There is nothing remarkable about mathematicians' achievements going unrecognised in a wider world changed by their discoveries.

The hidden history of Alan Turing is just a particularly bizarre example — one we can expect to become much better known during the 2012 centenary of his birth.

The mental checklist of things that make Turing remembered includes:

- Being just 24 years old when he came up with the idea of the "stored program" computer, basically the blueprint for every computer in existence today;
- His leading role at the secret decoding centre at Bletchley Park, helping shorten the Second World War by two years with his groundbreaking involvement in building and fully exploiting decoding machines;
- His seminal role in the actual designing and programming the early computers after the war, and his still important inﬂuence on how computer scientists see artiﬁcial intelligence;
- The innovative and original work in bringing mathematics to bear on important problems in biology and medicine
- And his disgraceful neglect, and prosecution for being gay, in 1950s Manchester.

But for many of us the peculiar resonance between the personal and the scientiﬁc makes Turing specially iconic. And the visionary form this interaction took gives Turing's writings a relevance and impact which continues to this day. There are many facets to Turing's life that echo through mathematics, computing, physics and biology, through philosophy, and through economics, the humanities and the creative arts.

**One of the true fathers of computing, Alan Turing also made many other advances that are only now becoming fully appreciated.** *Image credit: Wikipedia*

Why is that? Turing's life and science inhabited that mysterious region between the computable and the incomputable. Both in his research and in his life he persistently tried to make sense of things in what we can only describe as a computational sense.

His well-known eccentricities such as chaining his mug to the radiator at Bletchley Park or riding his bicycle in the summer wearing a gas mask, might be seen as the product of constructive thinking; and in his mathematics he was always wanting to bring the real world within the purview of computational mathematics.

But he was ever meeting and dealing in creative ways with the inevitable challenges to computability, sometimes succeeding, as with his universal computing machine; or his wartime cryptography; or his mathematical modelling of cows' spots; or zebras' stripes; or the moving patterns of tropical ﬁsh; or mapping out the limits of computation as with his unsolvable halting problem; or his hierarchy of theories and his oracle Turing machines in his insufﬁciently understood 1939 paper.

Of course, the interface between the computable and the incomputable is a hazardous area, as he found in the closing years of his life, bringing cruel uncertainties and a quite unpredictable end.

## 1. Computation Disembodied

The 17th century saw a dramatic change in the balance between computational and descriptive sway in science. Robert Hooke may have toyed with the inverse square law in physics, but it is Isaac Newton's mathematics which deliver not only persuasion but computational and predictive content to the intuitive descriptions.

The computational gives surety, gives ease of comparison between prediction and observation, and comes as a memetic package more easily passed between researcher and practitioner.

The Turing machine did for computational mathematics what Newton's computational mathematics did for his particle dynamics.

The Turing machine did for computational mathematics what Newton's computational mathematics did for his particle dynamics. The mathematics *disembodied* the science. It turned computation into *computer science*. Gone was the taxonomy of calculating machines built differently for different computational tasks. The hardware was trivial and did not need to be changed. The basic actions of the machine were as simple as could be. But the "machine" could compute anything a standard calculating machine could. While all the computing power lay in the *program*.

More generally, it enabled many to frame the familiar expectations of science encouraged by Newton — the so-called Laplacian model — within a precise *mathematical* model. Of course, the Newtonian model came with a "best before" date, one clear to the successors of the man who said (Albert Einstein, p. 54, *Out of My Later Years*, 1950):

*"When we say that we understand a group of natural phenomena, we mean that we have found a constructive theory which embraces them."*

Today, we take forward some of Turing's own questionings of the comprehensiveness of his disembodied computational model.

## 2. Universality, and Programs as Data

Of course, aspects of the 1936 Turing model were anticipated by others, such as Emil Post. The key extra ingredient was *universality*, based on the *coding* of machines as data. This essential feature of today's computer is often not understood — though was certainly recognised by John von Neumann, and implemented in his 1945 EDVAC report, which was so inﬂuential in the later development of the stored program computer.

Von Neumann later acknowledged Turing's role in his 1948 Hixon Symposium lecture. Although the practical impact of Turing's universal machine is difﬁcult to disentangle from the complexities of the early history of the computer, it established a hugely inﬂuential computing paradigm — that of the omnipotent computer.

It encouraged the development of the functionalist perspective on human cognition and artiﬁcial intelligence, as in Hilary Putnam's 'Minds and Machines' from 1960. The embodiment of human thinking is relegated to a subservient role, mirroring that of the Turing's universal machine.

Turing himself is said by Andrew Hodges to have spoken to Donald Bayley in 1944 of "building a brain". A more limited expression of the paradigm, in computing, is that of the *virtual machine* originally associated with IBM around 1965. The overriding concept is of varied computational environments being realisable independently of the particular hardware.

## 3. Programs as Data *Embodied*

Of course, a huge amount of work and ingenuity went into actually *building* universal machines, and Turing was very much part of this. The early programmable machines were...

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