:

Powers: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9,10 
     2: 1, 2, 4, 8. 5, 10,9, 7, 3, 6, 1 
     3: 1, 3, 9, 3, 4, 1, 3, 9, 5, 4, 1 
     4: 1, 4, 5, 9, 3, 1, 4, 5, 9, 3, 1 
     5: 1, 5, 3, 4, 9, 1, 5, 3, 4, 9, 1 
     6: 1, 6, 3, 7, 9, 10,5, 8, 4, 2, 1 
     7: 1, 7, 5, 2, 3, 10,4, 6, 9, 8, 1 
     8: 1, 8, 9, 6, 4, 10,3, 2, 5, 7, 1 
     9: 1, 9, 4, 3, 5, 1, 9, 4, 3, 5, 1 
    10: 1,10, 1, 10,1, 10,1, 10,1,10, 1

Now have some fun. Obviously, this action produces, from the open and growing geometric cycle, a regular structured periodicity. The question Fermat, Leibniz, and Gauss investigated was, “What generating principle produces this?” To begin to answer this, they looked for the paradoxes, within the seemingly regular structure.

First, it is clear that each period begins and ends with 1 and there are three types of periods. Those periods, such as for the powers of 2, 6, 7, and 8, that are 10 numbers long and include all the numbers between 1 and 10. Gauss called these numbers, 2,6,7 and 8 “primitive roots” of 11. The second type of period, for the powers of 3, 4, 5, and 9 are 5 numbers long. The third type of period is the powers of 10 which is only 2 numbers long. Thus, the residues of powers with respect to modulus 11 permits only certain size “orbits” so to speak, which are restricted to the size of 10 and the prime number factors of 10, that is, 2 and 5.

Now, even though each period puts the numbers in a different order, this order is highly determined. To see this, hunt through the whole chart and circle the primitive roots, 2, 6, 7, and 8, wherever they appear. You should discover that these numbers do not appear as residues, except in the periods that are 10 numbers long. Also, even though they appear in different places in each period, they always are residues of the powers 1, 3, 7, or 9, which are the numbers that are relatively prime to 10.

This begins to reveal the nature of the underlying generating principle of these “orbits”, as the characteristics of the number 10, specifically its prime factors and its relative primes, determine the ordering of the periods!

Keeping this in your mind’s eye, draw an alternating series of squares and rectangles, first by doubling, then by tripling, and label each according to the residue from the powers of 2 and 3 respectively to which it corresponds. This will reveal that the even powers correspond to squares and the odd powers correspond to rectangles. Notice how the residue 1 only appears on a square and the residue 10 only appears on a rectangle. Also, the squares always correspond to even numbered powers, while the rectangles correspond to odd numbered powers.

Thus, the quality of odd and even, reflect a geometric characteristic, not a numerical property of numbers. For these geometrical reasons, Gauss called the residues of the even powers, “quadratic residues” and the residues of the odd powers, “quadratic non-residues”. Gauss paid special attention to this characteristic, for its investigation opened the door to some of the most profound principles. In the next installment we will explore this more fully.

But, before closing, look at one more anomaly. Notice that the residue at the halfway point, that is the residue of the power 5, is always 1 or 10. Since 10 is congruent to -1, the residue of the middle power is always 1 or -1. While 5 is the arithmetic mean between 0 and 10, its residue, 1 or -1 is the geometric mean between 1 and 1! In other words, 1 and 10 are the square roots of 1 relative to modulus 11.

Look back over the preceding investigations from the perspective of a classical drama. Think of the foregoing as a drama of 10 characters. Each character has several roles, in which they wear the same costume, but do different things. The playwright has deliberately chosen this device so that the audience can be broken from judging these characters by naive sense-certainty. This helps convey an idea that could not be expressed in words by any of the characters, but only by the totality of all their actions taken as a whole, and the ironies revealed when the same actor does something completely different, without changing his costume. Each number from 1 to 10 has a different function, whether it’s a power, a residue, or a base. In some roles, the obvious characteristics of the number, such as odd or even, factor or relative prime, seem to affect its function, but in other cases, such as the primitive roots, these obvious characteristics seem to have no bearing. Only when all the roles are played out can we begin to taste the intention of the playwright.

Gauss could see that these anomalies could not be derived from a concept of number, in the naive sense of an object that counts things, but, rather, these anomalies revealed an underlying {geometric} generating principle, that shone through the numbers themselves. But, to bring out that light, required a complete revolution in the way people thought about number. As he said in the beginning of the first Treatise on Bi-quadratic Residues, “we soon came to the knowledge, that the customary principles of arithmetic, are in no way sufficient for the foundation of a general theory, and that it is very much necessary, that the region of higher arithmetic be, so to speak, infinitely much more extended.”

Gauss’ Attack on Deductive Thinking

In his 1854 habilitation dissertation, Bernhard Riemann referred to two “hints” as preliminary to his development of an anti-Euclidean geometry–specifically Gauss’ second treatise on bi-quadratic residues and Gauss’ essay on the theory of curved surfaces. It is but one more testament to the ignorance of all so-called experts today, (not to mention those who wish to qualify as educated citizens) that direct knowledge of these two works by Gauss, let alone a working understanding of Riemann himself, is virtually non-existent.

It should not be surprising that in a lecture focused on ridding science of “ivory tower” mathematics, Riemann would refer to the climactic conclusion of Gauss’ investigation of whole numbers. Riemann, like Gauss and Leibniz before him, began his scientific education by confronting the paradoxes that emerge from an anti-deductive investigation of whole numbers. At an early age, Riemann was given a copy of Legendre’s “Theory of Numbers”, and within one week he returned the 600 page book saying, “This is wonderful book. I know it by heart.”

Plato prescribed such investigations as necessary for the development of competent leadership, because it forced the mind out of realm of sense-certainty and into the realm of paradoxes where, “thought begins to be aroused within us, and the soul perplexed and wanting to arrive at a decision asks `Where is absolute unity?’ This is the way in which the study of the One has a power of drawing and converting the mind to the contemplation of true being.,,,we must endeavor to persuade those who are to be the principal men of our State to go and learn arithmetic, not as amateurs, but they must carry on the study {until they see the nature of numbers with the mind only;} nor again, like merchants or retail-traders, with a view to buying or selling, but for the sake of their military use, and of the soul herself; and because this will be the easiest way for her to pass from becoming to truth and being….”

And as LaRouche pointed out in “Marat, DeSade & Greenspin”:

“Since the beginning of the Nineteenth Century, Carl Gauss’s {Disquisitiones Arithmeticae}, inspired by his teacher, the great founder of anti-Euclidean geometry, Abraham K„stner, had been the standard for competent mathematics instruction. This masterpiece should be the recognized standard, even today, for basic secondary and higher education in mathematics. The result of replacing that standard with “the new math” program, should have reminded any literate professional of Jonathan Swift’s famous description of education as practiced on the allegorical floating island of Laputa.”

To ameliorate this pitiable condition of mankind, and save any readers of these pedagogicals from being condemned to perpetual flatulence on Laputa, the next installments of this series will begin to acquaint the reader with the basic conceptions of these two works by Gauss, as a prerequisite to looking more deeply into Riemann’s work itself.

To begin, you must, as Gauss does, give up all deductive notions of number. Instead of thinking of whole numbers as self-evident things in themselves, think of numbers as being generated by a principle. Gauss took an experimental approach to numbers, designing experiments that revealed paradoxes with respect to a known principle. The resolution of that paradox required the introduction of a new principle. Gauss himself described the approach to be taken in our investigations:

“The questions of higher arithmetic often present a remarkable characteristic which seldom appears in more general analysis, and increases the beauty of the former subject. While analytic investigations lead to the discovery of new truths only after the fundamental principles of the subject (which to a certain degree open the way to these truths) have been completely mastered; on the contrary in arithmetic the most elegant theorems frequently arise experimentally as the result of a more or less unexpected stroke of good fortune, while their proofs lie so deeply embedded in the darkness that they elude all attempts and defeat the sharpest inquiries…. These truths are frequently of such a nature that they may be arrived at by many distinct paths and that the first paths to be discovered are not always the shortest. It is therefore a great pleasure after one has fruitlessly pondered over a truth and has later been able to prove it in a round-about way to find at last the simplest and most natural way to its proof.”

The opening motivic idea of the Disquisitiones, is to identify numbers as being generated by an interval, or modulus, much the same way as musical notes are generated by intervals. If the interval between two numbers is divisible by the modulus, Gauss called those numbers, “congruent”. For example, 2, 7, 12, 17, 22, etc, are all congruent to each other relative to modulus 5. Relative to modulus 7, 2 is congruent to 9, 16, 23,etc.

Gauss’ use of the term congruence is consistent with Kepler’s use of that concept in Book II of his “Harmonies of the World”. For Kepler the word “congruentia” was the Latin equivalent to the Greek word, “harmonia”, which means to fit together. Thus, it is not the numbers on which the mind must focus, but the way they fit together.

Gauss’ concept of congruence reflects the actual nature of numbers more truthfully than the so-called “natural” ordering of numbers that seemed so commonsensical when you learned it in school. This is because, contrary to such common sense certainty notions, the concept of number does not arise from counting things. Rather, it arises from the juxtaposition of cycles, such as, for example, astronomical cycles. Each cycle is a One, but when juxtaposed to each other these cycles give rise to a multiplicity.

As Leibniz puts it in his doctoral dissertation, “On the Art of Combinations”:

“Furthermore, every relation is either one of union or one of harmony. In union the things between which there is this relation are called parts, and taken together with their union, is a whole. This happens whenever we take many things simultaneously as one. By one we mean whatever we think of in one intellectual act, or at once. For example, we often grasp a number, however large, all at once in a kind of blind thought, namely, when we read figures on paper which not even the age of Methuselah would suffice to count explicitly.

“The concept of unity is abstracted from the concept of one being, and the whole itself, abstracted from unities, or the totality, is called number.”

Any two cycles can be known in relation to each other only by a third. For example, the cycle discovered by the Greek astronomer Meton who attempted to resolve the lunar month and solar year cycles into a One. One solar cycle contains 12 lunar cycles, plus a small residue, so in Gauss’ words, the lunar cycle is {incongruent} with the solar one. However, Meton discovered that 19 solar years contains 235 lunar months with no residue. So, while one lunar month is not congruent to one solar year, one lunar month is congruent to 19 solar years. The relationship between the solar cycle and lunar cycle can be known with respect to this 19 year Metonic cycle, which defines the modulus under which the solar and lunar cycles are congruent.

To get familiar with this concept play with some more examples. Consider two cycles one of which is three times longer than the other. These cycles would be congruent to each other relative to modulus three. Examples of this relationship expressed in numbers would include: 3 is congruent to 9 relative to modulus 3, or 9 is congruent to 27 relative to modulus 3.

