A History of Mathematics From Mesopotamia to Modernity

244 A History ofMathematics

Well, who did invent the computer? Once one examines the history carefully, there is obviously
no single answer—unlike, say, the telephone. The claims of Charles Babbage and Ada Lovelace in
the mid-nineteenth century are attractive, and they have the advantage of being British; but the
machine which they designed was never built. Between their outlines and the actual machine-
building stands one undoubted landmark, Alan Turing’s 1937 paper ‘On computable numbers’.
This was Turing’s first paper, and it was pure mathematics; it dealt with the one part of Hilbert’s
programme which Gödel had not demolished—did there exist a procedure for determining which
formulae were provable (theEntscheidungsproblem)? The work was similar to Gödel’s, but with an
interesting difference of style. Turing’s famous image for computing real numbers (the problem he
aimed to prove undecidable) was via a machine, and the machine followed automatic instructions
to read and write.
A section of his paper is reproduced as Appendix B. It is easy, to begin with, to point out
that a Turing machine is not a computer, since it is infinite. (‘In general the arrangement of
the memory on an infinite tape is unsatisfactory in a practical machine’, Turing was to observe
10 years later—ironically? (Hodges 1985, pp. 318–9)) It is also not a physical machine, merely
a description of one. However, if one looks at its method of procedure, it is definitely ‘computer-
like’. The usual modern description—a slight variation on Turing’s paper—uses the following
specifications:

1. At a given time, the machine is in astate(one of a finite set) and in a givenpositionon the ‘tape’.


2. It nowreadsthe symbol at its current place on the tape.


3. Having done this, it then canchangethe symbol in its current position, ormoveleft or right, or
   a combination.


4. Which of the tasks specified in 3 is carried out is entirely determined by the machine’s state 1,
   and by the symbol which it has read 2.

The tasks initially assigned seem distinctly boring (Turing starts by designing a machine which

will compute the sequence ‘1010101...’), but since the ‘tape’ is infinite, it is easy to define the

universalTuring machine which will compute any computable number.

Turing was at that stage as innocent of engineering as any Cambridge mathematician might

be. ‘Mrs Turing [Alan’s mother] had a typewriter’, says Hodges. Definitely less sophisticated as

a machine than Ada Lovelace’s Jacquard loom, the typewriter seems to have served well enough

for what was at the time a theorem and a ‘thought-experiment’.

The next stage is well, if not always accurately known and somewhat contested. The Americans

are certain that they invented the computer (although a long-standing wrangle between von

Neumann and his collaborators leaves the balance of credit uncertain), the British in their more

modest way feel that they were responsible. Turing has an able advocate in Andrew Hodges; his

tragic end, prosecuted under Britain’s primitive homosexuality laws, victimized and finally driven

to suicide in 1954 certainly inspires sympathy. The case is a romantic one:

And it was thus that in this remote station of the new Sigint^7 empire, working with one assistant in a small hut, and

thinking in his spare time, an English homosexual atheist mathematician had conceived of thecomputer. (Hodges

1985, p. 295)

7. Signals intelligence.