Tetrahedra and Truth

Let’s say you have a tetrahedron (a polyhedron consisting of four conjoined triangles). You project each of its points onto a flat surface, along lines perpendicular to the surface. Depending on the tilt of the tetrahedron in relation to the surface, you will end up with either a triangle or a quadrilateral.

Now, both the triangle and the quadrilateral tell truth about the tetrahedron, but neither one tells the complete truth. However, if you rotated the tetrahedron and captured enough projections along the way, then you could determine the tetrahedron’s shape from the projections alone (if you already knew that it was a convex polyhedron). In other words, by considering the changes of the projections in time, you could see beyond the projections’ two-dimensional aspect to the tetrahedron’s three-dimensional shape. (You can try rotating a tetrahedron here.)

To even begin this project, you have to suspect that there’s something beyond the flat shapes that you see. You think: “Yesterday it had four sides. Today it has three. Something’s up with that.” Without such suspicion, you’re a prisoner in Plato’s cave, believing in the shadows on the wall because you’ve seen nothing else.

Now, suppose the tetrahedron were not stable in shape. Suppose it were crumbling or melting. Then you could not determine its shape from the projections. You could only approximate it—that is, by observing projections very close to each other in time and trying to spot abnormal changes. A sort of calculus would come into play. The more regular the tetrahedron’s disintegration, the more accurate your calculations would be. The projections would only pick up certain kinds of changes; they wouldn’t show concavities, for instance, if the edges were still intact.

Things get even more complicated if time itself is unstable: if it slows down, speeds up, loops around, breaks apart, or comes to an end (in relation to some other measure). We won’t get into that.

Imagine, now, that the phenomena in our lives are (at their very simplest) tetrahedral. Our instant impressions are limited, as they don’t capture the full shape of the phenomena. It takes time, knowledge, and insight to perceive their shapes.

We should not, then, place much value on the instant update or newest thing (the quick projection of part of the tetrahedron onto paper), except insofar as it adds to our knowledge and understanding. The latest projection is in itself no treasure; we must look to the old ones as well and—since we can’t spend all our time observing projections of tetrahedra—to other people’s interpretations of these shapes.

This is why we study history, literature, science, history of science, mathematics, philosophy, and music. It’s also why our current drive to collect instantaneous data on everyone (where we are, who our friends are, what our emotional reactions are to every possible product or classroom gesture) will do more harm than good. The purposes of such data-gathering are limited, even crude; the point is not to build wisdom or understanding, but to boost sales, test scores, and other quick results.

For example, developers and marketers have been considering the use of biometric bracelets not only in classrooms but in everyday life. Your bracelet will tell some subset of the world where you are, what you’re doing, and how you’re responding to that activity. Marketers and customers, then, can respond to you accordingly. But what happens, then, to friendship, which depends on voluntary disclosure and voluntary reserve?

Suppose I meet with a friend for dinner; what I do not say is as important as what I do, and both are my choice, to the extent that we choose such things. I learn about my friend through the things she chooses to tell me and the things that make her pause or stay quiet. Biometric bracelet data would ruin this. (“I see you were at the doctor’s office earlier today. Is everything OK? … Oh, is that so? I know you had a brain scan there. Why a brain scan, of all things?”) 

We can gather all sorts of data about people, but such data are little more than flat projections. Take that in stride, and those flat projections, maybe, can tell you something. Treat them like the real stuff, and you send your brains rolling down the hill.

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