CHAPTER 36

Introducing Turing’s

mathematics

robin whitty and robin wilson

A

lan Turing’s mathematical interests were deep and wide-ranging. From the begin-

ning of his career in Cambridge he was involved with probability theory, algebra (the

theory of groups), mathematical logic, and number theory. Prime numbers and the

celebrated Riemann hypothesis continued to preoccupy him until the end of his life.

Turing, master of all trades

As a mathematician, and as a scientist generally, Turing was enthusiastically omnivorous. His

collected mathematical works comprise thirteen papers,^1 not all published during his lifetime,

as well as the preface from his Cambridge Fellowship dissertation; these cover group theory,

probability theory, number theory (analytic and elementary), and numerical analysis. This

broad swathe of work is the focus of this chapter. But Turing did much else that was mathe-

matical in nature, notably in the fields of logic, cryptanalysis, and biology, and that work is

described in more detail elsewhere in this book.

To be representative of Turing’s mathematical talents is a more realistic aim than to be ency-

clopaedic. Group theory and number theory were recurring preoccupations for Turing, even

during wartime; they are represented in this chapter by his work on the word problem and

the Riemann hypothesis, respectively. A third preoccupation was with methods of statistical

analysis: Turing’s work in this area was integral to his wartime contribution to signals intel-

ligence. I. J. Good, who worked with Turing at Bletchley Park, has provided an authoritative

account of this work,^2 updated in the Collected Works. By contrast, Turing’s proof of the central

limit theorem from probability theory, which earned him his Cambridge Fellowship,^3 is less

well known: he quickly discovered that the theorem had already been demonstrated, the work

was never published, and his interest in it was swiftly superseded by questions in mathematical

logic. Nevertheless, this was Turing’s first substantial investigation, the first demonstration of