Now consider cycles that don’t fit exactly, such as a cycle of 4 and a cycle of 9. The smaller cycle of 4 will fit into the larger cycle of 9 twice with a residue of 1. Under Gauss’ concept, 9 is congruent to 1 relative to modulus 4. On the other hand, a cycle of 4 fits into a cycle of 10 twice with a residue of 2. Under Gauss’ concept, 10 is congruent to 2 relative to modulus 4. Continuing, a cycle of 4 fits into a cycle of 11 with a residue of 3. Thus, 11 is congruent to 3 relative to modulus 4. Further, a cycle of 4 fits into a cycle of 12 with 0 residue, and into a cycle of 13 with a residue of 1. Thus, a cycle of 4 will fit into any cycle with a residue of 0, 1, 2, or 3.

If you play around with this idea, trying cycles of different relationships, you will discover for yourself, that any modulus defines a period, from 0 to the modulus minus 1. This will probably strain your brain, as you will be forcing yourself to think in terms of relationships instead of things, but that is precisely why all great thinkers, from Plato onward, struggled to free themselves from the straight jacket of deductive relationships by investigating the nature of numbers.

This would probably be enough to get you started, but in order to speed up our pursuit of the concepts in Gauss’ second treatise of bi-quadratic residues, we should push ahead.

After developing the concept of congruence in the beginning of the Disquistiones, Gauss turns to an investigation of what he called, “residues of powers”. Here you must leave completely the world of sense certainty and deductive reasoning.

By “powers” Gauss meant the concept developed by Plato in the Theatetus dialogue. These are the magnitudes associated with action in what Riemann would call a doubly extended manifold. Shadows of these magnitudes are represented by Plato as the successive doublings, triplings, etc. of squares. When these magnitudes are expressed in whole numbers it generates a geometric series such as: 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, etc. or: 1, 3, 9, 27, 81, 243, 729, 2187, 6561, etc.

If you think of each number as a cycle, the series can be thought of as a cycle of cycles. This cycle of cycles doesn’t close, but gets bigger and bigger, according to a self-similar proportionality.

What may appear shocking to you, is that this open, growing, cycle of cycles generates a periodic, closed, cycle with respect to a modulus.

For example, take the geometric series formed from doubling squares and find the residues relative to modulus 3. This yields the period: 1, 2, 1, 2, 1, 2, 1, 2, etc. Now do the same for modulus 5. This yields the period: 1, 2, 4, 3, 1, 2, 4, 3, 1, 2, 4, 3, 1, etc. And for 7: 1, 2, 4, 1, 2, 4, 1, 2, 4, 1,etc. Try the same experiment with respect to the geometric series based on tripling. For modulus 5 it yields the period: 1, 3, 4, 2, 1, 3, 4, 2, 1, etc. Compare this with the period generated from the same modulus but the geometric series based on doubling. Modulus 7 for the same series yields the period: 1, 3, 2, 6, 4, 5, 1, 3, 2, 6, 4, 5, 1,etc.

This experiment is a simple example of what Gauss described as discovering certain truths by experiment. From where does this periodicity arise? What is its nature? What principles does it reflect?

Experiment with other geometric series and other moduli. Next week we will plunge ahead.

Doing the Impossible

“Nothing is fun but change,” is an apt transformation of Heraclites’ famous aphorism to convey the quality of mind required to grasp Leibniz’ calculus and its extension developed by Kaestner, Gauss, and Riemann. Inversely, one who is gripped by a bullheaded resistance to its import, and the corollary, “Without fun there is no change,” will be doomed to the dull, unchanging, “cult of Isaac Newton,” where, the only hope out, is, to change.

Like Leibniz’ original discovery of the calculus, the equally revolutionary breakthroughs of Kaestner, Gauss and Riemann were long in the making. As usual, the matter is most efficiently presented pedagogically, from the standpoint of Kepler.

Kepler’s discovery of the non-uniform motion of planetary orbits presented the paradox that while the characteristics of the planet’s orbit are knowable, the position and velocity of the planet were not susceptible to precise mathematical calculation. In his New Astronomy, Kepler introduced this paradox in the form of a challenge to future geometers. Kepler’s challenge confronted a mathematical system that confined itself to determining positions, only with respect to other positions. From this standpoint, it were impossible to determine a characteristic of change, which was always changing. But, for Kepler, Leibniz, Gauss and Riemann, change was a characteristic of mind, as well as the physical universe. Rather than measure position, as the existing mathematics insisted he do, Leibniz invented a calculus that measured the characteristics of change from which the position of the planet was produced. Gauss and Riemann extended this calculus, by measuring the characteristics of change, that produced the change, which was producing the positions. In other words, the orbit was measured by a total characteristic of change (integral) of which each momentary expression of it (differential) was a function. But, that orbit was itself a function of a characteristic of a higher process. It is to the characteristics of that process on which Gauss and Riemann focused, the which will be developed pedagogically in future weeks.

Investigations into the impossibility of mathematical solutions goes all the way back to classical Greece, as represented by the famous problems, of doubling the cube, trisecting the angle, the quadrature of the circle, and construction of the heptagon (7-gon). No methods were found by which these problems could be solved, in ways which were rigorously knowable, as Plato established the principle of “knowability” in his dialogues.

For example, as you know from the study of Cusa’s work on the quadrature of the circle (fn.1) the circle could not be measured precisely by rectilinear magnitudes. Cusa showed that the “unsovlability” of this problem was not due to an undiscovered method within the existing mathematics, rather, it was due to a deficiency in the entire system of mathematics, as long as that system did not admit of transcendental magnitudes. Such transcendental magnitudes were impossible in the domain of rectilinear magnitudes. Yet, these “impossible” magnitudes were reflected in a real principle, the principle of circular action. The existence of the circle could be known as a reflection of a distinct principle, but its measurement could not be accomplished by a mathematics that excluded that principle. There had to be a complete transformation of the system of mathematics, from a mathematics that included only one type of magnitude (rectilinear) (fn.2) to a mathematics encompassing two types of magnitudes (rectilinear and transcendental). It was not that Cusa made transcendental magnitudes possible, but that a system of mathematics without them, was proven to be impossible.

Like the quadrature of the circle, the difficulty of doubling the cube, trisecting the angle, and constructing the heptagon, resulted from the “impossibility” of constructing an appropriate magnitude, that was “knowable” within the given system of mathematics. This is most easily illustrated by the example of doubling of the cube, a problem that has been discussed in previous pedagogical discussions.

The issue involved is presented most effectively by the poetic report of the problem’s origination. It is said that the Delians were asked by the Gods to construct an altar double the size of the existing one. Plato told the Delians that Apollo posed this problem to them because, according to Kepler, “Greece would be peaceful if the Greeks turned to geometry and other philosophical studies, as these studies would lead their spirits from ambition and other forms of greed, out of which wars and other evils arise, to the love of peace and to moderation in all things.” In other words, “Change the system!”

It had been known by the Greeks how to find a magnitude that could double the area of a square, even though this magnitude was incommensurable with the side of the smaller square.(fn.3) But, they were unable to construct a magnitude that could produce a cube whose volume was doubled. Was this magnitude possible, or, was a system that could not produce it, “impossible”?

Renewed investigations into these questions emerged in the wake of Cusa’s revolution, which set the stage for the revolutionary ideas of Gauss and Riemann. This history is reported by Kaestner in his 1796, “History of Mathematics”, a much more reliable source than today’s generally available histories, which commit fraud by the fallacy of composition by lumping these investigations under the general topic of algebra. Kaestner distinguishes between the efforts to simply calculate with symbols, and those attempts to gain a deeper insight into proportions. For the former, Kaestner relates the story, from Cervantes, “Don Quixote”, about how the Aristotelean, Bachelor Sampson Carrasco, who, believing Don Quioxte was mad, pretends to be mad himself, in an effort to deceive Don Quioxte into giving up knight errantry. Having adopted Don Quioxte’s system, Carrasco gets caught in a joust with the mad knight and ends up receiving a thrashing himself. Having been left humiliated and with broken ribs, Carrasco is forced to seek out help from an algebraist. (fn.4) For the latter, Kaestner refers to the early efforts to investigate the deeper implications of the concept of powers, that had been expressed by Plato in the Theatetus dialogue. Such efforts were associated with Luca Pacioli and Girolamo Cardan (1501-1576), whose father is reported to have been a collaborator of Leonardo da Vinci.

In investigating the relations of squares and cubes, Cardan discovered magnitudes that were “impossible” according to the prevailing system of algebra. Cardan’s example was, “If someone says to you, divide 10 into two parts, one of which multiplied into the other shall produce 30 or 40 it is evident that this case is impossible. Nevertheless, we shall solve it in this fashion….” Cardan’s solution was to produce the results by the magnitudes, (5 + ?-15)(5 – ?-15). But, since the square root of a negative number were “impossible” in the algebraic system, Cardan concluded, “This subtlety results from arithmetic of which this final point is as I have said, as subtle as it is useless.”

And so, the question was posed again: were these magnitudes “impossible”, or was the system which could not produce them, “impossible”? How this question was approached, is an instructive marker that separates the true thinkers (those who know how to have fun) from the frauds.

For example, in his investigations of the same equations, Descartes maintained that the square root of a negative number was “impossible”. On the other hand, Leibniz, in a 1673 correspondence with Huygens, produced the following result:

?(1 + ?-3) + ?(1 – ?-3) = ?6

Of which he said, “I do not remember to have noted a more singular and paradoxical fact in all analysis: for I think I am the first one to have reduced irrational roots, imaginary in form, to real values….”

To which Huygens replied:

“The remark you make concerning inextractable roots and roots with imaginary magnitudes, the which, nevertheless, upon addition yield a real quantity, is surprising and completely new. One would never have believed that ?(1 + ?-3) + ?(1 – ?-3) could be equal to ?6, and there is something hidden there which is incomprehensible to us.”

Later, Leibniz would say, “The imaginary number is a fine and wonderful recourse of the divine spirit, almost an amphibian between being and non-being.”

In a 1702 letter to Varignon, Leibniz further reflects on this paradox:

“Without worry one can use infinitely small and large lines as ideal concepts — even though they do not exist as real objects in the metaphysically rigorous sense — as a means to shorten calculation, just as the imaginary roots in ordinary analysis, such as for example ?-2. Irregardless of whether one calls these ‘imaginary’, they are nonetheless useful and sometimes even indispensible, in order to express real magnitudes analytically; so, for example, it is impossible, without using them, to give an analytical expression for a line segment, which divides a given angle into three equal parts. Just so, one could not elaborate our calculus of transcendental curves, without talking about differences, which are in the act of vanishing, and introducing once and for all the concept of incomparably small magnitudes….

“Also the imaginary numbers have their {foundation in reality} (fundamentum in re). When I pointed out to the late Mr. Huygens, that ?(1 + ?-3) + ?(1 – ?-3) = ?6, he was so amazed, that he answered, for him there is something incomprehensible in this. But just so, one can say, that the infinite and infinitely small have such a solid basis, that all results of geometry, and even the processes of Nature, behave as if both were complete realities … because everything obeys the Rule of Reason.”

By contrast, Euler (held in such high esteem by today’s algebraists) said, “All such expressions, as square root of -1, or square root of -2, are consequently impossible or imaginary numbers, since they represent roots of negative quantities; and of such numbers we may truly assert that they are neither nothing, nor greater than nothing, nor less than nothing, which necessarily constitutes them imaginary or impossible.”

