The Beginnings of Digital Culture - Digital Culture - page 1

1. The Beginnings of Digital Culture

turing's conceptual machine

In the late s Alan Turing, a fellow of King's College, Cambridge,

published a paper entitled 'On Computable Numbers with Appli-
cation to the Entscheidungsproblem '. 1 It was a response to part of
one of a number of proposals by the German mathematician David
Hilbert. Hilbert wished to cast all mathematics in an axiomatic
structure, using the ideas of set theory. From the beginning of the
twentieth century on, he proposed a number of programmes that
would achieve his aim. In the

1930

s he put forward his most ambi-
tious of these programmes, in which all mathematics was to be put
into an axiomatic form, the rules of inference to be only those of
elementary logic. He proposed that a satisfactory system would be
one which was consistent, in that it would be impossible to derive
both a statement and its negation, complete, in that every properly
written statement should be such that either it or its negation should
be derivable from the axioms, and decidable, in that any statement
or its negation should be provable by an algorithm. The complete-
ness part of this programme was shown to be unworkable by the
German mathematician Kurt Gödel who, in

1920

, demonstrated that
there was no system of the type Hilbert proposed in which integers
could be defined and which was both consistent and complete. The
impact of this discovery, which is known as 'Gödel's incompleteness
theorem', was great and its importance goes beyond mathematics

1931