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](http://www.nct.anth.org.uk/counter.htm). And, well, how could I possibly resist that?

A Bit of Background

---------------------

There's this bunch of rather expensive private schools, called [the Waldorf schools](http://www.waldorfanswers.org). 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](http://dir.salon.com/story/mwt/feature/2004/05/26/waldorf/index.html). And then, it gets absolutely bizarre and hysterically funny.

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:](http://www.doyletics.com/arj/landarvw.htm)

>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.

Steiner is a serious literalist; he can't see the difference between abstractions/ideas and reality. If there's an abstraction that makes sense, according to Steiner, it *must* be reflected in reality; and everything in reality *must* be part of any abstraction. So regular math is very naughty, because

it doesn't include any way of describing "souls".

Steiner Math

---------------

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](http://georgeshollenberger.blogspot.com/).

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](http://en.wikipedia.org/wiki/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.

But that's not crazy enough. No sirree bob. It gets *much* loonier. You see, *some* things - like human beings - exist in both normal space *and* in counter space at the same time. And because of the fundamental strangeness of counter space, the "metric" of counter-space is only preserved if the objects *size* changes as it moves in counter-space. And if the size changes in counter-space, but not in normal space, then the sizes of the object in the metrics of counter-space and normal space become *different*.

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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Projective geometry is not all that strange, mathematically -- projective space is a natural setting for algebraic geometry and complex manifolds. It's especially nice for intersections: in the projective plane, curves defined by degree m and degree n polynomials intersect in mn points.

However, it's pretty absurd that Steiner makes a big deal about the line at infinity -- this term is just a colloquialism. In the projective plane, for example, you have coordinates [x:y:z] (with at least one coordinate nonzero), except for any tâ 0 [tx:ty:tz] is considered the same point. The points (x,y) of the real plane can be identified with the projective points [x:y:1].

The "line at infinity" is simply the collection of points [x:y:0] with at least one of x and y nonzero. From the perspective of someone living in projective space, there's nothing distinctive about these points!

You shouldn't confuse Anthroposophy with theosophy, more popularly associated with HP Blavatsky's work, although it is a general term with many other exemplars in the history of religious and spiritual thought.

There are all sorts of interesting things that happen on projective geometries, but I can see not seeing the direct practicalities.

There's one greatly useful idea there that you might like. Basically it's about finding the minimum number of experiments to compare various combinations of choices. For example, say you've got three types of grapes and three types of soil. How can you find the best wine you can grow? How many comparisons will it take? It turns out you can do this by counting the lines on a projective plane with finitely many points! The details escape me at the moment, but I know you can find it in one of Ian Stewart's columns in his compilation [i]Game, Set, Math[/i].

I wish I could get some of whatever that dude's on. Oh man, I bet that's some good shit!... if I can see the music, does that mean I must also be able to hear the light?

"There are all sorts of interesting things that happen on projective geometries, but I can see not seeing the direct practicalities."

As well as the whole of perspective drawing in art beeing applied projective geometry, it is also the maths of map projection in cartography and of physical optics. So as you can see there are some very practicle applications.

The fundamentals of projective geometry are taught as basic curriculum in maths to fourteen year old German school children.

Antroposophy, the woo offshoot of woo theosophy, has lead to a lot of pseudoscience: waldorf pedagogy, antroposophic medicin/anti-vaccine theories, bio-dynamic agriculture, and race theories.

I didn't know about these perversions of math and science though. On the "Ethers" page they try to cover everything from music to life. All vacuous, as usual.

Back to your original quest for elementary math articles, have you seen Math Fun Facts? It has won a few awards.

http://www.math.hmc.edu/funfacts/

Just wanted to chime in here as someone who has actually attended two Steiner schools (the last I left around 13 years ago), rather than just read about them.

I'll start by saying I'm from the UK, and have been to many different schools in my time (we moved around a lot) including several state-funded primaries and secondaries. I am also a hardline atheist and a Software Engineer.

There is a world of difference between the practical reality of Steiner education and the philosophy behind it. In fact, it seems that one of the core beliefs is to not expose the students to the Steiner philosophy - there is no indoctrination, or scary warping of tiny minds. In fact, it is quite possible to go through the Steiner system and leave knowing absolutely nothing about Anthroposophy or Steiner, except that in some lights he looks a little like Jeremy Irons. This is a good thing, because it's some far out crazy stuff. But there are some practical things that Steiner schools achieve that are, in my experience a distinct advantage over the mainstream education, so while the reasons are utter bollocks, the intentions and outcomes can often be very good. For example, Steiner schools start teaching you French and German from Kindergarten - I don't know what the States is like, but this is certainly unusual in the UK, and our language skills nationally are appalling. The education you get is very broad, given the lesson structure and the equal weighting given to both traditional academic subjects and the arts.

