# Nomenclature: Theorem or Law?

by thelema418
Tags: law, philosophy maths, terminology, theorem
 P: 100 From the perspective of mathematical philosophy, what is the difference between a "theorem" and a "law"? In particular, I'm wondering if there is a difference between what makes the Pythagorean Theorem a "theorem" and the Law of Cosines a "law". Thanks.
 Sci Advisor P: 3,319 Your question is about literary style and traditions. You'd have to study the history of mathematical textbooks to determine why some things are called "laws". In modern secondary school textbooks, where the purpose is drilling the facts into students without studying the flow of logic, it is often convenient to refer to fundamental facts as "laws" regardless of whether they are assumptions or theorems. For example, in a sophisticated axiomatic development of the integers, the existence of "a zero" is assumed and the theorem that there is one and only one zero can be proven. However when teaching elementary algebra to kids who wouldn't appreciate such a proof, it is convienient to teach all the simple properties of real numbers as "laws", regardless of whether they are assumptions or theorems.
 P: 100 So, you are saying that a "law" is NOT an axiom, but a type of theorem. As a type of theorem, it is essentially the same as a theorem, except for a historical tradition within pedagogy for various reasons --- such as the expedition / efficiency / conveniences of teaching young learners who are yet unprepared to appreciate "proof" -- that has determined it to be an essential fact. (?) I do read some very old mathematical texts. I seem to recall works even at the time of Newton using "The Law of Cosines" or the Latin equivalent of the phrase. And I still don't see why the "Pythagorean Theorem" wouldn't be called the "Pythagorean Law" because a) it is a fundamental fact and b) students don't usually prove it. The Law of Cosines is essentially the same thing as the Pythagorean Theorem, so the difference is name makes it unusual.