Turing

Alan Mathison Turing (1912-1954)

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 new-found 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 anti-war movement. He studied the Soviet system but concluded there was a totalitarian aspect to it and became identified with the middle-class 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:

1. Was mathematics complete, in the sense that every statement could be proved or disproved,
2. 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
3. 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 self-referential 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 mechanical 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 zeta-function (which he thought he might be able to prove false by finding a zero with the wrong value). The Riemann zeta-function 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 zeta-function.

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 Laboratory 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, been 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 half-eaten apple containing potassium cyanide. Whereas his mother maintained death was accidental the court decided that the poison was self-administered.

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

1. Hodges, Andrew, Alan Turing, the Enigma, Vintage Edition, Random House, London, 1992 (first published Burnett 1983).

This T Shirt, which celebrates Turing, is available from the AMT.

Peter Taylor

Australian Mathematics Trust

Canberra

14 May 2002