Question: Hello Lyn. I'm from Seattle, and I just read your "On the Subject of Metaphor" paper. And you talk a lot about, actually, Kepler's "Snowflake" paper. And you talk about the Golden Section, as being a reflection of actually, of negative curvature--or, actually of the process of development, and this is also reflected in negative curvature. You speak about one way of looking at things, as space being the close packing of spheres, with the envelope of all possible spaces being negative curvature. I was curious about that.

But, in relation to this, I was speaking to Leni, yesterday, and we were talking about the fact, that when people are developing, you have this negentropic growth; but, once you stop, you start to degenerate. You start to--like the snowflake begins to melt, because it has an entropic geometry. And I was thinking about this, in relationship to what we're speaking about now, with the nation. Because, I use this in the field a lot, what Benjamin Franklin said, "Now, you have a republic, if you can keep it." And I was thinking about this in relationship to this negative curvature, which I don't fully understand. So...

LaRouche: All right, fine [laughing]. Well, first of all, I haven't degenerated yet, and I don't plan to. As a matter of fact, I'm doing better than I ever did, before! So, we shouldn't assume that there's any automaticity in decadence and in systems. I do not expect I'll be biologically immortal--I'm not claiming that--but, I'm doing pretty well.

But, this goes to a very deep, underlying question, which I only touched upon in that piece on metaphor. But, let me just give it to you, because other people will probably have similar questions or the same question, and it's a very important one, today. Now, as I've emphasized many times, the human sense organs, the process of sense perception, as such, is a part of the biological organization internal to our bodies. Now, what you think you perceive, in sense perception, is not the world outside your skin, so to speak. What you're perceiving is the reaction of your sense organization to the impact of the world upon it, from outside your skin, so to speak.

So therefore, you do not know by sense perception, exactly what that world is. However, not being an animal, you have a mind, a human mind. And therefore, you're able to recognize that there're certain paradoxes, which are called "ontological paradoxes," which arise in the course of perception. And these paradoxes enable the mind to conjecture on what are called "hypotheses," as to what the mysterious, unseen principle is, out there, which is causing these paradoxes to appear within sense perception. If they're proven experimentally, that is, if man can not only hypothesize a principle, but show experimentally that it's valid, that you can control it, you can use it--then, you know it exists, and you're able to change the behavior of the sense perceptual world through the use of this principle. That's what we call "knowledge." And, that is the only form of knowledge.

Now, look at that from the standpoint of the ancient--say, the ancient Greeks: Thales and Pythagoras, who gave us the root of modern science, in their method of so-called "constructive geometry." The key thing, as was emphasized by the Pythagoreans, is a doctrine called "spherics." Now, spherics refers, essentially, to man's looking up at the nighttime sky--particularly the nighttime sky--and seeing a certain order in the nighttime sky. Now, this nighttime sky, then, typifies for man the envelope of sense perception, the universal envelope of sense perception. This sense perception envelope, at first, they assumed to be spherical, which is what they meant by the equation between spherics and astronomy in, say, the ancient Pythagoreans.

But, the point is, what kind of geometry do you have when you introduce the idea of principles, reality lying, so to speak, "outside" the sensorium, outside their spherical sensorium? Well, Gauss has an answer for that. And, the reason I've emphasized so strongly this Gauss 1799 paper, attacking Euler and Lagrange, in particular, is precisely because youth today need some standard of truth, in a time when mere opinion has been substituted for truth in human relations. So you need a yardstick, a standard of truth. What do you mean, by the word "truth"? Give me an example of what you mean by the word "truth"? And, this Gauss test does that for you.

So, anyway. Now, what does Gauss do? Gauss says, "Look at these points in the sky." And, the great example was Gauss's discovery of the nature of the Asteroid Belt; he discovered the Asteroid Belt. He didn't actually discover it, because Kepler had anticipated its necessary existence, a couple of hundred years earlier, with his description of this exploded missing planet, and had explained why it must have exploded, because of a discontinuity in that area. So, Gauss re-discovered and confirmed this.

