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Alan Turing’s ‘On Computable Numbers’


Alan Turing is best known for helping to crack the Nazi’s Enigma cipher machine at Bletchley Park during World War II. Turing’s work helped the Allies to shorten the conflict by at least two years. However, Turing was more than an incredible codebreaker. He was a brilliant mathematician, skilled logician and the forefather of modern computing. 

His 1936 study ‘On Computable Numbers’ proposed a machine capable of adapting to the instructions it was given. Furthermore, his 1950 study changed the way we think about machinery and intelligence.

‘On Computable Numbers’

Turing was allowed to stay on at Cambridge for an extra year, taking an advanced course in the foundations of mathematics, for which the lecturer was Max Newman. Newman was describing a problem which had the mathematicians of the early 1930s scratching their heads. The question was whether there is a mathematical litmus test to apply to mathematical theorems: in other words, can you know ahead of time if it’s worth spending the time sweating over a proof. This unsolved question was called the Entscheidungsproblem.

Newman explained that the issue was whether there was a mechanical process – ‘I may even have said, a machine can do it’ – to determine whether theorems can be proved. He hadn’t expected to be taken literally. But Turing, a mere student, decided to do just that. In his most famous mathematical paper ‘On Computable Numbers, with an Application to the Entscheidungsproblem’, Turing set out the idea of a multi-purpose machine, whose function would be changed according to the instructions it was given. This concept of a programmable computer became known as the ‘Turing machine’. And along the way, Turing demolished the Entscheidungsproblem – he proved there was no litmus test for mathematical theorems.

‘Computing Machinery and Intelligence’

Turing’s 1950 paper ‘Computing Machinery and Intelligence’ sets out a simple test for deciding whether a machine can think: you hide it behind a screen and ask it questions. If you think it’s a human, it’s probably thinking. The ‘Turing Test’ is now firmly established (and still criticised) in the study of artificial intelligence, and annual competitions are run to see if a computer program can beat the judges. Oddly, developments in technology now require internet shoppers and others to identify themselves as humans by deciphering wonky text (called a CAPTCHA, or Completely Automated Public Turing test to tell Computers and Humans Apart) – a task that is beyond the capabilities of computers; or, to put it another way, the computer is applying a reverse Turing Test to see if the user is a human or another program.

In the 1950s computer programs were still primarily about calculation. The idea of a machine ‘thinking’ was thus controversial, but it was also great fun: Turing, Max Newman and Sir Geoffrey were invited to have a debate about it on the radio; and even today there are debates about the effectiveness of machine learning and what machines can do, what humans can do, and where the dividing lines might be.

By Dermot Turing

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