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".
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
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.