The Boole Lecture A Very Mathematical Dilemma

Mathematics is facing a dilemma at its heart: the nature of mathematical proof. We have known since Church and Turing independently showed that mathematical provability was undecidable that there are theorems whose shortest proofs are enormous. Within the last half-century we have discovered practical examples of such theorems: the classification of all finite simple groups, the Four Colour Theorem and Kepler’s Conjecture. These theorems were only proved with the aid of a computer. But computer proof is very controversial, with many mathematicians refusing to accept a proof that has not been thoroughly checked and understood by a human. The choice seems to be between either abandoning the investigation of theorems whose only proofs are enormous or changing traditional mathematical practice to include computer-aided proofs. Or is there a way to make large computer proofs more accessible to human mathematicians?