In 1936, a twenty-three year old Cambridge mathematician named Alan Turing published a paper with the unpromising title On Computable Numbers, with an Application to the Entscheidungsproblem. Its purpose was narrow and technical: to answer a question posed by the mathematician David Hilbert about whether there exists a mechanical procedure that could decide, for any mathematical statement, whether it is provable. Turing's method for answering it created something far more consequential than the answer itself.
To make precise what a mechanical procedure even meant, Turing described an imaginary device, now called a Turing machine, consisting of an infinitely long tape divided into cells, a read-write head that could move along the tape one cell at a time, and a simple table of rules determining what the machine did next based on the symbol it read and its current internal state. Despite its extreme simplicity, Turing showed that such a machine could, in principle, carry out any calculation that could be carried out by following an explicit set of rules at all, a claim now known as the Church-Turing thesis after Turing and the logician Alonzo Church, who had reached a related result independently at almost the same time using a different mathematical approach.
Turing used the machine to prove that Hilbert's decision problem has no general solution, by showing that no algorithm can determine in advance, for every possible machine and input, whether that machine will ever stop running, the halting problem. The proof was itself an early triumph of computer science before any computer existed. Every general-purpose computer built since, from the earliest vacuum-tube machines to a modern laptop, is, in the strict mathematical sense Turing defined, equivalent to the abstract machine he described eleven years before the first stored-program computer ran.