Can Noether's Theorem Be Derived Without Induction?

[Mektrik]

Hi all,
I'm writing something on the philosophy of science and I was wondering if those of you more knowledgeable than me could lend a helping hand. What I want to know is whether Noether's theoerm can be derived without induction. Given the fact that it is a theorem as opposed to a theory, it seems intuitive that one can derive it without observation. However, I'd love to hear what you have to say.

Just so that we're on the same wavelength, by induction I mean any reference to the observable world. Newton's theory of gravitation is inductive whereas Pythagoras is clearly deductive.

Thanks

 

[jbriggs444]

Whether the laws of nature actually do possesses any particular symmetry is a matter of observation and experiment.

Whether the existence of a particular symmetry leads to a conserved quantity is proven by Noether's theorem. That fact is independent of observation and experiment.

If you want to deduce conservation of momentum from Noether's theorem you first have to get some experimental verification that the laws of nature are invariant under translation.

To reiterate what I said the last time you asked this same question, "No, you cannot deduce conservation of momentum without recourse to experiment".

Similarly, the truth of the Pythagorean theorem does not assure you that the sum of the squares of the lengths of two sides of a [large] right triangle drawn on the [spherical] surface of the Earth will equal the square of the length of the remaining side. Not all of the assumptions underlying the Pythagorean theorem hold true in such a case.

 

[lightarrow]

Given the fact that it is a theorem as opposed to a theory, it seems intuitive that one can derive it without observation.

What is a theory of physics? Said simplistically, it's an algorithm which allows to derive conclusions (theorems) from postulates, definitions and mathematical rules, provided all of these things are mutually consistent.

In mathematics, you can wake up tomorrow morning and invent a new set of postulates, definitions and rules, which are mutually consistent, and...voila', you have a new theory.

But that's not enough in physics, all of what you create also have not to contradict experimental results. For this reason, in physics you don't start from arbitrary postulates, you start from experimental laws, or the risk to create a useless theory is extremely high...

So, to answer your "intuition": yes, you derive theorems, from postulates, without experiments, BUT it's only because somene else has proved experimentally the validity of the postulates.