• "Four is the sign of the cosmos or of creation."

• "Seven is the number of perfection."

• "[A]ddition in itself is the arithmetical rule

particularly suited to phlegmatic children."

* "Subtraction ... is especially suited

to melancholic children."

— Rudolf Steiner

 

MYSTIC MATH

The Waldorf Curriculum:

Arithmetic and Beyond

 

Two plus two equals four. The rules and truths of mathematics do not change. Math is math, and therefore kids who learn math in a Waldorf school learn the same things that kids everywhere else are taught. Right? Well, not exactly.

As with most other subjects and activities, the study of math in Waldorf schools has an occult purpose. Math is not so much about learning how to handle numbers as learning to clairvoyantly perceive the gods' divine cosmic plan. That is, it is about becoming an occultist.

Here are statements made on the subject by Steiner and his followers. You'll find some truth in their statements — mathematics does indeed open doors to an understanding and appreciation of the structure of the universe. Whether any parts of mathematics ought to be considered sacred, however — and whether math leads us to occultism — may perhaps be debatable. The quotations take us somewhat far afield, but the ride is worth taking if you want to comprehend the Waldorf approach to math. As much as it may strain our credulity, all of the following comes within the context of mathematics as Steiner, using his marvelous clairvoyance, understood it.

“Mathematical science teaches the way to become independent of sense-perception, and at the same time it teaches the surest path; for though indeed its truths are acquired by supersensible means, they can always be confirmed in the realm of the senses ... No one can become an Occultist who is not able to accomplish within himself the transition from thought permeated with sense to thought emancipated from sense-perception. For this is the transition where we experience the birth of the ‘Higher Manas’ from the ‘Kama Manas.’" [1] 

“Through the schooling in the spirit of mathematics lies one of the paths to the purification from life in the senses. And just as the mathematician is consistent in life, just as he is able to construct bridges and bore tunnels by virtue of his training — that is to say, he is able to command the quantitative reality, in the same way, only he will be able to understand and rule the qualitative, who can make himself master in the ethereal heights of sense-free perception. This is the Occultist. Just as the mathematician builds the shapes of iron into machines according to mathematical laws, so does the Occultist shape life and soul in the world according to the laws of these realms which he has understood in the spirit of mathematical science.” [2]

“Teachers might give descriptions and tell stories that bring the inner qualities of numbers to light, and they will try to inspire their students to search for numbers and their revelations in the world of nature.” [3]

“Those who steep themselves in the right way in what, in the Pythagorean sense, we may call the 'study of numbers,' will learn to understand life and the world in this number symbolism.” [4] 

 

“[N]umbers and numerical proportions have a certain meaning for the cosmos and the world. It is in numbers, we might say, that the harmony that wells through space is expressed.” [5] 

“[N]umbers can give you a clue to what is called meditation if you have the key to plunge deeply enough.” [6]

“Those who draw attention to this remarkable law of numbers explain it in an altogether materialistic way; but the weight of the facts themselves is already compelling people today once again to recognise the spiritual, mathematical law prevailing in the things of the world. We see how deeply true it is that everything which comes to expression in personal form in the later course of human evolution is a shadow-image of what was present formerly in elemental, original grandeur, because the connection with the spiritual world was still intact.” [7]

“[F]acts compel people today to rediscover in the succession of outer events certain regularities, certain periodicities, which are connected with the old sacred numbers.” [8]

“Basic geometric concepts awaken clairvoyant abilities.” [9]

“Times is the fourth dimension. It remains invisible within the three dimensions of ordinary space and can be perceived only with clairvoyant powers." [10]

“In the astral world, urges and passions take on animal shapes ... Because these astral beings can also make use of other bodies, it is dangerous to allow mediums to go into a trance without the presence of a clairvoyant who can ward off evil ... [W]hen we enter the astral realm of numbers, time, and morality, we are dealing with a complete mirror image of everything we customarily think to do here on the physical plane." [11]

"[F]our-dimensional space is related to what the alchemists called transformation ... When we truly meditate on pure concepts [such as those in math], when we allow these concepts to work on our souls ... light arises out of concepts ... In the language of alchemy, water plus light equals mercury ... After we awaken out ability to create light out of our work with pure concepts, mercury comes about as the mingling of this light with our vision of water." [12]

Waldorf math teachers may not state their occult views openly in class. But sometimes they may. These are the views they embrace, the "meanings" they find behind their subject — and naturally, on occasion, they voice such meanings. As we will see presently, they may lead students to the view that mathematics proves occult or at least spiritualistic beliefs. And to do this, they may not always be above twisting the numbers.

