Alan Mathison Turing (19121954)
Alan Mathison Turing was born on 23 June 1912 in a nursing home in Paddington,
London. He was the second of two boys. His grandfather, John Robert
Turing had taken a mathematics degree in 1848 at Trinity College,
Cambridge, being placed 11th. However he gave up mathematics to take up the ministry.
He fathered ten children. The second son, Julius Mathison Turing, born
in 1873, was reputedly not as able as his father in mathematics,
and studied literature and history at Corpus Christi College, Oxford,
graduating in 1894. He became a member of the Indian Civil Service.
Julius Turing was to meet his future wife and Alan Turing's mother, Ethel
Stoney, on a voyage home in 1907. Ethel Stoney was descended from a family
originally from Yorkshire, but which had become Protestant landowners in
Ireland, and she had been born in Madras, where her father had gone as an engineer
in 1881. She had a distant relative who was the Irish
scientist responsible for coining the word "electron".
Both Alan Turing's parents had come from families which were resourceful,
although having funds.
Alan Turing was conceived in India but born in London and never actually
went to India. His parents decided he should not be exposed to the heat
and other risks of Madras and left him and his older brother in the care
of another family (later, when Turing was at school, his father did
prematurely retire and his parents returned to Britain).
In 1926 Turing sat the Entrance Exam for Sherbourne College and was accepted.
This was one of the original English Public Schools, based in Dorset. At
this school Turing showed a talent in mathematics and interest in some
of the sciences but little interest in the humanities.
Whereas his first choice had been to study at Trinity College,
Cambridge, Turing was unsuccessful in gaining a scholarship there. He
was however successful in his second choice, King's College, which he
entered in 1931, reading for the Tripos degree in mathematics. He enjoyed
his newfound freedom here, with a room to himself and the academic
climate enriched at Cambridge with the presence of GH Hardy (who returned
from Oxford in the same year to take up the Sadleirian Chair), Eddington,
Dirac and the eminent topologist MHA Newman. During this period
Turing remained a Pure Mathematician, although he dabbled with interests
in quantum physics. He was particularly interested in theoretical studies
which had interface with the physical world.
An example of this was his proof of the Central Limit Theorem in 1933. Turing
was inspired to investigate this by a course given by Eddington, where he noticed
that irrespective of the distribution of a set of n sample points,
the mean of these points always had the same mean as that of the original
sample, had a variance equal to the variance of the original distribution
but divided by n (making it smaller for large numbers of data points, as one
would expect) but furthermore was normally distributed. After
proving it, Turing was told that this very difficult and famous theorem had been
proved in 1922 by Lindeberg. However it seems that his original
work was acknowledged as such.
While at Cambridge Turing flirted with the antiwar movement. He
studied the Soviet system but concluded there was a totalitarian aspect
to it and became identified with the middleclass progressive
opinion espoused by such journals as the New Statesman. Turing
also thought deeply about religion and agnosticism.
In 1934 Turing wrote the Tripos examination and passed with Distinction.
He and eight others were made B* Wranglers. Turing was not particularly interested
in the exact status, regarding this as resulting only from an exam.
He was interested in further things and had been inspired by a course given
by Newman on the Foundation of mathematics. This course had largely been
given in the spirit of Hilbert's attempts to define mathematics. In 1900
at the International Congress of Mathematicians Hilbert had posed a number
of unsolved problems. Hilbert had continued to consolidate his position
as one of the world's leading mathematicians as Professor at Goettingen. Germany
had been forbiddden representation at the 1924 Congress but Hilbert did attend in 1928
and posed three fundamental questions:
 Was mathematics complete, in the sense that every statement
could be proved or disproved,
 Was mathematics consistent in the sense that no apparently
incorrect statement such as "1+1=3", could be arrived at by a sequence of valid steps
of proof, and
 Was mathematics decidable, in that did there exist a definite
algorithm which could, in principle, be applied to any assertion and which
was guaranteed to produce a correct decision as to whether that assertion
was true?
It is believed that Hilbert expected the answer
to each question to be "yes". As it happened the Czech logician Kurt Gödel
answered the first two questions, answering the first in the negative by
producing selfreferential statements such as "This statement is unprovable".
He did answer the second question in the affirmative. To answer the
third question in the negative one would need to find an assertion
for which no algorithm existed. Indeed, the definition
of algorithm itself needed to be carefully addressed.
Turing was appointed to a Fellowship at Cambridge after his graduation,
granting him considerable scope to continue research and in addition
to research in group theory, pondered Hilbert's third question. He found the
answer to be in the negative by devising a concept called a
computable number and mecanical processes (later
to be called Turing Machines).
A Turing machine (as it later became known) was an abstract mechanical process
which changed from state to state by a finite set of rules depending
on single symbols read from a tape (rather like the program on
a modern computer). A Turing machine could write a symbol
on to the tape or delete a symbol from it.
He defined a computable number as a real number whose decimal expansion
could be produced by a Turing machine starting with a blank tape.
He was able to demonstrate that the irational number pi was computable.
However he was able to produce a diagonalisation argument not
unlike that of Cantor when proving the uncountability of the real numbers
to show that there was an uncountable number of numbers which are not
computable. He was also able to describe such a number.
This result was also independently produced by the Princeton mathematician
Alonzo Church, but Turing's method, involving the idea of the Turing Machine
(a term later coined by Church) provided an avenue for more significant later
development.
After some lectures given at Cambridge by Princeton's John von Neumann
and discussion with Newman, Turing moved himself to Princeton in 1936 on a Fellowship
which enabled him to collaborate with Church and others. During his time
in Princeton Turing helped develop some machines to multiply numbers, and to compute
zeros of the Riemann zetafunction (which he thought he might be
able to prove false by finding a zero with the wrong value). The Riemann
zetafunction problem, which can establish the way in which
primes "thin out" for large numbers, was posed as one of
the important unsolved problems in mathematics by Hilbert in 1900.
It remains, in 2002, as the Everest of unsolved mathematics problems.
Turing also started to think of how to develop
high quality cipher which was easy to encrypt.
During 1937 Turing had his Fellowship renewed at Princeton for a second year.
His paper Computable Numbers was published while he was at Princeton and
he also completed a PhD further exploring Gödel's theories.
In 1938 he was offered an attractive position at Princeton by von Neumann
and also had his Fellowship at King's renewed. He decided he wished to return to King's,
and commenced lecturing there and working on further developments with
calculating zeros of the Riemann zetafunction.
After the outbreak of war Turing's life changed
from that of a University Pure Mathematician. He was recruited by HM Government's
Code and Cypher School, which had become concerned at its inability
to crack the German Enigma Code. Turing commenced a new phase of his career,
unable to publish or talk about new results.
Turing arrived at Bletchley Park as one of a team of mathematicians charged
with decoding the German Enigma messages. This was an immense mathematical
challenge, not only mastering the principles used by the Germans, but also
dealing with variations between the services and changes from time to time. It
is history that Turing not only met these challenges, but was in effect
the leader of the powerful group. As part of the process Turing and
associates produced a machine known as a Bombe, based on Polish ideas,
which was able to decode Enigma. From late 1940 these Bombes were able
to decode German Luftwaffe messages. The German Naval Enigma codes produced
more of a challenge, but they were able to decode them by mid 1941.
Of young mathematicians who later became more famous the mathematicians
at Bletchley Park included William Tutte (who later moved to Canada) and
Peter Hilton, who later worked in the US. Another noted
resident was Chess Master Harry Golombek, who was able to defeat Turing at
chess, turn the board around, and recover from the resigned position.
Later in the Bletchley Park operations
MHA Newman arrived himself to add his own expertise.
Turing rose to the point where he travelled to America to advise
on British progress and to exchange ideas. He had personal clearance
from Churchill and the White House and worked also for a few months at
the Bell Labs in New York. He also advised the Allies on safer encryption
of their own messages.
Later in the war the Germans improved their code and whereas Turing
was not involved in the later deciphering, his ideas were still used.
By late 1943 Bletchley had got on top of their work and Turing became
to some extent redundant, allowing him to transfer to nearby Hanslope
Park, where, under the secret service, he was able to commence a speech
encipherment project. This project, code named Delilah, after the "betrayer of men",
involved Turing and Don Bayley, the construction of a machine and the use of
a surprising amount of undergraduate mathematics, particularly
in the fields of Fourier and Numerical Analysis. They succeeded in
getting the machine to work by the end of the war, but it had made no impact
on the war effort and in fact had a little too much crackle
to have a future. It seemed other organisations had also made other plans
to deal with tasks which Delilah could potentially solve.
