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The Fourth Dimension
GA 324a

Questions and Answers XIV

11 March 1920, Stuttgart

FIRST QUESTION: The Question is, does such an understanding correspond to reality? Since what we did in simple geometry also would have to be possible in all domains of mathematics, could understanding mathematical objects as intermediary links between archetype and physical image perhaps serve as a foundation for the types of calculations needed to support the physics presented in this lecture?

SECOND QUESTION: Might this be a path to the so-called hyper-empirical realm that we reach by controlling and enhancing our thinking?

If I understand your first question correctly, you are asking whether we can approach the realm of mathematics as an intermediary stage between archetype and physical image. [Note 72] [Note 73] Let's look at the domains of mathematics from a purely spiritual and empirical perspective. What are the spatial and geometric domains of mathematics? Or were you thinking of arithmetic as well?

Alexander STRAKOSCH: I was thinking of geometry.

During this lecture series, I have already suggested parenthetically how we arrive at ordinary geometric figures. [Note 74] We do not discover them by abstracting from empirical ideas. Initially, mathematical and geometric figures are an intuition of sorts. They are derived from the will nature of the human being, so we can say that when we experience mathematical figures, it is always possible for us to be active and to relate to reality in the mathematical domain. Thus, such figures, even on an empirical level, already represent a type of intermediate state between external realities (which we can possess only in image form) and the direct contents of being (which we experience inwardly). A spiritually empirical perspective would show that when we understand geometry, we grasp an intermediate stage between archetype and physical image.

However, there is something we must still do in order to verify this train of thought. If geometric and mathematical figures are indeed intermediate states between archetype and image, they must have a certain non-material ideal attribute that images do not have, though it only becomes so non-material in the sphere of images.

An image also can be a combination,—it does not necessarily correspond to its archetype. Any mere image that we confront need not correspond to an archetype. But if we have an intermediate state that incorporates a certain amount of reality, we need to be able to discover a corresponding specific field of reality, and we cannot combine such domains arbitrarily. We can never combine archetypes in a living way,—we must seek them out in their own domains, where they are present as distinct experiences. Thus, in order to grasp this middle domain in the right way, what you called the domain of the perceived lawfulness of mathematical objects, we also must understand its construction as an intermediate state between absolute, fixed archetypes and a boundless number of images. That is, we would have to interpret all of mathematics, and especially geometry, as inherently mobile, as existing at least in latent form in all of reality. For example, we could not imagine a triangle as immobile but would have to visualize the full scope of the concept. What is a triangle? A triangle is an area bounded by straight lines, and the sum of its angles is 180°. We would have to imagine the lengths of its three sides as being infinitely variable, and our definition would yield an infinite number of triangles, or a triangle in flux. This way of looking at things would result in a fluid geometry. [Note 75] We would have to be able to prove that this fluid geometry has some significance for the natural kingdom—that it corresponds to an aspect of the law of crystallization, for example. So the answer to your question is yes, this view is indeed based on an idea that corresponds to reality, but a great deal remains to be done to make the entire concept clear. I must still touch on another subject that plays into all this. You see, in recent times people have made a habit of taking refuge in higher dimensions when they want to enter higher domains of reality. That was not always the case in the formalism that formed the basis of our conceptions of the occult. In earlier times, people said that while we must conceive of ordinary physical figures as three dimensional, figures belonging to astral space must be seen in the context of a two-dimensional plane. Note that I am now talking about the spheres or planes of existence, and therefore the term astral is used in a sense different from the one I used when talking with Mr. Blümel and describing the steps between the physical body and the "1." We must imagine the next level, the Rupa plane, as one-dimensional in scope, and when we imagine the Arupa plane, we arrive at a point. [Note 76]

In this way we can say that as we move toward more spiritual ideas, the number of dimensions must decrease rather than increase. We are subject to this phenomenon when we move from above to below, as we do, for example, when we attempt the following train of thought. We can distinguish quite well among spirit, soul, and body. But what is the spiritual element in a human being walking around on Earth? We must say that this spiritual element is present in an extremely filtered form. We humans owe our abstract thinking to the spirit; it is the spiritual element in us. On its own, it tends to perceive only sense-perceptible objects and events, but the means of perceiving is spiritual. When we trace the spirituality of thinking down into the bodily element, we find that it has an expression in the human physical body, while the more comprehensive spiritual element has no such expression. Crudely speaking, one-third of the spiritual world in which we humans take part has an expression in the physical human body.

Moving on to the soul, two-thirds of the spiritual world in which humans take part achieve expression in the physical human body. And when we move on to the physical body, three-thirds has achieved expression. As we move from above to below, we must imagine that in the progression from the archetype to its image, the archetype easily leaves aspects of its being behind, and this phenomenon provides the essential characteristic of our physical aspect. In contrast, as we move upward, we discover new elements that have not been incorporated into the image. As we move downward, however, what we encounter is not merely an image,—reality plays into it. It is not true that at night when the physical and ether bodies are lying in bed, the astral body and "I" simply pull out of the body and leave it empty. Higher forces enter the physical and ether bodies and enliven them while the astral body and "1" are gone. Similarly, an image contains elements that do not originate only in its archetype. These elements enter when the image becomes an image, when it belongs to the entity.