◊◊◊◊

The study of math as perceived by Steiner and his followers certainly takes us in unexpected directions. Here's a striking example, touching on a subject I've discussed on other pages at Waldorf Watch. Steiner said that a strictly mathematical comprehension of Copernicus's third law leads us to the unmistakable conclusion that the Earth does not orbit the Sun.

“[T]he third Copernican law has been disregarded. People preferred the easy assumption that the Earth rotates around its axis in twenty-four hours, progressing all the while so as to move around the Sun ... [But the] Sun moves; it would outrun the Earth if the Earth simply revolved around the Sun. 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." [13]

That's the sort of wisdom Steiner found in math.

Let's circle (dare I say orbit?) back to one of the quotations we saw, above. Here it is at greater length:

“[T]here is in Berlin an interesting doctor who has made remarkable observations based entirely on facts, apart from any theory. I will indicate it on the blackboard. — Let us suppose that this point represents the date of a woman's death. I am not speaking of a hypothetical case but of something that has been actually observed. — The woman is the grandmother of a family. A certain number of days before her death a grandchild is born, the number of days being 1,428. Strange to say, 1,428 days after the grandmother's death another grandchild is born, and a great-granddaughter 9,996 days after her death. Divide 9,996 by 1,428, and you have 7. [Steiner taught that 7 is the number of perfection.] After a period, therefore, seven times the length of the period between the birth of the first grandchild and the death of the grandmother, a great grandchild is born. And now the same doctor shows that this is not an isolated case, but that one may investigate a number of families and invariably find that in respect of death and birth absolutely definite numerical relationships are in evidence. And the most interesting point of all is that if, for example, you take the number 1,428, again you have a number divisible by seven. In short, the very facts compel people today to rediscover in the succession of outer events certain regularities, certain periodicities, which are connected with the old sacred numbers. And already today the number of findings in this direction collected by Fliess — such is the name of the doctor in Berlin — and his students, is a proof that the sequence of such events is regulated by quite definite numbers. These figures are already available in overwhelming quantity. The interpretation placed upon them is thoroughly materialistic, but the facts themselves compel belief in the factor of number in world-happenings. I must emphasize that the application of this principle by Fliess and his students is extremely misleading and erroneous. The way he applies his main numbers, especially 23 and 28 — 28 = 4 times 7 — will have to be amended in many respects. Nevertheless, in a study such as this we can see something like an instinctive emergence of ancient Babylonian culture in the age when mankind is on the path of ascent. Of course, such things are confined to small circles; the vast majority of people have no feeling for them.” [14]

The "small circle" Steiner addressed consisted of his followers, Anthroposophists. He sought to expand this circle through the work of Waldorf schools, among other efforts. Thereby, the occult significance of mathematics — and all other things — would become more widely apprehended. And yet did Steiner show us anything except numerical flummery? Here's another example (mercifully brief):

“The danger of succumbing to the realm of [the demon] Ahriman was at its greatest around the year 333 BC. This was the moment in time when humanity began to make use of mere intellect, mere logic. Then the mystery of Golgotha [i.e., Christ's Crucifixion] occurred and entered the life of humanity. And from AD 333 onwards the time began in which human beings must consciously strive to re-enter the realm of the higher hierarchies...

           " 333 BC

         [+] 333 AD

        666

                                                                                              

"If you add these two numbers together you get 666. That is the 'number of the beast'....”  [15] 

It is always possible to get the numerical answer you want if you are willing to play with the numbers that "lead" to it. Note that Steiner says Ahriman was at his peak "around" 333 BC. Perhaps. But the math gets mighty fuzzy — especially if the calculation also depends on the year 333 AD as being exactly the time when humans had to start consciously reentering "the realm of the higher hierarchies." All that Steiner has done is to get the answer he wanted, 666, by arbitrarily pinpointing two nebulous dates. Granted, 333 +333 equals 666. But then so does 200 + 466, or 665 + 1, or...

