Eliezer, I want to complement you on this post. But I would suggest that you apply it more generally, not only to mathematics. For example, it seems to me that any of us should be (or rather, could be after thinking about it for a while) more sure that 53 is a prime number than that a creationist with whom we disagree is wrong. This seems to imply that our certainty of the theory of evolution shouldn’t be more than 99.99%, according to your figure, definitely less than a string of nines as long as the Bible (as you have rhetorically suggested in the past.)
A string of nines as long as the Bible is really, really long.
But if we aren’t willing to assign probabilities over some arbitrary limit (other than 1 itself), we’ve got some very serious problems in our epistemology. I would assign a probability to the Modern Synthesis somewhere around 0.99999999999999 myself.
If proposition An is the proposition “the nth person gets struck by lightning tomorrow”, then consider the following conjunction, n going of course from 1 to 7 billion:
P(A1 & A2 & … & A7e9)
Now consider the negation of this conjunction:
P(~(A1 & ~A2 & … A7e9))
I had damn well better be able to assign a probability greater than 0.9999 to the negation, or else I couldn’t assign a probability lower than 0.0001 to the original conjunction. And then I’m estimating a 1/10000 chance of everyone on Earth getting struck by lightning on any given day, which means it should have happened several times in the last century. Also, I can’t assign a probability of any one person being struck as less than 1/10000, because obviously that person must get struck if everyone is to be struck.
Eliezer, I want to complement you on this post. But I would suggest that you apply it more generally, not only to mathematics. For example, it seems to me that any of us should be (or rather, could be after thinking about it for a while) more sure that 53 is a prime number than that a creationist with whom we disagree is wrong. This seems to imply that our certainty of the theory of evolution shouldn’t be more than 99.99%, according to your figure, definitely less than a string of nines as long as the Bible (as you have rhetorically suggested in the past.)
A string of nines as long as the Bible is really, really long.
But if we aren’t willing to assign probabilities over some arbitrary limit (other than 1 itself), we’ve got some very serious problems in our epistemology. I would assign a probability to the Modern Synthesis somewhere around 0.99999999999999 myself.
If proposition An is the proposition “the nth person gets struck by lightning tomorrow”, then consider the following conjunction, n going of course from 1 to 7 billion: P(A1 & A2 & … & A7e9) Now consider the negation of this conjunction: P(~(A1 & ~A2 & … A7e9))
I had damn well better be able to assign a probability greater than 0.9999 to the negation, or else I couldn’t assign a probability lower than 0.0001 to the original conjunction. And then I’m estimating a 1/10000 chance of everyone on Earth getting struck by lightning on any given day, which means it should have happened several times in the last century. Also, I can’t assign a probability of any one person being struck as less than 1/10000, because obviously that person must get struck if everyone is to be struck.