However, the principal advantage, to my mind, is that because Steiner schools are both private and very niche, class sizes are tend towards the small (my own class varied between 14 and 6 pupils over the few years I went there) and the teachers are highly enthusiastic.

I often say that one Steiner school I attended was the only school I've ever come across that had one (excellent) teacher who covered all of the sciences, and yet had two beekeepers.

The big downer is that they seem to exist in a bubble of unreality, so while it seems to give you an admirable and fully rounded education, if Steiner is all you've known it doesn't really prepare you for life in the real world, which is much harsher. The bottom line - if I am able to do so, I will definitely send my children to a Steiner school at some point in their life, although I would want them to attend state schools in equal measure in the same way I did.

Great, great woo!

As for the good math being discussed, I liked the treatment of projective geometry in Ian Stewart's Flatterland.

One upon a time, my son, currently eighteen years old in his 5th year at university (having passed the college entrance exam at age twelve), needed an elementary school near our home. There was a Waldorf school on a nice wooded lot, with a mile or so of our home here in the foothills of the San Gabriel Mountains.

My wife, currently a full-time Physics professor, attended some sort of holiday event/open house at the school. The staff were dressed in green like elves, or somesuch, for reasons related to the holiday. My wife spoke with the gentleman she (of Scottish origin) described as "headmaster."

He explained the Steiner theory of why children should not be taught too young to read. "Too late for my son," said my wife. "He started reading before he was two, and has already had a poem he wrote published in Rank and File, the newsletter of the Southern California Chess Federation."

He held out his hand and showed her what he described as "a crystal" that he'd had some children examine. My wife looked at it.

"Oh, yes," she said. This is calcite. Specifically, it is iceland spar. Noted for its birefringence. If you draw a line on paper with a pencil, place this on top of that, you'll see two lines, and as you rotate the crystal, the distance between them changes. Great, you teach them Optics."

"Calcite?" he said. "Iceland spar? I don't know anything about that. It's a crystal. It's natural."

"So is poison ivy," she said, left, and continued the search for a real school.

One upon a time, my son, currently eighteen years old in his 5th year at university (having passed the college entrance exam at age twelve), needed an elementary school near our home. There was a Waldorf school on a nice wooded lot, with a mile or so of our home here in the foothills of the San Gabriel Mountains.

My wife, currently a full-time Physics professor, attended some sort of holiday event/open house at the school. The staff were dressed in green like elves, or somesuch, for reasons related to the holiday. My wife spoke with the gentleman she (of Scottish origin) described as "headmaster."

He explained the Steiner theory of why children should not be taught too young to read. "Too late for my son," said my wife. "He started reading before he was two, and has already had a poem he wrote published in Rank and File, the newsletter of the Southern California Chess Federation."

He held out his hand and showed her what he described as "a crystal" that he'd had some children examine. My wife looked at it.

"Oh, yes," she said. This is calcite. Specifically, it is iceland spar. Noted for its birefringence. If you draw a line on paper with a pencil, place this on top of that, you'll see two lines, and as you rotate the crystal, the distance between them changes. Great, you teach them Optics."

"Calcite?" he said. "Iceland spar? I don't know anything about that. It's a crystal. It's natural."

"So is poison ivy," she said, left, and continued the search for a real school.

Sorry for the duplication. You are not reading through Iceland Spar. Both the previewing and posting took long enough so that I could go and get a cup of coffee from the kitchen. Then I got a server error message. Is there a problem at your end, or mine?

This follow-up, on the other hand, previewed with a couple of seconds.

That is pretty... out there. I'm slightly curious to know if the fellow has looked into the abstractions used in non-mathematical topics. If so, they would no doubt be just as "inspired" as what you have described, and could even be good for a laugh.

Jonathan Vos Post wrote:

Alan Alda's formulation was this:

Josiah:

The Waldorf schools official curriculum is based on Steiner's "science" of anthroposophy, and it's pretty much the same kind of woo all the way through it. As I said, Steiner has this strange kind of literalism, and it pervades just about everything. They've got the same kind of woo mixed into everything from the method they use to teach reading to what games they play outside.

A good example of some of the bizzare literalism is the connection of "eye teeth" and "eyesight"; some Waldorf educators following Steiner's lead maintain that tooth decay causes bad eyesight.

And what causes tooth decay? What else could it possibly be but poor finger dexterity! Here's a Steiner quote I found looking for some of their strange ideas about eye teeth; it's discussing why Waldorf schools spend so much of the school day learning knitting and/or crochet:

Is he the "Steiner" that invented Steiner systems? The Wikipedia article about Steiner systems doesn't have any links to say who they are named after. I wrote my PhD thesis on them but I'm sorry to say I don't know either. Although the mention of projective geometry and lines meeting at infinity does suggest the Steiner triple system of order 9.

Hmm, yeh, hilarious, little children being prevented from getting a good education is exchange for this. I just can't stop laughing.

