-”The more time passes, the more evidence we get that (4) is false from computational tests and also the more we should doubt any proof (3) as its complexity grows. Therefore, the logical probability of (1) and (2) is growing over time.”
The fact that the methods we use to check proofs are not perfectly reliable (and I think you greatly overstate the importance of this consideration, computer proof checkers are very reliable) does not at all imply that the probability that a proof exists is decreasing over time. You need to distinguish between the fact of a proof existing (which is what Godel’s theorem is about) and the fact of us being able to check a proof.
-”In short, coincidence theory has chances of being false 1 to 10^−3700 and any future formal proof has chances to be false around 1 to 100. Thus, I still should bet on the coincidence theory, even if the formal proof is convincing and generally accepted.”
Neither of these numbers are right. The ratio 1 to 10^−3700 is not the chance that coincidence theory is false, but rather the chance that GC is false given that either GC is false or coincidence theory is true. And I don’t know where you got the ratio 1 to 100 from, but it is not remotely plausible; out of the many thousands of formal proofs which have gained widespread acceptance since the advent of formality in mathematics, I am not sure if there has been even one which was wrong. (I suppose it may come down to exactly how you define your terms.)
Since this seems to be the core of your argument, I guess I have disproved your thesis?
-”(I assume here that there is only one correct proof, but actually could be many different valid proofs, but not for such complex and well-researched topics as GC.)”
Why would you assume that? There are many theorems with multiple non-equivalent valid proofs, even for “such complex and well-researched topics as GC”. This is leaving aside that “non-equivalent” for proofs is an ill-defined concept, and any proof can always be transformed into a new proof via trivial changes.
-”Fermat theorem is surprising, but GC is unsurprising from probabilistic reasons alone.”
Fermat’s theorem is also unsurprising from probabilistic reasons alone.
-”One may argue that GC is so strong for the numbers below 100 that we should assume that there is a rule.”
Actually, the probabilistic arguments already imply that GC is likely (although with less dramatic odds) even for numbers less than 100, especially since 3, 5, and 7 are all prime.
-”E.g. twin primes are an example of non-randomness.”
Actually, the standard random model of the primes predicts that there are infinitely many twin primes.
-”The more time passes, the more evidence we get that (4) is false from computational tests and also the more we should doubt any proof (3) as its complexity grows. Therefore, the logical probability of (1) and (2) is growing over time.”
The fact that the methods we use to check proofs are not perfectly reliable (and I think you greatly overstate the importance of this consideration, computer proof checkers are very reliable) does not at all imply that the probability that a proof exists is decreasing over time. You need to distinguish between the fact of a proof existing (which is what Godel’s theorem is about) and the fact of us being able to check a proof.
-”In short, coincidence theory has chances of being false 1 to 10^−3700 and any future formal proof has chances to be false around 1 to 100. Thus, I still should bet on the coincidence theory, even if the formal proof is convincing and generally accepted.”
Neither of these numbers are right. The ratio 1 to 10^−3700 is not the chance that coincidence theory is false, but rather the chance that GC is false given that either GC is false or coincidence theory is true. And I don’t know where you got the ratio 1 to 100 from, but it is not remotely plausible; out of the many thousands of formal proofs which have gained widespread acceptance since the advent of formality in mathematics, I am not sure if there has been even one which was wrong. (I suppose it may come down to exactly how you define your terms.)
Since this seems to be the core of your argument, I guess I have disproved your thesis?
-”(I assume here that there is only one correct proof, but actually could be many different valid proofs, but not for such complex and well-researched topics as GC.)”
Why would you assume that? There are many theorems with multiple non-equivalent valid proofs, even for “such complex and well-researched topics as GC”. This is leaving aside that “non-equivalent” for proofs is an ill-defined concept, and any proof can always be transformed into a new proof via trivial changes.
-”Fermat theorem is surprising, but GC is unsurprising from probabilistic reasons alone.”
Fermat’s theorem is also unsurprising from probabilistic reasons alone.
-”One may argue that GC is so strong for the numbers below 100 that we should assume that there is a rule.”
Actually, the probabilistic arguments already imply that GC is likely (although with less dramatic odds) even for numbers less than 100, especially since 3, 5, and 7 are all prime.
-”E.g. twin primes are an example of non-randomness.”
Actually, the standard random model of the primes predicts that there are infinitely many twin primes.