Formal Proof definition
Formal Proof. The Four-Color Theorem.” ▇▇▇▇▇▇▇ ▇▇▇▇▇▇▇▇. Notices of the American Mathematical Society. 2008 Vol. 55. p. 1382–1393 called a minimal criminal, is the smallest possible example that shows the proof to be false. In this instance, a minimal counter example would be a map that requires at least five separate colors, but if one country were removed, could be colored with only four colors. So, to prove the validity of the theorem, one needs only to show that no minimal counter examples exist. To do so, however, would require one to devise and check a near infinite number of different configurations of maps, a task unaccomplishable by a human. Accordingly, although it was first proposed in 1852, the problem remained unsolved until 1976. The proof of this theorem was undoable by hand. So, University of Illinois mathematicians ▇▇▇▇▇▇▇ ▇▇▇▇▇ and ▇▇▇▇▇▇▇▇ ▇▇▇▇▇ used a computer. This computer checked approximately two thousand reducible configurations – an arrangement of regions that cannot occur in a minimal counter example – one by one to show that a minimal counter example to the four-color theorem could not exist, thus through brute force proving the theorem by contradiction. Although this proof, as the first proof completed by a computer, was highly scrutinized at the time of its completion, it has withstood any challenges since.18 Today, computerized proofs of mathematical theorems are both widely accepted and routinely utilized.19 This computer clearly meets a stricter criteria for possession of episteme. In addition to running via the use deductive syllogisms in the form of programs, it solves mathematical proofs in a way indicative of episteme. Much as with ▇▇▇▇▇▇▇▇▇’s geometer, the four-color theorem computer began from basic indisputable first premises, properties of various shapes and their boundaries, and managed to prove deductively a greater conclusion, in this instance the four- color theorem. Further, the computer’s proof is demonstrable as it could, in theory, teach an eager student the proof line by line by breaking down each of the two thousand cases it checked.
Formal Proof. The Four-Color Theorem.” ▇▇▇▇▇▇▇ ▇▇▇▇▇▇▇▇. Notices of the American Mathematical Society. 2008 Vol. 55.