There is a famous saying, often attributed to Ludwig Mies van der Rohe, the German-born architect who established his fame in Chicago, that “God is in the Details.” Others in the know claim it was Gustave Flaubert, renowned author of Madame Bovary and other novels, who said it. Personally, I love this uncertainty as to whom the quote should be attributed; it links together a literary giant with a architectural giant, and the latter’s work would have been impossible without the proper application of mathematics. It is a joyful coincidence that literature and math are the two major themes of this blog.
I have a different slant on where God can be found, and I couldn’t be more serious when I say, for me, that God is in the vectors.
Let me explain.
My first exposure to linear algebra (LA) was in the early 1960s as a junior in high school. LA was an oddity back then, something that was useful if you needed a simultaneous solution to a given small number of equations with an equal number of unknowns. The trouble was, without the availability of computer power, once you got past systems of four or five such equations and unknowns, the machinations and iterations that were needed to be done manually became overwhelming. They also taught us about matrix determinants because if you calculate a determinant of a given matrix as zero it tells you immediately that there is no simultaneous solution for the unknowns of the given matrix.
Above is a typical example of a four-by-four matrix: four columns and four rows. Each column of numbers is called a column vector, and each row, a row vector. All of the sixteen numbers are the coefficients of the unknowns we hope to identify. Although not shown in this example, the real (or complex) numbers used can also be negative and/or include zero.
One of the wonders of LA is that no matter how big the matrix is or how long the vectors are that make it up, there are only three possibilities for its solutions: one solution; no solution; or an infinite number of solutions. This has to sink in for a while; whether the matrix is three by three, like one of the nine little boxes in a Sudoku game, or 5 million by 5 million, the potential solutions remain the same: one; none; or an infinite number. Somehow, this leads me to contemplate alpha, omega, and everything-in-between.
Now, on to truly large matrices. Google has reportedly worked with matrices having upwards of one billion column vectors. Netflix is close to joining them in this mathematical exosphere. I cannot picture one billion of anything in my minds eye, so let’s bring the number down to a matrix of 100 rows and columns for ease of illustration. Then, if we orthogonalize this matrix, we make the length of each vector = 1 and each of the one hundred vectors perpendicular (orthogonal) to the other 99. How can this be? How on God’s green earth can we end up with one hundred vectors with each one perpendicular to each of the others? We’re talking one hundred dimensions here, dimensions well beyond our ken, and yet these and much larger matrices are made to work every minute of every day to solve real word problems.
We can easily imagine 1, 2, and 3 dimensions: a point, a plane, a box or sphere, respectively. Einstein and others suggested that we could add a fourth, which would be time. When I plan to meet someone, it’s not enough to say, for example, “meet me on the 45th floor of the Hancock building.” GPS will get me to the correct latitude and longitude, and the elevators or stairway, to the height, but without the element of time we may never meet. Let’s choose to leave this last conjecture to the physicists and get on with physical dimensions.
There have been various attempts to drawn spaces of four dimensions and upward, but in my opinion they all fail. We just do not seem capable of visualizing physical spaces of more than three dimensions. An example of a three-dimensional space is easy to construct; three dowel rods, all at right angles to themselves, and we’re done. But now I give you a fourth rod, a fifth, a sixth. What can you show me now? The answer is nothing more that you showed me with the first three. Is it because we live in a three-dimensional world that our brains top out at three dimensions too? Or did our parents forget to select the nth-dimensional upgrade before putting their new-offspring order into the pipeline?
Whatever the reason, our brains seem to grasp three physical dimensions and no more. Yet Google does work in, give or take, one billion dimensions and the math works. If this seems out of reach, we can take it down to matrices of ten dimensions. And, yes, the information we derive from them results in a valid solutions to the problems at hand. We get lost in the vectors, but since the math works despite our shortcomings, it must be God who reigns over them. God is in the vectors.
These vectors humble us, just as they should. We are intelligent; we are the only being capable of altering the conditions of the world we live in and even of destroying it. Almost daily, technological discoveries, scientific breakthroughs, proofs of long-standing math conjectures or hypotheses and other achievements become responsible for our ever-swelling heads as a species. And yet we are lost in other things with little hope of finding our way out. Can we ever understand “infinity” (the infinitely large and the infinitely small) or “eternity” (of course, first we need to agree that time really exists).
When all is said and done, though, I posit that we will never understand the secrets of large matrices and their vectors that arrange themselves in unfathomable arrays. I like to think of it as God’s Last Stand: the line in the sand separating humanity from divinity for ever. It is there to confirm the shout out to us from Psalm 46: “Be still and know that I am God.”
Postscript: While linear algebra and some other math concepts are used here, this blog was never intended to be a math lesson. My goal was to give just enough information to make my theological point more easily understood by the reader. Those with a good understanding of math may say I left out certain operations and concepts. “Orthogonalize” is perhaps the word most easily picked on. I chose not to explain it and instead decided to let it be taken for granted.The long explanation required would have detracted from my main theme. Did I succeed; did I fail. I’ll leave that up to the reader. Finally, the classroom illustration used at the top of this blog shows Gilbert Strang, Professor of Mathematics at the Massachusetts Institute of Technology, whose books on linear algebra are generally considered the best.