Question: Hello Mr. LaRouche, my question has more to do with physics, so I'm sort of changing the subject a little bit. I've heard you refer to the inverse as being topologically finite. And Einstein said pretty much the same thing, when he referred to "finite, but unbounded." And, I'd like to open that up for discussion: See, if the universe is to be continuous, it has to be curved. Like, if you wanted to make a finite line continuous, you would curve it into a circle. If you wanted to make a plane, you would have to wrap it around a sphere to make it continuous. So, to make our universe continuous, that kind of implies an extra dimension of curvature.

And, I think you find pretty much the same idea in Gauss's method of squaring the circle, in that, every time you add an order of magnitude, you add an extra 360 degrees. And, so basically, my question is: How does the idea of continuous universe related to idea of a 360 degree time rotation of bidirectional symmetrical time-flow in our universe?

LaRouche: This is something, again, I just finished a paper, I started on and edited on May 30, on "Visualizing the Complex Domain." And, that's where the problem is. We've done a lot of things on Gauss, but I was not satisfied, that we had effectively addressed the epistemological implications of the Gauss-Riemann work. And therefore, I wrote this paper to, shall we say, "sharpen up" our epistemology, in dealing with this problem.

Now, the problem is essentially this--the epistemological problem--which is clear, I think, from Kepler, in part; it's clear with Plato; it's also clear with Gauss; and it's most clear with Riemann. But, it is not understood, generally, because of bad university ideology, where the university teaching of Gauss is ideologized to conform to Lagrange, LaPlace, Cauchy, and so forth. That's where the problem arises: fakery.

So, the problem is this: We do not know the universe by means of our senses. Our senses are organs of our body, which reflect the impact of the universe on that sense-organ. But, what we think we see, hear, and so forth, is not the universe; it is the shadow of the universe, upon our sense capabilities.

This, then, results in experiments, such as Kepler's great experiment on developing the Solar System conception. If you realize that there are paradoxes, in the perceived scheme of things. In other words, if you imagine that the universe, the astronomic universe, appears to our senses as a Sensorium, a kind of interior surface of a sphere of undetermined, but very large radius. And you're looking at this motion of the stars, and these wanderers called planets and moons, in this process. So, you find out, there are errors, in the attempt to find a principle, which can explain certain motions in terms of a continuous function.

For example, you have, in the case of the universe, the Solar System, you have elliptical functions as orbits, with the Sun at one foci of the ellipse, one of the foci. The rate of motion, along the orbits, is never continuous: It's always non-uniform. So, from the standpoint of continuous function, there is no simple principle, which accounts for--especially for the Solar System as a whole. Especially for Kepler's statement--proven by Gauss--that there was a missing planet, which had to have existed in this universe in the Solar System. But the planet had to explode, because of its harmonic characteristics. And then, Gauss, in 1803, 1805, 1806, demonstrated that the orbit of Ceres corresponded to an exploded planet, which is called the Asteroid Belt, today. And, with the characteristics attributed to that missing orbit, by Kepler.

So, that's the universe, hmm? Anomalies. Then, we had the anomaly of quickest action, as first discovered by Fermat; developed by Huyghens and Leibniz, and becomes the principle of universal least action, with Leibniz. There are similar things. The world is full of anomalies, which do not correspond to a sense-certainty interpretation of the universe.

Now, that means we have two curvatures. One of is the curvature of the apparent curvature, the sense-certainty curvature, which is best estimated by this looking at the interior surface of a very large sphere. That's one geometry, which gives us the first order of the complex domain. But, the action which occurs along the trajectories, within this sense-perceptual system, is not explained in that way. Something else is acting, something we don't see, but which we can measure, as a physical principle.

So now, you have an intersecting curvature, at every point, say, along a trajectory. For example: The concept of gravitation, by Kepler, is acting from outside the sense perceptual universe, upon every known part of the motion, of the orbit along its orbital trajectory--the planet. You don't see it, but it's there. You can measure it.

So, this is the curvature, we're actually measuring. We're measuring an intersection of each of these principles, which is a curvature, which is acting--in this case, gravitation--continuously along the orbit. There's another curvature operating, which is regulating the orbit, as a non-circular orbit.

Now, you find another principle: That one also intersects every part of that orbit, of the sensorium. So therefore, you have two geometries: You have one geometry, which is the sense-perceptual geometry, which is actually is an anti-Euclidean geometry, not a Euclidean geometry. It's just a geometry of sense-perception, as the ancient Greeks saw it with a constructive geometry, as Archytas, and Plato, and so forth, saw it. We have the second geometry, which is a geometry of the physical principles, which, invisible to the senses, are nonetheless efficiently acting at every point of motion in the universe. This is a [tech?] curvature.

Therefore, you have what is called a Riemann Surface Function. A Riemann Surface Function has two characteristics: One, a function, which is the perceptual function, in the simplest approximation. Then, you have a different curvature, which intersects that perceived function, at every point in the perceived function, which causes the net result.

So, that's the way in which it has to be understood. This then, becomes the complex domain. It's the complex domain of the physical domain, which is the so-called "pure" Riemannian domain, in which there's nothing but physical principles acting on the universe. The other is the apparent effect of that Riemannian domain, upon the perceptual domain, which defines a Riemann Surface Function.

Now, you find the Riemann Surface Functions also occur at a higher level, in terms of domain beyond the perceptual domain; that, in the higher domain, there are also Riemann Surface Functions. And the point that people have difficulty in, is understanding, that there is an efficient relationship between an unseen set of curvatures, but which are efficiently measurable, but unseen, sensorily, and the sense universe, in which these unseen principles are acting to produce. So, the two.

And, the problem is just exactly that, the difficulty of people of bridging, of getting away from the idea of trying to explain everything from a plausible sense-perceptual or Cartesian manifold, and trying to stretch a Cartesian manifold, in the way that, for example Lobatchevsky and Jonas Bolyai tried to do. They came up with a so-called non-Euclidean geometry, which was an attempt to "correct" and stretch a Euclidean geometry, as a non-Euclidean geometry. The Gauss-Riemann geometry is not non-Euclidean: It's anti-Euclidean, in the sense, that it's based on a pre-Euclidean, constructive geometry, as specified by Kaestner. And what this has developed, in a succession of phases, largely through experiment, by Riemann, as a generalized Riemann Surface Function.

But, it involves both of these concepts, and the mind must get used to this way of thinking about the universe. That's the usual problem.

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