models as suggesting blueprints for designing physical devices, then an analog
machine (such as planned by Turing
in 1939) would come much closer than
anything built along the lines of a UTM
to embodying specific, “fundamental
processes” associated with a particular
number theoretic problem, in the sense
suggested in his Ph.D. thesis.
When Turing in 1950 returned to
the task of calculating zeros of ζ(s) the
sea changes that had revolutionized the
world of automatic computation had
rendered all those pre-war considerations obsolete. The natural approach
to follow for Turing was now to write a
specific program to run in a stored-program, general-purpose machine. But the
Mark I, like all other similar machines at
the time, was not only stored-program.
It was also electronic, large-scale, high-speed, general purpose, and digital. In
1939, all these crucial components of the
machines that started to be built in the
late 1940s were far beyond the horizon.
This article is a somewhat belated elaboration of a talk I gave at the “Turing in
Context II” conference in Brussels, October 10–12, 2012 (
http://www.dijkstras-cry.com/Corry). I want to thank Edgar
Daylight for transcribing the talk and
for his continued encouragement to
publish my ideas on this topic. Thanks,
too, to three anonymous referees for
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Leo Corry ( firstname.lastname@example.org) is the Bert and Barbara
Cohn Professor of History and Philosophy of Science and
Dean of the Lester and Sally Entin Faculty of Humanities at
© 2017 ACM 0001-0782/17/08 $15.00
different sets of proofs, and by choosing a suitable machine one can approximate ‘truth’ by ‘provability’ better
than with a less suitable machine, and
can in a sense approximate it as well as
you please. The choice of a machine involves intuition,... or as [an] alternative
one may go straight for the proof and
this again requires intuition.” 6
The fact that still, in 1940, when the
classical debates on the foundations
of arithmetic had almost totally faded
away, Newman and Turing continued
their exchanges on such matters, is
worthy of attention in itself. But no less
interesting is the subtle twist Turing
introduced into this discussion when
he mentioned the possibility of having
various kinds of machines according to
different kinds of intuitions that are relevant to different mathematical situations. The U TM was not for Turing “
universal” in this important sense.
It seems that now—in those few opportunities when he could think about
the foundations of mathematics and
about questions of “truth” or “
provability”— Turing also incorporated new
directions (such as he had explored in
his Ph.D. dissertation). This included,
no doubt, the oracle, but also, so it
seems, alternatives to the basic “
machine” he had defined in 1936.
I conclude with a final, somewhat conjectural suggestion. By its very nature,
Turing’s oracle could not be a standard
Turing machine. “Solving a given number theoretic problem” is one of “its fundamental processes.” And the Riemann
Hypothesis is one such problem. Now,
it seems to me, Turing’s construction
of his analog machine and the variety of
machines he mentioned in his response
to Newman shed interesting light, retrospectively, on that passing, somewhat
unclear comment Turing advanced in
his thesis. From the letter we learn that
each mathematical situation calls for
the choice of a suitable machine and
these choices rely on the right intuition
to do so in each case. The U TM had been
a highly successful, specific choice for
dealing with the Entscheidungsproblem, but that would not mean—even in
principle—it would provide a model for
a physical universal machine, suitable
for all mathematical tasks. If we may
somehow think of these mathematical
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