The mathematics department at the University of Indiana had a new trumpet
to blow. They replaced their ‘largest discovered prime’ postage stamp with the
news that ‘four colours suffice.’ This was local pride but where was the general
applause from the world’s mathematics community? After all, this was a
venerable problem which can be understood by any person of Tiny Tim’s tender
years, but for well over a century had teased and tortured some of the greatest
mathematicians.
The applause was patchy. Some grudgingly accepted that the job had been
done but many remained sceptical. The trouble was that it was a computer-based
proof and this stepped right outside the traditional form of a mathematical proof.
It was only to be expected that a proof would be difficult to follow, and the
length could be long, but a computer proof was a step too far. It raised the issue
of ‘checkability’. How could anyone check the thousands of lines of computer
code on which the proof depended. Errors in computer coding can surely be
made. An error might prove fatal.
That was not all. What was really missing was the ‘aha’ factor. How could
anyone read through the proof and appreciate the subtlety of the problem, or
experience the crucial part of the argument, the aha moment. One of the fiercest
critics was the eminent mathematician Paul Halmos. He thought that a computer
proof had as much credibility as a proof by a reputable fortune teller. But many
do recognize the achievement, and it would be a brave or foolish person who
would spend their precious research time trying to find a counterexample of a
map which required five colours. They might well have done pre Appel and
Haken, but not afterwards.
After the proof
Since 1976 the number of configurations to be checked has been reduced by a
factor of a half and computers have become faster and more powerful. This said,
the mathematical world still awaits a shorter proof along traditional lines.
Meanwhile the four-colour theorem has spawned significant problems in graph
theory and had the subsidiary effect of challenging the mathematician’s very
notion of what constitutes a mathematical proof.