I don’t think it’s a fair deduction to conclude that Goldbach’s conjecture is “probably true” based on a estimate of the measure (or probability) of the set of possible counter examples being small. The conjecture is either true or false, but more to the point I think you are using the words probability and probable in two different ways (the measure theoretic sense, and in the sense of uncertainty about the truth value of a statement), which obfuscates (at least to me) what exactly the conclusion of your argument is.
There is of course a case to be made about whether it matters if Goldbach’s conjecture should be considered as true if the first counter example is larger than an number that could possibly and reasonable manifest in physical reality. Maybe this was what you are getting at, and I don’t really have a strong or well thought out opinion either way on this.
Lastly, I wonder whether there are examples of heuristic calculations which make the wrong prediction about the truth value of the conjecture to which they pertain. I’m spitballing here, but it would be interesting to see what the heuristics for Fermat’s theorem say about Euler’s sum of primes conjecture (of which Fermat’s last theorem is K = 2 case), since we know that the conjecture is false for K = 4. More specifically, how can we tell a good heuristic from a bad one? I’m not sure, and I also don’t mean to imply that heuristics are useless, more that maybe they are useful because they (i) give one an idea of whether to try to prove something or look for a counter example, and (ii) give a rough idea of why something should be true or false, and what direction a proof should go in (e.g. for Goldbach’s conjecture, it seems like one needs to have precise statements about how the primes behave like random numbers).
Note that the probabilistic argument fails for n=3 for Fermat’s last theorem; call this (3,2) (power=3, number of summands is 2).
So we know (3,2) is impossible; Euler’s conjecture is the equivalent of saying that (n+1,n) is also impossible for all n. However, the probabilistic argument fails for (n+1,n) the same way as it fails for (3,2). So we’d expect Euler’s conjecture to fail, on probabilistic grounds.
In fact, the surprising thing on probabilistic grounds is that Fermat’s last theorem is true for n=3.
There is of course a case to be made about whether it matters if Goldbach’s conjecture should be considered as true if the first counter example is larger than an number that could possibly and reasonable manifest in physical reality. Maybe this was what you are getting at, and I don’t really have a strong or well thought out opinion either way on this.
That would be sufficient, but there are more easily met conditions like:
1)
if you have a bunch of sequences of observations (100 or 1000 year’s worth)
and none of them include the counterexample as an observation with high probability
2)
The frequency is low enough that it isn’t worth accounting for.
For example, if you flip a coin it comes up heads or tails—is it worth bringing up another possibility?
I don’t think it’s a fair deduction to conclude that Goldbach’s conjecture is “probably true” based on a estimate of the measure (or probability) of the set of possible counter examples being small. The conjecture is either true or false, but more to the point I think you are using the words probability and probable in two different ways (the measure theoretic sense, and in the sense of uncertainty about the truth value of a statement), which obfuscates (at least to me) what exactly the conclusion of your argument is.
There is of course a case to be made about whether it matters if Goldbach’s conjecture should be considered as true if the first counter example is larger than an number that could possibly and reasonable manifest in physical reality. Maybe this was what you are getting at, and I don’t really have a strong or well thought out opinion either way on this.
Lastly, I wonder whether there are examples of heuristic calculations which make the wrong prediction about the truth value of the conjecture to which they pertain. I’m spitballing here, but it would be interesting to see what the heuristics for Fermat’s theorem say about Euler’s sum of primes conjecture (of which Fermat’s last theorem is K = 2 case), since we know that the conjecture is false for K = 4. More specifically, how can we tell a good heuristic from a bad one? I’m not sure, and I also don’t mean to imply that heuristics are useless, more that maybe they are useful because they (i) give one an idea of whether to try to prove something or look for a counter example, and (ii) give a rough idea of why something should be true or false, and what direction a proof should go in (e.g. for Goldbach’s conjecture, it seems like one needs to have precise statements about how the primes behave like random numbers).
Note that the probabilistic argument fails for n=3 for Fermat’s last theorem; call this (3,2) (power=3, number of summands is 2).
So we know (3,2) is impossible; Euler’s conjecture is the equivalent of saying that (n+1,n) is also impossible for all n. However, the probabilistic argument fails for (n+1,n) the same way as it fails for (3,2). So we’d expect Euler’s conjecture to fail, on probabilistic grounds.
In fact, the surprising thing on probabilistic grounds is that Fermat’s last theorem is true for n=3.
That would be sufficient, but there are more easily met conditions like:
1)
if you have a bunch of sequences of observations (100 or 1000 year’s worth)
and none of them include the counterexample as an observation with high probability
2)
The frequency is low enough that it isn’t worth accounting for.
For example, if you flip a coin it comes up heads or tails—is it worth bringing up another possibility?