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I read on Wikipedia that G. Spencer-Brown gave a non-computer based proof of the four color theorem. As I'm not an expert in the subject I'm unable to verify that claim. Does any one have an idea about the proof or a reference to a serious discussion of the subject?

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    $\begingroup$ The Wikipedia article has links to three papers that discuss the claim. $\endgroup$
    – András Salamon
    Commented Sep 29, 2010 at 20:17
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    $\begingroup$ I am not sure that a proof is available online but see lawsofform.org/gsb/nature.html $\endgroup$
    – Tony Huynh
    Commented Sep 29, 2010 at 20:35
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    $\begingroup$ Interesting typo in lawsofform.org/gsb/nature.html : "the difficulty of the foul colour problem" :-) $\endgroup$
    – Robin Chapman
    Commented Sep 29, 2010 at 20:40
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    $\begingroup$ I do not think MathOverflow should become the go-to place for evaluating other people's mathematical claims (especially if said people have made additional bold claims concerning the Goldbach conjecture and Fermat's last theorem with no evidence of prior mathematical work in that area). I encourage votes to close, and am contemplating deletion. $\endgroup$
    – S. Carnahan ♦
    Commented Sep 30, 2010 at 2:08
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    $\begingroup$ Scott, if deletion means that Cam McLeman's answer disappears, I'm against it. $\endgroup$
    – Gerry Myerson
    Commented Sep 30, 2010 at 5:11

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I spent some time with this a couple of years ago out of curiosity, but did not make it any farther than you're like to be able to find online. The water is definitely murky.

Here is a discussion of the work which makes it clear that there are substantial ideas in Spencer-Brown's work, though Kauffman makes it clear that he is not evaluating the work per se. Kauffman reports that Spencer-Brown definitely gives (with proof) a reformulation of the 4-Color Theorem into something called the Primacy Principle, that "A minimal planar (non-empty) uncolorable trail is prime." (You'll have to see the references for the terminology). Kauffman points out that Spencer-Brown has set up his logical foundations to have basically unintentionally axiomatized this Principle (hence the line on Wikipedia that Spencer-Brown's work "straddles the boundary between mathematics and of philosophy"), which is the source of Spencer-Brown's claimed proof. This seems to be far as Kauffman is willing to go in giving caution as to the validity of the proof.

The Wikipedia article on Laws of Form is decidedly less charitable.

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