- Here's something for Franky and other anthro-mathematicians:

I took a brief peek into the Abyss, holding my breath of course, with one hand on my nose and the other one tight around the shaft of Michael's sword, and there was a post from Diana, where she basks in the glory of a polemical article against Steiner's mathematics as taught in Waldorf schools. The blog looks like some sort of popular amateur science website, at least judging from this article, but there may be plenty of good stuff there too, elsewhere.

I posted a comment, and below is the original article plus my comment. Perhaps some of you guys are more conversant with math and geometry and can add something more:

http://scienceblogs.com/goodmath/2006/11/woo_math_steiner_and_theosophi.php

Category: ">bad math; ">bad math ">woo**Woo Math: Steiner and Theosophical Math**

Posted on: November 8, 2006 9:09 PM, by Mark C. Chu-Carroll

While waiting for I was innocently browsing around the net looking at elementary math curriculums. I want to be able to teach my kids some fun math, just like my dad did with me when I was a kid. So I was browsing around, looking at different ways of teaching math, trying to find fun stuff. In the process, I came across woo-math: that is, incredible crazy woo justified using crazy things derived from legitimate mathematics. And it's not just a bit of flakiness with a mathematical gloss: it's big-time, wacky, loonie-tunes grade woo-math: the Rudolph Steiner Theosophical version of Mathematics. And, well, how could I possibly resist that?

There's this bunch of rather expensive private schools, called the Waldorf schools. A first glance at them seems a little flakey, but kind of cute. They're very into nature; the toys for children are all made out of natural materials: solid wood, linen, cotton, etc. The kids go for walks in the woods every day when the weather is good. There's a lot of independent learning, with kids learning at their own pace. It all sounds sort of sweet... until you get to the details. And then, it gets absolutely bizarre and hysterically funny.**A Bit of Background**

The schools were started by a complete nutjob named Rudolph Steiner, who started a "new science" which he called*theosophy*. As usual for crackpots, it's a brilliant new approach that totally revolutionizes every single field of modern science and philosophy, and proves that pretty much everything that came before was wrong. The Waldorf schools are based on Steiner's theories of learning. And as you go deeper, many of the strange practices of the school start to move from looking silly to looking insane, or even sinister.

Just to give you an example of where it starts getting silly... The purpose of those walks in the woods every day? Rudy believed in Gnomes (which he always capitalized). The walks in the woods are to look for and commune with the Gnomes. And it's harmful to a child's soul to teach him or her to read before any of their adult teeth come in.

On the sinister side, it turns out that the reason why the schools like toys made out of natural materials is because Theosophy is based on a sort of bizzare mix of Christianity and Zoroastrianism; it features two devil figures, Lucifer and Ahriman, and technology is developed from the influence of Ahriman; therefore, you've got to protect children from its dark influence. Here's a nice Steiner quote about this that leads us into his crazy math:- But woe betide if this Copernicanism is not confronted by the
knowledge that the cosmos is permeated by soul and spirit. It is this
knowledge that Ahriman wants to withhold. He would like to keep people so
obtuse that they can grasp only the mathematical aspect of
astronomy.

Of course, along with completely reinventing education, philosophy, religion, medicine, and physics, Rudy (with help) devised his own twisted take on mathematics. It's actually a lot like a more well-developed version of our old friend George's math.**Steiner Math**

Like George, Rudy is obsessed with the idea of infinity. But instead of just having an obsession with infinity on a number-line, Rudy Steiner was obsessed with geometry. To him, geometry is the heart of everything: he's obsessed with geometric infinities. So naturally, he decided that all of reality was based on projective geometry.

Projective geometry is a rather strange non-Euclidean geometry. You can think of it as a kind of geometry derived from the idea of perspective art, where "parallel" lines appear to get closer together as they go off into the distance. In projective geometry, parallel lines converge to meet at infinity. But since there are parallel lines in different directions, they can't end up at the same place - so "infinity" on a plane is a line; in a 3-space, infinity is a plane.

I haven't ever studied projective geometry. It's not something that I find terribly interesting. But in the hands of Steiner, it's fascinating as an example of pathological thinking at work. In normal projective geometry, there's an interesting kind of duality, where you can take theorems involving lines and points and switch the lines and the points in the theorem, and the result is also a theorem. So, for example: given two distinct points, there is exactly one line that crosses through both of them. The dual statement of that is: given two distinct lines, there is exactly one distinct point that they both cross through.

Steiner insists on carrying duality to silliness, and that's where the really crazy math comes in. Since there's normal space where parallel lines converge and intersect at infinity, there must be a dual space where everything is at infinity, and things converge towards the finite. The dual space is what he calls counter-space. Counter-space is defined by his the combination of the fact that he believes that projective geometry is the "real* geometry, and his extreme belief in the fundamental duality of projective geometry.

Then he starts to mix it with his truly wacky woo. You see, counter-space is where consciousness lives:- Counter space is the space in which subtle forces work, such as those
of life, which are not amenable to ordinary measurement. It is the polar
opposite of Euclidean space. It was discovered by the observations of
Rudolf Steiner and described geometrically by George Adams and,
independently, by Louis Locher-Ernst. Instead of having its ideal
elements in a plane at infinity it has them in a "POINT at
infinity". They are lines and planes, rather than lines and points
as in ordinary space. We call this point the counter space infinity, so
that a plane incident with it is said to be an ideal plane or plane at
infinity in counter space. It only appears thus for a different kind of
consciousness, namely a peripheral one which experiences such a point as
an infinite inwardness in contrast to our normal consciousness which
experiences an infinite outwardness.

