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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 six times one is one.
I don't remember how it came up in conversation, but for whatever reason numbers became relevant and I clearly and directly stated my false belief. It was late, we were driving back from a long hard chess tournament, and I evidently wasn't thinking clearly. I said the words "because of course six times one is one." Everyone thought for a second and someone said "no it's not." Predictable reactions occurred from there.
The reason I like the anecdote is because I reacted exactly the same way I would today if someone corrected me when I said that six times one is six. I thought the person who corrected me must be joking; he knows math and couldn't possibly be wrong about something that obvious. A second person said that he's definitely not joking. I thought back to the sequences, specifically the thing about evidence to convince me I'm wrong about basic arithmetic. I ran through some math terminology in my head: of course six times one is one; any number times one is one. That's what a multiplicative identity means. In my head, it was absolutely clear that 6x1=1, this is required for what I know of math to fit together, and anything else is completely logically impossible.
It probably took a good fifteen seconds from me being called out on it before I got appropriately embarrassed.
This anecdote is now my favorite example of the important lesson that from the inside, being wrong feels exactly like being right.
One of the smartest people in my high school spent a class arguing that a there were 4^20 possibilities for a sequence of 4 amino acids, when in fact it was 20^4. Not quite as elementary as yours, but our brains all play tricks on us.
Wikipedia says there are 500 known amino acids. I take it you were talking about a domain involving fewer potential amino acids?
The 20 amino acids involved in protein creation.
Presumably the slightly greater than twenty that form the universal basis of biological protein polymers. In other words, the amino acids explicitly coded for by DNA.
I once spent about 20 minutes in class trying to justify the claim that a cubic meter of water weighs one ton, not 100 kilograms.
As you may note, this is not a false belief; it's relevant because I spent a significant fraction of this time wracking my brain trying to jog it free to see if I was being dense (pun intended) like you're describing (I have pretty strong reasons to believe I didn't remember this incident backwards as a case of self-justification by retcon)
You just made me sufficiently confused to quickly google the volume of a cubic meter and the weight of a litre of water. I suspect this is a good habit to reinforce but I can't help feeling silly.
How long do you think you had the wrong belief? Was it just something that happened in that moment or did you carry that believe around for you for longer?
Just that moment. I definitely didn't follow any of its implications. (Other than "if I say this then people will react as if I said an obvious true thing.")
In my case such "short term mistakes" are often caused by fatigue. It's as if my brain enters some kind of energy saving mode and sanity checks are deemed not quite as necessary as some other things. In one case I somehow managed not to notice a contradiction in the idea that a cube has four sides and because of that I failed to solve a problem in a school mathematics competition (it must have been one of the problems, as I must have been really tired by then). It seems to me that sanity checks are analogous to redundancy and duplication of components in engineering. Therefore it is not surprising that when the mental energy is very low my brain may decide that these safety measures are not necessary (of course, they aren't until they are).
In another case, another student asked me how to solve a particular exercise saying that he tried to use a certain lemma he thought might be useful but was unable to apply it. It was only after some time of trying to solve it myself I got the idea to check whether the statement of a lemma was correct (it wasn't). It seems that in this energy saving mode I did not to think about what exactly was the best thing to check given the fact that he tried and failed to solve it, and instead tried to solve it myself without a single thought that lemma's statement might be incorrect. In other words, my brain did not try to estimate conditional expectations of possible action to take given all the facts I had, it "calculated" only expectations for a general case, when lemmas printed in a textbook are usually stated correctly (in other words, I did not take all the information into account when deciding what should I do next). Even if it still wasn't more likely, the idea about wrongness of lemma should have at least occurred to me (and it would have been easier to check on a toy example). Of course, this seems to be a "hybrid" mistake as it seems to be caused by both failure of a heuristic (to trust mathematics textbooks) in this particular case and a fatigue induced tendency to avoid things that require mental energy.
However, it seems to me that such short term fatigue induced mistakes are quite different from long term mistakes and quite different methods are usually required to correct them.
Except when it doesn't.
I suspect you were saying six times one, but your brain was thinking of one to the sixth power, which indeed is one.