[ExI] Goldbach Conjecture
Anders Sandberg
anders at aleph.se
Sat Nov 28 11:18:02 UTC 2009
Giulio Prisco (2nd email) wrote:
> I think the Goldbach conjecture is probably false, with probability 1
> (that means, certainly false). Here is why:
>
> Apparently there is nothing in the laws of arithmetics that forces an
> even number to be the sum of two prime numbers. The conjecture is true
> for all even numbers on which it has been tested, but these are an
> infinitesimal part of the total (any finite number is infinitesimal
> wrt infinite). Hence, if there is no proof, the probability of he
> Goldbach conjecture being true is zero.
>
Well, being a Bayesian subjectivist about probability means I can't say
you are wrong since your probability is your own subjective estimate. I
assume "probability 1" above also means you actually mean "1 minus
something infinitesimal", since to actually use a probability of exactly
1 as a prior would force you to disregard any disconfirming evidence.
Otherwise you are at least irrational, if not wrong :-)
But the above argument doesn't seem that reliable to me. How do you know
there is nothing in the laws of arithmetic doing the forcing? A proof of
the conjecture would be exactly that, a demonstration that there is some
subtle property of how addition and multiplication works on natural
numbers that leads to this amazing coincidence. That we lack such a
proof is not strong evidence for its impossibility; after all, we lacked
proof of the Fermat conjecture for a long time too. Goldbach could be
wrong, having a counterexample somewhere. It might be right and
provable. Or it could be right and unprovable.
In general, speaking of the probability that a mathematical conjecture
is true is a tricky concept. We have a whole bunch of philosophers of
mathematics around here discussing what it means.
--
Anders Sandberg,
Future of Humanity Institute
Philosophy Faculty of Oxford University
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