It's true that Waldorf schools do try to avoid formal indoctrination into Steiner's philosophy, but the practical application of that philosophy can have serious practical consequences. Here in Boulder, the Shining Mountain school has been at the center of an outbreak of pertussis as a direct result of Steiner's belief that children should not be vaccinated. I heard Steiner referred to approvingly just last week on our local woo radio program,

Naturally,during a discussion of antibiotics, of all things. Steiner's is not the only bad advice going around, but every little bit hurts.Paul: Steiner systems are named after Jakob Steiner, an 19th-century Swiss geometer. So, different guy, but still a connection with projective geometry.

OTOH, if I'm remembering my history right, Steiner didn't exactly invent Steiner triple systems (from which the general systems are generalized); the question of existence was posed in the British Isles, solved by Kirkman, and then later posed on the continent by Steiner (and solved by someeone else).

Paul:

I honestly agree with you that it's a tragedy that children are being "educated" in schools that teach this nonsense.

To me, that's all the more reason to mock it mercilessly. I do know of parents who've looked into Waldorf schools, and a friend of a friend sends her kids to a Waldorf. The best defense against this kind of bullshit is to make the inanity and insanity of it as widely known as possible. And to treat this kind of stuff with the *dignity* of a non-sarcastic analysis is to give them far more credit and credibility than they deserve.

I want to see information on the net so that when a parent looks up Waldorf schools, right away, they see "This is a school that teaches children to believe in gnomes, that crotcheting prevents tooth decay, and that computers are the work of evil spirits." I want them to see how utterly ridiculous it is. Send your kid to Waldorf, and they're going to get anthroposophical math and science, not real math and science. They're going to be sitting and crocheting instead of learning to read and write. And I want them to see that even when it *sounds* credible, what Steiner followers believe and want to teach is just ridiculous. As a math geek, the math is the part that caught my attention, and that I'm qualified to talk about. And it's dreadful stuff; exactly the kind of thing you'd get from a literalist who doesn't really understand what he's talking about: this is a guy who believes that a point written in polar notation *is a fundamentally different point* than one written in rectangular notation: that there are *distinct spaces* in the real world for things that are measured in polar and rectangular.

jre wrote:

Have you passed this along to Orac? It sounds like a classic example of antivaccination woo, one of Respectful Insolence's favourite targets. Joint scienceblogger attack!

Blake:

Yes, I did pass it over to Orac. Anthroposophical medicine is a combination of homeopathy and the counter-space woo from Steiner's math. Quite strange stuff.

Right now my reflex to laugh like crazy is jostling for control of my attention with my reflex to learn more about this "projective geometry" thing that I've not heard of but sounds interesting, and I'm not sure which reflex to act on first.

Okay, but wait, although it is true that all [tx:0:0] are equal for any given t and [0:ty:0] are equal for any given t, [x:0:0] is not equal to [0:y:0], is it?

What exactly is the concept of "uniqueness" or "distinctness" that applies in this geometry, since it appears that one single point can be identified by many different coordinate triplets?

That... um.. hm. Is it in fact possible to axiomatize this as an (impractical and abstract, but self-consistent) extension of projective geometry? Or is this just pure gibberish?

If you're looking for some fun math for kids that's based on actual math....try the work of Alfred Possamentier, including a book called Math Charmers. Full of interesting tidbits, most of which require very little formal math

(mostly arithmetic and a sense of fun)

I'd be interested in your thoughts; some of his topics might even be worth a diary.

I attended a Steiner school in the UK from age 8 to 16, and I have to say Mark's description doesn't match what I experienced. While they were a bit new-agey, the staff certainly never claimed that gnomes were real; we learned to crochet (& knit & weave) but no-one claimed it had anything to do with tooth decay; and we used perfectly normal math & science textbooks. They may well have had some pretty odd ideas in the background, but they didn't expose us to many of them (perhaps sensing the merciless derision we'd have heaped on them).

There are a few practices common in Waldorf schools which actually make sense - such as the notion of a 'class teacher' who teaches a set of students core subjects through their entire time at school (rather than handing the class off to a new teacher after each school year).

Overall I feel it was a pretty positive experience for me personally, though of course YMMV.

Of course I agree that Steiner's philosophy is ludicrous, as well as harmful if it is taken seriously. However, I doubt most Waldorf students are even aware of it, or that most Waldorf teachers take it really seriously. Lots of things sound ridiculous but don't do as much harm as one might fear. For example, one could describe Catholic schools by saying they are run by a group that claims to participate in ritual cannibalism and decorates classrooms using an ancient torture device. Nevertheless, most of their students are unharmed by the experience.

Mark,

"Send your kid to Waldorf, and they're going to get anthroposophical math and science, not real math and science."

I don't know how it is in the states, and I'm sure there may well be some nutjobs that do this kind of thing to kids, just as there are freakish indoctrinating homeschoolers out there. But, speaking from first-hand, practical experience of two Steiner schools, this is a complete falsehood. We were taught the entirety of the standard UK curriculum, and generally to the highest standard of any schools in my area.

