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When a new theory is introduced and predicts phenomena that the previous theory couldn't should the previous theory be abandoned? For example Einstein's Special Relativity. We still use Newtonian mechanics while we know that the theory is wrong. It may be correct in the limit of low speeds but this doesn't mean that the theory is correct as a whole. We have evidence that disprove the theory. I understand that Newtonian mechanics are easier for calculations in everyday applications. That is they gave the same result but their based on different theories. In other words Newton's theory provide a different explanation for the same phenomenon than Einstein's theory. Consider a scenario where two theories A and B compete about the nature of matter. Their assumptions are different. For example in theory A atoms are assumed to be "small spheres" whereas in theory B atoms are assumed to be "small cubes". Theory B hasn't been disproved but is harder to do calculations with while theory A has disproved but it is easier to do calculations with. In a certain range of experiments (as in low limit speed in Newtonian mechanics) they both agree. In that range if we use theory A isn't like we lie to ourselves if theory B is the best model we have for reality?

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    Because it is correct to a sufficient precision and its predictions are much easier to compute. – Conifold Apr 23 at 21:22
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    Because there are many ways to "use" a theory, and one of them is exactly "to derive very accurate calculations in appropriate contexts." – Noah Schweber Apr 23 at 22:25
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    Newtonian mechanics weren't proven to be wrong, they were proven to be insufficient for specific cases. The metaphysics that underlies it, though, are mostly abandoned and replace with Einstein's. That is to say, Newton-Einstein is not the best example for what you're asking. If a scientific theory is to be proven wrong, i.e. all/most of its premises or predictions are incorrect or insufficient explanations, thus it will be abandoned. For example Phlogiston theory and early Galvanism theory. – Yechiam Weiss Apr 24 at 8:33
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    @YechiamWeiss Even phlogiston/caloric theory is not really abandoned, see What are the major flaws of the “caloric” theory of heat? The analogy of the spread of heat to a spread of fluid, as precisely expressed in the heat and transport equations, is very much in use, instead of expensive direct modeling of Brownian motion. I think it is the same effect you noted, metaphysics is abandoned but the technical apparatus persists. The same can be said about ether electrodynamics. – Conifold Apr 24 at 11:59
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    Early theory of electric current, due mostly to Biot (who tried to explain Galvani's experiments and Volta's pile by electrostatics), was closer to just "plain wrong". Even its apparatus was almost fully replaced by Ampere and Ohm, see What is the history of electric current and resistance? – Conifold Apr 24 at 12:12
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In the case of Newton's law vs. general relativity, physicists who do calculations with both are aware that Newton's law is but a useful and convenient approximation to the results of GR in the limit of weak fields. And yes, they are aware that Newton's law yields wrong results outside of this regime. But they are also aware that in that regime, doing derivations in GR is a waste of effort because the results are the same for both approaches to a satisfactory level of accuracy.

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If you look at this from the perspective of someone following Popper (say Lakatos or Toulmin), the focus is no longer on theories being proved wrong or overturned. From a more subtle approach, science does not determine the truth of theories, or even their falsehood. It determines their relative effectiveness, which it endeavors to continually improve. A theory that has limitations and is sometimes just wrong might still be used until another one proves more effective. Most theories are incomplete or incorrect, we are only aiming to minimize those aspects.

Every science, even when dominated by a single approach, then, always contains internal contradictions, sometimes referred to as 'anomalies', the resolution of which guides its 'research programme'. Over time, it works out whether it can or cannot resolve these anomalies. If it cannot resolve them, then they will become important issues to attend to in proposing new approaches to replacing the underlying theory which will give a different 'programme' to future research. In the meantime, the current theory develops a 'protective belt' of special conditions and workarounds that allow it to be used despite its weaknesses.

A lot of results within physics are computationally intractable if you include the effects of relativity. So they are addressed with classical mechanics. And their results are then combined with predictions that depend strictly upon relativity (like the energy equation.) But this incompatible amalgamation is still better than the available alternatives: either depending only on the old theory which has already failed in this circumstance, or making predictions too complex to test. So these results are used, creating anomalies, which we then hopefully remove in the future. But even if we need several different solutions that apply under different circumstances in order to evade the contradictions we know are there, it is better than simply not saying anything useful about the difficult cases.

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Consider proving that the relatively more narrative-orientated domain of chemistry, is completely reducible to the equations of physics. Do we consider chemistry to no longer be useful? No. It is a useful overlay for a domain of concern, even though rules from chemistry might not apply completely in a particle accelerator, say. Incorrect models of electron orbitals are even used, during the learning process, where they are 'correct enough'.

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