◊◊◊◊

I may have mentioned that I attended a Waldorf school. The math teacher there spoke, from time to time, about the Platonic nature of phenomena.

All "real" phenomena are actually just shadows of their ideal prototypes that exist in a supersensible realm. That realm is real, and indeed the ideal phenomena there are more real than the "real" things we see around us. A chair, for instance, or a geometric form, or indeed a mathematical result, is only a poor manifestation of the ideal chair, form, or result existing out there in the wild blue ether. (I'm not using his exact words, which I can no longer recall.)

The lesson I took away was that the "real" world is unreal, while the "ideal" or supernal world is reality. I didn't learn much math, but I absorbed this occult lesson, and I took it very much to heart. When I saw Jimmy Stewart in the play "Harvey" — about a man who believes he is accompanied everywhere by an invisible six-foot-tall rabbit — I took it seriously. Sure, invisible realities are all around us. You say you have a six-foot-tall spirit-rabbit pal? Who am I to gainsay it?

I left Waldorf a seriously befuddled lad.

◊◊◊◊

We find mathematical hocus-pocus in Waldorf instructional materials published quite recently — teachers' guides for Waldorf math courses. For instance, drawing on indications given by Steiner, Waldorf educator John Blackwood recommends the following math exercise. Determine the average respiration rate for children in a class. The result should be approximately 18 breaths per minute. Multiply this by 60 to get the number of breaths per hour, then multiply that result by 24 to get the number of breaths per day. The result should be 25,920. [16]

Now, the “Platonic Cosmic Year” (also called the “Great Year”) is the length of time required for the Sun to cycle through all 12 signs of the zodiac. (That is, the Great Year is the period of the precession of the equinoxes.) The length of this “Year” is 25,920 regular Earth years, more or less.

Now, calculate the number of days in a human life. Multiply 72 (years) by 360 (days) and what is the result? 25,920!

Wow! This surely indicates the great design of the universe, no? The number of breaths we take in a day is equal to the number of days we live, which in turn is equal to number of years in a “Great Year.”

Bingo. From this, we can plainly see that human beings are microcosmic representatives of the great macrocosm, the vast universe presided over by the divine powers! Here’s how the Waldorf teacher puts this: “Put these [results] side by side and it all gets interesting — the human being is surely the microcosm in the macrocosm. Man is made in the image of God.” [17]

Perhaps we were made in the image of God, but this mathematical sleight of hand does not prove anything of the sort. Break it down:

◊ How many breaths do we take per minute? This varies widely, depending on all sorts of factors. The results can be anything from about 10 to about 30. But, so as not to quibble, let’s try the calculation using 20 breaths per minute instead of 18. 20 x 60 x 24 = 28,800. If we accept this new result, our nice little paradigm wobbles: We want an answer of 29,520, not 28,800. So let’s ignore the new result.

◊ How long is the “Platonic Cosmic Year” or “Great Year”? According to many sources, it is 26,000 years, more or less. Other sources, aiming for greater precision, say it is 25,800 years, and some zero in on 25,765 years. [18] Sadly, none of these numbers is precisely what we want (25,920). So let’s ignore them.

◊ How many days do we will live? The Biblical span of three score years and ten is 70 years. A year is 365 days long. So, the average life should be 70 x 365 = 25,550 days. To be more precise, a year is really 365.256 days long (remember leap year), so the result is 70 x 365.256 = 25,568.2 days. Neither answer, sadly, is 25,920. So let’s ignore these answers, too.

Where does all this get us? Nowhere. We see one of Steiner's followers playing the same tricks Steiner played, cooking the numbers to get a desired result. In Waldorf schools, realities may be ignored in precisely this way while arbitrary “results” and “harmonies” and “symmetries” are offered to “prove” occult doctrines. [19] It is nonsense, but it is also something worse than nonsense. It is potentially very harmful to children, since it may lure them away from rationality and into an occultism. [20]  

◊◊◊◊

Not all Waldorf math instruction is suffused with occultism. Some of it is merely superficial. For example, H. v. Baravalle’s GEOMETRY  [21] sidesteps logical proofs of the kind that are central to geometry as taught in most schools. Instead, the book emphasizes looking at and creating pretty geometric designs. This is fun for kids, and it may have some educational value — but it is intellectually lightweight. The same holds for the same author’s TEACHING ARITHMETIC AND THE WALDORF SCHOOL PLAN [22], which dwells on such matters as “magic squares” (in essence, what today we might call simplified sudoku) — fun, but trivial.

