394 | 36 INTRODUCING TURING’S mATHEmATICS
that different types of customer have different spending patterns; there is a technical condi-
tion, now called the ‘Lindeberg condition’, that the measurements must obey. Turing derived a
stronger condition than Lindeberg’s, so his theorem is less general, but it was still better than
anything proved before the 1920s. It was a remarkable achievement, showing deep insight and
accomplished in isolation and at great speed. These are precisely the hallmarks of Turing’s land-
mark discoveries in logic just a year later.
Group theory and the word problem
Unlike the modern theory of probability, the theory of groups was mainstream fare for
British mathematicians of the 1930s. In 1897 William Burnside, like Turing a Cambridge
graduate who became a Cambridge Fellow and a fine sportsman, published the first English-
language book on the subject, The Theory of Groups of Finite Order (the second edition of
1911 became a standard reference work in the subject for many years and remains in print to
this day).^5 For reasons that are not clear, Burnside moved in 1885 to a position at the Royal
Naval College at Greenwich, from where he refused to be tempted back to Cambridge, even
to be Master of Pembroke College. But he was a central figure in British mathematics: a
President of the London Mathematical Society, a Fellow of the Royal Society, and a recipient
of its Royal Medal.
Despite Burnside’s influence, group theory did not immediately take Britain by storm. In his
presidential address to the London Mathematical Society in 1908, Burnside complained:
It is undoubtedly the fact that the theory of groups of finite order has failed, so far, to arouse
the interest of any but a very small number of English mathematicians; and this want of inter-
est in England, as compared with the amount of attention devoted to the subject both on the
Continent and in America, appears to me very remarkable.
But his text on the theory of groups had a major influence on a near contemporary of Turing’s
at King’s College. Philip Hall graduated in 1925, nine years before Turing, and like Turing he
proceeded to a Fellowship, although in Hall’s case his dissertation, in group theory, was to set
the mould for a lifetime of distinguished specialization. His Fellowship was renewed in 1933,
the year before Turing was elected, on the strength of a paper that is now considered a milestone
in the history of group theory.
So by the 1930s group theory was making an impact in Cambridge, and at King’s College in
particular. Indeed, the Turing Archive for the History of Computing includes a correspondence
between Turing and Hall (in April 1935) concerning Turing’s very first published paper:
I enclose a small-scale discovery of mine. I should be very grateful if you could advise me how to
get it published. Perhaps you referee group theory for the LMS yourself.
‘Small-scale’ it may have been, but Turing’s first paper, ‘Equivalence of left and right almost
periodicity’ was, as J. L. Britton has remarked,^6 ‘surely a promising beginning to have noticed
something that Von Neumann, already enormously successful, had missed’.
‘I am thinking’, Turing continued, ‘of doing this sort of thing’—and there follow, perhaps
tellingly, the words ‘more or less seriously’ erased. The correspondence continues amicably,
because in May 1935 Turing wrote to Hall thanking him for a dinner invitation. A sporadic
interchange continued after Turing’s move to Princeton, with only one archive letter including