# Alan Turing: The Enigma (1983) by Hodges

in the early days of computing, a number of terms for the practitioners of the field of computing were suggested in the Communications of the ACM — turingineer, turologist, flow-charts-man, applied meta-mathematician, and applied epistemologist.

In a man of his type, one never knows what his mental processes are going to do next.

There have been two big films about Turing (three if you count the uselessly fictionalised *Enigma* (2001)). All are dishonestly melodramatic to some degree; for instance they depict Turing’s relationship with his dead love Christopher as the driver of his work on machine intelligence. And more generally they depict him as tragic. But he wasn’t tragic: we were. In the 1950s we attacked a superlatively profound person, because we were certain it was the right thing to do.

Hodges, whose book began the great public rehabilitation of Turing and served as the source for the films, bears no blame for this: it’s one of the best biographies I’ve ever read (better even than Kanigel on Ramanujan and Issacson on Einstein). Hodges actually understands Turing’s work, not just its consequences, and not just the drama around it. And what work!

### Begin with his achievements:

- 1935:
*Mathematical statistics*:

An independent proof of the Central Limit Theorem. - 1935:
*Group theory*:

An extension to a theorem of von Neumann's. - 1936:
*Mathematical logic*:

One of the all-time great papers, an answer to Hilbert's halting problem and an elaboration of the incompleteness of all mathematics,*and*the formal statement of a single machine that can perform all computable work. - 1936:
*Computability theory*:

Same paper. Creator thereof. - 1936:
*Automata theory*:

Same paper. Creator thereof. - 1936:
*Computer engineering*:

Same paper. Inventor of the stored-program concept, used in all computers since 1950. - 1937:
*Group theory*:

Proof that general continuous groups cannot be approximated by finite groups. - 1938:
*Mathematical logic*:

Invention of ordinal logics, an attempt to handle incompleteness. - 1938:
*Analytic number theory*:

Algorithm ("Turing's method") for calculating values of the zeta-function. - 1938:
*Computer engineering and Mathematical methodology*:

Design of an analogue machine to approximate the zeroes of the zeta function. - 1939:
*Cryptanalysis*:

developed most of the logical methods used against Nazi Germany's naval cipher, Enigma. Including a new sort of indirect frequency analysis, "simultaneous scanning", search trees, an independent invention of Shannon's information entropy (as "Weight of evidence")... - 1940:
*Mechanical engineering*:

redesigned the Polish Bomba to handle the exponential explosion in the Enigma's state space. - 1941:
*Statistics*:

independent invention of sequential analysis, for "Banburismus". - 1940:
*Bayesian inference*:

independent reinvention of Bayes factors and the first approximation of what we would now call empirical Bayes estimation. IJ Good quite rightly calls Bayes factors, "Bayes-Turing factors". (Though it should be Laplace-Turing factors.) - 1942:
*Cryptanalysis*:

A hand-method for cracking the Lorenz cipher, "Turingery". - 1944:
*Crytography, audio engineering and electrical engineering*:

Design, proof and much of the construction of "Delilah", an electronic speech encipherment device. - 1945-6:
*Algorithmics*:

The discovery of the stack. A neglected but vast accomplishment. (Zuse had already implemented subroutines by then.) - 1945-6:
*Computer engineering*:

Design of the Automatic Computing Engine, the first complete design of a stored-program computer, including circuit diagrams, instruction set and cost estimate. (von Neumann's is incomplete.) - 1948:
*Computer music*:

The first computer music. Turing's handbook for the Mark I had a section on using it to produce notes, and they gave a demo for radio in 1951, also a first. Not really a synth (not real-time) and not real electronic music (produced by moving parts). - 1948:
*Linear algebra*:

Better ways of solving linear systems and inverting matrices. - 1949:
*Group theory*:

Proof that the 'word problem' is insoluable for cancellation semigroups. Computability mainstream in mathematics by then. - 1949:
*Formal verification*:

Paper on proving that computer programs will behave. - 1950:
*Philosophy of mind and artificial intelligence*:

His famous one, "Computing machinery and intelligence" is one of the top 100 set texts in philosophy, but*Computable Numbers*is deeper, outlining how computability places limits on what the brain can do, and how difficult it will be to redo. He sees machine learning coming very clearly. - 1951:
*Group theory*:

Another big result in the word problem for groups. (unpublished) - 1951:
*Chess engine*:

Published the first algorithm to play a full game of chess automatically. - 1952:
*Mathematical biology*:

a profound chemical theory of how life grows, now a textbook model of morphogenesis. - 1952:
*Number theory*:

Numerical evidence (computed on the Manchester Baby) for thousands of values of the zeta-function. - 1952:
*Pattern formation*:

Construction of the "Swift-Hohenberg" equation, 23 years before them.

But even more than that: Copeland guesses that breaking U-boat Enigma saved 14 million lives, a large fraction of which we can lay at Turing’s feet. If this is even roughly right this puts him in the top 50 life-savers ever.

But, outside of logic and engineering, where he was among the few most sophisticated people in the world, he was famously unsophisticated:

As at school, trivial examples of ‘eccentricity’ circulated in Bletchley circles. Near the beginning of June he would suffer from hay fever, which blinded him as he cycled to work, so he would use a gas mask to keep the pollen out, regardless of how he looked. The bicycle itself was unique, since it required the counting of revolutions until a certain bent spoke touched a certain link (rather like a cipher machine), when action would have to be taken to prevent the chain coming off. Alan had been delighted at having, as it were, deciphered the fault in the mechanism, which meant that he saved himself weeks of waiting for repairs, at a time when the bicycle had again become what it was when invented – the means of freedom. It also meant that no one else could ride it.