But hey, my evidence is anecdotal - I've never seen any kind of comparative study done.

"This is a school that teaches children to believe in gnomes, that crotcheting prevents tooth decay, and that computers are the work of evil spirits."

Again, news to me. I remember spending a chemistry lesson learning why homeopathic medicine can't actually be anything other than a placebo.

I think, just as there are some whacko Jebus schools that teach that eye contact is a sin and that Darwin shall burn in hell for his heresy, there are others that are balanced centres of learning excellence. The variation between individual schools is likely to make generalisation difficult, and some places perhaps succeed in spite of their philosophy, rather than because of it.

There is only one point of the form [x:0:0], since all such points are equivalent to [1:0:0]! (Set t=1/x.) Similarly for [0:y:0] -- it's really [0:1:0]. One single point *can* be identified many different ways, though it's common to normalize points by, say, writing them as [x/z:y/z:1] when zâ 0.

It's a bit easier to get a grasp on by first considering the projective line, where you have points [x:y] with the same identification. The way to think about the points of the projective line is to think of them as representing lines through the origin in the standard Euclidean plane. The projective point [a:b] (which is equivalent to the point [a/b:1]) with bâ 0 corresponds to the line of slope a/b; the point [1:0] corresponds to a vertical line.

It's gibberish. The whole "at infinity" business is just a colloquial way of describing the difference between real space and projective space. There is no meaningful sense in which the line at infinity is really "at infinity" when you're looking at things from the perspective of projective geometry. It's just some points, much like any other points. Trying to dualize to obtain a space where "everything is at infinity" is nonsense. (Side note: there is a notion of dual projective spaces, but this is not what it means.)

This does not sound much worse or more dangerous than what hundred of thousands of American kids homeschooled by Evangelical parents (on accasion without a high-school diploma) are learning in America.

Both my brother and sister went to a Waldorf school for middle school. The school did have a not too disagreeable approach to education. The worst I can say about the place is that it was kinda new-agey around the edges. They don't teach Steiner's actual babbling, as far as I know.

Davis, thanks.

Oh, and I should clarify that this point is the "point at infinity" on the projective line.

Unrelated Steiner [triple systems]?

I stumbled upon something called the Steiner system which is "a type of block design" in mathematics.

http://en.wikipedia.org/wiki/Steiner_system

Apparently this is related to Golay Codes for error correcting.

http://en.wikipedia.org/wiki/Binary_Golay_code

or

http://mathworld.wolfram.com/GolayCode.html

It is also apparently related to sporadic groups like Mathieu and has a Ternary form researched by JH Conway [Fractan post].

http://en.wikipedia.org/wiki/Ternary_Golay_code

I am speculating that this may assist in linking game theory with projective or differential geometries.

That wouldn't be surprising. Any problem in mathematics that can be expressed using equations is connected to some sort of geometry. If the equations are algebraic (i.e., polynomials) then the problem links to algebraic geometry, in which projective geometry plays an important role. If the equations are merely differentiable, then the problem links to differential geometry. And so on.

Doug:

Rudolf Steiner;

Jakob Steiner.

The first of these was the Austrian writer and esotericist from whose ideas the Waldorf Schools sprung. The second of these was the Swiss mathemetician after whom Steiner surfaces, Steiner trees and I assume Steiner systems were named.

If you look through the archives of the Panda's Thumb, you'll incidentally find a series of articles by Dave Thomas about an evolutionary algorithm he implemented for solving the Steiner tree problem. The Steiner referred to in these is Jakob Steiner.

Doug:

Rudolf Steiner was the Austrian writer and esotericist from whose ideas the Waldorf Schools sprung.

Jakob Steiner was the Swiss mathemetician after whom Steiner surfaces, Steiner trees and I assume Steiner systems were named.

If you look through the archives of the Panda's Thumb, you'll incidentally find a series of articles by Dave Thomas about an evolutionary algorithm he implemented for solving the Steiner tree problem. The Steiner referred to in these is Jakob Steiner.

I see real money in figuring this out. Here's my logic.

If you look at railroad tracks going out to the horizon, according to this math, they actually do meet somewhere. That somewhere actually exists. Now, everywhere you look at railroad tracks, when you go to that spot, they're always about the same distance apart. But now we know that somewhere they actually meet.

That must be the same place where the rainbow touches the ground. And, when you get there, there will be a pot of gold.

Clearly, this looks like the math that will get you there.

So, just send me some venture capital, and i'll organize the Steiner mathemeticians, and get an expedition mounted!

This reminds me of the time I went down to clean out the last chant saloon. There was a bunch o' theosophists down at the end of the bar raisin' th' devil--had him about two feet off the ground. I went up and I said to him, "Sam McKoo, there ain't room enough in this life-cycle for th' both of us."

