These works of Alan Mathison Turing (1912–1954) were relatively late in appearing, but the high quality of the editorial work justifies the wait. The works presented here are carefully annotated and reviewed with comments helpful to the nonspecialist reader. This work (with three supplementary volumes on mathematical logic, mechanical intelligence, and morphogenesis) should be not only in the regular science libraries, but also in special computer science reference collections. I hope Turing’s works will be available to all students, as Turing’s vision is perpetually relevant.

Some information can be gleaned from the table of contents alone. Turing’s published papers included here are “Equivalence of Left and Right Almost Periodicity” (1935), “Finite Approximation to Lie Groups” (1938A), “The Extensions of a Group” (1938B), “A Method for the Calculation of the Zeta-function” (1943), “Rounding-off Errors in Matrix Processes” (1948), “The Word Problem in Semi-groups with Cancellation” (1950), “Some Calculations of the  Riemann  Zeta-function” (1953), and “Solvable and Unsolvable Problems” (1954). The editor seems aware of the apparent inconsistency in omitting Turing’s watershed 1937 paper on the Entscheidungsproblem [1] while including his 1950 and 1954 papers.

Previously unpublished papers by Turing included in this book are “A Note on Normal Numbers,” “The Word Problem in Compact Groups,” “On Permutation Groups,” “The Difference ψ ( x ) - x ,” and (with S. Skewes) “On a Theorem of Littlewood.” The papers are supplemented by related papers published elsewhere: by W. W. Boone [2] explaining the 1950 paper, by A. M. Cohen and M. J. E. Mayhew expounding unpublished work on analytic number theory [3], and by I. J. Good explaining the otherwise unrecorded statistical work [4].

My vivid impression is that Turing was an all-purpose pure and applied mathematician. His preoccupation with analytic number theory may seem as far from the Turing machine as Newton’s preoccupation with curves (cubic and conic) was from natural philosophy, but it illustrates an intense background of pragmatism. The 1953 calculations for the range of Zeta-function zeros on the Mark I Manchester Computer are almost like a lab report, as the results were so strongly at the mercy of maintenance. The remarkably clear 1954 exposition of solvability is as thrilling to me today as when I first encountered it as a Penguin periodical on the newsstands. It is also edifying to know that Turing’s paper on matrix methods was meant to limit the expected error, as was done concurrently by von Neumann and Goldstine [5].

The legendary British loyalty to the Official Secrets Act is blamed for the delay in these volumes. Indeed, the unpublished paper on permutation groups is modeled after the wartime Enigma cryptographic machine. So is Turing’s statistical work, as decoding is a Bayesian process. With the words “bit” and “byte” still unknown in 1941, Turing introduced the words “ban” and “deciban” for units of information leading to certainty (see Good [4]). It is impossible to open this book without finding some significant and thought-provoking gem of nostalgia.