#### Alan Turing's Lost Notebook

Jack Copeland

Alan Turing's personal mathematical notebook went on display recently at
Bletchley Park near London, the European headquarters of the Allied
codebreaking operation in World War II. Until now, the notebook has been
seen by few—not even scholars specializing in Turing's work. It is on
loan from its current owner, who acquired it in 2015 at a New York auction
for over one million dollars.^{1}

The yellowing notebook—from Metcalfe and Son, just along the street from Turing's rooms at King's College in Cambridge—contains thirty-nine pages in his handwriting. The auction catalogue (which inconsequentially inflated the page count) gave this description:

Hitherto unknown wartime manuscript of the utmost rarity, consisting of 56 pages of mathematical notes by Alan Turing, likely the only extensive holograph manuscript by him in existence.

A question uppermost in the minds of Turing fans will be whether the notebook gives new information about his famous code-cracking breakthroughs at Bletchley Park, or about the speech-enciphering device named "Delilah" that he invented later in the war at nearby Hanslope Park. The answer may disappoint. Although most probably written during the war, the notebook has no significant connection with Turing's work for military intelligence. Nevertheless it makes fascinating reading: Turing titled it "Notes on Notations" and it consists of his commentaries on the symbolisms advocated by leading figures of twentieth century mathematics.

My interest in the notebook was first piqued more than 20 years ago. This was during a visit to Turing's friend Robin Gandy, an amiable and irreverent mathematical logician. In 1944-5 Gandy and Turing had worked in the same Nissen hut at Hanslope Park. Gandy remembered thinking Turing austere at first, but soon found him enchanting—he discovered that Turing liked parties and was a little vain about his clothes and appearance. As we sat chatting in his house in Oxford, Gandy mentioned that upstairs he had one of Turing’s notebooks. For a moment I thought he was going to show it to me, but he added mysteriously that it contained some private notes of his own.

In his will Turing left all his mathematical papers to Gandy, who eventually passed them on to King's College library—but not the notebook, which he kept with him up till his death in 1995. Subsequently the notebook passed into unknown hands, until its reappearance in 2015. Gandy's private notes turned out to be a dream diary. During the summer and autumn of 1956, two years after Turing's death, he had filled thirty-three blank pages in the centre of the notebook with his own handwriting. What he said there was indeed personal.

Only a few years before Gandy wrote down these dreams and his autobiographical notes relating to them, Turing had been put on trial for being gay. Gandy began his concealed dream diary: "It seems a suitable disguise to write in between these notes of Alan's on notation; but possibly a little sinister; a dead father figure and some of his thoughts which I most completely inherited."

Mathematical Reformer

Turing's own writings in the notebook are entirely mathematical, forming a
critical commentary on the notational practices of a number of famous mathematicians,
including Courant, Eisenhart, Hilbert, Peano, Titchmarsh, Weyl, and others.
Notation is an important matter to mathematicians. As Alfred North Whitehead—one
of the founders of modern mathematical logic—said in his 1911 essay "The Symbolism
of Mathematics", a good notation "represents an analysis of the ideas of the subject
and an almost pictorial representation of their relations to each other".^{2} "By
relieving the brain of all unnecessary work", Whitehead remarked, "a good notation
sets it free to concentrate on more advanced problems". In a wartime typescript
titled "The Reform of Mathematical Notation and Phraseology" Turing said that an
ill-considered notation was a "handicap" that could create "trouble"; it could even
lead to "a most unfortunate psychological effect", namely a tendency "to suspect the
soundness of our [mathematical] arguments all the time".^{3}

This typescript, which according to Gandy was written at Hanslope Park in 1944 or 1945, provides a context for Turing's notebook. In the typescript Turing proposed what he called a "programme" for "the reform of mathematical notation". Based on mathematical logic, his programme would, he said, "help the mathematicians to improve their notations and phraseology, which are at present exceedingly unsystematic". Turing's programme called for "An extensive examination of current mathematical … books and papers with a view to listing all commonly used forms of notation", together with an "[e]xamination of these notations to discover what they really mean". His "Notes on Notations" formed part of this extensive investigation.

Key to Turing's proposed reforms was what mathematical logicians call the "theory of types". This reflects the commonsensical idea that numbers and bananas, for example, are entities of different types: there are things that it makes sense to say about a number—e.g. that it has a unique prime factorization—that cannot meaningfully be said of a banana. In emphasizing the importance of type theory for day-to-day mathematics, Turing was as usual ahead of his time. Today virtually every computer programming language incorporates type-based distinctions.

Link to the Real Turing

Turing never displayed much respect for status and—despite the eminence of the mathematicians whose notations he was discussing—his tone in "Notes on Notations" is far from deferential. "I don't like this" he wrote at one point, and at another "this is too subtle and makes an inconvenient definition". His criticisms bristle with phrases like "there is obscurity", "rather abortive", "ugly", "confusing", and "somewhat to be deplored". There is nothing quite like this blunt candor to be found elsewhere in Turing's writings; and with these phrases we perhaps get a sense of what it would have been like to sit in his Cambridge study listening to him. This scruffy notebook gives us the plain unvarnished Turing.

References

^{1} www.bonhams.com/auctions/22795/lot/1/

^{2} Whitehead, A. N. *An Introduction to Mathematics,* Holt, 1911, pp. 39, 40.

^{3} Turing, A. M. 'The Reform of Mathematical Notation and Phraseology'. The typescript (which is incomplete) is in the library of King's College, Cambridge, with a digital facsimile at http://www.turingarchive.org/browse.php/C/12; and it also appears in Cooper, S. B. and van Leeuwen, J. (eds) *Alan Turing: His Work and Impact* (Elsevier, 2013) with an introduction by Floyd, J. 'Turing, Wittgenstein and Types: Philosophical Aspects of Turing's "The Reform of Mathematical Notation and Phraseology" (1944-5)'.