^{Waldorf schools often teach forms of math that are far from rigorous. But that is not the main point. Let's stipulate that making math fun is good, and let's stipulate that approximately correct answers are better than thoroughly wrong answers (call it fuzzy math and say it's okay). In other words, let's say that Waldorf math teachers emphasize correct answers as much as teachers in other schools do. None of this would remove the fundamental flaw in Waldorf-style math, which is that it purports to find occult significance in math. Waldorf math is a place of magic, "sacred" numbers, "sacred" geometry, and the like. It is a place where occultism is prepared. Anthroposophists may try to make this seem unobjectionable by indicating, for instance, that their beliefs are essentially Christian. [23] Steiner himself did this. But Waldorf "Christianity" is woven with un-Christian beliefs (polytheism, reincarnation, evolution, the Sun God, nature spirits, and so forth). Bear this in mind when reading Steiner's explicit admission that spiritualistic attitudes should be cultivated in all academic subjects, including math. "It is possible to introduce a religious element into every subject, even into math lessons. Anyone who has some knowledge of Waldorf teaching will know that this statement is true. A Christian element pervades every subject, even mathematics. This fundamental religious current flows through all of [Waldorf] education." [24] If you subscribe to the religious doctrines of Anthroposophy, you may approve. If not, you may want to look for a different sort of school for your children.} 

◊◊◊◊

You may find some benefit in considering what passes for reasoning among Steiner and his followers. They explicitly downplay "mere intellect, mere logic." Their opposition of rationalism leads them to the sorts of propositions we've seen, above.

What do Steiner and his follower turn to instead of rationality? Pass your paper to your neighbor. Hint: Consider the titles of Steiner texts I've quoted here:  “Mathematics and Occultism”, OCCULT SIGNS AND SYMBOLS, OCCULT HISTORY, and THE FOURTH DIMENSION: Sacred Geometry, Alchemy, and Mathematics. In brief: the answer is occultism. Did you get it? Excellent, A+. Now, for extra credit...

— Roger Rawlings

---

AFTERWORD

Steiner repeatedly stressed the importance of the wonderful number 25,920 for a while. His account was somewhat more sophisticated than Blackwood's, but not much. Here is one version (Steiner explained this wonderful matter more than once):

"The vernal point must travel along the whole Zodiac and it will then return to its point of departure. The time required for this will be about 25,920 years. These 25,920 years are also designated as the so-called PLATONIC YEAR [sic] ... Thus we may say: These 25,920 years are most important for the life of the sun, because during that period the sun's life passes through a unity, through a real unity, a complete whole. The next 25,920 years are a repetition. Thus we obtain a rhythmic repetition of this unity, consisting of 25,920 years.

"...On the average, a human being breathes 18 times a minute. This may, of course, vary, for our breathing is different in our youth, and in old age, but if we take an average, we obtain as a normal figure for the respiration, 18 breaths a minute. We thus renew our life rhythmically 18 times a minute. Let us now see how often we do this in one day. In one hour this would be equal to 18 x 60 = 1080. In 24 hours: 1080 x 24 = 25,920, that is to say, 25,920 times.

"...Now we might say: Something, therefore, breathes within us, yet it is another kind of breathing, it is something which rises and falls ... it breathes within us in the course of one day, in the same way in which something breathes within us during the 18th part of a minute. Something breathes within us in the course of one day. Let us now see if that which breathes within us in the course of one day, if the rising and falling of our etheric body, which thus breathes within us, also sets forth something which resembles a circular movement, a return to a point of departure. In that case, we would have to investigate what 25,920 days really are. For 25,920 of these breaths, in which the etheric rises and falls, would have to correspond, in their rise and fall, to a reproduction of the platonic year. Just as one day corresponds to 25,920 respirations, so 25,920 days should also correspond to something in human life. How many years are 25, 920 days? Let us see.

