I am not a mathematician, but I really would like to understand/know the following:
What is so special about Brouwer's "Fan Theorem"? Is there an easy to undestand proof somewhere? Why was Brouwer apparently so obsessed with the "Fan Theorem"?
Brouwer's fan theorem is important because:
The contrapositive of the fan theorem is known as Weak König's Lemma, and you can find a classical proof of that in many places (including Wikipedia).
However, Weak König's Lemma is not constructively valid, so a different justification is required for constructive mathematics. Brouwer's proof of the fan theorem was (IIRC) via introspection about the possible methods of constructive proofs. Brouwer later developed the notion of "bar induction" which can be used to prove the fan theorem.
In the past decade or so, the fan theorem and restrictions of it have occurred very often in constructive reverse mathematics, in a way that I find very analogous to the role of Weak König's Lemma in classical reverse mathematics.
Brouwer did accept some sorts of "metamathematical" proof methods, actually. He wasn't looking at formalized systems that could be analyzed metamathematically in the modern sense, but he was willing to reason about what he thought were all possible constructive proof methods. Unsurprisingly proofs of this sort were often unsatisfying, because constructive proof was not formalized and so was difficult to reason about. This is an integral part of his subjectivist philosophy of mathematics, though. As Douglas Bridges wrote ,
This "speculation" is the "metamathematical" analysis we are talking about.
I don't know that Brouwer was obsessed with the fan theorem more than other things. But, to take his side for a moment, he was trying to develop an entirely new philosophy of mathematics, relying on subjective understanding of the content of proofs rather than on any reference to objective truth. It's natural that a project like that would require him to go back and revisit previous results, or to attempt to provide more solid justifications for principles that he felt should be acceptable but didn't feel he had properly justified yet.
A little bit of googling yielded this http://www.cairn.info/revue-internationale-de-philosophie-2004-4-page-483.htm
I hope it's not too difficult to read for a non-mathematician.