Scenario 1: I have defined a sequence of numbers Xn, but these numbers are not computable. Nevertheless you give a proof that the limiting value of these numbers is 2, and then another, entirely different proof that the limiting value is 3. Therefore, 2 = 3. But since Xn is not computable, your proofs are necessarily non-constructive, so you haven't given me a physical recipe for turning 2 quarters into 3 quarters. I would sooner say that you had proved something false, and re-examine some of your nonconstructive premises.
Scenario 2: You prove that 2 = 3 constructively. This means you have given me a recipe for turning 2 quarters into 3 quarters. I wouldn't say you had proved something false but that you had discovered a new phenomenon, weird but true.
In both cases I would suspect my own mathematical ability, or even my sanity, before suspecting maths. Lcpwing those concerns away, I would observe that a certain set of statements had been proven not mutually consistent which in turn means they do not underpin our physics (granted this would be more surprising in one case than the other).
Update: Discussion has moved on to a new thread.
The hiatus is over with today's publication of chapter 73, and the previous thread is approaching the 500-comment threshold, so let's start a new Harry Potter and the Methods of Rationality discussion thread. This is the place to discuss Eliezer Yudkowsky's Harry Potter fanfic and anything related to it.
The first 5 discussion threads are on the main page under the harry_potter tag. Threads 6 and on (including this one) are in the discussion section using its separate tag system. Also: one, two, three, four, five, six, seven. The fanfiction.net author page is the central location for information about updates and links to HPMOR-related goodies, and AdeleneDawner has kept an archive of Author's Notes.
As a reminder, it's often useful to start your comment by indicating which chapter you are commenting on.
Spoiler Warning: this thread is full of spoilers. With few exceptions, spoilers for MOR and canon are fair game to post, without warning or rot13. More specifically: