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I have two questions which relate to two different subjects, Science and Mathematics have different meanings of "theory", first is based on ever-growing scientific evidence while other is based on logical deductions following from axioms that are assumed to be true, Now paradoxes often lead us to develop our theory, As was the case for "Set theory" when Russel's paradox was proposed

Taking the above in Context, Here are My questions

1)Do paradoxes always indicate that there is something wrong in the theory, And New theory must be proposed? Does the existence of paradoxes itself act as evidence/proof(Science/math) for incorrectness of the theory proposed

2)Should a complete theory in specifically science ideally have no paradox?

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  • 1) yes. 2) not exactly: QM has "paradoxical" aspects. Also relativity had. Feb 4 at 10:52
  • @MauroALLEGRANZA, I wanted to ask If a Scientific Theory that explains every empirical observation,Can be hold true Even If seems to have a Logical(philosophical) paradox? Feb 4 at 21:16

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A paradox is a situation which does not seems plausible, “a statement that runs contrary to one's expectation”. Further investigation may show that it is indeed an antinomy, i.e. it produces a logical contradiction.

Hence the classification as a paradox is subjective, while the classification as an antinomy is objective. Russell discovered even an antinomy in the concept of arbitrary comprehension in set theory.

My answer to your questions:

  1. A paradox is always the prompt for further investigation to decide whether it turns out as an antinomy. In case of an antinomy the theory has to be changed, possibly limited or in the worst case dismissed. In the case of a confirmed paradox we have to correct our own wrong expectation.

    The non-invariance of Newton’s mechanics with respect to Lorentz transformations indicates that the theory has to be limited to classical mechanics, presupposing low energy, low velocities, …

  2. A complete scientific theory does not exist. Said pointedly: Scientic theories are successful hypotheses.

Remark: Revised due the comment of @DheerajGujrathi

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  • Regarding your answer to 1.While it seems true to me regarding Science,I feel little intuition for this statement regarding mathematics,Can/Should we really throw out a consistent(except 1 paradox) Mathematical theory as false,Just because it has 1 paradox? Feb 4 at 21:12
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    @DheerajGujrathi I revised and clarified my answer. Anyhow: A mathematical theory which produces an antinomy has to be corrected or dismissed. Some errors in mathematical theories can be corrected, others cannot. - Thanks.
    – Jo Wehler
    Feb 4 at 21:45
  • I can't speak for biologists or chemists, but paradoxes in physics are there to point out a hard-to-spot problem with the physicist's understanding, choice of language, or simplifying assumptions. Philosophers and members of the general public often take them wrongly to be either mysterious dialetheia about reality, or antinomies in the theory.
    – g s
    Feb 5 at 8:23
  • @JoWehler,Thank you for the extension, Got my answer, Especially liked the semantics argument, Paradoxes might mean to indicate limitations of theory in mathematics, And can also can lead us to change our expectations in proposed theoretical framework for science.. Feb 5 at 12:02
  • @gs,Do you mean to say that a seemingly valid paradox for one person might not be a paradox for another one because of other one's better understanding of the subject? If that is the case, It is Only because of one's intuitive expectations, But The point I was interested in. was regarding paradoxes .that are generally agreed in the science community to be valid argument against proposed theory, Which might lead science community to believe the limitations of theory or might change minds of the community to somehow gulp that reality is not what we expected or predicted, Am I correct here? Feb 5 at 12:10
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A theory that leads to predictions that contradict with each other, or with reality, is likely to be incorrect or incorrectly applied. A theory in science ideally should agree with reality, so if reality is not paradoxical the theory should not be.

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  • what do you intend to say by "If reality is not paradoxical"? Feb 5 at 11:55
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The Barber Paradox in its various forms, for example, can easily be resolved using formal logic.

Version 1

The original form with an unspecified domain of discussion:

~EXIST(barber): ALL(a): [Shaves(barber, a) <=> ~Shaves(a, a)]

Where Shaves(x, y) means "x shaves y."

Version 2

We can also specify a particular domain:

ALL(barber): [Man(barber) => ~ALL(a): [Man(a) => [Shaves(barber, a) <=> ~Shaves(a, a)]]]

Where Man(x) means x is a man.

Version 3

More interestingly, by narrowing the domain even further, we can prove:

ALL(village): ALL(barber):ALL(men): [barber @ village & ALL(a): [a @ men => a @ village]

=> [EXIST(shave): ALL(a): [a @ men => [(barber,a) @ shaves <=> ~(a,a) @ shaves]] <=> ~barber @ men]]

Where men is a subset of village, and shaves is binary relation instead of a predicate as above.

In words: In an village with a resident barber, that barber can shave those and only those men in the village who do not shave themselves if and only if that barber is not a man.

See formal proofs here.

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