## MYSTIC MATH

### The Waldorf Curriculum:

### Arithmetic and Beyond

◊ "It is possible to introduce a religious element

into every subject, even into math lessons."

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

Seven is the number of perfection ... [etc.]"

◊ "Adding is related to the phlegmatic temperament,

subtracting to the melancholic, multiplying to the...."

◊ “[G]eometric concepts awaken clairvoyant abilities.”

— Rudolf Steiner

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. Waldorf 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 these 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, properly comprehended, 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 grasp 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]

◊ “Time 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 usually state their occult views openly in class. But sometimes they do state them. 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, Waldorf math teachers may lead students to the view that mathematics proves the truth of occult or at least spiritualistic beliefs. And to lead students in this direction, Waldorf math teachers may not always be above twisting the numbers.

### <<◊>>

For Steiner and his followers, studying numbers comes close to affirming numerology. They see mystical significance in numbers, and they affirm the ancient concept of "sacred numbers" — the notion that some numbers have special occult significance, being imbued with sacred power. Here's one example, taken from a lecture by Rudolf Steiner:

“[T]here is in Berlin an interesting doctor who has made remarkable observations ... I will indicate it on the blackboard ... [S]uppose that this point represents the date of a woman's death ... 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 ... 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.” [13]

Essentially, in the Waldorf belief system, math entails playing with numbers and geometric shapes in order to "discover" ideas that Anthroposophists already believe and are absolutely determined to affirm. Thus, they decide that 7 is the number of perfection. Then they make calculations (based in this case on unsubstantiated anecdotes) that enable them to arrive at the marvelous result: 7. So, golly! Seven!

But once we have recovered from our astonishment (if any) occasioned by such elaborate occult reasoning, we might ask whether anything has actually been proven. Have we learned, for instance, that 7 is really a sacred number? Have we learned that 7 actually represents perfection? Have we learned that there are "periodicities" related to "the old sacred numbers"? Have we, in fact, learned anything? Or have we just watched some people fool around with numbers in order to beguile themselves?

Here's another example:

“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'....” [14]

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 idea that 333 AD was exactly the year when humans needed 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... On the other hand, 330 BC + approximately 330-340 AD equals approximately 660-670, or thereabouts, give or take, more or less... The more we consider Steiner's numbers, the less impressive they become. And we haven't even asked ourselves whether Ahriman's power really peaked around 333 BC, or whether Ahriman even exists, or whether human beings actually began using logic in the year 333 BC (i.e., they hadn't used logic before and didn't begin in some year other than 333 BC), or whether mankind really turned a corner and needed to "strive to re-enter" the "higher hierarchies" precisely during the year 333 AD, or whether the "higher hierarchies" really exist, and so on. The more we consider Steiner's numbers, the less impressive they become. In fact, the more we examine Steiner's numbers, the more convinced we should be that his numbers mean nothing.`

### <<◊>>

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, he taught us. 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 my teacher's 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. Indeed, this was the lesson we were given in all sorts of classes, not just math. It was one of our central tenets. The unreal is real, and it is all around us. 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.

### <<◊>>

John Blackwood, MATHEMATICS IN SPACE AND TIME (Floris Books 2006).

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. [15]

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 (25,920) is equal to the number of days we live (25,920), which in turn is equal to number of years in a “Great Year” (25,920).

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 Waldorf teacher Blackwood 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.” [16]

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. The average, in other words, is about 20. Let’s try the calculation using 20 breaths per minute instead of 18. Multiply 20 x 60 x 24, and what do you get? 28,800. If we accept this new result, our nice little paradigm wobbles: We want an answer of 25,920, not 28,800. We're off by 2,880 (28,880 - 25,920). So let’s ignore the new result. Disregarding the facts, let's cling to the idea that we breathe 25,920 times a day.

◊ 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 specify 25,765 years. [17] Sadly, none of these numbers is precisely what we want (25,920). We're off by anywhere from 80 to 155 years (approximately one to one-and-a-half centuries). So let’s ignore these alternate results. Disregarding the facts, let's cling to the idea that a Great Year is 25,920 years long.

◊ 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. [19] 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. We're off by 370 or perhaps 352.08 days — in either case, approximately a whole year. So let’s ignore these results, too. Disregarding the facts, let's cling to the idea that a human life is 25,920 days long.

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. [18] 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. Intellect and logic are, of course, absolutely essential to math. The science called mathematics is really nothing more than the use of logic as applied to numbers. The Anthroposophical aversion to logic helps explain why math in Waldorf schools is so often shallow.

What do Steiner and his follower turn to instead of rationality? Write down the answer and 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 the right answer? Excellent, A+. Now, for extra credit...

