It was strong meat because it was in German, a language in which Alan had demonstrated his lack of aptitude for languages at Sherborne; but if you were serious about maths in the 1930s, you had to be able to get on with mathematical German. Another reason it was strong meat was that it was about the axioms underpinning the oddities of quantum mechanics: that there was a consistent logical system in mathematics, into which the problematic results in quantum physics could neatly fit. Logic, consistency, and truth.
Alan’s enthusiasm must have caught the eye of Richard Braithwaite, a young King’s mathematics don, who was himself exploring the cross-overs between mathematics, logic and philosophy. He was influential in the Moral Sciences Club – ‘moral sciences’ being Cambridge-speak for philosophy – and Alan was invited to present a paper on mathematics and logic at their meeting, in Alan’s rooms, on 1 December 1933. Alan said in a letter home, ‘I hope they don’t know it all already’.
A positive contribution to mathematical thought
Part of the problem in quantum mechanics is that observations mess things up. It is not surprising, then, to find that Alan chose to attend Professor Sir Arthur Eddington’s lectures on the methodology of science. Alan had chosen Eddington’s The Nature of the Physical World for the Morcom Prize in 1930, and Eddington was now trying to create a fundamental theory to unify quantum theory, gravitation and relativity. Eddington’s lectures mentioned that scientific observations tended to be distributed according to a normal, or Gaussian, curve, around the mean value. For Alan, this was a challenge; it was all very well to assert such a thing, but assertions needed to be proved.
So, by early 1934, in his trademark get-it-done-best-on-your-own way, Alan proved Eddington’s assertion. Here was a substantive result, one that would probably even surpass Champernowne (who, under the influence of Keynes and the other economic brains at King’s, had switched to economics), and might even give grounds for election to a fellowship, and a career in academia. Alas for Alan: the Central Limit Theorem, to give the problem its usual name, had been first posited in 1733 and thought about by many famous mathematicians and scientists, including Laplace and Galton; eventually, in 1922, the Finnish mathematician Jarl Waldemar Lindeberg had published a proof (in German, of course). Alan Turing’s first major achievement had been scooped.
So it was back to business, with final exams looming in May and the need to prove himself. The timing was terrible, with Julius Turing’s prostate operation right in the middle, as well as the duties of a college oarsman. But all went well, despite the best efforts of The Times to print a class-list suggesting Alan had got a B grade.
Dear Mother
I enclose list of Maths Tripos II in case you have been taking ‘the Times’ too seriously. I hope the aunts won’t see it & write congratulatory letters on getting a (b).
It is very kind of you all to send me these telegrams. It seems to me more extravagant than taking a taxi. […]
Yours
Alan.
Ethel Turing has annotated this letter with the remark ‘Reference to taxi is due to his father’s belief that to take a taxi was great extravagance’. Alan had in fact passed as a B-star wrangler, the obscure Cambridge maths tripos codeword meaning that he had achieved first-class honours (a ‘wrangler’) with distinction in additional papers. (The explanation on the class-list says, ‘The mark (b*) is attached to the names of those candidates who in the opinion of the Moderators and Examiners deserve special credit in subjects of Schedule B’.) There were nine other B-stars, from a total of 37 mathematicians ranked as wranglers in 1934. King’s was generous to its B-star wranglers, and allowed Alan a £200 research studentship to stay on and try for a fellowship. Meanwhile, in the summer vacation, Alan stood as best man to his brother John, who was getting married to Joan Humphreys, who was, as you would expect, a daughter of the Indian Civil Service.
Because Alan had not been aware of Lindeberg’s proof, his work on the Central Limit Theorem might, with a bit of polishing-up, be suitable as a fellowship dissertation.
I have a horrid recollection [wrote John] of ‘The Gaussian Error Function’ (whatever that might be), reputedly the subject of this monograph, for Alan had left it to the eleventh hour to sort the sheets, parcel and dispatch them. My mother and I spent a frantic half-hour on hands and knees putting them in order; Mother did up the parcel in record time and Alan sped with it to the [Post Office] on his bike, announcing on his return that there were at least twenty minutes to spare. This is my only positive contribution to mathematical thought.