It was the genius of Gauss, building on the work of Kaestner, to recognize that it was the system, not the magnitudes, which were impossible, which he demonstrated from his very earliest work. (fn.5)

Gauss’ concept of complex numbers has been treated extensively in previous pedagogicals, and will be the subject of the coming installments in this series. But, the standpoint from which he approached it is expressed in his “Second Treatise on Bi-Quadratic Residues”

“Thus we reserve for ourselves a more detailed treatment of these subjects for another opportunity. The difficulty, one has believed, that surrounds the theory of imaginary magnitudes, is based in large part to that not so appropriate designation (it has even had the discordant name impossible magnitude imposed on it). Had one started from the idea to present a manifold of two dimensions (which presents the conception of space with greater clarity), the positive magnitudes would have been called direct, the negative inverse, and the imaginary lateral, so there would be simplicity instead of confusion, clarity instead of darkness….

“It is this and nothing other, that for the true establishment of a theory of bi-quadratic residues, the field of higher arithmetic, that otherwise extends only to the real numbers, will be enlarged also to the imaginary, and these must be granted complete and equal civil rights, with the real. As soon as one considers this, these theories appear in an entirely new light, and the results attain a highly surprising simplicity.”

These concepts, however, are not limited to matters of arithmetic, as Gauss expressed in his 1811 letter to his friend Hansen:

“These investigations lead deeply into many others, I would even say, into the Metaphysics of the theory of space, and it is only with great difficulty can I tear myself away from the results that spring from it, as, for example, the true metaphysics of negative and complex numbers. The true sense of the square root of -1 stands before my mind (Seele) fully alive, but it becomes very difficult to put it in words; I am always only able to give a vague image that floats in the air.”

And, that’s where the fun begins.

NOTES

1. Cusa’s investigation and its successors by Huygens and Leibniz as reviewed by Bob Robinson in his pedagogical series a1096rar001; a1106rar001; a1146rar001; a1176rar001)

2. Here I include both rational and irrational (algebraic) magnitudes under rectilinear.

3. Euclid had given a general method for construction of such a magnitude. Draw a semi-circle and its diameter AB. Connect A and B to any point C on the circumference, forming right triangle ABC, with right angle ACB. Drop a perpendicular from C to AB, whose intersection call D. The length of line CD is the square root of the length of line DB. The reader can prove this using the Pythagorean theorem.

4. Kaestner here is pointing to a pun of Cervantes, as the Spanish word algebraist also meant “bone-mender”

5. See Gauss’ work on the division of the circle, Riemann for Anti-Dummies Parts 11 &12. Also see Gauss’ 1799 doctoral thesis, “New Proof of the Fundamental Theorem of Algebra”, in which he explicitly demolishes the mathematics of Euler, Lagarange and D’Alembert, which considered complex numbers to be “impossible”.

Science is not Concensus

Between 1818 and 1832, Carl Gauss undertook the primary responsibility for making a geodetic survey of the Kingdom of Hannover. The task was exemplary of a great project. Its execution presented major technological difficulties that could only be overcome by developing new technologies based on new scientific principles, and its success would lay the basis for a transformation, through economic development, of the physical universe. But, perhaps even more important to Gauss, was that it provided an opportunity for him to, once again, do science, that is, prove that what everyone was thinking about the universe was wrong.

In the course of his survey, Gauss conducted the following crucial experiment: He measured the angular height of the pole star over the horizon from his observatory in Goettingen. His collaborator, Schumacher, measured the angular height of the pole star from his observatory in Altona, which was on the same meridian as Goettingen. From the difference of these angular measurements, Gauss calculated the distance along the surface of the Earth between the two observatories.

Then, with great effort, Gauss created a triangular grid over the entire Kingdom of Hannover. From these triangles, he made a second calculation of the distance along the surface of the Earth between the two observatories. The difference between the two calculations was 16″ of an arc. A small error, by Baby Boomer standards, especially when compared to the Financial Times report yesterday that the “consensus” among financial experts concerning the prospects for tech stocks in 2001 was revised from 37% up to 30% down, without blushing. (That’s why Baby Boomers like consensus. Everyone agrees to change their opinion together, so no one will be embarrassed when the consensus is proven wrong.)

However, for Gauss, this 16″ of an arc discrepancy, like Kepler’s 8′ of an arc, was the opportunity to demonstrate that the way people were thinking was wrong. Not, that people were thinking SOMETHING wrong, but that the WAY they were thinking was wrong. (It is a characteristic of genius to be able to recognize when such small discrepancies are matters of principle, and not simply errors.)

What Gauss proved by this 16″ of arc, as did Kepler, Fermat, and Leibniz previously, was that the mind must be free from any a priori, or “ivory tower” set of assumptions, such as those axioms, postulates and definitions of Euclidean geometry. Not, simply free from the particular axioms, postulates and definitions of Euclid, but free from any a priori set of assumptions.

Look back over the above described experiment, and discover the assumptions. First, the angular height of the pole star, is measured from the horizon. But, what is the horizon? It is not mathematically determined, but, it is physically determined as the perpendicular to the pull of gravity, as measured by plane leveller or plumb bob. A horizon defined in this way, will be tangent to the surface of the Earth at the point of measurement. Thus, the direction of the pull of gravity, is itself a function of the shape of the Earth. To calculate the distance along the surface of the Earth, from this angular change, one has to make an assumption about what is the shape of the Earth; i.e. if it’s a sphere, the measurement is along a circle, if it’s an ellipsoid, the measurement is along an ellipse. (Draw a circle and an ellipse. Draw tangents at different places. Draw perpendiculars to the points of tangency. What direction do these perpendiculars point? On a circle they all point to the center. On an ellipse they don’t.)

Similarly, calculations from the triangular measurements, depend on what shape the triangles lie. (Compare triangles drawn on a sphere, an ellipsoid, or some irregular shape, like a watermellon.)

So, could some shape be found, on which the two different methods of measurement would agree?

Gauss rejected such “curve fitting” methods of thinking and made the revolutionary discovery that the shape of the Earth is that shape which is everywhere perpendicular to the pull of gravity, today called the “Geoid”. The Geoid is not a geometrical shape, but rather a physically determined one. And, such a shape is not only non-uniform, but it is irregularly non-uniform, even changing over time.

Such an irregular, non-uniform surface, was, as Kepler’s orbits, or the Fermat’s path of least-time, or Leibniz’ and Bernoulli’s catenary, physically demonstrable, but unrecognizable by the generally accepted mathematics of the day. Having already been provoked by his teacher, Abraham Kaestner, Gauss had long before ceased to let mathematics dictate his thinking. Rather, he, like Leibniz invented a new mathematics. This extension of Leibniz’ calculus did not rely on a priori assumptions about shape, but was a mathematics of transformations. Just as Leibniz’ calculus made position along a curve a function of change, Gauss, and later Riemann, made shape a function of transformations, and curves a function of shape. The change that determines position along a curve, is itself determined by the transformation that generated the surface. This new mathematics required a new type of number; complex numbers. (The next several installments will work through these concepts in more detail.)

Gauss’ method of inventing mathematics is rooted in Cusa’s (of whom Kaestner placed great importance in his “History of Mathematics”). In “De Ludo Golbi”, Cusa writes:

“And the soul invents branches of learning e.g. arithmetic, geometry, music, and astronomy and it experiences that they are enfolded in its power; for they are invented, and unfolded, by men….For only in the rational soul and in its power are the mathematical branches of learning enfolded; and only by its power are they unfolded. [This fact is true] to such an extent that if the rational soul were not to exist, then those branches of learning could not at all exist….”

“…the soul’s reason, i.e. its distinguishing power, is present in number, which is from our mind, and in order that you may better know that that distinguishing power is said to be composed of the same and different, and of one thing and another thing just as is number, because number is number by virtue of our mind’s distinguishing. And the mind’s numbering is its replicating and repeating the common one, i.e. is its discerning the one in the many and the many in the one and its distinguishing one thing from another. Pythagoras, noting that no knowledge of anything can be had except through distinguishing, philosophized by means of number, I do not think that anyone else has attained a more reasonable mode of philosophizing. Because Plato imitated this mode, he is rightly held to be great.”

What’s in a Moment?

We are now at the point in this series, where we can begin to dig directly into that rich vein of knowledge revealed by Bernhard Riemann’ s development of complex functions. However, it is necessary, before embarking on that leg of this journey, that you first contemplate this short, but important, pedagogical exercise. Its relevance will become increasingly apparent to you.

Take the case of a Keplerian orbit. At each moment the planet is changing its speed and trajectory. That change is being guided by the underlying hyper-geometry of the solar system, which has determined the shape of the orbit. That hyper-geometry, thus, requires the planet to change its speed and trajectory, at each moment, according to a principle, derived from the characteristics of the hyper-geometry. Kepler showed that the principle governing the change in speed and trajectory was expressed by the way equal portions of the planet’s period corresponded to equal areas swept out.

Now, here’s the paradox. In any interval of the orbit, no matter how small, the planet is doing something different at the beginning of that interval than at the end, with the exception of the maximum and minimum intervals. The maximum interval is the entire orbit. There, (at least in first approximation), at the beginning and end of this interval, the planet is doing the same thing. (For example, if we consider this interval to be from, say, perihelion to perihelion.) The minimum interval is the moment of change. In that moment, the planet ceases to do what it just did, and starts becoming what it will be. Paradoxically, the beginning and end of each moment, like the entire orbit, are also equal. However, the type of change the planet is undergoing at that moment is determined by the entire orbit. Thus, the maximum interval and the minimum interval coincide.

From the standpoint of Leibniz’ calculus, the integral is the maximum as seen from the minimum, while the differential, is the minimum as seen from the maximum.

This concept was expressed by Nicholas of Cusa in “On Learned Ignorance” Book II:

“In like manner, if you consider the matter carefully: rest is oneness which enfolds motion, and motion is rest ordered serially. Hence, motion is the unfolding of rest. In like manner, the present, or the now, enfolds time. The past was the present, and the future will become the present. Therefore, nothing except an ordered present is found in time. Hence, the past and the future are the unfolding of the present. The present is the enfolding of all present times; and the present times are the unfolding, serially, of the present; and in the present times only the present it found. Therefore, the present is one enfolding of all times. Indeed the present is oneness. In like manner, identity is the enfolding of difference; equality [the enfolding] of inequality; and simplicity [the enfolding] of divisions, or distinctions.

“Therefore, there is one enfolding of all things. The enfolding of substance, the enfolding of quality or of quantity, and so on, are not distinct enfoldings. For there is not only one Maximum, with which the Minimum coincides and in which enfolded difference is not opposed to enfolding identity. Just as oneness precedes otherness, so also a point, which is a perfection, [precedes] magnitude. For what is perfect precedes whatever is imperfect. Thus, rest precedes motion, identity precedes difference, equality [precedes] inequality, and so on regarding the other perfections. These are convertible with Oneness, which is Eternity itself (for there cannot be plurality of eternal things). Therefore, God is the enfolding of all things in that all things are in Him; and He is unfolding of all things in that He is in all things.”

