Nov 26, 2012
MATHEMATICAL PROOF 3_Documented by Marsigit
A probabilistic proof should mean a proof in which an example is shown to exist by methods of probability theory - not an argument that a theorem is 'probably' true.
The latter type of reasoning can be called a 'plausibility argument'; in the case of the Collatz conjecture it is clear how far that is from a genuine proof. Probabilistic proof is one of many ways to show existence theorems, other than proof by construction.
If we are trying to prove, for example, "Some X satisfies f(X)", an existence or nonconstructive proof will prove that there is a X that satisfies f(X), but does not tell you how such an X will be obtained. A constructive proof, conversely, will do so.
A statement which is thought to be true but hasn't been proven yet is known as a conjecture.
Sometimes it is possible to prove that a certain statement cannot possibly be proven from a given set of axioms; see for instance the continuum hypothesis.
In most axiom systems, there are statements which can neither be proven nor disproven; see Gödel's incompleteness theorem.