For instance, the idea of negative space is absurd, but, nevertheless, we can use it to great advantage. So what does this mean, that we are divine creatures? It took mankind centuries to take this giant leap for the first time, and it was not done without a lot of hand wringing and pain. Yet, we expect children to grasp it as a matter of course in between watching American Idol and playing Mortal Kombat.

Fortunately, I’ve finally found that we can understand the meaning of the idea as well as how to use it abstractly, but only if we are willing to ferret it out by thinking on our own, definitely not by reading textbooks. The key is to understand that space doesn’t exist as stuff that has properties, like cabbages, or fabric, and that the early Greeks were wise in keeping the ideas of magnitude and numbers separate.

It turns out that, if we introduced to children the difference in the ideas of operationally defined magnitudes as opposed to quantitatively defined magnitudes, and the proper use of real numbers in these respective definitions, then we could avoid a lot of grief, not only for naïve children, trying to learn mathematics, but also for the sophisticated adults they later turn into, who then try to formulate physical theories.

What you would normally call n cabbages, call n and 0 cabbages and then negative n cabbages just becomes 0 and n cabbages. No pesky minus signs to remember, just order your cabbages piles! That’s how I still think about it – helps with K-theory.

>the difference between imagining the existence of negative numbers and imagining the existence of extra dimensions of space is one of degree, not of kind

Care to explain or support this notion somehow?

Sean, you mentioned whether “our faculty of abstract mathematical reasoning is somehow related to the development of grammar.” I wonder what film has to do with it. In my personal experience its made me more flexible with alternate worlds and things. I can envision special effects and multi-dimensional images that make any abstract notion feasible.

Mm. I remember taking quantum mechanics and writing notes, the wave function is written like this, yes, hit it with another with another wavefunction, from a different space, squeeze a Hamiltonian in the middle if you must, square it and you get the probability that you’ll end up in the state represented by the second wavefunction. And this stuff is so amazing, but we were supposed to take notes and learn how to calculate probabilities. And that was cool, but at some point I would have loved the professor to just stop speaking and look at us. As if to say, can you believe this? Isn’t this wild? Because we were being asked to make a jump, to grow up in a way, and you need some space to realize what you are really doing. My sister the poet said Jennifer you guys need to have cots in the backs of your classrooms, so when this stuff hits you you can go lie down for a while and try to take it in. Because when I would explain what I was learning to her, she would make me stop so she could digest it – it makes her head whirl.

I’d always thought that people who need to lie down when making a leap were somehow…maybe a bit more advanced, as far as being a human being and really living life, understanding the immensity of what they are trying to take in. But, maybe we’re just a bit more in touch with our earliest ancestors…that would be cro-magnon and his ilk, not the fig-leafed ones…