However, not all “moments” are the same. In the case of a planetary orbit, while at all moments the planet’s speed and trajectory are changing, there are two unique moments, in which that change is a complete transformation, specifically, aphelion and perihelion. In the former, the planet’s action changes from slowing down to speeding up, while in the latter, the action changes from speeding up to slowing down. These two moments are called singular moments, or singularities. The change at all other moments of the planet’s action, is thus determined by these two singularities, aphelion and perihelion.

As Kepler showed, these singularities are determined by a higher principle of the hyper- geometry underlying the solar system. In other words, that two singularities exist, is a characteristic of eccentric orbits; that the orbit has this specific relationship to these singularities, is a characteristic of the “more basic principle” , i.e. the harmonic principle, that governs the whole solar system.

As we work through Riemann’s discoveries in future pedagogicals, we will present more examples of this same principle. For now, think back over this one, so you get used to this way of thinking.

The Solar System’s Harmonic Twist

Significant insight can be obtained, for those wishing to master the art of changing one’s own axioms, by re-living Kepler’s transformation of his own thinking, from his initial hypothesis connecting the planetary orbits to the five Platonic solids, to the supersession of that hypothesis, under his concept of “World Harmony.”

As presented in last week’s installment, Kepler sought the reason underlying the ordering of the solar system by investigating, “why things were such and not otherwise: (namely) the number, size, and the motion of the circles.” The anomaly between Mars and Jupiter, initially investigated in terms of the relationship of the distances between the planetary orbits, provoked Kepler to reject his initial attempts to answer this question using simple numerical values, or, geometrical relationships among plane figures. After much work, Kepler found that the underlying reason for the number and size of the planetary orbits corresponded to the ordering of five Platonic solids.

(An example of the this anomaly between Mars and Jupiter was recently pointed out by Jonathan Tennenbaum. If one extrapolates from the relationship of the distances between the orbits of Mars and Earth outward, one calculates two planetary orbits between Mars and Jupiter. On the other hand, extrapolating from the relationship of the distance between Jupiter and Saturn inward, only one planetary orbit between Mars and Jupiter is obtained.)

As to the motions of the planets, Kepler later discovered that his initial concept had to be superseded by a “more basic principle.” Specifically, the non-uniform motion, i.e., eccentric, motion of the planets showed that the planetary orbits were not fixed circles, such as those found on spheres circumscribing and inscribing Platonic solids. Rather, the orbits were regions in which the planets moved non-uniformly, getting closer and farther from the Sun as they moved around it. Thus, the solids were not sufficient to account for these eccentricities.

But, what principle was determining these eccentricities? This presented a far different problem than determining merely the distances between the planetary orbits. Specifically, what is the appropriate measure of a non-uniform orbit? Kepler’s equal area principle is such an appropriate measure within an orbit. But, what is an appropriate measure for determining intervals among eccentric orbits? Kepler’s so-called third principle, (that the mean distance from the Sun equals the 3/2 power of the periodic time), is a first approximation of such a measure, but it doesn’t express, “why these eccentricities and not others?”

Kepler recognized that the eccentricity of a planets’ orbit is uniquely determined by its singularities, specifically its fastest speed at perihelion and its slowest speed a aphelion. These extreme motions reflect the intention of the planet’s action in the intervening intervals. In other words, how much the planet speeds up, at each moment, from aphelion to perihelion, is a function of what it is to become at perihelion, and conversely, in the interval from perihelion back to aphelion. Thus, the principle determining these extremes, in turn determines the characteristic eccentricity of the whole orbit, which in turn, determines the distances between the planetary orbits.

As Kepler stated, “It was good, that for the formation of the distances the solid figures should give way to the harmonic relationships, and the greater harmonies between pairs of planets, to the universal harmonies of all, so far as this was necessary.”

Since Kepler is almost unique among scientific discovers in presenting to us not only his discovery, but also the change in his own thinking which brought him to it, we quote at length from the concluding “envoi” of his “Harmonies of the World”:

“For where there is a choice between different things which do not allow each other to have sole possession, in that case the higher are to be preferred, and the lower must give way, as far as is necessary, which the very word “cosmos,” which means “decoration,” seems to argue. But harmonic decoration is as far above the simple geometrical as life is above the body, or form above matter.

“For just as life completes the bodies of animate beings, because they were born to lead it, which follows from the archetype of the world, which is the actual divine essence, so motion measures out the regions allotted to the planets, to each its own, as a region has been assigned to a star so that it could move. But the five solid figures, in virtue of the word itself, relate to the spaces of the regions, and to the number of them and of the bodies; but the harmonies to the motions. Again, as matter is diffuse and unlimited in itself, but form is limited, unified, and itself the boundary of matter; so also the number of the geometrical proportions is infinite, the harmonies are few….Therefore, as matter strives for form, as a rough stone of the correct size indeed, strives for the Idea of the human form, so the geometrical proportions in the figures strive for harmonies; not so as to build and shape them, but because this matter fits more neatly to this form, this size of rock to this effigy, and also this proportion in a figure to this harmony, and therefore so that they may be built and shaped further, the matter in fact by its own form, the rock by the chisel into the appearance of an animate being, but the proportion of the spheres of the figures by its own, that is, by close and fitting harmony.

“What has been said up to this point will be made clearer by the story of my discoveries. When, twenty four years ago I had engaged in this study, I first enquired whether the individual circles of the planets were separated by equal distances from each other (for in Copernicus the spheres, are separated, and do not mutually touch each other),. Of course, I acknowledged nothing as more splendid than the relationship of equality. However, it lacks a head and a tail, for this material equality provided no definite number for the moving bodies, no definite size for the distances. Therefore, I thought about the similarity of the distances to the spheres, that is about their proportion. But the same complaint followed, for although in fact distances between the spheres emerged which were certainly unequal, yet they were not unequally unequal, as Copernicus would have it, nor was the size of proportion nor the number of the spheres obtained. I moved on to the regular plane figures; they produced the distances in accordance with the ascription of their circles, but still in no definite number. I came to the five solids; in this case they revealed both the number of the bodies and nearly the right size for the intervals so much so that I appealed over the remaining discrepancy to the state of accuracy of astronomy. The accuracy of astronomy has been perfected in the course of twenty years;’ and see! There was still a discrepancy between the distances and the solid figures, and the reasons for the very unequal distribution of the eccentricities among the planets were not yet apparent. Of course in this house of the cosmos I was looking for nothing but the stones of more elegant form, but of a form appropriate to stones not knowing that the Architect had shaped them into a fully detailed effigy of a living body. So little by little, especially in these last three years, I came to the harmonies, deserting the solid figures over fine details, both because the former were based on the parts of the form which the ultimate hand had impressed, but the figures from matter, which in the cosmos is the number of the bodies and the bare breadth of their spaces, and also because the former yielded the eccentricities, which the latter did not even promise. That is to say the former provided the nose and eyes and other limbs of the statue, for which these latter had only prescribed the external quantity of bare mass.

“Hence just as the bodies of animate beings have not been made, and a mass of stone is not usually made, according to the pure norm of some geometrical figure, but something is removed from the external round shape, however elegant (though the correct amount of bulk remains) so that the body can take on the organs necessary to life, and the stone the likeness of an animate being, similarly also the proportion which the solid figures were to prescribe for the planetary spheres, as lower, and having regard only in a body of a particular size and to matter, must have given way to the harmonies, as much as was necessary for the former as to be able to stand close and to adorn the motions of the globes.”

A Still More Basic Principle

So, the geometrical proportions of the solids give way to the more basic principle of the harmonic principles. What principle, therefore, determines the harmonies?

Kepler himself stated that the harmonic proportions are determined by the ear, not numerical values. To what does the ear turn? To the universal principles of Classical artistic composition, as exemplified by J.S. Bach’s well-tempered polyphonic compositions. It is in the domain of these compositions (Ideas) from which the values of the well-tempered intervals are derived, which, in turn, determine the harmonic proportions from which the planetary orbits derive their eccentricities.

With Piazzi’s discovery of the asteroid Ceres, Gauss’ subsequent determination of its orbit, and the follow up discoveries of the asteroids Pallas, Juno and Vesta, Kepler’s principles were confirmed anew. The motion of each asteroid conformed to Keplerian principles, moving in elliptical paths with equal areas measuring equal portions of their periods, and, their mean distances from the Sun equaling the 3/2 power of their periodic times.

But, there was a twist. It now became possible to measure, in these orbits, cyclical changes in the eccentricities, that were occurring, but were hitherto beyond measurement, in all the planetary orbits. Furthermore, unlike the orbits of the major planets, which enclosed one another, the asteroid orbits intertwined. For example, at perihelion Pallas was closer to the Sun than Ceres, but at aphelion, Ceres was closer to the Sun than Pallas. This intertwining suggests the asteroids’ orbits are both many individual Keplerian orbits, and one whole Keplerian orbit at the same time. What then, is the still more basic principle that governs the solar system which contains this new type of orbit represented by the asteroids?

The initial work on this was done by Gauss, whose investigation into the changing eccentricities and the intertwinings provoked his creation of new mathematical metaphors, which, like Leibniz’ calculus, had applications far beyond the original paradoxes that gave rise to them. The changing eccentricities provoked Gauss to conceive of the orbits as elliptical rings in which the mass of the planet, or asteroid, was distributed in the ring according to Kepler’s equal area principle. (For a more complete treatment of Gauss’ concept, see the pedagogical series, “Dance With the Planets” 98406bmd002; 98416bmd001; 98426bmd001 )

Gauss also considered the implications of the intertwining of the asteroid orbits, for the geometrical characteristics of the solar system as a whole. Gauss took this up in a preliminary way in an 1804 paper, “On the Determination of the Geocentric Positions of the Planets.” Here Gauss considered the inverse of the problem he confronted in the determination of the Ceres orbit. In that case, Gauss had a few geocentric positions of Ceres, from which he had to determine its heliocentric orbit. Now he considered the inversion. What characteristics of a heliocentric orbit govern the geometry of its geocentric positions. For this, he explicitly turned to Leibniz’ and Carnot’s “Geometry of Position.”

Each planet or asteroid makes a circuit through the zodiac. But, since the Earth is also moving, the zodiac changes its position with respect to an observer on the Earth. Consequently, the locus of all geocentric positions of a planet or asteroid form a zone on the celestial sphere, that Gauss called its “zodiacus.” The determination of the boundaries of that zone required the construction of function that mapped the changes of the heliocentric positions of the planet and the Earth, onto the celestial sphere. While Gauss was able to calculate specific values for this function, more importantly, he investigated its general characteristics. He showed that the nature of that zodiacus depends on the relationship of the planet’s orbit to the Earth’s. Either the planets’ orbit is completely inside the Earth’s, completely outside, or, it overlaps. Gauss showed that the first two situations determined a zodiacus with definite boundaries, but, in the third case those boundaries were indeterminable. He noted, ironically, that, the implications of this paradox had, until then, been avoided, because none of the known planets or asteroids, had ever appeared in strange places, such as near the poles of the ecliptic! Nevertheless, Gauss was pointing out a crucial principle on which Riemann would later rely. Specifically, that orbits that completely enclose one another defined completely different geometrical characteristics than those that overlapped.