Then the interesting question arises, How does a merely imaginatively combined image become a real image? That is when the other subject I mentioned enters in. Let me still comment that when we consider two dimensions, our initial train of thought leads directly to a second that can illuminate the first. All two-dimensional figures can be drawn in two dimensions, but figures that occupy three-dimensional space cannot. Suppose, however, that I begin to sketch a picture using colors instead of drawing in perspective or the like—that is, I copy colors,—I supply images of colors. Anyone will admit that I am then incorporating space directly into the plane to form the image. At this point I may ask, Does what expresses color in this image lie in any of the three dimensions of space? Is it possible to use colors to suggest something that can replace the three dimensions? Once we have an overview of the element of color, we can arrange colors in a specific way that creates an image of three-dimensionality in two dimensions. Anyone can see that all blues tend to recede, while reds and yellows advance. Thus, simply by supplying color, we express three dimensions. By using the intensive aspect of color to express the extensive aspect of three-dimensionality, we can compress three-dimensionality into two dimensions.

By linking other thoughts to this train of thought, we arrive at fluid geometry. And we may indeed be able to expand geometry to incorporate considerations such as this: In mathematics, we can construct congruent triangles A and B, but could we not also discover an expanded mathematical connection between red and blue triangles drawn in a plane? Is it really permissible for me simply to draw the simple lines that form a red triangle in the same way that I draw a blue triangle? Would I not have to state expressly that when I draw a red triangle and a blue one in the same plane, the red one would have to be small just because it is to represent red, while the blue one would have to be large simply because it is blue?

Now the question arises, Is it possible to incorporate an intensity factor into our geometry, so that we can perform calculations with intensities? This would reveal the full significance of how our right and left eyes work together. Stereoscopic vision depends on both eyes working together. In the domain of optics, this phenomenon is the same as grasping my left hand with the right. A being that could never touch one part of its body with another would be physically incapable of conceiving of the "I." This conception depends on being able to touch one part of my being with another. I can experience myself as an "I" in space only because of a phenomenon that is slightly hidden by ordinary empiricism, namely, the fact that my right and left vision crosses. This fact, though it does not encompass the reality of the "I," allows us to form a correct conception of the "I."

Now imagine how our physical ability to conceive of the "I" would be affected if our eyes were strongly asymmetrical instead of more or less symmetrical. What if your left eye, for example, was significantly smaller than the right, making your left and right stereoscopic images very different? Your left eye would produce a smaller image that it would constantly attempt to enlarge, while your right eye would have to attempt the opposite, namely, to reduce the size of its image. These efforts would add an enlivened form of vision to your static stereoscopic vision.

Real enlivened vision, however, must be achieved as soon as you even begin to approach imaginative perception. This perception results from constantly having to adapt asymmetrical elements to each other. The central figure in the Dornach sculpture had to be depicted as strongly asymmetrical in order to show that it is ascending to the spirit. It also suggests that every aspect of the human being—for example, our stereoscopic vision—is basically a state of balance that constantly deviates toward one or the other pole. We are human because we must continually create a state of balance between above and below, forward and back, and left and right.

  1. Question-and-answer session during the lecture cycle Geisteswissenschaftliche Impulse zur Entwickelung der Physik: Zweiter Naturwissenschaftlicher Kurs ("Spiritual Scientific Impulses for the Development of Physics: Second Scientific Course") (GA 321). Alexander Strakosch (1879-1958), railway engineer and teacher at the first Waldorf School in Stuttgart, asked these questions after giving a lecture on "Mathematical Figures as an Intermediate Link Between Archetype and Copy" in Stuttgart, March 11, 1920. To date no transcript of his lecture has been found.

  2. On the relationship between archetype and image in the context of mathematics, see also Rudolf Steiners essay on "Mathematics and Occultism" in Philosophy and Anthroposophy (GA 35).

  3. In the lecture of March 5, 1920 (GA 321). For further discussion of the evolution of geometric and mathematical views arising out of the will nature of the human being, see also Rudolf Steiner's lectures of January 3, 1920 (GA 320),—September 29, 1920 (GA 322),—March 16, 1921 (GA 324); and December 26, 1922 (GA 326).

  4. For a further discussion of fluid or mobile geometry, see also Rudolf Steiner's lecture of January 20, 1914 (GA 151).

  5. For more on the relationships between the planes or regions of the spiritual world and the higher dimensions, see also Rudolf Steiner's lectures of May 17 and June 7, 1905,—the question-and-answer sessions of April 7, 1921 (GA 76) and April 12, 1922 (GA 82), and the lectures of August 19, 20, 22, and 26, 1923 (GA 227).

    Ernst Blümel (1884–1952), mathematician and teacher. See Renatus Ziegler's Notizen zur Biographie des Mathematikers und Lehrers Ernst Blümel ("Notes on the Biography of Ernst Blümel, Mathematician and Teacher"), Dornach, 1995, in Arbeitshefte der Mathematisch-Astronomischen Sektion am Goetheanum, Kleine Reibe, Heft 1 ("Working Papers of the Section for Mathematics and Astronomy at the Goetheanum, Short Series, No. 1").