Soleron said:
ebw said:
Sorry, but this is my area, and no one I know states the Riemann Hypothesis as fact. Mathematicians by training have pretty rigorous standards for what they are willing to call a fact. However, being creatures of abstraction not bound to the real world, we are still happy to write speculative papers built on the premise of RH, which are clearly labeled as being conditional and not factual results.
A slightly better example from math would be the Axiom of Choice. Again, you won't find a mathematician who claims it as fact, but entire areas of mathematics would be unusable without this assumption (the effect is large enough that you wouldn't routinely call out the use of AC unless you were addressing a wider audience). Even still, operating in this way is pretty credibly justified by Gödel's and Cohen's consistency results (we essentially know that AC will never be proven or disproven in the future, not without taking down some more fundamental axioms).
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Oh, yeah, I'm not saying the papers state is as fact. I just mean a lot of work has been built on assuming it is true that would disappear if it's proven false.
Is it possible to quantify (even informally) how likely RH is to be false? Or AC?
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To be fair, if the consequences from papers written conditionally on RH turn out to deduce something false, this would comprise a legitimate proof of the falseness of RH. I'm not sure I would characterize this fate as "disappearing". You're right in that many of these papers would have not much value beyond historical interest, but then many well-founded math papers also become irrelevant over time.
Hmmm putting odds on RH is a pretty deep question. The answer really depends on how you choose to frame the question (for instance, RH has several equivalent formulations).
If you want to go by "wisdom of the masses", then a great majority of experts would bet on "true", although there are some notable exceptions (see http://arxiv.org/abs/math/0311162 ).
Naively one could apply statistical methods to the trillions of zeroes of zeta which have already been computed: while no counterexamples have been found, there is such a thing as a "close call". I haven't done the calculations but I'd guess that these are rare enough that if you tried to do some distribution fitting to estimate the "expected number" of counterexamples, it would be small and finite. The danger of this approach is that number theory is full of phenomena that only appear at log (or log-log-log) scales, so even trillions of data points may not be enough to fit a curve.
Others have approached this question from the viewpoint of framing the Riemann zeta function as one of a wider class of similar functions with many of the same special properties (Euler product, meromorphy, reflection formula). In some cases there are indeed zeta-like functions for which the analogous hypothesis is false, but I feel the Riemann zeta is special enough that I wouldn't be comfortable placing odds by treating it as a random specimen from this class.
For AC we are on much more solid ground. It's a basic working assumption that the standard axioms of set theory (called Zermelo-Fraenkel or ZF) are internally consistent (we also know since Gödel that there is no way to prove consistency of standard mathematical logic from within itself, nor is there any particular reason to trust such a proof). We now know since the 1950s that AC is logically independent of ZF: if ZF is consistent then so is ZF + "AC is true", and also so is ZF + "AC is false". Unless ZF is itself self-contradictory, there will never be a proof or a disproof of the Axiom of Choice.
