foundations of mathematics Computers and proof

While many mathematicians use computers only as word processors and for the purpose of communication, computer-assisted computations can be useful for discovering potential theorems. For example, the prime number theorem was first suggested as the result of extensive hand calculations on the prime numbers up to 3,000,000 by the Swiss mathematician Leonhard Euler (1707–83), a process that would have been greatly facilitated by the availability of a modern computer. Computers may also be helpful in completing proofs when there are a large number of cases to be considered. The renowned computer-aided proof of the four-colour mapping theorem by the American mathematicians Kenneth Appel (b. 1932) and Wolfgang Haken (b. 1928) even goes beyond this, as the computer helped to determine which cases were to be considered in the next step of the proof. Yet, in principle, computers cannot be asked to discover proofs, except in very restricted areas of mathematics—such as elementary Euclidean geometry—where the set of theorems happens to be recursive, as was proved by Tarski.

As the result of earlier investigations by Turing, Church, the American mathematician Haskell Brooks Curry (1900–82), and others, computer science has itself become a branch of mathematics. Thus, in theoretical computer science, the objects of study are not just theorems but also their proofs, as well as calculations, programs, and algorithms. Theoretical computer science turns out to have a close connection to category theory. Although this lies beyond the scope of this article, an indication will be given below.