Riemann’s Treatment of Abelian Functions
Several aspects of Bernhard Riemann’s Theory of Abelian Functions, published in 1857, are of crucial importance today for understanding economics and scientific method more broadly, by allowing the true nature of human economy to be considered and discussed without the sickening effects of using deadly mathematical language. While Abelian functions, named after the Norwegian Niels Abel, certainly are mathematical, Riemann saw in them an analog for the continuing development of human thought, and treated them, ironically, not as mathematical objects, but as geometrical relationships. In doing so, he defended the key concept of metaphor in language, including in mathematical language.
In this presentation, we’ll see how Riemann’s work helps us conceptualize qualitative transformations as the primary substance of economic progress, and lets us think of ourselves as embodying the continuous development that expresses itself in such shifts. This will take place in two broad sections. First, the concept of potential and Riemann’s use of what he called “Dirichlet’s Principle”. Second, Riemann’s treatment of Abelian functions. Then, we’ll conclude with applications to economics and theology.
I. Potential and Dirichlet’s Principle
In his work on studying the power of motion, Gottfried Leibniz introduced two important terms: vis viva (meaning living force) and vis mortua (meaning dead force). Living force is seen in the movement of an object, such as the power of a baseball bat to change the motion of a baseball, and Leibniz famously showed that, contrary to Descartes’s assertions, the measure of the power inhering in moving objects was proportional not to their speed, but to the square of their speed. In contrast to such living forces, Leibniz considered a dead force as one which could come to life. For example, a skier at the top of a slope can develop speed as he descends, and a bow ready to fire an arrow has a dead force that comes to life when the archer releases the string. For the case of raised objects, their elevation is a measure of their dead force. The farther they can fall, the greater the speed they’ll reach, and the higher they are raised, the more effort was required to get them there. To learn more about Leibniz’s work, see my video at larouchepac.com/visviva.
A topographic map is a measure of dead force, by measuring elevation. Take a look at this landscape. If we mark off lines for the paths that stay at a constant level, we can view it from overhead, and see what is known as a topographic map. The lines of constant elevation are known as contour lines, and they indicate elevation just like lines of latitude indicate how far north or south you are. If we were to set a ball on the ground on one of these contour lines, it would start rolling perpendicular to the line, since it wants to roll down, not stay at the same level. We’ll come back to this.
A major development of Leibniz’s concept of dead force was made by Karl Gauss, who re-introduced it by the name of potential, which is the name we use today. Its use can best be shown by an example. Here you have a simulation of a comet coming into a planetary system -- according to Newton, the comet must be thought of as being pulled on by the star, all the planets, and all the asteroids. As you can see, it gets pretty complicated to think about things this way. Gauss instead looked at the space of the planetary system as having a certain shape, and the comet simply moved around in the space the same way a ball would move on the terrain we saw earlier in the topographic map. Here, the bands represent the potential (the dead force) just like the contour lines on our topographic map. They are a generalized idea of height. Again, the comet just moves in this shaped space, without considering a huge number of individual forces. Moving the camera down, you see the star and planets making small gravity wells -- this may be familiar from other videos you’ve seen about Einstein and curved spacetime.
But there’s something missing -- contour curves are fine for a two-dimensional surface, like the surface of the earth, or the ecliptic of the planetary system. But what about all of three-dimensional space? Instead of contour lines, we’d have equipotential surfaces. All of the spots on each of the surfaces has the same potential -- the same dead force.
Now, let’s take a look at magnetism. Magnetic compasses were used before it was known that the earth itself has a magnetic field -- some people thought the north star was itself magnetic. The hypothesis that the earth is a giant magnet came from William Gilbert, whose work helped to inspire Kepler. Measurements of magnetism from around the planet revealed that the field didn’t correspond to a simple magnet. Tobias Meyer said that the earth’s field must be off-center and crooked, and Hansteen hypothesized that there were two magnets inside the earth. As he entered the field of earth magnetism himself, Gauss said that everyone was approaching the problem the wrong way. Rather than starting with a hypothesis of magnets, it were first required to figure out what the overall magnetic field was! Without such knowledge, hypothesizing magnets was just shooting in the dark. Gauss proved that it was actually impossible to know what was happening inside the earth: any magnetism inside would have the same effect outside the earth, as if the magnets were spread over the surface itself. That is, unless you make measurements deep inside the earth, you can’t actually know what’s going on inside: all you can know is the field itself. Freed from the need to think about a configuration of magnets, Gauss treated the magnetic field as a field, discovering how to use limited observations to map the field as a whole. Here is a world map Gauss produced of magnetic declination: the lines indicate difference between geographic north and the magnetic north shown on a compass.
Again, with potential as his tool, Gauss was able to look at entireties as entireties, as gestalts, as wholes, rather than as an aggregation of parts.
On the economic front, how might this be useful in thinking of NAWAPA as a new platform, rather than simply as infrastructure?
