Sunday, 11 November 2012

Why a flawed proof is worrying

In preparation for a class I have been looking at Alfred Bray Kempe's 1879 "proof" of the Four Colour Theorem.  Until Percy Heawood pointed out the flaw in Kempe's proof eleven years later it was accepted.  We had a clear, relatively straightforward proof of the result: anyone could check it for themselves (very different from the position now, when we all accept Appel and Haken's computer proof!)  The error in Kempe's proof is subtle: in preparing the false proof to present to my class, I found it very persuasive!  (The story is told in Robin Wilson's excellent book Four Colours Suffice, published as Four Colors Suffice in parts of the world where modern spelling hasn't yet arrived.)

False "proofs" are worrying.  As a teenager I was turned off geometry by a well-known fake proof that all triangles are isosceles.  The proof relies on an incorrect diagram.  After seeing that I found it hard to accept any geometric proof: I became suspicious of geometric arguments (which I think was good) and turned my back on geometry (which was not so good for an aspiring mathematician!)  I wasn't worried so much by trick algebraic proofs that 1=2, which relied on division by 0 or on an ill-based induction argument: I appreciated these jokes, but geometric "proofs" like E.A. Maxwell's "Fallacy of the empty circle" destroyed my enjoyment of geometry.

So Kempe's proof worries me because it is so plausible.  If a simple argument can be accepted by the leading mathematicians of the day, where do we stand with today's proofs which are accessible only to specialists and which the rest of us cannot take on trust?

Many proofs contain errors but these errors are usually fixable.  Kempe's wasn't (although a proof of the weaker Five Colour Theorem was salvaged) and that seems a warning against complacency.

Wilson's book records that Haken's son presented a seminar on the Appel-Haken proof at Berkeley which led to fierce discussion.  Those under 40 were reluctant to accept the computer's role in the proof: those under 40 were more suspicious of a proof relying on 700 pages of human calculation.  As I have grown older I have gone in the reverse direction. Whereas I once had concerns about computer proofs, I am much less sure now than I was thirty years ago that there is a gold standard for proofs.  If we cannot be sure of a proof we have checked for ourselves (and that is what the story of Kempe's proof suggests) then where is mathematical certainty?

1 comment:

  1. I think Kempe's proof is fixable. During the time of Kempe lack of structural and coloring properties of the planar graphs would not permit to fix his "proof". I would suggest my two articles in this direction:

    1.The Four-Color Map Theorem: "Kempe’s Fallacious Proof Repaired"

    and the first six pages of

    2.A Victorian Age Proof of the Four Color Theorem