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.
- wiki


In a man of his type, one never knows what his mental processes are going to do next.
- JAK Ferns, Turing's coroner



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 superlative 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 Isaacson 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. (Schmidhuber thinks that Goedel (1931) deserves the credit for the first universal coding. I see the implicit sense in this, but it seems a stretch.)

  • 1936: Computability theory:
    Same paper. Creator thereof.

  • 1936: Automata theory:
    Same paper. Creator thereof.

  • 1936: Computer engineering:
    Same paper. Inventor of (several variants of) the stored-program concept, the basis of 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: Cryptography, audio engineering and electrical engineering:
    Design, proof and much of the construction of "Delilah", a highly portable electronic speech encipherment device. This was never deployed and remained classified for decades. As such, we know that it was at least 10 years ahead of its time.

  • 1945-6: Algorithmics:
    The discovery of the stack. A neglected accomplishment. (Also subroutines, but Zuse had already implemented those.)

  • 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 the deeper contribution, outlining how computability limits 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
For geometric problems the touch-lines yield
... 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.


[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.



Highlights

Hofstadter:

"Is a mind a complicated kind of abstract pattern that develops in an underlying physical substrate, such as a vast network of nerve cells? If so... could something else be substituted for the tiny nerve cells, such as millions of small computational units made of arrays of transistors, giving rise to an artificial neural network with a conscious mind?... In short, can thinking and feeling emerge from patterns of activity in different sorts of substrate − organic, electronic, or otherwise?

...Could a language-using machine give the appearance of understanding sentences and coming up with ideas while in truth being as devoid of thought and as empty inside as a nineteenth-century adding machine or a twentieth-century word processor? ...Are understanding and reasoning incompatible with a materialistic, mechanistic view of living beings?

Could a machine ever be said to have made its own decisions? Could a machine have beliefs? Could a machine believe it made its own decisions? Could a machine erroneously attribute free will to itself?... Could creativity emerge from a set of fixed rules? Are we − even the most creative among us − but passive slaves to the laws of physics that govern our neurons?

...Could a machine be frustrated and suffer? Could a frustrated machine release its pent-up feelings by going outdoors and self-propelling ten miles? Could a machine learn to enjoy the sweet pain of marathon running? Could a machine with a seeming zest for life destroy itself purposefully one day, planning the entire episode so as to fool its mother machine into “thinking” that it had perished by accident?


- These are the sorts of questions that burned in the brain of Alan Mathison Turing, the great British mathematician who spearheaded the science of computation; yet if they are read at another level, these questions also reveal highlights of Turing’s troubled life.




Hodges:
...the sheer timelessness of pure mathematics transcends the limitations of his twentieth-century span. When Turing returned to the prime numbers in 1950, they were unchanged from when he left them in 1939, wars and superpowers notwithstanding. As GH Hardy famously said, they are so. This is mathematical culture, and such was his life, presenting a real difficulty to minds set in literary, artistic or political templates.

...It was difficult enough being a mathematician, this being the frightening subject of which even educated people knew nothing, not even what it was, and of which they might proudly boast ignorance.

Puzzled since childhood by the ‘obvious duties’, he was doubly detached from the imitation game of social life, as pure scientist and as homosexual. Manners, committees, examinations, interrogations, German codes and fixed moral codes – they all threatened his freedom. Some he would accept, some actually enjoy obeying, others reject, but in any case he was peculiarly conscious, self-conscious, of things that other people accepted ‘without thinking’; It was in this spirit that he enjoyed writing formal ‘routines’ for the computer, just as he enjoyed Jane Austen and Trollope, the novelists of social duty and hierarchy. He enjoyed making life into a game, a pantomime. He had done his best to turn the Second World War into a game.

What he had done was to combine such a naive mechanistic picture of the mind with the precise logic of pure mathematics. His machines – soon to be called Turing machines – offered a bridge, a connection between abstract symbols, and the physical world...

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.

It was a very remarkable fact that Emil Post’s [independently conceived] ‘worker’ was to perform exactly the same range of tasks as those of the Turing ‘machine’... Post’s paper was much less ambitious than Computable Numbers; he did not develop a ‘universal worker’ nor himself deal with the Hilbert decision problem... But he guessed correctly that his formulation would close the conceptual gap that Church had left. In this it was only by a few months that he had been pre-empted by the Turing machine, and Church had to certify that the work had been completely independent. So even if Alan Turing had never been, his idea would soon have come to light in one form or another. It had to. It was the necessary bridge between the world of logic and the world in which people did things.

[A corollary of Turing's discovery of the universal machine]: the law of information technology: all mechanical processes, however ridiculous, evil, petty, wasteful, or pointless, can be put on a computer.