Lejeune Dirichlet built on Gauss’s work in his own lectures on potential. He made another important discovery, very similar to Gauss’s, that has become known as “Dirichlet’s Principle.” Let’s take a look at a space filled by potential. Here, we are looking at an electric field, with two positive charges near the center, and a negative one near us. Now that we have drawn in the potential sheets, we’ll focus on a region of space that does not include the electric charges themselves. The change you’re seeing here is just restricting the region of sheets that we’re showing. Now, Dirichlet says that the shape of all of those sheets inside the sphere is completely determined if we know the potential of the surface of the sphere. That is, simply knowing the boundary, completely determines the shape of what lies inside the volume. Dirichlet gave the condition for the internal potential corresponding to the given surface: he said that the shape of the potential minimizes the total force inside. That is, least-action is a characteristic of potential fields.
This is a very important concept, because it changes the idea of what it means to know something. Do you have to know every possible detail about something to understand it fully? What are the determining characteristics, such that, if you knew them, you’d know everything about its behavior? Riemann applied this physical principle when studying mathematical complex functions, including those of Abel which we’ll come to in the next section.
While physicists, familiar with electricity and magnetism thought the concept made sense, mathematicians attacked Riemann’s later use of Dirichlet’s principle. They basically said that Riemann had to say what the minimum force was, not simply that it existed. But aren’t there some ideas that you only communicate by describing what they do, rather than giving a description? To not allow that, is the same as saying metaphors are not allowed. This is a thorny issue, and we’ll get at it using riddles. Don’t worry, Dirichlet’s principle will come back.
Put your thinking-cap on. Here’s the first riddle: You see it both in field and town. It cannot get up, but often falls down. You see it both in field and town. It cannot get up, but often falls down. Here is a second one: A barrel of water weighs 20 pounds. What must you add to make it weigh 15? And a third: Two legs it has, and this is quite neat: only at rest, with the ground will they meet. Two legs it has, and this is quite near: only at rest, with the ground will they meet. Last one: What is black when bought, red when used, and gray when thrown away?
Go ahead and hit pause to give yourself some time to work on these. It’s important that you try to figure them out, or the rest won’t make sense.
Okay, the first answer is rain, which falls all the time, but never gets up. For the second one, you’d have to add holes to a barrel to make it weigh less. A wheelbarrow has two legs that only touch the ground when it is at rest. And coal is black when bought, red when used, and gray when thrown away. So those answers weren’t themselves complicated, although the way they are described with riddles, is.
Now for another: What only runs when it is cold, and stops as soon as it is told? What only runs when it is cold, and stops as soon as it is told? Again, pause the video to think about it. The answer is not your nose, which runs when cold, but doesn’t stop when told. Although there are some complicated answers you might give (like a well-behaved sled-dog), there is no answer. This isn’t a real riddle, although it sounds like one. These conditions don’t correspond to any simple thing. I wanted to show that it is possible to phrase riddles, without actually having answers.
So, we heard several riddles with simple answers. In each case, the riddle isn’t necessary to describe the object in question. For example, if you wanted to borrow a wheelbarrow, you wouldn’t ask your neighbor “a thing with two legs which touch the ground only when it is at rest” (at least, I hope you wouldn’t ask that way). Instead, you’d just say you want a wheelbarrow.
Now, let me ask you a question: are there some things that you can only say as riddles? That is, are there some concepts that you can only describe by what they do, by saying the conditions they fulfill, rather than just saying what they are? (Pause)
If you’ve ever had the experience of sharing a big idea with someone to whom it is new, you know that you can’t just say it by name -- you have to cause the other person’s mind to generate the idea anew, in a process similar to telling riddles. The thought must be created from within their minds, provoked by your outside action, but not... inserted or given.
How about poems? Poems aren’t always straightforward, and a good poem says something that could not be transformed into simple prose. While bad poems are like riddles without answers, if you’ve worked on Shakespeare’s sonnets (for example) it is clear that Shakespeare does have a point, and that although you might think about it or discuss it in prose, the best way to convey the thought is through the sonnet itself. The same is the case with great music: it often doesn’t require words to make its point, but it is precise, saying something that could not be put into simple words.
Metaphor is a form of negative communication, where a specific idea is conveyed, but not by stating it directly. Instead, the conditions the idea must fulfill (which are usually paradoxical until the idea is formed) are used to impel the other person to develop a new thought. Truly new ideas are always like this: they require a new way of thinking, not just a new object-like thought. They are more like a new motion than a new thing.
Return to Dirichlet’s principle. Using our analogy here, the surface conditions have only one answer: the shape of the internal potential, just as the conditions of a riddle have an answer they imply. So, the surface is like the riddle, and the internal shape is the answer.
Riemann treats mathematical functions this way: not as formulas, but as transitive verbs, as that which does something, looking at what their characteristic activity is, rather than their description. And by doing so, he allows metaphors into his mathematics, rather than banishing them as uptight symbol-minded mathematicians do. Riemann’s approach to Abelian functions is to abandon formulas as much as possible.
Looking at metaphor in economics, the new discoveries of principle that shift human society and economy forward as if by leaps, are of this nature -- they are solutions to paradoxes presented by the senses, which lie outside the domain of the senses, in the domain of the causes of what later appear to our senses. They are the unseen movers whose effects present themselves to us as scientific riddles.
Economic and political forecasting and policy depends not on amassing huge amounts of data, although that may be required. Instead, a developed knowledge of the determining conditions is what is important.
Next, we’ll look at the most essential characteristic of economics -- the seemingly discontinuous jumps that make up economic progress -- in the next segment.