Software 253 mathematician, he managed to prove this conjecture and so, two years after first announcing that he had solved Fermat's Last Theorem he could finally lay it to rest. The Special Purpose Objection Before I declare mankind’s outright victory over computers, the Special Purpose Objection must be overcome. The objectors would argue that Wiles is a Special Purpose computer. Special Purpose computers are at no risk of breaking the Turing limit when they solve problems they have There is no algorithm which, for a given arbitrary Diophantine equation, would tell whether the equation has a solution or not. been programmed to answer. The objection misses the key point. I am not arguing having a solution to a given mathematical puzzle presents a difficulty to a computer; I am arguing a computer cannot discover one. Take, for example, the search engine Google. If I type “where can I find the proof of Fermat’s Last Theorem?” into the search box, it will retrieve a PDF of the proof as the third result. It appears this special purpose computer solved the problem. But you immediately see the difficulty. Google search already knew the answer, or more precisely had indexed the answer. The computer was not tackling a random problem from scratch. It was tackling a problem for which it knew the answer, or at least where an answer could be found. There is no sense in which the search engine discovered the proof. To really understand this objection we need to examine exactly what Turing and Matiyasevich proved. An arbitrary problem is one you do not already know the solution to when you write the algorithm. You can think of it as a variable. Is there an algorithm that can solve problem ‘X’? The alternative is a special program. It can solve problem Y. Y is a problem it knows. It must have the solution coded somewhere within it in a computably expandable way. You might think of this as a table of constants; problem Y has solution 1, problem Z has solution 2, and so