54 | 6 TURING’S GREAT INVENTION
short-term memory; for example, the moving scanner can ‘remember’ that the square it is
vacating contains the symbol ‘1’ (say), simply by setting the dial to position ‘1’ as it leaves the
square. In Turing-machine jargon, the dial’s current position is known as the current state.
Turing showed that the universal machine can carry out highly complex tasks by chaining
together large numbers of these five elementary actions: erase, print, move left, move right, and
change state. It is a remarkable fact that, despite the spartan austerity of Turing’s machines,
they are able to compute everything that any computer on the market today can. Indeed, by
reading and obeying instructions that the programmer stores on the memory-tape, the univer-
sal Turing machine—Turing argued—can perform every possible computation. This is called
‘Turing’s thesis’ (see Chapter 41).^18
In the universal machine, as in every modern computer, programs and data take the same
form: numbers stored in memory. Turing showed how to construct a programming code in
which all possible sequences of instructions—that is, all possible Turing-machine programs—
can be represented by means of numbers, and so ultimately by groups of zeros and ones. This
idea of storing programs of coded instructions in the computer’s memory was simple, yet pro-
found, and made the modern computer age possible.
In Turing’s 1936 article ‘On computable numbers’, often regarded as marking the birth of
theoretical computer science, Turing had brought machinery into discussions of the founda-
tions of mathematics. That innovation was one of the features that made his approach to math-
ematical foundations so novel—even daring. In a biographical memoir prepared for London’s
Royal Society shortly after Turing’s death, Max Newman wrote:^19
It is difficult to-day to realize how bold an innovation it was to introduce talk about paper tapes
and patterns punched in them, into discussions of the foundations of mathematics.
Ironically, however, today’s computer science textbooks usually present the universal Turing
machine as a purely mathematical entity, an abstract mathematical idea. Purely mathemati-
cal notions, such as sets of symbols, and functions from state–symbol pairs, replace Turing’s
scanner, tape, and punch-holes. (For example, according to one influential modern textbook,
a Turing machine is defined to be an ordered quadruple <K, s, ∑, ∂>, where K is a finite set of
states, s is a member of K (called the initial state), ∑ is a set of symbols, and ∂ is a transition func-
tion from K × ∑, the values of which are state–symbol pairs, including the special symbols L and
R, replacing the idea of left and right motion.^20 ) So Turing’s bold innovation has been purified
and rendered into the conventional coin of mathematics. Turing machines are no longer objects
located in time and space, and subject to cause and effect. The paper tape, and the punched
patterns that cause the machine to act in certain ways, are gone.
A ‘Turing-machine realist’ such as myself rejects this modern view of the Turing machine.
Turing-machine realists regard the mathematics of the preceding paragraph merely as a use-
ful formal representation of a Turing machine. But, just as a mathematical representation of
digestion should not be confused with the process of digestion itself, so too the mathematical
representation of a Turing machine must not be confused with the thing that is represented—
namely, an idealized physical machine.^21
It seems transparently clear that Turing himself was a Turing-machine realist. There is a posi-
tively industrial flavour to his account of his machines, with its references not only to punched
paper tape, but even to ‘wheels’ and ‘levers’ within the scanner.^22 Yet the realism of Turing’s own
account is lost in the modern purified version.