— Roger Rawlings

### <<◊>>

Above I referred to Platonic nature and Platonic years. Plato pops up pretty often in Waldorf schools. Plato said many things that are more or less mystical, so he is popular among Steiner's followers. Thus, for instance, Anthroposophists accept as reality the myths Plato spun about an ancient land: Atlantis. [See "Atlantis and the Aryan".] But Plato is big in Waldorf schools for another reason, as well. Waldorf teachers try not to talk about Steiner too much in front of their students — they know they are not supposed to "teach Anthroposophy" to the kids (not openly, anyway). So, instead, they often make use of Plato, Goethe, Wordsworth, Emerson, or any other famous figure whose words might be twisted to seem to agree with Steiner's. They use these proxies to stand in for Steiner. Why? In order to teach Anthroposophy to the kids.

### <<<◊>>>

**AFTERWORD**

**AFTERWORD**

### That Big Result, Again

Steiner repeatedly stressed the importance of the wonderful number 25,920. 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 ... [I]t 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.

### <<◊>>

◊ "In all occultism, the [number] One has always designated the indivisible unity of God in the universe. God is indicated by the number one." — Rudolf Steiner, OCCULT SIGNS AND SYMBOLS (Anthroposophic Press, 1972), Ibid., p. 32.

◊ "Two is called the number of revelation in occultism ... Everything appears in duality. Two, duality, is the number of appearance, of manifestation." — Ibid., pp. 32-34.

◊ "Three is the number of the Divinity revealing itself." — Ibid., p. 34.

◊ "Four is the sign of the cosmos or of creation." — Ibid, p. 43.

[Etc.]

### <<◊>>

◊ "The square is the symbol of the fourfold nature of man; physical body, ether-body, astral body and ego. The triangle is the symbol for Spirit-Self, Life-Spirit, Spirit-Man." — Rudolf Steiner, THE FESTIVALS AND THEIR MEANING, “Signs and Symbols of the Christmas Festival” (Rudolf Steiner Press, 1981), p. 39.

◊ "[E]theric streams are...active in man: earth either from the head to the right foot, from there water ether to the left hand, from there fire ether to the right hand, from there air ether to the left foot, and then thought ether back to the head. This is the occultist's sacred pentagram, the symbol of man." — Rudolf Steiner, FROM THE CONTENTS OF ESOTERIC CLASSES (SteinerBooks, 2007), p. 148.

◊ "Adding is related to the phlegmatic temperament, subtracting to the melancholic, multiplying to the sanguine, and dividing — working back to the dividend — to the choleric." — Rudolf Steiner, DISCUSSIONS WITH TEACHERS (Anthroposophic Press, 1997), p. 50.

### <<<◊>>>

Waldorfish math is even more mystical

than I've let on thus far.

To gaze upon other aspects of this subject, see

"Magic Numbers" and "Temperaments".

### <<<◊>>>

Use the following link to go to

the second part of "Mystic Math".

### <<<◊>>>

**ENDNOTES**

**ENDNOTES**

[1] Rudolf Steiner, "Mathematics and Occultism", THE ANTHROPOSOPHICAL MOVEMENT, Vol. V, No. 28, GA 35.

"Manas" is a Theosophical/Anthroposophical term derived from Sanskrit. 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] “Mathematics and Occultism”.

[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, GA 101.

[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] OCCULT HISTORY, p. 75.

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

In popular belief, 666 is the mark of the Antichrist. In Revelation 13:18 (King James version), we find

"Here is wisdom. Let him that hath understanding count the number of the beast: for it is the number of a man; and his number is Six hundred threescore and six."

[15]

"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: The astrological and other occult premises in Waldorf math are baseless. (A carpenter who needs to cut a board 4 feet long but cuts it 4.1 feet long will be off by almost an inch and a quarter; the board won't fit.)

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

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

[18] 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.

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. But forcing phenomena into preselected, arbitrarily delimited brackets proves nothing. You can always get the results you are determined to get, if you are willing to cut, trim, and paste to suit.

[19] Actually, in Germany today, the average lifespan is about 80 years. (Remember that Steiner delivered most of his lectures in Germany.) In Great Britain, it is about 80.4 years, in France about 81.4 years, in the USA about 78.2, in Japan about 82.9. [See www.google.com/publicdata.] Thus, the average Japanese lives about 30,280 days, or 4,360 days longer than the magic number, 25,920. Of course, lifespans were shorter in Steiner's day; but the point is that the wonderful pattern he claimed to spot has no basis in reality, and any apparent plausibility in Steiner's words has only declined over time as lifespans have lengthened.

[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, which is so important in Anthroposophical thinking. [See, e.g., "Astrology".] 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).