"Let us take the year with an average of 365? days, let us make a division and then we shall obtain as a result of the division 25,920 ÷ 365.25 = about 71  that is to say, about 71 years, which is the average duration of human. Life. Of course, the human being has his freedom and frequently he may grow much older. But you know that the patriarchal age is indicated as 70 years. Thus you have the duration of human life equal to 25,920 days, 25,920 of such great breaths! Once more, we obtain a cycle which reproduces microcosmically in a wonderful way the macrocosmic happenings. Thus we may say: If we live one day, we reproduce the platonic world-year with our 25,920 respirations; if we live 71 years, we again reproduce the platonic year with 25,920 great breaths, with the rising and falling pertaining to our waking up and our falling asleep.

"We may now pass on from this to something which would lead us too far, if I would explain it in detail to-day; but I shall indicate what may be felt occultly.

"...Let us now try to see if it is possible to speak of a similar breathing process when we place ourselves within the whole platonic year of the sun. In that case, we would have 25,920 years. Let us now consider these 25,920 years as ONE year and investigate its relationship to one day. If we wish to consider the whole platonic year as one year and if we then wish to discover what would constitute one of its days, we would have to divide it by 365 1/4, and this would give us one day. If the whole represents one year and if we then divide it by 365 1/4, we obtain ONE day. Let us see what result we reach when we divide 25,920 years by 365 1/4. We obtain 71 years, which is the duration of a human life.

"...If we consider our physical body, we have within this physical body which passes through its patriarchal age, one breath of that great Being, whose life is so long, that 25,920 years correspond to one year. Our patriarchal age (71 years) is in that case equivalent to one day of that Being. If we therefore think of a Being that lives together with our earth, alternating day and night in the course of 24 hours, this would represent one respiration for our etheric body; the true respiration of our astral body would be equivalent 1/18th part of a minute.

'...Let us now return to the earth. It breathes us in and out in the course of one day. And let us now go to the air, which forms part of the earth. It breathes us in and out in 1/18th of a minute; yet the number 25,920 always constitutes a return to the point of departure. This shows us a regular rhythm; we feel that we are standing within the universe; we learn to know that human life, and one day of human life, are, for greater and more encompassing Beings, equivalent to one of the breaths which we ourselves draw in our own life. And if we take up this knowledge through our feeling; the old saying, according to which we repose in the bosom of the universe, acquires an extraordinary significance." [25] By the way, on other occasions Steiner said that the earth breathes much more slowly than he indicates here.

---

"Four is the sign of the cosmos or of creation."

[Rudolf Steiner, OCCULT SIGNS AND SYMBOLS (Anthroposophic Press, 1972), p. 43.]

"Seven is the number of perfection."

[Ibid., p. 40.]

"[A]ddition in itself is the arithmetical rule particularly suited to phlegmatic children."

[Rudolf Steiner, DISCUSSIONS WITH TEACHERS (Anthroposophic Press, 1997), p. 49.]

"Subtraction ... is especially suited to melancholic children."

[Ibid., p. 49.]

— Rudolf Steiner

Waldorfish math is even more mystical than I've let on in this essay.

To gaze upon other aspects of this subject, see

"Magic Numbers" and "Temperaments".

---

"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.]

---

"THE PROBLEM OF DIVERSITY.

"From Kircher's Ars Magna Sciendi.

"In the above diagram Kircher arranges eighteen objects in two vertical columns and then determines he number of arrangements in which they can be combined. By the same method Kircher further estimates that fifty objects may be arranged in 1,273,726,838,815,420,339,851,343,083,767,005,515,293,749,454,795,408,000,000,000,000 combinations. From this it will be evident that infinite diversity is possible, for the countless parts of the universe may be related to each other in an incalculable number of ways; and through the various combinations of these limitless subdivisions of being, infinite individuality and infinite variety must inevitably result. Thus it is further evident that life can never become monotonous or exhaust the possibilities of variety." [Manly P. Hall, THE SECRET TEACHINGS OF ALL AGES, http://www.sacred-texts.com/eso/sta/sta03.htm]

---

[Geometric designs 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 I believe that 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.

