Quite a few key mathematical problems which resisted solution for a long time have eventually fallen. As far as I know, Andrew Wiles’ proof of Fermat’s last theorem is generally accepted as correct.
And the famous four-color theorem was apparently proved by a massive computer attack (not that I like that brute-force approach, but it seems to have gained acceptance).
However a lot of very bright mathematicians have been chipping away at the Riemann hypothesis for a long time now without much progress?
Possible? Sure. I think that the only way we can rule out any given proposition being undecidable is by either proving it, disproving it, or showing that the problem is finite. And of course, if it is truly undecidable, we’ll never know that.
Turing’s halting problem in a nutshell. The computer can run for many times the age of the universe, and perhaps it will eventually find a solution. But as you say, we have no way of predicting that.
Point of order: it is possible to prove that an undecidable proposition is undecidable. The most famous example was the demonstration by Gödel and Cohen that the Continuum Hypothesis (CH) is undecidable. Gödel demonstrated that the CH cannot be disproved from the Zermelo-Frankel set theory (even with the axiom of choice - ZFC). Cohen demonstrated, on his part, that CH cannot be proven from ZFC.
Sure, some statements can be proven undecidable (I was speaking too generally, up there), but the Riemann hypothesis isn’t one of them. If it’s false, then there exists a counterexample, and if there exists a counterexample, then it can be disproven simply by stating that counterexample. So the only way for it to be undecidable is for it to be true, and so proving it undecidable would prove it true.
I have only very superficial knowledge of this, but this seems relevant from the Wiki article:
Undecidability of a statement in a particular deductive system does not, in and of itself, address the question of whether the truth value of the statement is well-defined, or whether it can be determined by other means. Undecidability only implies that the particular deductive system being considered does not prove the truth or falsity of the statement. Whether there exist so-called “absolutely undecidable” statements, whose truth value can never be known or is ill-specified, is a controversial point among various philosophical schools.
What do you mean here? To be clear, we can write an explicit computer program that will run and stop when it detects a counterexample to the Riemann hypothesis (if it exists). So i.e. as @Chronos explains if the negation of the Riemann hypothesis is unprovable, then the hypothesis must be true.
I think what I’m finding logically confusing here is the difference between whether something is algorithmically provable and whether it’s actually true. So I’m not sure if Chronos’ reasoning here is correct:
I’m now thinking that it can be undecidable without being true. Decidability is not about truth, it is about algorithmic proof. If we can show that no algorithm exists that is guaranteed to find counterexamples if they exist, then it can be undecidable without being true.
Here’s the very oversimplified layman’s version of the background to this thread:
It has been proved (by Kurt Gödel, in 1931) that there must be mathematical statements which are true but which cannot be proved to be true.
The Riemann hypothesis is arguably the most famous currently unsolved problem in mathematics. It’s a proposition that, so far, no one has been able to figure out (i.e. prove) whether it is true or false.
So, the OP’s question is about whether it might be one of those statements that inherently cannot be proven true or false.
Another piece of background is that I’m an evolutionary biologist, not a mathematician. I chose Riemann’s name to use here on a whim because his work is so important, and I thought that would stimulate me to try to understand it better to avoid embarrassment.
I will no doubt nevertheless proceed to embarrass myself.
I am not a mathematician, just an interested spectator. But it seems to me that what Godel provided was an ‘existence proof’: that true but unprovable statements must exist within any sufficiently complete mathematical system.
He did not actually construct such a statement.
In fact, it is difficult to see how one could do so, since once stated, it ought to be testable.
The kicker here though is the difference between potential and actual infinities: Cantor etc.
And of course the Riemann hypothesis states that “all” the nontrivial zeros lie on the real 0.5 line.
To infinity and beyond (sorry).
Sure, a single counterexample would disprove it. I think a lot of computer time has already been used on this? But we could run the most powerful conceivable computer until the heat death of the universe without finding one, and that still wouldn’t constitute a logical proof.
That was the key point of the Hilbert program in the early 20th century: he wanted to show that, in principle, an algorithmic calculation could always determine the truth or falsity of any mathematical proposition.
