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I understand the value of falsifiable beliefs. Often they make predictions, and are classified as scientific ideas, useful in churning out predictions.

But do unfalsifiable beliefs have any value? Philosophically or mathematically or otherwise? I know value is a little vague, but consider for example the concept of eternal return. If true it would have a large impact. But, then it's unfalsifiable. Can't be verified at all.

So a sentence A, if true would have a large impact, but it's unfalsifiable at the same moment, would you assume any importance to such sentences?

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    Falsifiable and verifiable are two different things. "There is a unicorn" is verifiable but not falsifiable, "All unicorns are white" is falsifiable but not verifiable. Neither verifiable nor falsifiable beliefs can have pragmatic value, if they hold in most cases encountered in practice, heuristic value in developing new theories, ethical value in guiding behavior, etc. And if something does have an impact then that impact makes it testable in some way, even if not in the narrow verifiable/falsifiable sense. – Conifold Apr 25 '18 at 1:51
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    I can see that an unfalsifiable belief may have some value, perhaps therapeutic or psychological. But I'm not sure there are any unfalsifiable beliefs. It would depend on how we define 'unfalsifiable'. if it means 'not demonstrably false' then there are many of them. If it means 'unfalsifiable by any means' then I'd suggest there are no such beliefs. . – PeterJ Apr 25 '18 at 9:38
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In many epistemologies, unfalsifiable beliefs are going to have great value. In large part, this is because of the following:

universal skepticism fails.

A good way to grasp this is to think about Descartes' project (or perhaps better stated his supposed project). A common claim is that Descartes is this radical skeptic bent on doubting everything, but if you read the Meditations carefully, he's already said this is not going to work by the end of the first meditation. In fact, Descartes' project turns into one where he believes that things which are "clear and distinct" are unfalsifiable.

Now, I mention this not because I think you need to agree with Descartes but to point out a pattern -- for many types of epistemologies, including Humeanism empiricism, Cartesian rationalism, and Kant's, you are going to have some beliefs that of critical importance that are not falsifiable, because these are the beliefs that make other beliefs both verifiable, possible, and falsifiable.

So for instance, Kant has a theory of mind for which he argues, but the main value of it is that this theory enables him to move past skepticism about objects outside the self and to explain how the self comprehends objects. Then we can argue about particular objects and about sensibles and other things. But if we don't get that off the ground, then we can't do much.

A later example might be James' critique of Clifford (from what I understand James' critique is rather unfair but that's not the point). The basic thrust is that you cannot have a principle of universal doubt that stands up to itself -- so you have to have at least something you take to be non-falsifiable to do anything.

Are there ways of trying to avoid this? Yes, you can try to compose a set of things where every particular is falsifiable but the set as a whole (or even if some fail) provides this support.

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I understand the value of falsifiable beliefs. Often they make predictions, and are classified as scientific ideas, useful in churning out predictions.

But do unfalsifiable beliefs have any value? Philosophically or mathematically or otherwise? I know value is a little vague, but consider for example the concept of eternal return. If true it would have a large impact. But, then it's unfalsifiable. Can't be verified at all.

Verifiability and falsfiability are not the same. An idea is falsifiable if some experiment can be performed whose results could contradict that idea. The idea is verifiable if some experiment can be performed whose results could prove the idea, or make it more probable. For more explanation, see

Is falsificationism a reliable scientific methodology?

Now, you ask if unfalsifiable ideas are useful. Suppose that you say unfalisifiable ideas are useless. Is that a falsifiable idea? No. It's an idea about what you should do, not about what people actually do. Also, what about the standard by which you judge the success of falsification? More generally, all methodological or moral ideas are unfalsifiable. Since these ideas are useful, unfalsifiable ideas are useful.

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But do unfalsifiable beliefs have any value

Religion is often take to be a body of unverifiable and unfalsifiable beliefs; William James in the final chapter of his The Varieties of Religious Experience offers a defence of such in the context of the human community. I'm not going to be able to summarise his argument, but I'd urge you to have a look at it.

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The idea of eternal return might be falsifiable:

While the big bang theory in the framework of relativistic cosmology seems to be at odds with eternal return, there are now many different speculative big bang scenarios in quantum cosmology which actually imply eternal return... (Wikipedia, “Eternal return”)

So, observations available today leave the question open. But future data, gathered by improved instruments, might make eternal return an untenable idea; such a process eventually sank the theories of phlogiston and interplanetary ether. Or future data might show that eternal return is indeed verifiable, and, like relativity in 1905, waits only for the experiment that will confirm its truth.

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While I really like Virmaior's answer, I would like to add a few points, mainly regarding the ideas of verifiability and falsifiability as they relate to scientific theories.