He made a more explicit defence of his tea-mug (again irreplaceable, in wartime conditions) by attaching it with a combination lock to a Hut 8 radiator pipe. But it was picked, to tease him.

Trousers held up by string, pyjama jacket under his sports coat – the stories, whether true or not, went the rounds. And now that he was in a position of authority, the nervousness of his manner was more open to comment. There was his voice, liable to stall in mid-sentence with a tense, high-pitched ‘Ah-ah-ah-ah-ah’ while he fished, his brain almost visibly labouring away, for the right expression, meanwhile preventing interruption. The word, when it came, might be an unexpected one, a homely analogy, slang expression, pun or wild scheme or rude suggestion accompanied with his machine-like laugh; bold but not with the coarseness of one who had seen it all and been disillusioned, but with the sharpness of one seeing it through strangely fresh eyes. ‘Schoolboyish’ was the only word they had for it. Once a personnel form came round the Huts, and some joker filled in for him, ‘Turing A.M. Age 21’, but others, including Joan, said it should be ‘Age 16’...

It was demeaning, but the repetition of superficial anecdotes about his usually quite sensible solutions to life’s small challenges served the useful purpose of deflecting attention away from the more dangerous and difficult questions about what an Alan Turing might think about the world in which he lived. English ‘eccentricity’ served as a safety valve for those who doubted the general rules of society. More sensitive people at Bletchley were aware of layers of introspection and subtlety of manner that lay beneath the occasional funny stories. But perhaps he himself welcomed the chortling over his habits, which created a line of defence for himself, without a loss of integrity.

We have words for this now (“nerd”, “wonk”, “aspie”), and massive institutions, and even social movements, but at the time he had to make do with “don”, and hide inside academia. Again: the problem wasn’t him, it was us.

He gets called a mathematician most often, I suppose because people don’t want to be anachronistic. But scroll up: his most famous work is as a logician and a systems engineer, and the rest is statistics and algorithmics and cognitive science. He was falling between several chairs, until computer science caught up with him:

a pure mathematician worked in a symbolic world and not with things. The machine seemed to be a contradiction... For Alan Turing personally, the machine was a symptom of something that could not be answered by mathematics alone. He was working within the central problems of classical number theory, and making a contribution to it, but this was not enough. The Turing machine, and the ordinal logics, formalising the workings of the mind; Wittgenstein’s enquiries; the electric multiplier and now this concatenation of gear wheels – they all spoke of making some connection between the abstract and the physical. It was not science, not ‘applied mathematics’, but a sort of applied logic, something that had no name.

The philosopher-engineer. One of several moments in Hodge’s book that left me dumbstruck is Turing arguing with Wittgenstein about the foundations of mathematics. (In the spring of 1939 they were both teaching courses at Cambridge called that!) Bit awkward, and in my view Alan goes easy on Ludwig. But you still couldn’t make it up.

The government employed Turing for 9 years, paying him about £6000 over the duration (£300k in today’s money). In that time he produced 3 gigantically advanced systems (most of the Hut 8 system, the Delilah and the ACE design), about 10 or 20 years ahead of their time. Hodges sees this as a triumph of managerial socialism. Now, breaking naval enigma for £300k is an unbelievable deal (the savings from undestroyed shipping and cargo alone would be in the billions, let alone the loss of life, let alone the decisive tactical advantage). But the government suppressed Delilah and totally screwed up the ACE project. So I’m not sure if we can cheer too much. Keynes says somewhere that

The important thing for Government is not to do things which individuals are doing already, and to do them a little better or a little worse; but to do those things which at present are not done at all.

This is true of Bletchley. But instructive failures are only helpful if they occur in public. (As at least the ACE report was.)

The most annoying part of the films making up emotionally powerful unifying themes for Turing is that they are already there. But to grasp them, you’d have to actually display what was most wonderful and important about him, his technical work, and there goes the box office.

In an end-of-term sing-song [at Sherborne, when Turing was 12], the following couplet described him:

Turing’s fond of the football field... another verse had him ‘watching the daisies grow’ during hockey... although intended as a joke against his dreamy passivity, there might have been a truth in the observation.

For geometric problems the touch-lines yield

[20 years later] ...One day he and Joan were lying on the Bletchley lawn looking at the daisies... Alan produced a fir cone from his pocket, on which the Fibonacci numbers could be traced rather clearly, but the same idea could also be taken to apply to the florets of the daisy flower.

[30 years later] ...he was trying out on the computer the solution of the very difficult differential equations that arose when [one] followed the chemical theory of [plant] morphogenesis beyond the moment of budding...

...he also developed a purely descriptive theory of leaf-arrangement... using matrices to represent the winding of spirals of leaves or seeds round a stem or flower-head... The intention was that ultimately these two approaches would join up when he found a system of equations that would generate the Fibonacci patterns expressed by his matrices.

...Such observations reflected an insight gained from... [a program called] ‘Outline of Development of the Daisy’. He had quite literally been ‘watching the daisies grow’... on his universal machine.