"Is there a formula in numbers, or any other means by which we can express the respective relationships of force in the physical, the etheric and the astral bodies? ... I shall now sketch the relationship of these for you on the blackboard. If we consider deeply this figure on the blackboard...it is like a part of the occult script [i.e., the Akashic Record] on which we can meditate ... When, in meditation, we give ourselves up to this occult sign and evoke a certain inward feeling of the relationship of these...surfaces to one another, we obtain an impression of the mutual proportions of the physical body, etheric body and Ego."  — Rudolf Steiner, WONDERS OF THE WORLD (Kessinger, facsimile of 1929 edition), pp. 40-42. 

[Illustration from p. 40; I have added color to the b&w image.]

◊

[Design from Wil Stegenga's

PICTORIAL ARCHIVE OF GEOMETRIC DESIGNS

(Dover Publications, 1992).

I have added colors and spooky shadings

— R.R., 2010.]

Because Steiner said that geometry helps foster clairvoyance, this branch of math receives special emphasis in Waldorf schools. Students are often led to create dramatic geometric designs. Here is one I did recently, more or less capturing the feel of designs I remember creating as a Waldorf student. (But, if I recall, all of them consisted of straight, intersecting lines connecting points on a circle, and the choice of color was more programmatic. Still, the result was similar: swirling, concentric rings.)

[R. R., 2010.]

Clearly, such designs are related to mandalas, which are traditional aids to religious meditation.

[Design by Alberta Hutchinson;

I have tweaked it and altered the colors

— R.R., 2010.]

Anthroposophists show considerable interest in mandalas. Not long ago, for instance, a Rudolf Steiner Institute summer course bore the title "The Mandala: An Archetype of Self and World."

◊

[Anthroposophic Press, 2001.]

From the back cover: 

Steiner discusses, among other things, 

"The relationship between geometric studies 

and developing direct perception 

of spiritual realities."

For "direct perception" 

read "clairvoyant sight."

(As for alchemy...)

◊

Some people's math is other people's moonshine. Here is Steiner explaining, as he often did, that the planets move along geometric lines — but not ellipses or circles, since they do not orbit the Sun. 

"I have pointed out that the Earth moves behind the Sun in a screw-like line, the Earth moving along always with the Sun. And then if we view the line from above, we get a projection of the line and the projection shows a lemniscate." — Rudolf Steiner, MAN: HIEROGLYPH OF THE UNIVERSE (Rudolf Steiner Press, 1972), lecture 6, GA 201. 

The planets do indeed move along "screw-like" trajectories, since they orbit the Sun and the Sun is in motion (it orbits the center of the galaxy). SoSteiner would almost be correct, except that he denied (sometimes) that the planets orbit the Sun.

◊ "[I]t is not that the planets move around the Sun, but that these three, Mercury, Venus, and the Earth, follow the Sun [in a line], and these three, Mars, Jupiter, and Saturn, precede it." — Rudolf Steiner, FACULTY MEETINGS WITH RUDOLF STEINER (Anthroposophic Press, 1998), p. 30.

◊ “The Earth progresses, but exactly in a line behind the Sun ... [T]he Earth moves along behind the Sun.” — Rudolf Steiner, MAN - HIEROGLYPH OF THE UNIVERSE (Rudolf Steiner Press, 1972), p. 33.

◊ "The Earth cannot revolve around the Sun because meanwhile the Sun would move away from it. In reality, the Sun moves on, and the Earth and the other planets follow it. [They move in] a line like the thread of a screw, with the Sun at one point and the Earth at the other end. Our dual focus on the Earth and Sun and on their progressive, screwlike movement creates the illusion that the Earth is revolving around the Sun." — Rudolf Steiner, THE FOURTH DIMENSION (Anthroposophic Press, 2001), p. 128. 

Wrong. Wrong. Wrong.

And as for the Earth's path describing a lemniscate... I'd suggest you ask an astronomer.

[R.R. sketch, 2010.]

◊

The mathematics teacher at the Waldorf school I attended often said that if you enter a city from one direction, it will be a different city than if you enter it from another direction. I remember questioning him about this. He said he did not simply mean that you would see the city differently; he meant that the city would literally, truly, be different. This concept is consistent with Anthroposophy, which teaches that thoughts exist as real beings in the spirit realm. Such beings have power to shape and alter reality. And we humans (whose real home is the spirit realm) share this power. The universe is malleable — our subjective states make the universe different from what it might be if we had different subjective states. Even the import of mathematics would be different if we had a different mental attitude.

