What's the difference between principle, law, rule, theorem and equation?

[Alexandre]

Well, I do understand what mathematical theorem means, and I also know what differential equation is but I don't really get why sometimes certain things are called "equations" instead of "law" (Maxwell's equations, nobody calls it Maxwell's laws) and conversely some equations are called laws (Newton's second law, Hook's law, well people do refer to them as equations because they are, but officially they are called laws). Uncertainty principle, Pauli exclusion principle, Fermat principle, why not rule, why Hund's rule is a rule? Noether's theorem, why theorem, because she was a mathematician? LOL. Liouville's theorem (in Hamiltonian mechanics), why theorem again? Law of large numbers, now that's pure mathematics why call it a law when it has a proof? And etc.

 

[Drakkith]

An equation is a mathematical term for a formula in the form of A = B, where A and B are mathematical expressions that may contain one or more variables.

A scientific law is a statement based on repeated experimental observations that describes some aspects of the universe. A scientific law always applies under the same conditions, and implies that there is a causal relationship involving its elements. These laws may take the mathematical form of an equation, but not always.

A scientific principle and a rule are, as far as I can tell, the same thing as a law.

A mathematical theorem is a statement that has been proven on the basis of previously established statements, such as other theorems—and generally accepted statements, such as axioms. It is not the same as a scientific theory, which is not capable of being proven.

The reason for the use of several different terms is because of historical reasons. In the early days of science, there were no set rules like we have now days and different terms have been used over the years.

(All the above definitions are from wiki)

 

[phinds]

"law" vs "theory" is a particularly confusing one.

THE prominent example, is "Newton's Law of Gravity", which as it turns out is wrong (although it works great for all practical purposes on Earth) and has been superseded by Einstein's General Theory of Relativity, which while only a "theory" is more correct than Newton's "law". As Drakkith said, some of this is just historical context, and as he also pointed out, a "law" can have a limited context.

 

[Alexandre]

Now it's kinda clear, thanks!

 

[Calion]

As I usually think of it:

A principle/law/axiom is something that is assumed to be true based on observation or experience. There is no underlying reason or derivation. These are the building blocks of a logical system. As an example, Newton's law F=ma is simply taken to be true because that is what we observe.

A theorem is what is generated by combining axioms and other theorems.

Sometimes, you can switch around what is an axiom and what is a theorem, but the convention is that axioms are the most fundamental ideas. Usually, the idea is for a theory to depend on as few axioms as possible.

An equation describes a relationship between two quantities.

There are also definitions and theories. A definition is simply assigning a name to some set of quantities or operations. A theory is a collection of related theorems, axioms, and definitions.