Riemann II(b): Abelian Functions -- transcript
November 8, 2011 • 3:03AM

II. Abelian functions

Let’s continue to look at the most essential characteristic of economics: it seems to be fundamentally composed of discontinuities, of breaks, of jumps. A new creative discovery makes things possible that simply could not have existed before. As an example, consider the Apollo moon mission, launched by President Kennedy. Going to the moon cost quite a bit of money, and there are many estimates of the payback: the new technologies developed revolutionized many aspects of production. The cultural benefits seem almost unquantifiable. But let’s take, as a basis, the figure arrived at by Chase Econometrics -- they estimated a return of $14 for every dollar invested in the moon shot. But let me ask, even if this were a good estimate, were the $14 we got back the same as the dollars as we spent? No. What’s different?

To answer this question, let’s compare two kinds of computerized machine tools, first developed during and for the Apollo mission, known as CNC (computer numerical control) systems: a three-axis mill, and a five-axis mill.

For this three-axis mill, the cutting implement moves in the X and Y directions (back and forth) and also up-and-down in the Z direction. A computer coordinates these motions to create 3d metal shapes such as the one seen here. With a five-axis mill, two additional motions are added: in this case, the cutting head tilts and rotates. Such a machine is capable of making more complex forms, such as the part you see here. A piece like this, made from a hard alloy, could literally take days or weeks to machine without computer numerical control and a five-axis mill. Comparing the two shapes, they are both three-dimensional, but the two additional dynamic dimensions of action of the five-axis mill allow it to create a whole slew of shapes that are simply impossible with simpler equipment. How would an accountant represent the higher value of five-axis machining?

This is an excellent analogy for changes in the economy as a whole with the introduction of new principles, such as: electricity, the germ theory of disease, and fertilizer production, just to name a few.

Now, let’s examine this concept of incommensurable jumps in geometry, to get into Riemann’s treatment of Abelian functions.

Incommensurability

The first example of incommensurability comes from Plato’s Meno dialog, in which Socrates confronts an uneducated slave boy with what seems to be a simple geometric problem: how do you construct a square twice as large as a given square? The boy’s first guess is to double the sides of the square, but, as Socrates points out, that actually creates a square with four times the area. The boy’s next guess is to try a length of 1½, which adds on an area equal to a half plus a half and then some, so it is also too large. With a bit of prompting from Socrates, the boy considers this crooked square. It is made of four triangles, twice as many as the two such triangles making up the original square. That means it is twice as large.

Now, while Socrates certainly provoked the boy to think about this, the ability to recognize that the doubled square really is double already lay in the boy’s mind. It is as though he was simply remembering something he already knew. Now, since we didn’t make this crooked square by starting with a length, like the other ones, we don’t yet know how long its side is. So, let’s figure it out.

The longer side is made of the original side plus a certain remainder. That remainder goes into the side... two times, with a small remainder left over. But that remainder goes into the previous remainder two times with a bit left over. The construction continues similarly forever. If we write the doubled-square side as a fraction of the original side, we get 1 + 1/(2+1/(2+1/(2+......))). This is because it is one of the original side, plus a piece which goes into it two times, plus a little bit more, which itself goes into the first remainder two times, plus a little bit more, and so on. This continued fraction would go on forever. The side of the doubled square -- the square root of two -- lies beyond all fractions, resting literally infinitely far away.

The square has a simple area, but its length simply does not exist as a length. If you didn’t have a two-dimensional area to draw on, you could never make the side of the doubled square simply by working with length alone -- adding, subtracting, multiplying and dividing lengths won’t do it. None of the infinite number of fractions correspond to this length, just as a three-axis mill could never make the 5-axis-produced part we saw earlier. We’ve gone beyond the infinite of fractions, and the side of the doubled square is said to be incommensurable, since it cannot be measured as a length. When you write \sqrt{2}, you are writing a riddle: “What’s the length of a square of area two?”

Different Infinites

The higher power of area shows up in the smaller power of length as lying “in between the cracks” or “beyond the infinite.” That’s what happens when you try to look at a greater process using the language or understanding of a lesser one that’s unable to bring it about.

As our next example, consider a circle. You can try to consider it as a polygon with an increasing number of sides, but no matter how many sides you add, the polygon is still made out of little bits of straightness, while the circle is fundamentally curved. Polygons may say the circle has an infinite number of sides, but the circle knows better.

Let’s look at the circle some more. Here you see rotation along a circle, along with two lengths. The arc along the circumference of the circle will let us measure angle, as the distance along the circumference. The vertical line and horizontal line have lengths that change repeatedly. The horizontal line is known as the cosine, and the vertical line is the sine. Let’s try to write out algebra formulas for the sine and the cosine.

Here’s my first guess [4th power] -- you can see it doesn’t work that well.

Here’s another [11th power] -- hmm, that’s better, I actually got a bit of a circle.

Now let’s try this one [25th power] -- success? Not quite...

How about this one? [50th power] -- Aha, we got it! Oh, whoops, that broke too.

