MichaelVassar comments on What Cost for Irrationality? - Less Wrong

59 Post author: Kaj_Sotala 01 July 2010 06:25PM

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Comment author: wedrifid 03 July 2010 10:19:15AM *  11 points [-]

Overwhelming prior makes my claim more likely to be correct than the majority of claims made by myself or others. ;)

  • Mersenne primes are powers of two - 1.
  • There are 3 powers of two with 1 billion digits, and a 0.32 (that is, log10(2) -3) chance of a 4th.
  • It is proven that the powers of two for a mersenne prime must themselves be primes.
  • In order to have 1 billion decimal digits the power of two must have 1billion + 1digits.
  • There aren't all that many numbers with 1 billion + 1 digits that are prime.
  • Of the powers of two of those numbers - 1 a ridiculously smaller proportion will also be prime.
  • Given that there are only three possibilities... I'm confident to the point of not being able to conveniently express my confidence numerically that there are no mersenne primes with a billion digits.
  • Even if I made a couple of mistakes in the above reasoning the remainder would still give me cause to be confident in my assertion. (I have a suspicion about my expression of exactly which two ridiculously big numbers must be prime for one of up to the four candidates.)
Comment author: MichaelVassar 06 July 2010 04:44:35PM 2 points [-]

I think that the above only gives the odds that there are no such primes unless there is some good deep reason (presumably a set of symmetries, which doesn't seem at all likely since billion is an arbitrary seeming round decimal) for there to be some such prime or primes. Without that caveat, such statements would bite-in-the-ass far too many people historically who would have made overly confident mathematical claims. To clarify; I think you should be ridiculously confident, but not as confident as your reasoning by itself would justify.

Comment author: wedrifid 07 July 2010 04:53:17AM 1 point [-]

To clarify; I think you should be ridiculously confident, but not as confident as your reasoning by itself would justify.

I agree (and voted accordingly). The influence of the direct probability I calculated would be utterly overwhelmed in my confidence calculation compared to meta-uncertainty. I certainly wouldn't go as far as placing 1:10,000 odds, for example, even though my calculations would put it at 1^(-lots). In fact, I can't even assign extreme odds to something as obvious as there is no Jehova, except for signalling purposes. I know enough about the way me (and my species) think that assigning extreme probabilities would be ridiculously overconfident. (How this relates to things like Pascal's wager is a different and somewhat more philosophically difficult problem.)

Comment author: faul_sname 05 August 2012 04:09:34AM *  1 point [-]

1^(-lots)

This would be 1.

Comment author: wedrifid 05 August 2012 05:26:25AM 0 points [-]

Something does seem to be missing in that expression.

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