And Now For Something Completely Different . . .

Projective Geometry and Beyond!

Hi Friends,

I’d like to use this post to introduce you to what I’m thinking is coming up next in terms of new writing from yours truly.  The Mulling Over the Manifesto series was purely an intuition of mine that I really needed to do that. . . and I feel quite satisfied with the results.  But, it was a hefty chunk of thinking and writing!   Seven parts, each of which felt like a mini-chapter to me.  I found that, with that task plus with the holidays and other busy-ness, I didn’t have as much time to keep up with my Wondering Wednesdays series, in which I ply you with questions and answers about honey bees and science, interwoven with the ideas of Rudolf Steiner to boot.  I hope to get back to posting some more WW Q&A’s soon.

As I’ve explained in some of those WW Q&A posts, in his graduate studies, Steiner was inspired by the scientific work of Johann Wolfgang von Goethe. Goethe is much better known for his poetry and plays, for instance Faust.  But he was also an avid student of science, and approached it in a very different way from that of his contemporaries, such as Isaac Newton.  He especially studied the areas of optics (color and light) and botany.   Steiner retranslated all of Goethe’s scientific writing as a PhD student, and came to believe that Goethe was doing something uniquely valuable in science and culture, something worth sharing and developing.  For this reason, the approach to science that comes out of Goethe’s and Steiner’s work is sometimes called “Goetheanism,” or “Goethean science.”  Alternatively, it’s also sometimes called “phenomenology,” but that term can be confused with a number of other schools of thought that are unrelated. 

This is all leading us to what I’m going to start writing about next.  In a weeks’ time, I will start commuting to the Chicago Waldorf School to teach a course that I haven’t had the pleasure to teach for 9 years. It’s a course that is typically taught to 11th graders, and it is called Projective Geometry.  It is probably my absolute favorite subject to teach. I’m quite excited for it, and also excited to share some nuggets with all of you here as I do.  I’m so excited that I set up another section here on my Brian Does Whatever He Feels Like Doing, Gosh! Substack.  The new section is called “Projective Geometry and Beyond!” and you should be receiving this post through that new section right now, if I set it up right (if you just read my posts in your email it shouldn’t look much different at all; it’s only if you click on it and read it on the webpage).

Now, I know what you’re thinking: “Geometry?  Ugh.”  I am aware that for those who don’t have a natural love of mathematics, or worse, were damaged by their mathematics education, mentioning geometry might not raise the most positive feelings. But, if you bear with me, I can explain how this fits into many of the other things I’ve already been writing about.  Then you can decide whether you’re interested, or not!

Rudolf Steiner spent his entire adult life working to flesh out this new approach to science and philosophy.  That body of work came to be known as anthroposophy and spawned a number of sister movements, such as Waldorf schools and Biodynamic Agriculture, to name a few.  But Goethean science was always the core of anthroposophy for Steiner.  Steiner felt that the world deeply needed this novel approach to science to counterbalance the negative effects he saw all around him that were the unconscious outgrowth results of materialism.  One of the major negative effects was thinking of the earth and cosmos as dead matter, unconnected in any way from the living world or from the human being.  Another was industrial war and a kind of mechanization of our thinking and living. For this reason, Steiner referred to what he was doing as “spiritual science,” although I have come to call it “anti-materialism” in most of my writing.  Here is a post where I tried to introduce that to you.  

OK, so where does Projective Geometry come into all of this?  Well, you might imagine, lots of people who came to be intrigued by Steiner’s work had a hard time following him.  If you’ve followed any of my posts on what Steiner said in the honey bee lectures (Parts 1, 2, 3, 4, 5, 6 so far!), you can get a feel for how hard it is.  We are such materialists these days, it’s hard any more to think any other way. You ask somebody, “Why is the sky blue?” and you expect to get an answer having to do with diffraction scattering of photons of different wavelengths, and that’s supposedly all there is to it.  This despite the fact that almost no one other than Physics majors like myself actually understands anything through explanations like that . . . and these days I’d argue that most of the Physics majors don’t really understand, either!

 Steiner was trying to show people a reality to which they all had (and most of us still today have) a humongous blind spot.  To try to address this blind spot, Steiner (whose university education had been in mathematics and science) started to tell people that, if they wished to be able to practice thinking in ways that would loosen and open them up further for the things he was teaching them . . . they should study projective geometry.

We all know that physics today is entirely mathematical in nature.  The so-called “deepest secrets of nature and the universe” are expressed not in words but in mathematical formulae, and physicists who work with those formulae are supposed to be a modern sect of priests who possess insight into the deepest knowledge of “Life the Universe and Everything.”  (Any Hitchhiker’s Guide to the Galaxy fans out there?  The answer, of course, is 42. But what is the question? 🙂)

This was true even in Steiner’s time, and Steiner was aware of that.  This is a big reason why so many of us have been damaged by our math education, and why there is still a myopic obsession with math in schools today. It reminds me of how in early times when the church held greater authority over the masses, it was absolutely essential to memorize your bible verses . . . to the point that generations of kids came to hate memorization! Now today, a couple generations of us have been tortured with endless mathematical repetition out of some sense that all of nature’s secrets are at heart mathematical and folks that are good at math are somehow smarter than others (and that, little Mary-Jo, is why you have to pay attention in algebra class!).

