Czech physicist and translator
You can't define whole disciplines of science to agree with random constraints that you invented. Instead, the purpose of disciplines of science is to decide whether your assumptions about the Cosmos and other things are correct. If there is a disagreement between the best theory and your assumptions, it's your assumptions that are wrong according to science.
Weinberg says that "we want to know what's happening out there". But this is just the standard anti-quantum zealots' misunderstanding of the essence of quantum mechanics. Quantum mechanics says that all knowable things about the physical system must be obtained through a measurement whose outcomes are predicted via the Born rule. There is nothing else happening out there, at least nothing else knowable that is happening out there! So quantum mechanics does answer the question what is happening out there and the claim that quantum mechanics is incomplete in this sense is simply a lie.
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Lorentz symmetry might be broken but there doesn't really exist a glimpse of evidence or motivation why it should be broken. Such a breaking might be studied as a possibility, to some extent, but it would cause many problems, it would most likely return us philosophically before 1905, and the people who say that such a breaking is needed in quantum gravity are clearly wrong because string theory satisfies all the required properties of quantum gravity but it also respects the Lorentz invariance in flat space.