For instance, force was discovered to be proportional to mass and acceleration.
How are these proportional relationships discovered and proven to be true?
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Newton's second law is a bad example because any measuring of force presupposes its proportionality to acceleration, see How did Newton measure forces in his experiments to establish the laws of motion? With that convention about force, proportionality of it to (inertial) mass was inductively inferred from experiments (with pendula). So was proportionality of force to displacement in Hooke's law (for springs), see Did Hooke's law come from experiments, or was it mathemtically derived from Newtonian mechanics? – Conifold Oct 08 '21 at 12:06
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I think that the problem is that people think that $F=ma$ is a law (and needs a proof) while it is a definition of $F$ and $m$, so does not require a proof.
markvs
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Wrong and wrong. Anything that is a mathematical (or physics) "law" is not provable - it's a premise or axiom upon which you build your system. It's easy to define mass and acceleration, showing that there's a useful thing that is the product of those, and relating that to energy consumption, is what matters. – Carl Witthoft Oct 08 '21 at 14:16
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1@CarlWitthoft "It's easy to define mass..." I wouldn't go that far, try to provide a definition of mass that isn't dependent on force? That's like saying it's easy to define electric charge. I would say markvs is correct about Newton's law defining the concepts of mass and force, although because it's axiomatic, it's equivalent to also e.g. define momentum and force from them and to derive mass. It's also true that physical laws require proof (or rather, evidence), or must be derived from data. They must be based on experiment and observation. – Sam Gallagher Oct 08 '21 at 16:27
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@CarlWitthoft I believe there are no non-cyclical definitions of mass. If you have one, you are welcome to present it here. – markvs Oct 09 '21 at 00:14
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@SamGallagher: The question was "How are these proportional relationships discovered and proven to be true?" My answer: these cannot be proved because these are definitions and not laws (provable statements). – markvs Oct 09 '21 at 00:18
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@markvs I agree with you, Carl was way off with his comment, but that's what you get with SE sometimes – Sam Gallagher Oct 10 '21 at 01:09