I believed that the composition of a rational rotation of a sphere and another rational rotation of a sphere will be rational. (By a rational rotation I mean a rotation of a sphere around some axis which in radians is a rational multiple of pi, and thus will end up putting the sphere back where it started if you apply it enough.) Counterexample: Two 30 degree rotations each around a different axis with the two axies perpendicular to each other. I believed this because I was too used to thinking about the two-dimensional case, where it is trivially true.

Until very recently, I was convinced that it was extremely unlikely that any form of adiabatic quantum computing would have any chance at working at providing speedups, either asymptotically or practically. This belief came to a large extent as what was in retrospect an irrational reaction to the junk and bad hype that has been repeatedly coming from D-Wave. I changed my position when Scott Aaronson made this comment (comment number 25).

More mind-killing territory: Until about 3 days ago, I was convinced that claims that mass shootings were increasing in the US were due purely to media scare tactics and general human tendencies to see things as getting worse. This article made me strongly update against that. Since then, I've seen this response and this one which were both deeply unpersuasive as responses go.

Even more potential mindkilling: Having read more of Slatestarcodex, I've become convinced that he's correct that there really is a substantial fraction of what self-identifies as the "social justice" movement, primarily in an online context, that really is toxic, and that the rest of the left and the serious, sane part of the SJers aren't doing enough to call them out on it. On the flipside, "Gamergate" has convinced me that there's still a very real need for a vocal feminist movement, and that latent misogyny is still pretty common. Edit: To specify what this means in an operational sense, that there are a lot of SJers out there who are making personal attacks or calls for censorship against those with whom they disagree.

I was convinced in 2008 that Obama was going to be good for civil liberties. I don't think I need to discuss in any detail why that was wrong or how I got convinced otherwise, since the reasons should be pretty obvious.