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So in logic we have every line of a proof being either an axiom or a theorem -- but then why do we have concepts like the "law of sines" and the "law of cosines"? Are these technically "theorems" as well? How is "law" being used here? Is there a mathematical reason? A historical one?

zahbaz
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user525966
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    Actually in every other language that I know they are called theorems. You are right, strange that in English they are called different. – Mark Sep 21 '18 at 21:28
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    They are statements that can be proven by inference from other statements. Hence, they are theorems. I doubt there's any particular significance in calling them "laws" and this might even just be a remnant of their historical names. – Jam Sep 21 '18 at 21:32
  • If we posted an answer stating that they are theorems, would that answer be sufficient, or are you looking for a historical explanation of why they came to be known as "the law of sines" and "the law of cosines"? – Tanner Swett Sep 21 '18 at 21:51
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    Some theorems are called "formulas", some theorems are called "lemmas", some others are called "rules". I'd say linguistic variety is good, but since good things are bad and bad things are slightly better, let's say that as rule of thumb in the first case you want to stress on the computational aspect, in the second you want to say that it's a useful observation, but perhaps not a very deep cultural fact and in the third case you want to stress on some algorithmic/computative/(perhaps even qualitative?) aspect. –  Sep 21 '18 at 21:54
  • I would imagine a theorem (or an axiom) can be any type of expression (provided it is true and provable). A "law" I would assume is a specific type of theorem, possibly one the pertains to a charactoristic of sines or cosines that always applies. I guess. I personally wouldn't care myself as they are clearly names of theorems and semantically do not cause any difficulty to me personally. I guess that's not an answer but... yes, they are theorems. – fleablood Sep 21 '18 at 22:23
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    If you had encountered something you considered particularly profound —especially if you had derived it— wouldn't you want to call it a "Law"? :) "Law"s are everywhere. Laws of Sines and Cosines, Laws of Motion and Gravity, Laws of Large Numbers and Small, Laws of Murphy and Godwin. It's an honorific. The "Earliest Known Uses of Some of the Words of Mathematics" site has various "Law" entries; maybe some of those sources explain why the author chose to designate them as such. (Relatedly, there are "Fundamental Theorems" in many fields.) – Blue Sep 21 '18 at 22:24
  • Your question reads like a rant. Please give some references to give us something concrete to go on. – Rob Arthan Sep 21 '18 at 22:28
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    @RobArthan Then I'd say you have a rather unusual definition of "rant." – user525966 Sep 21 '18 at 22:32
  • @Blue I think that's probably what's going on (the honorific usage), if I had to guess – user525966 Sep 21 '18 at 22:35
  • Well one dictionary lists: "fulminate, sound off, spout, pontificate, trumpet, bluster, declaim" as definitions for "rant". Your question reads very aggressive. – Rob Arthan Sep 21 '18 at 22:52
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    Expanding on my parenthetical ... The "Earliest Known Uses ..." entries for "Fundamental Theorem"s note that Gauss himself may have been the first to attach that title to a result; writes Gauss (translated): "Since almost everything that can be said about quadratic residues depends on this theorem, the term fundamental theorem, which we will use from now on, should be acceptable." Some theorems just seem to deserve special names. – Blue Sep 21 '18 at 22:58
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    (For what it's worth: I didn't read your question as a rant.) – Blue Sep 21 '18 at 23:01
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    I'm voting to close this, not because it's a bad question in any way, but purely because I don't know whether you're asking for a mathematical reason why they're called "laws" or for a historical reason. By the way, I think that this question doesn't sound like a rant at all and I can't think of any way it could be edited to sound less like a rant. – Tanner Swett Sep 21 '18 at 23:38
  • @TannerSwett I'm asking about both, really. Just in general why they're laws instead of theorems. If there's a mathematical reason behind it, a historical one, etc. I edited the post to add some clarification. – user525966 Sep 21 '18 at 23:50
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    @TannerSwett math-history is a tag. I edited it in. This question falls within the scope of the site, though it may also be apt here. – zahbaz Sep 21 '18 at 23:59
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    It could be both a law and a theorem. Just as "the commutative law" could also be a theorem (or an axiom, depending on how you organize your deductions). A harder question ... Why is a certain theorem of Poncelet called a "porism"? – GEdgar Sep 22 '18 at 00:11
  • Trigonometry dates back to Ancient Greece (3rd century BC Hellenistic mathematicians such as Euclid and Archimedes, 2nd century AD, the Greco-Egyptian astronomer Ptolemy). Math logic is quite new... – Mauro ALLEGRANZA Sep 24 '18 at 06:54
  • @Blue The link for "Earliest Known Uses of Some of the Words of Mathematics" site is not working. – SARTHAK GUPTA Feb 28 '20 at 16:48
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    @SARTHAKGUPTA: I suppose it's inevitable that the links will break someday, but that day has not yet come; the links work for me. Perhaps you could inspect the associated URLs and try them directly. In any case, the site's table of contents is located at http://jeff560.tripod.com. – Blue Feb 28 '20 at 17:32
  • https://math.stackexchange.com/questions/24758/difference-between-a-theorem-and-a-law – fruitbat Mar 05 '20 at 14:39

2 Answers2

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One could be forgiven for thinking the terminology is a mess. There are other examples of laws/rules in mathematics, some trigonometric, but not in general. The term "identity" is also common (in fact, some of those links came from here). I could find nothing anywhere indicating when to use each term. However, based on my experience, the way people use the terms in the names of named results seems to go as thus (no doubt with plenty of exceptions):

  • Any result of the form $f(x)=g(x)$ with the same variables $x$ on each side tends to be called an identity (whereas $f(x)\ne g(x)$ would be an inequation, and $f(x)\ge g(x)$ and $f(x)>g(x)$ would be inequalities), unless it governs the arithmetical operations on such variables, e.g. $ab=ba$, in which case "law" is typically used;
  • Any equally simple to use result for calculations, which may use different variables on each side, is called a law or rule, sometimes interchangeably (e.g. $a=2R\sin A$ has been called both the law of sines and the sine rule, although for some reason you only seem to see "law" in trigonometry, e.g. no-one would refer to Cramer's law);
  • A result you can use procedurally to verify a claim tends to be called a criterion;
  • Other results tend to be called theorems: in particular, "for all $x$ there exists a $y$ such that..." would be a theorem.

Technically, all of these are theorems, but even if people acknowledge them as such, the name of the result might not reflect that.

J.G.
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This is a law because no other law can prove this. Though using basic mathematics can prove this. Like pythagoras theorem is a special case of cosine law hence it is called a theorem

Abbas Murtaza
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    This provides no justification against calling it a theorem. It is also untrue - there are numerous ways of proving the sine and cosine laws from other theorems. Why aren't they laws too? – Jam Mar 05 '20 at 14:24