Proof is a firmly attested and evident objective fact, or a coherent set of facts, which cannot be refuted, often an inescapable conclusion based on undeniable evidence. Proofs have been set aside by logical fallacies and prejudice.
A mathematical proof is a step-by-step demonstration of the truth of a mathematical theorem. Proofs build on axioms, which are statements that are assumed to be true without proof, as well as previously-proved theorems.
Main article: Scientific evidence
Unlike the theorems of mathematics, science does not seek to prove that its theories are true. Instead, the scientific method seeks to check whether the predictions implied by a theory are observed in nature. Therefore, as philosopher of science Karl Popper argued, science can only hope to show that a theory is false. But scientists recognize that science can never prove that a theory is true in the same sense that a mathematical theorem is true. Therefore, scientists never claim that their theories are facts. Instead, science searches for theories that are not disproved by currently-known experimental observations. Insofar as theories are consistent with nature, they may serve as a guide to improve technology for example and can be considered as true in laymen's terms.
A notable exception may be found in the field of biology, where educators and other proponents frequently contend that, "Evolution is a fact."
- John L. Synge:
"...when one examines some proofs in the Neo-cartesian spirit, too often they seem to dissolve completely away, leaving one in a state of wonder as to whether the author really thought he had proved something. Or is the reader stupid? It is hard to say."
In American courts, crimes are proved "beyond reasonable doubt" to a jury, based on the jury's own analysis of the admissible evidence. Other legal issues may be decided by clear and convincing evidence or by a preponderance of the evidence (more likely than not).
- Marco M. Capria, Aubert Daigneaut et al. (2005). "5. General Relativity: Gravitation as Geometry and the Machian Programme", Physics Before and After Einstein. IOS Press, 97, 114. ISBN 1-58603-462-6. “John L. Synge, who was the author of one of the classic reference books on relativity , wrote half a century after Einstein’s first formulation of general relativity: [...] when one examines some proofs in the Neo-cartesian spirit, too often they seem to dissolve completely away, leaving one in a state of wonder as to whether the author really thought he had proved something. Or is the reader stupid? It is hard to say. In any case I am still waiting for a rational treatment of the dynamics of the solar system according to Einstein’s theory [100, p. 14].”