This is the radical subjectivity promoted by Waldorf education, and it is clearly wrong. Facts are facts; they do not bend to our preferences. Truth is truth; we do not make something true by thinking it. (You do not alter the nature of New York City, for instance, by perceiving it from the east instead of from the west.) We find  truth by discovering it as it objectively is,  apart from any preferences, moods, viewpoints, or wishes of our own. Attaining objectivity is difficult, sometimes extremely so. Some would argue that it is unattainable. Nonetheless, if we want to deal with the universe as it is, not as we might wish or fear or dream or worry that it is — if we want to deal with reality, not fantasies, we must strive for objectivity. Waldorf schools, however, follow Steiner in stressing subjectivity.

"[F]acts are simply upon occasion quite different from the concepts we hold of them with our present-day abstract, theoretical grasp of things, so remote from life and reality. There is not even a true grasp of what it might mean to take in the sciences of arithmetic and geometry with quite another soul-constitution than we have to-day, with a mature soul-condition ... The mathematics of the universe, which have become so thoroughly abstract to us, revealed [in the ancient past] something really living, because the revelation found completion in what was brought to understand it." — Rudolf Steiner, THE TWO CHRISTMAS ANNUNCIATIONS (Anthroposophic News Sheet 1938, Supplement 5-6, GA 203. 

Steiner was speaking, here, about the mystical meaning of mathematics and geometry (or what is sometimes called sacred geometry). He found occult meaning in numbers, in geometrical design, and indeed in all orderly phenomena. This is what "a mature soul-condition" may find. But is it truly an apprehension, or merely a subjective desire? Is it found in phenomena, or is it read into them?

Our subjective states are, of course, important. How we feel about things is, of course, important. The spirit in which we act is, of course, important. But recognizing the importance of such things should not muddle us. Our inner states are important, but they are separate from — and do not control — outer, objective reality. Steiner's teachings result in such concepts as the following: 

"If these ideas [i.e., the doctrines of Anthroposophy] are not true, they should be true. What we believe shapes the reality. If we become conscious of these ideas and hold them, they will become true." — Dr. Ronald E. Koetzsch, an Anthroposophist connected with the Association of Waldorf Schools of North America.

What we believe certainly may shape reality if we act on our beliefs — but believing,  in and of itself, cannot make false ideas true any more than it can change the laws of mathematics or the objective facts about the universe. A city is what it is, no matter what street we drive along to enter it. Thinking otherwise does not create higher truths; it is self-deception. Training children to "think" in this manner does them a severe disservice.

◊

"Arithmetic and geometry, indeed all mathematics, occupy a unique position in education. Education can only be filled with the necessary vitality and give rise to a real interplay between the soul of the teacher and the soul of the child, if the teacher fully realizes the consequences of his actions and methods. He must know exactly what effect is made on the child by the treatment he receives in school, or anywhere else...

"Now it is the physical and etheric bodies which are affected when the child is taught arithmetic or geometry, or when we lead him on to writing from the basis of drawing and painting. All this remains in the etheric body and its vibrations persist during sleep. On the other hand, history and such a study of the animal kingdom...work only upon the astral body and Ego-organization. What results from these studies passes out of the physical and etheric bodies into the spiritual world during sleep...

"[T]he remarkable thing is that arithmetic and geometry work upon both the physical-etheric and the astral and Ego ... During sleep...our body of formative forces continues to calculate, continues all that it has received as arithmetic and the like. We ourselves are then no longer within the physical and etheric bodies; but supersensibly, they continue to calculate or to draw geometrical figures and perfect them... 

"Now the point is that everything conveyed in an external way to the child by arithmetic or even by counting deadens something in the human organism. To start from the single thing and add to it piece by piece is simply to deaden the organism of man. But if we first awaken a conception of the whole, starting from the whole and then proceeding to its parts, the organism is vitalised. This must be borne in mind even when the child is learning to count...."

— Rudolf Steiner, EDUCATION (Rudolf Steiner Publishing Co., 1943), lecture 9, GA 307.

[R. R., 2013.]