442 | 40 TURING’S mENTOR, mAX NEwmAN
what a fluke!
I conclude with some reflections about this very unusual history. As a result of Penrose’s early
interest in mathematical logic, and the unusual mixture of mathematics, logic, and philosophy
in Vienna, Newman changed direction: had he stayed in Cambridge in 1922–23, he would surely
have continued on the path indicated by his 1923 paper about avoiding axioms of choice—
namely, Hardy–Littlewood mathematical analysis. Even then his interest in logic could have
waned and he might not have set up a foundations course to be taken by Turing. The existence
of this course in 1935 was a highly improbable event, in that Newman might easily not have
accompanied Penrose to Vienna, and might never have had an inclination to teach logic. After
Turing’s graduation in 1934 he was himself another budding Hardy–Littlewood mathematical
analyst, working on ‘almost periodic’ functions and with interests in mathematical statistics
and group theory. One can imagine Turing remaining happily engaged in these branches of
mathematics—particularly in analysis, with the influential Hardy back at Cambridge from
1931 after twelve years at Oxford. Turing would not have come across recursive functions or
undecidability, and would not have invented his universal machine.
Could Newman and Turing have come to these topics by another route—via some of the
Cambridge philosophers, perhaps? Frank Ramsey, who visited the philosopher Ludwig
Wittgenstein in Austria in 1923 and 1924, and died in 1930, was along with other Cambridge phi-
losophers largely concerned with revising the logicism of Russell and the early Wittgenstein.^38
Wittgenstein himself, back in Cambridge from 1929, was philosophically a monist, and so dis-
tinguished between what can be said and what can only be shown. By contrast, the Hilbertians
depended centrally on the concept of a ‘hierarchy’, mathematics and metamathematics, upon
which Turing seized. (The logician Rudolph Carnap, another member of the Vienna Circle,
coined the term ‘metalogic’ in 1931, on the basis of reflecting upon Gödel’s procedures in his
proof of the incompleteness of arithmetic, and Tarski was starting to speak of ‘metalanguage’
at about the same time.) Russell was in a strange position on this issue. Back in 1921, in his
introduction to Wittgenstein’s Tractatus, he had advocated a hierarchy of languages to replace
the Wittgensteinian showing–saying distinction—a move that Wittgenstein rejected—but he
never envisaged a companion hierarchy of logics, and so neither understood (nor even stated)
Gödel’s theorems properly.^39 It thus seems highly unlikely that either Turing’s contacts with
Russell or the contacts over logic and logicism that he had with Wittgenstein would have led
him to metamathematics and decision problems.
If neither Turing nor Newman had found their way to those crucial topics in logic and meta-
mathematics, then—while they might have been recognized as clever analysts who were also
good at chess—they would not have been such obvious choices for Bletchley Park. If they had
got there, they would probably not have been as effective as the actual Newman and Turing
were, because they would not have known much, if any, of the key mathematics. So a crucial
part of the British wartime decoding effort, especially Turing’s Enigma-breaking bombe, came
about as a consequence of Turing’s change of direction years earlier, which was inspired by the
happenstance of Newman’s course on the foundations of mathematics, offered because of his
own earlier change of direction towards this unusual topic, a change in turn brought about as
an unintended consequence of some decisions that Penrose took to develop his own career. In
short, what a fluke!^40