Theorems are proofs away from being mere conjectures
A theorem is a statement proved from axioms by accepted inference—often silently Zermelo–Fraenkel set theory with choice. Conjectures wait for such proofs. Nineteenth-century crises, including non-Euclidean geometries, forced mathematicians to spell those rules more carefully.
Working mathematicians rarely state their ground rules; when unspoken, they are nearly always ZFC or a weaker system such as Peano arithmetic. Only significant results usually earn the name theorem, while smaller steps are called lemmas, propositions or corollaries. Many take the form if A, then B: the claim is not that B holds, only that it must follow whenever A does. In the statement that an even natural number halved is still a natural number, evenness is the hypothesis, a word here meaning a premise rather than a guess.
For centuries, results built on supposedly obvious postulates were treated as final truths, such as Euclid's proof that a triangle's angles total 180 degrees. Then geometers altered the fifth postulate and got consistent systems where the total differs, and naive reasoning about sets produced Russell's paradox. The repair was to make foundations explicit, so the triangle result now holds under Euclid's axioms, and a theorem's validity depends only on its proof, not on what the axioms mean in the physical world.
Treating theories as mathematical objects allows theorems about theorems. Gödel showed that any consistent theory containing the natural numbers has true statements about them that it cannot prove. Goodstein's theorem can be stated in Peano arithmetic yet cannot be proved there, though set theory proves it.
Some proofs cannot be written out by hand: the four color theorem and the Kepler conjecture rest on computer-checked searches, a method many mathematicians initially rejected. Fermat's Last Theorem is the classic case of a result easy to state but deep to prove. And unlike a scientific theory, which must be testable and can be refuted by experiment, no experiment can prove a theorem.
Source: Theorem