Sorry, had to say it--how often do theosophists come up in conversation to give me a chance to quote Firesign Theater? No flames for my errors, please; I'm quoting from memory, and mostly what I remember is other people quoting this, so I'm sure there are some.

It is hard to find a more natural branch of mathematics than projective geometry. Close one eye and look around: what you see is exactly a projective plane (that is, projection of the 3-dimensional space on the sphere -- retina of your eye).

I went to a Waldorf-Steiner school for some years, and I can attest that they didn't teach us to believe in gnomes, that crocheting prevents tooth decay, or that computers are the work of evil spirits. I did learn later that some of the teachers shared some of Steiner's odder beliefs, but they never gave even an indication that they did, much less present these things to us as facts.

Sure, there were some odd things. Hour-long welcome ceremonies with singing and dancing at the beginning of the day. An obsession with drawing geometrical shapes, colors, and later perspective. The mythologies of various ancient civilisations (Egyptian, Indian, Zoroastrian) told as exciting stories, but not presented as real.

But what mattered was that they genuinely cared about the children, and they filtered Steiner's beliefs through a praiseworthy amount of common sense. They also were very idealistic, teachers working for very little pay, and they certainly weren't expensive. Mostly they were just nice, kind parents who thought the school system too cold and uncaring, and who saw in Steiner pedagogics a potential way out, probably tempted by some of the genuinely good common-sense ideas in there (for instance that much of the day is devoted to the week's current topic). Just one of the teachers was formally educated in Steiner's pedagogics, a very patient and likeable music teacher.

One of my friends was a dyslectic, but he was good at maths, until the class had to be closed and he started at a regular school. There his main teacher was an unsympathetic drunkard who thought he was an idiot, and it didn't go long before his math skills were as poor as everybody elses, and he dropped out of school and into trouble.

I'm in a bit of a dilemma whether I would send my son to such a school. I've found out how crazy and downright heathen some of the things are "behind the scenes", but I feel it's more important that he gets teachers who genuinely, profoundly care about his upbringing, even if they are a bit misguided in many things. We will probably try a regular school first, but if there is any sort of bullying or "misopedic" attitudes from teachers, I know where I can find at least a slightly better alternative.

Thanks, Mark, for kicking off this entertaining thread. I hope you don't mind that I've put up a link to this blog from the articles section of the PLANS web site.

Since you refer to http://www.waldorfanswers.org in your original post, I'd like to post the URL of the site that so enraged Waldorf supporters that waldorfanswers was created as an attempt to rebut it. You be the judge as to whether the rebuttal is effective. Please see the web site of People for Legal and Nonsectarian Schools (PLANS): http://www.waldorfcritics.org

Thanks, Dan

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 ZÃ¼rich. After ten years of teaching in ZÃ¼rich 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.

Clarification, rectification asked.

Dear Mark C. Chu-Caroll,

You present a quote as being from Rudolf Steiner when it is obvously not. I infer tha it could be from Nick Thomas. In that case you are ridiculising some one else than Steiner and can yo clarify or rectify yo humour target?

You said:

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.

In any case Waldorf School if they can staff projective-geometry, wold only do some preliminary work on a three week basis, at the most enough to introduce geometry of imaginary numbers to upper school. Some form of hand drawinf, called 'form-drawing- may use curved lines

for miroring forms, but there it is more an aesthetic pursuit than leading to formal conceptualisation.

FranÃ§ois Gaillemin

This is a blog post, which is different. And I doubt that you will find any mathematician waste research time on woo.

Why do you think philosophy is important here? Epistemology doesn't tell you how to do science or math.

Btw, I can't find any research published by a Georg Unger in arxiv or Scholar. What I find is a book "Forming concept in Physics" that has such strange chapter titles as "So-called General Relativity Theory" and "Homeopathic Dilutions (Potencies) and Cosmic Influences as Examples".

FranÃ§ois:

Are you actually defending the use of Steiner's math (mirroring forms?) in schools?

Epistemology investigates the origin, nature, methods, and limits of human knowledge. Science is systematized knowledge. Science and mathematics are intimately connected and often inseparable, and together they represent the only field where intellectual proof is valid. Furthermore, Rudolf Steiner's doctoral thesis of 1992, Wahrheit und Wissenschaft ['Truth and Science' or 'Truth and Knowledge'] followed by his major work two years later, Die Philosophie der Freiheit, represents a radical departure from the mainstream epistemology that still influences orthodox thinking today, also in science

It can tell you how you already do science or math and even help you experiment with doing it differently. The primary tool in science and math is thinking, applying the intellect. Epistemology helps you to examine this tool and learn how it works. Many natural sciences are based upon the observation of phenomena through our physical senses or through the extention and amplification of these through instruments, technology. The observations are interpreted and contextualized through our thinking. Epistemology helps us understand how what we call knowledge is acquired in this manner and may also help us improve it by experimenting with different methods that we might not have thought of if we had not studied epistemology.

