Understanding the Role of Theorems in Physics

[Chris11]

Hello. I was listening to a podcast the other day when one of those involved in the discussion said the following: " there is a theorem in physics that says..." My question is this: what exactly is meant when one says there is a theorem in physics? Is it that, provide that certain assumptions are met, and event A occurs, event B will occurr? Is it a component of a theory that is supported by an incredibly large body of evidence? Is it like a mathematical theorem, being derivded from previous theorems that were ultimiatly deduced from a set of axioms?

 

[Feldoh]

Physics is similar to mathematics. Axioms are rationalized out by empirical measurements and from there we typically formulate laws from which everything else is derived.

Edit: I should add that theorems are generally tested for accuracy as well however some of the time empirical measurements lead to theorems that people would not have been able to come up with otherwise.

 

[Chris11]

Sure, physics is indeed similar to mathematics in some respects. However, I would beg do differ by arguing that the perceived similarity between mathematics and physics is due to the lingua de franca of physics is mathematics, and that similar reasoning processes are present in both disiplines. Calling something a theorem just has far too much mathematical conotation for me to approve of the words use in the context of a scientific disipline. For instance, the reasoning at the core of physics is inductive in nature--almost by definition; in contrast, mathematical reasoning is deductive. In science, you have to accept a certain degree of uncertanity with every idea; however, in mathematics, you have certainity after the acceptance of axioms. The word theorem connontes the level of certanity found only in mathematics, in addition to the deduction that can only truly be said to be found in mathematics alone; therefore, I don't think that the usage of the word 'theorem' in physics is ever justifiable.

 

[Academic]

So then what do you call some relation or idea in physics that is a necessary consequence of postulates/axioms? If you don't call it a theorem, what do you call it? And, what is to be gained by using a different definition than the community at large? Seems like a recipe for communication breakdown to me.

 

[Feldoh]

Okay that's great don't call any physics "theorems" theorems then. Problem solved.

As Academic said you're going to have a lot of communication problems if you start renaming things. Just think about it.

What if I decided that since the electron is the thing that actual is moving in a current that I should switch the sign of an electron to positive and a protons charge to negative. As you can see there would be some problems if I were to give a speech to the community at large.

 

[dulrich]

I don't think the word "theorem" is commonly used in physics, but "law" is. One has Newton's laws, the laws of thermodynamics, the laws of electromagnetism.

I do think the word "theorem" implies a mathematical procedure. Given Newton's laws (as axioms) one can prove the conservation of momentum (via a theorem), for example. But even here I think one would more normally say we have "derived" one from the other.

Physics (like all the sciences) has both inductive and deductive elements. The word theorem could be used for the second group. But even in mathematics, the certainty of the result is dependent on the certainty of the axioms. In physics, the axioms are emperical and don't carry the same certainty as mathematical axioms.

It sounds to me like the OP was simply listening to someone less familiar with the subject (and its vocabulary) than as they could have been...

 

[Klockan3]

Sure, physics is indeed similar to mathematics in some respects. However, I would beg do differ by arguing that the perceived similarity between mathematics and physics is due to the lingua de franca of physics is mathematics, and that similar reasoning processes are present in both disiplines. Calling something a theorem just has far too much mathematical conotation for me to approve of the words use in the context of a scientific disipline. For instance, the reasoning at the core of physics is inductive in nature--almost by definition; in contrast, mathematical reasoning is deductive. In science, you have to accept a certain degree of uncertanity with every idea; however, in mathematics, you have certainity after the acceptance of axioms. The word theorem connontes the level of certanity found only in mathematics, in addition to the deduction that can only truly be said to be found in mathematics alone; therefore, I don't think that the usage of the word 'theorem' in physics is ever justifiable.

There is no difference, with the specified set of axioms these theorems holds for sure, just like in maths. Just because nature might not agree with these axioms do not make the theorems any less true.

Theorems in physics are not found by experiments, they are worked out just like things in maths are worked out, they might however give new things to do experiments on to test the current axioms since they are a direct product of them.

Edit: Note however that most physicists do not do anything like this, only people being in right between maths and physics do.