American Scientist
Ahead of His Time
Martin Davis
The Man Who Knew Too Much: Alan Turing and the Invention of the Computer. David Leavitt. x + 319 pp. W.
W. Norton, 2006. $22.95.
To "compute" used to mean to perform a calculation, presumably with numbers. How is it then that our modern
computers can be used to carry out so many varied tasks that, at least on the face of it, seem to have little to do
with calculation? It is to Alan Turing that we owe our understanding of the full extent of the concept of
computation as well as the realization that a computing machine can be "all-purpose": There is no need for a
myriad of separate devices to perform the full variety of possible computational tasks.
Turing was conceived in India, where his father was a civil servant under the Raj, and was born in London in
1912. Much of his childhood was spent in England, while one or both of his parents stayed in India. His interest
in scientific matters developed early and began to flower while he was at the Sherborne School, a traditional
British "public" school (which is to say a private one) for boys. It was also there that he had an intense
schoolboy crush on a brilliant fellow student, whose death devastated him. Turing's mathematical ability, which
was noticed but rather disdained at Sherborne, led to a scholarship at King's College, Cambridge. After his
degree, he was elected a Fellow of the college and was on a well-worn path to a career as a journeyman
mathematician.
Lectures on the foundations of mathematics by Max Newman led to an abrupt shift in direction. In the early
1930s there was much excitement concerning the discovery by Kurt Gödel that no single formal system of logic
could decide all propositions of mathematics. Newman, a topologist, had no professional expertise in these
matters but evidently provided a thorough and captivating account, beginning with the basics and leading his
students up to the frontier of knowledge. In particular, he called attention to David Hilbert's
Entscheidungsproblem.
Hilbert had asked for an algorithm that, provided with some premises and a purported conclusion as input, all
written in the language of formal logic, would determine whether that conclusion was indeed a logical
consequence of the premises. In a sense, this was an old problem, going back at least to Leibniz, but it attained
its sharp form only with the 20th-century development of "symbolic" logic. Mathematicians were very familiar
with algorithms, and many important algorithms were part of the body of mathematics. But many doubted that
an algorithm existed that could do what Hilbert wanted. In fact, another theorem of Gödel's suggested very
strongly that no such algorithm could exist.
But how could one prove such a negative? Mathematicians certainly could recognize an algorithm when they
saw one. However, to prove there is no algorithm for some particular problem, this would not suffice. It seemed
one would need a precise definition of what constituted an algorithm, and when Turing began thinking about the
problem, no such definition was available to him. He began by analyzing and simplifying what it is that a
human being does when executing an algorithm with pencil and paper, and then presented a convincing
argument that such a process could, with no loss, be replaced by a very simple imagined device that did nothing
but move back and forth on a linear tape, noting and replacing symbols. To prove that the
Entscheidungsproblem is unsolvable, Turing proceeded by showing that no such device could solve it. This type
of device would later be known as a Turing machine.