Now, from that point on, and from his other work, there's a development in Gauss's work--from astronomy, especially; but also in geodesy, as well--in which, you say, "Well, what are you doing here?" And this became Gauss's conception of the general principles of curvature, a concept which became the basis for Riemannian physical geometry.

What you're seeing out there, this dot, this sensation, this object, on the sensorium, is actually--if it involves a principle, external to the so-called "Cartesian manifold," then it can be treated as a curvature, which is tangent to the sensory spherical universe of sense perception, at that point. So therefore, the comparison of the curvature of this motion, by observing several points, as Gauss did in discovering the orbit of Ceres: by comparing these several points, you could determine the curvature of something external to sense perception, which you know, by this kind of treatment of the subject of points within the areas there of sense perception.

This becomes, then, your sense of what is true. That there's a universe outside sense perception, which is the true universe, which you know by these experimental principles, and you know it in terms of the intersection, in terms of paradoxes between the points of observation of the sense perception. You look at this in one way, in astronomy; then the same thing is applied to things on the macro-physical scale of ordinary experience; and then, again to living processes, as distinct from non-living processes; and to domains such as microphysics. And that becomes the area of knowledge.

Now, there are several famous examples of this kind of process, in ancient Platonic knowledge. This includes the definition of: How do you define a line? As opposed to what's called a Euclidean definition, or Cartesian. How do you define a surface, with respect to a line, like doubling the square, for example? How do you define a solid, with respect to a surface: doubling the cube, for example? How do you explain the fact, that, from a spherical system, there are precisely five regular Platonic solids?

So, these indicate, as Kepler recognized this, later, in his work: That, from knowledge of principles, we can prove and develop the notion, of a real universe, which exists external to mere sense perception. We have access to that, because the powers of the human mind, which demonstrate themselves to be efficient, because we are able to increase our power in nature through these powers; that, through these powers, we're able to discover universal physical principles, and thus, gradually, to get an image of the real universe, outside the range of sense perception.

Now, the way this looks, then, from the standpoint of the geometer or mathematician, is that you have three kinds of curvature, that you're looking at, in terms of the sensorium, that is, the spherical sensorium. Two are external tangencies; a tangency which is a simple curvature, outside the curvature of the spheroid; secondly, a negative curvature, which is tangent to the sensorium. And internally, a positive curvature, which is tangent to the sensorium. And so forth and so on.

So therefore, you get a conception of different kinds of negative and positive curvature, which is what we get in, for example, the geometry of the Periodic Table of Mendeleyev, as Mendeleyev understood this; and as our dear friend, late departed friend Dr. Moon understood it.

So, we have these examples, and when we think about these in this geometric way, rather than a Cartesian, statistical way, then the universe becomes much more comprehensible to us. In particular, we're able to understand a point which was emphasized by Vernadsky, in his definition of the Noösphere: That we deal in physical chemistry, with three types of experimental evidence. In one case, the principles involved, that is, the curvature, the real physical curvature, corresponds to what we call an "abiotic," or non-living system. Then, we have--the same kind of experimental methods, show us that there exist, also, certain changes, which would not otherwise exist, which exist only in the efficient presence of living processes. This is the Pasteur-Curie line, for example. Then, thirdly, we recognize, that the ability of the human mind, to discover principles, and willfully apply them to the universe--rather than just watching the universe operate on them--willfully applying them to the universe, by our will, we change the universe. And therefore, man's relationship to the planet, for example, is such that man's power in the planet, and over the planet, has been increasing through man's existence. And thus, we have a third category of phenomena, where we see a universal principle of mind, of the human mind, of cognition, existing, in addition to living processes.

Therefore, you have these three types of systems: One, that we call the abiotic, because we define it that way experimentally. Second, living processes, which we define as distinct from the standpoint of experiment. And thirdly, by the great experiment of human progress, there is the third principle, the principle of cognition, which is sometimes called spiritual.

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