How to know if something is a primitive concept, a law, a definition or a theorem

Question

Some basic Physics books are often misguiding in the sense that they don't make clear whether something is a primitive concept, a law, a definition or a theorem. This is often a little confusing. I've asked here some moments ago if there's a definition of force, and I've been redirected to a question about whether Newton's Laws are laws or definitions of mass and force.

This is just an example, but there are many others, like for instance, the conservation of energy, which some books present as a theorem, and some places says that this is a law of Physics. Time on the other hand, is something that according to Feynman's lectures on Physics, cannot be defined, so we just say how to measure it and leave it as a primitive concept.

In this context: how in general can we identify whether something is a primitive concept, a fundamental law of Physics, a definition or a theorem?

Answer 1

how in general can we identify whether something is a primitive concept, a fundamental law of Physics, a definition or a theorem?

You can't, because it depends completely on context and on the taste of whoever has written the presentation you're using.

For example, some people like to consider Gauss's law to be a definition of charge, some people call it a law of physics, and some people call it a theorem. None of these choices are right or wrong in and of themselves.

Here's an example from mathematics. You can take Euclid's postulates and prove the Pythagorean theorem. But you can also take Euclid's first four postulates, get rid of the parallel postulate, and add in the Pythagorean "theorem" as a postulate. Then the parallel postulate becomes a theorem.

So applying this to physics, nobody can say whether the Pythagorean theorem is a law (in the sense that it's an empirical fact about how space behaves) or a theorem.

The same thing happens with primitive notions. For example, a line can be a primitive notion (as in Euclid) or a defined one (in the Cartesian approach where you define it as a set of points).