Seems pretty clear from this pattern that we’d need a formula that is infinitely long to make the circle. Now you see the result as we add more and more terms, up to 100. It keeps going a little bit further before diverging. To go around forever, the formula would also have to go on forever: not a very satisfying definition!

And while the sine cannot be calculated in terms of the arc, it is possible to say exactly how the sine would change if the angle were doubled. There are expressible internal relationships in the domain of circular functions, even though the functions themselves are transcendental.

One final shape: the lemniscate. This is made by inverting a hyperbola through a circle, and is a higher transcendental than the circle or the catenary. As we move along the lemniscate, there is absolutely no formula for the blue length in terms of the green one, although the relationships between their changes is expressible. Nonetheless, some constructions are possible. Using the projected length on the right, we can create a complementary point on the lemniscate, whose green distance from the right is the same as the other point from the center. It is even possible to add, subtract, and multiply arc-lengths, and know exactly how the blue length would change.

Whew! So, let’s review! We had the normal numbers -- fractions. Then we had the square root of two, which was incommensurable to it. Then we saw the circle and the catenary, which are beyond the infinites of simple arithmetic. And finally, the lemniscate, which cannot be approached even with the help of circles and catenaries. The Norwegian mathematician Niels Abel showed that the lemniscate was just the beginning of a whole series of transcendentals, each inexpressible from what came before, just as the $14 payback from Apollo was not the same kind of dollar as the $1 invested in it. Nonetheless, Abel showed that there were characteristics of the higher functions that could be foreseen.

Riemann and Topology

Riemann brought something startlingly new to Abel’s work -- he looked at functions not as symbols written on a piece of paper, or even the mathematical relationships between them. Going further, Riemann employed what was known to Leibniz as Analysis Situs to see the hidden, dynamic geometry behind relations. This is seen in Riemann’s habilitation dissertation, where he develops the different possible curvatures space could have, saying that only physics, not mathematics, can justifiably give the shape of space.

For our first example, look back at the circle, and watch the vertical length, the sine. Here we’ll trace it out as we move around the circle. Because it repeats itself, instead of drawing this out on a plane, we could have put it on a cylinder. The sine has one periodic, and its geometry is implicitly cylindrical.

While the circle has a simple periodicity, the lemniscate has two distinct periods. This double periodicity is similar to a variety of early computer games, like you see see here. This is like the sine in its repetitive nature. Let’s just twist it into a cylinder... But we still have the problem with the top and bottom. What if we connect them too? The unusual behavior on the plane is totally natural when we use a torus. It was the geometry implied by the action. That’s analysis situs. So the dynamic domain of the lemniscate or the game are toroidal.

Riemann showed that the domains of the higher functions developed new geometries in the same way that we went from a plane to a cylinder to a torus. The unseen dynamic geometry of their internal relationships could be understood topologically by these surfaces, in a way that would never occur by simply looking at the formulas. The shapes we see here, are the shapes of action. Riemann’s approach allowed the fundmamental incommensurability of each of the series of Abelian functions to be represented topologically as a new surface characteristic.

Think about these changes in terms of the ongoing development of the universe and the increasing power of man’s creative reason. Using the freedom afforded him by Dirichlet’s principle, and the surface characteristics he discovered, Riemann was able to look at the higher transcendentals not as mathematical symbols on a piece of paper, but as increasingly complex geometric spaces, becoming more dense in their interconnectedness. The characteristic of ongoing changes, each of which seems incommensurable to what came before (expressible only as a metaphor), began to develop a higher continuity lying outside the series of changes themselves.

Economics Conclusion

This ongoing series of development, seen as the characteristic of the universe as a whole, the increasing power of the biosphere of the earth, and the continuing progress of the human species, is itself the truest substance. Individual human beings, unlike all of life outside us, have the opportunity to participate in that creative motion. Such an immortal life, like that of the greatest inventors, scientists, discoverers, musicians, playwrights, engineers, and poets of the past, is the highest responsibility of government. The greatest mission of the nation-state is to ensure its citizens the opportunity to have not only the standard of living required to pursue such aims with physical dignity, but the cultural and scientific trajectory to allow its people to play what they can know will be a role of lasting value in history.

Without the baggage of oligarchy and monetarism, a credit system oriented economy can bring us to such a matured state of culture, throwing off the childish ways of the past, and looking forward to the beautiful future glimpsed in such promises as NAWAPA, space exploration, and fusion research. With such an outlook, we can even see a rebirth of development in music, largely stalled since the death of Brahms.

A final thought: how does the continually developing nature of creation, as expressed in two recent LPAC videos, square with the idea of a constant and eternal Creator? A simplistic view of an unchanging God like a stubborn old man must give way to the continuity that lies behind and determines the creative shifts we see and create. As we move forward with real economic development, as we see what is actually constant through change, we’ll discover more about the true nature of the human personality, and about the universe whose essential character we embody. One last riddle: While we see discoveries as discrete events, as shifts and jumps, we also know that this continuity exists. What is the nature of this continuous change? Do you have a name for the answer to that riddle?

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