This idea is not new.  Pythagoras was one of the early thinkers in the West who came to believe that the mystical and secret workings of Number ruled the universe.  Those of us who have studied and enjoyed mathematics know that it can be a real high, a mind trip, to understand a new mathematical proof for the first time. We humans are pattern-recognizing beings, and finding a new, deeper pattern can be a thrill unlike any other I can think of.  You don’t have to love math to appreciate this.  You can have this same experience studying music, knitting, sports, cooking, the stock market, whatever.  We humans love to build and understand patterns, and this truly does allow us to see aspects of our world and our lives that we could not see before.  I’m reminded of my post about Matthew Crawford’s book, The World Beyond Your Head, in which he has an inspired chapter about the concept of a jig (the tool, not the Irish dance!).  A jig is any social or physical tool that a human who recognizes a pattern makes, to extend themselves beyond their own bodily and mental limitations. So, for example a loom is a complex jig to make weaving possible. Mathematics, then, is another kind of jig, a symbolic language that allows us to succinctly demonstrate patterns and connections and expands our human understanding much farther.

But . . . we live in a time now that the mathematical equations of Physics have led to so many startling discoveries and technological innovation, that many people start to believe that the mathematical models are the science, or are the fabric of reality itself.  It’s a kind of neo-Pythagoreanism today. Or, perhaps a neo-Platonism, in which we think that the abstract world of form is what is “really real”, and the everyday physical world we live in is only a shadow of the real thing.  Rather than remembering that the mathematical equations are only models, and that the models were built on observations and assumptions, we start to think that the math itself is the core of the universe.  In this view, the universe mathematizes and we are just equation variables. That is yet another version of materialism. 

While not at all disputing the beautiful mathematical symmetry that has become core to our studies of science, especially Physics, Rudolf Steiner was also aware that our obsession with prioritizing certain kinds of mathematical knowing was a part of our big materialistic blind spot.  So, he made this recommendation:  If you want to “free your mind” from being too caught up in the “objective reality” of mathematics, try some projective geometry.  See how you feel when you’ve had a few meals of that, and then let’s talk!

Because Projective Geometry is, truly, mind-bending. It is a kind of math that I bet most of you’ve never done before.  And, what’s more, it’s entirely accessible even if you haven’t done any math in years, and even if you hated it the last time you did it.  It has the potential to awaken your unspoken and unrecognized assumptions and make you curious again about things you probably seldom think about, like space, and number, and the human thinking activity that happens unconsciously in the background when one conceives of such things. And, while I could teach you Projective Geometry in the same soul-crushing mind-numbing way you were taught algebra, none of that is necessary, because PG can be taught entirely synthetically.  That means, solely with drawings and imagination, and without any numbers!  This doesn’t mean it doesn’t require you to put on your thinking cap, it certainly does.  Specifically, when you do PG, you find yourself trying to imagine forms and shapes as they move in your mind.  It’s a fun kind of exercise that you might like to try.  

I have no intention of adopting a systematic approach here. If you want that, then I’d suggest you buy yourself some great workbooks from an excellent PG teacher, Henrike Holdrege. My job is only going to be to give you some samplings, to whet your appetite, and weave in other ideas that I want to write about, gosh!

Here I want to give a shout out to my buddy Jeremy, who is on this Substack with us and also teaches Projective Geometry.  A not-so-secret motivation I have for starting this new line of inquiry is to entice Jeremy to chime in with his super deep knowledge of the subject.  We will see if he takes the bait!

This explanation about what I’m doing and why I’m doing it has become long, and so I will wait a couple days before making a first post. But, I look forward to starting to walk you through this fantastical mathematical world together, while I’m also teaching it to some cool kids.  The nature of the upcoming posts will be that I may often suggest you get out a piece of paper, a pencil, and a ruler or some other straight edge.  You never have to do this, but these can serve as an effective jig to get your imagination moving!

In the meantime, let me give you this pop-culture reference to prep you a bit:  It’s from the first Matrix movie.  Maybe it will make sense given what I’ve written so far, and maybe it won’t.  I hope it will soon!  Here is the link if you want to watch the clip.

Neo is visiting the Oracle for the first time, and he’s waiting in a room full of children who are other “Potentials” (humans who might have the ability someday to become the One, the person who can manipulate the Matrix at will, and thus defeat the machines).  Neo sees a boy bending a spoon with his mind, and has this conversation:

Potential: Don’t try to bend the spoon, that’s impossible.  Instead, only try to realize the truth. 

Neo: What truth?  

Potential: There is no spoon.  

Neo: There is no spoon?  

Potential: Then you will see that it is not the spoon that bends, it is only yourself.

Cheers, Everyone. As always, you can click “Like” if you want, and leave comments about all of this. I love to get that kind of interactive feedback!

Brian