In 1933 Provost Brooke – the uncle of the poet Rupert Brooke – retired, to be replaced by John Sheppard, who was not the bookies’ favourite, but who was nonetheless to hold office for the next 21 years. Sheppard’s great characteristic was that he loved the young men of his college, and he possessed in abundance the ability to cross social boundaries to put them at ease. Sometimes this could be disconcerting, as noted in a history of the college:
His fine head of hair had gone white when he was in his thirties, and he early adopted the pose of a benevolent old gentleman scattering ‘blessings’. Many of those who came into residence he got to know, sometimes by the simple method of stopping them in the court and saying, ‘Who are you, dear boy?’ (‘I’m not your dear boy,’ one replied, ‘I’m at Selwyn.’)
The bachelor existence. Half out-of-shot, Alan is a bystander as best man beside his brilliantined brother at John’s wedding in August 1934.
To the great good fortune of Alan Turing, Provost Sheppard chaired the fellowship election committee, and Provost Sheppard used his personal knowledge of the candidates to ensure that the right man was elected, even if, on occasion, academic evidence pointed in favour of a different candidate. Turing was making his mark on the college – unusual, maybe, but not invisible. Complaining that he had never had a teddy bear in childhood, Alan asked for one for Christmas in 1934; it was duly provided by a bemused Ethel Turing, and Porgy the bear was installed in Alan’s rooms to greet visitors. (Porgy is still receiving visitors in the museum at Bletchley Park.) Basileon, the King’s College satirical magazine, noted in its 1936 edition a pageful of Sayings of the Year, including two aphorisms of Alan Turing: ‘I think the College is going to have more kittens’, and ‘I am prepared to admit that there are other people besides myself’.
Porgy, Alan’s teddy bear, given to him in adulthood because, as a foster-child, nobody had given him one in infancy.
Those ‘other people’ had to pronounce on the Gaussian Error Function. On 12 November 1934, a dissertation by A.M. Turing on that subject had been referred by the Electors to Fellowships to Professor R.A. Fisher and Mr A.S. Beskovitch for review.
Mr A.S. Beskovitch’s Report:
The dissertation is not to be judged from the point of view of its scientific value, as its main results were established long ago and even the fundamental idea of the method is not new.
Prof. R.A. Fisher’s Report:
The subject chosen for the thesis is one which I have thought decidedly unattractive, and which has been worked over, from various points of view, by continental and especially Scandinavian writers to the point of making it positively repellent.
Oh dear. But:
Mr A.S. Beskovitch’s Report:
The development of Mr Turing’s method is very much different from that of Lindeberg, which makes me completely confident that the work has been done in a genuine ignorance of Lindeberg’s work. Mr Turing’s proof is somewhat more complicated than the Lindeberg proof, but all the same it is an excellent success and it would be so not only for a beginner but also for a fully developed scientist. If the paper were published fifteen years ago it would be an important event in the mathematical literature of that year. In Mr Turing’s case we see a display of very exceptional abilities at the very start of his research work, which makes me to recommend him as a very strong candidate for a Fellowship.
Prof. R.A. Fisher’s Further Report:
I have
no hesitation in judging Turing’s thesis the work of a first-class candidate. He seems to have thought out his own methods. Finally in reading through his paper, I formed a very high opinion of his taste, virtuosity would not be too strong a term, in the art of framing conclusive mathematical demonstrations.
Going up in the world. From the dingy Q2 (1932) to the riverside X8 (1935).
So the lack of priority on the Central Limit Theorem could be overlooked, and on 16 March 1935, ‘at a Meeting of the Electors to Fellowships holden in the Combination Room’, presided over by Mr Provost, Alan Turing was elected to a fellowship of King’s at the ripe old age of 22. Alan Turing moved from the grimness of his earlier rooms on ‘A’ and ‘Q’ staircases to a top-floor set of rooms in the airy Bodley’s Court, with a pleasant view of the river, and his friend Fred Clayton downstairs. Fitting for a mathematician, this was on Staircase X, and his rooms would from now on sport the dignified label ‘Mr Turing’. Basileon rounded off the achievement with a clerihew:
Turing
must be very alluring
to be made a don
so early on.
Notes
1 Roughly taking the following form:
PROTAGONIST: His cigar is a sight and he stays up all night.
