A mathematician’s contribution

Perhaps you had not heard of G.H. Hardy before you read Chapter One of this book. Very likely, however, you will have heard of Alan Turing. One likely reason for knowing about Turing is that he worked at Bletchley Park during the Second World War as one of the code breakers in that secret establishment. The story of how the Nazi war communication code was cracked, of the Enigma machine, and Turing’s part in it, has been told a number of times, not least in a movie starring Benedict Cumberbatch. But Turing was much more than a successful cryptanalyst. He was a mathematician every bit as capable as G.H. Hardy of deep and beautiful mathematics. The difference was that Hardy worked in a field that has been studied at least since Pythagoras, whereas Turing founded an entirely new discipline which we now call computing science.

Turing enrolled as an undergraduate in King’s College Cambridge in 1931, the same year that Hardy returned from Oxford to take up the Sadleirian Chair of Pure Mathematics. Upon graduating he was immediately, at the age of 22, elected as a Fellow of King’s, on the basis of his final-year dissertation. In 1936 he read a paper to the London Mathematical Society (published in its Proceedings the next year) with the title ‘On computable numbers, with an application to the Entscheidungsproblem’. This is the seminal paper of computing science. It is also deeply rooted in mathematics.

To understand what computable numbers are, and what the Entscheidungsproblem is, requires a bit of explanation. For this purpose, it is best to go back to a speech made by another mathematician, David Hilbert, at the beginning of the 20th century. Hilbert was the most eminent mathematician of his time, and therefore a natural choice to give a keynote address at the Second International Congress of Mathematicians, in Paris in 1900. In his speech he outlined 23 mathematical problems which were unsolved at that time, and which he hoped would be solved during the coming century.