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*To*: fom@math.psu.edu*Subject*: FOM: Appel on 4CT proof*From*: "Carl G. Jockusch" <jockusch@math.uiuc.edu>*Date*: Thu, 10 Dec 1998 10:41:36 -0600 (CST)*Cc*: appel@math.uiuc.edu*Sender*: owner-fom@math.psu.edu

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@oregano.unh.edu Thu Dec 10 09:16:22 1998 From: Kenneth Appel <kia@oregano.unh.edu> 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 "theses" 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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