ASSEMBLY: Vive la compagnie!
PROTAGONIST: And his blood sweat and tears will give Hitler a fright.
ASSEMBLY: Vive la compagnie!
Vive-la, vive-la, vive la reine,
Vive-la, vive-la, vive le roi,
Vive-la, vive-la, vive l’esprit,
Vive la compagnie!
The Assembly then calls out their guess as to the subject of the Vive-la (which ought to be obvious, though more acidic wit is expected than in this feeble example).
1 King’s encouraged students to return to college for a few weeks’ informal study in a ‘Long Vacation Term’
2 Second-year exams
5
MACHINERY OF LOGIC
A FORMER PUPIL of Sherborne School, who achieves a top-class degree in mathematics at Cambridge University, is like as not to be interested in the rules which govern mathematical reasoning. It would be necessary to discuss the subject with philosophy professors who specialise in logic. And a very clever Old Shirburnian might even go so far as to write a world-famous work on the subject.
Alfred North Whitehead went to Sherborne School in 1875. His older brother Henry was also at Sherborne and went on to become Bishop of Madras in 1899, where he was most likely cultivated by Ethel Turing, who had a soft spot for bishops. Unlike Alan Turing, A.N. Whitehead did well on the games field, became head of school, and went to Trinity College, Cambridge to study maths, coming fourth in the list of wranglers. Until Alan came onto the scene, Whitehead was Sherborne’s star intellectual alumnus; even when I attended the school in the 1970s his name was spoken with reverence. Whitehead had, with Bertrand Russell as junior author, written the Principia Mathematica. To give a book, published over the years 1910–1913, a Newtonian title in Latin would seem an act of blazing arrogance were it not for its ambition and achievement. For Whitehead’s work set out to do no less than codify the rules of mathematical thought.
Rules, unlike mathematicians, ought to be logical and orderly. They ought to allow for any mathematical proposition to be deduced from a handful of basic precepts, without contradiction or uncertainty. That cleanliness, after all, is what distinguishes maths from metaphysics. And who better to develop notions of tidiness in mathematics than the Germans? In 1920, David Hilbert, a mathematics professor working in Göttingen, posed three objectives about the rules mathematicians need in order to carry out proofs:
1. Completeness. Every proposition which is true, can be proved, using the rules. Eg: I assert that the angles of a triangle add up to 180 degrees; I can prove this from Euclid’s axioms.
2. Consistency. No contradictions will arise, when applying the rules. Eg: having proved that the angles of all triangles add up to 180 degrees, I cannot produce from my hat a super-triangle whose angles add up to 200 degrees.
3. Decidability. There is a method for deciding whether any proposition is true or false. Eg: you assert that the angles of a square add up to 360 degrees. Neither of us knows whether this is true or false, but we can find out. Hilbert put it in Latin thus: ‘In mathematics there is no ignorabimus’. There should be no unknowable unknowns in maths.
It would all be so neat if these things could be shown to be true. Alas, in 1931 an Austrian mathematician, Kurt Gödel, proved that mathematics was incomplete. There were theorems which could neither be proved nor disproved. Worse, he went on to demonstrate, using a modern rerun of one of Zeno’s paradoxes (‘this proposition is unprovable’), that a mathematical system cannot be proved to be consistent either, and it absolutely cannot be both complete and consistent.
For students of higher mathematics all this was tremendous fun, and in 1935 Cambridge was running a course of lectures for its advanced students on the foundations of mathematics. The lecturer was M.H.A. Newman, and Alan Turing was one of his students. Newman had done war service in the Army and some teaching before he came to Cambridge. He had taken his degree in 1921 – as a B-star wrangler, needless to say. By 1935 he was 38 years old, newly married, and a fellow of St John’s College where he was working up his ideas to write a textbook on topology. He was musical, had an impish dry wit, inspired great loyalty, and was destined to become Alan Turing’s lifelong mentor. In 1935 his job was to explain the Hilbert plan, and its partial debunking, to students like Alan. The one piece of Hilbert’s architecture which might still be intact was the issue of decidability, in German called the Entscheidungsproblem. The Entscheidungsproblem captured Alan’s imagination. It was tailor-made for his way of thinking, which always, even in relation to the most abstract mathematical ideas, sought links back to the real, practical world. Newman explained:
Alan’s lifelong mentor, M.H.A. Newman. It was Newman’s lecture invoking a ‘mechanical process’ which set Alan Turing off on a lifetime of work and ideas about machines, logic, computing and shape.
