In 1936, Alan Turing created the abstract mathematical device that is now known as the Turing Machine. With it, he was able to solve the “Hilbert Entscheidungsproblem,” one of the mathematical challenges put forward by David Hilbert. This alone would have been a significant achievement, but the consequences of the Turing Machine are far greater than that.

The impact and power of Turing Machines is summed up in the sentence “Anything a real computer can compute, a Turing machine can also compute.” What has always astounded me about this is that a mathematician invented the abstract model for computing years before there were anything that we would consider a computer was operational. There have been notational improvements in how the Turing Machine is represented, but the original model is largely unaltered. Score one for the math people!

The Turing Machine is also effective in providing a foundation for what is meant by the word “computable” in the broad mathematical sense. There is a wide variety of content in the papers in this collection about Turing’s ideas and what they meant for mathematics and computer science. Some of them deal with the formal mathematical ideas of computation, including transfinite computation. For many readers, this may be a formula too far, but they are still interesting.

There are other papers that deal with history, philosophy and the precise definitions of concepts like computability and algorithms. These are words whose definitions and application are subject to interpretation. These papers would make excellent talking points in math, computer science and even philosophy classes. Their contents go to the very heart of algorithms and decision theory.

My favorite paper in the collection was “Turing, von Neumann, and the Universal Electronic Computer.” One of the primary threads in this paper was an exploration as to who exactly originated the concept of the stored-program. I found it a bit amusing, for it is definitely in the category of “academic only.” Like so many concepts, for this one there was no one person having an “Aha!” moment when the stored-program idea was born.

People of all mathematical and computer science backgrounds will find something of interest in this collection. Educators that would like to go deeper into the meaning of computability and algorithms can find material that will set the stage for discussions and research papers on the meaning and impact of the life and work of Alan Turing.

Charles Ashbacher splits his time between consulting with industry in projects involving math and computers, and teaching college classes. In his spare time, he reads about these things and helps his daughter in her lawn care business.