Rudolf Steiner was a prolific Austrian author and philosopher of the late 19th and early 20th centuries. He felt a strong connection to mysticism and spiritualism, ever since he supposedly communicated with the ghost of a recently deceased aunt at the age of 9. Steiner is well-known for having led a group that split off from the popular circle of European mystics known as the Theosophical Society, which seemed heavily inclined to regard the religions of East Asia as somehow providing the keys to understanding spirituality. Steiner called his new group the Anthroposophical Society, and this competing group believed that Western science and culture were just as strongly connected to the spiritual-- it was just a matter of intepreting them properly. One particular Western idea that Steiner was fond of was the concept of a fourth physical dimension, another mathematically-defined direction that we cannot percieve but is just as real as length, width, and height. Steiner believed that our consciousness extended into this fourth dimension, and that phenomena like ghosts and ESP resulted from activity in this hidden dimension. And most interestingly, he believed he had a simple philosophical proof that this fourth dimension really does exist, and our human minds really do extend into this additional dimension.
Here's how Steiner's proof goes. We all know that a creature of a particular dimension, if it looks out at its world, really only sees a view that is one dimension smaller. For example, a one-dimensional creature living in Lineland, a universe that exists entirely on a single straight line, can only perceive a single point on either side of himself: a zero-dimensional view. Similarly, a two-dimensional Flatlander, living on a plane, really only sees a line; it is only us three-dimensional creatures, looking down on the plane from above, who can truly comprehend its full two-dimensional world. And in real life, when we look out with our eyes, we are only seeing a plane. Yet somehow we do believe we fully understand and perceive the three dimensions of our world. Steiner draws what he believes is a natural conclusion from this: "The fact that we can delineate external beings in three dimensions and manipulate three-dimensional spaces means that we ourselves must be four-dimensional... We float in a sea of the fourth dimension just like ice cubes on water." In other words, our ability to fully perceive our three-dimensional space shows that our minds must extend beyond those three dimensions.
It's a fun thought, but you can see something fishy there right away, if you think about the world of modern computing. I can think of all sorts of situations in which an object in three dimensions is represented by a model in fewer dimensions. For example, most computer memories and circuits that power modern three-dimensional computer games are essentially stored in flat two-dimensional circuit boards. While these are technically 3-D like all physical objects, the memory storage can be thought of as truly two-dimensional in some sense, as each (x,y) coordinate on the circuit board only stores one encoded value at any given time. More basically, you may recall the concept of a Turing Machine discussed in some earlier podcasts: this is a theoretical model of computing, based on writing and reading values from a long, essentially one-dimensional, tape. It has been shown that any modern computer can be modelled by a very slow, but 100% accurate, Turing machine equivalent. So even the 3-D models in a modern computer game could, with enough work, be represented in one dimension.
I think the main flaw in Steiner's argument is his fundamental premise, that a creature of n dimensions can only perceive n-1 dimensions. It is true that through the sense of sight, a creature can only see one dimension lower, but our senses are not limited to sight. Think about a blind man, who perceives the world mainly by walking around and tapping items with his cane to understand their form: he can walk forward, back, right, or left, and even climb ladders up and down. He is truly perceiving the full three dimensions of his world, travelling within all three of those dimensions and building a mental model based on his real experiences. This applies to the lower-dimensional examples as well: the flatlander can move around and perceive his full plane, and even the poor Linelander can move back and forth on his line. Thus, the idea that perceiving your full dimensionality requires capabilities from a greater dimensionality does not really seem to ring true. You need to think of perception much more generally than simple line-of-sight.
Naturally, this does not fundamentally prove that Steiner was wrong about our minds extending into the fourth dimension; it just means that the proof of such an idea is not so simple. So it's still entirely possible that the concept of our mystical four-dimensional minds is correct but unproven, and the rest of Steiner's Anthroposophical Society ideas might still be valid. This philosophy of the fourth dimension was just a launching point for a variety of mystic concepts, related to traveling along this fourth dimension to the astral plane where you could encounter ghosts, life after death, etc. Some of Steiner's lectures get amusingly specific on details of the astral plane-- apparently he believed that his meditation and similar activities had actually taken him to this place, so he could talk about how astral dimensions mirrored our own, and writing there would appear backwards. Personally, I'm a bit of a skeptic on this topic, but these kinds of ideas do seem to have a lasting appeal, as shown by the New Age sections you can find in many modern bookstores. If you're into that stuff, try meditating hard enough, and maybe you too can follow Steiner's path into the astral plane through the fourth dimension. While you're there. see if you can track down Steiner's spirit, to discuss the flaws in his philosophical proofs.
And this has been your math mutation for today.