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During the last half of the nineteenth century the need for formal methods of proof became evident to mathematicians who were making such confidence-shaking discoveries as non-Euclidean geometry. The demand is not to be denied; every jump must be barred from our deductions. That it is hard to satisfy must be set down to the tediousness of proceeding step by step. Every proof which is even a little complicated threatens to become inordinately long. [M1] G. Frege, 1884 This general desire for rigor has persisted since that time, and a great deal has been learned about formal methods. But, for the reason noted by Frege, very little of real mathematics has been done with full formal treatment. Our present hope is to use computers to take the drudgery out of formal demonstrations, just as they are taking it out of accounting. Toward this end, several programs are under way. They vary in purpose; the Proofchecker [H8, H9] is to be capable of filling the gaps of a proof; the work of Mott et. al. [H10] aims to achieve the equivalent of a desk calculator ability as an aid to a mathematician doing formal proofs. The most intriguing prospect, however, is that computers can eventually be made to both devise and prove interesting non-trivial theorems wholly on their own. The first of these desires, the devising of interesting conjectures, has not even been attempted. I believe, however, that we are on the verge of achieving the second of these ends, the mechanical proof of non-trivial theorems, a belief which I hope I can justify in the sequel. |
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