The first time machines out-argued mathematicians

In July 1976 a problem that had held for more than a century finally gave way. No more than four colours are needed to colour any map so that no two neighbours share one. Kenneth Appel and Wolfgang Haken proved it at the University of Illinois.

They did not do most of the work by hand. A computer ran for 1,200 hours and made roughly 10 billion decisions. The proof survived by checking a very large number of special cases, which is the part that made people angry.

Appel and Haken knew what they had built. They wrote that the system began to teach them things about how to proceed that they never expected, and conceded it had surpassed its creators at some intellectual tasks. John Koch helped with the algorithmic side.

The reaction from the field was not admiration. Herbert Wilf of the University of Pennsylvania said God would not let the theorem be proved by a method as terrible as that. Paul Halmos of UC Santa Barbara objected to relying on what he called an Oracle and said down with Oracles. Frank Bonsall at Edinburgh argued that if the solution involved computer verification of special cases, it did not belong to mathematical science at all. Daniel Cohen at Hunter College said admitting the shenanigans would leave everyone intellectually unfulfilled.

George Spencer-Brown at Cambridge went further and denied a proof had been found. He called it the most ridiculous case of the Emperor's new clothes in the history of mathematics. One department head banned Appel and Haken from meeting his graduate students, on the theory that no first-rate mathematician would work on a solved problem, so a decent proof might be delayed forever.

The generational split was blunt. After a talk by Haken's son, who had helped with the proof, the room divided. The over-forties could not be convinced that a proof by computer was correct. The under-forties could not be convinced that a proof running to 700 pages of hand calculation was correct.

Endorsements came slowly and from people who had earned them. Bill Tutte, who had once publicly doubted the method, accepted it and marked the occasion with a poem about Haken smiting the Kraken. Heinrich Heesch accepted it begrudgingly after four decades on the problem, having built the algorithmic methods Appel and Haken actually used. Ted Swart had been closing on his own computer-aided proof and was beaten to it.

The New York Times refused to cover the breakthrough, afraid it was a false alarm. By the time editors considered it legitimate, months later, they judged it no longer newsworthy. It reached the paper of record only through an op-ed by Lipman Bers, then president of the American Mathematical Society.

In 1979 the philosopher Thomas Tymoczko made the sharpest version of the objection in the Journal of Philosophy. He did not doubt that the proof proved what it claimed. He argued an unsurveyable proof was not a mathematical proof in the traditional sense. No mathematician had seen a proof of the four colour theorem, he wrote, and it was very unlikely any ever would.

Swart answered him in the American Mathematical Monthly. He regarded computer-assisted proof as an extension of pencil and paper. Human beings get tired, he noted, their attention wanders, and they are prone to slips. Machines do not wander.

The argument was never really about arithmetic. It was about what counts as knowing something when the chain of reasoning is longer than a mind can hold. Every automated system in a studio today inherits that question. You trust a process you cannot read end to end, or you do not ship.

The full account, with the original quotations and the illustrations, is here: https://newsletter.pessimistsarchive.org/p/the-1st-time-machines-mogged-mathematicians

■ Filed from Swarm Yard: the Insects, Air wing of the city.