Alan Turing Gives Computation a Mathematical Model

In 1936, the English mathematician Alan Turing submitted a thirty-six-page paper to the London Mathematical Society that redefined the study of logic. Titled "On Computable Numbers, with an Application to the Entscheidungsproblem", the work tackled a famous challenge issued by David Hilbert in 1928, which asked whether a universal mechanical procedure could determine whether an arbitrary statement of first-order logic is logically valid. Rather than approaching the problem purely through abstract symbolic manipulation, Turing began by analyzing the physical steps a human clerk followed when calculating numbers on paper.

To formalize this process, Turing introduced hypothetical mechanisms later termed Turing machines. Each model possessed a discrete set of internal states and moved along a divided strip of tape, scanning, altering, or printing symbols according to an explicit schedule of rules. Turing argued that such an arrangement could perform any mathematical calculation capable of being expressed by finitary means as an algorithm. He showed that despite its basic operational structure, this theoretical apparatus established clear, definitive boundaries for mathematical calculation.