At the end of the war Turing's King's fellowship had been extended and he
had the prospect of an eventual Lectureship there. However the War had more
than interrupted his career. He now wanted to construct a "brain" and set
about thinking of how this could be done. The brain was to draw on the abstract
Turing machines and his experience built up with decryption and machines as
acquired during the war, and was to become the modern computer. In
his remaining time at Hanslope Park his ideas were developed.
In 1945 he was recruited to the National Physical Laboatory to
provide the mathematical direction to development of a computing machine
known as ACE (Automatic Computing Engine). Whereas there were other computing
projects in the US (involving von Neumann) and there had been others, including
those with which Turing had been involved for special purposes, this project
planned the nearest thing to a true computer able to perform programmed
tasks. Other mathematicians working at the NPL at the time included Leslie Fox and
JH Wilkinson, both Numerical Analysts. The ACE program progressed rapidly
in its early stages, and during Turing's
stay had rather complete plans, but the production
of such a machine seemed a long way off in 1947, when Turing, in some frustration,
returned to King's College to resume his Fellowship.
In the meantime, Newman had been appointed after the war to a Chair in
Pure Mathematics at Manchester, a promotion from his Lectureship at Cambridge.
He had developed an interest in computing machines through his time at Bletchley.
Newman also had funding from the Royal Society
to develop computing equipment. Also, a key electronics
expert, FC Williams, had been appointed to a Chair in Engineering at Manchester.
Newman offered Turing a Readership in Pure Mathematics at Manchester
to enable him to supervise the mathematical work on a computer
being developed by Williams and he took up this appointment, resigning
formally from the NPL in 1948. Williams actually got the first electronic
computer working before Turing arrived, but Turing kept busy writing all the
mathematical subroutines for this machine, which was eventually devloped
for manufacture by Ferranti.
Whereas the Manchester computer was the first, it should be acknowledged
that Wilkes had also developed a computer at Cambridge. This became the first
one really available for serious mathematical work. It should also be noted that the
NPL's ACE eventually was completed, consistent with
one of Turing's plans, in 1950, with
Wilkinson still involved. And of course computers were being developed in the US
also.
In 1950 Turing published in Mind the highly significant
Computing Machinery and Intelligence. In this paper he posed a test
for whether a machine had acquired intelligence, a test still used 52 years later.
In 1951 Turing was elected a Fellow of the Royal Society.
In 1952 Turing was arrested and convicted for homosexual activity. This
had a major effect on him. He had still, unknown to his colleagues at
Manchester, being doing secret work for the government. As a result
of his conviction his security cover was stopped and police started investigating
his activities and his associates around the world.
Turing's last two or three years at Manchester were spent on developing
his ideas to biological forms, in particular a field called morphogenesis,
the development of pattern and form in living organisms. He also developed
ideas on quantum mechanics, particle theory and relativity.
Turing was found dead on 7 June 1954 during a period in which he was conducting
electrolysis experiments. Beside him was a halfeaten apple
containing potassium cyanide. Whereas his mother maintained death was accidental
the court decided that the poison was selfadministered.
A note should be made about Turing's athletic fitness. He was a dedicated
cyclist, always commuting around cities by bicycle rather
than cars. In his earlier years he
was also a competitive rower. In his later career he took up
athletics seriously, showing proficiency as a distance
athlete. He almost made the British Olympic team in
the marathon, having finished fifth in
the 1947 AAA Marathon. He was also successful at distances from 800 metres and
up and was a stalwart of the Walton Athletic Club.
Certainly, despite his short life, Turing achieved much, and as much
as anyone in the twentieth century, inspired the development of
the modern computer.
References
 Hodges, Andrew, Alan Turing, the Enigma, Vintage Edition, Random House,
London, 1992 (first published Burnett 1983).
Turing is the subject of the Australian Mathematics
Trust T Shirt in 2002.
This is available for sale through
AMT Publishing.
Peter Taylor
Australian Mathematics Trust
Canberra
14 May 2002
