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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?

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    $\begingroup$ Actually in every other language that I know they are called theorems. You are right, strange that in English they are called different. $\endgroup$ – Mark Sep 21 '18 at 21:28
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    $\begingroup$ 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. $\endgroup$ – Jam Sep 21 '18 at 21:32
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    $\begingroup$ 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.) $\endgroup$ – Blue Sep 21 '18 at 22:24
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    $\begingroup$ (For what it's worth: I didn't read your question as a rant.) $\endgroup$ – Blue Sep 21 '18 at 23:01
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    $\begingroup$ 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"? $\endgroup$ – GEdgar Sep 22 '18 at 0:11

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