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Mathematical proof is foolproof, it seems, only in the absence of fools

Brian Hayes

I was a teenage angle trisector. In my first full-time job, fresh out of high school, I trisected angles all day long for $1.75 an hour. My employer was a maker of volt-meters, ammeters and other electrical instruments. This was back in the analog age, when a meter had a slender pointer swinging in an arc across a scale. My job was drawing the scale. A technician would calibrate the meter, recording the pointer's angular deflection at a few key intervals—say 3, 6, 9, 12 and 15 volts. When I drew the scale, using ruler and compass and a fine pen, I would fill in the intermediate divisions by interpolation. That's where the trisection of angles came in. I was also called upon to perform quintisections and various other impossible feats.

I joked about this with my coworker and supervisor, Dmytro, who had been drawing meter scales for some years. We should get extra pay, I said, for solving one of the famous unsolvable problems of antiquity. But Dmytro was a skeptic, and he challenged me to prove that trisection is impossible. This was beyond my ability. I did my best to present an outline of a proof (after rereading a Martin Gardner column on the topic), but my grasp of the mathematics was tenuous, my argument was incoherent, and my audience remained unconvinced.

On the other hand, Dmytro himself quickly produced visible evidence that the specific method of trisection we employed—drawing a chord across the angle and dividing it into three equal segments—gave incorrect results when applied to large angles. After that, we made sure all the angles we trisected were small ones. And we agreed that the whole matter was something we needn't discuss with the boss. Our circumspect silence was a little like the Pythagorean conspiracy to conceal the irrationality of Ö—2.

Looking back on this episode, I am left with vague misgivings about the place of proof in mathematical discourse and in everyday life. Admittedly, my failure to persuade Dmytro was entirely a fault of the prover, not of the proof. Still, if proof is a magic wand that works only in the hands of wizards, what is its utility to the rest of us?

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