[R. R., 2010, based on a design by Alberta Hutchinson.]

 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 "clairvoyance."

(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. R.R. sketch, 2010.]

---

[OLD-TIME CUTS AND ORNAMENTS (Dover, 2001), p. 32.]

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. 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; what we think comes to pass, literally, because we think it. 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. We find truth by discovering it as it objectively is, apart from any preferences, moods, or wishes of our own. Objectivity is difficult, but it is the goal we should aim for. 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.

---

Each of us is a microcosm.

A pretty idea, perhaps.

But...

[R.R., 2010.]

---

A non-Anthroposophical version of the micro/macro concept

[Lewis Spence, AN ENCYCLOPEDIA OF OCCULTISM (Dover, 2003), facing p. 256.]

The occultism found behind and in Waldorf schools

differs from other forms of occultism is many ways —

it derives directly from Rudolf Steiner’s doctrines.

But there are also many links between Anthroposophy

and occultism at large.

---

Waldorfish art, as it were

[R. R., 2010].

---

ENDNOTES

^{[1] Rudolf Steiner, “Mathematics and Occultism” [THE ANTHROPOSOPHICAL MOVEMENT, Vol. V, No. 28; GA 35.]}

"Manas" is a Theosophical/Anthroposophical term. Here is a portion of the definition as given in the ENCYCLOPEDIC THEOSOPHICAL GLOSSARY (1999, Theosophical University Press.) "Manas (Sanskrit) [from the verbal root man to think] The seat of mentation and egoic consciousness; the third principle in the descending scale of the sevenfold human constitution. Manas is the human person, the reincarnating ego, immortal in essence, enduring in its higher aspects through the entire manvantara. When imbodied, manas is dual, gravitating toward buddhi in its higher aspects and in its lower aspects toward kama. The first is intuitive mind, the second the animal, ratiocinative consciousness, the lower mentality and passions of the personality." Manvantara is a period of manifestation. Buddhi is an advanced stage of spiritual consciousness, the transformed etheric body. Kama is desire, which is "colorless" and either good or bad, depending.

[2] Ibid.

[3] Robert Trostli, RHYTHMS OF LEARNING: What Waldorf Education Offers Children, Parents & Teachers (SteinerBooks, 1998), p. 123.

[4] Rudolf Steiner, OCCULT SIGNS AND SYMBOLS (Anthroposophic Press, 1972), lecture 3, “The Symbolism of Numbers”.

[5] Ibid.

[6] Ibid.

[7] Rudolf Steiner, OCCULT HISTORY (Rudolf Steiner Press, 1982),  p.77.

[8] Ibid., p. 75.

Steiner spoke of sacred numbers as well as sacred geometry. For more on the occult significance of numbers, see "Magic Numbers".

[9] Rudolf Steiner, THE FOURTH DIMENSION: Sacred Geometry, Alchemy, and Mathematics (Anthroposophic Press, 2001), p. 92.

[10] Ibid., p. 74.

[11] Ibid., pp. 24-25.

For more on mediums, see "seances".

[12] Ibid., pp. 39-40.

[13] Ibid., p. 128.

Kepler promulgated a third law; so far as I can determine, Copernicus did not. The editor of THE FOURTH DIMENSION says Steiner was referring to "movement in declination" as Copernicus described it. [THE FOURTH DIMENSION, p. 206.] Here is a more complete indication: "Generally we hear that Copernicus taught only two things: that the Earth revolves on its own axis and that the Earth moves round the Sun. It is seldom noticed that he taught also a third form of movement — that the whole solar system moves onward in a spiral." [Rudolf Steiner, AT THE GATES OF SPIRITUAL SCIENCE (Rudolf Steiner Publishing Co., 1986), lecture 11, GA 95.] 

[14] OCCULT HISTORY,  p.75.

[15] Rudolf Steiner, GUARDIAN ANGELS (Rudolf Steiner Press, 2000), pp. 96-97.

