# Difference between a theorem and a law

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?

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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 3 '11 at 5:39
@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. – KronoS Mar 3 '11 at 5:40
@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 3 '11 at 5:43
@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 3 '11 at 5:55
@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 3 '11 at 17:16

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.

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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 3 '11 at 6:13
Also, there is De Morgan's Laws. – Eric Naslund Mar 3 '11 at 6:20
Have a very, merry Christmas, Jasper Joy...oops, Jasper Loy! – amWhy Dec 25 '12 at 0:03
@Jasper Perhaps a Christmas gift? ;-) – amWhy Dec 25 '12 at 0:38
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: pvs.csl.sri.com/doc/pvs-language-reference.pdf – Ben Hocking Aug 17 '15 at 17:24

A theorem is a statement proved by a chain of reasoning. The theorem is not self evident. It is derived after considering the results of several logical statements (often including other theorems). A famous example of this is the Pythagorean Theorem, which has nearly 400 proofs.

Laws are statements which are inferred by observation. Laws are not proved. Laws are demonstrated based on repeated observations. It expresses a causal relationship between entities under certain conditions, and is often expressed mathematically. The Laws of Thermodynamics (so often quoted without the mathematical context and misstated for that very reason) are good examples.

A theory is an established and respected explanation of a natural phenomenon, acquired through confirmation of its principles through the scientific method - testing, confirmation, and observation and experimentation.

A fact is a true statement under the set conditions. A thing that is indisputably the case. Facts can generally be reproduced to be verified. A single counter example will immediately demote a fact to a false claim.

A proof in science a successful demonstration of a hypothesis under study using evidence and analysis. It is a misconception in many ways, since science actually can't prove anything for certain. In this sense, "proof" is a simplified way of saying "greatly enhanced confidence".

A rule is an informal axiom that expresses a philosophical point. A rule governs how a subject behaves under certain circumstances. A rule indicates increased chance of a certain outcome when a subject is in a certain state. Newton's "rules" or Occam's Razor are examples.

A Conjecture is a statement not (yet) proven, but which intuitively seems to be true, or for which the author has some more complex reasons to believes to be likely to be true. Some deceivingly simple of those resist centuries of attacks by brightest mathematicians.

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Please note that this Math.SE, and the question was asked within the context of mathematics. While your comments on the scientific method are factual and not unwelcome, please restrict your scope to the usage of these terms within mathematics. For example, a proof in math is a series of logical inferences starting from a set of assumptions and arriving at a conclusion(s), and it is certainly not "greatly enhanced confidence." – Tyg13 May 12 '15 at 3:14

"Law" is most likely a religious euphemism. In civil philosophy, a law is a decided decree of governors. To refer to a natural phenomenon as a law is therefore probably a simple bit of poetic license: scientific and mathematical results as "decrees" of God-of-your-choice.

Consequently

1. It is very old fashioned to refer to theorems as laws and
2. The term "law" carries a connotation of absolute truth
3. You might want to refrain from referring to your own results as laws, as it would sound unnecessarily presumptuous or arrogant to many.
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