(If you want to have some fun, take two rings, one bigger than the other. Study the relationship between positions on the two rings when the smaller ring is inside the larger. Now, compare these relationships with two rings that are interlinked. I leave it to the reader to discover the difference on your own.)

Finally, think of the implications of these intertwining asteroid orbits for Kepler’s harmonic proportions. Preliminary calculations performed by this author for 10 asteroids show that when the extreme speeds of each asteroid are individually compared with the extreme speeds of Jupiter and Mars, similar harmonic proportions to those Kepler found for the major planets occur. The diverging and converging intervals each asteroid makes with Jupiter correspond to intervals Kepler would consider consonant. With Mars, the diverging intervals are consonant, while the converging intervals correspond to the deisis that Kepler found between Jupiter and Mars.

The twist comes up in forming intervals among the asteroids themselves. Since their orbits overlap the very meaning of converging and diverging intervals is different than in the intervals between the major planets. For example, when is Ceres converging towards Pallas or diverging away from it? When both are moving away from the Sun, Ceres is getting closer to Pallas, while, when both are moving closer to the Sun, they become divergent. Unlike the major planets, however, the point of divergence and convergence does not occur at the extreme positions. And since the eccentricities of the asteroids’s orbits are changing, where these orbits cross over from diverging to converging is itself changing. Now, think of the connectivity involved when thinking of this relationship among many asteroids, not just two, as in this example!

This braided, overlapping characteristic is not limited to orbits within the range of the asteroid belt. In fact, the solar system is filled with orbits that similarly overlap, including asteroids whose orbits overlap the Earth’s. Such overlapping orbits, suggests a new set of harmonic relationships, akin to the transformation of Bach’s well-tempered polyphony by Beethoven in his late quartets.

The Dissonance that Smiled

By all accounts, Descartes, Newton, Euler, and Kant all shared one common trait: they were grouchy old farts. As such, these poor souls fled from the dissonance and tension by which the universe presents its development to the mind of man. Like their Venetian brethren, who only desired forms of music devoid of Lydian intervals, these minds would not conceive that God would present to them a challenge, by which their own cognitive capacity would be improved. Never could they know that mixture of woefulness and joyfulness that Schiller associates with the sublime. Yet, there is no need to distinguish whether they were grouchy because they hated dissonance, or whether that hatred came because they were grouchy. The lesson to be learned is the same: Grouchy old farts can’t know the minds of Kepler, Leibniz, Kaestner, Gauss, and Riemann, and people who can’t comprehend these great thinkers, become grouchy old farts.

It is in this spirit, that we turn our pedagogical attention to that dissonance in the solar system that today we recognize as the asteroid belt, and the corresponding cognitive transformations that its discovery and investigation produced.

That our solar system would contain an orbit with the characteristics of the asteroid belt, was already affecting human cognition, even before the first asteroid presented itself to human eyes. In his earliest work, the Mysterium Cosmographicum, Kepler had already noticed an anomaly in the organization of the planets in the solar system, with respect to the distances of the known planets from the Sun. While Kepler found that the orbits of the planets, in first approximation, were consistent with the five regular Platonic solids, this ordering produced an anomaly between Mars and Jupiter. This anomaly had impinged on Kepler’s thinking, even before he discovered his polyhedral hypothesis, and, according to his own description, provoked him to arrive at that discovery.

In trying to determine, “why things were such and not otherwise: [namely] the number, size, and the motion of the circles [of the planets],” Kepler first looked for some ratio of numbers that corresponded to the observed distances between the planetary orbits. When this failed, “I tried an approach of remarkable boldness. Between Jupiter and Mars, I placed a new planet and also another between Venus and Mercury, which were to be invisible perhaps on account of their tiny size, and I assigned periodic times to them…. Yet the interposition of a single planet was not sufficient for the huge gap between Jupiter and Mars.” Failing to find a numerical ratio that corresponded to the distances, Kepler tried to find a sequence of inscribed and circumscribed polygons that would correspond to the observed distances. This, too, failed, in the interval between Jupiter and Mars, provoking him to discover the correspondence between the size, number, and motion of the planetary orbits, with the five Platonic solids.

The anomaly between Jupiter and Mars was still indicated under the polyhedral hypothesis, by the placement of the tetrahedron in this gap, as the tetrahedron is the one solid which is its own dual.

This anomaly provoked Kepler to further investigate, and, upon closer examination of the orbit of Mars, he discovered the non-uniform nature of the planets’ orbits. Now, he had a further dissonance. Circumscribing and inscribing spheres around the Platonic solids gave the distances between the circles of the orbits, but the orbits were not circular. They were eccentric. The question posed by this dissonance was: what governed the eccentricities, or, in other words, why was each planet’s eccentricity “this way and not other”?

As he wrote in the Harmonies of the World, “As far as the proportion of the planetary orbits is concerned, between pairs of neighboring orbits, indeed it is always such as to make it readily apparent that in each case, the proportion is close to the unique proportion of the spheres of one of the solid figures; that is to say the proportion of the circumscribed sphere of the figures to the inscribed sphere. However, it is not definitely equal, as I once dared to promise for eventually perfected astronomy….

“From that fact it is evident that the actual proportions of the planetary distances from the sun have not been taken from the regular figures alone; for the Creator, the actual fount of geometry who, as Plato wrote, practices eternal geometry, does not stray from his own archetype. And that could certainly be inferred from the very fact that all the planets change their intervals over definite periods of time, in such a way that each one of them has two distinctive distances from the Sun, its greatest and its least; and comparison of distances from the Sun between pairs of planets is possible in four ways, either of greatest distances or of the least or of the distances on opposite sides when they are furthest from each other, or when they are closest. Thus, the comparisons between pair and pair of neighboring planets are twenty in number, whereas on the other hand there are only five solid figures. However, it is fitting that the Creator, if He paid attention to the proportion of the orbits in general, also paid attention to the proportion between the varying distances of the individual orbits in particular, and that that attention should be the same in each case, and that one should be linked with another. On careful consideration, we shall plainly reach the following conclusion, that for establishing both the diameters and the eccentricities of the orbits in conjunction more basic principles are needed in addition to the five regular solids.”

The “more basic principles” that Kepler discovered concerned the harmonic relationship among the extreme speeds of neighboring planets. The planet’s speed at any moment is a function of its distance from the Sun at that moment; the slowest speed of the planet is at its maximum distance from the Sun (aphelion) and its fastest speed is at its minimum distance from the Sun (perihelion). These extremes are themselves a reflection of the planet’s eccentricity. The solar system chose, so to speak, those eccentricities for the planets that produced the speeds, according to a “more basic principle”. That principle was reflected in the harmonic relationships among those speeds.

Kepler measured those speeds by the arc the planet traversed at perihelion and aphelion, as seen from the Sun, during one Earth day. The results were:

  
Saturn  at aphelion-  1'30"; at perihelion-  2'15";
Jupiter at aphelion-  4'30"; at perihelion-  5'30";
Mars    at aphelion- 26'14"; at perihelion- 38' 1";
Earth   at aphelion- 57' 3"; at perihelion- 61'18";
Venus   at aphelion- 94'50"; at perihelion- 97'37";
Mercury at aphelion-147' 1"; at perihelion-384' 0";

When these speeds are compared between neighboring planets, their ratios correspond to harmonic musical intervals. Each pair of planets makes two intervals: a converging interval between the perihelion speed of the outer planet with the aphelion speed of the inner one; a diverging interval between the aphelion speed of the outer planet with the perihelion speed of the inner one. These intervals according to Kepler are:

  
Saturn - Jupiter diverging 1/3;  converging 1/2;
Jupiter- Mars    diverging 1/8;  converging 5/24;
Mars   - Earth   diverging 5/12; converging 2/3;
Earth  - Venus   diverging 3/5;  converging 5/8;
Venus  - Mercury diverging 1/4;  converging 3/5;

These intervals correspond to those Kepler derived for musical intervals with one exception. The converging interval between Jupiter and Mars, deviated from Kepler’s musical intervals by a diesis (otherwise called a quarter-tone). The diesis, Kepler said, is “the smallest interval, by which the human voice in figured melody is almost perpetually out of tune. However, in the single case of Jupiter and Mars, the discrepancy is between a diesis and a semitone. It is therefore evident that this mutual concession on all sides hold exceedingly good.”

That small dissonance would later reveal itself to be a reflection of a further “more basic principle”, whose expression Gauss and Riemann would later provide.

The Finer Art of Science

How often have you heard, after briefing someone on the strategic situation and LaRouche’s unique role in leading mankind out of this crisis, the retort, “I just don’t think one man can have the answer.” Such a response, not only indicates a narrow, petty, and small minded way of thinking, but, it actually displays a gross illiteracy concerning the history of ideas. In fact, all scientific (and artistic) discoveries were made by one person, who, when that discovery was made, was the only person in the universe who, “had the answer”. In truth, the human race has existed to date, because, one person, “had the answer”, when no one else did.

Such genius is characterized by the ability, and the willingness, to find principles, in small deviations from the expected, where everyone else finds either no deviations, or excuses such deviations as mere errors. For example, Kepler’s discovery that the planetary orbits were elliptical, was provoked by a deviation of 8 minutes of an arc, between the observations, and the results he expected from his hypothesis that the planets moved on eccentric circles. Or, Gauss’ determination that the shape of the Earth was both non-uniform and irregular, was provoked by the 16 seconds of an arc deviation, between the measurements of the height of the pole star and the measurements of his geodetic triangulation from Goettingen to Altona. Or, LaRouche’s determination of the trajectory of the economy, from the small changes in the mental activity of the population.

In each example, as in all scientific discoveries, it was these small deviations from the “normal” from which new revolutionary concepts were derived.

This process was described by B. Riemann in some philosophical fragments, published for the first time in English in the Winter 1995-1996 issue of Twenty First Century Science and Technology:

“Natural science is the attempt to understand nature by means of exact concepts.

“According to the concepts through which we comprehend nature our perceptions are supplemented and filled in, not simply at each moment, but also future perceptions are seen as necessary. Or, to the degree that the conceptual system is not fully sufficient, future perceptions are determined beforehand as probable; according to the concepts, what is “possible” is determined ( thus what is “necessary” and conversely, impossible). And the degree of possibility (of “probability”) of each individual even which is seen as possible, in light of these concepts, can be mathematically determined, if the concepts are precise enough.

“To the extent that what is necessary or probable, according to these concepts, takes place, then this confirms the concepts, and the trust that we place in these concepts rests on this confirmation through experience. But, if something takes place that is unexpected according our existing assumptions, i.e. that is impossible or improbable according to them, then the task arises of completing them or, if necessary reworking the axioms, so that what is perceived ceases to be impossible or, improbable. The completion or improvement of the conceptual system forms the “explanation” of the unexpected perception. Our comprehension of nature gradually becomes more and more complete and correct through this process, simultaneously penetrating more and more behind the surface of appearances.

“The history of causal natural science, in so far as we can trace it back, show that this is, in fact, the way our knowledge of nature advances. The conceptual systems that are now the basis for the natural sciences, arose through a gradual transformation of older conceptual systems, and the reasons that drove us to new modes of explanation can always be traced back to contradictions and improbabilities that emerged from the older modes of explanation.

