[FOM] Gap in the original Hales-Ferguson proof of Kepler
Timothy Y. Chow
tchow at alum.mit.edu
Tue Dec 29 21:38:38 EST 2015
A piece of old news that I only recently became aware of was that about
halfway through the Flyspeck project (i.e., about five years ago), they
discovered that there was a non-trivial gap in the original version of the
proof. Details here:
I'm guessing that many FOM readers were not aware of this either, because
nobody mentioned it back in October 2014, when Hendrik Boom and Adriano
Palma asked a related question.
To quote Palma: "Are there cases in which the check in terms of syntax
done by the machines revealed a real flaw in a proof, in the informal or
intuitive style?" Admittedly, the gap above did not cause anyone to doubt
the truth of the Kepler conjecture, and it was fixed pretty quickly, but
it was a real gap, and would probably not have been discovered if Hales et
al. had not pursued the Flyspeck project.
This incident also prompted me to consider something that had not occurred
to me before. To explain, let me first back up a step. Critics of formal
proofs sometimes complain that formal proofs promote the wrong attitude
towards mathematics; proof should be about *understanding*, they argue,
and an unintelligible proof that has been mechanically verified has very
little value. This criticism has always struck me as misguided, because
there is nothing to stop us from generating a humanly understandable proof
in addition to the formal proof. Indeed, by eliminating any worries we
might have about correctness, it frees us to spend more time on
understanding, as opposed to tedious verification.
However, the aforementioned gap in the proof of Kepler got me thinking.
In some sense, the proof wasn't *wrong*. It was just assumed that a
certain step was straightforward---"without loss of generality we may
assume..." The step was in fact correct, but there was a subtlety that
had been overlooked. We can even bill this as a success story, because it
demonstrates that the process of formalization can not only detect gaps
but it can draw our attention to subtleties and thereby increase our
But suppose that the proof assistant had been a lot more "intelligent"
than today's proof assistants are, and had *figured out on its own how to
plug the gap*. Then we humans might never have become aware that there
was a subtle point that had been overlooked. The critics would then have
been at least partially vindicated: The machinery of formal proofs would
have stolen away an opportunity for deeper understanding.
Admittedly, in this case, even if a proof assistant had plugged the gap,
the "lost insight" would not have been a big loss to mathematics. After
all, the argument amounted to showing that a certain weird situation was
not a real possibility, so who really cares? As proof assistants become
more intelligent, however, perhaps their ability to fill gaps in reasoning
could "hide" bigger and bigger insights.
In one sense, this issue is not new. There is already a danger that we
could become over-reliant on Groebner basis algorithms and miss geometric
insights, or over-reliant on WZ-style identity provers and miss
combinatorial insights. Atiyah is reported to have said: "Algebra is the
offer made by the devil to the mathematician. The devil says: `I will
give you this powerful machine; it will answer any question you like.
All you need to do is give me your soul: give up geometry and you will
have this marvellous machine.'" Still, proof assistants potentially take
this Faustian bargain up to a new level. I, for one, would still welcome
our new theorem-proving overlords, but with more sympathy for the critics
that I previously dismissed.
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