Last time, in “Theory of Dots,” we talked about differences between perceptions and the phenomena they are about. We noticed that solids only appear solid, liquids only appear continuous, television images appear to be made up of many colors but in fact contain only three colors, and films and videos appear to show motion but do not. By the twenty-first century, most people learn at any early age in school that the sun appears to move through the sky but in fact it’s the earth that’s turning, and the stars don’t really revolve around the earth every twenty-four hours. I have been trying to loosen the firm conviction we bring to daily life that the way things look are the way they actually are. It is this conviction which most confirms the realist’s simple belief in a simple reality, and which most prevents a clear understanding of our situation in reality.
Let’s look at a very simple diagram of a spatial relationship:
In the figure, a and b are dots. The arc labeled e is an edge. In the vocabulary of geometry, edges connect points. Now, these two dots can represent any two locations in space, for example we could let a stand for the city of New York, and b stand for the city of London. The edge e then might be the distance between them. (It could also represent some other kind of relation between them, but for our purposes, distance will serve as a generic kind of relationship).
The distance between the two cities is an important fact, because we experience it whenever we buy an airplane ticket and fly between New York and London. Not only does it cost money, but it also takes time to traverse that edge. The distance between them impacts our lives in a way that we have to deal with. Even if we only mean to call a friend in b from our phone in a, the cost of the call will depend on e, and so will the time zone difference, and hence our friend’s irritation if we call at 3am.
But look at the diagram again. The two dots a and b are there. In reality, they take up space. The cities comprise matter. They age, having duration in time. You can bomb them, and destroy them. (Do not attempt.) The situation with the edge e is entirely different. It isn’t made of matter. You can search all around London without finding the line through space that goes to New York. You can’t bomb the distance or damage it. You can’t put any kind of screen between the two cities that will block the distance; they will continue to be separated by the same distance no matter what you put between them. In other words, the distance is intangible. It’s only an abstraction, and not physically present in the universe.
This, for those of us who have studied Wittgenstein’s Tractatus, is what he means by
1.1 The world is the totality of facts, not of things.
The world, at least our world with London and New York in it, could not exist as we know it without the meaning of this abstract relationship of distance lying between the two cities. In logical terms, we would say their positions imply the distance between them. That is, the meaning of the distance is constructed with logic, and its relationship to us is mathematical.
How so? There may not be a way to change the distance without moving the points, but the implications of the distance can be altered by choosing a faster airplane. Traversing the edge faster will reduce its apparent length; i.e., the time required for traversal.
This is the foundation of the application of mathematics to physics. It asserts that the abstract relationships mathematics allows us to form, do have experiential and phenomenal impact on us, despite their intangibility.
I want to show one more diagram.
The diagram shows a system of four dots making a pyramid with three sides and a base. One of the edges of the pyramid is labeled e. The edge connects two of the points, a and b. But notice, now, three of the dots define a face f. The face is a plane structure that occupies two dimensions, unlike a point or a line. The combination of plane surfaces or faces on the four sides of this pyramid make it a three-dimensional structure, also called a solid.
No doubt you didn’t really need the geometry lesson, but I wanted to make clear the source of apparent solidity. Just as with the line earlier, the face f in the figure is not physically present. It’s a geometric relationship having mathematical form, just as with the line. In other words, our perception of solidity in matter is geometrical and mathematical in nature; it’s a construction based on points, but not made of points. It lies between the points, just as meaning lies between words. This ability to perceive mathematical forms which don’t exist physically is a unique quality of our minds.
(to be continued)