In the original dustsheet. Black board binding with silver title on the spine.

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Turing’s fascinating and remarkable theory, which now forms the basis of computer science, explained for the general reader. In 1936, when he was just twenty-four years old, Alan Turing wrote a remarkable paper in which he outlined the theory of computation, laying out the ideas that underlie all modern computers. This groundbreaking and powerful theory now forms the basis of computer science. In Turing’s Vision, Chris Bernhardt explains the theory, Turing’s most important contribution, for the general reader. Bernhardt argues that the strength of Turing’s theory is its simplicity, and that, explained in a straightforward manner, it is eminently understandable by the nonspecialist. As Marvin Minsky writes, “The sheer simplicity of the theory’s foundation and extraordinary short path from this foundation to its logical and surprising conclusions give the theory a mathematical beauty that alone guarantees it a permanent place in computer theory.” Bernhardt begins with the foundation and systematically builds to the surprising conclusions. He also views Turing’s theory in the context of mathematical history, other views of computation (including those of Alonzo Church), Turing’s later work, and the birth of the modern computer. In the paper, “On Computable Numbers, with an Application to the Entscheidungsproblem,” Turing thinks carefully about how humans perform computation, breaking it down into a sequence of steps, and then constructs theoretical machines capable of performing each step. Turing wanted to show that there were problems that were beyond any computer’s ability to solve; in particular, he wanted to find a decision problem that he could prove was undecidable. To explain Turing’s ideas, Bernhardt examines three well-known decision problems to explore the concept of undecidability; investigates theoretical computing machines, including Turing machines; explains universal machines; and proves that certain problems are undecidable, including Turing’s problem concerning computable numbers.

Review: This is an interesting introduction to some of the key mathematical work of Alan Turing (mostly his 1936 results on Computable Numbers and Turing Machines) relating it to both contemporary developments in logic (the lambda calculus) and later developments in computer science and related ideas like finite automata. The book is thus mostly an introduction to mathematical and computer science concepts but serves to give them some intellectual context also and even a little bit of social context for Turing’s life and legacy.

The book begins with some discussion of mathematical logic, Hilbert’s program (the Entscheidungsproblem) and the Godel results that Turing’s work built on, responded and contributed to. Things like finite automata and the lambda calculus are worked though in preparation to lead the reader through Turing’s laying out of his model of computing in terms of Turing machines and proof that the computable numbers are not effectively innumerable. Also other mathematical work such as Cantor’s diagonalization argument are sketched out and worked through to the extent necessary to relate them to Turing’s work. Some key elements of Turing’s life and other work are summarized as are a few seminal instances from the origin of the computer, some philosophical issues around artificial intelligence and a final reflection on Turing’s life and legacy.

The book is a fairly readable and understandable exposition of the mathematics. However at times I found the examples and work through of the problems a bit brief. I had some rather haphazard knowledge of the mathematics in Turing’s work including my own attempt to read Turing’s paper, but much of the mathematical logic and machine theory was new to me. There is a difficult balance between being comprehensible to the general reader (or at least the interested neophyte) and avoiding tedium in explaining the very simple but elaborate procedures involved. Also the depth and breadth of the discussion is limited. Careful reading would no doubt be rewarded with greater comprehension but I found I profitably read it without having to master all the technical nuance presented.

The history presented is often very schematic outside of tracing the pedigree of certain mathematical techniques there are just a few bits of greatest hits. The reader may come away with a somewhat inflated sense of Turing’s role in the origin of actual computing machines (Turing’s exact contributions to various elements is a matter of some dispute), a few interesting early machines are left unmentioned in the cursory round up of early computers but otherwise I did not notice any serious errors.

Chris Bernhardt is Professor of Mathematics at Fairfield University and the author of Turing’s Vision: The Birth of Computer Science (MIT Press).