Is Scott's response available to see?

I'm sorry Mike, but statements like "According to this, heuristic arguments are the enemy of existence for a proof – most (?) true statements have one or the other, not both" are just froth.

I don't know of any sense in which "most" true statements about the natural numbers are unprovable from the Peano axioms, but even if such a thing were true, consider that (1) the axioms of ZF set theory are much, much stronger than PA, so many statements unprovable in PA are still theorems of ZF; and (2) theorems of mathematical interest probably don't look much like "random theorems", in much the same way that special functions of interest usually don't behave much like "random functions" unless they're constructed to be such an example.

"Because whatever happens will probably happen at the small numbers, and this happened to be one tree and this happened to be three... After a little while there just isn't room for anything else interesting to happen." As a counterexample to this statement, the Pólya conjecture seemed to be true for all integers until a counterexample was found around $1.845 × 10^{361}$. You can never assume something is true just because it holds for the first $N$ integers.

@DanielP, I didn't (mean to) say that it's true "just because it holds for the first N integers." I said that in this case, the heuristics imply that: it becomes less and less likely to find a counterexample, even if you sum to infinity. Was the case for the Polya conjecture?

@DanielP In any event, that doesn't affect my argument. If it's not true, it's not true. If it is true, it could still be because it was very likely to be so.

No heuristics imply anything. No conjectures imply anything. Seriously, this is a waste of time. You can believe what you want, this is not even close to a proof. I'm voting to close.

@MikeArrh You could absolutely say that the counterexample was extremely unlikely to find. Up to the found point, over half of all numbers checked had an odd number of prime factors. So for this pattern to break at $1.845×10^{361}$, you could say the chance is $1/2^{1.845×10^{361}}$.

@freakish en.wikipedia.org/wiki/…

@MikeArrh seriously, dude, I don't give a damn about "heuristic justification". Either you have a proof or not. End of story for me.

@JairTaylor Here's on my Google Drive: docs.google.com/document/d/…

@DanielHast These seem like reasonable points; could be Scott Aaronson was saying something like that too. Maybe I should reframe (3) above as a question, maybe a stand-alone question: Is there any way to quantify (say, by order of the length of the characters in the theorem) what fraction of theorems are provable? It sounds like you think that "most" theorems - those that actually sound like theorems! - have proofs. Is anything known about this? I asked Aaronson that as a followup question and he didn't answer.

@DanielP It turns out there's a counterexample to Polya less than a trillion. In any case that isn't the kind of heuristic I was using; that one's really more inductive reasoning en.wikipedia.org/wiki/Inductive_reasoning

There is no randomness involved here, and no probabilities other than zero and one. I disagree with your very first premise in #1 "it is well known..."

@aschepler This has been proved by Terence Tao and others. One example: quantamagazine.org/…

Tao's relationship to random walks shows that some properties are true of "most" natural numbers in a carefully defined sense. This won't help prove or disprove the actual conjecture, which claims a property of all natural numbers. "The bigger the starting point gets, the less likely it is to escape the only known cycle" is not an accurate summary of these types of results and is not correct as stated since every starting point either does or does not lead to $1$ and does not have any other "likelihood".

I posted the wikipedia page where they discuss the heuristic likelihood of Goldbach's Conjecture. These kind of results are common; it's the same idea. You find the likelihood for numbers of a certain size, then sum them to infinity.

Yes, there are many results for natural numbers where we can find bounds on the proportion of numbers satisfying the property in sets of large size. I'd avoid the word "likelihood" entirely. But note it's also true in this same sense that almost all natural numbers satisfy the property $n \neq 10$, and it's obviously false that all natural numbers satisfy that property.

I feel bad that this question was closed, but it is, sadly, not really MSE material. Is there a... philosophy of math/science SE that this would belong in? I feel like there's some value in discussing the question "Is this provable?" But it's a philosophical question.

@aschepler So would you apply that to Goldbach's Conjecture as well? Even though it is trivial to construct counterexamples, it makes sense to me to say that the Conjecture is obviously going to be true, even if no proof is ever found.

No problem, Eric. But there's a story that Euler said about Goldbach's Conjecture (I may have those details wrong), that he too could think of numerous theorems that were true but that it was hopeless to prove. I was suggesting here that this is the kind of thing that Euler had in mind.

@EricSnyder I did just ask (part "3" of my question here) mathoverflow.net/questions/423731/… and got a partial answer.

@MichoelR Yes, Goldbach's Conjecture and related "heuristics" are much the same. I don't consider it "obviously" true.

@MikeArrh It's an interesting and fairly compelling intuitive argument. But of course the prime numbers aren't actually random, so it's still not that convincing to me.