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Nate_Gabriel comments on What false beliefs have you held and why were you wrong? - Less Wrong Discussion
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I once believed that most people have a basic understanding of how Solar system works. My belief in humanity shattered when during a discussion with with several graduates of physics (!) I discovered that most of them do not know what is the orbital period of Moon. An impromptu survey revealed that about 8 people from 10 thought it was one day or one week. One knew, one even asked if I want to know sideric or synodic period.
I once walked around a university campus convincing people that it's impossible to see the Moon during daylight hours. I think it was about 2/3 who believed me, at least until I pointed up.