“The formation of new concepts, in so far as this process is accessible to observation, therefore takes place in this way.”

In that same fragment, Riemann goes on to say:

“1. When is our comprehension of the world true?

“`When the relations among our conceptions correspond to the relations of things.’

“The elements of our picture of the world are completely distinct from the corresponding elements of the reality which they picture. They are something within us; the elements of reality are something outside of ourselves. But, the connections among the elements in the picture, and among the elements of reality which they depict, must agree, if the picture is to be true.’

“The truth of the picture is independent of its degree of fineness; it does not depend upon whether the elements of the picture represent larger or smaller aggregates of reality. But, the connections must correspond to one another; a direct action of two elements upon each other may not be assumed in the picture, where only an indirect one occurs in reality. Otherwise the picture would be false and would need correction. If, however, an element of the picture is replaced by a group of finer elements, so that its properties emerge, partly from the simpler properties of the finer elements, but partly from their connections, and thus become in part comprehensible, then this increases our insight into the connection of things, but without the earlier understanding having to be declared false.”

Consequently, the healthy mind seeks the ever finer elements that reveal those, yet undiscovered, cycles governing action in the universe. Such cycles had been there all along, but, once discovered, the universe, in which they were acting, changes, by virtue of their now becoming an object of human cognition. This, in turn, enables man to begin a new quest for even finer elements, a search whose possibility depends on the just discovered new cycles. It is the intention, at this point in this series, “Riemann for Anti-Dummies”, to investigate those finer discoveries, on which the new concepts of Gauss and Riemann are based. Significantly, the deeper implications of those concepts were not fully recognized, even by Gauss and Riemann, until LaRouche’s discoveries in physical economy.

The finer elements that gave rise to these new concepts of Gauss and Riemann were centered in the investigations of the inter-related areas of astrophysics, geodesy, electromagnetism and life. The most efficient path to grasp the Gauss/Riemann theory of functions is through a pedagogical presentation of them, which defines the intended trajectory of this series.

1. Astrophysical– Kepler had left open for discovery, a planetary orbit between Mars and Jupiter based on a dissonance between the angular speeds between those two planets, which dissonance, Kepler notes, was evidenced by the smallest deviation perceptible. Gauss’ determination of the orbit of Ceres, and the subsequent discoveries of other asteroids, confirmed Kepler’s hypothesis. The orbits of these finer elements were consistent with the principles Kepler had discovered for the six visible planets. The eccentricities, inclinations, and interweavings, of these orbits made hitherto unobserved, but suspected, orbital irregularities, measurable. Gauss’ investigation showed that these irregularities, were, in fact, not irregularities, but evidence of finer cycles that permeated the whole solar system. This extended colligation of cycles gave rise to a new concept of manifold in the minds of Gauss and Riemann.

2. Geodesic– Earlier measurements of the Earth had shown that its shape was non- uniform (ellipsoidal) rather than uniform (spherical). Gauss spent nearly 20 years making and supervising careful physical measurements of the Earth’s gravitational and magnetic characteristic, and relating those measurements to astronomical ones. Gauss’ meticulous effort revealed that these characteristics deviated slightly from the concept of simple non-uniformity, as, for example, in an ellipsoid, and he showed the error of assuming any shape a priori. Instead, Gauss developed the idea of the shape of the Earth as a non-uniform and irregular manifold of measurement, today called the “Geoid”. To measure this concept, Gauss, and later Riemann, extended Leibniz’ calculus from concerning the characteristic of action along pathways, to the characteristic of action in the surfaces on which those pathways exist.

3. Electromagnetism and Light– The work of Ampere and Fresnel posed the paradox that the assumed characteristics of action in space in the macroscopic realm became discontinuous in the microphysical realm. Such paradoxes led to Gauss’ and Riemann’s development of retarded potential, and Riemann’s concepts of complex functions.

4. Life– The functional relationship between living and non-living processes were investigated by Riemann, notably in his researches into the characteristics of a sound wave, in the human ear, and in the air.

But, it would be wrong to leave the suggestion that a thinking mind would be content with existing concepts, until knocked on the head by some physical deviation, instead of actively seeking out such paradoxes. Think of this process as a type of a well-composed fugue, in which the theme and counter-theme become indistinguishable as to cause, and only the whole composition remains in the mind as a One. For example, the discoveries of Kepler, Fermat and Leibniz had already provoked Abraham Kaestner to knock down the remaining pillars of ivory tower mathematics with his attacks on the a priori acceptance of Euclidean geometry. Once that was initiated, Kaestner shifted life’s trajectory of the young Gauss, by (as discussed in the last two installments) provoking that young man to draw out the deeper implications of his first discovery, the constructability of the 17-gon. That small shift, which Kaestner induced into Gauss’ mind, contained the insights that emerged, years later, in the investigations of the physical paradoxes just described.

That’s where this series is headed.

Gauss’ Division of the Circle

The pursuit of a discovery of a universal principle always requires the pursuer to follow the Socratic method of negation, or, as Cusa called it, “Learned Ignorance”. This is the method by which Kepler ascended from the tangle of observed motions of the planets on the inside of an imaginary sphere, to the “hypergeometric” function, whose characteristics we’ve been exploring throughout this series. While the cause of these observations is determined from the top down by that function, like the shadows on Plato’s Cave, we cannot know that function directly. Rather, we must look between the gaps in the observations, and discern the hypergeometry of which those observations are a function.

It is the nature of this method, that, as our knowledge of the underlying hypergeometry increases, new gaps appear, through which new characteristics of the hypergeometry become discernable. These new characteristics, in turn, recast the previous discoveries, in a new light. For example, the principle that light travels the shortest path, discovered by the Greeks as a characteristic of reflection, becomes a special case of the principle that light travels the path of least time, discovered by Fermat as a characteristic of refraction. In both cases, light acts according to a minimizing principle. But, in the former, that principle is with respect to a manifold of space, while the latter is with respect to a manifold of space-time. It is the manifold from which the characteristic of the path is determined, but it is the characteristics of the path, by which the manifold is discovered.

Another example may help illustrate this point. What is the meaning of the question, “When did human life begin?”? If your conception of the universe is a reductionist/Darwinian one, in which non-living processes give rise to living ones, which, in turn, give rise to cognitive processes, the answer to that question, is, a search for a mystical point in time, in which something, becomes what it’s not. (i.e., non-living processes become living, or living processes becomes cognitive.) However, from the standpoint of a multiply-connected universe, which, as LaRouche most recently wrote, is organized from the top down, cognitive, to living, to non- living, the answer to the question, as Plato, Philo, and Augustine have said, is “from the beginning.” (cognition created man) Under a conception of the universe organized from the bottom up, the trajectory of evolution is a line. (Whether that line is curved or straight, it is still a one-dimensional magnitude.) Whereas, under the top-down conception, the trajectory is of a quite different nature.

To grasp the nature of that trajectory requires the metaphors developed by Riemann, toward which this series is heading. To get there, we must first traverse the intervening steps. This is what brings us to Gauss’ division of the circle. What follows may be at times dense, as it brings together discoveries that span more than 2500 years of cognition. But roll up your sleeves and work it through.You will be well rewarded with learned ignorance.

Kepler, in the first book of the “Harmonies of the World”, showed that the divisions of the circle generate a hierarchy of types of magnitudes. Following the Greek tradition, these magnitudes are ranked according to “knowability”. Those magnitudes that are measurable directly by the diameter of the circle, have the first degree of knowability, those measurable by a part of the diameter, have the second degree, and those whose squares are measurable by the diameter, have the third. All others are incommensurable. It is these incommensurable magnitudes, on which non-uniform motion depends, as Kepler had already shown in his Mysterium Cosmographicum and The New Astronomy. Therefore, to grasp the principles of non-uniform motion, one had to grasp the principles of generation of these incommensurables.

Magnitudes of the first three degrees of knowability, are also called constructable, because they can be constructed from the circle itself, or to put it colloquially, by straight-edge and compass. (The straight-edge being the diameter and compass being the circumference of the circle.) It is in the investigation of what is constructable, that we discover a gap, an anomaly, through which we ascend to the idea, that, the seemingly uniform circle, is, in fact, not uniform at all, but a special case of non-uniform action!

Ancient Greek philosophers had fully investigated, that, while the circle is uniform in all its parts, it does not divide uniformly. The circle can be divided into two parts by folding in half. By folding in half again, and again, the circle can be divided into 2,4,8,16,etc. parts. But to divide it into three parts, that is, to inscribe a triangle in it, requires the generation of a magnitude of the second degree of knowability. (Half the diameter, is the side of a hexagon.). Once that magnitude is created, it too can be doubled, to produce a division by 6, 12, 24, etc. parts. But, to divide the circle into five parts, requires the generation of a magnitude of the third degree of knowability, specifically, the golden section. The golden section is not commensurable with the diameter or a part of the diameter, but its square is. Hence, it is constructable.

However, it had been believed for more than 2000 years, that all other divisions of the circle were non-constructable. In other words, those divisions of the circle based on prime numbers greater than five, depended on magnitudes that were beyond Kepler’s third degree of knowability. This boundary condition, suggests that something outside the circle, a higher principle, is governing. It is that principle, that Gauss sought.

To discover this principle, Gauss, in effect, inverted the problem all together. Instead of thinking of two different types of magnitudes, constructable and non-constructable, he investigated a general principle governing the generation of magnitudes, of which the constructable ones were a special case. (A suggestive example of this concept was supplied by LaRouche several years ago, when he posed that the number 5 associated with the hypotenuse of a 3-4-5 right triangle, should be considered a special type of irrational number.)

Additionally, Gauss, using a unique application of geometry of position, which, at first may seem obscure. He inverted the conception of the circle. Instead of beginning with the circle and trying to find those positions that divided it, he sought the functions that created a divided circle. Thus, the positions were produced by the division, not the division by the positions.

Gauss showed that both the above principles relied on two, seemingly unrelated, conceptions that were at the heart of Greek science, and, that had been extended by the work of Kepler, Leibniz, Bernoulli, and Fermat: the geometric generation of incommensurables, and the more shrouded principles governing the generation of prime numbers. And, spurred by Kaestner’s prodding, Gauss showed that these principles extended beyond the circle.

As Gauss described it in the opening of the final chapter of “Disquistiones Arithmeticae”, “Among the most splendid developments contributed by modern mathematicians, the theory of circular functions without doubt holds a most important place. We shall have occasion in a variety of contexts to refer to this remarkable type of quantity, and there is no part of general mathematics that does not depend on it in some fashion…. I will speak of the theory of trigonometric functions as related to arcs that are commensurable with the circumference, or of the theory of regular polygons….The reader might be surprised to find a discussion of this subject in the present work which deals with a discipline apparently so unrelated; but the treatment itself will make abundantly clear that there is an intimate connection between this subject and higher Arithmetic.

“The principles of the theory which we are going to explain actually extend much farther than we will indicate. For they can be applied not only to circular functions but just as well to other transcendental functions….”