[16] "This correspondence was pointed out by Rudolf Steiner.”  [John Blackwood, MATHEMATICS IN SPACE AND TIME (Floris Books 2006), p. 100.] Both Steiner and Blackwood acknowledge that there is some imprecision in the calculations, so the number 25,920 is an approximation. Whether this is helpful in math class is questionable (is 2 + 2 = 4.1 approximately correct or simply wrong?), and it does not remove the central problem, that the astrological and other occult premises behind finding significance in these calculations are baseless.

[17] MATHEMATICS IN SPACE AND TIME, p. 102.

[18] See the following sources, among others: ENCYCLOPEDIA BRITANNICA, library.thinkquest, NASA (www-istp.gsfc.nasa.gov), and absolute astronomy.com.

[19] To put this another way: We can always get the results we want if we play fast and loose with numbers. So, if we say that an average life is 72 years long instead of 70, and that a year is 360 days long instead of 365, we can get the result we wanted from the beginning: 25,920. But this is not a real result, it is merely the number we were determined to get, come hell or high water.

There’s a larger point, too. Let’s say that we don’t fiddle with any numbers — let's say that scrupulous calculation really shows that the number of breaths in a day is equal to the number of days in a lifetime, and this in turn is equal to the length of a Great Year. Have we proven anything? Or have we simply found a coincidence? Specifically, have we proven that we are microcosms of the universe? Or have we made a huge, illogical leap? Consider the following, for instance. What if instead of using breaths per minute we use the average number of heartbeats per minute? (Like breaths, heartbeats are extremely variable, but let’s pretend that they aren’t.) Some sources give an average of 50 beats per minute. 50 x 60 x 24 = 72,000. This is nowhere close to 25,920, so our nice little paradigm is knocked to pieces. For this reason, a Waldorf teacher will insist on breaths per minute, not heartbeats per minute. To get a predetermined answer, you must take care to select only the data that will produce that answer. In other words, you must fudge.

(Anyway, the heart is relatively unimportant. Steiner taught that it does not pump blood. Science “sees the heart as a pump that pumps blood through the body. Now there is nothing more absurd than believing this, for the heart has nothing to do with pumping the blood.” [Rudolf Steiner, FREUD, JUNG, AND SPIRITUAL PSYCHOLOGY, pp. 124-125.] I repeat this quotation pretty often, as you may have noticed. It helps us gauge the degree of truth in Steiner's teachings.)

Steiner fudged with numbers incessantly. For occult reasons, he was determined to categorize phenomena in groups of seven and twelve, for instance. He very often succeeded, and this impresses some people. See the design of the universe? But forcing phenomena into pre-selected, arbitrarily delimited brackets proves nothing. You can always get the results you are determined to get, if you are willing to cur, trim, and paste to suit.

[20] The exercise suggested in MATHEMATICS IN SPACE AND TIME amounts to Anthroposophical indoctrination or at least softening. At first blush, members of mainstream Western religions may think the exercise seems okay: We are created in God's image; who wants to deny this? But consider whether your faith includes astrology, as in the concept of the Great Year. Christianity, Islam, and Judaism do not embrace astrology. Anthroposophy does (albeit an odd astrology reworked by Steiner).

[21] H. v. Baravalle, GEOMETRY (Publications of the Waldorf School, Adelphi College, 1948). The book is a junior high school teachers' guide, so we should cut it some slack. However, most guides and textbooks for junior high geometry in regular schools are considerably more substantial.

[22]  H. v. Baravalle, TEACHING ARITHMETIC AND THE WALDORF SCHOOL PLAN, (Publications of the Waldorf School, Adelphi College, 1950).

[23] Christians, Hindus, and Zoroastrians may be attracted to Anthroposophy, since it contains elements of their faiths. But they should realize that Anthroposophy diverges far from their faiths in many of its others doctrines. Meanwhile, of course, Muslims, Jews, secularists, and most others should realize that Anthroposophy explicitly rejects their viewpoints.

[24] Rudolf Steiner, THE CHILD's CHANGING CONSCIOUSNESS AS THE BASIS OF PEDAGOGICAL PRACTICE (Anthroposophic Press, 1996), p. 94.

[25] Rudolf Steiner, “Man’s Position in the Cosmic Whole, the Platonic World-Year” (ANTHROPOSOPHIC NEWS SHEET,  Jan. 8, 1940, No. 1-2).