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The trigger for this question is from "University Physics with Modern Physics" (by Young & Freedman) when they mentioned that Ohm's law is not actually a law; this sentiment was echoed in "Physics for Scientists & Engineers – A Strategic Approach with Modern Physics" (by Knight). The rationale given was that Ohm's law is not a general description of all matter; same was argued for Hooke's law as well.

By their rationale, shouldn't Newton's laws of motion not be considered a law as well since it will break down at the high-energy regime (I think)? From what I understand, all laws have their limitations and will break down at some point; so why do these authors single out Ohm's law and Hooke's law? What exactly constitute a "fundamental law of nature"?

EDIT: While this question has a somewhat similar heading to this post, the essence of the questions differ. My question heads in the direction of "when can an empirical relationship be considered a fundamental law"; on the other hand, the other post has already made the assumption that the empirical relations is a law and is interested in "which law is more fundamental".

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It is indeed a fundamental law of nature that every "fundamental law of nature" is only valid in its region of validity. The only fundamental law that could evade this fundamental law would be the fundamental law that explains everything, but despite many attempts and arguable some glimpses of what it could be, such a law has not yet been found.

As to your question, it is really very much a matter of opinion/choice. Usually one would consider a law to be fundamental if one does not wish to extend the theory to other areas. For example newton's law are perfectly acceptable to describe classical mechanics (low speed, low gravity and large distances). If you step outside of these boundaries you are considering special relativity, general relativity and quantum mechanics respectively. In what sense are they fundamental? For example momentum conservation follows in classical field theory from translational symmetry.

So it is really all in the eye of the beholder.

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Fundamental law of nature is an idealized concept of course. It is naive to think we will ever truly know them despite the progress science has made in narrowing them down.

The difference in the examples you mention though is that Newton's laws were an attempt at describing the fundamental. Readers of the Principia knew that they would one day be superseded but not by what. Hooke and Ohm, on the other hand, from the moment they wrote down their "laws" already knew of several phenomena not captured by them. They were attempts at modelling materials or devices that were sufficiently linear.

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