Difference between a theorem and a law

Question

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?

Answer 1

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.

Answer 2

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.