A History of Mathematics From Mesopotamia to Modernity

AChaoticEnd? 245

John von Neumann, the other mathematician involved, a Hungarian refugee who turned Cold

Warrior, has his supporters, particularly in the US military and business establishment; and a bio-

graphy which is sympathetic to his role in both is available (Macrae 1992); naturally, he appears

less the outsider than Turing.

While all the other computer makers were generally heading in the same direction, von Neumann’s genius clarified

and developed the paths better than anyone else. (Shurkin, cited in Macrae 1992, p. 287)

Certainly large teams were involved on both sides of the Atlantic in the years following Second

World War; the introduction of fast electronic components was crucial in making it possible to

build machines which would perform the required tasks. John von Neumann was employed by the

US Army from 1937 (the year of ‘Computable Numbers’) to advise on problems in ballistics. These

were some of the difficult problems which he mentions in our second quote, but it was not until the

end of the war that he learned of machines under construction which could help. These machines

could, essentially, solve a single problem extremely fast; they belong to engineering history. They

resembled the Turing machine which computes just one number.

There is no evidence that von Neumann had read ‘Computable Numbers’, although he was in

Princeton and writing a letter of recommendation for Turing when it was published. In any case,

the step which he and Turing took (and then had to get engineers to implement) was equivalent

to replacing the one-task Turing machine by the universal one. While they were not abstract

mathematicians in the post-1930 mould, they were both influenced by the logical ‘revolution’ of

the early part of the century; in terms of this revolution,instructionslike ‘add’ or ‘move right’

had the same status as signs on the paper, tape, or whatever asnumbers. Hence we arrive at the idea

of the ‘programme’, a stream of instructions and numbers which areencodedas numbers.

Minor cycles fall into two classes:Standard numbersandorders...These two categories should be distinguished from
each other by their respective first units...i.e. by the value ofi 0. We agree accordingly thati 0 =0 is to designate
a standard number andi 0 =1 an order. (Von Neumann 1945, p. 45)
The engineering work of producing various machines for various jobs is replaced by the office work of ‘programming’
the universal machine to do these jobs. (Turing in Hodges p. 293)

The two quotations express the same idea, arrived at by Turing and von Neumann almost simul-
taneously in the year 1945. Hidden from us, it happens all the time inside our machines, and it is
probably second nature to us. To write (in old-fashioned programmers’ language):

X=1

FORI=1TO100

X=1+1/X

NEXT I

PRINT X

is to write a sequence of instructions to the machine. In von Neumann’s terms, ‘1’ and ‘100’ are

numbers; while all the other signs (‘=’, ‘for’, ‘next’) are instructions, when suitably read. Here,

with what Turing is already calling (if in quotes) ‘programming’, is the decisivemathematicalinput

into the computer. The question of invention, for mathematicians, goes no further (fortunately).

Development, under the relentless pressure of late capitalism’s understanding of how much time