**Figure 4.** Alan Turing on a 2000 "millennium" stamp commemorating his 1937 theory of digital computing. Turing was based at Princeton University throughout 1937. (Source: MacTutor History of Mathematics Archive)

Alan Turing’s viewpoint can be derived from one telling statement in his initial paper. He wrote, “The behaviour of the computer at any moment is determined by the symbols which he is observing, and his ‘state of mind’ at that moment” [**12**, p. 250]. To the modern reader the association of “he” with “computer” might sound strange but one must remember that back in the 1930s the term *computer* referred to a human being who performed computations. Moreover, the expression “state of mind” referred to distinct states in the course of the computation. In this sense Turing was developing a precursor to the modern computer.

Tellingly, Turing wrote that “the computation is carried out on one-dimensional paper, i.e., on a tape divided into squares” [**12**, p. 249]. That “tape” is equivalent to what we refer to as coded instructions, or stored programs today, such as those stored in FORTRAN. With this in mind, a Turing machine consists of a finite set of quintuples of the form \( p \alpha \beta X q \). Here the machine encounters the symbol \( \alpha \) in state \( p,\) transforms it into the symbol \( \beta \) in state \( q,\) and the machine then moves to a state to the right, left, or same position as the state where it began according to whether \(X\) is R, L, or N. Turing called a real number “computable” if such a machine could produce a sequence of binary numbers representing the expansion of that number into 0s and 1s starting with an empty tape.

This whole approach, even as seen in this brief description, appears to be entirely theoretical. But there was a practical side to Turing as well. This universal machine provided a model for what we call a “stored program” computer today. While at Princeton he even designed an electro-mechanical binary multiplier and gained entrance to the graduate-student machine shop in the physics department to try to build the relays himself. Although he was unable to complete this project before returning to England, once back in Cambridge he was awarded funds to construct a special-purpose analog computer to calculate the zeros of the Riemann zeta-function. However, World War II broke out and that machine too was never built.