The first principle of Greek geometry that Gauss re-worked, is described by Plato in the Meno, Theatetus and Timaeus dialogues, and concerns the generation of incommensurable magnitudes as a consequence of a change in dimension. The reader can construct a geometrical representation of this by drawing a square, then drawing the diagonal, then, drawing a new square using the diagonal for its side. As Plato demonstrated in the Meno, the second square will have twice the area of the first, but the side of the second square, will be incommensurable to the side of the first. (In Kepler’s terms, the diagonal will be the third degree of “knowability”.) If you continue this drawing, you will produce a spiral sequence of squares, whose sides are the diagonals of the previous squares, and whose areas are double, the previous squares. The Greeks called the diagonals, the “geometric mean” between the two squares.

However, something new develops if you try and replicate this process in 3 dimensions, as in the case of doubling a cube. The diagonal of the cube does not correspond to the side of a cube with double the volume. This is the famous problem, the Delian priests brought to Plato. Eratosthenes reports Plato’s famous rebuke, that the Gods had posed this problem to the Greeks, because they wanted to chide the Greeks into studying geometry in order to improve their thinking. Hippocrates of Chios had shown that the incommensurable associated with the doubling of the cube, was of a different species, than the incommensurable associated with doubling the square. In other words, a change in dimensionality, produced a different species of incommensurable. In the Timeaus, Plato reports Hippocrates discovery:

“But it is not possible that two things alone be joined without a third; for in between there must needs be some bond joining the two…. Now if the body of the All had had to come into being as a plane surface, having no depth, one mean would have sufficed to bind together both itself, and its fellow-terms; but now it is otherwise for it behoved it to be solid in shape, and what brings solids into harmony is never one mean, but always two.”

By inversion, if two or more geometric means are required to double a magnitude, the doubling of that magnitude, is an action, that originates in a dimensionality greater than two.

Gauss’ insight rested on these Platonic principles, with a crucial extension supplied by Bernoulli’s discovery of the equiangular spiral, which he called, “spiral mirabilus”. Bernoulli showed that this spiral was an exemplar of geometric growth. For example, a line extending from the center of the spiral outward, will be cut at different intervals by each spiral arm. These intervals will be the same proportion to one another, as the areas and diagonals of Plato’s squares. Similarly, lines emanating from the center of the spiral at equal angles from each other, will cut the spiral arms in geometric proportion. Thus, equal divisions of the spiral, cut the spiral into parts that are in geometric proportion.

Now, if we think of the circle as a special case of the spiral, then the division of the circle by lines (radii) emanating from the center at equal angles, cuts the circumference in arcs that are in geometric proportion. The intersection of these lines with the circumference correspond to the vertices of an inscribed polygon. Thus, to divide the circle into “n” parts, corresponds to finding “n-1” geometric means. Those divisions, that can be accomplished by finding one mean between two others, are constructable; and those requiring two or more means are not. Thus, the different divisions of the circle are actually projections, of action originating in manifolds, of higher dimensionality, than the seemingly two dimensions of the circle. Again, like the shadows on Plato’s Cave, or Learned Ignorance, we can only ascend to knowledge of those manifolds, from the anomalies embedded in their reflection.

Gauss showed that these anomalies can be discovered, if we think of the circle, not in a Euclidean/Cartesian plane, but in the complex domain. A simple example is, perhaps, the easiest way to illustrate the point. To divide a circle into 4 parts, first think of the circle in the complex domain. Pick a point on the circumference for the first vertex, and call it 1. To divide the circle into four parts we would mark off three other points that are 90 degrees from each other. According to what we said above from Bernoulli and Plato, these points are all in geometric proportion to each other. Expressed in numbers, these points form a cycle of geometric means from 1 to 1. Using the letter i to denote the square root of -1, that series is, 1, i, -1, –i. These four numbers produce a cycle, such that if you multiply each one by itself 4 times, you get 1.

Gauss’ insight was based on the following: Dividing the circle into “n” parts requires finding “n-1” geometric means between 1 and 1. Each “n” divisions requires a function with specific characteristics. Gauss discovered a general principle that governed the nature of those characteristics.

Again, it is easiest to demonstrate this by example. Put five dots on a page, roughly in a circle and number the dots counter-clockwise, 1, 2, 3, 4, 5. Now, connect dot 1-3, then 3-5, then 5- 2, then 2-4, then 4-1. In this action, you went around the circle twice, connecting 2 dots in each turn. You also drew a pentagonal star. Perform the same action, connecting 1-4; 4-2; 2-5; 5-3; 3- 1. This produces the same result as above. Try the same with 7 dots. Connect 1-3; 3-5; 5-7; 7-2; 2- 4; 4-6; 6-1. This action required 2 cycles, connecting 3 dots in each cycle (3, 5, 7) and (2, 4, 6). Notice the shape of the heptagonal star. Make another configuration of 7 dots, and connect them in the following sequence: 1-4; 4-7; 7-3; 3-6; 6-2; 2-5; 5-1. This action required 3 cycles, connecting 2 dots in each cycle. Notice that the shape of this heptagonal star is different than the previous one. If you experiment around, using different cycles, (e.g. 1-5; 5-2;, etc. ) the result will be identical to one of the two produced above. Now, continue these types of experiments with 7, 11, 13 and 17 dots. In each case, you will find that the combinations of cycles and dots, is constrained by the prime number factors of 10, 12, and 16 respectively, ( n-1).

Gauss showed that these combinations of cycles corresponds to Plato’s principle of means. Those divisions that can be resolved completely into cycles of 2, correspond to inserting 1 geometric mean between two others, as in doubling of the square, and are therefore constructable. Those prime number divisions that cannot be so resolved, correspond to inserting two or more geometric means, and, like the doubling of the cube, are not constructable. Thus, it is possible to construct figures of 2, 3, 5, 17, 257, 65,537 and any other prime number divisions of the form (2²)^{n + 1}. All other divisions cannot be constructed because they are reflections of actions of a dimensionality higher than two.

It is the thinking underlying the above discovery of Gauss, that is at the heart of Gauss’ and Riemann’s development of the theory of functions.

Transcending Euclid

It is crucial for anti-dummies to always bear in mind the groundwork for all modern science, that Nicholas of Cusa teaches us in “On Learned Ignorance”:

“Wherefore it follows that, except for God, all positable things differ. Therefore, one motion cannot be equal to another; nor can one motion be the measure of another, since, necessarily, the measure and the thing measured differ. Although these points will be of use to you regarding an infinite number of things, nevertheless if you transfer them to astronomy, you will recognize that the art of calculating lacks precision, since it presupposes that the motion of all the planets can be measured by reference to the motion of the Sun. Even the ordering of the heavens with respect to whatever kind of place or with respect to the risings and settings of the constellations or to the elevation of a pole and to things having to do with these is not precisely knowable. And since no two places agree precisely in time and setting, it is evident that judgments about the stars are, in their specificity, far from precise. If you subsequently adapt this rule to mathematics, you will see that equality is {actually} impossible with regard to geometrical figures and that no thing can precisely agree with another either in shape or in size. And although there are true rules for describing the equal of a given figure as it exists in its definition, nonetheless equality between different things is {actually} impossible. Wherefore, ascend to the recognition that truth, freed from material conditions, sees, as in a definition, the equality which we cannot at all experience in things, since in things equality is present only defectively.”

Such was the approach of Kepler, Fermat, and Leibniz, who rejected the method of imposing mathematical definitions on the physical universe, but, in doing so, discovered a new higher principle, that led to the creation of a new mathematics. Be it the motion of the planets, the path of refracted light, or the shape of a hanging chain, physical action does not conform to perfect mathematical rules. For example, the planets don’t move in perfect ellipses, nor does a ray of refracted light move in a perfect cycloid, nor does a hanging chain form a perfect catenary. It is in this deviation from the perfect, that the higher, harmonic, principles governing the action can be discovered. Furthermore, these deviations are not expressible in perfect mathematical ratios, such as, for example, the dependence of the planet’s or light’s path on the sine of the angle. These higher principles define the characteristics of what Riemann called “multiply-connected functions”. In other words, it is not the geometry that defines the physical action, it is the physical action that defines the geometry. That geometry, so defined, does not conform to perfect mathematical rules, but is knowable, yet, one must first grasp what is meant by knowable. Cusa’s cited imperfection of the physical universe, thus provoked the discovery of the universe’s perfectability.

The irony is, that if the universe conformed to perfect mathematical rules, it would not be discoverable. As Kepler describes this process in the “Harmonies of the World”:

“… [U]nless the Earth, our domicile, measured out the annual circle, midway between the other spheres changing from place to place, from station never would human cognition have worked its way to the true intervals of the planets, and to the other things dependent from them, and never would it have constituted astronomy.”

Cusa’s method had opened the door, not only to Kepler’s revolutionary discoveries, but, to the revolutionary discoveries of Fermat, who abandoned the path of shortest distance for the path of least time; J.S. Bach, who rejected abstract mathematical notions of musical intervals, creating the well-tempered system through his compositions; and Leibniz, who created the mathematics of non-constant curvature, that he called the infinitesimal calculus, to name but a few.

It was through this doorway, that the 18-year old Gauss walked on March 30, 1796, when he made his first entry into his scientific notebook — his discovery of the constructability of the 17-sided polygon. For more than 2000 years, it was believed that it were impossible to construct a 17-sided regular polygon with straight-edge and compass. On this day, Gauss recalled, after much hard work, the possibility of such a construction appeared to him all at once.

Gauss always considered this one of his most important discoveries, which, accompanied by a provocation from Kaestner, determined the trajectory of his entire creative life. So much so, that he asked that a 17-gon be engraved on his tombstone. (Because an engraved 17-gon would look so much like a circle, a 17-sided star was engraved instead.)

In the next installment, we will work through Gauss’ construction in detail. However, some historical background will be useful to set the stage. From 1792-1795, Gauss received his early education at the classically-oriented Collegium Carolineum in Brunswick-Wolfenbuettel. The curriculum consisted of ancient and modern languages, classical sciences, aesthetics, poetry, music, and art. On Oct. 11, 1795, he left for Goettingen, where he was attracted to the study of philology, under Christian Gottlob Heyne. The philology seminar at Goettingen had been founded by Johann Matthais Gesner, a former colleague of J.S. Bach at the St. Thomas School in Leipzig. Heyne was Gessner’s successor. Also teaching at Goettingen was Leibniz’ defender and Benjamin Franklin’s host, Abraham Gotthelf Kaestner, who had come from Leipzig in 1750, and under whom Gauss studied mathematics.

In a later letter to his Collegium professor, E.A.W. Zimmerman, Gauss said he was particularly attracted to the philology lectures of Heyne and wished to concentrate on the subject. Initially, he thought Kaestner was a dull old man, but, “I have since realized I was in error and that he is quite an extraordinary man.”

In May 1796, Gauss told Zimmerman in a letter, that he had decided to devote himself to mathematics after discovering the constructability of the 17-gon. When he first showed his discovery to Kaestner, the teacher was not at all impressed, and Gauss interpreted this as hostility to something new. He persevered, and when he discussed the concept further, Kaestner was astounded, but responded that the discovery would be of no use, and he (Kaestner) had already developed the basis for the discovery in his, “Beginning foundations of the analysis of finite magnitudes.” Gauss persisted further, obtaining an agreement from Kaestner, to have him review Gauss’ paper. After further contemplation, Gauss realized that he had to separate Kaestner’s criticism concerning the discovery’s practical use, from the rest of Kaestner’s reaction, “but, if I (Gauss) were to be able to give a more general treatment of the subject, it would be of pleasing curiosity and perhaps produce a brighter insight into this area of mathematics.” Gauss told Zimmerman that he took Kaestner’s judgment entirely to heart.

