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There are plenty of theorems out there as well as laws within mathematics. For example, in Boolean algebra:

Theorems

  • Idempotent
  • Involution
  • Theorem of Complementarity

Laws

  • Commutative
  • Associative
  • Distributive

There are countless other examples out there, but my real question is this: What makes a theorem a theorem and a law a law?

James Mertz
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    In these examples, "laws" are really identities that are satisfied/imposed on the algebra a priori, while the theorems are propositions that are deduced from those identities and other axioms. – Arturo Magidin Mar 03 '11 at 05:39
  • @ArturoMagidin Perhaps these are bad examples... this question arose as I was studying for my embedded circuits class... but I'm looking for a "in general" answer... if one exists. – James Mertz Mar 03 '11 at 05:40
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    @KronoS: It's nomenclature; what is the difference between "Theorem", "Proposition", "Lemma", and "Corollary"? Answer: the importance the author places on them. But generally speaking, "laws" are a priori restrictions/rules/identities, while "Theorems" are invariably "derived truths": conclusions that follow logically from whatever you are starting from. That is, "laws" are a priori, "theorems" are a posteriori. "Generally" speaking, anyway. – Arturo Magidin Mar 03 '11 at 05:43
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    @Arturo: I think "law" is used much more generally, e.g. law of exponents, law of sines, parallelogram law, quadratic Reciprocity Law, Sylvester's law of inertia. "Law" is a bit old-fashioned, and tends to be used more frequently in applied math, e.g. physical laws. – Bill Dubuque Mar 03 '11 at 05:55
  • @Bill: Fair point; certainly, very common in Physics (Hooke's, Fermat's, etc). Still, much like the difference between Theorem and Lemma. (Though, while there are a lot of famous "Lemmas", like Zorn's, Fatou's, etc., is there any equally famous "Corollary"?) – Arturo Magidin Mar 03 '11 at 05:57
  • @Arturo Magidin Isn't arguably Fermat's Last Theorem just a simple corollary of Taniyama-Shimura? – Uticensis Mar 03 '11 at 07:43
  • @Billare: Well, T-S came after FLT; and my use of quotations was meant as quote: "Zorn's Lemma" and "Fatou's Lemma" are called lemmas. FLT is not called a "corollary". Is there any famous result that is routinely refered to as "xxx Corollary"? – Arturo Magidin Mar 03 '11 at 14:08
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    @KronoS: As you see in comments, the distinction between “law” and other notions is not precise, unlike other things in mathematics. I even collected a list of 7 synonyms of the word “theorem”. It's going mad. :) The real distinction exists between a logical formula and a theorem. A theorem is a proven logical formula. – beroal Mar 03 '11 at 17:16

2 Answers2

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Theorems are results proven from axioms, more specifically those of mathematical logic and the systems in question. Laws usually refer to axioms themselves, but can also refer to well-established and common formulas such as the law of sines and the law of cosines, which really are theorems.

In a particular context, propositions are the more trivial theorems, lemmas are intermediate results, while corollaries are results deduced easily from others. However, lemmas and corollaries may be major results on their own.

Note that a system may be given axioms in more ways than one. For example, we can use the least upper bound axiom to define the real numbers, or we can consider this axiom as a theorem if we were to construct the reals from the rationals using Dedekind cuts and prove it instead. The difference here lies in which axioms we choose to start with.

  • I am not so sure that laws "refer to axioms themselves." Doesn't Bills comment give at east 5 examples otherwise? To add a couple more, how about the law of large numbers, or the law of cosines. – Eric Naslund Mar 03 '11 at 06:13
  • Also, there is De Morgan's Laws. – Eric Naslund Mar 03 '11 at 06:20
  • Have a very, merry Christmas, Jasper Joy...oops, Jasper Loy! – amWhy Dec 25 '12 at 00:03
  • @Jasper Perhaps a Christmas gift? ;-) – amWhy Dec 25 '12 at 00:38
  • If laws are axioms, then the makers of PVS have made a mistake in their labeling (which I think is possible, but very unlikely), since they have laws being essentially identical to theorems, lemmas, propositions, corollaries, et al. See top of page 26 of this PDF: http://pvs.csl.sri.com/doc/pvs-language-reference.pdf – Ben Hocking Aug 17 '15 at 17:24
  • See also https://en.wikipedia.org/wiki/Theorem: "A law or a principle is a theorem that applies in a wide range of circumstances. Examples include the law of large numbers, the law of cosines, Kolmogorov's zero-one law, Harnack's principle, the least upper bound principle, and the pigeonhole principle.[" – Ben Hocking Aug 17 '15 at 17:25
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Theorems are deducted, laws are induced.

Theorems are deducted from axioms. Laws are induced from observations.

A law can be used as an axiom or as a theorem of a theory.

Leonardo Castro
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