Formal and Informal Proof
The informal proofs of everyday mathematical practice are unlike the formal proofs of proof theory. They are rather like high-level sketches that would allow an expert to reconstruct a formal proof at least in principle, given enough time and patience. For most mathematicians, writing a fully formal proof is too pedantic and long-winded to be in common use.
Formal proofs are constructed with the help of computers in interactive theorem proving. Significantly, these proofs can be checked automatically, also by computer. (Checking formal proofs is usually simple, whereas finding proofs (automated theorem proving) is generally hard.) An informal proof in the mathematics literature, by contrast, requires weeks of peer review to be checked, and may still contain errors.
Read more about this topic: Proof Theory
Famous quotes containing the words formal, informal and/or proof:
“The bed is now as public as the dinner table and governed by the same rules of formal confrontation.”
—Angela Carter (1940–1992)
“We as a nation need to be reeducated about the necessary and sufficient conditions for making human beings human. We need to be reeducated not as parents—but as workers, neighbors, and friends; and as members of the organizations, committees, boards—and, especially, the informal networks that control our social institutions and thereby determine the conditions of life for our families and their children.”
—Urie Bronfenbrenner (b. 1917)
“Ah! I have penetrated to those meadows on the morning of many a first spring day, jumping from hummock to hummock, from willow root to willow root, when the wild river valley and the woods were bathed in so pure and bright a light as would have waked the dead, if they had been slumbering in their graves, as some suppose. There needs no stronger proof of immortality. All things must live in such a light. O Death, where was thy sting? O Grave, where was thy victory, then?”
—Henry David Thoreau (1817–1862)