- Barrington's theorem. I've read it, talked to Barrington about it, and even taught it. I still don't believe that (say) the set of strings that have the number of 1's equivalent to 0 mod 101 can be done by a width 5 branching program.
- Banach Tarski Paradox A CS grad students who knows some math says that it shows that mathematics is broken. I would prefer to say it casts doubt on the axiom of choice.
- The classification of finite simple groups. Does any one person even know the proof? Couldn't they have missed some group? Counter argument: the list is on Wikipedia so it has to be correct.
- The rationals and naturals are the same size. I know someone who knows the proof and is happy to say they are the same cardinality but refuses to say they are the same size. (I think they are wrong and this is important- using the term size DOES matter.)
- A well known theorist told me that he used to believe both P ≠ BPP and there were problems in DTIME(2O(n)) that require circuits of size 2&Omega(n);. Oh well.
- Lance Fortnow tells me he has a hard time believing the Recursion Theorem. Perhaps because the proof is completely uninformative. (Ted Slaman, a well known recursion theorists, agrees that the proof is uninformative. Bob Soare thinks the proof is quite intuitive- a failed diag argument.)
- Probability has a few of these: The Central Limit Theorem says that stuff is all normal. That can't be true! I've done the calculations for Birthday Paradox but it still seems suspect to me. And don't get me started on The Monty Hall Paradox.
- Local Lovasz Lemma has gone from being something I didn't believe to something I now understand and believe. The original proof just looked like symbols being pushed around, but Moser's and later Moser-Tardos's constructive versions makes sense to me.
- We all know that Godel's theorem surprised people- but were there people who did not believe it? This theorem does not surprise Generation Xers who are not at all surprised to find out certain problems cannot be solved. Their response: Whatever.
- The existence of Geometries that are as valid as Euclidean but not Euclidean. Again, this surprised people, but were there those who did not believe it? In this age of moral relativism people have no problem with different geometries that are all valid.
Thursday, March 11, 2010
Theorems that you simply don't believe
There are some theorems that are surprising. I've already blogged on that (I can't seem to find the link). However, there are some theorems that some people simply do not believe. I mean people who understand the proofs and still don't believe them. Let me give you a contrast- I DO believe that NSPACE(n) is closed under complementation because, while surprising, the proof really does tell you why its true. For the following surprising results the proof does not help. Or at least does not help the people who were surprised by it.