  1. Prior to the turn of the last century, it was nearly universally believed that scientific theories could be proven in the same way as theorems in mathematics could. The surprising collapse of Newtonian mechanics, Newtonian gravity, and classical electrodynamics caused philosophers to realize that no amount of verification can lead to a guarantee that the next experiment won't produce a null result.

  2. Karl Popper recognized that theories were indeed not provable, but maintained that they (or at least the good ones) were falsifiable. This idea is still deeply entrenched in popular opinion and is still commonly offered as a solution to the problem of demarcation. However, the claim that scientific theories are falsifiable does not hold up under scrutiny.

Supporting Details

As Thomas Kuhn points out, in his book 'The Structure of Scientific Revolutions,' the scientific community will go to great lengths to prevent an accepted theory from being falsified. Take the famous example of the observation of perturbations in planet Uranus' orbit, which were not predicted by Newtonian mechanics. These discrepancies between observation and theory were known for nearly 70 years. As Kuhn points out, the scientific community does not reject a theory the first time there is a null result. In the case of the orbit of Uranus, the scientific community, rather than rejecting Newtonian mechanics and Newton's law of gravity, instead hypothesized the existence of a yet undiscovered planet: Neptune. They even calculated the exact location where the new planet would have to exist in order to explain the inconsistencies, which lead to the discovery of Neptune—a great achievement for Newtonian physics. However, never during that 70 year period (between the discovery of the discrepancy and the discovery of Neptune) did the scientific community ever consider rejecting Newtonian mechanics.

A few years after that, the existence of the planet Vulcan was hypothesized to explain persistent irregularities in the orbit of Mercury (irregularities that had first been observed a century earlier). However, unlike Neptune, the planet Vulcan was never discovered, and Mercury's orbit was only explained after Einstein published his theory of general relativity in 1916.

A third example, from antiquity, was the inclusion of epicycles by Ptolemy in the Aristotelian system of astronomy to account for irregularities in the observed orbit of the planets. All three of these examples highlight the scientific communities ability to defend a theory in the face of inconsistent experimental data.

The point of these three examples is to show that the scientific community can always tweak the theory or tweak auxiliary hypotheses (i.e posit a yet undiscovered planet) in the face of incongruent data.

Moreover, even when a theory is 'falsified,' there is no guarantee that it won't come back to life a century later. Take, for example, the particle (corpuscular) theory of light, which was supposedly falsified in 1819 when the French physicist Dominique-François-Jean Arago observed a bright spot at the center of the shadow of a circular disk (a bizarre prediction of Fresnel's wave-theory put forth to discredit the theory). Resistance to the wave theory collapsed and by all accounts, the particle theory of light was completely and utterly destroyed. However, fast-forward to 1905, and the particle theory is resurrected by Albert Einstein to explaining the photoelectric effect and ultraviolet catastrophe.

The point is that the scientific community can always explain away a null result by challenging one or more of the auxiliary hypotheses, and/or by adjusting the theory to account for the results. And even if the scientific community agrees that a theory is falsified, it still might be resurrected at some point in the future new and unforeseen reasons.

Conclusion

  1. The only conclusion you can draw is that just as you can not prove scientific theories, you also cannot disprove or falsify them either. A 'proven' theory might turn out to be wrong just as a 'falsified' theory might turn out to be correct in some way we could never imagine at the time.
  • I think you're misusing the term "falsified", it does not mean to be proved wrong, rather the existence of some hypothetical scenario under which it(the theory) might be proved wrong. – novice Apr 28 '18 at 23:09
  • @novice, your comment makes no sense to me. It sounds like you are saying 'falsified' does not mean proven wrong, but rather has the potential to be proven wrong? Is this what you are saying? – njspeer Apr 29 '18 at 0:14
  • Yes. Correct me if I am wrong. – novice Apr 29 '18 at 1:35
  • 'Falsified' means to prove false. 'Falsifiable' means that it could be proven false. I'm using these terms the way that Popper used them. According to Popper, good scientific theories are falsifiable and are often falsified (e.g. Newtonian mechanics and classical electrodynamics). He claimed that falsifiability was the key to the problem of demarcation: good theories, make bold (and risky) claims about the world, claims that are testable—special relativity was the perfect example for him. Pseudoscientific theories, on the other hand, explain everything but are never falsified by any evidence. – njspeer Apr 29 '18 at 2:20
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    I may edit my original answer to make it more readable to someone not familiar with the subject matter. Basically, the point is that scientific claims (or beliefs) are neither provable nor falsifiable. These terms are really only applicable to math, logic, and geometry. In science, it's a lot messier: evidence strengthens or weakens a theory but (in general) never proves or disproves it. This had to be learned the hard way: theories that were once thought to be proven turned out to be false, and theories that were considered disproven turned out to be true at a later point. – njspeer Apr 29 '18 at 3:34

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