What Defines a Theorem as Opposed to a Law in Mathematics?

[thelema418]

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.

 

[Stephen Tashi]

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.

 

[thelema418]

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.

 

[Stephen Tashi]

So, you are saying that a "law" is NOT an axiom, but a type of theorem.

No, I'm saying that both assumptions (i.e. "axioms", "postulates") and things that are proven using assumptions ( "theorems", "corollaries", "lemmas") are sometimes called "laws".

There is great difference in outlook between ancient and modern mathematics. In ancient times, mathematical facts were thought to have the same status as physical priniciples. The assumptions (axioms and postulates) of Euclidean geometry were thought to be the "true" properties of geometry. The purpose of stating the axioms wasn't to admit "we can't prove this and we don't know it's true, but we're going to assume it". The purpose was merely to provide a common efficient terminology for things every person "knew" were true already. From the modern perspective, assumptions are merely assumptions. There is no assertion that assumptions ("axioms", "postulates") state facts that are known to "true" in any objective or obvious way.

Since things that are thought to be objectively true about the physical world are often called "laws", is isn't surprising that when people regarded mathematics as a study of objective truth, some mathematical assertions were called "laws". As I said, the terminology "law" is still useful when textbooks wish to lump together a set of facts that combines both assumptions and theorems.