Now, if you've been following our discussions of topology, you'd probably say "so what?" A metric is just a way of describing something in terms of the structure of a particular metric space. Of course we can impose different metrics on the space space, and the fact that the size of an object measured in the metric of one space changes in one metric imposed on a space doesn't mean anything about the object itself or how it's measured in the other metric.

But just as he took the duality principle and insisted that the mathematical concept must be reflected in reality, he does the same thing here. The difference in metrics must have some real concrete meaning in the physical universe. Steiner-physics says that the metric difference created by Steiner-math means that there's a stress on the object because of the difference in its size in real space and counter space. And all of the fundamental forces of the universe come from this strain.

According to Steiner, this duality of existence in normal space and counter space is defined by linkages, and linkages are what makes reality work. The different ways that things can be linked in the two spaces defines what the strain means and what effect it has. So, for example, according to Steiner solids are things that are linked in real and counter space by a "euclidean metric" linkage, and strain is gravity.

Is that loony enough? It gets worse. He carries that ridiculous idea of the mathematical concepts having physical reality to an even loonier degree. You see, you can use either rectangular or polar coordinates to describe things. And which one you use means something. Some things can only be measured in polar, and some in rectangular, and the choice of polar vs. rectangular coordinates has deep physical meaning. So, for instance, affine linkages in the rectangular coordinate system describe the behavior of gases: gases are fundamental governed by rectangular coordinate systems, and the strain on gasses are reflected as pressure. But take the same thing and measure it in polar coordinates, and it's no longer gasses - it's light. The only difference between light and air is that air is measured in rectangular coordinates, and light is measured in polar coordinates.

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I am not a matematician, so my comment on this subject may be of limited value, but from the looks of it, the article by Mark C. Chu-Carroll is not professional, nor is it objective, which would be a prerequisite for any article dealing with scientific subjects. It is full of polemics and opinions, and much of its content does not address the subject itself; on the contrary, it looks like an attempt to contextualize it in a manner that invites ridicule. What is missing here is at least some reference to a peer review or other professional research that thallenges Dr. Steiner's approach to mathematics.

For further study, there is one professional mathematician who may be worth a good look if one is interested in academic discussions instead of polemical propaganda and subjective bigotry. I am referring to physicist Georg Unger, PhD, who was the head of the Department of Mathematics and Astronomy at the Goethanum in Dornach. (The Goethanum is the HQ of the Anthroposophical Society, which is based upon Dr. Rudolf Steiner's works.)

Georg Unger was born in Stuttgart, Germany, and he was also a pupil of the first Waldorf School, which opened in 1919. He finished extensive studies in mathematics, physics, and philosophy with the doctorate in Zürich. After ten years of teaching in Zürich he became a Swiss citizen and went in 1955 as visiting fellow to M.I.T.. Cambridge, Mass., for studies in cybernetics with N. Wiener and as visiting guest to the Institute for Advanced Studies in Princeton which gave occasion to meet R.J. Oppenheimer and to visit J.v. Neumann in Washington for scientific philosophic discussions. His publications range from the epistemological foundations of mathematics and physics to symptematic discussions of developments in the field of science.

In addition to Dr. Unger's books, I would also recommend studies in epistemology, especially related to scientific research and mathematics.

Dr. Unger shares with Dr. Steiner what Mark C. Chu Carroll calls "pathological thinking", because Steiner's works are often his point of departure. Interestingly, a discussion among at least half a dozen Nobel laureates in science hosted in Stockholm by BBC revealed that such pioneers are for the most part "pathological thinkers" of different kinds in the eyes of more mediocre contemporaries because they think "outside

the box" and refuse to be influenced by ridicule.

Mark C. Chu-Carroll's article also reveals ignorance. For instance, he finds it curious that Dr. Steiner always capitalized the word "Gnomes", not knowing that all nouns are capitalized in German, and that such capitalization in English is due to sloppy translation.

Posted by: Tarjei Straume | February 14, 2007 10:44 AM

- But woe betide if this Copernicanism is not confronted by the
knowledge that the cosmos is permeated by soul and spirit. It is this
knowledge that Ahriman wants to withhold. He would like to keep people so
obtuse that they can grasp only the mathematical aspect of
astronomy.
- --- kmlightseeker <kmlightseeker@...> wrote:

> I said:

Actually, I wasn't trying to be incorrect, but was

>

> > Some corrections (Assuming you didn't mean the

> above figures to be

> > incorrect :) [Ok, i'm guessing the latter is more

> likely, but I think

> > the correct representations need to be made here.

> :) ]):

> >

>

> I was remiss in mentioning that the latter

> possibility was that Frank

> was intentionally being incorrect. Sorry about that.

> :)

using Occram's Razor method, just counting the

letters, therefore fts=3. But in any which way, I see

a problem because of different languages. Numerology

may work in Hebrew, fe, but when translated into a

different language, it becomes nonsense (I assume).

But if something is true, it must be so in any

language - a concept, for example. Can a concept be

numerologized?

Frank

Frank Thomas Smith

http://SouthernCrossReview.org

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