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Provability, in mathematics, is the study of what results can be proved (under various conditions). This study is reliant on an understanding of the nature of what proof actually involves, which is still a point of controversy in mathematics. The followers of different schools of thought (formalism, intuitionism and logicism) have different views of what a proof is and, since proof is fundamental to mathematics, any different ideas of what the nature of mathematics really is will affect the ideas of what a proof really is, and vice versa. Intuitionists, for example, would deny that many results proved using the classical mechanisms of mathematics (proof by contradiction in particular) are really proved, and would set about to prove then by the restricted means at their disposal.
Within particular mathematical systems, more can be said about what can be proved than can in general. In any particular mathematical system which is strong enough for the idea of provability to be encoded within it (as is the case with Peano arithmetic), then Gödel\'s incompleteness theorem shows that there are results which are true but which are not provable within that particular system. This result itself has had an impact on the concepts of what mathematics really involves: see Hilbert\'s programme.
In recent years, a further facet of proof has become a controversial issue. In topology, there is a famous result, the four colour theorem (which states that with only four colours it is possible to colour the regions of a map on a flat surface in such a way that no adjacent regions share the same colour), which was finally proved fairly recently after many years. The problem with the proof is that it involved splitting the problem up into a large number of possible cases (thousands of them) and then checking each possibility with a computer. To check them all by hand would be beyond the lifespan of any person. So does this count as a proof, when it requires trust in the work of the computer? This problem becomes more acute as the problems to be solved become more complex, and as mathematicians rely ever more increasingly on computers to help them solve them. SMcL |
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