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Two scientists independently try to solve a problem to predict a certain phenomenon. The two scientists come up with different answers, but both of their solutions seem logical to each other. How do they know who is "right"? They run an experiment and see whose prediction is more accurate.
Two mathematicians independently try to solve a problem. The two mathematicians come up with different answers, but both of their solutions seem logical to each other. How do they know who is "right"?
The situation you describe can occur when certain problems turn out to be 'independent' of the axioms used. If ZFC is used as the foundational axioms, then it's possible to prove (and indeed, has been proven, famously) that both the Continuum Hypothesis and the negation of the Continuum Hypothesis are consistent with the axioms of ZFC: that is, there's a model of ZFC in which CH is true, and another model in which it's false. Such situations can be 'resolved' (in some sense) by adding further foundational axioms, but that requires agreement on whether those axioms are 'reasonable' (again in some sense).
This has led some set theorists, such as Joel David Hamkins, to suggest that we should be talking of "the set-theoretic multiverse" [PDF].
"Right" is not an adequate idiom for technical evaluations. The issue at hand is rational assessing the truth value of claims.
Falsifiable and verifiable hypotheses are rationally assessed a truth value inasmuch as they correspond to the observable.
Axiom, on the other hand (to paraphrase Ayer): "is true simply because we do not allow it to be otherwise" (from Language, Truth and Logic, ch.4 "The A Priori" p. 41)
In the instance of your mathematician example, whether Q.E.D. or Q.E.F., proof is merely a means of convincing a sympathetic audience.
There is uniqueness in the Ultimate Truth. We can rely on it always. Other truths--'Conventional truths', will certainly vary after some periods (long or short). Even in the case of the Ultimate Truth, sages call it by different names.
The truths we understand by means of our senses must certainly have some limitations for we have only five low-quality sense organs. And also our intellectual abilities are different (You may compare the abilities of the sense organs of humans and other creatures taking them separately (I mean, for sight, sound, smell, taste and touch.). Why do police use dogs for identifying culprits? You may think of...) That is why we get different answers while solving the same problem.
If you are badly in need of one right answer, experts (mathematicians or scientists, often with the help of philosophers) could verify all the parameters that badly affect the progress and decide which one is more useful to the progress of the whole world. If logical, we MUST give ALMOST equal consideration to both the answers. So, if both are correct, without discarding any of them, take the more suitable one (as I mentioned) as the better one (right one).
I can't see any other feasible solution for your question.
In the mathematical case, two mathematicians take a problem, use the same axiom system, and come up with two contradictory results. Both proofs look logical and correct.
Both proofs will be carefully scrutinized, on the grounds that at least one of them is invalid. A serious flaw in a proof is likely to be easy to confirm once pointed out.
It is in fact possible that the axiom system in use is inconsistent. In that case, both proofs may be valid, and mathematicians who suspect that will look for a simpler proof that that is the case.
The possibilities are that one proof is invalid, both proofs are invalid, or the system is inconsistent. There are no other possibilities.
This means that there will be some time before mathematicians in the field know the truth, and it's always possible for them to make a mistake, so it isn't quite as clean in practice as in theory.
CH and ¬CH both being consistent with ZFC doesn't imply that there's something 'wrong' with ZFC (save on the level of our desire for metatheoretic realism), but it would if they were both entailed by it.
Dec 14, 2018 at 15:15