Summer of Math Exposition

The film runs two minutes, states the halting problem, and stages the proof as a dispute between two people. No background is assumed: the film is built so that a viewer who has never met the problem can state it at the end, and say why running a program settles nothing.

The film opens and closes on AI, for the question presses on AI now. An AI is a procedure, and no design escapes what holds of procedure itself.

I met the halting problem writing my submission to the John Locke Institute’s Global Essay Prize, not in a computer science course: the essay turned on whether the reason behind an act can be decided from what a person does, and I turned to undecidability.

Everything was made by hand: no animation library, every motion keyframed in DaVinci Resolve, one property at a time.

The machine’s tape is a cable, on screen for almost the whole film. I photographed it coiled, then straightened it, fixed it to a sheet of white paper with double-sided tape, laid it on my bed, and moved it around the room for better light. Each move changed the white from shot to shot; I graded the shots until they matched.

The film starts from a photographed object, its metal, its wear, its place; the narration then says to ignore all of it, and the grade obeys. The physical goes, depth and tone with it, and everything on screen is one ink, the tape, the symbols written on it, the words. The picture makes the move the argument makes.

The sound took the longest. I recorded the effects on the microphone of a pair of earbuds, held in my hand, standing inside a wardrobe, for the clothes deaden the room; I recorded thirty seconds of silence in the same place, in case I needed it. The typing is mine, in two versions, loud and soft, so each moment could carry the weight it needed; a t-shirt over the keyboard gave the dead thud for the stops. When a line is struck through on screen, what you hear is a pencil dragged across paper.

Nothing was usable raw. The voice went through noise reduction, a leveler, a de-esser, equalization, compression, and a limiter; I recorded four takes and used one. I set the overall level wrong three times, too quiet, then distorting, then too quiet again, before it sat where it should.

I am seventeen and had no equipment. The style is a decision, not a consequence: white background, black text, nothing on screen that is not doing work. Nothing appears at once; every line is written in front of you.

In the proof, the program that breaks any predictor is built on its own text. In the film, the program is a person being petty: told what they will do, they do the other thing. That much anyone has already done, and it takes no notation.

Formal statements run under the narration in gray; nothing in the argument depends on reading them.

The machine halts because no rule matches the symbol under the head, not because it reaches a state marked accept or reject. The convention carries a trap: a machine that halts at the edge of the tape looks like one that ran out of tape, not one that ran out of rules. My first version was worse: I traced it by hand and found it looped forever. So this machine blanks a cell it has already visited, steps back onto it, and halts there, on a blank it wrote itself.

Before I called the film finished, I showed it to someone who had never met the problem and asked what the film said. The answer was right; that was the test I built the film to pass.