The point was that Hilbert had announced that he would find really a way of doing mathematics once and for all by putting it into purely symbolic form and analysing the grammar of the propositions of mathematics as put out in symbols, and that he would try to find a decision method, that is to say, a process for finding the answer ‘yes’ or ‘no’ to the question ‘Can this be proved false, can it be proved …?’ And the question, of course, as Turing saw, was what do you mean by ‘process’?
Newman had talked of Hilbert’s surviving idea as a mechanical process:
I believe it all started because he attended a lecture of mine on foundations of mathematics and logic. I think I said in the course of this lecture that what is meant by saying that a process is constructive is that it’s purely mechanical, a machine – and I may even have said, a machine can do it. But he took the notion and really tried to follow it right up, and did produce this extraordinary definition of a perfectly general what he called ‘computable function’, thus giving the first idea, really, of a perfectly general computing machine.
So Alan Turing started thinking about machines. The machine had to be perfectly general, because it had to be able to decide whether any proposition was true or false. It couldn’t, therefore, be like regular 1930s mechanical calculating machines, which could do boring adding or subtracting, or – to take the most sophisticated type of machine at the very forefront of technology – solve differential equations. These were single-purpose machines, whereas the ‘machine’ which Alan Turing needed would be all-purpose. And Alan’s all-purpose machine could be put to one special use. It would pull the last keystone out of Hilbert’s edifice, and bring the whole thing crashing down.
Mulling over these things took some time. Meanwhile, Alan had been elected to his King’s fellowship and produced a short paper taking forward the ‘group theory’ invented by John von Neumann, the mathematician whose book had ins
pired Alan’s talk to the Moral Sciences Club. (Von Neumann, who visited Cambridge and lectured briefly in 1935, had also been at Göttingen with Hilbert and was destined to float in and out of Alan’s life.) By early 1936 Alan’s idea had matured into a draft paper, and by Easter it was ready for review.
Dear Mother, […]
I saw Mr Newman four or five days after I came up. He is very busy with other things just at present and says he will not be able to give his whole attention to my theory for some week or so yet. However, he examined my note for C.R1 and approved it after some alterations. I have had no acknowledgment of it, which is rather annoying. I don’t think the full text will be ready for a fortnight or more yet. It will probably be about fifty pages. It is rather difficult to decide what to put into the paper now and what to leave over till a later occasion. […]
Yours
Alan
Newman did give his whole attention to Alan’s theory, and it was groundbreaking. Alan had demolished Hilbert’s plan, and had presented his demolition work with clarity and vision. The centrepiece of Alan’s idea was a ‘universal’ computing machine, a fantastic idea: an imaginary machine which had only a limited range of physical functions, but which could imitate the behaviour of any single-purpose machine. Alan had created a generic description of algorithms.
We may compare a man in the process of computing a real number to a machine which is only capable of a finite number of conditions, which will be called m-configurations. The machine is supplied with a ‘tape’ running through it, and divided into sections (called ‘squares’) each capable of bearing a ‘symbol’. At any moment there is just one square which is in the ‘machine’. We may call this square the ‘scanned square’. The symbol on the scanned square may be called the ‘scanned symbol’. The ‘scanned symbol’ is the only one of which the machine is, so to speak, ‘directly aware’. However, by altering its m-configuration the machine can effectively remember some of the symbols which it has ‘seen’ (scanned) previously. The possible behaviour of the machine at any moment is defined by the m-configuration and the scanned symbol. In some of the configurations in which the scanned square is blank (i.e. bears no symbol) the machine writes down a new symbol on the scanned square: in other configurations it erases the scanned symbol. The machine may also change the square which is being scanned, but only by shifting it one place right or left. In addition to any of these operations the m-configuration may be changed. Some of the symbols written down will form the sequence of figures which is the decimal of the real number which is being computed. The others are just rough notes to ‘assist the memory’. It will only be these rough notes which will be liable to erasure.
Prof Page 7
