I was talking today to my buddy George Ure about a strange way of looking at the world as a natural/spiritual duality. It was the key concept in a dream he had Sunday night/Monday morning.
On Monday in his “wujo” section, he related how his dream addressed the nature of our reality and God as being two complementary parts of the same larger “thing.”
I’m not sure if I ever told him about it (probably did), but I wrote a paper in college on exactly that topic for a seminar called, appropriately enough, “Physics for Poets.” That’s not to be confused with the “gut” overview course offered by the Physics department that jocks used to take, even though it was known by the same name among the students.
George was adamant that I had to write it up in my words today, so he could include it in the cosmic cartoon (my words, not his) stream-of-consciousness section of his wide-ranging blog.
The first order of business in our conversation was to let George know that Banach, Hilbert and other inner-product functional spaces have a word they use instead of “complement,” which I think fails to emphasize the fact that the two pieces together equal everything.
In math world, we call the part you’re not looking at the “dual space,” and that ever-so-nicely fits with the philosophers’ use of the word “duality.”
So let’s crawl (the thing we do before walking) through a very simple example.
Picture a 3-dimensional universe. We represent any point in that universe by expanding into three dimensions. For sighted people, good old Descartes gave us the most used version — Point A is defined as (x,y,z) — the measurements of distances along three mutually perpendicular axes from an origin point defined as (0,0,0). Distances between points are measured using Euclid’s geometry and Pythagoras’ Theorem.
I specified sighted because I had a friend and fellow math major who was blind when I was an undergrad. One day we were walking through an archway that had a soda machine. He asked me to guide him to a spot in front of the machine.
When I told him he was standing in front of it, he reached out and felt the space in front of him until he located the coin slot. Then he put his change in the slot and (from memory) moved to the button for a regular Coke. I noticed that he was not, as I would, moving along the row of buttons to pick the one he wanted (Cartesian coordinates), but rather, he was positioning himself and then reaching out at the spherical angle and length (declination, right ascension, length) that took his finger to the correct button.
The figurative light went on in my mind. Of course, if I lived in a dark world, it would feel like that to me — that everything in it had a latitude angle, a longitude angle and a distance from my location, rather than picturing myself moving about inside a grid the way we Euclidean types see the world, especially in cities and buildings.
Anyway, back to the problem of finding God, or something to that effect.
Picture a consciousness that can only perceive two dimensions of that universe like the cute story “Flatland.” In that story, the narrator, a square, could only see other two-dimensional geometric figures in its two-dimensional world.
The social commentary of the piece would have doomed it to limited interest in its own Victorian world, in my opinion, if Square hadn’t been allowed to see the one-dimensional world (Lineland) and even see the three-dimensional world inhabited by the super-cognizant Sphere who opened Square’s eyes, so to speak.
That led to the little novella having lasting power, such that even Isaac Asimov recommended it as a way to learn about dimensions.
In the book, Square gets to know the omniscient Sphere that floats above the plane of his world. Being of three dimensions, Sphere would appear as point (when he first touched down), and then a circle of increasing radius as he intersected the plane. Amusingly, circles are thought to be perfect beings in the Flatland heirarchy of social castes.
Why do I care?
Because I was reading very cool book by the “Father of Modern Computing” (John von Neumann) for entertainment at the time I took that college seminar called Physics for Poets. That book was the 1932 classic, Mathematical Foundations of Quantum Mechanics, which solved a problem that bedeviled everyone following Neils Bohr into the strange world of subatomic particle/waves.
That problem was that Werner Heisenberg had come up with a set of matrices to describe the experimental results he was seeing. Think of it as the (x,y,z, …) of Euclid’s geometry, taken to higher dimensions. Except to capture and compute all the results, Heisenberg had to allow the matrices to be infinitely large “squares” sometimes. In other words, his solution was infinite dimensional.
Another set of predictive solutions to the very same observations was made by Shrodinger, of cat-killing fame. His solution involved waves that corresponded to Newton’s classical degrees of freedom, where an n-particle system could be described by three dimensions to define locations and three dimensions of rotational momentum for each particle. A two-body system could be described using 12 dimensions of Shrodinger’s waves. (With the one small problem that occasionally the waves got infinitely “dense” — think of it as being multiplied by a bell curve that has the normal weight of one, but an infinitely small base to the curve.)
Von Neumann’s elegant solution was to note that we weren’t really looking at the objects in the system that represented the experiments (Heisenberg’s potentially infinite-dimensional matrices, or Schrodinger’s not-quite sinusoidal waves). What we were looking at was how the functions in those representation spaces interacted with each other — their functional spaces and the operators on those spaces. That infinite-dimensional inner product space that unified these two wildly different formulations for quantum mechanics is called Hilbert Space.
Hilbert Space was named after the brilliant 19th and 20th century mathematician who finished Euclid’s work nearly 2,000 years after the old guy laid down his first postulates and axioms. (Hilbert presented and proved the minimal set of axioms required to create Euclid’s geometry in 1899).
To me, a college sophomore drawn as much to mysticism as mathematics, this was cosmic.
Since the universe was obviously infinite dimensional (you had to use infinite dimensions to model even simple bits of it), yet I could only experience four dimensions (three normal plus time), I thought God might just be hanging out in those other [infinity minus four] dimensions.
Every time I interacted with God, of course the result would be zero, just the way a Flatland inhabitant who can only see (x,y,0) would be frustrated trying to interact with the orthogonal Lineland inhabitant that can only see (0,0,z). In that arrangement, Flatland and Lineland happen to add up to a three-dimensional space, but each of the two worlds is the dual space of the other.
Bingo! God could be real, just in the dual space to my observed reality. Everywhere and nowhere. Unseen but omnipresent.
Fill in the the rest with your favorite cosmology and/or religion and/or science fiction.