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JackV comments on Yes, Virginia, You Can Be 99.99% (Or More!) Certain That 53 Is Prime - Less Wrong
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Comments (68)
Yeah, that's interesting.
I agree with Eliezer's post, but I think that's a good nitpick. Even if I can't be that certain about 10,000 statements <i>consecutively</i> because I get tired, I think it's plausible that there's 10,000 statements simple arithmetic statements which if I understand, check of my own knowledge, and remember seeing in a list on wikipedia, (which is what I did for 53), that, I've only ever been wrong once on. I find it hard to judge the exact amount, but I definitely remember thinking "I thought that was prime but I didn't really check and I was wrong" but I don't remember thinking "I checked that statement and then it turned out I was still wrong" for something that simple.
Of course, it's hard to be much more certain. I don't know what the chance is that (eg) mathematicians change the definition of prime -- that's pretty unlikely, but similar things have happened before that I thought I was certain of. But rarely.
If mathematicians changed the definition of "prime," I wouldn't consider previous beliefs about prime numbers to be wrong, it's just a change in convention. Mathematicians have disagreed about whether 1 was prime in the past, but that wasn't settled through proving a theorem about 1's primality, the way normal questions of mathematical truth are. Rather, it was realized that the convention that 1 is not prime was more useful, so that's what was adopted. But that didn't render the mathematicians who considered 1 prime wrong (at least, not wrong about whether 1 was prime, maybe wrong about the relative usefulness of the two conventions.)
I emphatically agree with that, and I apologise for choosing a less-than-perfect example.
But when I'm thinking of "ways in which an obviously true statement can be wrong", I think one of the prominent ways is "having a different definition than the person you're talking to, but both assuming your definition is universal". That doesn't matter if you're always careful to delineate between "this statement is true according to my internal definition" and "this statement is true according to commonly accepted definitions", but if you're 99.99% sure your definition is certain, it's easy NOT to specify (eg. in the first sentence of the post)