The Four Colour Theorem: The first time machines mogged mathematicians
The Four Color Theorem: When Machines Outclassed Mathematicians
In July 1976, a century-old mathematical enigma was finally cracked, but the solution sparked a firestorm of controversy. The event didn't just solve a problem; it fundamentally challenged the definition of a "mathematical proof."
The Core Mystery: The Four Color Theorem
The theorem itself is deceptively simple:
The Four Color Theorem states that no more than four distinct colors are required to color the regions of any map such that no two adjacent regions share the same color.
In mathematical terms, if we consider a map as a graph , the chromatic number for any planar graph is:
The Unconventional Method
The breakthrough was achieved by Kenneth Appel and Wolfgang Haken, who relied on a collaborator that many of their peers found abhorrent: a computer. With the algorithmic assistance of programmer John Koch, the duo utilized a level of processing power previously unseen in pure mathematics.
| Metric | Value |
|---|---|
| Compute Time | 1,200 hours |
| Decisions Processed | 10 billion |
Appel and Haken admitted that the machine became a teacher, revealing paths of logic they hadn't anticipated. They conceded that in specific intellectual capacities, the system had effectively surpassed its creators.
The Great Divide: Reception and Resistance
The mathematical community—a group that prides itself on rigorous rationality—reacted with a mixture of shock, denial, and outright hostility. To many, this was not a discovery, but a heresy.
The "Anti-Oracle" Camp
Several prominent scholars viewed the computer's role as an "Oracle" rather than a tool of reasoning.
- Herbert Wilf (University of Pennsylvania): "God wouldn’t let the theorem be proved by a method as terrible as that!"
- Paul Halmos (UC Santa Barbara): He denounced the reliance on "Oracles," stating, "They are not mathematics."
- Frank Bonsall (University of Edinburgh): Argued that solutions involving computer verification of special cases simply "does not belong to mathematical science at all."
- Daniel Cohen (Hunter College NYC): Claimed that accepting these "computer shenanigans" would leave the field "intellectually unfulfilled."
Accusations of Fraud
Some went further, claiming the proof was a total fabrication. George Spencer-Brown of Cambridge University compared the publication to the fable of The King’s New Clothes, asserting that there was nothing in the work that even resembled a proof.
Institutional Gatekeeping
The resistance was so strong that one department head forbade Appel and Haken from speaking to his graduate students. His reasoning was a task list of academic concerns:
- Prevent students from losing the incentive to find an "elegant" proof.
- Ensure "first-rate" mathematicians didn't abandon the problem.
- Avoid delaying a "decent" (human-verifiable) proof indefinitely.
The Generational Gap
Interestingly, the divide was often demographic. Following a presentation by Haken’s son, the audience split along a clear age line:
| Age Group | Primary Skepticism |
|---|---|
| Over 40s | Could not trust a proof generated by a computer. |
| Under 40s | Could not trust a proof consisting of 700 pages of manual calculations. |
Path to Credibility
Despite the noise, the proof eventually gained traction through high-profile endorsements.
- Bill Tutte: Initially a skeptic, Tutte eventually conceded. He celebrated the victory with a poem:
Wolfgang Haken Smote the Kraken One! Quoth he: “The monster is no more.”
- Heinrich Heesch: A German mathematician who had spent 40 years on the problem. Though he begrudgingly accepted the result, he was the one who developed the key algorithms Appel and Haken used.
- Ted Swart: A mathematician who had been pursuing a similar computer-aided approach and was beaten to the finish line.
The Media's Hesitation
The New York Times famously missed the boat. Editors initially refused to cover the story, fearing it was a "false alarm." By the time they deemed it legitimate, they claimed it was no longer "newsworthy." The breakthrough only made it into the paper via an op-ed by Lipman Bers, President of the American Mathematical Society.
Philosophical Aftershocks
The debate evolved into a deeper philosophical inquiry regarding the nature of truth and verification.
The Skeptic's View: In 1979, philosopher Thomas Tymoczko argued in The Journal of Philosophy that a proof that cannot be surveyed by a human mind is not a traditional proof.
"No mathematician has seen a proof of the 4CT, nor has any seen a proof that it has a proof."
The Pragmatist's View: Ted Swart countered this in the American Mathematical Monthly, arguing that:
- Computer-assisted proofs are merely an
extension of pencil and paper. - Humans are fallible; they get tired and lose focus.
- Computers do not get tired.
This rebuttal eventually earned Swart an award for expository writing.
A Final Note of Despair
As noted in Robin Wilson's Four Colors Suffice, Bill Tutte once wondered about the future of the profession in an age of machine-led discovery. He imagined a future mathematician crying out in despair: "What shall I do?"
