A proof is a rigorous mathematical argument that unequivocally demonstrates the truth of a given proposition. A mathematical statement
that has been proven is called a theorem.
According to Hardy (1999, pp. 15-16), "all physicists, and a good many quite respectable mathematicians, are contemptuous about proof. I have heard Professor Eddington, for example, maintain that proof, as pure mathematicians understand it, is really quite uninteresting and unimportant, and that no one who is really certain that he has found something good should waste his time looking for proof.... [This opinion], with which I am sure that almost all physicists agree at the bottom of their hearts, is one to which a mathematician ought to have some reply."
To prove Hardy's assertion, Feynman is reported to have commented, "A great deal more is known than has been proved" (Derbyshire 2004, p. 291).
There is some debate among mathematicians as to just what constitutes a proof. The four-color theorem is an example of this debate
because its proof relies on exhaustive computer testing of many individual cases
that is impractical to reproduce by hand. Computer-assisted and automated proofs
can nevertheless supply explicit computations or
formal derivations whose individual steps can be checked.
Machine checking separates two questions. It can verify that a formal derivation establishes a formal statement from a stated axiom system,
but this does not itself show that the formal statement faithfully expresses the
intended informal claim. Checking the formalization is therefore distinct from checking
the derivation (Wolfram 2026).
A page of proof-related humor is maintained by Chalmers.