Correction: The year was 1892.

Philosophy in general, and epistemology especially, has practically no influence on practical or theoretical science. The last contribution I can think of is Popper explaining fully why testability is important.

That didn't seem to affect how science was done however, since tests was done before him.

The primary tool in (natural) science is observation. Also important is making theories and predictions, because an observations only method would not make much progress.

Epistemology has never contributed to either of these two activities. If you have examples that contradict this claim, it would be interesting to see them.

In that case, science was done without thinking, in the absence of any cognitive processes.

No theorizing or predicting can be done without thought as the primary tool. Otherwise, a being with no brain, i.e. a worm or an amoeba, could be a theorizing and predicting scientist.

It's a matter of common sense. Epistemology includes the study of how our senses work, and what happens inside our cognitive processes, our thinking, when we perceive and interpret sense-impressions, how and why we create theories and so on. By arguing that epistemology is irrelevant to these processes, you're saying that understanding how thinking and cognition works contributes no value or usefulness for the future of scientific research.

I'm not going to argue with that - Over and out.

I teach projective geometry to eleventh graders in a Waldorf school; during our accreditation process, one of the accreditation team came to visit this class. A former math teacher at one of the most prestigious private schools in the USA, he is presently head of curriculum at an equally renowned private school (neither of these schools has the slightest connection to Waldorf education).

He came up to me afterwards to say that he found the material inspiring, and that he used to teach projective geometry in his previous school, but that it had been pushed out by the stifling influence of standardized testing and standardized textbooks. He added that, having being reminded of the power of the approach, he was resolved to bring it into the curriculum in his current school - for the gifted students.

I find that it awakens the student's creative capacities in mathematics in a way complementary to Euclidean geometry (and best taught after the latter).

Dear TorbjÃ¶rn,

You ask: FranÃ§ois: Are you actually defending the use of Steiner's math (mirroring forms?) in schools? Posted by: TorbjÃ¶rn Larsson | February 14, 2007 03:38 PM

It is difficult to ask a question of a text that already brings the answer to your question: I can only

1-repeat my self: Some form of hand drawinf, called 'form-drawing- may use curved lines

for miroring forms, but there it is more an aesthetic pursuit than leading to formal onceptualisation. FranÃ§ois Gaillemin Posted by: FranÃ§ois Gaillemin | February 14, 2007 11:08 AM

2-wonder if the only virtue of your question is a polemic tone.

Before doing maths you have to think, as Tarjei pointed out, before asking a question on a text you have to read it.

FranÃ§ois

What a weird line of argument. You take an obvious fact (science and math require thinking) and draw the conclusion that epistemology has relevance to the subjects. Would you mind putting forth what, specifically, epistemology contributes to science and math? How is epistemology useful in theorizing?

I've studied some epistemology, and I do math; nothing in the theory of knowledge has been useful in the process of generating interesting mathematics. Like much of the philosophy I studied, epistemology was mostly really interesting mental masturbation.

Tarjei:

I am claiming that you don't understand how science is practiced. Nothing you say in your comment address that but instead supports my claim, as here.

Oh, and it is well known that "common sense" is a particularly dangerous criminal in science, to be shot at sight. Relativity (which protects invariance) is only common sense after it is learned, and quantum mechanics not even then. :-) (OK, I jest a little, since the use of linearity, superposition, complex variables, probability, the exclusion principle and nowadays decoherence makes certain sense. But most of the results does not.)

This is a straw man. Understanding this helps science. The question is, does epistemology help understanding? Not according to its results in science. As Davis says: "epistemology was mostly really interesting mental masturbation." I can't say it better!

I would also like to see an answer to my problem of finding any research from Unger. You have now made the fallacy to appeal to two authorities. And neither Unger nor Steiner seems to have done science.

FranÃ§ois:

It wasn't obvious to me, and since you never gave any references I couldn't find your definition of "miroring [sic] forms", making it a guess, albeit somewhat supported, that it was about the posts dual spaces.

In any case, the post explains why Steiner's discussion of math is woo, and it is not an argument against that to mention that it is used in education. If the post is correct it is an argument against its use.

Uups, sorry, the Anonymous comment was me.

I forgot to say that it is interesting to gain insight into theosophical dogma and Waldorf pedagogy. I also hope that it goes both ways; it should be interesting for theosophists to see what science and math practitioners think about the issues.

You seem unaware of the vital difference between theosophy and anthroposophy. The former has nothing to do with science, nor with Waldorf education.

I did say "over and out" because I have no more time for this, but if you are interested in pursuing discussions along these lines, you should try the public forum Anthroposophy Tomorrow . I believe there are some Waldorf teachers there, mathematicians maybe, science buffs, religion freaks, and whatever your heart desires.

Cheers,

Tarjei

Tarjei:

The "vital difference" between theosophy and anthroposophy is that theosophy is a form of interesting mental masturbation, whereas anthroposophy is a pile of goofy nonsense with delusions of grandeur.