WITTGENSTEIN: … Think of the case of the Liar. It is very queer in a way that this should have puzzled anyone – much more extraordinary than you might think. … Because the thing works like this: if a man says ‘I am lying’ we say that it follows that he is not lying, from which it follows that he is lying and so on. Well, so what? You can go on like that until you are black in the face. Why not? It doesn’t matter. … it is just a useless language-game, and why should anybody be excited?
TURING: What puzzles one is that one usually uses a contradiction as a criterion for having done something wrong. But in this case one cannot find anything done wrong.
WITTGENSTEIN: Yes – and more: nothing has been done wrong... where will the harm come?
TURING: The real harm will not come in unless there is an application, in which a bridge may fall down or something of that sort.
WITTGENSTEIN: ...The question is: Why are people afraid of contradictions? It is easy to understand why they should be afraid of contradictions in orders, descriptions, etc., outside mathematics. The question is: Why should they be afraid of contradictions inside mathematics? Turing says, ‘Because something may go wrong with the application.’ But nothing need go wrong. And if something does go wrong – if the bridge breaks down – then your mistake was of the kind of using a wrong natural law...
TURING: You cannot be confident about applying your calculus until you know that there is no hidden contradiction in it.
WITTGENSTEIN: There seems to me to be an enormous mistake there. … Suppose I convince Rhees of the paradox of the Liar, and he says, ‘I lie, therefore I do not lie, therefore I lie and I do not lie, therefore we have a contradiction, therefore 2 × 2 = 369.’ Well, we should not call this ‘multiplication’, that is all...
TURING: Although you do not know that the bridge will fall if there are no contradictions, yet it is almost certain that if there are contradictions it will go wrong somewhere.
WITTGENSTEIN: But nothing has ever gone wrong that way yet...


So in the summer of 1940, Alan Turing found himself in the position of telling other people what to do, for the first time since school. It was like school inasmuch as the WRNS and the ‘big room girls’ played the role of ‘fags’... one notable difference from school was that it brought him for the first time into contact with women... he specifically told [Joan] that he was glad he could talk to her ‘as to a man’. Alan was often lost when dealing with the Hut 8 ‘girls’, not least because he was unable to cope with the ‘talking down’ which was expected. But Joan’s position as cryptanalyst gave her the status of an honorary male.

It was the first time in his life that he had mixed with ordinary people for any length of time, people picked out neither by social class nor by a special kind of intellect. It was a typical Turing irony that this should happen at an establishment working for the secret service. [He was 30 at this point.]

Alan’s own youthfulness much endeared him to the younger recruits... it was hard to decide whether one so ‘schoolboyish’ could be as much as thirty, or whether one carrying so much intellectual standing could be so young. A conversation with him was like being invited into some older boy’s study where House Colours and Chapel Parade gave way to illicit jazz and D.H. Lawrence novels, but where the housemaster had to turn a blind eye because a precious scholarship was being won.

In 1941 everyone had to knit and glue and make their own entertainments... the siege mentality suited Alan rather well, with matters of social protocol that in the 1930s seemed so important now falling into abeyance. He always liked making things for himself, be they gloves, radio sets or probability theorems. At Cambridge he had a way of telling the time from the stars. Now the war was on his side. In a more self-sufficient England, everyone had to live in a more Turingesque way, with less waste of energy.

His high-pitched voice already stood out above the general murmur of well-behaved junior executives grooming themselves for promotion within the Bell corporation. Then he was suddenly heard to say: ‘No, I’m not interested in developing a powerful brain. All I’m after is just a mediocre brain, something like the President of the American Telephone and Telegraph Company.’ The room was paralysed, while Alan nonchalantly continued to explain how he imagined feeding in facts on prices of commodities and stock, and asking the machine the question ‘Do I buy or sell?’

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.

Glennie sometimes thought of Alan as Caliban, with his dark moods, sometimes gleeful, sometimes sulky, appearing in the laboratory on a somewhat random basis. He could be absurdly naive, as when bursting with laughter at a punning name that Glennie made up for an output routine: 'RITE'. To Cicely Popplewell he was a terrible boss, but on the other hand, there was no question of having to be polite or deferent to him – it was impossible. He was regarded as a local authority on mathematical methods; those who wanted a suggestion would just have to ask him straight out, and if they could keep his interest and patience, they might get a valuable hint... he was no world-standard mathematician, and it was often more amazing to the professional mathematician what he did not know, than what he did... indeed he had read very little mathematics since 1938.

Alan Turing presumably thought that eventually a machine would be capable of writing a book such as this [Hodge's biography of Turing]. In his 1951 radio talk, set against the opening of the Festival of Britain, he commented that ‘It is customary... to offer a grain of comfort, in the form of a statement that some peculiarly human characteristic could never be imitated by a machine. I cannot offer any such comfort, for I believe that no such bounds can be set.’


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 For geometric problems the touch-lines yield
... 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.


[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... it also required some rather sophisticated applied mathematics, which involved the use of ‘operators’ rather as in quantum mechanics. Numerical analysis was also important... In this it was like a private atomic bomb, the computer in both cases following the development of interacting fluid waves.

...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.


Gödel:
[Tarski and I both stress] the great importance of the concept of... Turing's computability... this importance is largely due to the fact that, with this concept, one has for the first time succeeded in giving an absolute notion to an interesting epistemological notion, i.e., one not depending on the formalism chosen

Going even further, modern papers sometimes employ the usage of 'turing machine'. Sinking without a capital letter into the collective mathematical consciousness (as with the 'abelian group', or the 'riemannian manifold') is probably the best that science can offer in the way of canonisation.

Why listen to me on this topic?

Nonfiction book reviews by nonspecialists are hazardous. It is just not easy to detect pseudo-empirical bullshit without

  1. immersion in the field and/or good priors for what makes for an extraordinary claim in it;

  2. incredible amounts of fact-checking gruntwork, at least 5x the time it takes to just read something; or

  3. incredible amounts of argument-checking, which doesn't need domain knowledge.

I always try to do (3) but surely often fail.



In this case: I am a computer scientist, and I've studied the early history of computing quite closely. I understand many of Turing's original papers, besides his group theory.



Cross-posted from Goodreads.

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Tags: bio, computers, effective-altruism