The more general treatment to which Kaestner was guiding Gauss, was toward the solution of the “Kepler challenge”, which concerned the divisibility of the ellipse, a curve of non- constant curvature. In fact, Gauss’ method for the division of the circle was based on the discovery, that the constantly-curved circle, is, actually, a special case of non-constant curvature!

Gauss’ investigation into the division of the circle, was an extension of the study done by Kepler in the first book of the “Harmonies of the World”: “On the regular figures, the harmonic proportions they create, their source, their classes, their order, and their distinction into knowability and representability.”

It is here that Kepler started his elaboration of the harmonic ordering principle that governs the physical universe. “We must seek the causes of the harmonic proportions in the geometrical and knowable divisions of a circle into equal number of parts,” he began.

To divide the circle by geometrical means, according to Kepler, is to determine the ratio of the side of the figure to the diameter. It is in the efforts to divide the circle, that the human mind discovers the different types of harmonic proportions, which Kepler ranked according to degrees of knowability.

The first degree of knowability pertains to those quantities which can be proven equal to the diameter. The second degree of knowability pertains to those quantities that can be proven to be equal to parts of the diameter. The third degree of knowability pertains to those quantities that are inexpressible in length but expressible in square. From this follows those quantities that are inexpressible, or as the Greeks called them, incommensurable. However, the ranking continues with the incommensurables. The fourth degree of knowability pertains to those lengths that are not expressible by squares, but are expressible as rectangles. Kepler continued, following the tenth book of Euclid, the further degrees of knowability of the incommensurable.

What is significant for us, is that Kepler is re-asserting the knowabiblity of incommensurable magnitudes, not only in the context of geometry, but as the very magnitudes by which the physical universe is characterized. Here Kepler carried out a strenuous polemic against Petrus Ramus, a leading Aristotelean of the day, who had sought to ban incommensurable magnitudes, not only from the physical universe, but from geometry as well!

As we will see in the next installment, it is in the division of the circle, that we encounter these different degrees of knowability. This raises the question, if the circle is, at it appears to the eye, a perfectly uniform, constantly curved figure, why, when one attempts to divide it, does one encounter magnitudes of different degrees of knowability? The reader should think of the different types of magnitudes necessary to construct a triangle, square, pentagon, hexagon, and septagon, for example. Each different division of the circle gives rise to different degrees of knowability, and some divisions, such as seven, seem to be unknowable altogether.

This is the question that Kepler investigates in the Harmonies. The question the young Gauss investigated was, “What is the principle that governs the principle of knowability?”

Bernouilli’s Brachistichrone:

An Exemplary Case of the “Science of the Moments of Becoming”

In response to Kepler’s call for the development of a mathematics appropriate to non-uniform motion, Leibniz invented a new form of geometry of position, that he called, the “infinitesimal calculus”. While a horror may well up in the minds of some at these words, such terrors can be calmed, were one to realize, that the source of this consternation, is due entirely to the Aristotelean assault on Leibniz, by Newton, Euler, Lagrange, Cauchy and their mindless adherents, who imposed on Leibniz’ beautiful invention, the scowling, constipated formalism of his enemies.

What Kepler’s discovery required, was a geometry that measured the position of the planet, with respect to the principle of change governing the planet’s motion. The achievements of the Greeks proved insufficient, as those investigations sought to determine positions with respect to other positions. What Leibniz supplied was a geometry of position that determined position with respect to a principle of change.

To attune our minds to Leibniz’ invention, turn to another investigation of the geometry of position, developed during the same time, under the leadership of J.S. Bach. As Bach’s compositions demonstrate, musical notes are not positions, that in turn determine intervals, which in turn determine scales and then keys, and then the whole well-tempered system. As any listener to a Bach composition can easily recognize, the position of any note, is an ambiguity, that becomes less ambiguous, as the composition unfolds, and the intervals so generated, and their inversions, are heard with respect to the well-tempered system of bel canto polyphony as a whole. It is the change, with respect to the whole well-tempered system, that determines the notes, not the notes that determine the change.

So too, with a planet in a Keplerian orbit. The position of the planet at any given moment, is a function of the harmonic ordering of the solar system. Two positions of a planet mark off an interval of an orbit, but that interval is not defined by the positions, rather, the positions are defined by the change that occurs in that interval of action. Since in a non-uniform orbit, the speed and trajectory of the planet is always changing, Kepler demanded a means to measure that change at each moment. Leibniz delivered, developing his new geometry of position, i.e., the infinitesimal calculus.

As mentioned in the last installment of this series, a good pedagogical example of Leibniz’ discovery, is its application in John Bernoulli’s discovery of the brachistichrone curve. (What follows is a summary of the concepts of Bernoulli’s construction. It will require some work on the part of the reader, and is intended to be read in conjunction with Bernoulli’s original essay, an English translation of which can be found in D.E. Smith’s, “Source Book of Mathematics”.

In 1697, Bernoulli put out a challenge in Leibniz’ Acta Eruditorum, to all mathematicians in the world. The problem was stated:

“Mechanical Geometrical Problem on the Curve of Quickest Descent.”

“To determine the curve joining two given points, at different distances from the horizontal and not on the same vertical line, along which a mobile particle acted upon by its own weight and starting its motion from the upper point, descends most rapidly to the lower point.”

The prize promised was not gold or silver, “for these appeal only to base and venal souls, for which we may hope for nothing laudable, nothing useful for science. Rather, since virtue itself is its own most desirable reward and fame is a powerful incentive, we offer the prize, fitting for the man of noble blood, compounded of honor, praise, and approbation; thus we shall crown, honor and extol, publicly and privately, in letter and by word of mouth the perspicacity of our great Apollo.”

As Bernoulli pointed out, the problem posed could not be solved, even by the method of maxima and minima of Fermat. For in those cases, Fermat sought the maximum and minimum from among a given set of quantities or loci, such as the point of maximum curvature of a conic section. Instead, Bernoulli’s problem was to find a minimum curve, among an infinity of possible paths. Every position on this sought after curve, was determined by a principle of change. So, what had to be discovered was, from a given a principle of change, i.e., least-time, how are the positions of the body determined. This is the equivalent of finding the correct orbit of a planet, not merely a possible one. Or to put it in metaphysical terms, “How can we know, how a falling body knows, to find the path of least descent?”

As you will see, Bernouilli was not posing an abstract mathematical puzzle, for the mere sake of befuddling others, the solution to this problem led to important discoveries in mechanics, as well as metaphysics.

Bernoulli’s attack on this problem began with what he called “Fermat’s metaphysical principle”, that light always seeks out the path of least time. It was a discovery of the ancient Greeks, that when light was reflected from a mirror, the path it took was the shortest length. However, when light was refracted by traveling through different media, such as water and air, the path of the light was not the shortest length. Fermat, discovering that the velocity of light was slower in denser media, demonstrated that the light changed its direction at the boundary between the two media, so as to follow the path of least time. This, of course, was consistent with the Greek discovery. In reflection, since the light travels through only one medium and therefore doesn’t change velocity, the shortest path, is also the path of least time. But, when there’s a change in medium, the light travels the shortest path in space-time, or the path of least-time.

Bernoulli’s approach was to follow the light, so to speak, to the path of least time. If the path of a ray of light traveling through a medium, whose density is continuously changing, according the same principle as that of a body falling under gravity, the the least time path of the light, will be the same as the least time path of the body.

But, how to discover the path, when we only know the principle of change, and have no positions to which to orient? At each moment along the light’s path, the light would be changing its speed and direction, such that its overall travel took the least time. Thus, similar to the motion of a planet, at each such moment, the light was ceasing to be what it was, and becoming what it will be. At each moment, the position of the light was a function of the principle of maintaining the least-time path.

Fermat had shown, that as light moved from a rarer to a denser medium, it slowed down, and its path became more vertical. For example, if light were traveling through air to water, the angle its path made with a vertical line, changed at the boundary between the air and water. If the angle its path made with the vertical in the air changed, the angle it made with the vertical under the water changed accordingly. However, the two angles did not change proportionally. Rather, they changed such that the sines of the angles were always in the same proportion.

So, at each “moment of becoming” along the light’s path, the light’s velocity and trajectory were changing, such the sine of the angle the light’s path made at that moment, was always proportional to the sines of the angles at all such “moment’s of becoming.”

To find the brachistichrone, Bernoulli thought of the medium in the following way:

“If we now consider a medium which is not uniformly dense but is as if separated by an infinite number of sheets lying horizontally one beneath another, whose interstices are filled with transparent material of rarity increasing or decreasing according to a certain law; then it is clear that a ray which may be considered as a tiny sphere travels not in a straight but instead in a certain curved path. This path is such that a particle traversing it with velocity continuously increasing or diminishing in proportion to the rarity, passes from point to point in the shortest time.”

Under this idea, at each horizontal sheet, the speed and direction of the light changes. The principle under which its speed and direction changes at each horizontal sheet, Leibniz called, the differential. The totality of all such differentials, (what Leibniz called the integral), is the sought after brachistichrone curve.

From one “moment of becoming” to the next, the position of the light changes, as it passes vertically from one density to the next. Each such vertical change in position is accompanied by a horizontal change in position, that corresponds to the sine of the angle of inclination at each “moment of becoming”. (Bernoulli’s geometrical construction of the above can be found on p. 652 of Smith.) Bernoulli adopted Leibniz’ notation for these ideas, calling the vertical change, dy, the horizontal change, dx, and the resulting change in the path of the light, dz. The proportion between the vertical and the horizontal, dy:dx, and the resulting change in the path, dz, is a function of the rate at which the density of the medium is changing.

Bernoulli shows, that if the density of the medium is changing according to the rate at which a body falls under its own weight, (specifically, that the velocity changes according to the square root of the vertical distance) then the resulting curve is a cycloid. “…you will be petrified with astonishment when I say that this cycloid, the tautochrone of Huygens is our required brachistochrone…” he declared.

But, Bernoulli noted that this was not a discovery of a particular physical phenomenon, but a metaphysical discovery of a universal principle:

“Before I conclude, I cannot refrain from again expressing the amazement which I experienced over the unexpected identity of Huygen’s tautochrone and our brachistochrone. Furthermore, I think it is noteworthy that this identity is found only under the hypothesis of Galileo so that even from this we may conjecture that nature wanted it to be thus. For, as nature is accustomed to proceed always in the simplest fashion, so here she accomplishes two different services through one and the same curve, while under every other hypothesis two curves would be necessary the one for oscillations of equal duration the other for quickest descent. If, for example, the velocity of a falling body varied not as the square root but as the cube root of the height falalen through, then the brachistochrone would be algebraic, then tautochrone on the other hand transcendental; but if the velocity varied as the height fallen through then the curves would be algebraic, the one a circle, the other a straight line.”