Anthroposophy claims to represent a revolution in mathematics, physics, medicine, education, philosophy, religion, etc.... Any yet, after 100 years or so, there's not

onedemonstrable effect of anthroposophy on the practice ofanythingthat it purportedly influences.In particular, the math of anthroposophy is a pile of gibberish. As I said in the original post - anthroposophical math is a jumble created by taking a ridiculous ultra-literal interpretation of a misunderstanding of mathematical principles.

Duality is not a statement of physical reality - it's a statement about a mathematical abstraction. Metrics in a space are not statements of reality - they're an assignment of an abstract measure to a space.

Projective geometry is a useful form of math for certain applications. That doesn't mean that the physical universe necessarily has an infinity point

Steiner's work is littered with that kind of ultra-literalism, where abstract concepts are taken as absolute literal properties of the concrete universe.

I ignored it due to the post discussing theosophical math. The Waldorf connection was actualized by Waldorf math teacher.

The distinction and dependencies between theosophy, anthroposophy and waldorf pedagogy is interesting when it it is actualized as here. But I'm not particularly interested in the specifics of dogmatic disciplines since they are anti-thesis to reliable knowledge.

'Ten Precepts for Freedom of Thought':

*

Do not feel absolutely certain of anything.

*

Do not think it is worthwhile to proceed by concealing evidence, for the evidence is sure to come to light.

*

Never try to discourage thinking, for you are sure to succeed.

*

When you meet with opposition, even if it should be from your partner or your children, endeavour to overcome it by argument and not by authority, for a victory dependent upon authority is unreal and illusory.

*

Have no respect for the authority of others, for there are always contrary authorities to be found.

*

Do not use power to suppress opinions you think pernicious, for if you do the opinions will suppress you.

*

Do not fear to be eccentric in opinion, for every opinion now accepted was once eccentric.

*

Find more pleasure in intelligent dissent than in passive agreement, for, if you value intelligence as you should, the former implies a deeper agreement than the latter.

*

Be scrupulously truthful, even if the truth is inconvenient, for it is more inconvenient when you try to conceal it.

*

Do not feel envious of the happiness of those who live in a fool's paradise, for only a fool will think that it is happiness.

Bertrand Russell-1951

Thats all.

Davis wrote, "Like much of the philosophy I studied, epistemology was mostly really interesting mental masturbation."

... or mental exercise, I'd offer. And like any form of exercise, it can have certain beneficial results. The ideal result of philosophical analysis is the ability to question one's own deepest assumptions, an ability obviously pertinent to scientific theorizing.

That said, philosophical depth is not science, but more like a fundamental prerequisite for good scientific thinking.

Who cares... modern or Anthropsophical, it all boils down to how much the student (me) is interested. I don't like math all that much anyway, although it's better at Waldorf than at most schools. It's history that matters. Math may have formulas and equations, but it hasn't got revolutions. How can your arithmetic and theorems even compare to the likes of Nero, Rob Roy, King Henry VIII, King David, Lenin, George Washington, Boudicca, or Eirik the Red (they all had ONE thing in common. Figure it out, join the cause, and march on Newcastle). That's my off-topic contribution.

All fields have revolutions. Axiomatic math was one example.

If you have read the "science" of Lenin, you would be ashamed to mention him as an example on a science blog.

How does your puny list compare to the Plimpton 322, the Rhind papyrus, Laghada, the Shulba Sutras, Pythagoras, PÄnini,Eudoxus, Aristotle, Euclid, Archimedes, Eratosthenes, Hipparchus, Diophantus, Aryabhata, Brahmagupta, Al-Khawarizmi, Omar KhayyÃ¡m, Bhaskara Acharya, Madhava, Gerolamo Cardano, John Napier, RenÃ© Descartes, Isaac Newton, Carl Friedrich Gauss, ... well, why not stop at the prince of mathematics.

Each and every one has revolutionized the field and contributed immensely to society.

I'm mostly just amused by this one because I'm not finding it very hard to think of mathematicians who also qualify as political revolutionaries. Off the top of my head I would mention Galois, or Russell, or Grothendieck...

(Double credit since each of those three also started or headed a mathematical revolution at some point!)

Perhaps some of you posters would care to talk to some of the admissions team members for colleges like Harvard and Columbia, where Waldorf students are very welcome because they are not only smart, but considered to be creative and independent thinkers and problem solvers, as well as well-rounded and mature young people, as compared to their public school counterparts. They are often described as more compassionate, more curious and more interested about life than the average student of the public school system. My dearest friend's daughter is a professor at USC's medical school, and a Harvard graduate. She attended Waldorf schools from kindergarten through grade twelve, and has co-written the first medical textbook on PTSD, which is going into its second printing. She is always eager to encourage her friends to send their children to Waldorf schools, as her experience was very positive. A young friend of one of my daughters works for Nasa and is highly regarded there...also a Waldorf graduate. And, as I recently read, one of the top executive officers...CEO or President, if I'm not mistaken, of the American Express company, is Waldorf graduate and quite proud of the fact. Waldorf schools are often chosen, by diplomats and high ranking political officials in Europe and by the wealthiest people in this and other countries, for their children because they are considered to be schools where the best education is to be found. Perhaps Mr. Chu-Carroll should try actually DOING a bit of projective geometry......

