FOM: Appel on 4CT proof

Carl G. Jockusch jockusch at
Thu Dec 10 11:41:36 EST 1998

I am pleased to forward (with his permission) the following message from
Ken Appel, who has read some of the recent fom discussion on the proof
of the four-color theorem.

Carl Jockusch

>From kia at  Thu Dec 10 09:16:22 1998
From: Kenneth Appel <kia at>
Subject: fom

One of the things that I find lacking from most discussions of the
proof of the Four Color Theorem is what I might call the
"inevitability" of the argument.  I think that many proofs in
mathematics find easier acceptance because of the intuitive certainty
on the point of most mathematicians that they are true.  Thus, I would 
like to describe the proof from the point of view of what might
legitimately dismissed as "semi-religious" reasoning but what really,
to my mind, motivates the belief that there is a proof in Erdos'
"God's Book".

The proofs, ours and the more recent ones, depend on the following two 

Thesis 1.  There are many acceptable classes of "reducible
configurations" on which such proofs can be based (for historical
reasons only Kempe's C and D reducible configurations that essentially 
date back directly to Birkhoff's work have been used), and these
configurations appear to be relatively dense among those that satisfy
Heesch's criteria and that we call "geographically good".

Thesis 2.  Looking at the intuitive electrical model, due in its most
sophisticated form to Haken, in a large dual triangulation there must
be many localities of positive charge and in many of them there will
be reducible configurations, many of which will be very unpleasant to
actually show reducible.

These theses are really what gives one confidence that if there are
errors in the presented arguments these errors are just errors of
presentation and not errors that lead to the invalidity of the
underlying understanding of the problem.

It is totally maddening that none of us seem to understand
reducibility well enough to prove good general theorems about useful
enough classes of reducible configurations and thus computers must be
used to show each individual configuration reducible.  It is totally
frustrating that it is becoming intuitively clear that almost any
reasonable use of the discharging procedures will work and that 
the collection of reasonable unavoidable sets is huge.

With this as background, it is almost as frustrating to depend on
specific verifications of unavoidable sets of reducible configurations 
as it would be to insist on finding ten spots in the Dutch dikes to
use pressure gauges to show that the Netherlands would be under water
if there were no structure of dikes.

I know of no other area in mathematics that the proof of a theorem has 
had to be made by such artificial means and the true intuition of why
the theorem is true has been so poorly communicated.

As a member of ASL for 42 years I am totally embarrassed to make such
a contribution to the discussion.  I hope that I am not drummed out as 
a result.

Ken Appel

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