Turing made his proof of the decision problem or entscheidungsproblem in
1936, *before* there were any programmable computers. He was
concerned with what can be computed or calculated by a human who is
following definite rules, i.e. by a human doing mathematics as the
formalists say it should be done.

He first had to formalise the notion of a ``definite method''. He did
this by constructing a mathematical model of somebody following a
definite set of rules. This model was the *Turing machine*. It:

- used a finite alphabet of symbols. Clearly, this is what
mathematicians use when working.
- wrote these on tape divided into squares, each square capable of
holding one symbol. This is an idealisation of the 2-d piece of paper
that the mathematician works on. Its two-dimensionality adds nothing
essential, so without loss of generality, we can model it as a 1-d tape.
- had a finite number of possible ``states of mind''. If the number
of states of mind were infinite, then some states would have to be
arbitrarily close, and hence could not be distinguished. (This also
applies to the alphabet of symbols.)
- can at any time inspect one symbol, that which is in the tape
square under its ``eye''. (The ``eye'' is usually termed a
*read head*.) - has a behaviour which is determined solely by the current symbol
under its ``eye'', plus its current ``state of mind''. In effect, the
machine contains a ``state table'' which maps every combination of these
two to a new state plus some behaviour. This is the assumption of
physical determinism. You can think of many machines which embody such
state tables: e.g. a coffee machine, whose current ``state of mind'' is
determined by the value of the coins inside it, or a washing machine,
whose current ``state of mind'' is determined by the signals from its
timer.
- builds its behaviour from very simple operations. It can write a symbol, change its state of mind, move the tape left or right. We lose no generality by restricting ourselves to such simple operations, because we can combine them to produce arbitrarily complex behaviours.

See the essay by Andrew Hodges in *The Universal Turing Machine*
edited by Herken (PSY KD:H 42) for a summary of the history behind
Turing's work. Other essays describe the mathematical background, but
may be too technical. Penrose pages 47-48 explains why the Turing
machine is an adequate model of what human mathematicians do when
following rules.

Wed Feb 14 23:47:23 GMT 1996