Perhaps some of you would be interested to know that Harvard and Columbia, among many other fine schools around the world, consider Waldorf students to be outstanding students and human beings, and are eager to have them apply to their institutions. The daughter of a dear friend is a professor at USC's medical school and was a Waldorf student from kindergarten through grade 12. she encoursges all her friends with children to send them to Waldorf schools. Another young man, a friend of my daughter, works for NASA and is highly regarded there. He appreciated and enjoyed his education in a Waldorf school. The CEO of American Express is also a former Waldorf student, and quite proud of the fact. Many high ranking political officials in Europe, as well as many of the wealthiest people in the world, send their children to Waldorf schools because they consider them to be where their children will get the best education available. Regarding some of the "rascist" comments, there will always people who make mistakes, and it is easy to point out the few and magnify those mistakes. Perhaps you might want to someday visit the Waldorf School of the Peninsula, in one of the wealthiest areas of the country, particularly the first grade, where there are children representing many ethnic backgrounds, speaking eight different languages and belonging to a variety of religions, including Buddhism, Eastern Orthodox, Zoroastrianism, Islam, Presbyterianism, and Judaism. Even atheists love sending their children to Waldorf schools, because their children get such a good education. Most Waldorf parents are happy sending their children to these schools, where they are accepted and honored for their humanity, and where their beliefs are not only tolerated, but welcomed. Compared to the many thousands of very contented Waldorf parents, you seem to me to be but the disgruntled few. What a pity that most of you posters seem to feel that "being spiritual" is a wicked thing! And perhaps Mr. Chu-Carroll should try actually DOING projective geometry instead of just bad-mouthing something he doesn't truly understand...

Why do you doubt the ability of Dr Chu-Carroll to do math? A computer scientist usually knows a whole lot of it by education or application as a bit browsing around here would show.

The whole point of the post is that Steiner tries to make physics out of math on a philosophical basis, which is pure rubbish and has nothing to do with the math of projective geometry. Something that you should know when discussing math or physics, and which underscores that Waldorf pedagogy and theosophy is harmful to a proper education.

The difference between a creative science education and an education that promotes creativity in general is that the former takes into account that science is constrained to be about actual knowledge.

This will probably end up being just a note to Mark, seen as how this article was originally posted over two years ago, but ah what the hell. :) The internet as a whole is much more colourful to a non-native english speaker, as more than anything else it shows how a particular culture can influence and bias even the fiercest freethinkers. I'm not posting here to flame, so I'll admit first thing I have my own biases, and I'm under the influence of just as many moral prejudices as anyone else.

That being said, just wanted to share a slightly different experience. I'm brazilian. My culture has taught me and my compatriots to deal with things in a largely different way than anglophones. This of course has serious practical implications in how foreign social structures and institutions manifest themselves in Brazil.

While my fellow Waldorf students from the US and UK who posted here learnt how to knit and had walks in the woods and whatnot, down under we had a largely different curriculum, designed both to be compatible with Brazil's oddball national curriculum and to adapt Steiner's whacky precepts to the local cultures. Our basic education consisted in three pilars: language (rhetoric in particular), science (maths and logic in particular) and the arts (fairly liberal with materials and techniques, and with a focus on personal production).

We were taught the history and criticism of Steiner's philosophy all the way... having read in the process Schopenhauer and Nietzsche, and then going to refutations and new uses of his ideas (I'm particularly interested in a philosopher called Peter Sloterdijk, where you might find an interesting example of mathematical abuse, cf. SphÃ¤ren). Overall, our mathematical education was brilliant. I don't know many people from other schools who actually learnt how an axiomatic system works pre-high school, and there were countless occasions where we were reminded (with a strong mention to Tarski) about how mathematical systems cannot ever set the criterion of their own veracy/sanity.

I'd say the course was kickass. I'm sure painting a very nice picture of a school which --- like any other --- had just so many qualities in contrast with a sea of flaws and quirks. But to be completely honest, I think it's a big mistake to establish such a strong belief in the duality between a theory and its many practises. :) I hope not to have been too rude, just wanted to share my experience and my two cents.

You really ought not to write something that you know nothing about. It becomes immediately apparent that you are entirely ignorant of the subject when you mention that Rudolf Steiner invented Theosophy. Also, just because we don't understand things, doesn't mean they can't be true; to believe otherwise would be to consider our own powers of perception to be perfected and infallible--an incredibly callous thing for a person to do.