[ExI] Goldbach Conjecture

[Anders Sandberg]

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